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1305.0714v1.Co2_FeAl_thin_films_grown_on_MgO_substrates__Correlation_between_static__dynamic_and_structural_properties.pdf | Co 2FeAl thin films grown on MgO substrates: Correlation between static, dynamic
and structural properties
M. Belmeguenai1, H. Tuzcuoglu1, M. S. Gabor2, T. Petrisor jr2, C. Tiusan2;3,
D. Berling4, F. Zighem1, T. Chauveau1, S. M. Chérif1and P. Moch1
1Laboratoire des Sciences des Procédés et des Matériaux, CNRS - Université Paris 13, France
2Center for Superconductivity, Spintronics and Surface Science, Technical University of Cluj-Napoca, Romania
3Institut Jean Lamour, CNRS-Université de Nancy, France and
4Institut de Science des Matériaux de Mulhouse, CNRS-Université de Haute-Alsace, France
Co2FeAl (CFA) thin films with thickness varying from 10 nm to 115 nm have been deposited on
MgO(001)substratesbymagnetronsputteringandthencappedbyTaorCrlayer. X-raysdiffraction
(XRD) revealed that the cubic [001]CFA axis is normal to the substrate and that all the CFA films
exhibit full epitaxial growth. The chemical order varies from the B2phase to the A2phase when
decreasing the thickness. Magneto-optical Kerr effect (MOKE) and vibrating sample magnetometer
measurements show that, depending on the field orientation, one or two-step switchings occur.
Moreover, the films present a quadratic MOKE signal increasing with the CFA thickness, due
to the increasing chemical order. Ferromagnetic resonance, MOKE transverse bias initial inverse
susceptibility and torque (TBIIST) measurements reveal that the in-plane anisotropy results from
the superposition of a uniaxial and of a fourfold symmetry term. The fourfold anisotropy is in
accord with the crystal structure of the samples and is correlated to the biaxial strain and to the
chemical order present in the films. In addition, a large negative perpendicular uniaxial anisotropy
is observed. Frequency and angular dependences of the FMR linewidth show two magnon scattering
and mosaicity contributions, which depend on the CFA thickness. A Gilbert damping coefficient as
low as 0.0011 is found.
I. INTRODUCTION
The performances of spintronic devices depend on the
spin polarization of the current. Therefore, half metallic
materials should be ideal compounds as high spin polar-
ized current sources to realize a very large giant magne-
toresistance, a low current density for current induced
magnetization reversal, and an efficient spin injection
intosemiconductors. Theoretically, differentkindsofma-
terials, such as Fe 3O4[1, 2], CrO 2[3], mixed valence per-
ovskites [4] and Heusler alloys [5, 6], have been predicted
to be half metals. Moreover, the half metallic proper-
ties in these materials have been experimentally demon-
strated at low temperature. However, oxide half metals
havelowCurietemperature( TC)andthereforetheirspin
polarization is miserably low at room temperature. From
this point of view, Heusler alloys are promising materials
for spintronics applications, because a number of them
have generally high TC[7] and therefore they may offer
an alternative material choice to obtain half metallicity
even at room temperature. Furthermore, their structural
and electronic properties strongly depend on the crystal
structure. Recently, Heusler compounds have attracted
considerable experimental and theoretical interest, not
only because of their half metallic behaviour but also due
to magnetic shape memory and inverse magneto-caloric
properties that they exhibit. One of the most important
Co-based full-Heusler alloys is Co 2FeAl (CFA). It has a
highTC(TC= 1000K) [7] and, therefore, it is promis-
ing for practical applications. Indeed, it can provide
giant tunnelling magnetoresistance ( 360%at RT) [8,9]
when used as an electrode in magnetic tunnel junctions.
Furthermore, as we illustrate in our present study, CFApresents the lowest magnetic damping parameter among
Heuslers. This low damping should provide significantly
lower current density required for spin-transfer torque
(STT) switching, particularly important in prospective
STT devices. However, the integration of CFA as a
ferromagnetic electrode in spintronic devices requires a
good knowledge allowing for a precise control of its mag-
netic properties, such as its saturation magnetization, its
magnetic anisotropy, the exchange stiffness parameter,
the gyromagnetic factor and the damping mechanisms
monitoring its dynamic behaviour. In this paper we
used X-rays diffraction (XRD), ferromagnetic resonance
in microstrip line (MS-FMR) under in-plane and out of
plane applied magnetic field, combined with transverse
biased initial inverse susceptibility and torque (TBIIST)
method, in order to perform a complete correlated analy-
sis between structural and magnetic properties of epitax-
ial Co 2FeAl thin films grown on MgO(001) substrates.
In addition, a detailed study of the different relaxation
mechanisms leading to the linewidth broadening is pre-
sented.
II. SAMPLES PREPARATION AND
EXPERIMENTAL METHODS
CFA films were grown on MgO(001) single-crystal sub-
strates using a magnetron sputtering system with a base
pressure lower than 310 9Torr. Prior to the deposi-
tion of the CFA films, a 4 nm thick MgO buffer layer was
grown at room temperature (RT) by rf sputtering from
a MgO polycrystalline target under an Argon pressure
of 15 mTorr. Next, the CFA films, with variable thick-arXiv:1305.0714v1 [cond-mat.mtrl-sci] 3 May 20132
30 40 50 60 70050100150200250300
42.5 45.0 47.50100200
115 nm
70 nm
45 nm
20 nm
2(degrees)Intensity (arb. units)(002) CFA
(004) CFA
10 nm(a) (220) CFA
30 40 50 60 70 80 90050100150200250300350400450500(b)
20 nm50 nmCFA(002)
CFA(004)Intensity (arb. units)
2 (degrees)
Figure 1: (Colour online) (a) X-ray 2 !(out-of-plane)
diffractionpatternusing(CuX-rayssource)fortheCr-capped
and (b) 2pattern (Co X-rays source) for the Ta-capped
Co2FeAl of different thicknesses. The inset shows selected
area in plane diffraction patterns around (220) Co 2FeAl re-
flection.
nesses (10 nmd115nm), were deposited at RT by dc
sputtering under an Argon pressure of 1 mTorr, at a rate
of 0.1 nm/s. Finally, the CFA films were capped with
a MgO(4nm)/Cr(10nm) or with a MgO(4nm)/Ta(10nm)
bilayer. Afterthegrowthofthestack, thestructureswere
ex-situ annealed at 600oC during 15 minutes in vacuum
(pressure lower than 310 8Torr). The structural prop-
erties of the samples have been characterized by XRD us-
ing a four-circle diffractometer. Their magnetic dynamic
properties have been studied by microstrip ferromagnetic
resonance (MS-FMR).
The MS-FMR characterization was done with the help
of a field modulated FMR setup using a vector network
analyzer (VNA) operating in the 0.1-40 GHz frequency
range. The sample (with the film side in direct con-
tact) is mounted on 0.5 mm microstrip line connected
to the VNA and to a lock-in amplifier to derive the field
modulated measurements via a Schottky detector. Thissetup is piloted via a Labview program providing flexi-
bility of a real time control of the magnetic field sweep
direction, step and rate, real time data acquisition and
visualization. It allows both frequency and field-sweeps
measurements with magnetic fields up to 20 kOe applied
parallel or perpendicular to the sample plane. In-plane
angular dependence of resonance frequencies and fields
are used to measure anisotropies. The complete analy-
sis of in-plane and perpendicular field resonance spectra
exhibiting uniform precession and perpendicular stand-
ing spin wave (PSSW) modes leads to the determination
of most of the magnetic parameters: effective magneti-
zation, gyromagnetic factor, exchange stiffness constant
and anisotropy terms. In addition, the angular and the
frequency dependences of the FMR linewidth are used
in order to identify the relaxation mechanisms responsi-
ble of the line broadening and allow us for evaluating the
parameters which monitor the intrinsic damping (Gilbert
constant) and the extrinsic one (two magnon scattering,
inhomogeneity, mosaïcity).
Magnetization at saturation and hysteresis loops for
each sample were measured at room temperature using a
vibrating sample magnetometer (VSM) and a magneto-
optical Kerr effect (MOKE) system. Transverse biased
initialinversesusceptibilityandtorquemethod(TBIIST)
[10] has been used to study the in-plane anisotropy for
comparison with MS-FMR. In this technique both a lon-
gitudinal magnetic sweep field HL(parallel to the inci-
denceplane)andastatictransversefield HB(perpendic-
ulartotheincidenceplane)areappliedintheplaneofthe
film and the longitudinal reduced magnetization compo-
nentmLis measured versus HLfor various directions of
HLwithconventionalmagneto-opticalKerrsetup. From
the measured hysteresis loops mL(HL)under transverse
biased field, the initial inverse susceptibility ( 1) and
the field offset ( H) which are related to the second and
first angle-derivative of the magnetic anisotropy, respec-
tively, are derived. Fourier analysis of 1andHversus
the applied field direction then easily resolves contribu-
tions to the magnetic anisotropy of different orders and
gives the precise corresponding values of their amplitude
and of their principal axes.
In order to obtain the desirable accuracy or even sim-
ply meaningful results higher-order nonlinear in mLcon-
tributions (quadratic or Voigt effect) as well as polar or
other contributions to the Kerr signal are carefully deter-
mined and corrected [10]. TBIIST method surely does
not have the same recognition than FMR techniques but
seems to be complementary, especially for samples with a
weak magnetic signal detectable with difficulty by FMR
methods.
III. STRUCTURAL CHARACTERIZATION
Figure 1 shows the X-rays 2 !diffraction patterns
for CFA of different thicknesses. These XRD patterns
show that, in addition to the feature arising from the3
Figure 2: (Colour online) Pole figures around the Co 2FeAl
(022) type reflection, for the 45 nm thick film, indicat-
ing the growth of Co 2FeAl on MgO with the Co 2FeAl
(001)[110]kMgO (001)[100] epitaxial relation. The 0 and 90
degrees axis of the graph correspond to the MgO [100]and
[010] crystalline directions.
(002) peak of the MgO substrate, the Cr-capped samples
(Fig. 1a: Cu X-rays source ( = 0:15406nm)) exhibit
only two peaks which are attributed to the (002) and
(004) diffraction lines of CFA. The Ta-capped films (Fig.
1b: Co-X-rays source ( = 1:7902)) show an additional
peak (around 2= 63 °) arising from the (002) line is-
sued from the Ta film. Pole figures (Fig. 2) allow to
assert an epitaxial growth of the CFA films according
to the expected CFA(001)[110]//MgO(001)[100] epitax-
ial relation. Using scans of various different orientations
we evaluated the out-of-plane ( a?) and the in-plane ( ak)
lattice parameters (Fig. 3). A simple elastic model al-
lowed us for deriving the unstrained a0 cubic parameter
aswellasthein-plane "kandtheout-of-plane "?strains:
a0=
C11a?+ 2C12ak
(C11+ 2C12);
"k=C11
(C11+ 2C12)
ak a?
a0;
"?=2C12
(C11+ 2C12)
ak a?
a0(1)
where the values of the elastic constants C11= 253
GPa andC12= 165GPa have been calculated previously
[11]. Introducing the Poisson coefficient =C12=(C11+
C12)the above parameters write as:
05 0 1 0 00.5650.5700.575
out-of-plane
in-planelattice parameter (nm)
thickness (nm)-10-50510152025 A002/A004
ratio (%)Figure 3: (Colour online) Evolution of the out-of-plane and
in-plane lattice parameters and of the ratio of the integral in-
tensitiesofthe (002)and(004)Co2FeAlpeaksA(002)/A(004)
with respect to the Co 2FeAl films thickness.
a0=
(1 )a?+ 2ak
+ 2ak
(1 +);
"?=(1 )
(1 +)
ak a?
a0;
"k= 2
1 +
ak a?
a0(2)
Thecubiclatticeconstant a0doesnotdependuponthe
thickness, except for the thinner 10 nm film (Fig. 4a),
which shows a significant reduction; its value, 0:5717
0:0005nm, is slightly smaller than the reported one in
the bulk compound with the L2 1structure (0.574 nm).
The in-plane strain "kreveals a tensile stress originat-
ing from the mismatch with the lattice of the MgO sub-
strate: however, its value does not exceed a few°/°°, well
below the Heussler/MgO mismatch, thus excluding an
efficient planar clamping. The strain "kdecreases versus
the thickness, at least above 40 nm (Fig. 4b).
Odd Miller indices (e.g.: (111);(311);...) are allowed
for diffraction in the L2 1phase [12]. In contrast, they
are forbidden in the B2 phase, which is characterized by
a total disorder between Al and Fe atoms but a regular
occupation of the Co sites. In the A2 phase the chemical
disorder between Fe, Co and Al sites is complete: (hkl)
diffraction is only allowed for even indices subjected to
h+k+l= 4n. We do not observe (111)or(311)lines
and then conclude to the absence of the L2 1phase in
the studied films. In contrast, a (002)peak is observed,
thus indicating that the samples do not belong to the A2
phase. However, the ratio I002=I004of the integrated in-
tensities of the (002)and of the (004)peaks increases ver-
sus the film thickness (Fig. 3). This ratio is proportional
to(1 2c)2, wherecis the chemical disorder. Assuming
that the thickest film belongs to the B2 phase ( c= 0)
the dependence of cupon the film thickness is shown in4
0 2 04 06 08 0 1 0 0 1 2 00.00.20.40.0050.0060.0070.5700.5710.572
(c)c
Film thickness (nm)(b)(a)a0 (nm)
Figure 4: (Colour online) Thickness dependence of (a) the
lattice cubic parameter a0, the in-plane strain "kand (c) the
chemical order cof Co 2FeAl thin films.
figure 4c: the A2 phase ( c= 0:5) is almost completely
achieved for the 10 nm thick sample. The reduction of
a0in the thinner sample is probably due to its previously
noticed [13] smaller value in the A2 phase compared to
the B2 one.
IV. MAGNETIC PROPERTIES
The experimental magnetic data have been analyzed
considering a magnetic energy density which, in addition
toZeeman, demagnetizingandexchangeterms, ischarac-
terized by the following effective anisotropy contribution
[14]:
Eanis: = 1
2(1 +cos(2('M 'u))Kusin2M+
K?sin2M 1
8(3 + cos 4('M '4))K4sin4M(3)
In the above expression, Mand'Mrespectively rep-
resent the out-of-plane and the in-plane (referring to the
substrate edges) angles defining the direction of the mag-
netizationMS.'uand'4define the angles between an
easy uniaxial planar axis or an easy planar fourfold axis,
respectively, with respect to this substrate edge. Ku,K4
andK?are in-plane uniaxial, fourfold and out-of-plane
uniaxialanisotropyconstants, respectively. Weintroduce
the effective magnetization Meff=Heff=4obtained
by:
4Meff=Heff= 4MS 2K?
MS= 4MS H?(4)
As experimentally observed, the effective perpendicu-
lar anisotropy term K?(and, consequently, the effective
perpendicular anisotropy field H?), is thickness depen-
dent:K?describes an effective perpendicular anisotropyterm which writes as:
K?=K?V+2K?S
d(5)
whereK?Srefers to the perpendicular anisotropy term
of the interfacial energy density. Finally we define Hu=
2Ku=MSandH4= 4K4=Msas the in-plane uniaxial and
the fourfold anisotropy fields respectively. The resonance
expressions for the frequency of the uniform and PSSW
modes assuming in-plane or perpendicular applied mag-
neticfieldsaregivenbyequations(6)and(7)respectively
[14, 15].
Fn:=
2(Hcos('H 'M) +2K4
MScos 4('M '4)
+2Ku
MScos 2('M 'u) +2Aex:
MSn
d2
)
(Hcos('H 'M) + 4Meff+K4
2MS(3 + cos 4('M '4))
+Ku
MS(1 + cos 2('M 'u)) +2Aex:
MS(n
d)2)(6)
F?:=
2(H 4Meff+2Aex:
MSn
d2
)(7)
In the above expressions
=2=g1:397106
s 1.Oe 1is the gyromagnetic factor, nis the index of
the PSSW and Aexis the exchange stiffness constant.
The experimental results concerning the measured
peak-to-peak FMR linewidths HPPare analyzed in
this work taking account of both intrinsic and extrinsic
mechanisms. Therefore, in the most FMR experiments,
the observed magnetic field linewidth ( HPP) is usu-
ally analyzed by considering four different contributions
as given by equation (8) [16-21].
HPP= HGi+ (Hmos+ Hinh+ H2mag)(8)
When the applied field and the magnetization are paral-
lel, the intrinsic contribution is not angular dependent;
it derives from the Gilbert damping and is given by:
HGi=2p
3
2f (9)
(9) wherefis the driven frequency and is the Gilbert
coefficient.
The relevant mechanisms [16] describing the extrinsic
contributions are:
1- Mosaicity: the orientation spread of the crystallites
contributes to the linewidth. Its contribution is given by:
Hmos= @Hres
@'H'H = @H
@'H'H
res(10)5
Where 'His the average spread of the easy axis
anisotropy direction in the film plane. It is worth to men-
tion that for frequency dependent measurements along
the easy and hard axes the partial derivatives are zero
and thus the mosaicity contribution vanishes. The suffix
“res” indicates that equation (10) should be evaluated at
the resonance. Therefore, using equation (6) for uniform
mode (n= 0), the expression of@H
@'His found and then
calculated using the corresponding value of Hand'M
at the resonance.
2- Inhomogeneous residual linewidth Hinhpresent
at zero frequency. This contribution is frequency and
angle independent inhomogeneity related to various local
fluctuations such as the value of the film thickness.
3- Two magnon scattering contribution to the
linewidth. This contribution is given by [22-24]:
H2mag= 0+ 2cos 2('H '2)+
4cos 4('H '4) arcsin
fp
f2+f2
0+f0!
(11)
with:f0=
Meff. The expected fourfold symmetry
induces the 0and 4coefficients; the coefficient 2is
phenomenogically introduced.
Theanalysisofthevariationoftheresonancelinewidth
HPPversus the frequency and the in-plane field ori-
entation allows for evaluating ,'H,Hinh, 0, 2
(and'2) and 4(and'4which, from symmetry consid-
erations, is expected to have a 0°or45°value, depending
upon the chosen sign of 4).
A. Static properties
The magnetization at saturation measured by VSM,
averaged upon all the samples has been found to be
MS= 100050emu/cm3, thus providing a magnetic
moment of 5.05 ±0.25 Bohr magneton ( B) per unit for-
mula, in agreement with the previously published values
for the B2 phase [7]. For all the studied films the hystere-
sis loops were obtained by VSM and MOKE with an in-
plane magnetic field applied along various orientations.
Figure 5 shows representative behaviors of different CFA
films. The observed shape mainly depends on the field
orientation, in agreement with the expected characteris-
tics of the magnetic anisotropy. As confirmed below, in
all the studied samples this anisotropy consists into the
superposition of a fourfold and of a uniaxial term show-
ing parallel easy axes: this common axis coincides with
one of the substrate edges and, consequently, with one
of the<110>crystallographic directions of the CFA
phase. It results that if an orientation (say 'H= 0re-
lated to [110]) is the easiest, the perpendicular direction
(('H= 90) related to [110]) is less easy. A similar sit-
uation was studied and interpreted previously [25]: it is
expected to provide square hysteresis loops for 'H= 0,
-60 -40 -20 0 20 40 60-1.0-0.50.00.51.0(a)
d=115 nmM/Ms
Magnetic field (oe) H=0°
H=45°
H=90°
-60 -40 -20 0 20 40 60-1.0-0.50.00.51.0(b)
d=50 nmM/Ms
Magnetic field (oe) H=0°
H=45°
H=90°
0.00 0.02 0.04 0.06 0.08 0.1001020304050607080Coercive field (Oe)
1/d(nm) Cr-capped
Ta-capped(c)Figure 5: (Colour online) MOKE hysteresis loops of the (a)
115 nm Cr-capped and (b) 50 nm Ta-capped Co 2FeAl thin
films. Themagneticfieldisappliedparalleltothefilmsurface,
at various angles ( 'H) with a respect to edges of the MgO
substrate ( [100]or[010]). (c) Thickness dependence of the
coercive field, deduced from hysteresis loops along the easy
axis, of Co 2FeAl Cr- and Ta-capped thin films.6
0 50 100 150 200 250 300 350-2.50.022.525.0-505253035
ML2-MT2MLMTML
CFA(50nm)/MgOKerr rotation (mdeg)
Sample orientation (degrees) Fit
FitMLMT
ML2-MT2ML
CFA(115nm)/MgOKerr rotation (mdeg) Fit
Fit
0 2 04 06 08 0 1 0 0 1 2 002420222426283032Amplitude, offset (mdeg)
Film thickness (nm) ML
Amplitude of MLMT
Amplitude of ML2-MT2
Offset of MLMT
Figure 6: (Colour online) (a) Separated quadratic MOKE
contributions as a function of the sample orientation at 46°
incidence. The fits are obtained using equation (12). (b) The
MLcontribution (at angle of incidence of 46°), the amplitudes
andoffsetofthe MLMTcontributionandtheamplitudeofthe
(M2
L M2
T) as a function of the Co 2FeAl thickness.
as evidenced in figure 5, while in contrast, for 'H= 90
, it leads to a two steps reversal, as can be seen in figure
5. The intermediate step leads to a magnetization nearly
perpendicular to the applied field. For all the studied
films a two steps loop is observed for 'Hranging in the
f55 130°ginterval. In figure 5c the deduced coercive
fields (HC) from hysteresis loops along the easy direction
('H= 0) are compared for different thicknesses (10, 20,
45, 50, 70, and 115 nm). For both Cr-capped and Ta-
capped films HC increases linearly with the inverse of the
filmthickness. TheCr-cappedsamplespresenthigherco-
ercive fields due to the different interface quality.
One can also observe that MOKE hysteresis loops are
not strictly centrosymmetrical (see for example Fig. 5b
for'H= 90) indicating the superposition of symmet-
rical (even function of applied sweep field HL) and anti-
symmetrical (odd in HL) components in the Kerr signal.
It has been shown and confirmed [26, 27] that, for in-
plane magnetized thin films, the antisymmetrical part
observed in the mL(HL)loops arises from the second or-der magneto-optical effects quadratic in magnetization.
Therefore, the present study was not limited to the usual
linear MOKE. We have also investigated this quadratic
contribution through the study of the Kerr signal depen-
dence upon the film orientation under a saturating in-
plane field. Within the cubic approximation for a (001)
surface, the Kerr rotation angle writes as [27]:
K=a1ML+a2(M2
L M2
T) sin(4 )+
(b2+ 2a2cos(4 ))MLMT(12)
WhereMLandMTstand for the longitudinal (i.e.:
within the incidence plane) and the transverse (i.e.: nor-
mal to the incidence plane) component of the magneti-
zation, respectively, and where is the angle of a cubic
<110>axis with the plane of incidence. The first term
describes the usual linear contribution while the follow-
ing ones correspond to the quadratic MOKE (QMOKE).
The experimental study was performed under an angle
of incidence of 46°using a field magnitude large with re-
spect to the anisotropy field. The different contributions
to the Kerr signal, as functions of the film orientation
are extracted by applying a rotating field technique [10].
Representative results obtained with 115 Cr- and 50 nm
Ta-capped films are shown in figure 6. Beside the lon-
gitudinal component ( ML) of the Kerr rotation, which
is dominant, the QMOKE signal, which is most proba-
bly due to the second order spin-orbit coupling [26], is
present. The derived ( M2
L M2
T) andMLMTangular
variations show the behaviour expected from the above
equation.
The values for the amplitudes of the 2MLMTand of
the (M2
L M2
T) contributions are the same within the
experimental error for each sample suggesting that the
applied cubic model is correct. The offset of the MLMT
contributionissmallerthantheamplitudes,butgenerally
it follows the same trend as the amplitudes. As the thick-
ness decreases the amplitudes and the offset decrease,
suggesting that the chemical order progressively changes
from the B2 to the A2 phase, as discussed above. More-
over, the amplitudes and offset values of CFA are compa-
rable to those measured for Co 2MnSi, which presents the
L21phase [28]. The TBIIST results are discussed in the
following section, in order to allow for a comparison with
the data derived from the FMR study of the dynamic
properties.
B. Dynamic properties
1. Exchange stiffness and effective magnetization
TheuniformprecessionandthefirstPSSWmodeshave
been observed in perpendicular and in-plane applied field
configurations for samples thicknesses down to 50 nm.
For the thickest film (115 nm) it was even possible to ob-
serve the second PSSW. For lower sample thickness, the7
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6246810121416d=115 nm
Fit
Fit
Fit
FitFrequency (GHz)
Magnetic field (kOe) Uniform mode: H=90°
PSSW mode: H=90°
Uniform mode: H=45°
PSSW mode: H=45°
14 15 16 17 18 1924681012141618Frequency (GHz)
Magnetic field (kOe) Uniform mode
PSSW1
PSSW2
Fit
Fit
Fitd=115 nm
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6246810121416182022
Fit
Fit
Fitd=50 nmFrequency (GHz)
Magnetic field (kOe) Uniform mode: H=0°
Uniform mode: H=45°
PSSW mode: H=0°
14 15 16 17 18 19 202468101214161820d=50nmFrequency (GHz)
Magnetic field (kOe) Uniform mode
PSSW mode
Fit
Fit
Figure 7: (Colour online) Field dependence of the resonance frequency of the uniform precession and of the two first perpendic-
ular standing spin wave excited (PSSW) mode of 115 nm Cr-capped and 50 nm Ta-capped Co 2FeAl films. The magnetic field
is applied perpendicular or in the film plane. The fits are obtained using equations (6) and (7) with the parameters indicated
in the Table I.
0.00 0.02 0.04 0.06 0.08 0.101213141516171819204Meff (kOe)
1/d (nm-1) Cr-capped films
Ta-capped films
Fit
Figure 8: (Colour online) Thickness dependence of the effec-
tive magnetization ( 4Meff) extracted from the fit of FMR
measurements. The solid lines are the linear fits.
PSSW modes are not detected due their high frequencies
over-passing the available bandwidth (0-24 GHz). Typ-
ical in-plane and perpendicular field dependences of theresonance frequencies of the uniform and PSSW modes
are shown on figure 7 for the 115 nm Cr- and the 50
nm Ta-capped films. By fitting the data in figure 7 to
the above presented model, the gyromagnetic factor (
),
the exchange stiffness constant ( Aex) and the effective
magnetization (4Meff) are extracted. The fitted
and
Aexvalues are 2.92 GHz/kOe and 1.5 µerg/cm, respec-
tively: they do not depend of the studied sample. The
derived exchange constant is in good agreement with the
reported one by Trudel et al. [7]. Meff=Heff=4
Figure 8 plots out the extracted effective magnetiza-
tion 4Meffversus the film thickness 1=d. It can be
seen thatMefffollows a linear variation. This allows
to derive the perpendicular surface anisotropy coefficient
K?S:K?S= 1:8erg/cm2. The limit of 4Meffwhen
1=dtends to infinity is equal to 12.2 kOe: within the
above mentioned experimental precision about the mag-
netization at saturation it does not differ from 4MS.
We conclude that the perpendicular anisotropy field de-
rives from a surface energy term; being negative, it pro-
vides an out-of-plane contribution. It may originate from
the magneto-elastic coupling arising from the interfacial
stress due to the substrate.8
0 50 100 150 200 250 300 35051015203690200400600800
F=6GHz
MeasurementsF=9GHzHPP (Oe)
Applied field direction H (degrees) MeasurementsH=255OeH=597Oe
Fit Measurements Measurements
F(GHz) FitF=6GHzF=9GHz
Measurements Measurementsd=50 nmHr (Oe)Fit
Fit
Fit Fit
0 50 100 150 200 250 300 3500100200300-40-2002040
Measurements
Fit(Oe)
Applied field direction H (degrees)d=50 nm
Measurements
FitH (Oe)
0 50 100 150 200 250 300 35012151821681012200400600
F=8GHzd=20 nmHPP (Oe)
Applied field direction H (degrees) Measurements H=828 Oe H=591 OeMeasurements Fit
FitF(GHz)Measurements F=8GHz
FitHr (Oe)
Measurements
Fit
0 50 100 150 200 250 300 3500200400600-50050
Measurements
Fit(Oe)
Applied field direction H (degrees) Measurements
FitH (Oe)d=20 nm
Figure 9: (Colour online) Angular dependence of the resonance frequency ( Fr), resonance field ( Hr), peak to peak field FMR
linewidth ( HPP), inverse susceptibility ( 1) and the field offset ( H) of 50 nm and 20 nm thick Co 2FeAl Ta-capped thin
films. The TBIIST measurements were obtained using transverse static bias field HB= 200Oe and 225 Oe respectively for 50
nm and 20 nm thick Co 2FeAl films. The solid lines refer to the fit suing the above mentioned models.
2. Magnetic anisotropy
Figure 9 shows the angular dependences of the reso-
nance field (at fixed frequency) and of the resonance fre-
quency (at fixed applied field) compared to the static
TBIIST measurements for three different CFA films.
Both FMR and TBIIST data show that the angular be-
havior is governed by a superposition of uniaxial and
fourfold anisotropy terms with the above-mentioned easy
axes. As noticed above, the symmetry properties of the
epitaxial observed films agree with the principal direc-
tions of the fourfold contribution. The fourfold and uni-
axialanisotropyfieldsextractedfromthefitoftheexperi-
mentalTBIISTandFMRdatausingtheabove-presented
model are drawn on figure 10 and summarized in Table I:
the compared results issued from the two techniques are
in excellent agreement. For all the samples the fourfold
anisotropy is dominant. While the uniaxial anisotropy
field (H2) of the Cr-capped films is small and does notseem to depend upon the thickness, in the Ta-capped
filmsH2is higher, maybe due to interface effects, and is a
decreasing function of the thickness (Figure 10). As sug-
gested previously, we believe that the uniaxial anisotropy
is induced by the stepping of the substrates, probably
resulting from a small miscut along their [100]crystallo-
graphicdirectioncorrespondingtothe [110]studiedfilms.
The reduced effect of the steps of the substrate when the
thickness increases could then explain the thickness de-
pendence of H2. However, up to now we have no com-
pletelysatisfyinginterpretationofthepresenceof H2and
of its variations versus the nature of the film capping.
The fourfold anisotropy fields ( H4) are comparable for
Cr- and Ta-capped films and decrease when their thick-
ness increases, as seen in figure 10. For large values of
d(45nm or higher) H4lies around 200 Oe and shows
a small linear variation versus the in-plane strain "k, as
shown in the insert of figure 10. This evolution confirms
a direct correlation between the H4 field and the in-plane
biaxial strain for the films with thicknesses above 45 nm.9
0.00 0.02 0.04 0.06 0.08 0.10 0.120102030402004006008001000
%H4 (Oe)Anistropy fields (Oe)
1/d(nm-1) H2:Cr-capped
H4:Cr-capped
H2:Ta-capped
H4:Ta-capped
B2 A20.5 0.6 0.7200205210215220225
Figure 10: (Colour online) Thickness dependence of the uni-
axial (H2) and the fourfold anisotropy fields ( H4) extracted
from the fit of FMR measurements. The solid lines are the
linear fits. The inset shows the evolution of the H4field, for
the 45, 70 and 115 nm thick samples, with the in-plane biaxial
strain.
At smaller values of d(10 or 20 nm) a large increase
ofH4, up to 920 Oe, is observed. It is presumably re-
lated to the B2)A2 phase transition observed through
X-rays diffraction. The observed symmetry argues for
a magneto-crystalline contribution, which, as previously
observed [29, 30], would be higher in phase A2 than in
phase B2.
3. FMR linewidth
In figure 9, the FMR peak to peak linewidth (( HPP)
is plotted as a function of the field angle 'Hfor the 50
nm and 20 nm Ta-capped CFA films using three driv-
ing frequencies: 6, 8, and 9 GHz. HPPis defined as
the field difference between the extrema of the sweep-
field measured FMR spectra. All the other samples show
a qualitatively similar behaviour to one of the samples
presented here. The positions of the extrema depend on
the sample. The observed pronounced anisotropy of the
linewidth cannot be due to the Gilbert damping contri-
bution, which is expected to be isotropic, and must be
due to additional extrinsic damping mechanisms. In the
50 nm thick sample, the HPPangular variation shows
a perfect fourfold symmetry (in agreement with the vari-
ation of the resonance position). Such behaviour is char-
acteristic of two magnon scattering. This effect is cor-
related to the presence of defects preferentially oriented
along specific crystallographic directions, thus leading to
anasymmetry(seeequation(11)). Concerningthe20nm
thick film, the in-plane angular dependence of HPPis
less simple and shows eight maxima, that is expected
from a mosaicity driven linewidth broadening. It can be
4 6 8 1 01 21 41 61 82 02 22 410203040506070
10 nm,
H=90°10 nm,
H=45°20 nm
50 nm70 nmFMR linewidth HPP (Oe)
Frequency (GHz)Figure 11: (Colour online) Frequency dependence of the easy
axis ('H= 0) peak to peak field FMR linewidth ( HPP) for
Co2FeAl thin. The solid lines refer to the fit using equations
(8-11).
seen that a smaller fourfold symmetry (four maxima) is
superimposed on the eight maxima, indicating that two
magnon scattering is still present. Therefore, the entire
angular dependence of the FMR linewidth in our samples
can be explained as resulting of the four contributions
appearing in equation (8).
In figure 11, HPPfor the field parallel to an easy axis
and a hard axis ( 'H= 45 °for 10 nm thick sample) of
the fourfold anisotropy is plotted as a function of driving
frequency for all samples. An apparently extrinsic contri-
bution to linewidth was observed, which increased with
decreasing film thickness. It should be mentioned that
the observed linear increase of the linewidth with fre-
quency in figure 11 maybe due to Gilbert damping but
other relaxation mechanisms can lead to such linear be-
haviour. Therefore, only an effective damping parameter
effcan be extracted from the slope of the curves and
ranges between about 0.00154 for the easy axis of the 50
nm thick film and 0.0068 for easy axis the thinnest film.
The pertinent parameters could thus be, in principle de-
rived from the conjointly analysis of the frequency and
angular dependence of HPP. However, due to the lim-
ited experimental precision, some additional hypotheses
are necessary in order to allow for a complete determi-
nation of the whole set of parameters describing the in-
trinsic Gilbert damping and the two magnon damping.
A detailed analysis is presented in the appendix. Using
the previously reported value: = 1:110 3[31], which
is in agreement with our experimental results, we were
able to 0for each film. 0, 2, 4,'2,'4are listed in
Table II which also contains the parameters describing
the damping effects of the mosaïcity ( 'H) and of the
inhomogeneity contribution ( Hinh).
The two magnon and the mosaïcity ( 'H) contribu-
tions to HPPincrease when the thickness decreases,
probably due to the progressive above reported loss of10
chemical order. The increase of the residual inhomo-
geneities linewidth ( Hinh) with the thickness is most
probably due the increase of defects and roughness. The
uniaxial term 2is observed only in the thinnest (20 and
10 nm) samples. As expected, '4= 0, but the sign of
4is sample dependent. Finally, it is important to no-
tice that the very low value of the intrinsic damping in
the studied samples allows for investigating the extrinsic
contributions.
V. CONCLUSION
Co2FeAl films of various thicknesses (10 nm d115
nm)) were prepared by sputtering on a (001) MgO sub-
strate. They show full epitaxial growth with chemical
order changing from B2 to A2 phase as thickness de-
creases. MOKE and VSM hysteresis loops obtained with
different field orientations revealed that, depending on
the direction of the in-plane applied field, two or one
jump switching occur, due to the superposition of uni-
axial and fourfold anisotropies. The samples present a
quadratic MOKE contribution with decreasing ampli-
tudes as the CFA thickness decreases. The microstrip
ferromagnetic resonance (MS-FMR) and the transverse
biased initial inverse susceptibility and torque (TBIIST)
methods have been used to study the dynamic proper-
ties and the anisotropy. The in-plane anisotropy presents
two contributions, showing a fourfold and a twofold ax-
ial symmetry, respectively. A good agreement concern-
ing the relevant in-plane anisotropy parameters deduced
from the fit of MS-FMR and TBIIST measurements has
been obtained. The fourfold in-plane field shows a thick-
ness dependence behavior correlated to the thickness
dependence of the chemical order and strain in sam-
ples. The angular and frequency dependences of the
FMR linewidth are governed by two magnon scattering,
mosaïcity and by a sample independent Gilbert damping
equal to 0.0011
Appendix
In the section dealing with the discussion of the FMR
linewidth measurements we stated that the conjointly
analysis of the frequency and angular dependence of
HPPdoes not allow for the determination of all the
parameters given in equation (8) and additional hypoth-
esisshouldbedone. Theaimofthisappendixistoclarify
themannerinwhichtheparameterssummarizedinTable
II is done.
For most of the exploitable measurements the mi-
crowave frequency f during the HPPmeasurements is
not larger than f0and generally smaller ( f0varies from
18.5 to 28.5 GHz, depending on the film thickness). It
then results that the two magnon damping is practically
proportional to f and that the sum of the Gilbert and ofthe two magnon damping terms reads as (see equations
(9) and (11)):
HGi+2mag=(( p
3+ 0
2Heff)+ 2
2Heffcos 2('H '2)
+ 4
2Heffcos 4('H '4))4
f(13)
It is not possible to completely identify the respec-
tive contributions of the Gilbert and of the two magnon
damping, only according to equation (13). The quasi-
linear variation versus the frequency (Fig. 11) observed
forHPPallows for defining an effective damping pa-
rameter eff, which, is angle dependent due to two
magnon scattering. The experimentally derived coeffi-
cient eff, from the linear fit of data presented in figure
11, varies from 0.0068 to 0.00154. Furthermore, the mea-
sured angular variation of the linewidth allows for evalu-
ating ( 2,'2) and ( 4,'4) but, concerning the isotropic
terms appearing in equation (13), only the sum +p
3 0
2Heff
can be derived. However, remembering that cannot be
negative, the maximal available value of 0(correspond-
ing to = 0) is easily found. Moreover, a lowest value
can be obtained for 0noticing that equation (13) can
also be written:
HGi+2mag=(( p
3+ 0 j 2j j 4j
2Heff)+
j 2j
2Heff(1cos 2('H '2))+
j 4j
2Heff(1cos 4('H '4)))4
f(14)
where the adequate third and the fourth terms rep-
resent the twofold and the fourfold contributions, which
take into account that both of them are necessarily non-
negative for any value of 'H. The additional residual
two magnon isotropic contribution cannot be negative.
Hence: 0>j 2j+j 4j.
Introducingthisminimalaccessiblevalueof 0, (j 2j+
j 4j), the maximal value of the Gilbert coefficient is
then easily obtained. To summarize, for each sample the
experimental data provide the allowed intervals for and
for 0, respectively [0, min] and [(j 2j+j 4j), 0Max],
and indeed, the chosen value of within [0, min] allows
for deducing 0. The smallest calculated interval for ,
equal to [0, 1:410 3] is obtained for the 70 nm film.
A previous publication by Mizukami el al. [31] has con-
cluded to a Gilbert coefficient equal to: = 1:110 3.
We then stated that = 1:110 3and, consequently,
we were able to deduce 0for each film. 0, 2, 4,'2,
'4are listed in Table II which also contains the parame-
ters describing the damping effects of the mosaïcity and
of the inhomogeneity.11
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1812.08404v1.Laser_Controlled_Spin_Dynamics_of_Ferromagnetic_Thin_Film_from_Femtosecond_to_Nanosecond_Timescale.pdf | 1
Laser Controlled Spin Dynamics
of Ferromagnetic Thin Film
from
Femtosecond to Nanosecond Timescale
Sucheta Mondal and Anjan Barman
*
Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for
Basic Sciences, Block JD, Secto
r III, Salt Lake, Kolkata 700 106, India.
*
abarman@bose.res.in
Key words: (
Thin Film Heterostructures, Ultraf
a
st Demagnetization, Gilbert Damping, Time
-
resolved
Magneto
-
optical Kerr Effect
)
Laser induced modulatio
n of the
magnetization dynamics
occurring over various time
-
scales
have been unified
here
for
a
Ni
80
Fe
20
thin
film excited
by
amplified
femtosecond laser pulses.
The weak correlation between demagnetization time and pump fluence with substantial
enhancemen
t in remagnetization time is demo
n
strated using three
-
temperature model
considering the temperatures of electron, spin
and
lattice.
The
picosecond
magnetization
dynamics is modeled using
the L
andau
-
Lifshitz
-
Gilbert equation.
W
ith increasing pump fluence
th
e Gilbert damping parameter shows significant enhancement
from its intrinsic value
due to
increment in the
ratio
of electronic temperature to Curie temperature within very short time
scale. The precessional frequency experiences
noticeable
red shift with i
ncreasing pump fluence.
The changes in the local magnetic properties due to accumulation and dissipation of thermal
energy within the probed volume are described by the
evolution of
temporal chirp parameter in a
comprehensive manner.
A unification of ultra
fast magnetic processes and its control over broad
timescale would enable the integration of various magnetic processes in a single device and use
one effect to control another.
2
I. INTRODUCTION
Recent development in
magnetic
storage
[1]
and
memory
[2
]
device
s
heavily relies
up
on
increasing
switching
speed and
coherent
switching
of
magnetic states
in
ferromagnetic thin films
and patterned structures
.
O
perating speeds of information storage devices have progressed into
the
sub
-
gig
ahertz
regime
and controlled switching in
individual
layers of magnetic
multilayers
and hetero
structures
has become
important
.
The relaxation processes
involved in magnetization
dynamics set
natural limit
s
for
these
switching times and data transfer rates.
In the context of
precessional
magnetization dynamics
the
natural
relaxation
rate
against the small perturbation is
expressed as Gilbert damping
(
α
)
according to the Landau
-
Liftshiz
-
Gilbert (LLG) equation
[3,
4]
.
This
is analogous to viscous damping of the mechanical frictional torque
and
leads to the
direct dissipation of energy from the uniform precessional mode to thermal bath in case of zero
wav
e
-
vector excitation.
Gilbert damping
originates from spin
-
orbit coupling and depends on the
coupling strength and
d
-
band width of the
3
d
ferromagnet
[5]
. Th
e damping
can be
varied
by
various
intrinsic and extrinsic
mechanisms including
phonon drag
[6]
, Edd
y current
[7],
doping
[8]
or
capping
[9]
with other material
, injection of spin current
[10]
, magnon
-
magnon scattering
[11]
and
controlling
temperature of
the system
[12]
.
Intri
n
sic and extrinsic
nature of Gilbert
damping
were primarily studied by using fe
rromagnetic resonance (FMR) technique. When the
magnetization is aligned with either in
-
plane or
out
-
of
-
plane
applied magnetic field, the
linewidth is proportional to the frequency with a slope determined by
damping co
ef
f
icient. This
is the homogeneous or
intrinsic contribution to the FMR linewidth. However, experiments show
an additional frequency
-
independent contribution to the linewidth
corresponds to
inhomogeneous line broadening
[13, 14]
.
However
,
state
-
of
-
the
-
art technique based on
pump
-
probe geomet
ry
has been developed and rigorously exploited for measuring ultrafast
magnetization dynamics of ferromagnetic thin films during last few decades
[15, 16]
.
Using
time
-
resolved magneto
-
optical Kerr effect (TR
-
MOKE)
technique
one can directly address the
pro
cesses which are responsible for the excitation and relaxation of a magnetic system on their
characteristic time scales
[17
-
19]
.
Generally
during
the pump
-
probe measurements pump fluence
is kept low to avoid nonlinear effects and sample surface degradation
. Some recent experiments
reveal that nonlinear spin waves play a
crucial
role in high power thin film precession
al
dynamics by introducing spin
-
wave instability
[20]
similar to
FMR
experiments
by
appl
ication
of
high rf power
[21]
.
The coercivity and aniso
tropy of the ferromagnetic thin films
can
also
be
lowered by pump fluence
,
which
may
have potential application in heat assisted
magnetic
recording
(HAMR)
[22]
.
Recent
report
s
reveal
that damping
coefficient
can be increased or
decreased noticeably in the
higher excitation regime due to opening of further energy dissipation
channels
beyond a threshold
pump power
[23
-
25]
.
Not only relaxation parameters but also
frequency
shift
due to enhancement in pump power
has been
observed
[20]
.
However,
the
experimental
evidence for
large
modulation of Gilbert damping along with frequency
shift
and
temporal chirping of the uniform precessio
n
al
motion
is absent
in the literature
.
This
investigation demands suitable choice of material,
and here we have chosen
Permalloy
(Ni
80
Fe
20
3
or Py here
on)
because
of its
high permeability,
negligible magneto
-
crystalline anisotropy,
very
low coercivity, large anisotropic magnetoresistance with reasonably low damping. Also,
due to
its
negligible
magnetostriction P
y
is less sensitive to st
rain and stress exerted during the thermal
treatment in
HAMR
[22]
.
In this
article
,
we have
used
femto
-
second amplified laser
pulses
for excitation and detection of
ultrafast magnetization
dynamics in
a
P
y
thin
film. Pump fluence dependent ultrafast
demag
netization is
investigated
along with fast
and slow
remagnetization
.
Our comprehensive
study
of
the
picosecond
dynamics
reveals transient nature of enhanced Gilbert damping in
presence of high pump fluence
. Also
,
the time
-
varying precession is subjected to
temporal
chirping
which occurs due to enhancement of temperature of the probed volume within a very
short time scale
being followed by
successive
heat dissipation.
This fluence dependent
modulation of magnetization dynamics will undoubtedly found suitable
application
in spintronic
and magnonic devices
.
II. SAMPLE PREPARATION AND CHARACTERIZATION
20
-
nm
-
thick Permalloy (Ni
80
Fe
20
, Py hereafter) film was deposited by using electron
-
beam
evaporation technique (SVT Associates, model: Smart Nano Tool AVD
-
E01) (ba
se pressure = 3
× 10
−8
Torr, deposition rate = 0.2 Å/S) on 8 × 8 mm
2
silicon (001) wafer coated with 300
-
nm
-
thick SiO
2
. Subsequently, 5
-
nm
-
thick SiO
2
is deposited over the Ni
80
Fe
20
using rf sputter
-
deposition technique (base pressure = 4.5 × 10
−7
Torr, Ar pressure = 0.5 mTorr
, deposition r
ate =
0.2 Å/S, rf power = 60 W).
This capping layer protects the surface from environmental
degradation, oxidation and laser ablation during the pump
-
probe experiment using femtosecond
laser pulses.
From the vibrating sample magnetometry (VSM
) we have obtained the saturation
magnetization (M
s
) and Curie temperature (T
c
) to be 850 emu/cc and 86
3
K
,
respectively
[26]
.
To study the ultrafast magnetization dynamics of this sample, we have used a custom
-
built time
resolved magneto optical Kerr eff
ect (TRMOKE) magnetometer based on optical pump
-
probe
technique as shown in Fig. 1 (a). Here, the second harmonic (λ = 400 nm, repetition rate = 1 kHz,
pulse width > 40 fs) of amplified femtosecond laser pulse generated from a regenerative
amplifier system
(Libra, Coherent) is used to excite the dynamics while the fundamental laser
pulse (λ = 800 nm, repetition rate = 1 kHz, pulse width ≈ 40 fs) is used as probe to detect the
time
-
resolved polar Kerr signal from the sample. The temporal resolution of the me
asurement is
limited by the cross
-
correlation between the pump and probe pulses (
≈120 fs). The probe beam
having diameter of about 100 µm is normally incident on the sample whereas the pump beam is
kept slightly defocused (spot size is about 300 µm) and is
obliquely (
≈ 30
◦
with normal to the
sample plane) incident on the sample maintaining an excellent spatial overlap with the probe
spot. Time
-
resolved Kerr signal is collected from the uniformly excited part of the sample and
slight misalignment during the
course of the experiment does not affect the pump
-
probe signal
significantly. A large magnetic field of 3.5 kOe is first applied at a small angle of about 10° to 4
the sample plane to saturate its magnetization. This is followed by reduction of the magnetic
field to the bias field value (
H
= in
-
plane component of the bias field), which ensures that the
magnetization remains saturated along the bias field direction. The tilt of magnetization from the
sample plane ensures a finite demagnetizing field along the
direction of the pump pulse, which is
further modified by the pump pulse to induce a precessional dynamics within the sample
[17]
. In
our experiment a 2
-
ns time window has been used, which gave a damped uniform precession of
magnetization. The pump beam is
chopped at 373 Hz frequency and the dynamic signal in the
probe pulse is detected by using a lock
-
in amplifier in a phase sensitive manner. Simultaneous
time
-
resolved reflectivity and Kerr rotation data were measured
and no significant breakthrough
of one
into another has been found
[26]
.
The probe fluence is kept constant at 2 mJ
/
cm
2
during
the measurement to avoid additional contribution to the modulation of spin dynamics via laser
heating. Pump fluence (
F
) was varied from 10 to 55 mJ
/
cm
2
to study the fl
uence dependent
modulation in magnetization dynamics. All the experiments were performed under ambient
condition and room temperature.
III. RESULTS AND DISCUSSIONS
A.
Laser
induced ultrafast demagnetization
When a femtosecond laser pulse
interacts
with
a
ferromagnetic
thin film in its saturation
condition
,
the magnetization of the system is partially or fully lost within hundreds of
femtosecond as measured by the time
-
resolved Kerr rotation or ellipticity.
This is known as
ultrafast
demagnetization of the
ferromagnet
and was first observed by Beaurepire et al. in 1996
[
27
]
.
This is generally followed by a fast recovery of the magnetization within sub
-
picosecond to
few picosecond
s
and a slower recovery within tens to hundreds of picoseconds, known as the fa
st
and slow remagnetization
.
In many cases the slower recovery is accompanied by a coherent
magnetization precession and damping [
17
].
In
our
pump
-
probe experiment,
the sample
magnetization is maintained in
the saturated state by application of a magnetic
field
H
=
2.4 kOe
before zero delay
.
Right after the zero
-
delay
and the
interaction
of the pump pulse
with the
electrons in the
ferromagnetic
metal, ultrafast demagnetization takes place
.
The local
magnetization is immediately quenched within first few hun
dreds of fs
followed by a subsequent
fast
remagnetization
in next
few ps
[27]
.
Figure
1
(b) shows ultrafast demagnetization
obtained
for
different pump fluences.
Several models have been proposed over
two decades to explain the
ultrafast demagnetization
[16
, 28
-
31]
.
Out of those
a phenomenological thermodynamic model
,
called three temperature m
odel
[
27,
32, 33]
has been most widely used
,
where the dynamics of
these spin fluctuations can be describes as:
)
(
)
)(
(
0
)
(
2
1
)
(
1
2
1
t
M
e
A
A
e
A
A
A
t
M
lat
el
sp
el
t
lat
el
sp
el
lat
el
t
sp
el
lat
el
sp
el
lat
el
(1)
.
This is an approximated
form based on the assumption that the electron temperature rises
instantaneously upon laser excitation
and can be applied to fit time
-
resolved
Kerr rotation
data
taken within few picoseconds timescale
.
T
he whole system i
s divided into three subsystems: 5
el
ectron, spin and lattice system. On laser excitation the hot electrons are created above Fermi
level. Then during energy rebalancing between the subsystems
,
quenched magnetization relaxes
back to the initial state.
The two exponential functions
in the abov
e equation
mirror the
demagnetization given by demagnetization time (τ
el
-
sp
) for energy transfer between electron
-
spin
and the decay of electron temperature
(
τ
el
-
lat
)
owing to the tr
ansfer of energy to the lattice. In
addition to these characteristics time
constants, the spin
-
lattice relaxation time also can be
extracted
by including another exponential term in the above equation
if the spin specific heat is
taken into account [
34
]
.
θ
is the Heaviside step function and
Γ(t)
stands for
the Gaussian function
to be convoluted
with the laser pulse envelope determining the temporal resolution
(showing the
cross c
orrelation between the probe an
d pump pulse)
.
The constant,
A
1
indicates the ratio
between amount of magnetization after equilibrium between electrons, s
pins, and lattice is
restored and the initial magnetization. A
2
is proportional to the initial electronic temperature rise.
We have plotted A1 and A2, normalized with their values at the highest fluence, as a function of
pump fluence in Fig.
3S
of the supp
lemental material
which shows that magnitude of both
parameters increases with
laser
fluence
[26]
.
We have
observed
that with increasing fluence the
demagnetization time has been
negligibly varied within a range of 250
±40
fs.
The weak or no
correlation bet
ween the pump fluence and the demagnetization rate describes the intrinsic nature
of the spin scattering
, governed by various mechanisms including Elliott
-
Yafet mechanism
[
35
]
.
Another
important
observation here is that
the delay of demagnetization process
es
which is the
time delay between pump pulse (full width at half maxima, FWHM
≈ 130±20 fs) and starting
point of the ultrafast demagnetization,
becomes shorter due to increase in pump fluences. A
plausible explanation for this is the dependence of
delay o
f
demagnetization on the electron
-
thermaliz
ation
time which is eventually proportional to electron density or pump fluences
[
3
6
]
.
On the other hand
,
fast
remagnetization
time has been found to be increased noticeably from
0.40
±
0.05 ps to 0.8
0
±
0.05 ps w
ithin the experimental fluence range
of 10
-
55 mJ/cm
2
. The
larger is the pump fluence, the higher is the electron temperature or further the spin temperature.
Therefore, it is reasonable that magnetization recovery time increases with the pump fluence.
B.
Pump fluence dependent modulation in Gilbert damping
F
ig
ure
1
(c) shows the representat
ive Kerr rotation data for
F
= 25
mJ/cm
2
consisting of three
temporal
regions
,
i.e.
ultrafast
demagnetization
, fast remagnetization and slow
remagnetization
superposed
with
damped
precession within
the time window
of 2 ns
. We
process
the
magnetization
precession part
after subtracting
a
bi
-
exponential background
to estimate the
damping and its modulation
.
The slower remagnetization is
mainly
due to heat diffusion from th
e
lattice to the substrate and surrounding. Within our experimental fluence range the slow
remagnetization time has increased from
≈0.4 ns to ≈1.0 ns.
The precessiona
l dynamics is
described
by phenomenological Landau
-
Lifshitz
-
Gilbert
(LLG)
equation,
dt
M
d
M
M
H
M
dt
M
d
s
eff
(2)
6
where
γ
is the gyromagnetic ratio,
M
is magnetization
,
α
is Gilbert damping constant
and
H
eff
is
the effective magnetic field consisting of
several field components.
The
variation
of precessional
frequency with the angle between sample plane and bias
magnetic
field direction is plotted in
F
ig.
1
(d
), which
reveals that there is no uniaxial anisotropy present in this sample.
The energy deposit
ed by the pump pulse, in terms of heat within the probed volume, plays a very
crucial role in modification of local magnetic properties
,
i.e. magnetic moment, anisotropy,
coercivity, magnetic susceptibility
,
etc. With increasing fluence the precessional fr
equency
experienced a red shift
[20, 25]. Thus, at the onset of the precessional dynamics (about 10 ps
from zero delay), for relatively high fluence, the initial frequency (
f
i
) will be smaller than its
intrinsic value (in absence of any significant heat di
ssipation). As time progresses and the sample
magnetization gradually attains its equilibrium value, the precessional frequency continuously
changes, causing a temporal chirping of the damped oscillatory Kerr signal.
The frequency shift
can be
estimated
fr
om the amount of temporal chirping
[
3
7
].
Figure
2
(a) shows the background
subtracted
time
-
resolved Kerr rotation data (
precessional
part)
for different pump fluences fitted
with
a
damped sinusoidal function with added temporal chirping
,
)
)
(
2
sin(
/
t
bt
f
Ae
i
t
k
where
A
,
τ
,
f
i
, b
and
Φ
are the amplitude of the magnetization precession, the relaxation time, the
initial precessional frequency,
chirp parameter
and
initial
phase, respectively.
At this point, we
are unsure of the exact nature of the damping,
i.e.
it may consis
t of both intrinsic and extrinsic
mechanisms and hence we term it as
effective damping parameter
(
α
eff
)
which can be
extracted
using the following formula
[3
8
]
,
)
2
4
(
1
eff
eff
M
H
(
3
)
γ
= 1.
83
×10
7
Hz/
Oe
for Py
an
d
M
eff
is the effecti
ve magnetization including pump
-
induced
changes
at
H
= 2.4 kOe
.
This formula is exploited to extract effective damping parameter
precisely
in the moderate bias fi
eld regime.
The variation of relaxation time and effective
damping
are
plotted with pump fluence in
F
ig.
2
(b) and (c).
Here,
τ
decreases with fluence while
damping increases
significantly with respect to
its
intrinsic value within this fluence range. We
h
ave repeated the experiment for two different field values (2.4 and 1.8 kOe). The slope of
fluence dependent damping remains unaltered
for both the field values
.
We have also observed
increase in relative amplitude
s
of precession with pump fluence as shown
in the inset of Fig. 2
(c).
To verify the transient nature of damping we have performed another set of experiment
where the probed area is exposed to different pump fluences
(
F
i
)
for several minutes. After the
irradiation, the precessioanl dynamics is mea
sured from that area
with fixed probe and pump
fluences 2 and 10 mJ/cm
2
, respectively. We found that damping remains almost constant for all
the measurements
(as shown in Fig.
2 (d))
. These results demonstrate that the enhancement of 7
damping is transient a
nd only exists in the presence of high pump fluence but dropped to its
original value when the pump laser was set to initial fluence.
The bias field dependence of precessional dynamics at four different pump fluence
s
is studied to
gain more insight about
the origin of fluence dependent damping.
First, we plotted the average
frequency
(
f
FFT
)
with bias field which is obtained from the fast Fourier transformation (FFT) of
the precessional data in
F
ig.
3
(a). The experimental data points are fitted with the Ki
ttel formula,
)
4
(
2
eff
FFT
M
H
H
f
(
4
)
M
eff
is the effective magne
tization of the sample. Figure
3
(b) shows that effective magnetization
does not v
ary much within the applied fluence range. So
,
we infer that
with increasing fluence
there is no
induced
anisotropy
developed in the system
which can modify the effective damping
up to this extent
[23]
. The variation of relaxation time with bias field for
four different pump
fluences are plotted in
F
ig.
3
(c). Relaxation
time is increased with decreasing
field for each case
but for the higher fluence regime, those value
s seem
to be fluctuating.
This depend
ence of τ on
field
was fitted with eq
uation
3 to extract damping coefficient at different fluence values. We
have further plotted the damping coefficient as a function of precession frequency (
f
FFT
)
[see
supplementa
l
material
,
F
ig.
4
S
]
[26]
, which shows a
n invariance of
α
eff
with
f
FFT
. From that we
can infer that the damping coefficient in our sample within the experimental field and fluence
regime are intrinsic in nature and hence, we may
now
term it as the intrinsic damping coefficient
α
0
.
The extrinsic
contributions to damping mainly come from magnetic anisotropy field, two
-
magnon scattering, multimodal dephasing for excitation of several spin
-
wave modes, etc, which
are negligible in our present case.
F
igure
3
(d)
shows the variation of
α
0
with pump flue
nce, which shows that even the intrinsic
damping is significantly increasing with pump fluence
[20,
3
9
]
.
For generation of perpendicular
standing spin
-
wave modes the film needs to be thick enough. Though the film thickness is 20 nm
here, but within the app
lied bias field range we have not found any other magnetic mode
appearing with the uniform Kittel mode within the frequency window of our interest
(as shown
in
F
ig.
5
S of suppleme
n
tal material
)
[26]
.
Also
,
for 20
-
nm
-
thick
Py
film,
the effect of eddy
curre
nt will be negligible
[
40
]
.
The overlap between spatial profile of focused probe and pump
laser spot may lead to the generation of magnons that propagate away from the region that is
being probed.
Generally
,
enhancement of
nonlocal damping by spin
-
wave emi
ssion becomes
significant
when the excitation area is less than 1
µm
. Recently
J.
Wu
et al.
showed that
propagation of magnetostatic spin waves could be significant even for probed regions of tens of
microns in size
[
4
1
]
.
Also
,
by generating spin
-
wave trap
in the pump
-
probe experiment
modification of precessional frequency in ferromagnetic thin film due to accumulation and
dissipation of thermal energy within the probed volume has been reported
[
4
2
]
. D
uring
our
experiment
the
overlap between
probe
and pump
spot
is
maintai
ned carefully
and
Kerr signal is 8
collected from the uniformly excited part of the sample so that slight misalignment during the
course of experiment does not
introduce
any
nonlocal effects
.
We will now substantiate our
results with some theo
retical arguments which involve the calculation of electronic temperature
rise in the system due to
application of higher
pump fluence. The electronic temperature (
T
e
) is
related to absorbed laser energy per unit volume (
E
a
) according to the following
equa
tion
[
4
3
]
,
2
/
)
(
2
0
2
T
T
E
e
a
(
5
)
where,
ξ
is the electronic specific heat of the system and
T
0
is
the
initial electronic temperature
(room temperature here).
First,
we have estimated
E
a
according to the optical parameters of the
sample
by using the following equation,
]
/
)
1
(
)
1
[(
d
R
F
e
E
d
a
(
6
)
where
,
d
is sample thickness,
Ψ
is optical penetration depth (
~
17 nm for
400
-
nm pump
laser
in
20
-
nm
-
thick Py
film
),
R
is the reflectivity of the sample
(0.
5
measured
for
the
Py
film
)
and
F
is
applied pump fluence.
By solving equations (
5
) and (
6
)
we have
observed
that
T
e
increases from
≈
1800
to
4
5
00 K within our experimental fluence range
of
10
to
55 mJ/cm
2
.
Decay time of the
electron temperature and other r
elevant parameters (i.e.
E
a
, T
e
at various flue
nce
s
) are described
in the
supplementa
l
material
[26]
.
The sample remains in its magnetized state even if the
electronic temperature exceeds the
Curie temperature
T
c
.
Importantly, ratio of the system
temperat
ure
,
T
(as decay of electronic temperature is strongly correlated with rise of lattice
temperature)
to
T
c
is affecting the magnetization relaxation time
which
fundamentally depends
on susceptibility
. Accordingly damping
should
be proportional to susceptibi
lity
which
is
strongly
temperature dependent
[
40
]
.
Various procedures for exciting precessional dynamics in
ferromagnets show the different mechanisms to be responsible for exploration of different energy
dissipation channels. The spin
-
phonon interaction m
echanism, which historically has been
thought to be the main contribution to magnetization damping, is important for picosecond
-
nanosecond applications at high temperatures such as spin caloritronics. But for laser
-
induced
magnetization dynamics, where spi
n
-
flips occur mainly due to electron scattering, quantum
Landau
-
Lifshitz
-
Bloch equation is sometimes exploited to explain the temperature dependence
of damping by considering a simple spin
-
electron interaction as a source for magnetic relaxation
[4
4
]
. This
approach suggests that increasing ratio between
system
temperature and Curie
temperature
induces electron
-
impurity
led
spin
-
dependent scattering. Even slightly below
T
c
a
pure change in the magnetization magnitude oc
c
urs
which causes
the
enhancement of
da
mping
.
Also our experimental results revea
l that the precession amplitude and
damping have been
subjected to a sudden change for F > 30 mJ/cm
2
. Energy density deposited in the probed volume
is proportional to pump fluence. For higher fluence, the temperatu
re dependence of the electronic
specific heat plays major role
.
The
increase in the electronic
specific heat
value
with temperature 9
may lead
to
longer thermal
-
relaxation time
. We infer that relative balance between the energy
depo
sited
into the lattice and electron system is
also
different for higher fluence regime
compared to that in the lower fluence regime. Thus
,
the system temperature remains well above
Curie temperature for F > 30 mJ/cm
2
, during the onset of precession for t
≥
10 ps. This may open
up additional energy dissipation channel for the magnetization relaxation process over
nanoseconds time scale.
Sometimes within very short time scale the spin temperature can go
beyond the Curie temperature leading towards formation o
f paramagnetic state but that is a
highly non
-
equilibrium
case
[
45
]
.
However
we believe that
in our experiment,
even for the
high
fluence limit and
in
local thermal
equilibrium
the ferro
magnetic
to paramagnetic tra
n
sition is not
observed
.
R
epetitive measur
ements established
the
reversibility of the damping parameter
and
bias
-
magnetic
-
field dependence of precessional frequency confirms ferromagnetic
nature
of the
sample
.
C.
Frequency modulation and temporal chirping
Pump fluence also eventua
lly modulates the precessional frequency by introducing temporal
chirping in the uniform precession. After immediate arrival of pump pulse, due to enhancement
of the surface temperature, the net magnetization is reduced in
picosecond
time scale which
resul
ts in chirping of the precessional
oscillation
. The initial frequency
(
f
i
)
is reduced with
respect to its intrinsic value at a constant field. But when the probed volume cools with time, the
spins try to retain their original precessional frequency. Thus
,
within a fixed time window, the
average frequency (
f
FFT
)
also undergoes slight modification. In the high fluence regime,
significant red shift is observed in both
f
FFT
and
f
i
. For
H
= 2.4 and 1.8 kOe, modulation of
frequency is found to be 0.020 GHz.cm
2
/mJ
for
f
FFT
and 0.028 GHz.cm
2
/mJ for
f
i
, from the slope
of linear fit (as shown in
F
ig. 4(a)). The
f
FFT
is redu
ced by 7.2
% of the extrapolated value at zero
pump fluence for both the fields.
On the other hand,
f
i
is decreased
by
8.7% of its zero pump value
f
or the highest pump fluence
.
The temporal chirp parameter
,
b
shows giant enhancement within the experimental fluence range
(
F
ig.
4
(b)).
For
H
= 2.4 kOe,
b
has increased up
to
ten
times (from 0.03 GHz/ns to 0.33 GHz/ns)
in this fluence limit which implies a
n increase in frequency of 0.66 GHz. Within our
experimental
scan
window (2 ns)
, the maximum frequency
shift
is found to be 4.5% for
F
= 55
mJ/cm
2
.
For another bias field (
H
= 1.8 kOe), the enhancement of chirp parameter follows the
similar
trend.
This
ult
rafast
modulation is attributed to the
thermal effect
on the
local magnetic
properties within
the
probe
d
volume
and
is inferred to be reversible
[
3
7
]
.
We
have also
plotted
the variation of
b
with applied bias field for four different pump
fluencies
.
I
t see
ms to be almost
constant for all the
field values
in moderate fluence regime
(as shown in
F
ig.
4
(c))
.
But for
F
=
40 mJ/cm
2
, data points are relatively scattered and large errors have been considered to take care
of those fluctuations.
10
IV. CONCLUSION
In
essence,
fluence dependent study
of ultrafast magnetization dynamics
in
Ni
80
Fe
20
thin film
reveals very weak correlation between ultrafast demagnetization time and Gilbert damping
within our experimental fluence range. W
e have reported
large
enhancement o
f damping with
pump fluence.
F
rom the bias field
as well as pump fluence
dependence of
experimentally
obtained
dynamic
al
parameters we have excluded all the possible extrinsic contributions
and
observed a pump
-
induced modulation of intrinsic Gilbert dampin
g. Also
,
from repetitive
measurements with different pump irradiat
ion we have shown that the pump
-
induced changes are
reversible in nature.
Enha
n
cement of the system temperature to Curie temperature ratio is
believed to be responsible for increment in
rema
gnetization
times and damping.
The temporal
chirp parameter has been found to be increased
by
up to
ten
times within the experimental
fluence range
,
while the frequency experiences a significant red
shift.
F
rom application point of
view,
as increasing dema
nd for faster and efficient magnetic memory devices, has led the
scientific community in the extensive research field of ultrafast magnetization dynamics, our
results will further enlighten the understanding of modulation of magnetization dynamics in
ferro
magnetic systems in presence of higher pump fluence.
Usually l
ow damping materials are
preferred because it is easier to switch their magnetization in expense of smaller energy
, lower
write current in STT
-
MRAM devices and longer propagation length of spin
waves im magnonic
devices
.
On the other hand,
higher damping is also required to
stop
the post switching ringing of
the signal.
The results also have important implications on the emergent field of
all
-
optical
helicity dependent
switching [4
6
-
4
8
].
In thi
s context, the
transient
modulation of Gilbert
damping
and other dynamical parameters
in ferromagnetic materials is of fundamental interest
for characterizing and controlling ultrafast responses in magnetic structures.
Acknowledgements:
We gratefully ack
nowledge the financial support from S. N. Bose National
Centre for Basic Sciences (grant no.:
SNB/AB/12
-
13/96
and SNB/AB/18
-
19/211
) and
Department of Science and Technology (DST), Government of India (grant no.:
SR/NM/NS
-
09/2011
). We also gratefully ackno
wledge the technical assistance of Dr. Jaivardhan Sinha and
Mr. Samiran Choudhury for preparation of the sample. SM acknowledges DST for INSPIRE
fellowship.
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ump fluences (
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25 mJ/cm
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φ is presented
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Figure 2
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F
having unit of mJ/cm
2
is mentioned in numerical figure.
Solid lines are fit
ting lines
.
Pump f
luence
dependen
ce of
(b)
relaxation time (
τ
) and (c) effective damping (
α
eff
).
Black
and
blue
symbols represent the variation of these
parameters at two different field values,
H
= 2.4 and 1.8 kOe, respectively.
A
mplitude of precession is also plotted
with pump
fluence for
H
= 2.4 kOe
,
(d) Variation of effective damping with irradiation fluence
(
F
i
)
at
H
= 2.4 kOe.
In order to check the possible damage in the sample as high fluence values the pump fluence was taken up to the
targeted value of
F
i
for several minut
es followed by reduction of the pump fluence to a constant value of 10 mJ/cm
2
and the pump
-
probe measurement was performed. The damping coefficient is found to be unaffected by the
irradiation fluence as shown in (d).
17
Figure 3
: (a) Bias
field dependence of precessional frequency for
F
= 10 mJ/cm
2
. The red solid line indicates the
Kittel fit. (b) Pump fluence dependence of effective magnetization (M
eff
) of the probed volume. (c) Bias field
dependence of relaxation time (
τ
) for four differ
ent fluences.
F
having unit of mJ/cm
2
is mentioned in numerical
figures. Solid lines are the fitted data. (d) Variation of intrinsic Gilbert damping (
α
0
) with pump fluence.
18
Figure 4
: (a) Pump
-
fluence dependence of precessional fre
quencies for
H
= 2.4 and 1.8 kOe. Red and black symbols
represent the variation of average frequency (
f
FFT
) and initial frequency (
f
i
) respectively. (b) Variation of temporal
chirp parameter ‘
b’
with pump fluence for two different magnetic field values. (c
) Variation of temporal chirp
parameter with bias field for four different pump fluences.
F
having unit of mJ/cm
2
is mentioned in numerical figure.
Dotted lines are guide to eye.
|
1502.00176v1.Bases_and_Structure_Constants_of_Generalized_Splines_with_Integer_Coefficients_on_Cycles.pdf | arXiv:1502.00176v1 [math.RA] 31 Jan 2015BASES AND STRUCTURE CONSTANTS OF GENERALIZED
SPLINES WITH INTEGER COEFFICIENTS ON CYCLES
NEALY BOWDEN, SARAH HAGEN, MELANIE KING, AND STEPHANIE REIN DERS
Abstract. Aninteger generalized spline is a set of vertex labels on an edge-
labeled graph that satisfy the condition that if two vertices are join ed by an edge,
the vertex labels are congruent modulo the edge label. Foundationa l work on these
objects comes from Gilbert, Polster, and Tymoczko, who generaliz e ideas from
geometry/topology (equivariant cohomology rings) and algebra (a lgebraic splines)
to develop the notion of generalized splines . Gilbert, Polster, and Tymoczko prove
that the ring of splines on a graph can be decomposed in terms of splin es on
its subgraphs (in particular, on trees and cycles), and then fully an alyze splines
on trees. Following Handschy-Melnick-Reinders and Rose, we analyz e splines on
cycles, in our case integer generalized splines.
The primary goal of this paper is to establish two new bases for the m odule
of integer generalized splines on cycles: the triangulation basis and t he King ba-
sis. Unlike bases in previous work, we are able to characterize each b asis element
completely in terms of the edge labels of the underlying cycle. As an ap plication
we explicitly construct the multiplication table for the ring of integer g eneralized
splines in terms of the King basis.
1.Introduction
Aninteger generalized spline is a set of vertex labels on an edge-labeled graph that
satisfy the condition that if two vertices are joined by an edge, the vertex labels are
congruent modulo the edge label. (See Definition 2.1 for a precise sta tement.) Figure
1 shows examples of splines on a three-cycle.
Theterm“spline”comesfromthenameofthethinstripsofwooduse dbyengineersto
model larger constructions like ships or cars. Mathematicians later adopted the term
We are extremely grateful to Julianna Tymoczko, Elizabeth Drellich, and Yue Cao for their
insight and contributions to this paper. We would also like to thank Rut h Haas and Joshua
Bowman for valuable discussions on these topics, and Michael DiPasq uale for his thorough review
and comments. This work was supported by Smith College and the Nat ional Science Foundation
through the Center for Women in Mathematics [DMS-1143716].
12 BOWDEN, HAGEN, KING, AND REINDERS
25
3
111
25
3
0212
25
3
0015
Figure 1. The edge labels are t2,5,3uand the sets of vertex labels
t1,1,1u,t0,2,12u, and t0,0,15ueach form a spline on the cycle.
torefertopiecewisepolynomialsonpolytopeswiththepropertytha tthepolynomials
on the faces agree at their shared edges up to a given degree of sm oothness. These
mathematical splines are also used for object-modeling purposes, hence the use of
the name.
Billera pioneered the algebraic study of splines, especially looking into q uestions
regarding thedimension ofthe moduleof splines [2]. Many peoplecont inued Billera’s
work, including among others, Rose [12, 13] and Haas [7] who worked on identifying
dimension and bases for the module of splines.
Splinetheorydevelopedindependently intopologyandgeometry. Go resky, Kottwitz,
and MacPherson [6], Payne [11], and Bahri, Franz, and Ray [1] constr ucted equivari-
ant cohomology rings using splines, although they did not use that na me.
Gilbert, Polster, and Tymoczko generalize the notion of splines that we use here to
what they call generalized splines [4] . These generalized splines are built on the
dual graph of the polytopes found in classical splines. The work of B illera and Rose
shows that the two constructions (on polytopes or their duals) ar e equivalent in most
cases, including the cases of classical interest [3].
Cycles turn out to be a particularly important family of graphs to stu dy. Indeed
Gilbert, Polster, and Tymoczko show that the ring of generalized sp lines on a graph
Gcan be decomposed in terms of splines on certain trees and cycles in G[4]. They
completely describe splines on trees, while leaving open the investigat ion of splines
on cycles. Similarly, Rose showed that cycles play a key role in the relat ions defining
modules of splines [13].
Handschy, Melnick, and Reinders begin analysis of integer generalize d splines on
cycles [9]. They prove the existence of a certain flow-up basis (see D efinition 2.3),
what we call the smallest-value basis, for splines on cycles, and thus prove that suchBASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 3
spline modules are free. They define their basis for arbitrary cycles , but only have
formulas for the leading nonzero elements.
Inthispaperwe introducetwo newbases forthemoduleofinteger g eneralized splines
on cycles: the triangulation basis and the King basis. Each of these b ases is fully
expressible in terms of the edge labels of the cycle, and each has its o wn strengths.
The triangulation basis, so called because it is constructed from tria ngulated cycles,
is useful because it exists on arbitrary cycles (Theorem 4.2). The a dvantage of the
King basis lies in the fact that it is relatively simple to calculate, with the e ntries
almost constant (Definition 5.1). Although the King basis only exists o n cycles with
a pair of relatively prime adjacent edge labels, this restriction is not u ncommon in
applications. Infactanevengreaterrestrictionthatalledgelabe lsberelativelyprime
is commonly used [5, 10]. The results of our work naturally generalize t o principle
ideal domains, which include classical univariate splines and Pr¨ ufer d omains; see
forthcoming work [8].
As an application we present the multiplication table of splines on cycles where the
products of splines are expressed in terms of the King basis. Finding multiplica-
tion tables of equivariant cohomology rings in terms of Schubert bas es is the central
problem of Schubert calculus. We view this work as a step in that geom etric direc-
tion.
The rest of this paper is organized as follows. In Section 2 we summar ize the im-
portant definitions and theorems that we use in our work. In Sectio n 3 we provide
a criterion for the existence of flow-up bases. Sections 4 and 5 are dedicated to
proving the existence of the triangulation basis and King basis respe ctively. In the
final section we give the multiplication table for the King basis and end w ith an open
question.
2.Preliminaries
2.1.Results from Handschy, Melnick, and Reinders. Handschy, Melnick, and
Reinders proved a number of results about splines on cycles [9]. Many of their
propositionsandtheorems play key rolesinour proofsregarding tr iangulationsplines
and King splines. We also use their notation, which we describe in this se ction.
2.1.1.Basic Definitions. The foundational combinatorial object we study is an edge-
labeled graph, defined here:4 BOWDEN, HAGEN, KING, AND REINDERS
Definition 2.1 (Edge-Labeled Graphs) .LetGbe a graph with kedges ordered
e1,e2,...,e kandnvertices ordered v1,...,vn. Letℓibe a positive integer label on
edgeeiand letL“ tℓ1,...,ℓkube the set of edge labels. Then pG,L qis an edge-
labeled graph.
Withthisnotationforedge-labeledgraphswehavetheformaldefin itionofsplines:
Definition 2.2 (Splines).A spline on the edge-labeled graph pG,L qis a vertex-
labeling as follows: if two vertices are connected by an edge eithen the two vertex
labels are equivalent modulo ℓi. We denote a spline G“ pg1,...,gnqwheregiis the
label on vertex vifor1ďiďn.
In this paper we assume the labels giPZ.
2.1.2.Flow-Up Classes and the Smallest-Value Basis. Flow-up classes are a partic-
ularly nice class of splines on cycles. They arise geometrically ([5], [10], [1 4]) and are
an analogue of upper triangular matrices.
Definition 2.3 (Flow-Up Classes) .Fix a cycle with edge labels pCn,Lqand fixk
with1ďkăn. A flow-up class GkonpCn,Lqis a spline with kleading zeros.
We say that a basis whose elements are flow-up classes is a flow-up basis . The
simplest flow-up class is the trivial spline; It exists on any edge-labele d cycle.
Proposition 2.4 (Trivial Splines [9, Prop 2.5]) .Fix a cycle with edge labels pCn,Lq.
The smallest flow-up class on pCn,LqisG0“ p1,...,1q. Moreover, any multiple of
G0is also a spline. We call the multiples of G0trivial splines.
The following theorem establishes that flow-up classes exist on any e dge-labeled
cycle.
Theorem 2.5 (Flow-Up Classes on n-cycles [9, Thrm 4.3]) .Fix a cycle with edge
labels pCn,Lq. Letně3and1ďkăn. There exists a flow-up class GkonpCn,Lq.
The next definition introduces smallest flow-up classes.
Definition 2.6 (Smallest Flow-Up Class) .Fix a cycle with edge labels pCn,Lq. The
smallest flow-up class Gk“ p0,...,0,gk`1,...,gnqonpCn,Lqis the flow-up class whose
nonzero entries are positive and if G1
k“ p0,...,0,g1
k`1,...,g1
nqis another flow-up class
with positive entries then g1
iěgifor all entries. By convention we consider
G0“ p1,...,1qthe smallest flow-up class G0.BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 5
The following theorem gives an explicit formula for the smallest leading e lement of
flow-up classes.
Theorem 2.7 (Smallest Leading Element of Gk[9, Thrm 4.5]) .Fix a cycle with edge
labels pCn,Lq. Fixně3andksuch that 2ďkăn. LetGk´1“ p0,...,0,gk...,gnqbe a
flow-up classon pCn,Lq. Theleadingelement gkis amultiple of lcmpℓk´1,gcdpℓk,...,ℓnqq
and there is a flow-up class Gk´1withgk“lcmpℓk´1,gcdpℓk,...,ℓnqq.
The smallest flow-up classes exist and form a basis for the set of splin es given any
edge-labeled cycle.
Theorem 2.8 (Basisfor n-Cycles [9, Thrm4.7]) .Fix a cycle with edge labels pCn,Lq.
The smallest flow-up classes G0,G1,...,Gn´1exist on pCn,Lqand form a basis over
the integers for the Z-module of splines on pCn,Lq.
2.2.Useful Computational Tool. For reasons related to finding an explicit basis
for splines on cycles, we want to find a formula for the value of the va riablexin the
following pair of congruences:
#
x”ymoda
x”0 modb
We note the conditions for when such a solution exists and we give an e xplicit
formulation for xin terms of y,a, andbprovided a solution does exist.
Proposition 2.9. Consider the system of congruences
#
x”ymoda
x”0 modb.
If this system has a solution then one solution is given by the following formula:
‚Ifa
gcdpa,bq“1thenx“bis a solution to the system.
‚Ifa
gcdpa,bq‰1then
x“yˆb
gcdpa,bq˙ ˆb
gcdpa,bq˙´1
modpa
gcdpa,bqq
is a solution to the system.6 BOWDEN, HAGEN, KING, AND REINDERS
Proof.The Chinese Remainder Theorem tells us that this system of congrue nces is
satisfied if and only if y”0 mod gcd pa,bq. In what follows we will assume that a
solution exists, and thus that y”0 mod gcd pa,bq.
Case 1: Let’s deal first with the case wherea
gcdpa,bq“1. This condition implies
that gcd pa,bq “aand sob“anfor some nPZ. Because y”0 mod gcd pa,bqby
assumption and gcd pa,bq “awe have y”0 moda. In other words, y“amfor
somemPZ. Thenx“bsatisfies the system of congruences because bis congruent
to zero modulo bandb“anis congruent to y“ammoduloa.
Case 2:Now supposea
gcdpa,bq‰1. We can rewrite the system of congruences as
#
x“y`as
x“bt
Equate both expressions.
bt“y`as
Recall that y”0 mod gcd pa,bq. This allows us to divide both sides by gcd pa,bqand
get an integer as the result.ˆb
gcdpa,bq˙
t“y
gcdpa,bq`ˆa
gcdpa,bq˙
s
Putting this back into modular form we haveˆb
gcdpa,bq˙
t“y
gcdpa,bqmodˆa
gcdpa,bq˙
.
The integers´
b
gcdpa,bq¯
and´
a
gcdpa,bq¯
are relatively prime so we can take the inverse
of the first modulo the second.
t”y
gcdpa,bqˆb
gcdpa,bq˙´1
modˆa
gcdpa,bq˙
.
Plug this expression for tinto the equation x“bt:
x“yˆb
gcdpa,bq˙ ˆb
gcdpa,bq˙´1
modpa
gcdpa,bqq.
This value is a solution to the original system of congruences. /square
Notice that this second case simplifies enormously if gcdpa,bq “1. In this situation
xreduces to:
x“ybrb´1smodaBASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 7
3.Basis Condition
LetpG,L qbe an arbitrary graph on nvertices with an arbitrary edge-labeling. Con-
sider a set of flow-up classes G0...Gn´1onpG,L q. In this section we give a necessary
and sufficient condition for this set to form a basis for the module of t he splines on
pG,L q. Any set G0,...,Gn´1that meets this basis condition is called a flow-up basis .
Such a basis is useful because linear independence is trivially verified.
LetG0...Gn´1be a set of flow-up classes and for each idenote
Gi“ p0,...,0,gpiq
i`1,...,gpiq
nq.
The subscript of each gpiqindicates the entry-position of gpiqin the spline Gi. The
superscript piqis to keep track of the fact that we are working with the flow-up clas s
Gi. In much of this paper and in previous work the superscript is suppr essed when
the flow-up class in question is obvious.
Theorem 3.1 (Basis Condition) .The following are equivalent:
‚The set tG0,...,Gn´1uforms a flow-up basis.
‚For each flow-up spline Ai“ p0,...,0,ai`1,...,a nqthe entry ai`1ofAiis an
integer multiple of the entry gpiq
i`1ofGi.
Proof.Suppose that G0,...,Gn´1forms a flow-up basis for the module of splines on
a graph pG,L q. Suppose that Ai“ p0,...,0,ai`1,...,a nqis a spline on pG,L qwith
exactlyileading zeros. We will show that ai`1“cgpiq
i`1for some cPZ.
SinceG0,...,Gn´1form a basis, we can write Aias a linear combination of the
splinesG0,...,Gn´1. The fact that Aihasileading zeros implies that the coeffi-
cients of G0,...,Gi´1must be 0. Thus we have Ai“ciGi`...`cn´1Gn´1for some
ci,...,c n´1PZ. Consider the pi`1qthentry of the splines on the right-hand side of
this equation. Note that Giis the only element of Gi,...,Gn´1with a nonzero entry
in this position. Considering the pi`1qthentry on each side of the equation, we have
ai`1“cigpiq
i`1`ci`10`...`cn´10“cigpiq
i`1.
Now we prove the converse. Let A“ pa1,...,a nqbe an arbitrary spline on pG,L q.
We prove by induction that
A“A1
j`j´1ÿ
k“0ckGk8 BOWDEN, HAGEN, KING, AND REINDERS
for all 1 ďjďnwhereA1
jis a spline with (at least) jleading zeros.
For our base case, note that by hypothesis we have
AҬ
˚˚˚˝an´c0gp0q
n
...
a2´c0gp0q
2
0˛
‹‹‹‚`c0G0
sincea1“c0gp0q
1. Letting A1
1“ p0,a2´c0gp0q
2,...,a n´c0gp0q
nqgivesA“A1
1`ř0
k“0ckGk.
Thus our claim holds for j“1.
Suppose as our induction hypothesis that we have A“A1
i`ři´1
k“0ckGkfor some
1ďiďn´1. We can write this as
AҬ
˚˚˚˚˚˚˚˝a1
n...
a1
i`1
0
...
0˛
‹‹‹‹‹‹‹‚`i´1ÿ
k“0ckGk.
By hypothesis we have that a1
i`1“cigpiq
i`1for some ciPZ. So we can write
AҬ
˚˚˚˚˚˚˚˚˚˚˝a1
n´cigpiq
n
...
a1
i`2´cigpiq
i`2
0
0
...
0˛
‹‹‹‹‹‹‹‹‹‹‚`iÿ
k“0ckGk.
LettingA1
i`1“ p0,...,0,0,a1
i`2´cigpiq
i`2,...,a1
n´cigpiq
nqgivesusA“A1
i`1`ři
k“0ckGk.
By induction we have A“A1
j`řj´1
k“0ckGkfor all 1 ďjďn. In particular we have
A“A1
n`řn´1
k“0ckGk. ButA1
nis a spline with nleading zeros. So A1
n“ p0,...,0q.
ThusA“řn´1
k“0ckGk. We conclude that every spline can be written as a linear
combination of G0,...,Gn´1as desired. /squareBASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 9
One important observation is that the basis condition is only a conditio n on the first
nonzero entry of each spline in a set of flow-up classes G0,...,Gn´1. This gives us
the following useful corollary:
Corollary 3.2. Suppose the set of flow-up classes tG0,...,Gn´1uforms a basis for
the module of splines. Suppose tG1
0,...,G1
n´1uis a set of flow-up classes for which
for eachithe first nonzero entry of G1
iequals the first nonzero entry of Gi. Then the
settG1
0,...,G1
n´1ualso forms a basis for the module of splines.
4.The Triangulation Splines
Triangulationsplinesformanotherbasisofflow-upclassesforcycle s. Theyaresimilar
toHandschy, Melnick, andReinders’ smallest-valueflow-upclasses inthattheleading
nonzero elements of both are the same. However we give a formula f or every entry
of the triangulation splines, unlike the smallest-value flow-up classes .
Definition 4.1 (Triangulation Splines) .Fix an edge-labeled cycle pCn,Lq. For
1ďkďn´1the vector Hk“ p0,...,0,hk`1,...,hnqhas entries as follows:
‚hk`1“lcmpℓk,gcdpℓk`1,...,ℓnqq
‚Fork`1ăiďnifℓi´1
gcdpℓi´1,...,ℓnq“1thenhi“gcdpℓi,...,ℓnq.
‚Fork`1ăiďnifℓi´1
gcdpℓi´1,...,ℓnq‰1then
hi“hi´1ˆgcdpℓi,...,ℓnq
gcdpℓi´1,...,ℓnq˙ ˆgcdpℓi,...,ℓnq
gcdpℓi´1,...,ℓnq˙´1
modℓi´1
gcdpℓi´1,...,ℓnq
The next theorem establishes that triangulation splines exist on any edge-labeled
cycle.
Theorem 4.2 (ExistenceofTriangulationSplines) .Fix an edge-labeledcycle pCn,Lq.
For1ďkďn´1the vector Hkis a spline on pCn,Lq.
Proof.Start with an edge-labeled cycle pCn,Lq. For 3 ďkďn´1 add an edge
between vertices v1andvkas shown in Figure 2. Label the edge between v1andvk
with gcd pℓk,...,ℓnq. We will show the vector Hksatisfies all of the edge conditions
represented by this graph, which implies it satisfies the cycle’s edge c onditions in
particular.10 BOWDEN, HAGEN, KING, AND REINDERS
ℓ1ℓ2ℓ3ℓn´1
ℓn
gcdpℓn´1,ℓnq
gcdpℓ4,...,ℓnq
gcdpℓ3,...,ℓnq
(a)Add edgesℓ1ℓ2ℓ3ℓn´1
ℓn
gcdpℓn´1,ℓnq
gcdpℓ4,...,ℓnq
gcdpℓ3,...,ℓnq
0h2h3
(b)Base case
Figure 2. Triangulated Cycle
Label vertices v1,...,vkzero. Label vertex vk`1with
hk`1“lcmpℓk,gcdpℓk`1,...,ℓnqq.
The integer hk`1satisfies the edge conditions on the downward edges (edges with
lower-indexed vertices) at vertex vk`1by construction:
#
hk`1”0 modℓk
hk`1”0 mod gcd pℓk`1,...,ℓnq
This is our base case, and we will label vertices from hk`2tohn´1inductively.
Our induction hypothesis is that hk`1,...,hifork`1ďiďn´1 satisfy the edge
conditions for downward edges. Consider the system of congruen ces at vertex vi`1
represented by the edges labeled ℓiand gcd pℓi`1,...,ℓnq:
#
hi`1”himodℓi
hi`1”0 mod gcd pℓi`1,...,ℓnq
By the Chinese Remainder Theorem a solution hi`1exists if and only if hi”0 mod
gcdpℓi,gcdpℓi`1,...,ℓnqq. In other words a solution exists if and only if hi”0 mod
gcdpℓi,...,ℓnq. By our induction hypothesis hisatisfies the downward edge conditions
at vertex viso in particular hi”0 mod gcd pℓi,...,ℓnq. Thus a solution hi`1exists.BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 11
This means
hi`1“$
&
%hi´
gcdpℓi`1,...,ℓnq
gcdpℓi,...,ℓnq¯ ´
gcdpℓi`1,...,ℓnq
gcdpℓi,...,ℓnq¯´1
modℓi
gcdpℓi`1,...,ℓnqifℓi
gcdpℓi,...,ℓnq‰1
gcdpℓi`1,...,ℓnq ifℓi
gcdpℓi,...,ℓnq“1
is a solution by Proposition 2.9.
In conclusion we can label each vertex vifork`1ăiďn´1 with
hi“$
&
%hi´1´
gcdpℓi,...,ℓnq
gcdpℓi´1,...,ℓnq¯ ´
gcdpℓi,...,ℓnq
gcdpℓi´1,...,ℓnq¯´1
modℓi´1
gcdpℓi´1,...,ℓnqifℓi´1
gcdpℓi´1,...,ℓnq‰1
gcdpℓi,...,ℓnq ifℓi´1
gcdpℓi´1,...,ℓnq“1
andhiwill satisfy the edge conditions represented by the edges labeled ℓi´1and
gcdpℓi,...,ℓnq.
Lastly for an integer hnto satisfy the edge conditions at vertex vnit must satisfy
the following system of congruences:
#
hn”hn´1modℓn´1
hn”0 modℓn
The Chinese Remainder Theorem tells us that a solution hnexists to this system if
and only if hn´1”0 mod gcd pℓn´1,ℓnq. We showed by induction that our choice of
hn´1satisfies the edge conditions of the downward edges at the pn´1q-th vertex. In
particular this means hn´1”0 mod gcd pℓn´1,ℓnqbecause this is the edge condition
represented by the edge labeled gcd pℓn´1,ℓnq. Therefore
hn“$
&
%hn´1´
ℓn
gcdpℓn´1,ℓnq¯ ´
ℓn
gcdpℓn´1,ℓnq¯´1
modℓn´1
gcdpℓn´1,ℓnqifℓn´1
gcdpℓn´1,ℓnq‰1
ℓn ifℓn´1
gcdpℓn´1,ℓnq“1
satisfies the vertex vnedge conditions by Proposition 2.9. Choose this integer to
label the n-th vertex.
All of the congruences represented by the graph are accounted for so the vector
Hk“ p0,...,0,hk`1,...,hnqis a spline on the graph. In particular Hkis a spline on
the cycle pCn,Lqas desired.
/square12 BOWDEN, HAGEN, KING, AND REINDERS
The Corollary to the Basis Condition Theorem allows us to succinctly co nclude that
the set of triangulation splines H0,...,Hn´1forms a basis for the set of splines on an
edge-labeled cycle.
Theorem 4.3. Fix an edge-labeled cycle pCn,Lq. The set of triangulation splines
H0,...,Hn´1form a basis for the set of splines on pCn,Lq.
Proof.Thesetofsmallestflow-upclasses G0,...,Gn´1formabasisforthesetofsplines
onpCn,Lqby Theorem 2.8. The leading entry of Hkequals the leading entry of Gk
by construction for 0 ďkďn´1. Thus the set of triangulation splines H0,...,Hk
forms a basis for the set of splines on pCn,Lqby Corollary 3.2. /square
As an example, we calculate the triangulation basis for the 4-cycle wit h edge labels
t2,6,10,15u.
2615
10
The first basis element H0is, as always, the trivial spline p1,1,1,1q. The nonzero
entries of the second basis element H1are calculated as follows:
hp1q
2“lcmp2,gcdp6,10,15qq “2
hp1q
3“2ˆgcdp15,10q
gcdp6,15,10q˙ ˆgcdp15,10q
gcdp6,15,10q˙´1
mod6
gcdp6,15,10q“2¨5¨ p5q´1
mod 6 “50
hp1q
4“50ˆgcdp10q
gcdp15,10q˙ ˆgcdp10q
gcdp15,10q˙´1
mod15
gcdp15,10q“50¨2¨ p2q´1
mod 3 “200
The nonzero entries of the third basis element H2are calculated as follows:
hp2q
3“lcmp6,gcdp10,15qq “30
hp2q
4“30ˆgcdp10q
gcdp15,10q˙ ˆgcdp10q
gcdp15,10q˙´1
mod15
gcdp15,10q“50¨2¨ p2q´1
mod 3 “120
The only nonzero element of the final basis element H3ishp3q
4“lcmp15,10q “
30. Thus we have the following triangulation basis for the 4-cycle with edge la-
bels t2,6,10,15u:H0“ p1,1,1,1q,H1“ p0,2,15,200q,H3“ p0,0,30,120q, and
H4“ p0,0,0,30q.BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 13
5.The King Splines
In this section we define King splines on n-cycles and prove that they form a basis
for the set of splines.
Definition 5.1 (King splines) .Fix a cycle with edge-labels pCn,Lqand assume ℓn´1
andℓnrelatively prime. The King splines on pCn,Lqare the vectors
K0Ҭ
˚˚˚˚˚˚˝1
1
...
1
1
1˛
‹‹‹‹‹‹‚,K1“¨
˚˚˚˚˚˚˝k1
ℓ1...
ℓ1
ℓ1
0˛
‹‹‹‹‹‹‚,K2“¨
˚˚˚˚˚˚˝k2
ℓ2...
ℓ2
0
0˛
‹‹‹‹‹‹‚,...,K n´1“¨
˚˚˚˚˚˚˝kn´1
0
...
0
0
0˛
‹‹‹‹‹‹‚
where
ki“#
ℓi¨ℓnrℓ´1
nsmodℓn´1for1ďiďn´2
ℓn´1ℓn fori“n´1.
By convention, we call K0the trivial King spline.
As our terminology suggests, the King splines are in fact splines.
Theorem 5.2. Letně3. Fix a cycle with edge-labels pCn,Lqwithℓn´1andℓn
relatively prime. The King splines K0,...,K n´1are splines on pCn,Lq.
Proof.First we note that the trivial King spline K0is the same as the trivial spline
G0which is indeed a spline on pCn,Lqby Proposition 2.4.
ConsideranarbitraryKingspline Ki“ p0,...,0,ℓi,...,ℓ i,kn´1qwhere1 ďiďn´2.
It has zero for its first ientries,ℓifor entries i`1 ton´1, andkn´1for its last
entry. We want to show that Kiis a spline on pCn,Lq. Note that zero is congruent
to itself modulo any integer, so in particular the following congruence s are satisfied:
!
0”0 modℓjfor 1 ďjďi´1 (1)
Also, since the integer ℓiis congruent to zero modulo ℓiwe have14 BOWDEN, HAGEN, KING, AND REINDERS
ℓi”0 modℓi (2)
The integer ℓiis congruent to itself modulo any integer, so in particular the following
congruences are satisfied:
!
ℓi”ℓimodℓjfori`1ďjďn´2 (3)
Finally we know ki“ℓi¨ℓnrℓ´1
nsmodℓn´1satisfies the following two congruences
#
ki”ℓimodℓn´1
ki”0 modℓn(4)
by Proposition 2.9. Collect the congruences in 1, 2, 3, and 4 into a sing le system of
congruences. This system represents the edge conditions on pCn,Lq. The vector Ki
satisfies all of these congruences so Kiis a spline on pCn,Lq.
Now consider the vector Kn´1“ p0,...,0,kn´1q. Zero is congruent to itself modulo
any integer, so the following system of congruences is satisfied:
!
0”0 modℓjfor 1 ďjďn´2. (5)
Sincekn´1“ℓn´1ℓnwe know
#
kn´1”0 modℓn´1
kn´1”0 modℓn(6)
Collect the congruences in 5 and 6 into a single system. This system re presents the
edge conditions on pCn,Lq. The vector Kn´1satisfies all of these congruences so
Kn´1is a spline on pCn,Lq.
Thus we have that Kiis a spline for all 0 ďiďn´1 as desired.
/square
Now that we know the King splines are splines, we confirm that they fo rm a ba-
sis.
Theorem 5.3. Fix a cycle with edge labels pCn,Lqwithℓn´1andℓnrelatively prime.
The set of King splines K0,...,K n´1forms a basis for the set of splines on pCn,Lq.BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 15
Proof.The set of smallest flow-up classes G0,...,Gn´1form a basis for the set of
splines on pCn,Lqby Theorem 2.8. We constructed the King splines so that the
leading entry Kiequals the leading entry of Gifor 0 ďiďn´1. Thus the set of
King splines K0,...,K n´1forms a basis for the set of splines on pCn,Lqby Corollary
3.2. /square
6.Multiplication Tables
The fact that we have simple explicit formulas for the entries of the K ing basis is
a powerful computational tool. In this section we use the King basis to write the
product of any pair of basis elements as a linear combination of basis e lements. This
kind of calculation is important in geometry and topology, which use sp lines over
polynomial rings to describe cohomology rings.
6.1.Multiplication Tables for n-Cycles on the King Basis. When multiplying
splines the operation is performed component-wise. Consider the K ing basis on a
given n-cycle.
Since the entries in the trivial spline K0are all ones, multiplying any spline Ki(with
0ďiďn´1) byK0simply yields Ki. The following theorem gives us the product
of any pair of non-trivial King splines.
Theorem 6.1. For arbitrary Ki,Kjwithi,j‰0andiďj, we have the product
KiKj“liKj`kjpki´liq
kn´1Kn´1.
Proof.We give a proof by construction.
Consider arbitrary basis elements KiandKjwithi,j‰0 andiďj. Their product
KiKjhas zeros up to the jthentry. The entries numbered j`1 through n´1 are
ℓi¨ℓj. The last entry is ki¨kj.
Note that ℓi¨Kjhas zeros for the first jentries,ℓi¨ℓjfrom entries j`1 ton´1,
andℓi¨kjfor thenthentry. This is almost exactly the product KiKj. However we
want this last entry to be ki¨kj. Addingkjpki´liq
kn´1Kn´1gives the desired result.
Thus for KiKjwithi,j‰0 andiďjwe have16 BOWDEN, HAGEN, KING, AND REINDERS
KiKj“ℓiKj`kikj´likj
kn´1Kn´1“ℓiKj`kjpki´liq
kn´1Kn´1
Since we are working in the integers, our last step is to prove that th e coefficient
kjpki´ℓiq
kn´1
is indeed an integer. We know ki”ℓimodℓn´1because Kiis a spline. Say
ki´ℓi“pℓn´1for some pPZ. Similarly, we know kj”0 modℓnbecauseKjis
a spline. Say kj“qℓnfor some qPZ. By definition we have kn´1“ℓn´1ℓn.
Plugging these values into the expressionkikj´likj
kn´1yields the following:
kjpki´ℓiq
kn´1“pqℓnqppℓn´1q
ℓn´1ℓn“pq
Thuskjpki´ℓiq
kn´1is always an integer.
/square
Note that the product KiKn´1for anyiďn´1 simplifies significantly.
Corollary 6.2. Choose any i‰0. ThenKiKn´1“kiKn´1.
Proof.We apply the formula for the product KiKjto the particular case where
j“n´1 and simplify:
KiKn´1“ℓiKn´1`kn´1pki´ℓiq
kn´1Kn´1“kiKn´1
/square
For example consider the 5-cycle with edge labels t3,4,8,2,5u. The King basis on a
5-cycle with these labels looks like the following:
K0
5348
2
1111
1K1
5348
2
033
3
15K2
5348
2
004
4
20K3
5348
2
000
8
40K4
5348
2
000
0
10BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 17
Let’s multiply the elements K1andK3. We obtain
K1K3“ K1
5348
2
033
3
15ˆ K3
5348
2
000
8
40“
5348
2
000
24
600
By the formula given above
K1K3“3K3`40p15´3q
10K4“3K3`48K4.
Pictorially this solution is shown below.
3K3`48K4“3K3
5348
2
000
8
40`48 K4
5348
2
000
0
10“
5348
2
000
24
600
Remark 6.3. The same argument can be used to give the multiplication tabl e for
arbitrarily labeled 3-cycles using the triangulation basi s (Def 4.1, Thrm 4.3). Given
the basis elements H0,H1,andH2we have the following table
H0Ҭ
˝1
1
1˛
‚,H1“¨
˝hp1q
3
hp1q
2
0˛
‚,H2“¨
˝hp2q
3
0
0˛
‚
H0H1 H2
H0H0H1 H2
H1H1hp1q
2H1`ΦH2hp1q
3H2
H2H2hp1q
3H2hp2q
3H2
whereΦ“hp1q
3php1q
3´hp1q
2q
hp2q
3.
Unlike with the King basis, we do not have nice formulas for entries of t he triangu-
lation basis. This leads to the following open question.18 BOWDEN, HAGEN, KING, AND REINDERS
Question 6.4. Is there a positive or combinatorial formula for the multipl ication
table of general n-cycles (i.e.not alternating sums from successively correcting each
spline entry)?
References
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Math. Proc. Cambridge Philos. Soc. 146(2009), no. 2, 395-405. MR 2475973
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[4] S.Gilbert, S.Polster,andJ.Tymoczko, Generalized splines on arbitrary graphs , arXiv:1306.0801
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rings and Pr¨ ufer domains. In process.
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(2014)
[10] A. Knutson and T. Tao, Puzzles and (equivariant) cohomology of Grassmannians . Duke Math.
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Kottwitz, and MacPherson , arXiv:math/0503369 (2005) |
2111.00586v1.Thermally_induced_all_optical_ferromagnetic_resonance_in_thin_YIG_films.pdf | 1
Thermally induced all-optical ferromagnetic resonance in thin YIG films
E. Schmoranzerová1*, J. Kimák1, R. Schlitz3, S.T. B. Goennenwein3,6, D. Kriegner2,3, H. Reichlová2,3, Z. Šobáň2,
G. Jakob5, E.-J. Guo5, M. Kläui5, M. Münzenberg4, P. Němec1, T. Ostatnický1
1Faculty of Mathematics and Physics, Charles University, Prague, 12116, Czech Republic
2Institute of Physics ASCR v.v.i , Prague, 162 53, Czech Republic
3Technical University Dresden, 01062 Dresden, Germany
4Institute of Physics, Ernst-Moritz-Arndt University, 17489, Greifswald, Germany
5Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany
6 Department of Physics, University of Konstanz, 78457 Konstanz, Germany
Laser-induced magnetization dynamics is one of the key methods of modern opto-spintronics which aims
at increasing the spintronic device speed1,2. Various mechanisms of interaction of ultrashort laser pulses
with magnetization have been studied, including ultrafast spin-transfer3, ultrafast demagnetization4,
optical spin transfer and spin orbit torques 5,6,7 , or laser-induced phase transitions8,9. All these effects can
set the magnetic system out of equilibrium, which can result in precession of magnetization. Laser-induced
magnetization precession is an important research field of its own as it enables investigating various
excitation mechanisms and their ultimate timescales2. Importantly, it also represents an all-optical analogy
of a ferromagnetic resonance (FMR) experiment, providing valuable information about the fundamental
parameters of magnetic materials such as their spin stiffness, magnetic anisotropy or Gilbert damping10.
The “all-optical FMR” (AO-FMR) is a local and non-invasive method, with spatial resolution given by the
laser spot size, which can be focused to the size of few micrometers. This makes it particularly favourable
for investigating model spintronic devices.
Magnetization precession has been induced in various classes of materials including ferromagnetic
metals11, semiconductors10, 12, or even in materials with a more complex spin structure, such as non-
collinear antiferromagnets13. Ferrimagnetic insulators, with Yttrium Iron Garnet (YIG, Y 3Fe5O12) as the
prime representative14, are of particular importance for spintronic applications owing to their high spin
pumping efficiency15 and the lowest known Gilbert damping16. However, inducing magnetization dynamics
in ferrimagnetic garnets using optical methods is quite challenging, as it requires large photon energies 2
(bandgap of YIG is Eg ≈ 2.8 eV)17. This spectral region is rather difficult to access with most common ultrafast
laser systems, which are usually suited for near-infrared wavelengths. Therefore, methods based mostly
on non-thermal effects, such as inverse Faraday18,19 and Cotton-Mouton effect20 or photoinduced magnetic
anisotropy21, 22 have been used to trigger the magnetization precession in YIG so far. For these phenomena
to occur, large laser fluences of tens of mJ/cm2 are required23. In contrast, laser fluences for a thermal
excitation of magnetization precession usually do not exceed tens of J/cm2 (Refs. 12, 21, 13). Using the
low fluence excitation regime allows for the determination of quasi-equilibrium material parameters, not
influenced by strong laser pulses. In magnetic garnets, an artificial engineering of the magnetic anisotropy
via the inclusion of bismuth was necessary to achieve thermally-induced magnetization precession21.
In this paper, we show that magnetization precession can be induced thermally by femtosecond laser
pulses in a thin film of pure YIG only by adding a metallic capping layer. The laser pulses locally heat the
system, which sets the magnetization out of equilibrium due to the temperature dependence of its
magnetocrystalline anisotropy. This way we generate a Kittel (n = 0, homogeneous precession) FMR mode,
with a precession frequency corresponding to the quasi-equilibrium magnetic anisotropy of the thin YIG
film10. We thus prove that the AO-FMR method is applicable for determining micromagnetic parameters
of thin YIG films. Using the AO-FMR technique we revealed that at low temperature the Kittel mode
damping is significantly faster than at room-temperature, in accord with previous FMR experiments24,25.
Our experiments were performed on a 50 nm thick layer of pure YIG grown by pulsed-laser deposition on
a gadolinium-gallium-garnet (GGG) (111)-oriented substrate. One part of the film was covered by 8 nm of
Au capping layer, the other part by Pt capping, both being prepared by ion-beam sputtering. Part of the
sample was left uncapped as a reference. X-ray diffraction confirmed the excellent crystal quality of the
YIG film with a very low level of growth-induced strain, as described in detail in Ref. 26. The magnetic
properties were further characterized using SQUID magnetometry and ferromagnetic resonance
experiments, showing the in-plane orientation of magnetization (see Supplementary Material, Part 1 and
Figs. S1 and S2). The deduced low-temperature (20 K) saturation magnetization µ0Ms 180 mT is in
agreement with results published on qualitatively similar samples27 again confirming a good quality of the
studied YIG film. Magnetic anisotropy of the system at 20 K was established from an independent
magneto-optical experiment (Ref. 28), the corresponding anisotropy constants for cubic anisotropy of the
first and second order are Kc1 = 4680 J/m3 and Kc2 = 223 J/m3, while the overall uniaxial out-of-plane
anisotropy is vanishingly small. 3
Laser-induced dynamics was studied in a time-resolved magneto-optical experiment in transmission
geometry, as schematically shown in Fig. 1(a). An output of a Ti:Sapphire oscillator generating 200 fs laser
pulses was divided into a strong pump beam, with fluences tuned between 70 and 280 µJ/cm2, and a 20-
times weaker probe beam. The beams were focused on a 30 m spot on the sample, which was placed in
a cryostat and kept at cryogenic temperatures (typically 20 K). An external magnetic field (up to 550 mT)
generated by an electromagnet was applied in y direction (see Fig. 1). The wavelength of pump pulses (800
nm) was set well below the absorption edge of the YIG layer, as indicated in the transmission spectrum of
the sample in Fig. 1(b). The wavelength of probe pulses (400 nm) was tuned to match the maximum of the
magneto-optical response of bulk YIG [see inset in Fig. 1(b) and Ref. 29].
The detected time-resolved magnetooptical (TRMO) signal corresponding to the rotation of polarization
plane of the probe beam Δβ, was measured as a function of the time delay Δt between pump and probe
pulses. In Fig. 1(c), we show an example of TRMO signals observed in uncapped YIG and two YIG/metal
heterostructures. Clearly, in the presence of the metallic capping layer an oscillatory TRMO signal is
observed, whose amplitude depends on the capping metal used. Frequency and damping of the
oscillations, on the other hand, remain virtually unaffected by the type of the capping layer, while no
oscillations are observed in the uncapped YIG sample.
The TRMO signals can be phenomenologically described by a damped harmonic function after removing a
slowly varying background (see Supplementary Material, Part 2 and Fig. S3),12
∆𝛽(Δ𝑡)=𝐴cos(2𝜋𝑓𝛥𝑡+𝜑)exp(−𝛥𝑡 𝜏⁄ ), (1)
where A is the amplitude of precession, f its frequency, φ the phase and τ the damping time. The fits are
shown in Fig. 1(c) as solid lines.
In order to demonstrate that the TRMO signals result from (laser-induced) magnetization dynamics, we
varied the external magnetic field Hext and extracted the particular precession parameters by fitting the
detected signals by Eq. (1). As depicted in Fig. 2(a), the experimentally observed dependence of the
precession frequency on the applied field is in excellent agreement with the solution of Landau-Lifshitz-
Gilbert (LLG) equation, using the free energy of a [111] oriented cubic crystal [see Supplementary, section
5, Eq. (S5) and Ref. 28]. This correspondence with the LLG model proves that our oscillatory signals reflect
indeed the precession of magnetization in uniform (Kittel) mode in YIG. We stress that the precession
frequency is inherent to the YIG layer and does not depend on the type of the capping layer. 4
The detection of the uniform Kittel mode can be further confirmed by comparing the frequency of the
oscillatory TRMO signal with the frequency of resonance modes observed in a conventional, microwave-
driven ferromagnetic resonance (MW-FMR) experiment. The MW-FMR experiment was performed in the
in-plane ( H = 0°) and out-of-plane ( H = 90°) geometry of the external field. We measured the TRMO signals
in YIG/Au sample in a range of magnetic field angles H and modelled the angular dependency of f by LLG
equation with the same parameters that were used in Fig. 2(a). The output of the model is presented in
Fig. 2(b), together with precession frequencies obtained from TRMO and FMR experiments. The MW-FMR
data fit well to the overall trend, confirming the presence of uniform magnetization precession [Ref. 30]
To find the exact physical mechanism that triggers laser-induced magnetization precession in our
YIG/metal bilayers, we measured the TRMO signals at different sample temperatures T. For comparison
we calculated also the dependence of f on the first order cubic anisotropy constant Kc1 from the LLG
equation, which is shown in the inset of Fig. 2(c). This graph reveals that f should be directly proportional
to Kc1 in the studied range of temperatures . In Fig 2(c) we plot f as a function of T, together with the
temperature dependence of Kc1 (T) obtained from Ref. 28 and Ref. 32. Clearly, both Kc1 and f show a similar
trend in temperature. Considering also the temperature dependence of the precession amplitude [see
Fig. S5 (a) and Section 4 of Supplementary Material], we identify the pump pulse-induced heating and
consequent modification of the magnetocrystalline anisotropy constant Kc1 as the dominant mechanism
driving laser-induced magnetization precession.
In order to estimate the pump-induced increase in quasi-equilibrium temperature of the sample, we first
fit the temperature dependence of the parameter Kc1 reported in literature by a second order polynomial
[Fig. 2(c)]. Owing to the linear relation between f and Kc1 and the known temperature dependence of f,
the measured dependence of f on pump fluence I can be converted to the intensity dependence of the
temperature increase T(I), which is shown Fig. 2(d). As expected, higher fluence leads to more
pronounced heating, which results in a decrease of the precession frequency. Note that for the highest
intensity of 300 J/cm2, the sample temperature can increase by almost 80 K.
Nature of the observed laser-induced magnetization precession was further investigated by comparing
samples with different capping layers. In Fig. 3(a) we show the amplitude A of the oscillatory signal in the
YIG/Pt and YIG/Au layers as a function of I. The difference between the samples is apparent both in the
absolute amplitude of the precession and in its increase with I, the YIG/Pt showing stronger precession.
Furthermore, precession damping is stronger in YIG/Au than in YIG/Pt, as apparent from Fig 3 (b) where
effective Gilbert damping parameter eff is presented as a function of Hext. These values of eff were 5
obtained by fitting the TRMO data by the LLG equation, as described in the Supplementary Material
(Section 5). Despite the relatively large fitting error, we can still see that YIG/Pt shows slightly lower
0.020, while the YIG/Au has 0.025. To understand these differences, we modeled the propagation of
laser-induced heat in GGG/YIG/Pt and GGG/YIG/Au multilayers by using the heat equation (see
Supplementary Material, Section 7). In Fig. 3(c), T is presented as a function of time delay t after pump
excitation for selected depths from the sample surface. In Fig. 3(d), the same calculations are presented
for variable depths and fixed t. The model clearly demonstrates that a significantly higher T can be
expected in the Pt-capped layer simply due to its smaller reflection coefficient as compared to Au-capping
(see Supplementary Material, Section 7). This in turn leads to a higher amplitude of the laser-induced
magnetization precession in YIG/Pt compared to the YIG/Au, as apparent in Fig. 3(a).
According to our model, an extreme increase in temperature is induced in the first few picoseconds after
excitation, which acts as a trigger of magnetization precession. After approximately 10 ps, precession takes
place in quasi-equilibrium conditions. The system returns to equilibrium on a timescale of nanoseconds,
which shows also in the TRMO signals as the slowly varying background (Fig. S3). The precession frequency
we detect reflects the quasi-equilibrium state of the system. Therefore, the temperature increase T
deduced from the TRMO signal can be compared with our model for large time delays after the excitation
(t 10 ps). In YIG/Au sample, the experimental values of T = (25 10) K for excitation intensity of 150
J/cm2 [see Fig. 2(d)], while the model gives us T 5K [Fig. 3 (c)]. Clearly, the values match in the order
of magnitude but there is a factor of 5 difference. This difference results from the boundary conditions
of the model that assumes ideal heat transfer between the sample and the holder, which is experimentally
realized using a silver glue with less than perfect performance at cryogenic conditions.
From Fig. 3(d) it also follows that large thermal gradients are generated across the 50 nm layer. This could
lead to significant inhomogeneity in magnetic properties of the layer, that would increase the damping
parameter by an extrinsic term. In our TRMO measurements, is indeed very large for a typical YIG
sample (TRMO 2-2.5 x10-2) and exceeds the value obtained from room-temperature MW-FMR by almost
an order of magnitude ( FMR 1x10-3, see Supplementary Material, Section 1b). As the modeled thermal
gradient alone cannot account for such a large change in Gilbert damping (see Supplementary Material,
Section 6), we attribute this increase in Gilbert damping to the difference in the ambient temperatures.
Large change of Gilbert damping (by a factor of 30) between room and cryogenic (20 K) temperature has
recently been reported on a seemingly high quality YIG thin film24. It was explained in terms of the presence
of rare earth or Fe2+ impurities that are activated at cryogenic temperatures. It is likely that the same 6
process occurs in our sample. Even though other mechanisms related to the optical excitation can also
contribute to the increase in TRMO (see Supplementary Material, Section 6), the all-optical and standard
FMR generated Kittel modes correspond very well [see Fig. 2(b)]. Furthermore, also the observed sample-
dependent Gilbert damping is consistent with this explanation. The YIG/Pt sample is heated to higher
temperature by the pump laser pulse [Fig. 3(c), (d)] than the YIG/Au sample, which according to Ref. 24
corresponds to a lower Gilbert damping. It is worth noting that damping parameter can be increased also
by spin-pumping from YIG to the metallic layer. However, this effect is expected to be significantly higher
when Pt is used as a capping, which does not agree with our observations.
In conclusion, we demonstrated the feasibility of the all-optical ferromagnetic resonance method in 50-
nm thin films of plain YIG. Magnetization precession can be triggered by laser-induced heating of a metallic
capping layer deposited on top of the YIG film. The consequent change of sample temperature modifies
its magnetocrystalline anisotropy, which sets the system out of equilibrium and initiates the magnetization
precession. Based on the field dependence of precession frequency, we identify the induced magnetization
dynamics as the fundamental (Kittel) FMR mode, which is virtually independent of the type of capping and
reflects the quasi-equilibrium magnetic anisotropy. The Gilbert damping parameter is influenced by line-
broadening mechanism due to low-temperature activation of impurities, which is an important aspect to
be taken into account for low-temperature spintronic device applications.
Regarding the efficiency of the optical magnetization precession trigger, it was found that the type of
capping layer strongly influences the precession amplitude. The precession in YIG/Pt attained almost twice
the amplitude of that in YIG/Au under the same conditions. This indicates that a suitable choice of capping
layer should be considered in an optimization of this local non-invasive magnetometric method.
Acknowledgments:
This work was supported in part by the INTER-COST grant no. LTC20026 and by the EU FET Open RIA
grant no. 766566. We also acknowledge CzechNanoLab project LM2018110 funded by MEYS CR for the
financial support of the measurements at LNSM Research Infrastructure and the German Research
Foundation (DFG SFB TRR173 Spin+X projects A01 and B02 #268565370).
7
LITERATURE
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[11] V. N. Kats et al., PRB 93, 214422 (2016)
[12] Y. Hashimoto, S. Kobayashi, and H. Munekata, Phys. Rev. Lett. 100, 067202 (2008).
[13] S. Miwa et al., Small Sci. 1, 2000062 (2021)
[14] A. A. Serga et al., J. Phys. D: Appl. Phys. 43, 264002 (2010)
[15] Y. Kijiwara et al., Nature 464, 262 (2010)
[16] V. Cherepanov, I. Kolokolov and V. L’vov, Phys. Rep.—Rev. Sec. Phys. Lett. 229 81 (1993)
[17] D. D. Stancil, A. Prabhakar: Spin waves – theory and applications (Springer, New York, 2009).
[18] F. Hansteen et al., Phys. Rev. Lett. 95, 047402 (2005)
[19] A. Stupakiewicz et al., Appl. Phys. Lett. 101, 262406 (2012)
[20] M. Montazeri et al., Nat. Comm. 6, 8958 (2015)
[21] L. A. Shelukhin, et al., Phys. Rev. B 97, 014422 (2018)
[22] A. Stupakiewicz et al., Nature 542, 71 (2017)
[23] F. Atoneche et al., Phys. Rev. B 81, 214440 (2010)
[24] C. L. Jermain et al., PRB 95, 174411 (2017) 8
[25] H. Maier-Flaig et al., Phys. Rev. B 95, 214423 (2017)
[26] B. Bhoi et al.: J. Appl.Phys. 123, 203902 (2018)
[27] J. Mendil et al.: Phys. Rev. Mat. 3, 034403 (2019)
[28] E. Schmoranzerova et al., ArXiv XXX (2021)
[29] E. Lišková Jakubisova et al., Appl. Phys. Lett. 108, 082403 (2016)
[30] We note that the FMR data were obtained at room temperature while the TRMO experiment was
performed at 20K. However, as apparent from Fig. 2(c), the precession frequency varies by less than
10% between 20 K and 300 K, which is well below the experimental error of (H). This justifies
comparison of the precession frequencies obtained from the TRMO experiment with the FMR data.
[31] M. Haider et al., J. Appl. Phys. 117, 17D119 (2015)
[32] N. Beaulieu et al., IEEE Magnetics Letters 9, 3706005 (2018)
FIGURES
9
Fig. 1: (a) Schematic illustration of the pump&probe experimental setup, where Eprobe is the probe beam linear
polarization orientation which is rotated by an angle after transmission through the sample with respect to the
orientation E’probe. An external magnetic field H ext is applied at an angle H. (b) Absorption spectrum of the studied
YIG sample, where OD stands for the optical density defined as minus the decadic logarithm of sample
transmittance. The red arrow indicates the wavelength of the pump beam PUMP = 800 nm. Inset: Spectrum of Kerr
rotation K of bulk YIG crystal29. The blue arrow shows the wavelength of the probe beam PROBE = 400 nm. (c)
Typical time-resolved magneto-optical signals of a plain 50 nm YIG film (black dots), YIG /Pt (green dots) and
YIG/Au bilayer (blue dots) at 20 K and 0Hext = 100 mT, applied at an angle H = 40°. Lines indicate fits by Eq. (1).
The data were offset for clarity.
Fig. 2: (a) Frequency f of magnetization precession as a function of magnetic field applied at an angle H = 40°, for
YIG/Pt (blue dots) and YIG/Au (green triangles) at T = 20 K and I = 150 J/cm2. The line is calculated from LLG equation
(Eq. S3) with the free energy given by (Eq.S5) (b) Field-angle dependence of f in YIG/Au sample for 0Hext = 300 mT
(blue dots), compared to a model by LLG model (line) and to frequencies measured by MW-FMR (red stars)32. (c)
Temperature dependence of f in YIG/Au sample (black points), where 0Hext = 300 mT was applied at H = 40°. The
temperature dependence of cubic anisotropy constant Kc1 was obtained from Ref. 28 (red dots) and Ref. 32 (red star,
T = 20 K). The data were fitted by an inverse polynomial dependence 𝐾ଵ(𝑇)= ଵ
(ା்ା்మ), with parameters: a = 0.18
m2/kJ; b= 9 x 10-4 m2/kJ.K; c = 9 x 10-6m2/kJ.K2. Inset: Dependence f(Kc1) obtained from the LLG equation. (d) f as a
function of pump pulse fluence I, from which the increase of sample temperature T for the used pump fluences was
evaluated using the f(T) dependence.
10
Fig. 3: Comparison of magnetization precession in YIG/Pt and YIG/Au samples. (a) Precession amplitude A as a
function of pump fluence I (dots) with the corresponding linear fits 𝐴 = 𝑠∙𝐼. The parameter s Pt = (1.05 0.09)x10-2
rad.cm2/J in the YIG/Pt, and s Au = (0.50.1)x10-2 rad.cm2/J in YIG/Au. These dependencies were measured for
0Hext = 300 mT and T 0 = 20 K. In YIG/Pt sample the as-measured data obtained for H = 40° are shown. In the YIG/Au
sample, the A(I) dependence was originally measured for H = 21° and recalculated to H = 40° according to the
measured angular dependence, as described in detail in Supplementary Material, Section 3. (b) Gilbert damping eff
for Hext applied at an angle H = 40°. The values of eff result from fitting the TRMO signals to LLG equation; I = 140
J/cm2. (c) and (d) Increase in lattice temperature as a function of time delay between pump and probe pulses for
selected depths from the sample surface (c) and as a function of depth for fixed time delays (d). I = 140 J/cm2, T0 =
20 K. The heat capacities and conductivities of individual layers are provided in the Supplementary Material, Section
7.
11
Thermally induced all-optical ferromagnetic resonance in thin YIG films:
Supplementary Material
E. Schmoranzerová1*, J. Kimák1, R. Schlitz3 , S.T. B. Goennenwein3, D. Kriegner2,3, H. Reichlová2,3, , Z.
Šobáň2, G. Jakob5, E.-J. Guo5, M. Kläui5, M. Münzenberg4, P. Němec1 , T. Ostatnický1
1Faculty of Mathematics and Physics, Charles University, Prague, 12116, Czech Republic
2Institute of Physics ASCR v.v.i , Prague, 162 53, Czech Republic
3Technical University Dresden, 01062 Dresden, Germany
4Institute of Physics, Ernst-Moritz-Arndt University, 17489, Greifswald, Germany
5Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany
6 Department of Physics, University of Konstanz, 78464 Konstanz, Germany
1. Magnetic characterization
A. SQUID magnetometry
A superconducting quantum device magnetometer (SQUID) was used to characterize the magnetic
properties of the thin YIG film at several sample temperatures. The magnetic hysteresis loops, detected
with magnetic field applied in [2-1-1] crystallographic direction of the YIG layer, are shown in Fig. S1. As
expected [t26], the saturation magnetization increases at low temperatures, which is accompanied by a
slight increase in coercive field. At room temperature, the effective saturation magnetization is estimated
to be Ms = 95 kA/m. This value is in good agreement with the effective magnetization Meff obtained from
the ferromagnetic resonance (FMR) measurement (see Section 1b), which indicates only a weak out-of-
plane magnetic anisotropy [s1]. However, as discussed in detail in Ref. 26, the Ms from our SQUID
measurement is burdened by a relatively large error. Therefore, mere comparison of SQUID and FMR
experiment is not sufficient to evaluate the size of the out-of-plane magnetic anisotropy. An additional
experiment such as static magneto-optical measurement [28] is needed in order to get more precise
estimation of the out-of plane magnetic anisotropy.
B. FMR measurement
The SQUID magnetometry was complemented by so-called broad band ferromagnetic resonance
measurements using a co-planar waveguide to apply electromagnetic radiation of a variable frequency f
=/2 to the sample. The measurement was performed at room temperature and further details on the
method can be found in Ref. s2. An exemplary set of spectra showing the normalized microwave
transmission | S21|norm obtained at different external fields magnitudes applied in the sample plane, is
shown in Fig. S2(a). The set of Lorentzian-shape resonances can be fitted by the equation: 12
|𝑆ଶଵ|୬୭୰୫=ቀഘ
మቁమ
ቀഘ
మഏିഘబ
మഏቁାቀഘ
మቁమ+𝑦 (S1)
Where f0 =0/2 is the FMR resonance frequency, /2 is the half width half maximum line width, B the
amplitude of the FMR line and y0 a frequency independent offset. From an automated fitting of the set of
lines obtained at different Hext, we extract the magnetic field dependence of the resonance frequency
0/2 (Hext) [Fig. S2(b)] and linewidth (Hext) [Fig. S2(c)]. Clearly, the resonance frequencies correspond
to the fundamental (Kittel) mode, and can correspondingly be fitted by the Kittel formula [s3]:
ఠబ
ଶగ=ఊ
ଶగඥ𝜇𝐻ୣ୶୲(𝜇𝐻ୣ୶୲+𝜇𝑀ୣ) (S2)
Where Meff is the effective saturation magnetization that includes the out-of-plane anisotropy term, and
is gyromagnetic ratio. From this fit, it is possible to evaluate Meff , Kittel = 94.9 kA/m
From the linewidth dependence (Hext)=2 + 0 we can extract both the inhomogeneous line
broadening and the Gilbert damping parameter, as shown in Fig. S2(c) [s2]. In our experiment, the
inhomogeneous linewidth broadening is 0 = 55.8 MHz, and the Gilbert damping parameter = 0.001.
Both values are on a higher side compared e.g. with YIG prepared by liquid phase epitaxy [s8] but in good
agreement with typical YIG thin films similar to our layers, which were prepared by pulsed laser deposition
[27]. This again confirms the good quality of the studied thin YIG films.
2. Processing of time-resolved magneto-optical data
In order to extract the parameters describing the precession of magnetization correctly from the time-
resolved magneto-optical (TRMO) signals, it is first necessary to remove the slowly varying background on
which the oscillatory signals are superimposed. For this purpose, we fitted the measured data by the
second-order polynomial. The fitted curve was then subtracted from the measured signals, as
demonstrated in Fig. (S3).
From the physical point of view, the background can be attributed to a slow return of magnetization to its
equilibrium state after the pump beam induced heating, which can take place on the timescale of tens of
nanoseconds [10]. Since both saturation magnetization Ms and magnetocrystalline anisotropy Kc are
temperature-dependent, their temporal variation can in principle contribute to the background signal.
However, as explained later in Section 4, the variation of Ms is very weak at cryogenic temperatures. The
heat-induced modification of Kc, and the resulting change of the magnetization quasi-equilibrium
orientation, is, therefore, more probable origin of the slowly varying background, which is detected in the
MO experiment by the Cotton-Mouton effect [28].
3. Angular dependence of precession amplitude
In order to mutually compare the values of precession amplitudes measured in YIG/Pt and YIG/Au samples
at different angles of the external magnetic field H, it is necessary to correct their values for the value of
H. The following procedure was used to correct the data presented in Fig. 3 of the main text .
13
First, we measured in detail angular dependence of the precession amplitude in the YIG/Au layer, which is
presented in Fig. S4. Amplitude of the oscillatory signal detected in our experiment does not depend solely
on the amplitude of the magnetization precession but also on the size of the magneto-optical (MO) effect.
In our experimental setup, the change of H was achieved by tilting the sample relative to the position of
electromagnet poles [see Fig. 1(a)]. The MO response, however, varies also with the angle of incidence
which is modified simultaneously with a change of H [see Fig. 1(a)] . Therefore, it is not straightforward to
describe the A(H) analytically. Instead, we fitted the measured dependence A(H) by a rational function in
a form of y = 1/(A+Bx2), which is the lowest order polynomial function that can describe the signal properly.
From the fit we derived a correction factor of 1.7 by which the amplitudes A measured at H =21° has to
be multiplied to correspond to that measured at H =40°. This factor was then used to recalculate the
intensity dependence of the precession amplitude A(I) in YIG/Au measured at H =21° to the A(I) at H
=40°, which could be directly compared to the A(I) dependence detected at YIG/Pt for H =40° - see Fig.
S4(b).
4. Temperature dependence of precession amplitude
In order to further investigate the origin of the laser-induced magnetization precession, the amplitude of
the oscillatory MO signal was measured as a function of the sample temperature in YIG/Pt sample, see Fig.
S5(a). In Fig. S5 (b), we show temperature dependence of saturation magnetization Ms, as obtained from
Ref. 32
The only parameter changed within this experiment was the sample temperature. It is reasonable to
expect that the size of the magneto-optical effect is not strongly temperature dependent in the studied
temperature range between 20 and 50 K (see Ref. 28) Therefore, the dependence A(T) presented in Fig S5
corresponds directly to the temperature dependence of magnetization precession amplitude. By
comparing the Ms(T) and A(T) data, it is immediately apparent that the laser-induced heating would not
modify Ms enough to account for the large change of the magnetization precession amplitude with the
sample temperature. Even assuming the most extreme laser-induced temperature increase T 80 K
shown in Fig. 3(c), the laser-induced Ms variation would be less than 5%, while the precession amplitude
changes by more than 50% between 20 and 50 K. In contrast, the magnetocrystalline anisotropy Kc1
changes drastically even in this relatively narrow temperature range [see Fig. 2(c)]. Consequently, the
change of Kc1, which leads to a significant change of the position of quasi-equilibrium magnetization
orientation in the studied sample (see Section 5) provides a more plausible explanation for the origin of
the laser-induced magnetization precession in the YIG/metal layer.
5. LLG equation model
The data were modelled by numerical solution of the Landau-Lifshitz-Gilbert (LLG) equation, as defined in
[s9]:
ௗ𝑴(௧)
ௗ௧= −𝜇𝛾ൣ𝑴(𝑡)×𝑯𝒆𝒇𝒇(𝑡)൧+ఈ
ெೞቂ𝑴(𝑡)×ௗ𝑴(௧)
ௗ௧ቃ, (S3) 14
where is the gyromagnetic ratio, is the Gilbert damping constant, and MS is saturated
magnetization.The effective magnetic field Heff is given by:
𝑯𝒆𝒇𝒇(𝑡)=డி
డ𝑴 (S4)
where F is energy density functional that contains contributions from the external magnetic field Hext,
demagnetizing field and the magnetic anisotropy of the sample. We consider the form of F including first-
and second-order cubic terms as defined in Ref. [t24]. The polar angle is measured with respect to the
crystallographic axis [111] and the azimuthal angle = 0 corresponds to the direction [21ത1ത], with an
appropriate index referring to the magnetization position (index M) or the direction of the external
magnetic field (index H). The resulting functional takes the form (in the SI units):
𝐹= −𝜇𝐻𝑀[sin𝜃ெsin𝜃ு+cos𝜃ெcos𝜃ுcos(𝜑ு−𝜑ெ)]+ቀଵ
ଶ𝜇𝑀ଶ−𝐾uቁsinଶ𝜃ெ
+𝐾c1
12ൣ7cosସ𝜃ெ−8cosଶ𝜃ெ+4−4√2cosଷ𝜃ெsin𝜃ெcos3𝜑ெ൧
+c2
ଵ଼ൣ−24cos𝜃ெ+45cosସ𝜃ெ−24cosଶ𝜃ெ+4−2√2cosଷ𝜃ெsin𝜃ெ(5cosଶ𝜃ெ−
2)cos3𝜑ெ+cos𝜃ெcos6𝜑ெ൧ , (S5)
where 0 is the vacuum permeability and we consider the following values of constants: magnetization M
= 174 kA/m, first-order cubic anisotropy constant Kc1=4.68 kJ/m3, second-order cubic anisotropy constant
Kc2 = 222 J/m3 [t24].
For modelling the dependence of precession frequency on the external magnetic field Hext [Fig. 2 (a)] and
on the angle H [Fig. 2 (a)], we assumed that in a steady state magnetization direction is parallel to Hext, i.e.
M = H, and M = H. This is surely fulfilled for large enough magnitude of Hext. Since the coercive field is
very small, we can assume the procedure to be correct. Further correspondence to experimental data
Evaluation of the Gilbert damping factor from the as-measured magneto-optical oscillatory data was done
by fitting signals by a theoretical curve calculated by solving numerically LLG equation [Eq. (S3)]. We
considered the magnetization free energy density in a form of Eq. (S5) using magnitude and direction of
the external magnetic field from the experiment. The electron g-factor was set to 2.0 and then the Gilbert
factor and five parameters of the fourth-order polynomial to remove the background MO signal were the
fitting parameters. The resulting dependence of fitted effective Gilbert factors αeff on external magnetic
field is displayed in Fig. 3(b) in the main text, from which the field-independent Gilbert factor α can be
evaluated.
6. Comparison of Gilbert damping parameter from MW-FMR and TRMO experiments
The Gilbert damping from the room-temperature FMR measurement on the YIG film 1∙10-3 and the
results from fits of the low-temperature pump&probe data 2∙10-2, differ by an order of magnitude. As
detailed in the main text, we attribute this difference to the different sample temperatures in the AO-FMR
and MW-FMR measurements. However, one might also argue that the increased damping in the optical
experiments is caused either by a spatial inhomogeneity of the magnetization oscillations or it is the result
of the perturbation of the YIG surface. 15
In the former case, we expect that the spatial inhomogeneity of the temperature distribution [see Fig.
3(d)] causes the magnetization to oscillate in a form of a superposition of harmonic waves with well-
defined in-plane wavevectors. Considering the dispersion of the allowed oscillatory modes [s4] and
including the relevant value of the exchange stiffness [s5], we revealed that neither the inhomogeneity
due to the finite cross section of the excitation laser beam nor the temperature gradient perpendicular to
the sample surface can cause such a strong decrease of the Gilbert damping factor that is observed
experimentally. Here, we provide an estimate on which time scales the mode dispersion influences the
decay of the signal if the exchange stiffness is taken into account. Following [s5], the mode dispersion is
described by the additive exchange field in the form:
𝜇𝐻ex=𝐷ቈ𝜋ଶ
𝑑ଶ𝑛ଶ+𝑘∥ଶ ,
where D ≈ 5∙10-17 T.m2 is the exchange stiffness, n is the order of the confined magnon mode, d is the YIG
layer thickness and k‖ is the in-plane magnon wave vector. We consider here only the n = 0 case since this
is the only visible harmonic mode observed in the experimental MO data, as proven by the numerical
fitting. Note that the frequency shift ∆𝜔/2𝜋=|𝛾|𝜇𝐻ex/2𝜋, where the symbol γ stands for the electron
gyromagnetic ratio, of the n = 1 mode would be 5.5 GHz, which would be then clearly distinguishable from
the basic n = 0 mode in the lowest external magnetic fields. The in-plane wave vector k‖ can be calculated
from the FWHM (full width at half maximum) width of the laser spot on the sample L, which is about 30
µm in our case, that leads to the order of magnitude k‖ ≈ (2π/L) ≈ 105 m-1. The frequency increase due to
the finite laser spot size can be estimated as ∆𝜔/2𝜋=|𝛾|𝐷𝑘∥ଶ/2𝜋≈14 kHz. Inverse of this value ( 0.1
ms) determines the typical time scale at which the magnon dynamics is influenced by their dispersion due
to the finite laser spot size, which is clearly out of the range of the experimental time scale.
The presence of a metallic layer on the top of the YIG sample surface can result into two significant
damping processes. First, the magnetization oscillations (and thus oscillations of the macroscopic magnetic
field) are coupled to electromagnetic modes which penetrate the surrounding material and can be
eventually radiative for small magnon wave vectors. Penetration into conductive material in turn causes
energy dissipation through finite conductivity of such material. We checked the magnitudes of the
additional damping caused by the radiative field and energy dissipation in a thin metallic layer and we
found that these processes exist but the additional energy loss cannot explain the observed magnitude of
the Gilbert damping parameter. The second possible explanation of the increased precession damping due
to the presence of the metal/YIG surface may be that there is an additional perturbation to otherwise
homogeneous sample due to some inhomogeneity through surface roughness or spatially inhomogeneous
local spin pinning. Since both the surface roughness and spin pinning can depend on the composition of
the capping layer, it can also cause a minor difference in the resulting damping factor, as observed in Fig.
3(b).
Overall, we attribute the experimentally observed difference in Gilbert damping measured by FMR and
pump&probe techniques to the difference in ambient temperatures that were used in these experiments,
which is in accord with the results of Ref. t22.
7. Heat propagation in YIG/Pt and YIG/Au 16
Heat propagation in our sample structures was modelled in terms of the heat equation:
డ்
డ௧=ఒ
∆𝑇 , (S4)
where T is the local temperature, λ is the local thermal conductivity, c is the heat capacity and the symbol
Δ denotes the Laplace operator. The spatio-temporal temperature distribution in the studied sample has
been calculated by a direct integration of Eq. (S4) in a time domain, assuming excitation of the metallic
layer by an ultrashort optical pulse [with a temporal duration of 100 fs (FWHM)]. We have taken the whole
structure profile of vacuum/metal/YIG/GGG into consideration, assuming that the GGG substrate had a
perfect heat contact with the cold finger of the cryostat, which has been held on a constant temperature.
The respective heat conductivities ( λ) and heat capacities ( c) were set to the following values. Au: λ = 5
W/m K [s6], c = 1.3∙104 J/cm3, Pt: λ = 10 W/m K [s7], c = 1.2∙104 J/cm3, YIG: λ = 60 W/m K, c = 6.7∙103 J/cm3,
GGG: λ = 300 W/m K, c = 2.1∙104 J/m3.
To evaluate the initial heat transfer from the optical pulses to the capping metallic layer, we considered
the proper geometry of our experiment, i.e. a 8 nm thick metallic layer deposited on the YIG sample, the
incidence angle of the laser beam of 45 degrees and its p-polarization. We then used optical constants of
gold and platinum in order to calculate transmission and reflection coefficients of a nanometer-thick
metallic layers by means of the transfer matrix method. From those, we estimated the efficiency of power
conversion from the optical field to heat to be 3% for gold and 6.5% for platinum. The total amount of heat
density was then calculated by multiplication of the pump pulse energy density and the above-mentioned
efficiency.
The data shown in Fig. 3(c)-(d) were then extracted from the full spatio-temporal temperature distribution.
Clearly, the temperature increase in the YIG/Pt sample is approximately twice larger than that of the
YIG/Au sample as a consequence of twice larger efficiency of the light-heat energy conversion in favour of
platinum. Correspondingly, also the amplitudes of the MO oscillations in Fig. 3(a) reveal the ratio 2:1.
LITERATURE
[s1] B. Bhoi et al.: J. Appl.Phys. 123, 203902 (2018)
[s2] H. Maier-Flaig et al,.PRB 95, 214423 (2017)
[s3] Ch. Kittel: “Introduction to solid state physics (8th ed.)”. New Jersey: Wiley. (2013).
[s4] D. D. Stancil, A. Prabhakar: Spin waves – theory and applications (Springer, New York, 2009).
[s5] S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko, B. Hillebrands,
and A. Conca, J. Phys. D: Appl. Phys. 48, 015001 (2014).
[s6] G. K. White, Proc. Phys. Soc. A 66, 559 (1953).
[s7] X. Zhang, H. Xie, M. Fujii, H Ago, K. Takahashi, T. Ikuta, T. Shimizu, Appl. Phys. Lett. 86, 171912 (2005) 17
[s8] C. Dubs et al., Phys. Rev. Materials 4, 024416 (2020)
[s9] J. Miltat, G. Albuquerque, and A. Thiaville, An introduction to micromagnetics in the dynamic regime,
in Spin dynamics in confined magnetic structures I, edited by B. Hillebrands and K. Ounadjela, Springer,
Berlin, 2002, vol. 83 of Topics in applied physics.
FIGURES
Fig. S1: Magnetic hysteresis loops measured by SQUID magnetometry with magnetic field Hext applied in
direction [2-1-1] at several sample temperatures. The saturation magnetization obtained from SQUID
magnetometry measurement at room temperature is roughly Ms = 95 kA/m , assuming a YIG layer thickness
of 50 nm.
18
Fig. S2: (a) Ferromagnetic resonance spectra measured at room temperature for several different external
magnetic field magnitudes µ0Hext from 0 to 540 mT applied in the sample plane. Resonance peaks were
fitted by Eq. (S1) and the obtained resonance frequencies and linewidths are plotted as points in panels
(b) and (c), respectively. The lines correspond to fit by Kittel formula [Eq. (S2)], which enables to evaluate
effective magnetization Meff = 94.9 kA/m and Gilbert damping parameter of = 0.001.
19
Fig. S3: Removal of slowly varying background from time-resolved magneto-optical signals. The red dots
correspond to as-measured signals, line indicates the polynomial background that is subtracted from the
raw signals. Black dots show the signal after background subtraction, black line representing the fit by Eq.
(1) of the main text. The data were taken at external field of 0Hext = 300 mT, temperature 20 K and pump
fluence I = 140 J/cm2..
Fig. S4: (a) Dependence of the amplitude A of oscillatory magneto-optical signal on the sample tilt
(different field angles of magnetic field H ) measured in YIG/Au sample . 0Hext = 300 mT, temperature T
= 20 K and pump fluence I = 150 J/cm2.. (b) Pump intensity dependence of A measured for YIG/Au
sample at H =21° (red points), the same dependence recalculated to correspond to H =40 (blue
points) where A(I) was measured for YIG/Pt sample (green points); T = 20 K.
Fig. S5: (a) Temperature dependence of amplitude of the time-resolved magneto-optical signals measured
for external field 0Hext = 300 mT applied at an angle H = 30°. (b) Temperature dependence of saturation
magnetization Ms obtained from Ref. 32.
|
1505.07248v1.Logarithmic_stability_in_determining_a_boundary_coefficient_in_an_ibvp_for_the_wave_equation.pdf | arXiv:1505.07248v1 [math.AP] 27 May 2015LOGARITHMIC STABILITY IN DETERMINING A BOUNDARY COEFFICIE NT IN
AN IBVP FOR THE WAVE EQUATION
KA¨IS AMMARI AND MOURAD CHOULLI
Abstract. In [2] we introduced a method combining together an observab ility inequality and a spectral de-
composition to get a logarithmic stability estimate for the inverse problem of determining both the potential
and the damping coefficient in a dissipative wave equation fro m boundary measurements. The present work
deals with an adaptation of that method to obtain a logarithm ic stability estimate for the inverse problem
of determining a boundary damping coefficient from boundary m easurements. As in our preceding work,
the different boundary measurements are generated by varyin g one of the initial conditions.
Keywords : inverse problem, wave equation, boundary damping coefficie nt, logarithmic stability, boundary
measurements.
MSC: 35R30.
Contents
1. Introduction 1
1.1. The IBVP 2
1.2. Main result 2
2. Preliminaries 3
2.1. Extension lemma 3
2.2. Observability inequality 4
3. The inverse problem 5
3.1. An abstract framework for the inverse source problem 5
3.2. An inverse source problem for an IBVP for the wave equation 7
3.3. Proof of Theorem 1.1 7
Appendix A. 9
Appendix B. 10
References 11
1.Introduction
We are concerned with an inverse problem for the wave equation whe n the spatial domain is the square
Ω = (0,1)×(0,1). To this end we consider the following initial-boundary value problem (abbreviated to
IBVP in the sequel) :
(1.1)
∂2
tu−∆u= 0 in Q= Ω×(0,τ),
u= 0 on Σ 0= Γ0×(0,τ),
∂νu+a∂tu= 0 on Σ 1= Γ1×(0,τ),
u(·,0) =u0, ∂tu(·,0) =u1.
Here
Γ0= ((0,1)×{1})∪({1}×(0,1)),
Γ1= ((0,1)×{0})∪({0}×(0,1))
12 KA¨IS AMMARI AND MOURAD CHOULLI
and∂ν=ν·∇is the derivative along ν, the unit normal vector pointing outward of Ω. We note that νis
everywhere defined except at the vertices of Ω and we denote by Γ = Γ0∪Γ1. The boundary coefficient ais
usually called the boundary damping coefficient.
In the rest of this text we identify a|(0,1)×{0}bya1=a1(x),x∈(0,1) anda|{0}×(0,1)bya2=a2(y),
y∈(0,1). In that case it is natural to identify a, defined on Γ 1, by the pair ( a1,a2).
1.1.The IBVP. We fix 1/2<α≤1 and we assume that a∈A, where
A={b= (b1,b2)∈Cα([0,1])2, b1(0) =b2(0), bj≥0}.
This assumption guarantees that the multiplication operator by aj,j= 1,2, defines a bounded operator on
H1/2((0,1)). The proof of this fact will be proved in Appendix A.
LetV={u∈H1(Ω);u= 0 on Γ 0}and we consider on V×L2(Ω) the linear unbounded operator Agiven
by
Aa= (w,∆v), D(Aa) ={(v,w)∈V×V; ∆v∈L2(Ω) and∂νv=−awon Γ1}.
One can prove that Aais a m-dissipative operator on the Hilbert space V×L2(Ω) (for the reader’s
convenience we detail the proof in Appendix B). Therefore, Aais the generator of a strongly continuous
semigroup of contractions etAa. Hence, for each ( u0,u1), the IBVP (1.1) possesses a unique solution denoted
byua=ua(u0,u1) so that
(ua,∂tua)∈C([0,∞);D(Aa))∩C1([0,∞),V×L2(Ω)).
1.2.Main result. For 0<m≤M, we set
Am,M={b= (b1,b2)∈A∩H1(0,1)2;m≤bj,/ba∇dblbj/ba∇dbl2
H1(0,1)≤M}.
LetU0given by
U0={v∈V; ∆v∈L2(Ω) and∂νv= 0 on Γ 1}.
We observe that U0×{0} ⊂D(Aa), for anya∈A.
LetCa∈B(D(Aa);L2(Σ1)) defined by
Ca(u0,u1) =∂νua(u0,u1)|Γ1.
We define the initial to boundary operator
Λa:u0∈ U0−→Ca(u0,0)∈L2(Σ1).
ClearlyCa∈B(D(Aa);L2(Σ1)) implies that Λ a∈B(U0;L2(Σ1)), when U0is identified to a subspace of
D(Aa) endowed with the graph norm of Aa. Precisely the norm in U0is the following one
/ba∇dblu0/ba∇dblU0=/parenleftBig
/ba∇dblu0/ba∇dbl2
V+/ba∇dbl∆u0/ba∇dbl2
L2(Ω)/parenrightBig1/2
.
Henceforth, for simplicity sake, the norm of Λ a−Λ0inB(U0;L2(Σ1)) will denoted by /ba∇dblΛa−Λ0/ba∇dbl.
Theorem 1.1. There exists τ0>0so that for any τ > τ0, we find a constant c >0depending only on τ
such that
(1.2) /ba∇dbla−0/ba∇dblL2((0,1))2≤cM/parenleftBig/vextendsingle/vextendsingleln/parenleftbig
m−1/ba∇dblΛa−Λ0/ba∇dbl/parenrightbig/vextendsingle/vextendsingle−1/2+m−1/ba∇dblΛa−Λ0/ba∇dblL2(Σ1)/parenrightBig
,
for eacha∈ Am,M.
We point out that our choiceofthe domain Ω is motivated by the fact t he spectral analysisofthe laplacian
under mixed boundary condition is very simple in that case. However t his choice has the inconvenient that
the square domain Ω is no longer smooth. So we need to prove an obse rvability inequality associated to this
non smooth domain. This is done by adapting the existing results. We n ote that the key point in establishing
this observability inequality relies on a Rellich type identity for the doma in Ω.
The inverse problem we discuss in the present paper remains largely o pen for an arbitrary (smooth)
domain as well as for the stability around a non zero damping coefficien t. Uniqueness and directional
Lipschitz stability, around the origin, was established by the author s in [3].LOGARITHMIC STABILITY IN DETERMINING A BOUNDARY COEFFICIE NT 3
The determination of a potential and/or the sound speed coefficien t in a wave equation from the so-called
Dirichlet-to-Neumann map was extensively studied these last decad es. We refer to the comments in [2] for
more details.
2.Preliminaries
2.1.Extension lemma. We decompose Γ 1as follows Γ 1= Γ1,1∪Γ1,2, where Γ 1,1= (0,1)× {0}and
Γ1,2={0}×(0,1). Similarly, we write Γ 0= Γ0,1∪Γ0,2, with Γ 0,1={1}×(0,1) and Γ 0,2= (0,1)×{1}.
Let (g1,g2)∈L2((0,1))2. We say that the pair ( g1,g2) satisfies the compatibility condition of the first
order at the vertex (0 ,0) if
(2.1)/integraldisplay1
0|g1(t)−g2(t)|2dt
t<∞.
Similarly, we can define the compatibility condition of the first order at the other vertices of Ω.
We need also to introduce compatibility conditions of the second orde r. Let (fj,gj)∈H1((0,1))×
L2((0,1)),j= 1,2. We say that the pair [( f1,g1),(f2,g2)] satisfies the compatibility conditions of second
order at the vertex (0 ,0) when
(2.2) f1(0) =f2(0),/integraldisplay1
0|f′
1(t)−g2(t)|2dt
t<∞and/integraldisplay1
0|g1(t)−f′
2(t)|2dt
t<∞.
The compatibility conditions ofthe second orderat the other vertic es ofΩ aredefined in the same manner.
The following theorem is a special case of [4, Theorem 1.5.2.8, page 50 ].
Theorem 2.1. (1) The mapping
w−→(w|Γ0,1,w|Γ0,2,w|Γ1,1,w|Γ1,2) = (g1,...,g 4),
defined on D(Ω)is extended from H1(Ω)onto the subspace of H1/2((0,1))4consisting in functions (g1,...,g 4)
so that the compatibility condition of the first order is sati sfied at each vertex of Ωin a natural way with the
pairs(gj,gk).
(2) The mapping
w→(w|Γ0,1,∂xw|Γ0,1,w|Γ0,2,∂yw|Γ0,2w|Γ1,1,−∂yw|Γ1,1,w|Γ1,2,−∂xw|Γ1,2) = ((f1,g1),...(f4,g4))
defined on D(Ω)is extended from H2(Ω)onto the subspace of [H3/2((0,1))×H1/2((0,1))]4of functions
((f1,g1),...(f4,g4))so that the compatibility conditions of the second order are satisfied at each vertex of Ω
in a natural way with the pairs [(fj,gj),(fk,gk)].
Lemma 2.1. (Extension lemma) Let gj∈H1/2((0,1)),j= 1,2, so that (g1,g2),(g1,0)and(g2,0)satisfy
the first order compatibility condition respectively at the vertices(0,0),(1,0)and(0,1). Then there exists
u∈H2(Ω)so thatu= 0onΓ0and∂νu=gjonΓ1,j,j= 1,2.
Proof.(i) We define f1(t) =/integraltextt
0g2(s)dsandf2(t) =/integraltextt
0g1(s)ds. Then (f1,g1) and (f2,g2) satisfy the com-
patibility conditions of the second order at the vertex (0 ,0).
(ii) Let/tildewideg1∈H1/2((0,1)) be such that/integraltext1
0|/tildewideg1(t)|2
tdt<∞. Let/tildewidef1(t) =/integraltextt
0g2(s)ds. Hence, it is straightfor-
ward to check that ( /tildewidef1,/tildewideg1) and (0,g2) satisfy the compatibility conditions of the second order at (0 ,0).
(iii) From steps (i) and (ii) we derive that the pairs [( f1,g1),(f2,g2)], [(f1,g1),(0,g2)] and [(0,g1),(f2,g2)]
satisfy the second order compatibility conditions respectively at th e vertices (0 ,0), (1,0) and (0,1). We
see that unfortunately the pair [(0 ,g1),(0,g2)] doesn’t satisfy necessarily the compatibility conditions of the
second order at the vertex (1 ,1). We pick χ∈C∞(R) so thatχ= 1 in a neighborhood of 0 and χ= 0 in a
neighborhood of 1. Then [(0 ,χg1),(0,χg2)] satisfies the compatibility condition of the second order at the
vertex (1,1). Since this construction is of local character at each vertex, t he cutoff function at the vertex
(1,1) doesn’t modify the construction at the other vertices. In othe r words, the compatibility conditions of
the second order are preserved at the other vertices. We comple te the proof by applying Theorem 2.1. /square4 KA¨IS AMMARI AND MOURAD CHOULLI
Corollary 2.1. Leta= (a1,a2)∈Aandgj∈H1/2((0,1)),j= 1,2, so that (g1,g2),(g1,0)and(g2,0)
satisfy the first order compatibility condition respective ly at the vertices (0,0),(1,0)and(0,1). Then there
existsu∈H2(Ω)so thatu= 0onΓ0and∂νu=ajgjonΓ1,j,j= 1,2.
Proof.It is sufficient to prove that ( a1g1,a2,g2) and (ajgj,0),j= 1,2, satisfy the first order compatibility
condition at (0 ,0) witha1(0) =a2(0) for the first pair and without any condition on ajfor the second pair.
Usinga1(0) =a2(0), we get
t−1|a1(t)−a2(t)|2≤2t−1|a1(t)−a1(0)|2+2t−1|a2(t)−a2(0)|2
≤2t−1+2α([a1]2
α+[a2]2
α)
≤2([a1]2
α+[a2]2
α).
This estimate together with the following one
|a1(t)g1(t)−a2(t)g2(t)|2≤2|a1(t)−a2(t)|2|g1(t)|2+2|a2(t)|2|g1(t)−g2(t)|2
yield
/integraldisplay1
0|a1(t)g1(t)−a2(t)g2(t)|2dt
t≤4([a1]2
α+[a2]2
α)/ba∇dblf/ba∇dblL2((0,1))+2/ba∇dbla2/ba∇dblL∞((0,1))/integraldisplay1
0|g1(t)−g2(t)|2dt
t.
Hence/integraldisplay1
0|g1(t)−g2(t)|2dt
t<∞=⇒/integraldisplay1
0|a1(t)g1(t)−a2(t)g2(t)|2dt
t<∞.
If (gj,0) satisfies the first compatibility at the vertex (0 ,0). Then
/integraldisplay1
0|gj(t)|2dt
t<∞.
Therefore/integraldisplay1
0|ajgj(t)|2dt
t≤ /ba∇dblaj/ba∇dbl2
L∞((0,1))/integraldisplay1
0|gj(t)|2dt
t<∞.
Thus (ajgj,0) satisfies also the first compatibility at the vertex (0 ,0). /square
2.2.Observability inequality. We discuss briefly how we can adapt the existing results to get an obs erv-
ability inequality corresponding to our IBVP. We first note that
Γ0⊂ {x∈Γ;m(x)·ν(x)<0},
Γ1⊂ {x∈Γ;m(x)·ν(x)>0},
wherem(x) =x−x0,x∈R2, andx0= (α,α) withα>1.
The following Rellich identity is a particular case of identity [5, (3.5), pag e 227]: for each 3 /2< s <2
andϕ∈Hs(Ω) satisfying ∆ ϕ∈L2(Ω),
(2.3) 2/integraldisplay
Ω∆ϕ(m·∇ϕ)dx= 2/integraldisplay
Γ∂νϕ(m·∇ϕ)dσ−/integraldisplay
Γ(m·ν)|∇ϕ|2dσ.
Lemma 2.2. Let(v,w)∈D(Aa). Then
2/integraldisplay
Ω∆v(m·∇v)dx= 2/integraldisplay
Γ∂νv(m·∇v)dσ−/integraldisplay
Γ(m·ν)|∇v|2dσ.
Proof.Let (v,w)∈D(Aa). By Corollary 2.1, there exists /tildewidev∈H2(Ω) so that/tildewidev= 0 on Γ 0and∂ν/tildewidev=−aw
on Γ1. In light of the fact that z=v−/tildewidevis such that ∆ z∈L2(Ω),z= 0 on Γ 0and∂νz= 0 on Γ 1, we get
z∈Hs(Ω) for some 3 /2< s <2 by [5, Theorem 5.2, page 237]. Therefore v∈Hs(Ω). We complete the
proof by applying Rellich identity (2.3). /square
Lemma2.2athand, wecan mimic the proofof[7, Theorem7.6.1,page2 52]in orderto obtainthe following
theorem:LOGARITHMIC STABILITY IN DETERMINING A BOUNDARY COEFFICIE NT 5
Theorem 2.2. We assume that a≥δonΓ1, for some δ >0. There exist M≥1andω >0, depending
only onδ, so that
/ba∇dbletAa(v,w)/ba∇dblV×L2(Ω)≤Me−ωt/ba∇dbl(v,w)/ba∇dblV×L2(Ω),(v,w)∈D(Aa), t≥0.
An immediate consequence of Theorem 2.2 is the following observability inequality.
Corollary 2.2. We fix0<δ0<δ1. Then there exist τ0>0andκ, depending only on δ0andδ1so that for
anyτ≥τ0anda∈Asatisfyingδ0≤a≤δ1onΓ1,
/ba∇dbl(u0,u1)/ba∇dblV×L2(Ω)≤κ/ba∇dblCa(u0,u1)/ba∇dblL2(Σ1).
Moreover,Cais admissible for etAaand(Ca,Aa)is exactly observable.
We omit the proof of this corollary. It is quite similar to that of [7, Coro llary 7.6.5, page 256].
3.The inverse problem
3.1.An abstract framework for the inverse source problem. In the present subsection we consider
an inverse source problem for an abstract evolution equation. The result of this subsection is the main
ingredient in the proof of Theorem 1.1.
LetHbe a Hilbert space and A:D(A)⊂H→Hbe the generator of continuous semigroup ( T(t)). An
operatorC∈B(D(A),Y),Yis a Hilbert space which is identified with its dual space, is called an admiss ible
observation for ( T(t)) if for some (and hence for all) τ >0, the operator Ψ ∈B(D(A),L2((0,τ),Y)) given
by
(Ψx)(t) =CT(t)x, t∈[0,τ], x∈D(A),
has a bounded extension to H.
We introduce the notion of exact observability for the system
z′(t) =Az(t), z(0) =x, (3.1)
y(t) =Cz(t), (3.2)
whereCis an admissible observation for T(t). Following the usual definition, the pair ( A,C) is said exactly
observable at time τ >0 if there is a constant κsuch that the solution ( z,y) of (3.1) and (3.2) satisfies
/integraldisplayτ
0/ba∇dbly(t)/ba∇dbl2
Ydt≥κ2/ba∇dblx/ba∇dbl2
H, x∈D(A).
Or equivalently
(3.3)/integraldisplayτ
0/ba∇dbl(Ψx)(t)/ba∇dbl2
Ydt≥κ2/ba∇dblx/ba∇dbl2
H, x∈D(A).
Letλ∈H1((0,τ)) such that λ(0)/ne}ationslash= 0. We consider the Cauchy problem
(3.4) z′(t) =Az(t)+λ(t)x, z(0) = 0
and we set
(3.5) y(t) =Cz(t), t∈[0,τ].
We fixβin the resolvent set of A. LetH1be the space D(A) equipped with the norm /ba∇dblx/ba∇dbl1=/ba∇dbl(β−A)x/ba∇dbl
and denote by H−1the completion of Hwith respect to the norm /ba∇dblx/ba∇dbl−1=/ba∇dbl(β−A)−1x/ba∇dbl. As it is observed
in [1, Proposition 4.2, page 1644] and its proof, when x∈H−1(which is the dual space of H1with respect to
the pivot space H) andλ∈H1((0,T)), then, according to the classical extrapolation theory of semig roups,
the Cauchy problem (3.4) has a unique solution z∈C([0,τ];H). Additionally ygiven in (3.5) belongs to
L2((0,τ),Y).
Whenx∈H, we have by Duhamel’s formula
(3.6) y(t) =/integraldisplayt
0λ(t−s)CT(s)xds=/integraldisplayt
0λ(t−s)(Ψx)(s)ds.6 KA¨IS AMMARI AND MOURAD CHOULLI
Let
H1
ℓ((0,τ),Y) =/braceleftbig
u∈H1((0,τ),Y);u(0) = 0/bracerightbig
.
We define the operator S:L2((0,τ),Y)−→H1
ℓ((0,τ),Y) by
(3.7) ( Sh)(t) =/integraldisplayt
0λ(t−s)h(s)ds.
IfE=SΨ, then (3.6) takes the form
y(t) = (Ex)(t).
LetZ= (β−A∗)−1(X+C∗Y).
Theorem 3.1. We assume that (A,C)is exactly observable at time τ. Then
(i)Eis one-to-one from HontoH1
ℓ((0,τ),Y).
(ii)Ecan be extended to an isomorphism, denoted by /tildewideE, fromZ′ontoL2((0,τ);Y).
(iii) There exists a constant /tildewideκ, independent on λ, so that
(3.8) /ba∇dblx/ba∇dblZ′≤/tildewideκ|λ(0)|e/bardblλ′/bardbl2
L2((0,τ))
|λ(0)|2τ/ba∇dbl/tildewideEx/ba∇dblL2((0,τ),Y).
Proof.(i) and (ii) are contained in [1, Theorem 4.3, page 1645]. We need only t o prove (iii). To do this, we
start by observing that
S∗:L2((0,τ),Y)→H1
r((0,τ);Y) =/braceleftbig
u∈H1((0,τ),Y);u(τ) = 0/bracerightbig
,
the adjoint of S, is given by
S∗h(t) =/integraldisplayτ
tλ(s−t)h(s)ds, h∈H1
r((0,τ);Y).
We fixh∈H1
r((0,τ);Y) and we set k=S∗h. Then
k′(t) =λ(0)h(t)−/integraldisplayτ
tλ′(s−t)h(s)ds.
Hence
[|λ(0)|/ba∇dblh(t)/ba∇dbl]2≤/parenleftbigg/integraldisplayτ
t|λ′(s−t)|
|λ(0)|[|λ(0)|/ba∇dblh(s)/ba∇dbl]ds+/ba∇dblk′(t)/ba∇dbl/parenrightbigg2
≤2/parenleftbigg/integraldisplayτ
t|λ′(s−t)|
|λ(0)|[|λ(0)|/ba∇dblh(s)/ba∇dbl]ds/parenrightbigg2
+2/ba∇dblk′(t)/ba∇dbl2
≤2/ba∇dblλ′/ba∇dbl2
L2((0,τ))
|λ(0)|2/integraldisplayt
0[|λ(0)|/ba∇dblh(s)/ba∇dbl]2ds+2/ba∇dblk′(t)/ba∇dbl2.
The last estimate is obtained by applying Cauchy-Schwarz’s inequality .
A simple application of Gronwall’s lemma entails
[|λ(0)|/ba∇dblh(t)/ba∇dbl]2≤2e2/bardblλ′/bardbl2
L2((0,τ))
|λ(0)|2τ/ba∇dblk′(t)/ba∇dbl2.
Therefore,
/ba∇dblh/ba∇dblL2((0,τ);Y)≤√
2
|λ(0)|e/bardblλ′/bardbl2
L2((0,τ))
|λ(0)|2τ/ba∇dblk′/ba∇dblL2((0,τ);Y).
This inequality yields
(3.9) /ba∇dblh/ba∇dblL2((0,τ);Y)≤√
2
|λ(0)|e/bardblλ′/bardbl2
L2((0,τ))
|λ(0)|2τ/ba∇dblS∗h/ba∇dblH1r((0,τ);Y).LOGARITHMIC STABILITY IN DETERMINING A BOUNDARY COEFFICIE NT 7
The adjoint of S∗, acting as a bounded operator from [ Hr((0,1);Y)]′intoL2((0,τ);Y), gives an extension
ofS. We denote by /tildewideSthis operator. By [1, Proposition 4.1, page 1644] S∗defines an isomorphism from
[Hr((0,1);Y)]′ontoL2((0,τ);Y). In light of the fact that
/ba∇dbl/tildewideS/ba∇dblB([Hr((0,1);Y)]′;L2((0,τ);Y))=/ba∇dblS∗/ba∇dblB(L2((0,τ);Y);Hr((0,1);Y)),
(3.9) implies
(3.10)|λ(0)|√
2e−/bardblλ′/bardbl2
L2((0,τ))
|λ(0)|2τ≤ /ba∇dbl/tildewideS/ba∇dblB([Hr((0,1);Y)]′;L2((0,τ);Y)).
On the other hand, according to [1, Proposition 2.13, page 1641], Ψ p ossesses a unique bounded extension,
denoted by/tildewideΨ fromZ′into [Hr((0,1);Y)]′and there exists a constant c>0 so that
(3.11) /ba∇dbl/tildewideΨ/ba∇dblB(Z′;[Hr((0,1);Y)]′)≥c.
Consequently, /tildewideE=/tildewideS/tildewideΨ gives a unique extension of Eto an isomorphism from Z′ontoL2((0,τ);Y).
We end up the proof by noting that (3.8) is a consequence of (3.9) an d (3.11). /square
3.2.An inverse source problem for an IBVP for the wave equation. In the present subsection we
are going to apply the result of the preceding subsection to H=V×L2(Ω),H1=D(Aa) equipped with its
graph norm and Y=L2(Γ1).
We consider the the IBVP
(3.12)
∂2
tu−∆u=λ(t)w inQ,
u= 0 on Σ 0,
∂νu+a∂tu= 0 on Σ 1,
u(·,0) = 0, ∂tu(·,0) = 0.
Let (0,w)∈H−1andλ∈H1((0,τ)). From the comments in the preceding subsection, (3.12) has a un ique
solutionuwso that (uw,∂tuw)∈C([0,τ];V×L2(Ω)) and∂νuw|Γ1∈L2(Σ1).
We consider the inverse problem consisting in the determination of w, so that (0,w)∈H−1, appearing in
the IBVP (3.12) from the boundary measurement ∂νuw|Σ1. Here the function λis assumed to be known.
Taking into account that {0}×V′⊂H−1, whereV′is the dual space of V, we obtain as a consequence
of Corollary 2.1:
Proposition 3.1. There exists a constant C >0so that for any λ∈H1((0,τ))andw∈V′,
(3.13) /ba∇dblw/ba∇dblV′≤C|λ(0)|e/bardblλ′/bardbl2
L2((0,τ))
|λ(0)|2τ/ba∇dbl∂νu/ba∇dblL2(Σ1).
3.3.Proof of Theorem 1.1. We start by observing that uais also the unique solution of
/braceleftbigg/integraltext
Ωu′′(t)vdx=/integraltext
Ω∇u(t)·∇vdx−/integraltext
Γ1au′(t)v,for allv∈V.
u(0) =u0, u′(0) =u1.
Letu=ua−u0. Thenuis the solution of the following problem
(3.14)/braceleftbigg/integraltext
Ωu′′(t)vdx=/integraltext
Ω∇u(t)·∇vdx−/integraltext
Γ1au′(t)v−/integraltext
Γ1au′
0(t)v,for allv∈V.
u(0) = 0, u′(0) = 0.
Fork,ℓ∈Z, we set
λkℓ= [(k+1/2)2+(ℓ+1/2)2]π2
φkℓ(x,y) = 2cos((k+1/2)πx)cos((ℓ+1/2)πy).
We check in a straightforward manner that u0= cos(√λkℓt)φkℓwhen (u0,u1) = (φkℓ,0).
In the sequel k,ℓare arbitrarily fixed. We set λ(t) = cos(√λkℓt) and we define wa∈V′by
wa(v) =−/radicalbig
λkℓ/integraldisplay
Γ1aφkℓv.8 KA¨IS AMMARI AND MOURAD CHOULLI
In that case (3.14) becomes
/braceleftbigg/integraltext
Ωu′′(t)vdx=/integraltext
Ω∇u(t)·∇vdx−/integraltext
Γ1au′(t)v+λ(t)wa(v),for allv∈V.
u(0) = 0, u′(0) = 0.
Consequently, uis the solution of (3.12) with w=wa. Applying Proposition 3.1, we find
(3.15) /ba∇dblwa/ba∇dblV′≤Ceλkℓτ2/ba∇dbl∂νu/ba∇dblL2(Σ1).
But
(3.16) a1(0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
Γ1(aφkℓ)2dσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle=1√λkℓ|wa((a1⊗a2)φkℓ)| ≤1√λkℓ/ba∇dblwa/ba∇dblV′/ba∇dbl(a1⊗a2)φkℓ/ba∇dblV,
where we used a1(0) =a2(0), and
(3.17) /ba∇dbl(a1⊗a2)φkℓ/ba∇dblV≤C0/radicalbig
λkl/ba∇dbla1⊗a2/ba∇dblH1(Ω).
HereC0is a constant independent on aandφkℓ.
We note (a1⊗a2)φkℓ∈Veven ifa1⊗a2/ne}ationslash∈V.
Now a combination of (3.15), (3.16) and (3.17) yields
a1(0)/parenleftBig
/ba∇dbla1φk/ba∇dbl2
L2((0,1))+/ba∇dbla2φℓ/ba∇dbl2
L2((0,1))/parenrightBig
≤C/ba∇dbla1/ba∇dblH1(0,1)/ba∇dbla2/ba∇dblH1(0,1)eλkℓτ2/2/ba∇dbl∂νu/ba∇dblL2(Σ1),
whereφk(s) =√
2cos((k+1/2)πs). This and the fact that m≤aj(0) and/ba∇dblaj/ba∇dblH1((0,1))≤Mimply
/ba∇dbla1φk/ba∇dbl2
L2((0,1))+/ba∇dbla2φℓ/ba∇dbl2
L2((0,1))≤CM2
meλkℓτ2/2/ba∇dbl∂νu/ba∇dblL2(Σ1).
Hence, where j= 1 or 2,
/ba∇dblajφk/ba∇dbl2
L2((0,1))≤CM2
mek2τ2π2/ba∇dbl∂νu/ba∇dblL2(Σ1).
Let
ak
j=/integraldisplay1
0aj(x)φk(x)dx, j= 1,2.
Since
|ak
j|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1
0aj(x)φk(x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ /ba∇dblajφk/ba∇dblL1((0,1))≤ /ba∇dblajφk/ba∇dblL2((0,1)),
we get
(ak
j)2≤CM2
mek2τ2π2/ba∇dbl∂νu/ba∇dblL2(Σ1).
On the other hand
/ba∇dbl∂νu/ba∇dblL2(Σ1)=/ba∇dblΛa(φkl)−Λ0(φkl)/ba∇dblL2(Σ)≤Ck2/ba∇dblΛa−Λ0/ba∇dbl.
Hence
(3.18) ( ak
j)2≤CM2
mek2(τ2π2+1)/ba∇dblΛa−Λ0/ba∇dbl.
Letq=M2
mandα=τ2π2+2. We obtain in a straightforward manner from (3.18)
/summationdisplay
|k|≤N(ak
j)2≤CqeαN2/ba∇dblΛa−Λ0/ba∇dbl.LOGARITHMIC STABILITY IN DETERMINING A BOUNDARY COEFFICIE NT 9
Consequently,
/ba∇dblaj/ba∇dbl2
L2((0,1))≤/summationdisplay
|k|≤N(ak
j)2+1
N2/summationdisplay
|k|>Nk2(ak
j)2
≤C/parenleftBigg
qeαN2/ba∇dblΛa−Λ0/ba∇dbl+/ba∇dblaj/ba∇dbl2
H1((0,1))
N2/parenrightBigg
≤C/parenleftbigg
qeαN2/ba∇dblΛa−Λ0/ba∇dbl+M2
N2/parenrightbigg
≤CM2/parenleftbigg1
meαN2/ba∇dblΛa−Λ0/ba∇dbl+1
N2/parenrightbigg
.
That is
(3.19) /ba∇dblaj/ba∇dbl2
L2((0,1))≤CM2/parenleftbigg1
meαN2/ba∇dblΛa−Λ0/ba∇dbl+1
N2/parenrightbigg
.
Assume that /ba∇dblΛa−Λ0/ba∇dbl ≤δ=me−α. Let thenN0≥1 be the greatest integer so that
C
meαN2
0/ba∇dblΛa−Λ0/ba∇dbl ≤1
N2
0.
Using
1
meα(N0+1)2/ba∇dblΛa−Λ0/ba∇dbl ≤1
(N0+1)2,
we find
(2N0)2≥(N0+1)2≥1
α+1ln/parenleftbiggm
/ba∇dblΛa−Λ0/ba∇dbl/parenrightbigg
.
This estimate in (3.19) with N=N0gives
(3.20) /ba∇dblaj/ba∇dblL2((0,1))≤2C√
α+1M/vextendsingle/vextendsingleln/parenleftbig
m−1/ba∇dblΛa−Λ0/ba∇dbl/parenrightbig/vextendsingle/vextendsingle−1/2.
When/ba∇dblΛa−Λ0/ba∇dbl ≥δ, we have
(3.21) /ba∇dblaj/ba∇dblL2((0,1))≤M
δ/ba∇dblΛa−Λ0/ba∇dbl.
In light of (3.20) and (3.21), we find a constants c>0, that can depend only on τ, so that
/ba∇dblaj/ba∇dblL2((0,1))≤cM/parenleftBig/vextendsingle/vextendsingleln/parenleftbig
m−1/ba∇dblΛa−Λ0/ba∇dbl/parenrightbig/vextendsingle/vextendsingle−1/2+m−1/ba∇dblΛa−Λ0/ba∇dbl/parenrightBig
.
Appendix A.
We prove the following lemma
Lemma A.1. Let1/2<α≤1anda∈Cα([0,1]). Then the mapping f/ma√sto→afdefines a bounded operator
onH1/2((0,1)).
Proof.We recall that H1/2((0,1)) consists in functions f∈L2((0,1)) with finite norm
/ba∇dblf/ba∇dblH1/2((0,1))=/parenleftbigg
/ba∇dblf/ba∇dbl2
L2((0,1))+/integraldisplay1
0/integraldisplay1
0|f(x)−f(y)|2
|x−y|2dxdy/parenrightbigg1/2
.
Leta∈Cα([0,1]). We have
|a(x)f(x)−a(y)f(y)|2
|x−y|2≤ /ba∇dbla/ba∇dbl2
L∞(0,1)|f(x)−f(y)|2
|x−y|2+|f(y)|2[a]2
α
|x−y|2(1−α),
where
[a]α= sup{|a(x)−a(y)||x−y|−α;x,y∈[0,1], x/ne}ationslash=y}.10 KA¨IS AMMARI AND MOURAD CHOULLI
Using that 1 /2<α≤1, we find that x→ |x−y|−2(1−α)∈L1((0,1)),y∈[0,1], and
/integraldisplay1
0dx
|x−y|2(1−α)≤1
2α−1, y∈[0,1].
Henceaf∈H1/2((0,1)) with
/ba∇dblaf/ba∇dblH1/2((0,1))≤1
2α−1/ba∇dbla/ba∇dblCα([0,1])/ba∇dblf/ba∇dblH1/2((0,1)).
Here
/ba∇dbla/ba∇dblCα([0,1])=/ba∇dbla/ba∇dblL∞((0,1))+[a]α.
/square
Appendix B.
We give the proof of the following lemma
Lemma B.1. Leta∈AandAabe the unbounded operator defined on V×L2(Ω)by
Aa= (w,∆v), D(Aa) ={(v,w)∈V×V; ∆v∈L2(Ω)and∂νv=−awonΓ1}.
ThenAais m-dissipative.
Proof.Let/an}b∇acketle{t·,·/an}b∇acket∇i}htbe scalar product in V×L2(Ω). That is
/an}b∇acketle{t(v1,w1),(v2,w2)/an}b∇acket∇i}ht=/integraldisplay
Ω∇v1·∇v2dx+/integraldisplay
Ωw1w2dx,(vj,wj)∈V×L2(Ω), j= 1,2.
For (v1,w1)∈D(Aa), we have
/an}b∇acketle{tAa(v1,w1),(v1,w1)/an}b∇acket∇i}ht=/an}b∇acketle{t(w1,∆v1),(v1,w1)/an}b∇acket∇i}ht (B.1)
=/integraldisplay
Ω∇w1·∇v1dx+/integraldisplay
Ω∆v1w1dx
Applying twice Green’s formula, we get
/integraldisplay
Ω∇w1·∇v1dx=−/integraldisplay
Ωw1∆v1dx+/integraldisplay
Γ1w1∂νv1dσ, (B.2)
/integraldisplay
Ω∆v1w1dx=−/integraldisplay
Ω∇v1·∇w1dx−/integraldisplay
Γ1aw1w1dσ. (B.3)
We take the sum side by side of identities (B.2) and (B.3). Using that ∂νv1=−aw1on Γ1we obtain
/integraldisplay
Ω∇w1·∇v1dx+/integraldisplay
Ω∆v1w1dx=−/integraldisplay
Ωw1∆v1dx−/integraldisplay
Ω∇v1·∇w1dx−2/integraldisplay
Γ1a|w1|2dσ
=−/an}b∇acketle{t(v1,w1),Aa(v1,w1)/an}b∇acket∇i}ht−2/integraldisplay
Γ1a|w1|2dσ.
This and (B.1) yield
ℜ/an}b∇acketle{tAa(v1,w1),(v2,w2)/an}b∇acket∇i}ht=−/integraldisplay
Γ1a|w1|2dσ≤0.
In other words, Aais dissipative.
We complete the proof by showing that Aais onto implying that Aais m-dissipative. To this end we are
going to show that for each ( f,g)∈V×L2(Ω), the problem
w=f,−∆v=g.
has a unique solution ( v,w)∈D(Aa).LOGARITHMIC STABILITY IN DETERMINING A BOUNDARY COEFFICIE NT 11
In light of the fact ψ→/parenleftbig/integraltext
Ω|∇ψ|2dx/parenrightbig1/2defines an equivalent norm on V, we can apply Lax-milgram’s
lemma. We get that there exists a unique v∈Vsatisfying/integraldisplay
Ω∇v·∇ψdx=/integraldisplay
Ωgψdx−/integraldisplay
Γ1awψdσ, ψ∈V.
From this identity, we deduce in a standard way that −∆v=gand∂νv=−awon Γ1. The proof is then
complete /square
References
[1]C. Alves, A.-L. Silvestre, T. Takahashi and M. Tucsnak, Solving inverse source problems using observability. Appl i-
cations to the Euler-Bernoulli plate equation , SIAM J. Control Optim. 48 (2009), 1632-1659.
[2]K. Ammari andM. Choulli ,Logarithmic stability in determining two coefficients in a di ssipative wave equation. Exten-
sions to clamped Euler-Bernoulli beam and heat equations , to appear in J. Diff. Equat.
[3]K. Ammari andM. Choulli ,Determining a boundary coefficient in a dissipative wave equa tion: uniqueness and directional
Lipschitz stability , arXiv:1503.04528.
[4]P. Grisvard ,Elliptic problems in nonsmooth domains , Pitman Publishing Inc., 1985.
[5]P. Grisvard ,Contrˆ olabilit´ e des solutions de l’´ equation des ondes en pr´ esence de singularit´ es , J. Math. Pures Appl. 68
(1989), 215-259.
[6]V. Komornik andE. Zuazua ,A direct method for the boundary stabilization of the wave eq uation, J. Math Pures Appl.
69 (1990), 33-54.
[7]M. Tucsnak and G. Weiss ,Observation and control for operator semigroups. Birkh¨ auser Advanced Texts, Birkh¨ auser
Verlag, Basel, 2009.
UR Analysis and Control of PDEs, UR 13ES64, Department of Mathem atics, Faculty of Sciences of Monastir,
University of Monastir, 5019 Monastir, Tunisia
E-mail address :kais.ammari@fsm.rnu.tn
Institut ´Elie Cartan de Lorraine, UMR CNRS 7502, Universit ´e de Lorraine, Boulevard des Aiguillettes, BP
70239, 54506 Vandoeuvre les Nancy cedex - Ile du Saulcy, 57045 Metz cedex 01, France
E-mail address :mourad.choulli@univ-lorraine.fr |
1906.01042v1.Magnon_phonon_interactions_in_magnetic_insulators.pdf | Magnon-phonon interactions in magnetic insulators
Simon Streib,1Nicolas Vidal-Silva,2, 3, 4Ka Shen,5and Gerrit E. W. Bauer1, 5, 6
1Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
2Departamento de Física, Universidad de Santiago de Chile, Avda. Ecuador 3493, Santiago, Chile
3Center for the Development of Nanoscience and Nanotechnology (CEDENNA), 917-0124 Santiago, Chile
4Departamento de Física, Facultad de Ciencias Físicas y Matemáticas,
Universidad de Chile, Casilla 487-3, Santiago, Chile
5Department of Physics, Beijing Normal University, Beijing 100875, China
6Institute for Materials Research & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan
(Dated: June 4, 2019)
We address the theory of magnon-phonon interactions and compute the corresponding quasi-
particle and transport lifetimes in magnetic insulators with focus on yttrium iron garnet at inter-
mediate temperatures from anisotropy- and exchange-mediated magnon-phonon interactions, the
latter being derived from the volume dependence of the Curie temperature. We find in general weak
effects of phonon scattering on magnon transport and the Gilbert damping of the macrospin Kittel
mode. The magnon transport lifetime differs from the quasi-particle lifetime at shorter wavelengths.
I. INTRODUCTION
Magnons are the elementary excitations of magnetic
order, i.e. the quanta of spin waves. They are bosonic
andcarryspinangularmomentum. Ofparticularinterest
are the magnon transport properties in yttrium iron gar-
net (YIG) due to its very low damping ( <10 4), which
makes it one of the best materials to study spin-wave or
spin caloritronic phenomena [1–6]. For instance, the spin
Seebeck effect (SSE) in YIG has been intensely studied
in the past decade [7–13]. Here, a temperature gradi-
ent in the magnetic insulator injects a spin current into
attached Pt contacts that is converted into a transverse
voltage by the inverse spin Hall effect. Most theories ex-
plain the effect by thermally induced magnons and their
transport to and through the interface to Pt [7, 14–19].
However, phonons also play an important role in the SSE
through their interactions with magnons [20–22].
Magnetoelastic effects in magnetic insulators were ad-
dressed first by Abrahams and Kittel [23–25], and by
Kaganov and Tsukernik [26]. In the long-wavelength
regime, the strain-induced magnetic anisotropy is the
most important contribution to the magnetoelastic en-
ergy, whereas for shorter wavelengths, the contribution
from the strain-dependence of the exchange interaction
becomes significant [27–29]. Rückriegel et al.[28] com-
puted very small magnon decay rates in thin YIG films
due to magnon-phonon interactions with quasi-particle
lifetimes qp?480 ns;even at room temperature. How-
ever, these authors do not consider the exchange interac-
tion and the difference between quasi-particle and trans-
port lifetimes.
Recently, it has been suggested that magnon spin
transport in YIG at room temperature is driven by
the magnon chemical potential [3, 30]. Cornelissen et
al. [3] assume that at room temperature magnon-
phonon scattering of short-wavelength thermal magnons
is dominated by the exchange interaction with a scat-
tering time of qp1 ps, which is much faster than
the anisotropy-mediated magnon-phonon coupling con-sidered in Ref. [28] and efficiently thermalizes magnons
and phonons to equal temperatures without magnon de-
cay. Recently, the exchange-mediated magnon-phonon
interaction [31] has been taken into account in a Boltz-
mann approach to the SSE, but this work underestimates
the coupling strength by an order of magnitude, as we
will argue below.
In this paper we present an analytical and numeri-
cal study of magnon-phonon interactions in bulk ferro-
magnetic insulators, where we take both the anisotropy-
and the exchange-mediated magnon-phonon interactions
into account. By using diagrammatic perturbation the-
ory to calculate the magnon self-energy, we arrive at a
wave-vector dependent expression of the magnon scat-
tering rate, which is the inverse of the magnon quasi-
particle lifetime qp. The magnetic Grüneisen parameter
m=@lnTC=@lnV[32, 33], where TCis the Curie tem-
perature and Vthe volume of the magnet, gives direct
access to the exchange-mediated magnon-phonon inter-
action parameter. We predict an enhancement in the
phonon scattering of the Kittel mode at the touching
points of the two-magnon energy (of the Kittel mode and
a finite momentum magnon) and the longitudinal and
transverse phonon dispersions, for YIG at around 1:3 T
and4:6 T. We also emphasize the difference in magnon
lifetimesthatbroadenlightandneutronscatteringexper-
iments, and the transport lifetimes that govern magnon
heat and spin transport.
The paper is organized as follows: in Sec. II we briefly
review the theory of acoustic magnons and phonons in
ferro-/ferrimagnets, particularly in YIG. In Sec. III we
derive the exchange- and anisotropy-mediated magnon-
phonon interactions for a cubic Heisenberg ferromagnet
with nearest neighbor exchange interactions in the long-
wavelength limit. In Sec. IV we derive the magnon decay
rate from the imaginary part of the magnon self-energy
in a diagrammatic approach and in Sec. V we explain
the differences between the magnon quasi-particle and
transport lifetimes. Our numerical results for YIG are
discussed in Sec. VI. Finally in Sec. VII we summarizearXiv:1906.01042v1 [cond-mat.str-el] 3 Jun 20192
and discuss the main results of the present work. The va-
lidity of our long-wavelength approximation is analyzed
in Appendix A and in Appendix B we explain why sec-
ond order magnetoelastic couplings may be disregarded.
In Appendix C we briefly discuss the numerical methods
used to evaluate the k-space integrals.
II. MAGNONS AND PHONONS IN
FERROMAGNETIC INSULATORS
Without loss of generality, we focus our treatment on
yttrium iron garnet (YIG). The magnon band structure
of YIG has been determined by inelastic neutron scatter-
ing [34–36] and by ab initio calculation of the exchange
constants [37]. The complete magnon spectral function
hasbeencomputedforalltemperaturesbyatomisticspin
simulations [38], taking all magnon-magnon interactions
into account, but not the magnon-phonon scattering.
The pure phonon dispersion is known as well [29, 39]. In
the following, we consider the interactions of the acoustic
magnons from the lowest magnon band with transverse
and longitudinal acoustic phonons, which allows a semi-
analytic treatment but limits the validity of our results
to temperatures below 100 K. Since the low-temperature
values of the magnetoelastic constants, sound velocities,
and magnetic Grüneisen parameter are not available for
YIG, we use throughout the material parameters under
ambient conditions.
A. Magnons
Spinsinteractwitheachotherviadipolarandexchange
interactions. We disregard the former since at the energy
scaleEdip0:02 meV [28] it is only relevant for long-
wavelength magnons with wave vectors k.6107m 1
and energies Ek=kB.0:2 K, which are negligible for
the thermal magnon transport in the temperature regime
we are interested in. The lowest magnon band can then
be described by a simple Heisenberg model on a course-
grained simple cubic ferromagnet with exchange interac-
tionJ
Hm= J
2X
hi6=jiSiSj X
igBBSz
i;(2.1)
where the sum is over all nearest neighbors and ~Siis the
spin operator at lattice site Ri. The lattice constant of
the cubic lattice or YIG is a= 12:376Aand the effective
spin per unit cell ~S=~Msa3=(gB)14:2~at room
temperature [28] ( S20forT.50 K[40]), where the
g-factorg2,Bis the Bohr magneton and Msthe sat-
uration magnetization. The parameter Jis an adjustable
parameter that can be fitted to experiments or computed
from first principles. Bis an effective magnetic field that
orients the ground-state magnetization vector to the z
axis and includes the (for YIG small) magnetocrystallineanisotropyfield. The 1=Sexpansionofthespinoperators
in terms of Holstein-Primakoff bosons reads [41],
S+
i=Sx+iSyp
2S[bi+O(1=S)];(2.2)
S
i=Sx iSyp
2Sh
by
i+O(1=S)i
;(2.3)
Sz
i=S by
ibi; (2.4)
whereby
iandbiare the magnon creation and annihilation
operators with boson commutation ruleh
bi;by
ji
=i;j.
Then
Hm!X
kEkby
kbk; (2.5)
where the magnon operators by
kandbkare defined by
bi=1p
NX
keikRibk; (2.6)
by
i=1p
NX
ke ikRiby
k; (2.7)
andNthe number of unit cells. The dispersion relation
Ek=gBB+ 4SJX
=x;y;zsin2(k a=2) (2.8)
becomes quadratic in the long-wavelength limit ka 1:
Ek=gBB+Eexk2a2; (2.9)
whereEex=SJ. WithEex=kB40 K = 3:45 meV
the latter is a good approximation up to k0= 1=a
8108m 1[34]. The effective exchange coupling is
thenJ0:24 meV. The lowest magnon band does not
depend significantly on temperature [38], which implies
thatEex=SJdoes not depend strongly on temper-
ature. The temperature dependence of the saturation
magnetization and effective spin Sshould therefore not
affect the low-energy exchange magnons significantly. By
using Eq. (2.9) in the following, our theory is valid for
k.k0(see Fig. 1) or temperatures T.100 K. In this
regime the cut-off of an ultraviolet divergence does not
affect results significantly (see Appendix A). We disre-
gard magnetostatic interactions that affect the magnon
spectrum only for very small wave vectors since at low
temperatures the phonon scattering is not significant.
B. Phonons
We expand the displacement Xiof the position riof
unit cellifrom the equilibrium position Ri
Xi=ri Ri; (2.10)
into the phonon eigenmodes Xq,
X
i=1p
NX
q;e
qXqeiqRi;(2.11)3
where 2 fx;y;zgandqa wave vector. We define
polarizations 2f1;2;3gfor the elastic continuum [42]
eq1= (cosqcos q;cosqsin q; sinq);(2.12)
eq2=i( sin q;cos q;0); (2.13)
eq3=i(sinqcos q;sinqsin q;cosq);(2.14)
where the angles qand qare the spherical coordinates
of
q=q(sinqcos q;sinqsin q;cosq);(2.15)
which is valid for YIG up to 3 THz(12 meV) [29, 39].
The phonon Hamiltonian then reads
Hp=X
qP qPq
2m+m
2~2"2
qX qXq
;
=X
q"q
ay
qaq+1
2
; (2.16)
wherethecanonicalmomenta Pqobeythecommutation
relations [Xq;Pq00] =i~q; q00and the mass of the
YIG unit cell m=a3= 9:810 24kg[27]. The phonon
dispersions for YIG then read
"q=~cjqj; (2.17)
wherec1;2=ct= 3843 m=sis the transverse sound ve-
locity andc3=cl= 7209 m=sthe longitudinal velocity
at room temperature [27]. In terms of phonon creation
and annihilation operators
Xq=aq+ay
qp
2m"q=~2; P q=1
irm"q
2
aq ay
q
;
(2.18)
andh
aq;ay
q00i
=q;q0;0.
In Fig. 1 we plot the longitudinal and transverse
phonon and the acoustic magnon dispersion relations for
YIG at zero magnetic field. The magnon-phonon inter-
action leads to an avoided level crossing at points where
magnon and phonon dispersion cross, as discussed in
Refs. [27] and [28].
III. MAGNON-PHONON INTERACTIONS
We derive in this section the magnon-phonon interac-
tions due to the anisotropy and exchange interactions for
a cubic lattice ferromagnet.
A. Phenomenological magnon-phonon interaction
In the long-wavelength/continuum limit ( k.k0) the
magnetoelastic energy to lowest order in the deviations
of magnetization and lattice from equilibrium reads [23–
26, 28]
0.0 0.5 1.0 1.5
k[109m−1]024681012Ek[meV]magnon model
parabolic approximation
longitudinal acoustic phonon
transverse acoustic phononFigure 1. Dispersion relations of the acoustic phonons and
magnons in YIG at zero magnetic field.
Eme=n
M2sZ
d3rX
[B M (r)M (r)
+B0
@M(r)
@r @M(r)
@r
X (r);(3.1)
wheren= 1=a3. The strain tensor X is defined in
terms of the lattice displacements X ,
X (r) =1
2@X (r)
@r +@X (r)
@r
;(3.2)
with, for a cubic lattice [28],
B = Bk+ (1 )B?; (3.3)
B0
= B0
k+ (1 )B0
?: (3.4)
B is caused by magnetic anisotropies and B0
by the
exchange interaction under lattice deformations. For
YIG at room temperature [27, 33]
Bk=kB47:8 K = 4:12 meV;(3.5)
B?=kB95:6 K = 8:24 meV;(3.6)
B0
k=a2=kB2727 K = 235 meV ;(3.7)
B0
?=a20: (3.8)
We discuss the values for B0
kandB0
?in Sec. IIIC.
B. Anisotropy-mediated magnon-phonon
interaction
The magnetoelastic anisotropy (3.1) is described by
the Hamiltonian [28],4
Han
mp=X
q
qb qXq+
qby
qXq
+1p
NX
q;k;k0k k0 q;0X
an
kk0;by
kbk0Xq
+1p
NX
q;k;k0k+k0+q;0X
bb
kk0;bkbk0Xq
+1p
NX
q;k;k0k+k0 q;0X
bb
kk0;by
kby
k0Xq;(3.9)
with interaction vertices
q=B?p
2Sh
iqzex
q+qzey
q
+ (iqx+qy)ez
q
; (3.10)
an
kk0;=Uk k0;; (3.11)
bb
kk0;=V k k0;; (3.12)
bb
kk0;=V
k k0;; (3.13)
and
Uq;=iBk
Sh
qxex
q+qyey
q 2qzez
qi
;(3.14)
Vq;=iBk
Sh
qxex
q qyey
qi
+B?
Sh
qyex
q+qxey
qi
: (3.15)
The one magnon-two phonon process is of the same
order in the total number of magnons and phonons as
the two magnon-one phonon processes, but its effect on
magnon transport is small, as shown in Appendix B.
C. Exchange-mediated magnon-phonon interaction
The exchange-mediated magnon-phonon interaction is
obtained under the assumption that the exchange inter-
actionJijbetween two neighboring spins at lattice sites
riandrjdepends only on their distance, which leads to
the expansion to leading order in the small parameter
(jri rjj a)
Jij=J(jri rjj)J+J0(jri rjj a);(3.16)
whereais the equilibrium distance and J0=@J=@a.
With ri=Ri+XRi;the Heisenberg Hamiltonian (2.1)
is modulated by
Hex
mp= J0X
iX
=x;y;z
X
Ri+ae X
Ri
SRiSRi+ae ;
(3.17)where e is a unit vectors in the direction. Expanding
the displacements in terms of the phonon and magnon
modes
Hex
mp=1p
NX
q;k;k0k k0 q;0X
ex
kk0;by
kbk0Xq;(3.18)
with interaction
ex
kk0;= 8iJ0SX
e
k k0;sink a
2
sink0
a
2
sin(k k0
)a
2
iJ0a3SX
e
k k0;k k 0(k k0
);(3.19)
where the last line is the long-wavelength expansion. The
magnon-phonon interaction
bb
k;k0;= ex
k;k0;+ an
k;k0; (3.20)
conserves the magnon number, while (3.12) and (3.13) do
not. Phonon numbers are not conserved in either case.
The value of J0for YIG is determined by the magnetic
Grüneisen parameter [32, 33]
m=@lnTC
@lnV=@lnJ
@lnV=J0a
3J;(3.21)
whereV=Na3is the volume of the magnet. The only
assumption here is that the Curie temperature TCscales
linearly with the exchange constant J[43]. mhas been
measured for YIG via the compressibility to be m=
3:26[32], and via thermal expansion, m= 3:13[33],
so we set m= 3:2. For other materials the magnetic
Grüneisen parameter is also of the order of unity and in
many cases m 10=3[32, 33, 44]. A recent ab initio
study of YIG finds m= 3:1[45].
Comparing the continuum limit of Eq. (3.17) with the
classical magnetoelastic energy (3.1)
B0
k= 3 mJS2a2=2; (3.22)
whereforYIG B0
k=a2235 meV . Wedisregard B0
?since
it vanishes for nearest neighbor interactions by cubic lat-
tice symmetry.
The coupling strength of the exchange-mediated
magnon-phonon interaction can be estimated from the
exchange energy SJ0aEex=SJ[31, 46] following
Akhiezer et al.[47, 48]. Our estimate of SJ0a= 3 mSJ
is larger by 3 m, i.e. one order of magnitude. Since the
scattering rate is proportional to the square of the in-
teraction strength, our estimate of the scattering rate is
a factor 100larger than previous ones. The assumption
J0aJis too small to be consistent with the experi-
mental Grüneisen constant [32, 33]. Ref. [3] educatedly
guessedJ0a100J;which we now judge to be too large.5
Figure2. Feynmandiagramsofinteractionsbetweenmagnons
(solid lines) and phonons (dashed lines). The arrows indicate
the energy-momentum flow. (a) magnon-phonon interconver-
sion, (b) magnon number-conserving magnon-phonon inter-
action, (c) and (d) magnon number non-conserving magnon-
phonon interactions.
D. Interaction vertices
The magnon-phonon interactions in the Hamiltonian
(3.9) are shown in Fig. 2 as Feynman diagrams. Fig. 2(a)
illustrates magnon and phonon interconversion, which
is responsible for the magnon-phonon hybridization and
level splitting at the crossing of magnon and phonon dis-
persions [27, 28]. The divergence of this diagram at the
magnon-phonon crossing points is avoided by either di-
rect diagonalization of the magnon-phonon Hamiltonian
[42] or by cutting-off the divergence by a lifetime param-
eter [31]. This process still generates enhanced magnontransport that is observable as magnon polaron anoma-
lies in the spin Seebeck effect [22] or spin-wave excitation
thresholds [49, 50], but these are strongly localized in
phase space and disregarded in the following, where we
focus on the magnon scattering rates to leading order in
1=Sof the scattering processes in Fig. 2(b)-(d).
IV. MAGNON SCATTERING RATE
Here we derive the magnon reciprocal quasi-particle
lifetime 1
qp=
as the imaginary part of the wave vector
dependent self-energy, caused by acoustic phonon scat-
tering [28],
(k) = 2
~Im(k;Ek=~+i0+):(4.1)
This quantity is in principle observable by inelastic neu-
tron scattering. The total decay rate
=
c+
nc+
other(4.2)
is the sum of the magnon number conserving decay rate
cand the magnon number non-conserving decay rate
nc, which are related to the magnon-phonon scattering
time mpand the magnon-phonon dissipation time mr
by
mp=1
c; mr=1
nc: (4.3)
otheris caused by magnon-magnon and magnon disorder
scattering, thereby beyond the scope of this work.
The self-energy to leading order in the 1=Sexpansion
is of second order in the magnon-phonon interaction [28],
2(k;i!) =1
NX
k0~2 bb
k;k0; 2
2m"k k0;nB("k k0;) nB(Ek0)
i~!+"k k0; Ek0+1 +nB("k k0;) +nB(Ek0)
i~! "k k0; Ek0
1
NX
k0~2 bb
k;k0; 2
2m"k k0;1 +nB("k+k0;) +nB(Ek0)
i~!+"k+k0;+Ek0+nB("k+k0;) nB(Ek0)
i~! "k+k0;+Ek0
; (4.4)
where the magnon number conserving magnon-phonon
scattering vertex bb
k;k0;= ex
k;k0;+ an
k;k0;and the
Planck (Bose) distribution function nB(") = (e " 1) 1
with inverse temperature = 1=(kBT). The Feynman
diagrams representing the magnon number conserving
and non-conserving contributions to the self-energy areshown in Fig. 3.
We write the decay rate in terms of four contributions
(k) =
c
out(k) +
nc
out(k)
c
in(k)
nc
in(k);(4.5)
whereoutandindenote the out-scattering and in-
scattering parts. The contributions to the decay rate
read [28]6
k
qk-q
k kk
qq-k
k k(a) (b)
Figure 3. Feynman diagrams representing the self-energy
Eq. (4.4) due to (a) magnon number-conserving magnon-
phonon interactions and (b) magnon number non-conserving
magnon-phonon interactions.
c
out(k) =~
mNX
q; bb
k;k q; 2
"q[(1 +nB(Ek q))nB("q)(Ek Ek q+"q)
+ (1 +nB(Ek q))(1 +nB("q))(Ek Ek q "q)]; (4.6)
c
in(k) =~
mNX
q; bb
k;k q; 2
"q[nB(Ek q)(1 +nB("q))(Ek Ek q+"q)
+nB(Ek q)nB("q)(Ek Ek q "q)]; (4.7)
nc
out(k) =~
mNX
q; bb
k;q k; 2
"q[nB(Eq k)(1 +nB("q))(Ek+Eq k "q)]; (4.8)
nc
in(k) =~
mNX
q; bb
k;q k; 2
"q[(1 +nB(Eq k))nB("q)(Ek+Eq k "q)]; (4.9)
where the sum is over all momenta qin the Brillouin
zone. Here the magnon/phonon annihilation rate is pro-
portional to the Boson number nB, while the creation
ratescaleswith 1+nB. Forexample,intheout-scattering
rate
c
out(k)theincomingmagnonwithmomentum kgets
scattered into the state k qand a phonon is either ab-
sorbedwithprobability nBoremittedwithprobability
(1 +nB). The out- and in-scattering rates are related
by the detailed balance
c
in(k)=
c
out(k) =
nc
in(k)=
nc
out(k) =e Ek:(4.10)
For high temperatures kBT Ek, we may expand the
Bose functions nB(Ek)kBT=E kand we find
in
outT2and
=
out
inT. For low temperatures
kBT Ek, the out-scattering rate
out!const:and
the in-scattering rate
ine Ek!0. The scattering
processes (c) and (d) in Fig. 2 conserve energy and linear
momentum, but not angular momentum. A loss of an-gular momentum after integration over all wave vectors
corresponds to a mechanical torque on the total lattice
that contributes to the Einstein-de Haas effect [51].
V. MAGNON TRANSPORT LIFETIME
Inthissectionwecomparethetransportlifetime tand
the magnon quasi-particle lifetime qpthat can be very
different [52–54], but, to the best of our knowledge, has
not yet been addressed for magnons. The magnon decay
rate is proportional to the imaginary part of self energy,
as shown in Eq. (4.1). On the other hand, the transport
is governed by transport lifetime tin the Boltzmann
equation that agrees with qponly in the relaxation time
approximation. The stationary Boltzmann equation for7
the magnon distribution can be written as [3, 42]
@fk(r)
@r@Ek
@(~k)= in[f] out[f];(5.1)
wherefk(r)is the magnon distribution function. The in
andoutcontributions to the collision integral are related
to the previously defined in- and out-scattering rates by
in[f] = (1 +fk)
in[f]; (5.2)
out[f] =fk
out[f]; (5.3)
where the equilibrium magnon distribution nB(Ek)is re-
placed by the non-equilibrium distribution function fk.
The factor (1 +fk)corresponds to the creation of a
magnon with momentum kin the in-scattering process
and the factor fkto the annihilation in the out-scattering
process. The phonons are assumed to remain at thermal
equilibrium, so we disregard the phonon drift contribu-
tion that is expected in the presence of a phononic heat
current.
Magnon transport is governed by three linear response
functions, i.e. spin and heat conductivity and spin See-
beck coefficient [42]. These can be obtained from the ex-
pansion of the distribution function in terms of temper-
ature and chemical potential gradients and correspond
to two-particle Green functions with vertex corrections,
that reflect the non-equilibrium in-scattering processes,
captured by a transport lifetime tthat can be different
from the quasi-particle (dephasing) lifetime qpdefined
by the self-energy. We define the transport life time of
a magnon with momentum kin terms of the collision
integral
out[f] in[f] =1
k;t[f](fk(r) f0;k);(5.4)
withf0;k=nB(Ek)and we assume a thermalized quasi-
equilibrium distribution function
fk(r) =nBEk (r)
kBT(r)
; (5.5)
whereis the magnon chemical potential. We linearize
the function fkin terms of small deviations fkfrom
equilibrium f0;k,
fk=fk f0;k: (5.6)
leading to [3]
fk= k;t[f]@f0;k
@Ek@Ek
@(~k)
r+Ek
TrT
;(5.7)
where the gradients of chemical potential rand tem-
perature rTdrive the magnon current. In the relax-
ation time approximation we disregard the dependence
of k;t[f]onfand recover the quasi-particle lifetime
k;t! k;qp.Tofirstorderinthephononoperatorsandsecondorder
in the magnon operators the collision integral for magnon
number non-conserving processes,
nc
out[f] nc
in[f]
=~
mNX
qj bb
k;q k;j2
"q(Ek+Eq k "q)
[(1 +nq)fkfq k nq(1 +fq k)(1 +fk)];
(5.8)
where the interaction vertex bb
k;k0;is given by Eq. (3.12)
andnq=nB("q). By using the expansion (5.6) in the
collision integral that vanishes at equilibrium,
out[f0] in[f0] = 0; (5.9)
we arrive at
1
nc
k;t=~
mNX
qj bb
k;q k;j2
"q(Ek+Eq k "q)
nB(Ek q) nq+fq k
fk(nB(Ek) nq)
:
(5.10)
For the magnon number conserving process the deriva-
tion is similar and we find
1
c
k;t=~
mNX
qj bb
k;k q;j2
"q"
(Ek Ek q+"q)
nq nB(Ek q) fk q
fk(nB(Ek) +nq+ 1)
+(Ek Ek q "q)
1 +nB(Ek q) +nq+fk q
fk(nB(Ek) nq)#
;
(5.11)
with interaction vertex bb
k;k0;given by Eq. (3.20). Due
to thefk q=fkterm this is an integral equation. It
can be solved iteratively to generate a geometric series
referred to as vertex correction in diagrammatic theo-
ries. By simply disregarding the in-scattering with terms
fk q=fkthetransportlifetimereducestothethequasi-
particle lifetime of the self-energy. We leave the general
solution of this integral equation for future work, but
argue in Sec. VID that the vertex corrections are not
important in our regime of interest.
VI. NUMERICAL RESULTS
A. Magnon decay rate
In the following we present and analyze our results
for the magnon decay rates in YIG. We first consider8
0.0 0.2 0.4 0.6 0.8 1.0 1.2
k[109m−1]01020304050γc(k) [106s−1]
(100)
(001)
(110)
(111)
(011)0.0 0.5 1.0−0.10.00.10.2
Figure 4. Magnon decay rate in YIG due to magnon-phonon
interactions for magnons propagating along various directions
atT= 50 KandB= 0. We denote the propagation direction
by(lmn), i.e.lex+mey+nez. The inset shows the relative
deviation
c=
cfrom the (100) direction.
0.00 0.01 0.02 0.03 0.04 0.05
kx[109m−1]0.000.050.100.150.200.250.300.350.40γc(k) [103s−1]γc, total
γc, anisotropy
γc, exchange
Figure 5. Comparison of the contributions from exchange-
mediated and anisotropy-mediated magnon-phonon interac-
tions to the magnon number conserving scattering rate
cat
T= 50 KandB= 0.
the case of vanishing effective magnetic field ( B= 0)
and discuss the magnetic field dependence in Sec. VIC.
Since our model is only valid in the long-wavelength ( k<
8×108m 1) and low-temperature ( T.100 K) regime,
we focus first on T= 50 Kand discuss the temperature
dependence in Sec. VIB.
InFig.4weshowthemagnonnumberconservingdecay
rate
c(k), which is on the displayed scale dominated by
the exchange-mediated magnon-phonon interaction and
is isotropic for long-wavelength magnons.
In Fig. 5 we compare the contribution from the
exchange-mediated magnon-phonon interaction (
c
k4) and from the anisotropy-mediated magnon-phonon
interaction (
ck2). We observe a cross-over at
k4107m 1: for much smaller wave numbers, the
exchange contribution can be disregarded and for larger
wave numbers the exchange contribution becomes domi-
nant.
The magnon number non-conserving decay rate
ncin
Fig. 6 is much smaller than the magnon-conserving one.
0.0 0.2 0.4 0.6 0.8 1.0 1.2
k[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1](100)
(001)
(110)
(111)
(011)Figure 6. Magnon decay rate in YIG due to magnon num-
ber non-conserving magnon-phonon interactions for magnons
propagating along various directions at T= 50 KandB= 0.
This is consistent with the low magnetization damping
of YIG, i.e. the magnetization is long-lived. We observe
divergent peaks at the crossing points (shown in Fig. 1)
with the exception of the (001) direction. These diver-
gences occur when magnons and phonons are degenerate
atk= 0:48109m 1(1:2 meV) andk= 0:9109m 1
(4:3 meV), respectively, at which the Boltzmann formal-
ism does not hold; a treatment in the magnon-polaron
basis [42] or a broadening parameter [31] would get rid
of the singular behavior. The divergences are also sup-
pressed by arbitrarily small effective magnetic fields (see
Sec. VIC). There are no peaks along the (001) direc-
tion because in the (001) direction the vertex function
Vq;(see Eq. (3.15)) vanishes for q= (0;0;kz). For
k >~cl=(D(p
8 2)) = 1:085109m 1the decay rate
ncvanishes because the decay process does not conserve
energy ((Ek+Eq k "q) = 0).
B. Temperature dependence
Above we focused on T= 50 Kand explained that we
expect a linear temperature dependence of the magnon
decay rates at high, but not low temperatures. Fig. 7
showsourresultsforthetemperaturedependenceat kx=
108m 1. Deviations from the linear dependence at low
temperatures occurs when quantum effects set in, i.e. the
Rayleigh-Jeans distribution does not hold anymore,
1
e"=(kBT) 16kBT
": (6.1)
C. Magnetic field dependence
The numerical results presented above are for a mono-
domain magnet in the limit of small applied magnetic
fields. A finite magnetic field Balong the magnetization
directioninducesanenergygap gBBinthemagnondis-
persion, which shifts the positions of the magnon-phonon9
0 2 4 6 8 10
T[K]0.00.10.20.30.40.50.60.7γ[103s−1]γc
γnc
Figure7. Temperaturedependenceofthemagnondecayrates
ncand
catB= 0,kx= 108m 1andky=kz= 0, i.e.
along (100).
0.0 0.2 0.4 0.6 0.8 1.0 1.2
kx[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1]B= 0 T
B= 0.1 T
B= 0.5 T
B= 1 T
B= 2 T
Figure 8. Magnetic field dependence of the magnon number
non-conserving magnon decay rate in YIG at T= 50 Kwith
magnon momentum along (100).
crossingpointsto longerwavelengths. Themagneticfield
suppresses the (unphysical) sharp peaks at the crossing
points (see Fig. 8) that are caused by the divergence of
thePlanckdistributionfunctionforavanishingspinwave
gap.
In the magnon number conserving magnon-phonon
interactions, the magnetic field dependence cancels in
the delta function and enters only in the Bose func-
tion vianB(magnetic freeze-out). Fig. 9 shows that
the magnetic field mainly affects magnons with energies
.2gBB= 0:23(B=T) meV.
As shown in Fig. 10 the magnon decay by phonons
does not vanish for the k= 0Kittel mode, but only
in the presence of a spin wave gap E0=gBB. Both
magnon conserving and non-conserving scattering pro-
cesses contribute. The divergent peaks at B1:3 Tand
B4:6 Tin
ncare caused by energy and momentum
conservation in the two-magnon-one-phonon scattering
process,
(Ek=0+Eq "q) =(2gBB+Eexq2a2 ~cq);(6.2)
when the gradient of the argument of the delta function
0.0 0.1 0.2 0.3 0.4 0.5
kx[109m−1]0246810δγc/γcB= 1 T
B= 10 TFigure 9. Relative deviation
c=
cfrom theB= 0result of
the magnon number conserving magnon decay rate in YIG at
T= 50 Kwith magnon momentum along (100).
vanishes,
rq(Ek=0+Eq "q) = 0; (6.3)
i.e., the two-magnon energy Ek=0+Eqtouches either the
transverse or longitudinal phonon dispersion "q. The
total energy of the two magnons is equivalent to the en-
ergy of a single magnon with momentum qbut in a field
2B, resulting in the divergence at fields that are half of
those for the magnon-polaron observed in the spin See-
beck effect [31, 42]. The two-magnon touching condition
can be satisfied in all directions of the phonon momen-
tumq, which therefore contributes to the magnon decay
rate when integrating over the phonon momentum q. For
k6= 0this two-magnon touching condition can only be
fulfilled for phonons along a particular direction and the
divergence is suppressed.
The magnon decay rate is related to the Gilbert damp-
ing kas~
k= 2 kEk[55]. We find that phonons
contribute only weakly to the Gilbert damping, nc
0=
~
nc
0=(2E0)10 8atT= 50 K, which is much smaller
than the total Gilbert damping 10 5in YIG, but
the peaks at 1:3 Tand4:6 Tmight be observable. The
phonon contribution to the Gilbert damping scales lin-
early with temperature, so is twice as large at 100 K. At
low temperatures ( T.100 K) Gilbert damping in YIG
has been found to be caused by two-level systems [56]
and impurity scattering [40], while for higher tempera-
tures magnon-phonon [57] and magnon-magnon scatter-
ing involving optical magnons [34] have been proposed to
explain the observed damping. Enhanced damping as a
function of magnetic field at higher temperatures might
reveal other van Hove singularities in the joint magnon-
phonon density of states.
D. Magnon transport lifetime
We do not attempt a full solution of the integral equa-
tions (5.10) and (5.11) for the transport lifetime. How-
ever, we can still estimate its effect by the observation10
0 1 2 3 4 5 6
Magnetic field [T]01020304050γ[103s−1]γc(k= 0)
γnc(k= 0)
Figure 10. Magnetic field dependence of the magnon decay
rates in YIG at k= 0andT= 50 K.
that the ansatz 1
k;tkncan be an approximate solu-
tion of the Boltzmann equation with in-scattering.
Our results for the magnon number conserving interac-
tion are shown in Fig. 11 (for rT= 0and finite rjjex),
where
t= 1
t. We consider the cases n= 0;2;4,
wheren= 0or k;t= const:would be the solution for
a short-range scattering potential. For very long wave-
lengths (k.4107m 1) the inverse quasi-particle life-
time 1
k;qpk2and for shorter wavelengths 1
k;qpk4.
n= 2is a self-consistent solution only for very small
k.4107m 1, while 1
k;qpk4is a good ansatz up
tok.0:3109m 1. We see that the transport life-
time approximately equals the quasi-particle lifetime in
the regime of the validity of the n= 4power law.
For the magnon number non-conserving processes in
Fig. 12 the quasi-particle lifetime behaves as 1
k;qpk2.
The ansatz n= 2turns out to be self-consistent and we
see deviations of the transport lifetime from the quasi-
particlelifetimefor k&5107m 1. Theplotonlyshows
our results for k<1108m 1because our assumption
of an isotropic lifetime is not valid for higher momenta
in this case.
We conclude that for YIG in the long-wavelength
regime the magnon transport lifetime (due to magnon-
phonon interactions) should be approximately the same
as the quasi-particle lifetime, but deviations at shorter
wavelengths require more attention.
VII. SUMMARY AND CONCLUSION
We calculated the decay rate of magnons in YIG
induced by magnon-phonon interactions in the long-
wavelength regime ( k.1109m 1). Our model
takes only the acoustic magnon and phonon branches
into account and is therefore valid at low to intermedi-
ate temperatures ( T.100 K). The exchange-mediated
magnon-phonon interaction has been recently identified
as a crucial contribution to the overall magnon-phonon
interaction in YIG at high temperatures [3, 29, 45]. We
emphasize that its coupling strength can be derived from
0.0 0.1 0.2 0.3 0.4 0.5
kx[109m−1]0.00.51.01.52.0γc
t(k) [106s−1]quasi-particle
1/τ∼k0
1/τ∼k2
1/τ∼k4Figure 11. Inverse of the magnon transport lifetime in YIG
(with magnon momentum along (100)) due to magnon num-
ber conserving magnon-phonon interactions at T= 50 Kand
B= 0for magnons along the (100) direction.
0.00 0.02 0.04 0.06 0.08 0.10
kx[109m−1]0.00.20.40.60.81.01.21.41.6γnc
t(k) [103s−1]quasi-particle
1/τ∼k0
1/τ∼k2
1/τ∼k4
Figure 12. Inverse of the magnon transport lifetime in YIG
(with magnon momentum along (100)) due to magnon num-
ber non-conserving interactions at T= 50 KandB= 0.
experimental values of the magnetic Grüneisen parame-
ter m=@lnTC=@lnV[32, 33]. In previous works this
interaction has been either disregarded [28], underesti-
mated [29, 46], or overestimated [3].
In the ultra-long-wavelength regime the wave vector
dependent magnon decay rate
(k)is determined by the
anisotropy-mediated magnon-phonon interaction with
(k)k2, while for shorter wavelengths k&4107m 1
the exchange-mediated magnon-phonon interaction be-
comes dominant, which scales as
(k)k4. The magnon
number non-conserving processes are caused by spin-
orbit interaction, i.e., the anisotropy-mediated magnon-
phonon interaction, and are correspondingly weak.
In a finite magnetic field the average phonon scatter-
ingcontribution, fromthemechanismunderstudy, tothe
Gilbert damping of the k= 0macrospin Kittel mode is
about three orders of magnitude smaller than the best
values for the Gilbert damping 10 5. However, we
predict peaks at 1:3 Tand4:6 T, that may be experi-
mentally observable in high-quality samples.
The magnon transport lifetime, which is given by the
balance between in- and out-scattering in the Boltz-11
mann equation, is in the long-wavelength regime approx-
imately the same as the quasi-particle lifetime. However,
the magnon quasi-particle and transport lifetime differ
more significantly at shorter wavelengths. A theory for
magnon transport at room temperature should therefore
include the “vertex corrections”.
A full theory of magnon transport at high temperature
requires a method that takes the full dispersion relations
of acoustic and optical phonons and magnons into ac-
count. This would also require a full microscopic descrip-
tion of the magnon-phonon interaction, since the magne-
toelastic energy used here only holds in the continuum
limit.
ACKNOWLEDGMENTS
N. V-S thanks F. Mendez for useful discussions. This
work is part of the research program of the Stichting voor
Fundamenteel Onderzoek der Materie (FOM), which is
financially supported by the Nederlandse Organisatie
voor Wetenschappelijk Onderzoek (NWO) as well as a
Grant-in-Aid for Scientific Research on Innovative Area,
”Nano Spin Conversion Science” (Grant No. 26103006),
CONICYT-PCHA/Doctorado Nacional/2014-21140141,
Fondecyt Postdoctorado No. 3190264, and Fundamen-
tal Research Funds for the Central Universities.
Appendix A: Long-wavelength approximation
The theory is designed for magnons with momen-
tumk < 0:8109m 1and phonons with momen-
tumq < 2:5109m 1(corresponding to phonon en-
ergies/frequencies 12 meV/3 THz), but relies on high-
momentum cut-off parameters kcbecause of the assump-
tion of quadratic/linear dispersion of magnon/phonons.
We see in Fig. 13 that the scattering rates only weakly
depend onkc.
The dependence of the scattering rate on the phonon
momentum cut-off qcis shown in Fig. 14. qc= 3:15
109m 1corresponds to an integration over the whole
Brillouin zone, approximated by a sphere. From these
considerations we estimate that the long-wavelength ap-
proximation is reliable for k.8108m 1. Opti-
cal phonons (magnons) that are thermally excited for
T?100 K (300 K) are not considered here.
Appendix B: Second order magnetoelastic coupling
The magnetoelastic energy is usually expanded only to
first order in the displacement fields. Second order terms
can become important e.g. when the first order terms
vanish. Thisisthecaseforone-magnontwo-phononscat-tering processes. The first order term
X
q
qb qXq+
qby
qXq
(B1)
onlycontributeswhenphononandmagnonmomentaand
energies cross, giving rise to magnon polaron modes [42].
In other areas of reciprocal space the second order term
should therefore be considered. Eastman [58, 59] derived
the second-order magnetoelastic energy and determined
thecorrespondingcouplingconstantsforYIG.Inmomen-
tum space, the relevant contribution to the Hamiltonian
is of the form
H2p1m=1p
NX
k;q1;1;q2;2
q1+q2+k;0 b
q11;q22Xq11Xq22bk
+q1+q2 k;0 b
q11;q22Xq11Xq22by
k
;(B2)
where the interaction vertices are symmetrized,
b
q11;q22=1
2
~ b
q11;q22+~ b
q22;q11
;(B3)
and obey
b
q11;q22=
b
q11; q22
: (B4)
The non-symmetrized vertex function is
~ b
q11;q22=1
a2p
2S[B144(iI1 I1;x$y)
+B155(iI2 I2;x$y)
+B456(iI3 I3;x$y)]; (B5)
with
I1=a2ex
q11qx
1h
ey
q22qz
2+ez
q22qy
2i
;(B6)
I2=a2h
ey
q11qy
1+ez
q11qz
1i
h
ey
q22qz
2+ez
q22qy
2i
; (B7)
I3=a2
ex
q11qz
1+ez
q11qx
1
h
ex
q22qy
2+ey
q22qx
2i
; (B8)
andx$ydenotes an exchange of xandy. The relevant
coupling constants in YIG are [58, 59]
B144= 648 meV; (B9)
B155= 446 meV; (B10)
B456= 328 meV: (B11)
The magnon self-energy (see Fig. 15) reads
2p1m(k;i!) = 2
NX
q1;1;q2;21
X
q1+q2+k;0
b
q11;q22 2F1(q1;
)F2(q2;
!):
(B12)12
0.0 0.2 0.4 0.6 0.8 1.0
kx[109m−1]0102030405060γc(k) [106s−1](a)
kc= 3.15×109m−1
kc= 0.8×109m−1
0.0 0.2 0.4 0.6 0.8 1.0 1.2
kx[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1](b)
kc= 3.15×109m−1
kc= 0.8×109m−1
Figure 13. Dependence the magnon decay rate along (100) on the high magnon momentum cut-off kcfor the (a) magnon
number conserving (
c) and (b) non-conserving (
nc) contributions at T= 50 KandB= 0.
0.0 0.2 0.4 0.6 0.8 1.0
kx[109m−1]0102030405060γc(k) [106s−1](a)
qc= 3.15×109m−1
qc= 2.5×109m−1
qc= 2×109m−1
0.0 0.2 0.4 0.6 0.8 1.0 1.2
kx[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1](b)
qc= 3.15×109m−1
qc= 2×109m−1
Figure 14. Dependence the magnon decay rate along (100) on the high phonon momentum cut-off qcfor the (a) magnon number
conserving (
c) and (b) non-conserving (
nc) contributions at T= 50 KandB= 0.
with phonon propagator
F(q;
) =~2
m1
~2
2+"2
q: (B13)
and leads to a magnon decay rate
nc
2p(k) = 2
~Im 2p1m(k;i!!Ek=~+i0+)
=~3
m2NX
q1;1;q2;2q1+q2+k;01
"1"2 b
q11;q22 2
f2(Ek+"1 "2) [n1 n2]
+(Ek "1 "2) [1 +n1+n2]g; (B14)
where
n1=nB("q11); n2=nB("q22); (B15)
"1="q11; "2="q22: (B16)
The first term in curly brackets on the right-hand-side
of Eq. (B14) describes annihilation and creation of a
phonon as a sum of out-scattering minus in-scattering
contributions,
n1(1 +n2) (1 +n1)n2=n1 n2;(B17)while the second term can be understood in terms of
out-scattering by the creation of two phonons and the
in-scattering by annihilation of two phonons,
(1 +n1)(1 +n2) n1n2= 1 +n1+n2:(B18)
For this one-magnon-two-phonon process the quasi-
particle and the transport lifetimes are the same,
t= qp; (B19)
since this process involves only a single magnon that is
either annihilated or created. The collision integral is
then independent of the magnon distribution of other
magnons and the transport lifetime reduces to the quasi-
particle lifetime.
The two-phonon contribution to the magnon scatter-
ing rate in YIG at T= 50 Kand along (100) direction
as shown in Fig. 16 is more than two orders of magni-
tude smaller than that from one-phonon processes and
therefore disregarded in the main text. The numerical
results depend strongly on the phonon momentum cutoff
qc, even in the long-wavelength regime, which implies
that the magnons in this process dominantly interact13
k k
q'q
Figure 15. Feynman diagram representing the self-energy
Eq. (B12) due to one-magnon-two-phonon processes.
0.0 0.2 0.4 0.6 0.8 1.0 1.2
kx[109m−1]0.00.10.20.30.40.5γnc
2p(k) [103s−1]qc= 3.15×109m−1
qc= 2.5×109m−1
qc= 2×109m−1
Figure 16. Two-phonon contribution to the magnon number
non-conserving magnon scattering rate with magnon momen-
tum along (100) for different values of the phonon momentum
cutoffqcatT= 50 KandB= 0.
with short-wavelength, thermally excited phonons. In-
deed, the second order magnetoelastic interaction (B5) is
quadratic in the phonon momenta, which favors scatter-
ingwithshort-wavelengthphonons. Ourlong-wavelength
approximation therefore becomes questionable and the
results may be not accurate at T= 50 K, but this should
not change the main conclusion that we can disregard
these diagrams.
Our finding that the two-phonon contributions are
so small can be understood in terms of the dimension-
ful prefactors of the decay rates (Eqs. (4.8-4.9) and
(B14)): The one-phonon decay rate is proportional to
~=(ma2)7106s 1, while the two-phonon decay
rate is proportional to ~3=(m2a4")33 s 1, where
"1 meVis a typical phonon energy. The coupling con-
stants for the magnon number non-conserving processes
areBk;?5 meVwhile the strongest two phonon cou-
pling which enhances the two-phonon process by about
a factor 100, but does not nearly compensate the pref-
actor. The two phonon process is therefore three orders
of magnitudes smaller than the contribution of the one
phonon process. The physical reason appears to be the
large mass density of YIG, i.e. the heavy yttrium atoms.
Appendix C: Numerical integration
The magnon decay rate is given be the weighted den-
sity of statesI=Z
BZd3qf(q)("(q)); (C1)
that contain the Dirac delta function (")that can be
eliminated to yield
I=X
qiZ
Aid2qf(q)
jr"(q)j; (C2)
where the qiare the zeros of "(q)andAithe surfaces
inside the Brillouin zone with "(q) ="(qi). The calcu-
lation these integrals is a standard numerical problem in
condensed matter physics.
For aspherical Brillouin zone of radius qcand spherical
coordinates (q;; ),
I=Z
0dZ2
0d Zqc
0dqq2sin()f(q;; )("(q;; )):
(C3)
When"(qi;; ) = 0
("(q;; )) =X
qi(; )(q qi(; ))
j"0(qi(; );; )j;(C4)
where"0=@"=@qand
I=Z
0dZ2
0d X
qi(; )<qcq2
i(; ) sin()
f(qi(; );; )
j"0(qi(; );; )j;(C5)
which is particularly useful when the zeros of "(q;; )
can be calculated analytically for linear and quadratic
dispersion relations.
We can also evaluate the integral Ifully numerically
by broadening the delta function [60] e.g. replacing it by
a Gaussian [60],
(")!1pexp
"2
2
; (C6)
whereis the broadening parameter. An alternative is
the Lorentzian (Cauchy-Lorentz distribution),
(")!1
2
"2+2; (C7)
which has fat tails that are helpful in finding the zeros of
the delta function for an adaptive integration grid. Here
we use the cubature package by Steven G. Johnson [61],
which implements an adaptive multidimensional integra-
tion algorithm over hyperrectangular regions [62, 63].14
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2108.07676v1.Spectral_enclosures_for_the_damped_elastic_wave_equation.pdf | arXiv:2108.07676v1 [math.SP] 17 Aug 2021Spectral enclosures for the damped elastic wave equation
Biagio Cassano1, Lucrezia Cossetti2and Luca Fanelli3
1Dipartimento di Matematica e Fisica, Universit` a degli Stu di della Campania “Luigi Vanvitelli”, Viale Lincoln 5,
81100 Caserta, Italy; biagio.cassano@unicampania.it
2Fakult¨ at f¨ ur Mathematik, Institut f¨ ur Analysis, Karlsr uher Institut f¨ ur Technologie (KIT), Englerstraße 2, 7613 1
Karlsruhe, Germany; lucrezia.cossetti@kit.edu
3Ikerbasque &Departamento de Matematicas, Universidad del Pa´ ıs Vasco/ Euskal Herriko Unibertsitatea (UPV/EHU),
Aptdo. 644, 48080, Bilbao, Spain; luca.fanelli@ehu.es
17 August 2021
Abstract
In this paper we investigate spectral properties of the damp ed elastic wave equation. Deducing a correspon-
dence between the eigenvalue problem of this model and the on e of Lam´ e operators with non self-adjoint
perturbations, we provide quantitative bounds on the locat ion of the point spectrum in terms of suitable
norms of the damping coefficient.
1 Introduction
This paper is concerned with the damped elastic wave equation
utt+a(x)ut−∆∗u= 0,(x,t)∈Rd×(0,∞), (1)
Herea:Rd→Cd×ddenotes the damping coefficient assumed to be a (possibly) non herm itian matrix. We shall
make the standard assumption of a bounded damping, i.e.a∈L∞(Rd)d.The symbol −∆∗is used to denote
the Lam´ e operator of elasticity which is a matrix-valued differential operator acting, w.r.t. the spacial variable
x∈Rdon smooth vector fields as
−∆∗u=−µ∆u−(λ+µ)∇divu, u ∈C∞
0(Rd)d:=C∞
0(Rd;Cd). (2)
The material-dependent Lam´ e parameters λ,µ∈Rare assumed to satisfy the ellipticity condition
µ>0, λ+µ≥0. (3)
It is customarily to write the second-order evolution system ( 1) as a doubled first-order system introducing the
vector field U= (u,ut)T.Then (1) can be rewritten as Ut=A∗
aU,whereA∗
ais the 2d×2dmatrix-valued damped
elastic wave operator defined as
A∗
a:=/parenleftbigg0 1
∆∗−a/parenrightbigg
,D(A∗
a) :=H2(Rd)d×˙H1(Rd)d. (4)
The damped elastic wave equation ( 1) and the corresponding damped operator ( 4) have attracted considerable
attention in the last decades. In the constant coefficient case, na melya(x) =α, α >0,Bocanegra-Rodr´ ıguez
et al. [10] considered the longtime dynamics of this semilinear model in the pres ence of nonlinear structural
forcing terms and external forces: they proved well-posedness ` a laHadamard and established the existence of
finite dimensional global attractors together with the upper semic ontinuity thereof. Energy decay results in
relation with stability properties of solutions to this elastic model hav e been also deeply investigated. In [ 6]
1Bchatnia and Daoulatli obtained a general energy decay estimate in a three dimensional bounded domain in the
presence of localized nonlinear damping and an external force. By a dding viscoelastic dissipation of memory
type Bchatnia and Guesmia [ 8] established a more general energy decay. Different viscoelastic d issipations have
been considered in [ 31,32]. Strong stability of Lam´ e systems with fractional order boundar y damping were
studied by Benaissa and Gaouar in [ 9].
For the undamped elastic wave equation, more commonly known as Navier equation , a more varied bibli-
ography is available. In [ 3] Barcel´ o et al. proved uniform resolvent estimates (Limiting Absor ption Principle)
for this model. With this stationary tool at hands they also proved a priori averaged estimates for the corre-
sponding Cauchy problem. The resolvent estimates in [ 3] were generalized in [ 16] and then improved in [ 26],
where a sharp result (analogous to the one available for the Laplacia n [25]) was proved. Surprisingly, differently
from the Laplacian, in [ 26] the authors also showed the failure of uniform Sobolev and Carlema n inequalities
for the Lam´ e operator. In [ 23] it was proved that if spacial lower-order perturbations are repla ced by temporal
ones,i.e.if one considers the damped equation, then those estimates becom e available again. In [ 2] the authors
generalized the results in [ 3] proving Agmon-H¨ ormander type estimates of the Navier equatio n when this is
perturbed by small 0-th order matrix-valued potential. From thes e results Strichartz estimates for the evolution
equation followed (in the same manner as for classical wave equation , see [11,12]). These Strichartz estimates
were then generalized in [ 23,24]. In particular in [ 23] the endpoint case is deduced.
The Navier equation got also attention of the inverse problem’s comm unity. In particular, inverse scat-
tering was studied in [ 4,5], whereas inverse boundary problems were considered in [ 5,7,18,22,33]. Boundary
determination of Lam´ e parameters has been studied in [ 13,30,35].
In this paper we are interested in spectral properties of the damp ed elastic wave equation ( 1), or equivalently
of the elastic wave operator ( 4). More precisely, we aim at deducing quantitative bounds on the loca tion of the
point spectrum of A∗
ain terms of suitable norms of the damping coefficient. In order to do t hat we establish a
correspondence (see Lemma 2.1) between the eigenvalue problem associated to ( 4) and the one corresponding
to suitable Lam´ e operators with non self-adjoint perturbations, that is operators of the form
−∆∗+V, (5)
whereVdenotes the operator of multiplication by a (possibly) non hermitian m atrix-valued function V:Rd→
Cd×d.
The study of the spectrum of ( 5) has already a bibliography. It is well known that the free Lam´ e ope rator
−∆∗is self-adjoint on H2(Rd)dandσ(−∆∗) =σac(−∆∗) = [0,∞).It is a natural question [ 14–16,26] to ask
whether and how the spectrum changesunder 0th-orderpertur bations,i.e.considering the operator( 5). In [15],
adapting to the elasticity setting the method of multipliers developed for non self-adjoint Schr¨ odinger operators
in [20] (see also [ 21] for similar problems on the plane), the author showed that the poin t spectrum of the
perturbed Lam´ e operator ( 5) remains empty (as in the free case) under suitable variational sma ll perturbations
(inverse-square Hardy potential with small coupling constant is co vered). Later, in [ 14] we showed that full
spectral stability, i.e.σ(−∆∗+V) =σ(−∆∗) = [0,∞),can be proved in three dimensions d= 3 under
perturbations which satisfy a smallness condition of Hardy-type (s ee [14, Thm. 1.4]). Focusing on the point
spectrum only, if no stability can be proved a priori, an interesting qu estion is related to provide quantitative
bounds on the location in the complex plane of this part of the spectr um which, in the perturbed setting, is
possibly no longer empty. In this direction, some preliminary result va lid for the discretespectrum can be found
in [16] (see also [ 26]). Later, these results have been extended in [ 14] to cover embedded eigenvalues as well.
More precisely in [ 14] the following result was proved.
Theorem 1.1 (Thm. 1.1, [ 14]).Letd≥2,0< γ≤1/2ifd= 2and0≤γ≤1/2ifd≥3andV∈
Lγ+d
2(Rd;Cd×d).Then there exists a universal constant cγ,d,λ,µ>0independent on Vsuch that
σp(−∆∗+V)⊂/braceleftbigg
z∈C:|z|γ≤cγ,d,λ,µ/ba∇dblV/ba∇dblγ+d
2
Lγ+d
2(Rd)/bracerightbigg
. (6)
In the self-adjoint case, namely for real-valued perturbations, t he result above holds for a larger class of indices
γ.More precisely, the following result holds true.
2Theorem 1.2 (Thm. 3.1, [ 16]).Letd≥2, γ >0ifd= 2andγ≥0ifd≥3andV∈Lγ+d
2(Rd;R).Then
there exists a universal constant cγ,d,λ,µ>0independent of Vsuch that any negative eigenvalue z(if any) of
the self-adjoint perturbed Lam´ e operator −∆∗+VIRdsatisfies
|z|γ≤cγ,d,λ,µ/ba∇dblV−/ba∇dblγ+d
2
Lγ+d
2(Rd), (7)
whereV−is the negative part of V,i.e.V−(x) := max {−V(x),0}.
Making use of Theorem 1.1and Theorem 1.2and the correspondencebetween the eigenvalueproblem associat ed
to the damped elastic wave operator and the one of the perturbed Lam´ e operator ( 5) (see Lemma 2.1below)
we shall prove the following two results valid in the self-adjoint and th e non self-adjoint setting.
Theorem 1.3. Letd≥2and assume γsatisfies the hypotheses of Theorem 1.2anda∈L∞(Rd;R).Then there
exists a universal constant cγ,d,λ,µ>0independent of the damping asuch that for any positive (respectively
negative) eigenvalue zof the damped elastic wave operator A∗
aanda−∈Lγ+d
2(Rd)(respectively a+∈Lγ+d
2(Rd))
satisfies
(±z)γ−d
2≤cγ,d,λ,µ/ba∇dbla∓/ba∇dblγ+d
2
Lγ+d
2(Rd), (8)
Settingγ=d/2 in (8), the previous theorem provides sufficient condition on the size of t he damping coefficient
to guarantee absence of positive (respectively negative) eigenva lues.
Corollary 1.1. Ifd≥2and
cd
2,d,λ,µ/ba∇dbla∓/ba∇dbld
Ld(Rd)<1,
thenA∗
ahas no positive (respectively negative) eigenvalues.
In the non self-adjoint setting we shall prove the following result.
Theorem 1.4. Letd≥2and assume γsatisfies the hypotheses of Theorem 1.1anda∈L∞(Rd;Cd×d)is a
(possibly) non hermitian matrix. Then there exists a univer sal constant cγ,d,λ,µ>0independent of the damping
asuch that
σp(A∗
a)⊂/braceleftBig
z∈C:|z|γ−d
2≤cγ,d,λ,µ/ba∇dbla/ba∇dblγ+d
2
Lγ+d
2/bracerightBig
. (9)
Remark 1.1.Notice that in the non self-adjoint case, due to the more restrictiv e class of indices for which
Theorem 1.1is valid compared to Theorem 1.2, no analogous of Corollary 1.1holds true ( γ=d/2 is not
admissible).
The main motivation behind our project relies on the following simple obs ervation: the ellipticity condition ( 3)
allows taking λ+µ= 0 in the definition of the Lam´ e operator ( 2). This choice turns the Lam´ e operator ( 2) into
a vector Laplacian and consequently the damped elastic wave equat ion (1) into a system of classical damped
wave equations. For the (scalar) damped wave equation, results in the spirit of Theorem 1.3and Theorem 1.4
have been recently proved in [ 27]. Thus, Theorem 1.3and Theorem 1.4can be seen as a generalization of the
results in [ 27, Thm. 1, Thm. 5 and Thm. 6] in the sense that they recover∗them when λ+µ= 0.
Theorem 1.3and Theorem 1.4are not stated for d= 1,as a matter of fact the one dimensional case is rather
special and it is treated separately. In d= 1 the Lam´ e operator −∆∗turns into a scalar differential operator,
more precisely it is simply a multiple of the Laplacian
−∆∗:=−µd2
dx2−(λ+µ)d2
dx2=−(λ+2µ)d2
dx2.
As a straightforward consequence of the celebrated result of Ab ramov, Aslanian and Davies for 1D-Schr¨ odinger
operators (see [ 1, Thm. 4]), in [ 16] the following result for the one dimensional non self-adjoint Lam´ e operator
was proved.
∗the constants involved slightly differ due to the presence of the coefficient µof the vector Laplacian and due to the vectorial
form of the wave equation once λ+µ= 0 in (1).
3Theorem 1.5 (Thm. 1.1, [ 16]).Letd= 1andV∈L1(R;C).Then
σp(−∆∗+V)⊂/braceleftBig
z∈C:|z|1/2≤1
2√λ+2µ/ba∇dblV/ba∇dblL1(R)/bracerightBig
.
Remark 1.2.We stress that Theorem 1.1 in [ 16] was stated only for eigenvalues outside the essential spectrum,
namely for z∈C\[0,∞).Nevertheless, it is easy to show that embedded eigenvalues can be c overed as well
(see [17, Cor. 2.16]).
In the self-adjoint case, as an immediate consequence of the Lieb- Thirring inequalities ([ 28,29]) valid for the
Schr¨ odinger operators, one has the following result.
Theorem 1.6. Letd= 1andV−∈L1(R;R).Then
σp(−∆∗+V)⊂/braceleftBig
z∈C:|z|1/2≤1
2√λ+2µ/ba∇dblV−/ba∇dblL1(R)/bracerightBig
. (10)
Theorem 1.5and Theorem 1.6together with Lemma 2.1below allow to deduce properties on the point spectrum
of the one dimensional damped elastic wave operator A∗
a.Differently from the higher dimensional setting, in
d= 1Theorem 1.5doesnotentailanyquantitativeboundonthe locationinthe complex planeoftheeigenvalues
ofA∗
a,on the other hand it provides an explicit smallness condition on the size of theL1-norm of the damping
such thatA∗
adoes not have eigenvalues. More precisely we have the following resu lt.
Theorem 1.7. Letd= 1anda∈L1(R;C).If/ba∇dbla/ba∇dblL1(R)<2√λ+2µ,thenσp(A∗
a) =∅.Moreover, the constant
2√λ+2µis optimal.
In the self-adjoint situation it holds true a slightly different result co mpared to the ones introduced so far.
Theorem 1.8. Letd= 1and assume that ais real-valued and satisfies
/integraldisplay
R|x||a(x)|dx<∞and lim
R→∞/ba∇dbla/ba∇dblL∞(R\BR(0))= 0. (11)
Letzbe a real eigenvalue of A∗
a.Ifz>0and/integraltext
Ra<−4√λ+2µ(orz <0and/integraltext
Ra>4√λ+2µ), then
|z| ≥(λ+2µ)/parenleftBigg/integraldisplay
R|x||a(x)|dx/parenrightBigg−1
.
Moreover the following quantitative bound on the location of eigenva lues holds.
Theorem 1.9. Letd= 1and assume that ais real-valued and satisfies (11). Moreover, assume
|z|<(λ+2µ)/parenleftBigg/integraldisplay
R|x||a(x)|dx/parenrightBigg−1
.
Ifz>0and/integraltext
Ra<0(respectively, z <0and/integraltext
Ra>0), then there exists exactly one α>0satisfying
2/parenleftBigg/integraldisplay
Ra−(x)dx/parenrightBigg−1
≤α≤ −4/parenleftBigg/integraldisplay
Ra(x)dx/parenrightBigg−1/parenleftBigg
respectively, 2/parenleftBigg/integraldisplay
Ra+(x)dx/parenrightBigg−1
≤α≤4/parenleftBigg/integraldisplay
Ra(x)dx/parenrightBigg−1/parenrightBigg
such thatz/αis an eigenvalue of A∗
a.
The rest of the paper is divided as follows. In the next Section we pro vide the proof of the preliminary
Lemma2.1establishing the correspondence between the eigenvalue problem a ssociated to the damped elastic
waveoperatorand the perturbed Lam´ eoperator. Afterwards , in Section 2.1we showthe validity of Theorem 1.3
and Theorem 1.4which hold in higher dimension d≥2.The one dimensional case, that is Theorem 1.7-
Theorem 1.9, is treated separately in Section 2.2.
42 Proofs
As a starting point we show how the eigenvalue problem associated to the damped elastic wave operator A∗
a
defined in ( 4) is related to the one of a perturbed Lam´ e operator of the form ( 5).
Lemma 2.1. Letd≥1and assume a∈L∞(Rd;Cd×d).For everyz∈C,
z∈σp(A∗
a)⇐⇒ −z2∈σp(−∆∗+za).
Proof.Assumez∈σp(A∗
a),then there exists a non-trivial Ψ = ( ψ1,ψ2)T∈ D(A∗
a) such that A∗
aΨ =zΨ.In
other words, ψ1∈H2(Rd)d, ψ2∈˙H1(Rd)dandψ2=zψ1,∆∗ψ1−aψ2=zψ2.Plugging the first equation in
the second one gives −∆∗ψ1+zaψ1=−z2ψ1.Sinceψ1/\e}atio\slash= 0,then−z2∈σp(−∆∗+za).Conversely, assume
−z2∈σp(−∆∗+za),then there exists a non-trivial ψ∈H2(Rd)dsuch that ( −∆∗+za)ψ=−z2ψ.Defining
Ψ := (ψ,zψ)T,then Ψ∈ D(A∗
a) and (A∗
aΨ)T= (zψ,∆∗ψ−zaψ) =z(ψ,zψ) =zΨT.Therefore,z∈σp(A∗
a).
Remark 2.1.From the validity of Lemma 2.1, one has that 0 /∈σp(A∗
a) as the spectrum of the unperturbed
Lam´ e operator −∆∗+0a=−∆∗is purely continuous.
2.1 Higher dimensions d≥2 :Proof of Theorem 1.3and Theorem 1.4
With Lemma 2.1at hands we now show that Theorem 1.3and Theorem 1.4are consequence of Theorem 1.2
and Theorem 1.1, respectively.
Proof of Theorem 1.3.From Lemma 2.1we know that z∈σp(A∗
a) if and only if −z2∈σ(−∆∗+za).From
Theorem 1.2there exists cγ,d,λ,µ>0 such that
|z|2γ≤cγ,d,λ,µ/ba∇dbl(za)−/ba∇dblLγ+d
2
Lγ+d
2(Rd), (12)
where (za)−is the negative part of za,i.e.(za)−=za+ifz∈(−∞,0) and (za)−=za−ifz∈(0,∞).Using
this fact in ( 12) and dividing both sides of ( 12) by|z|γ+d/2(z/\e}atio\slash= 0,see Remark 2.1) we obtain ( 8).
Now we consider the non self-adjoint situation.
Proof of Theorem 1.4.The proof of Theorem 1.4is analogous to the one of Theorem 1.3. Letz∈σp(A∗
a),then
by Lemma 2.1−z2∈σp(−∆∗+za).Using the eigenvalue bound ( 6) then one has
|z|2γ≤cγ,d,λ,µ|z|γ+d
2/ba∇dbla/ba∇dblγ+d
2
Lγ+d
2(Rd),
which gives ( 9) and concludes the proof.
2.2 1D: Proof of Theorem 1.7, Theorem 1.8and Theorem 1.9
We startconsideringtheself-adjointsituation. Let z∈Randlet{λ∗
n(za)}N
n=1denotethe sequenceofeigenvalues
of−∆∗+za,then the following preliminary lemma on the sum of the square root of t he eigenvalues holds.
Lemma 2.2. Letd= 1.Then
N/summationdisplay
n=1|λ∗
n(za)|1/2≥ −z
4√λ+2µ/integraldisplay
Ra(x)dx. (13)
Moreover if/integraltext
R|x||a(x)|dx<∞,then the following bound on the number Nof eigenvalues λ∗
n(za)
N≤1+|z|
λ+2µ/integraldisplay
R|x||a(x)|dx (14)
holds.
5Proof.Ifλ∗
n(za) is an eigenvalue of −∆∗+za,then there exists ψ∈H2(R) such that −(λ+2µ)∆ψ+zaψ=
λ∗
n(za)ψor, equivalently,/parenleftBig
−∆+za
λ+2µ/parenrightBig
ψ=λ∗
n(za)
λ+2µψ. (15)
Denoting by λn(V) the eigenvalues of the Schr¨ odinger operator −∆ +V,then we conclude that λ∗
n(za) is
an eigenvalue of −∆∗+zaif and only if there exists n∈Nsuch thatλ∗
n(za) is a multiple of an eigenvalue
λn(za/(λ+2µ))ofthe Schr¨ odingeroperator −∆+za/(λ+2µ),moreprecisely λn(za/(λ+2µ)) =λ∗
n(za)/(λ+2µ).
In particular the number of eigenvalues coincides. The Buslaev-Fad deev-Zakharov trace formula ( cf.[19]) for
1D-Schr¨ odinger operator −∆+Vstates that
N/summationdisplay
n=1|λn(V)|1/2≥ −1
4/integraldisplay
RV(x)dx,
this and the correspondence above give immediately ( 13).
The Bargmann bound [ 34, Pb. 22] provides a control from above of the number of eigenvalu es of the 1D-
Schr¨ odinger operator −∆+Vunder the condition/integraltext
R|x||V(x)dx<∞.More precisely,
N≤1+/integraldisplay
R|x||V(x)|dx. (16)
Similarly as above (that is using the correspondence between eigenv alues of the Lam´ e operator −∆∗+zaand
of the Schr¨ odinger operator −∆+za/(λ+2µ)) from ( 16) one easily gets ( 14). This concludes the proof.
Proof of Theorem 1.8.Letzbe a real eigenvalue of A∗
a,in order to prove Theorem 1.8we will show that if
|z|<(λ+ 2µ)/parenleftBig/integraltext
R|x||a(x)|dx/parenrightBig−1
then/integraltext
Ra≥ −4√λ+2µforz >0 and/integraltext
Ra≤4√λ+2µforz <0.First
of all notice that ( 13) is non-trivial only if z/integraltext
Ra(x)dx <0.This last condition, in particular is known to be
a sufficient condition which guarantees that inf σ(−∆∗+za)<0.From the decay assumption ( 11), then it
follows that −∆∗+zaposses at least one negative eigenvalue. From the upper bound ( 14) it follows that if
|z|<(λ+2µ)/parenleftBig/integraltext
R|x||a(x)|dx/parenrightBig−1
then−∆∗+zahas exactly one negative eigenvalue λ∗
1(za).Thus, from ( 13)
and the correspondence in Lemma 2.1one has
|z|=|λ1(za)|1/2≥ −z
4√λ+2µ/integraldisplay
Ra(x)dx. (17)
This implies/integraltext
Ra(x)dx≥ −4√λ+2µforz>0 and/integraltext
Ra(x)dx≤4√λ+2µforz <0.
Proof of Theorem 1.9.From the hypotheses, as above, one has that −∆∗+zaposses exactly one negative
eigenvalue. The Lieb-Thirring type bound ( 10) in Theorem 1.6and the estimate in ( 17) give
−z
4√λ+2µ/integraldisplay
Ra(x)dx≤ |λ1(za)|1/2≤z
2√λ+2µ/integraldisplay
Ra−(x)dx,
/parenleftBig
respectively −z
4√λ+2µ/integraldisplay
Ra(x)dx≤ |λ1(za)|1/2≤|z|
2√λ+2µ/integraldisplay
Ra+(x)dx/parenrightBig
.
Using the correspondence in Lemma 2.1the result follows.
Proof of Theorem 1.7.Ifz∈Cis an eigenvalue of A∗
a,then from Lemma 2.1−z2∈σp(−∆∗+za).Thus, from
Theorem 1.5we have
|z| ≤1
2√λ+2µ|z|/ba∇dbla/ba∇dblL1(R).
Dividing by |z|,which cannot be zero (see Remark 2.1), one has 1 ≤1
2√λ+2µ/ba∇dbla/ba∇dblL1(R).If theL1-norm ofais
small, namely if /ba∇dbla/ba∇dblL1(R)<2√λ+2µ,then we get a contradiction. Thus, σp(A∗
a) =∅.The optimality of the
result can be proved as in [ 27, Thm. 4].
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8 |
0810.4633v1.The_domain_wall_spin_torque_meter.pdf | The domain wall spin torque-meter
I.M. Miron, P.-J. Zermatten, G. Gaudin, S. Au ret, B. Rodmacq, and A. Schuhl
SPINTEC, CEA/CNRS/UJF/GINP,
INAC, 38054 Grenoble Cedex 9, France
(Dated: September 15, 2021)
Abstract
We report the direct measurement of the non-adiabatic component of the spin-torque in domain
walls. Our method is independent of both the pinning of the domain wall in the wire as well as
of the Gilbert damping parameter. We demonstrate that the ratio between the non-adiabatic and
the adiabatic components can be as high as 1, and explain this high value by the importance of
the spin-
ip rate to the non-adiabatic torque. Besides their fundamental signi cance these results
open the way for applications by demonstrating a signi cant increase of the spin torque eciency.
PACS numbers: 72.25.Rb,75.60.Ch,75.70.Ak,85.75.-d
1arXiv:0810.4633v1 [cond-mat.other] 25 Oct 2008The possibility of manipulating a magnetic domain wall via spin torque e ects when
passing an electrical current through it opens the way for conceptually new devices such as
domain wall shift register memories[1]. Early spin-torque theories[2, 3, 4] were based on a
so called adiabatic approximation which assumed that the incoming electron's spin follows
exactly the magnetization as it changes direction within the domain wall. Nevertheless, the
observed critical currents needed to trigger the domain wall motion were lower than the
value predicted within this framework[5]. As rst predicted by Zhang[6], the existence of a
non-adiabatic term in the extended Landau-Lifshitz-Gilbert equation leads to the vanishing
of the intrinsic critical current. The action of this non-adiabatic torque on a DW is expected
to be identical to that of an easy axis magnetic eld. Micromagnetic simulations have been
used to predict the velocity dependence on current for a DW submitted to the action of the
two components of the spin-torque[7]. The quantitative measurement of this non-adiabatic
torque can be achieved either by demonstrating the equivalence of eld and current in
a static regime, or by observing the complex dynamic behavior[7]. The main diculty
of these measurements comes from the pinning of the DW by material imperfections. It
masks the existence of the intrinsic critical current, and in addition, above the depinning
current, obscures the DW velocity dependence on current. Moreover, most of the DW
velocity measurements were done using materials with in-plane magnetization[5, 8, 9, 10,
11], where the velocity can also depend on the micromagnetic structure of the wall[12]
(transverse wall or vortex wall). Despite the simpler micromagnetic structure of the DWs,
very few results were reported[13] for Perpendicular Magnetic Anisotropy (PMA) materials.
In this case the intrinsic pinning is much stronger, probably due to a local variation of
the perpendicular anisotropy. Up to now, none of the measurements were able to clearly
evidence the equivalence between eld and current, nor to reproduce the predicted dynamic
behavior; hence the value of the non-adiabatic torque is still under debate.
In this letter we use a novel approach for the measurement of the non-adiabatic component
of spin-torque. Instead of measuring the DW velocity, we perform a quasistatic measurement
of its displacement under current and magnetic eld. In principle this method is similar to
any quasi-static force measurement: a small displacement is created, rst with the unknown
force and then with a known reference force. In our case the unknown force is caused by
the electric current passing through the DW while the reference force is due to an applied
magnetic eld. By comparing the two displacements one directly compares the applied
2FIG. 1: Schematic representation of the experimental setup. The inset shows an SEM picture of a
sample.
forces. Due to the high sensitivity of our method (able to detect DW motion down to
10 2nm[14]) we can study the displacement of the DW inside its pinning center. Since the
measurement relies on the comparison to a reference force, the method is independent of
the strength of the pinning. Moreover, as the eld and current are applied quasi-statically,
the damping parameter does not play any role.
According to recent theories[6, 15, 16] that derived the value of the spin torque, (the
ratio between the non-adibatic and adiabatic torques) is given by the ratio between the
rate of the spin-
ip of the conduction electrons and that of the s-d exchange interaction.
Generally, two conditions must be ful lled to obtain a high spin-
ip rate. First it is necessary
to have a strong crystalline eld inside the material. The electric elds will yield a magnetic
eld in the rest frame of the moving electrons. Second, a breaking of the inversion symmetry
is needed. Otherwise the total torque of the magnetic eld on the electron spin averages
out, and the spin-
ip may only occur during momentum scattering[17].
In order to highlight these e ects we have patterned samples from a Pt 3nm/
Co0.6nm/(AlO x)2nmlayer[18]. In this case the symmetry is broken by the presence of the
3AlO xon one side of the Co layer, and of the heavy Pt atoms on the other[19, 20]. We
will emphasize the importance of the spin-
ip interaction to spin torque by comparing
results from these samples with those for samples fabricated from a symmetric Pt 3nm/
Co0.6nm/Pt 3nmlayer[21], where a much smaller spin-
ip rate is expected. As the only di er-
ence between the two structures is the upper layer, we expect similar growth properties for
the Co layer. Both samples exhibit PMA and a strong Anomalous Hall E ect (AHE)[22].
The lms are patterned into the shape depicted in Figure 1. This shape is well suited for a
quasi static measurement as a constriction is created by the presence of the four wires used
for the AHE measurement ( gure 1 inset). This way a DW can be pinned in a position
where changes in the out of plane component of the magnetization (i.e. DW motion) can
be detected by electrical measurements. A current is passed through the central wire. This
current will serve to push the domain wall as well as to probe the eventual displacement.
In the case where the DW does not move under the action of the current, the transverse
resistance remains unchanged and the voltage measured across the side wires (AHE) will be
linear with the current. If the DW moves due to the electric current, the exciting force will
create resistance variations, causing a nonlinear relationship between the measured voltage
and the applied current. A simple way to detect such nonlinearities is to apply a perfectly
harmonic low frequency (10 Hz) ac current, and look at the rst harmonic in the Fast Fourier
Transform (FFT) of the measured voltage. Its value is a measure of the amplitude of the
DW displacement at the frequency of the applied current. To quantitatively compare the
action of a magnetic eld to that of an electric current, the magnetic eld is applied at
the same frequency and in phase (or opposition of phase) with the electric current. By
applying current and eld simultaneously, we ensure that their corresponding torques act on
the same DW con guration. In addition to the displacement provoked by the current, the
eld induced displacement will add to the value of the rst harmonic, which can be either
increased if the eld and current push the wall in the same direction, or decreased if they
act in opposite directions.
Figure 2 shows the dependence of the resistance variation at the frequency of the current
(R f) on the current amplitude for di erent values of the eld amplitude. First, at low
current and eld amplitudes the displacement is almost linear ( 107A/cm2), but for higher
values, the R fvaries more rapidly. A simple estimation based on the value of the resistance
variation compared to the total Hall resistance of a cross (1
) yields 1 nm for the
4FIG. 2: (a) Dependence of the resistance variation on the current amplitude for several eld
amplitudes (Pt/Co/AlO xsample). The inset shows a possible nonlinear and asymmetric
potential well. The energy landscape can be modeled by an e ective out of plane magnetic
eld that has negative values on one side of the equilibrium position and positive values
on the other. (b) A zoom on the small amplitude regime. The inset shows the perfect
superposition obtained by shifting the curves horizontaly with 1.25 105Acm 2Oe 1.5maximum amplitude of the DW motion in the rst regime and 7 nm for the second regime.
This behavior can be explained by the anatomy of the local pinning. The local potential
well trapping the DW can be considered as a superposition of the geometric pinning [23]
and intrinsic pinning caused by defects randomly distributed inside the material[13, 21].
Because the potential well for the small scale displacements (below 10nm) is dominated by
the random intrinsic pinning rather than geometric pinning (the increase of the length of
the DW is small 1%) in the general case it should be asymmetric. We have veri ed the
supposed asymmetry of the e ective potential well by applying alongside the ac current and
eld, a dc bias eld that changes the local potential well (inset of gure 3). By varying
this eld we observed a reduction of the current amplitude needed to access this strongly
non-linear regime ( gure 3). When the magnetic bias eld was reversed this second regime
was no longer attained with the available current densities (not shown).
The observed dependence of R fon current and eld ( gures 2 and 3) is in perfect
agreement with the characteristic features of the non-adiabatic component of the spin-torque.
First, we do not observe any critical current down to the lowest current value (106A/cm2-
gure 4 in [14]). Futhermore, by extrapolating the amplitude of the DW displacement ( gure
2), when the current is reduced, the displacement goes to zero as the current goes to zero,
in agreement with the absence of the critical current.
However, the most important feature of the R fbehavior is that the curves obtained
for any eld amplitude can be obtained from the curve corresponding to zero eld just by
shifting it horizontally (in current): towards the lower current values when the eld and
current act in the same direction on the DW and towards higher values when their actions
are opposed. This means that any displacement of the DW can also be achieved with a
di erent current if a magnetic eld is added. The di erence in current is compensated by
the magnetic eld. The value of this horizontal shift gives the eld to current correspondence.
The inset of gure 1b shows that all the curves corresponding to di erent eld amplitudes
have the same shape; by shifting them horizontally (using the eld-current correspondence),
they all collapse on the zero eld amplitude curve. This shows that independently of the
direction or strength of the applied current and eld, as predicted by the theories, their e ect
on the DW is fundamentally similar. Moreover, further evidence that this correspondence is
intrinsic and not in
uenced by pinning is that its value remains the same within the di erent
amplitude regimes as well as when the local potential well is tuned by a constant bias eld.
6FIG. 3: The nonlinear regime (Pt/Co/AlO xsample). When an external bias eld is added,
the e ective pinning eld changes (inset) and the nonlinear regime is reached for di erent
current and eld amplitudes. However, this does not cause any change in the eld to current
correspondence: the horizontal distance between the curves remains the same.
7Since the motion of the DW is quasi-static the magnetization can be considered to be
at equilibrium during motion. In this case the sum of all torques must be zero. In order
for the DW to remain at rest, the torque from the applied current must be compensated
by the torque generated by the magnetic eld. The upturn observed on the -60 Oe curve
( gure 2b) determines the position of the zero amplitude point. Note that the position of
this point is in perfect agreement with the eld to current correspondence obtained from the
horizontal shifting of the curves. By taking into account the micromagnetic structure of the
DW (very thin 5nm Bloch wall) the two torques are integrated over the width of the wall,
and by comparing their values (the eld torque is easily calculated; [14]) the non-adiabatic
term of the spin-torque is determined. In the case of Pt/Co/AlO xstacks the current- eld
correspondence is approximately 1.25 105A/cm2to 1Oe, corresponding to a value of = 1.
Similar measurements ( gure 1 in [14]) were also performed in the saturated state (with-
out the DW). They con rm that there is no contribution to the signal from the ordinary
Hall e ect, but indicate a small contribution from thermoelectric e ects - the Nernst-
Ettingshausen E ect(NEE)[24]. The contribution from DW motion to R fis much higher
than the NEE for the Pt/Co/AlO xstack. In the case of Pt/Co/Pt layers we nd that the
amplitude of the current induced DW motion is much smaller and entirely masked by the
NEE. When a DW is moving inside the perfectly harmonic region at the bottom of the po-
tential well, its displacement depends linearly on the applied force. In such a scenario, the
current induced DW motion and the NEE are indistinguishable. They both lead to a linear
dependence of the R fresponse on current. The only possibility to separate these e ects,
for the Pt/Co/Pt layer, is to attain the high amplitude nonlinear regime of DW motion.
This is done by keeping the current amplitude constant and varying the eld amplitude.
When the current and eld push the wall in the same direction, the nonlinear regime should
be reached for smaller eld amplitudes, than if their actions were opposed.
In the presence of current induced displacements, the nonlinearities observed in the
R fversus eld amplitude curve should be asymmetric. Moreover the asymmetry should
depend on the current value. Such an asymmetry is observed (inset of gure 4) in the case
of Pt/Co/AlO xsamples. In contrast to this behavior, a fully symmetric dependence that
does not depend on the current amplitude is measured for the Pt/Co/Pt samples ( gure 4).
We conclude that in this case the spin torque induces DW displacements smaller than the
resolution limit of this method. This limit value leads to (supplementary notes) 0.02.
8FIG. 4: The nonlinear response of a DW to magnetic eld. (a) R fvs. the amplitude of the eld
for three di erent current densities in the case of Pt/Co/Pt layers (inset Pt/Co/AlO x).(b)
Derivative of R fvs. the eld amplitude for a Pt/Co/Pt sample (inset Pt/Co/AlO x).
Theoretical estimations[6] based on a spin-
ip frequency of 1012Hz yield a value =0.01.
To clarify the di erence of the spin-torque eciency in the two samples, the symmetry
breaking due to the presence of the AlO xsurface must be taken into account. As a metallic
lm gets thinner, the conduction electron's behavior resembles more and more to that of a
two-dimensional electron gas. When such a gas is trapped in an asymmetric potential well,
9the spin-orbit coupling is much stronger than in the case of a symmetric potential due to
the Rashba interaction[25]. This e ect was rst evidenced in nonmagnetic materials where
this interaction leads to a band splitting (0.15 eV for the surface states of Au (111)[26]).
In the case of ferromagnetic metals this e ect was already proposed to contribute as an
e ective magnetic eld[27] for certain DW micromagnetic structures, but should not have
any e ect for Bloch walls in PMA materials. The simple 1D representation used in this
case[27] to model the DW accounts for the coherent rotation of the spins of the incoming
electrons around the e ective eld, but excludes any de-coherence between electrons having
di erent k-vector directions on the Fermi sphere (di erent directions of the Rashba e ective
eld) as well as possible spatial inhomogeneities of this eld (surface roughness). Since the
spin-torque is caused by the cumulative action of all conduction electrons [6], the relevant
parameter is not the spin-
ip rate of a single electron but the relaxation rate of the out
of equilibrium spin-density [6]. In a more realistic 2D case, in the presence of the above
mentioned strong decoherence e ects, the relaxation rate of the out of equilibrium spin-
density approaches the rate of spin precession around the Rashba e ective eld. The above
value of the measured spin-orbit splitting (0.15 eV) will yield in this case an e ective spin-
ip rate of 30 1012Hz, which is in excellent agreement with the order of magnitude of the
measured non-adiabatic parameter, supporting this scenario.
In summary, a technique that allows the direct measurement of the torque from an
electric current on a DW was developed. We have pointed out the importance of spin-
ip interactions to spin torque by comparing its eciency between two di erent systems.
We show that the Pt/Co/AlO xsample with the required symmetry properties to increase
the spin-
ip frequency (breaking of the inversion symmetry) shows an enhanced spin torque
e ect. A value of the order of 1 was measured for the parameter approaching the maximum
value predicted by existing theories. This value can be explained by order of magnitude
considerations on the Rashba e ect observed on surface states of metals. Obtaining a high
eciency spin torque in a low coercivity material would make possible the development
of nanoscale devices whose magnetization could be switched at low current densities. The
order of magnitude of the current densities would be similar to the one observed for magnetic
10semiconductors[28], but, as the resistance is smaller, the supplied power will be lower.
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12 |
1207.6686v1.Ultrafast_optical_control_of_magnetization_in_EuO_thin_films.pdf | 1
Ultrafast optical control of magnetization in EuO thin films
T. Makino1,*, F. Liu2,3, T. Yamasaki4, Y. Kozuka2, K. Ueno5,6, A. Tsukazaki2, T. Fukumura6,7, Y. Kong3, and
M. Kawasaki1,2
1 Correlated Electron Research Group (CERG) and Cr oss-Correlated Materials Research Group (CMRG), RIKEN Advanced Science Institut e,
Wako 351-0198, Japan
2 Quantum Phase Electronics Ce nter and Department of Applied Physics, Un iversity of Tokyo, Tokyo 113-8656, Japan,
3 School of Physics, Nankai Un iversity, Tianjin 300071, China
4 Institute for Materials Research, T ohoku University, Sendai, 980-8577, Japan
5Graduate School of Arts and Sciences, University of Tokyo, Tokyo 153-8902, Japan
6PRESTO, Japan Science and Technology Agency, Tokyo 102-0075, Japan,
7Department of Chemistry, Univers ity of Tokyo, Tokyo 113-0033, Japan,
All-optical pump-probe detection of magnetization precession has been performed for ferromagnetic EuO
thin films at 10 K. We demonstrate that the circ ularly-polarized light can be used to control the
magnetization precession on an ultrafast time scale. This takes place within the 100 fs duration of a single
laser pulse, through combined contribution from two nonthermal photomagnetic effects, i.e., enhancement
of the magnetization and an inverse Faraday effect. From the magnetic field dependences of the frequency
and the Gilbert damping parameter, the intrinsic Gilbert damping coefficient is evaluated to be α ≈ 3×10-3.
PACS numbers: 78.20.Ls, 42 .50.Md, 78.30.Hv, 75.78.J
2 Optical control of the spin in magnetic materials has
been one of the major issues in the field of spintronics,
magnetic storage technology, and quantum computing1.
One type of the spin controls is based on the directional manipulation in the spin moments
2. This yields in
observations of spin precession (reorientation) in
antiferromagnets and ferromagne ts when magnetization is
canted with respect to an external field3–14. In many
previous reports, the spin precession has been driven with
the thermal demagnetization induced with the photo-irradiation. Far more intriguing is the ultrafast
nonthermal control of magnetization by light
8,10,14, which
involves triggering and suppression of the precession. The precession-related anisotropy is expected to be
manipulated through laser-induced modulation of
electronic state because the anisotropy field originates
from the magnetorcrystalline anisotropy based on the
spin-orbit coupling. Recently, the spin precession with the
non-thermal origin has been observed in bilayer manganites due to a hole-concentration-dependent
anisotropic field in competing magnetic phases
15. Despite
the success in triggering the reorientation by ultrafast laser pulses, the authors have not demonstrated the possibility
of the precessional stoppage.
On the other hand, photomagnetic switch of the
precession has been reported in ferrimagnetic garnets with
use of helicity in light
8,10. The authors attributed the
switching behavior to long-lived photo-induced
modification of the magnetocrystalline anisotropy16
combined with the inverse Faraday effects17,18. The
underlying mechanism for the former photo-induced effect
is believed to be redistribution in doped ions16. This is too
unique and material-dependent, which is not observed in wide variety of magnets. For establishing the universal
scheme of such “helicity-controllable” precession, it
should be more useful to rely on more generalized mechanisms such as the carrier-induced ferromagnetism
and the magnetic polarons
19. A ferromagnet should be a
better choice than a ferrimagnet or an antiferromagnet, e.g., for aiming a larger-amplitude modulation by making
use of its larger polarization-rotation angle per unit length .
We have recently reported the optically-induced
enhancement of magnetization in ferromagnetic EuO
associated with the optical transition from the 4 f to 5 d
states
20. This enhancement was attributed to the
strengthened collective magnetic ordering, mediated with
the magnetic polarons. The helicity-controllable
precession is expected to be observed in EuO by combining the photo-induced magnetization
enhancement20 with the inverse Faraday effect17,18 because
the magnetization is related to the magnetic anisotropy.
The occurrence of the inverse Faraday effects is expected
because of the high crystalline symmetry in EuO17,18. The
magnetic properties of EuO are represented by the
saturation magnetization of 6.9 μB/Eu, the Curie
temperature of 69 K, and the strong in-plane
anisotropy21,22.
In this article, we report observation of the
photomagnetic switch of the spin precession with the
nonthermal origin in a EuO thin film for the first time to
the best of our knowledge. Due to the above-mentioned
reasons, our findings deserve the detailed investigations
such as the dependence on the circularly polarized lights, the frequency of precession, the Gilbert damping constants,
and the magnitudes of the photo-induced anisotropic field.
EuO films were deposited on YAlO
3 substrate using a
pulsed laser deposition system with a base pressure lower
than 8×10-10 Torr22. The EuO films were then capped with
AlO x films in-situ . EuO and AlO x layers have thicknesses
of 310 and 30 nm, respectively. The film turned out to be
too insulating to be quantified by a conventional transport measurement method. The all-optical experiments have
been performed using a standard optical set-up with a
Ti:sapphire laser combined with a regenerative amplifier (accompanied with optical parametric amplifier). The
wavelength, width, and repetition rate of the output pulse
were 650 nm, ≈100 fs, and 1 kHz, respectively. The
pump and probe pulses were both incident on the film at
angles of θ
H ≈ 45 degree from the direction normal to the
film plane as shown in inset of Fig. 1. The direction of the probe beam is slightly deviated from that of the pump so
as to ensure the sufficient spatial separation of the
reflected beams. The angle between the sample plane and the external field is approximately 45 degree. The
polarization rotation of the reflected probe pulses due to
the Kerr effect was detected using a Wollaston prism and a balanced photo-receiver. The pump fluence was
approximately 0.5 mJ/cm
2. A magnetic field was applied
using a superconducting electromagnet cryostat. The maximum applied magnetic field was μ
0H ≈ 3 T. All the
measurements were performed at 10 K.
Figure 1 shows a magneto-optical Kerr signal as a
function of the pump-probe delay time for a EuO film at
μ
0H = 3.2 T under the irradiation of right-circularly 3 polarized ( σ+) light. Its time trace is composed of
instantaneous increase and d ecay of the Kerr rotation, and
superimposed oscillation20. The oscillatory structure
corresponds to the precession of magnetization. A solid (black) curve in Fig. 1 shows the result of fit to the
experimental data using an exponentially decaying
function and a damped oscillatory function. The precession is observed even wi th the linearly polarized
light, which is consistent with the fact that EuO is a
ferromagnet at this temperature.
FIG. 1 (color online). Time-resolved Kerr signals recorded
for a EuO thin film at a magnetic field of 3.2 T, and a
temperature of 10 K for ri ght circularly-polarized ( σ
+)
light. The inset schematically shows the experimental arrangement. Experimental data are shown by (red)
symbols, while the result of fit was shown by a full (blue)
line.
For the detailed discussion of the precession properties,
we subtracted the non-oscillatory part from the Kerr signal as a background. The results are shown in Fig. 2 for nine
magnetic fields and for σ
+ and left-circularly polarization
(σ¯). The subtracted data were then fitted with the damped
harmonic function in the form of Aexp(−t/τ) sin(2πft+φ),
where A and φ are the amplitude and the phase of
oscillation, respectively. The amplitude of the precession was not found to depend on the plane of the linear
polarization of the pump pulse. There is a linear
relationship between the amplitude of precession and the pump fluence for the excitation intensity range measured.
It is also noticed in Fig. 2 that the precession amplitudes
are different each other for the two helicities ( σ
+ and σ¯)
even at the same magnetic fields. The magnetic field
dependence of the amplitude is summarized in Fig. 3(d). The minimum precession amplitude appears at around μ0H
= +0.4 T for the σ¯, while the minimum is observed at
μ0H = −0.4 T for the σ+ as indicated by the shaded regions.
To explain such disappearance of the precession and the triggering of precession even with a linearly-polarized
light, it is necessary to take two effects into account. One
of the effects that we seek should be odd with respect to the helicity of light. An effective magnetic field through
the inverse Faraday effect is plausible to interpret this
phenomenon because this satisfies the above requirements
[H
F
// (black arrows) in Figs. 3(a) and 3(b)]. While the
normal Faraday effect causes difference in the refractive
indices for the left and right circularly polarized lights propagating in a magnetized medium, it is also possible to
induce the inverse process where circularly polarized
lights create a magnetization or an effective field
17,18. The
field associated with the inverse Faraday effect changes its
sign when the circular polarization is changed from
left-handed to right-handed.
FIG. 2 (color online). A series of precession signals under various
magnetic fields for right- and left-circularly polarized ( σ+ and σ¯)
lights. Solid circles show the e xperimental data for which the
non-oscillatory background is s ubtracted, while solid curves
represent the calculated data as described in the text.
The other effect involved is considered to be the
photoinduced enhancement of the anisotropic field
(magnetization) associated with the 4 f →5d optical
transition [ ΔM (purple arrows) in Figs. 3(a) and 3(b)]20.
Our previous work quantified the photoinduced
4 enhancement of the magnetization to be ΔM/M ≈ 0.1%20.
The amplitude of precession is determined from
combination of ΔM with the component of the
inverse-Faraday field ( HF
//) approximately projected onto
the easy-axis direction. For example, no precession is
triggered for μ0H of +0.4 T ( −0.4 T) and σ¯ (σ+), which is
due to the balance of these two effects [Fig. 3(a)]. On the other hand, constructive contribution of these effects leads
to a change in the direction of the magnetization [two
dashed lines and a red arrow in Fig. 3(b)], which enhances the precession amplitude. The strength of the
photoinduced field H
F can be estimated to be
approximately 0.2 T at the laser fluence of 0.5 mJ/cm2.
The derivation was based on Eq. (17) of Ref. 10. For more
quantitative discussion for the suppression and
enhancement of precession, the effect of the perpendicular component of inverse Faraday field is necessary to be
taken into account. Such analysis is not performed here
because this goes beyond the scope of our work.
FIG. 3 (color online). Graphical illustrations of the magnetic
precession; its suppression (a) and enhancement (b). M is a
magnetization (green), H the external magnetic field (blue), Heff the
effective magnetic field (red), ΔM a photo-induced magnetization
enhancement (purple), and the HF
// the inverse Faraday field (black).
The situations of suppression correspond to the conditions of 0.4 T for σ¯ and −0.4 T for σ+. The situations of enhancement are for
opposite cases. Magnetic field depe ndences of the magnetization
precession related quantities for σ+ and σ¯; precession frequency f
(c), amplitude (d), and effective Gilbert damping αeff (e) (f).
For the derivation of the precession-related parameters,
we plot the frequency ( f) and the amplitude of the
magnetization precession for two different helicities as a
function of H with closed symbols in Figs. 3(c) and 3(d).
To deduce the Landé g-factor g, we calculated f(H) using
a set of Kittel equations for taking the effect of tilted
geometry into account as12,23:
12 f HH ( 1 )
2
1e f f cos( ) cosH HH M ( 2 )
2e f f cos( ) cos 2H HH M ( 3 )
Here, γ is the gyromagnetic ratio ( gμB/h), μB the Bohr
magneton, h Planck’s constant, and θH an angle between
the magnetic field and direction normal to the plane. Meff
is the effective demagnetizing field given as Meff = MS-2K
⊥/MS, where MS is the saturation magnetization and K⊥ is
the perpendicular magnetic anisotropy constant. θ is an
equilibrium angle for the magnetization, which obeys the
following equation:
eff sin 2 (2 / )sin( )H HM ( 4 )
A solid (black) line in Fig. 3( c) corresponds to the result
of the least-square fit for the frequency f. The values of
parameters are g ≈ 2 and μ0Meff ≈ 2.4 T. The g value is
consistent with the one derived from the static ferromagnetic resonance measurement
24.
Having evaluated the precession-related parameters
such as g and Meff, we next discuss H dependence of an
effective Gilbert damping parameter αeff. This quantity is
defined as:
eff1
2f ( 5 )
Figures 3(e) and 3(f) show the effective Gilbert
damping parameter αeff derived from the decay time
constant ( τ) for σ+ and σ¯, respectively. Despite relatively
strong ambiguity shown with ba rs in Figs. 3(e) and 3(f),
the damping parameters αeff is not independent of the
magnetic field. It is rather a ppropriate to interpret that for
5 αeff for low fields are larger than those at higher fields.
Such dependence on magnetic field is consistent with
those in general observed for a wide range of the
ferrimagnets and ferromagnet s. Two-magnon scattering
has been adopted for the explanation of this trend25. When
the magnitude or direction of the magnetic anisotropy
fluctuates microscopically, magnons can couple more efficiently to the precessional motion
25. Such may cause
an additional channel of relaxation. Due to the suppressed
influence of the abovementioned two-magnon scattering, the higher-field data correspon d to an intrinsic Gilbert
damping constant α ≈ 3×10
-3, as shown with a dashed
(black) line in Figs. 3(e) and 3(f). This value is
comparable with that reported in Fe26,27,28,29 and
significantly larger than that of yttrium iron garnet, which
is known for intrinsically low magnetic damping8,10,14.
In conclusion, we have reported the observation of
magnetization precession and the dependence on light helicity in ferromagnetic EuO films. We attribute it to the
photo-induced magnetization enhancement combined with
the inverse Faraday effect. The magnetic field dependence of the precession properties al lowed us the evaluation of
the Gilbert damping constant to be ≈3×10
-3.
Acknowledgements—the authors thank K. Katayama,
M. Ichimiya, and Y. Takagi for helpful discussion. This
research is granted by the Japan Society for the Promotion
of Science (JSPS) throug h the “Funding Program for
World-Leading Innovative R&D on Science and
Technology (FIRST Program),” initiated by the Council
for Science and Technology Policy (CSTP) and in part supported by KAKENHI (Grant Nos. 23104702 and
24540337) from MEXT, Japan (T. M.).
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|
1410.0439v1.Investigation_of_the_temperature_dependence_of_ferromagnetic_resonance_and_spin_waves_in_Co2FeAl0_5Si0_5.pdf | 1
Investigation of the temp erature-dependence of fe rromagnetic resonance and
spin waves in Co 2FeAl 0.5Si0.5
Li Ming Loong1, Jae Hyun Kwon1, Praveen Deorani1, Chris Nga Tung Yu2, Atsufumi
Hirohata3,a), and Hyunsoo Yang1,b)
1Department of Electrical and Computer Engine ering, National University of Singapore, 117576
Singapore
2Department of Physics, The Univers ity of York, York, YO10 5DD, UK
3Department of Electronics, The Unive rsity of York, York, YO10 5DD, UK
Co
2FeAl 0.5Si0.5 (CFAS) is a Heusler compound th at is of interest for sp intronics applications, due
to its high spin polarization a nd relatively low Gilbert dampi ng constant. In this study, the
behavior of ferromagnetic resonance as a functi on of temperature was investigated in CFAS,
yielding a decreasing trend of damping constant as the temperature was increased from 13 to 300
K. Furthermore, we studied spin waves in CF AS using both frequency domain and time domain
techniques, obtaining group velocities and atte nuation lengths as high as 26 km/s and 23.3 m,
respectively, at room temperature.
a) Electronic mail: atsufumi.hirohata@york.ac.uk
b) Electronic mail: eleyang@nus.edu.sg 2
Half-metallic Heusler compounds with low Gilbert damping constant ( ) are promising
candidates for spin transfer torque-based (STT) spintronic devices,1-3 spin-based logic systems,4
as well as spin wave-based data comm unication in microelectronic circuits.5 Hence, a deeper
fundamental understanding of the magnetiza tion dynamics, such as the behavior of
ferromagnetic resonance (FMR) and spin waves in Heusler compounds, could enable better
engineering and utilization of these compounds fo r the aforementioned applications. In previous
work, FMR has been investigated in several Heusler compounds, such as Co 2FeAl (CFA),6
Co2MnSi (CMS),7 and Co 2FeAl 0.5Si0.5 (CFAS).8 In addition, the variation of with temperature
has been studied for other material s, such as Co, Fe, Ni, and CoFeB.9-11 However, the
temperature-dependence of in Heusler compounds has not be en reported yet. Furthermore,
while there have been some studies of spin waves in Heusler compounds, such as CMS and
Co2Mn 0.6Fe0.4Si (CMFS),7,12 these studies have focused on frequency domain measurements.
Thus, time domain measurements remain scarce, and mainly consist of time-resolved magneto-optic Kerr effect (TR-MOKE) experiments.
13 In this work, we investigate the temperature-
dependence of in CFAS, a half-metallic Heusler compound.14,15 Moreover, we utilize both
frequency domain and pulsed inductive micr owave magnetometry (PIMM) time domain
measurements to study the magnetiza tion dynamics in CFAS. We obtain of 0.0025 at room
temperature, which is 6 times lower than the va lue at 13 K. In addition, we evaluate the group
velocity ( vg) and the attenuation length ( ) in CFAS, leading to values as high as 26 km/s and
23.3 m respectively, at room temperature.
CFAS (30 nm thick) was grown by ultrah igh vacuum (UHV) molecular beam epitaxy
(MBE) on single crystal MgO (001) substrates and capped with 5 nm of Au. The base pressure
was 1.210-8 Pa and the pressure during deposition was typically 1.6 10-7 Pa. The substrates 3
were cleaned with acetone, IPA and deionised wate r in an ultrasonic bath before being loaded
into the chamber. After the film growth, the samples were in-situ annealed at 600 °C for 1 hour.
CFAS alloy and Au pellets were used as targ ets for electron-beam bombardment. Figure 1(a)
shows the vibrating sample magnetometry (V SM) results, from which the saturation
magnetization ( Ms) was extracted. The measurement was also repeated at different temperatures
to extract the corresponding values of Ms for subsequent data fitting. The Ms value increases
from 1100 emu/cc at 300 K, to 1160 emu/cc at 13 K. From the VSM data, we verify a hard axis
along [100] and an easy axis along [110] , consistent with earlier reports.3,8 In addition, the -2
XRD data shown in Fig. 1(b) verified the presen ce of the characteristic (004) peak, indicating
that the CFAS film wa s at least B2-ordered.1,14 As shown in Fig. 1(c), the film was patterned into
mesas, which were integrated with asymme tric coplanar waveguides (ACPW). The ACPWs
were electrically isolated from the mesa by 50 nm of Al 2O3, which was deposited by RF
sputtering. Vector network analy zer (VNA) and PIMM techniques were used to excite and detect
ferromagnetic resonance (FMR) as well as spin waves in CFAS. The former technique allows
frequency domain measurements, while the la tter technique was us ed for time domain
measurements. The experimental setup enable d the excitation of Damon-Eshbach-type (DE)
modes, as the external magnetic field was applied along the ACPWs, shown in Fig. 1(c).16
A VNA was connected to the AC PWs, and reflection as well as transmission signals were
measured to study the FMR and spin wave pr opagation, respectively. Background subtraction
was performed to obtain the resonance peaks. Figure 2(a) shows th e FMR frequency as a
function of applied magnetic fiel d at different temperatures, with the corresponding fits using the
Kittel formula,17
݂ൌఊ
ଶగඥሺܪܪሻሺܪܪ4ܯߨ ௦ሻ, (1)4
where f is the resonance frequency, is the gyromagnetic ratio, H is the applied magnetic field,
and Ha is the anisotropy field. The g factor, which was extr acted using the equation ߛ ൌ
2ߤ݃ߨ/݄ ,where B is the Bohr magneton and h is Planck’s constant, was found to be 2.03 0.02,
while Ha generally decreased from 130 Oe at 13 K to 70 Oe at 300 K. The ( g – 2) value is lower
than those of Co and Ni, but comparable to th ose of other Heusler comp ounds, such as CMS and
Co2MnAl (CMA).18 The deviation of the g factor from the free electron value of 2 is correlated
with the spin-orbit interaction in a material, where a smaller deviation indicates weaker spin-
orbit interaction, and lower .18 The inset of Fig. 2(a) show s the resonance frequency at H =
1040 Oe as a function of temperature, with a Bloch fitting, indicating a Curie temperature of
approximately 1000 K. The Bloch fitting was perf ormed by substituting the following equation19
into Eq. (1):
ܯ௦ൌܽ൫1െܽ ଵܶଷ/ଶെܽଶܶହ/ଶെܽଷܶ/ଶ൯, ( 2 )
where T is temperature, and a0, a1, a2, and a3 are positive coefficients.
As shown in Fig. 2(b), the extracted FMR field linewidths were fitted with the linear
equation20 ΔH ൌ ΔH0 4αf/, where H is the field linewidth and H0is the extrinsic field
linewidth . This enabled the extraction of the intrinsic Gilbert damping ( ) from the fit line slopes.
Figure 2(c) shows that increases as the temperature decreases. The value of at room
temperature was found to be 0.0025, which is comp arable with the previously reported room
temperature value for CFAS.8 The trend of with temperature is consistent with previous first-
principle calculations,9 and could be attributed to longe r electron scattering time at lower
temperatures, due to a reduction in phonon-elec tron scattering. Consequently, the angular
momentum transfer at low temperatures occu rs predominantly by direct damping through
intraband transitions.11 Similar temperature-dependence of has also been observed 5
experimentally. For example, the of Co 20Fe60B20 has been found to increase by a factor of 3
from 0.007 at 300 K, to 0.023 at 5 K.11 This is comparable to our results, where increases by a
factor of almost 6 from 0.0025 at 300 K to 0.014 at 13 K. It shoul d be noted that spin pumping
into the Au cap layer could have contributed to the measured resonance linewidth, thus causing
the extracted to be higher than its actual value ( CFAS). Thus, = CFAS + sp, where sp
denotes the spin pumping c ontribution to the damping.21 While an investigation of sp in the
CFAS/Au system would exceed the scope of this work, sp values for a Fe/Au system21,22 have
nonetheless been included in Fig. 2(c) to prov ide a gauge of the temperature dependence of sp,
as well as a rough estimation of the magnitude of sp in the CFAS/Au system. Figure 2(d) shows
an increase in H0 as temperature increases. This could be due to the effect of temperature on the
interaction between magnetic precession and sample inhomogeneities, or on magnon-magnon
scattering, as these f actors contribute to H0.20,23 In both Fig. 2(c) and 2(d), room temperature
values of and H0 for sputter-deposited CFAS were in cluded, for comparison with the MBE
sample. It can be seen that the is higher for the sputter-deposite d sample, consistent with lower
half-metallic character due to greater structural disorder.1,6
We have also measured the time domain PI MM data at 300 K as shown in Fig. 3(a),
where SW15 and SW30 denote edge-to-edge signal line separations of 15 and 30 m,
respectively. The width of all the signal lines was fixed at 10 m. Using the temporal positions
of the centers of the Gaussian wavepackets ( t15 and t30, respectively), the group velocity ( vg) was
calculated with the equation5,24 vg ൌ 1 5 m/ሺt30 – t15ሻ. Fast Fourier transform (FFT) was
performed on the PIMM data, as shown in Fig. 3(b), verifying the presence of multiple modes,
where each mode manifested as a dark-light-dark oscillation. The vg decreases from 26 km/s at
50 Oe to 11 km/s at 370 Oe, as shown in Fig. 3( c). Moreover, from the VNA transmission data, 6
which is another measure of spin wave propagation, attenuation length ( ) and were extracted
as a function of magnetic fiel d at room temperature, using the method reported elsewhere.24,25
The spin wave amplitude was extracted from Lorentzian fittings of the VNA transmission
resonance peaks, which were measured using wa veguides with different center-to-center signal
line-signal line (S-S) spacings. Then, was extracted using the equation24 A1expሺ x1/ሻ ൌ
A2expሺ x2/ሻ, where A1 and A2 denote the measured spin wave amplitudes, while x1 and x2 denote
the different S-S spacings for the corresponding waveguides. The decreases from 23.3 m at
460 Oe, to 12.1 m at 1430 Oe, as shown in Fig. 3(c). Using the following equation,25 was
calculated at different magnetic fields, as shown in Fig. 3(d)
ߙൌఊሺଶగெೞሻమௗషమೖ
ଶగሺுାଶగெ ೞሻ ( 3 )
where d is the film thickness and k is a spin wave vector, which can be estimated by 2 /(signal
line width).5 The values (0.0026 – 0.0031) are consistent with the room temperature value
(0.0025) obtained from the FMR measurements.
As shown in Fig. 3(c), and vg decreased as the applie d magnetic field increased,
consistent with previous experimental11 and theoretical5 results. This trend can be understood in
terms of the following equation,5,26
ݒൌఊమఓబమெೞమௗ
଼గ݁ିଶௗ, ( 4 )
where μ0 is the permeability of free space. As the a pplied magnetic field increases, the resonance
frequency increases, thus vg decreases. In addition, for a given value of , the magnetic
precession will decay within a ce rtain amount of time. Hence, the distance travelled by the
precessional disturbance within that amount of time depends on its propagation velocity, vg.
Consequently, the higher the vg, the longer the distance travelled, and thus, the higher the .The 7
obtained values of and vg are comparable to those of other ferromagnetic materials for the
Damon-Eshbach surface spin wave mode.5,11,12 For example, of 18.95 m was extracted for
CFA by micromagnetic simulations,5 while and vg values as high as 23.9 m and 25 km/s,
respectively, were experimentally observed in CoFeB.11 Furthermore, as high as 16.7 m was
experimentally observed in CMFS.12
In conclusion, we have found a decreasing trend of with increasing temperature for
MBE-grown Co 2FeAl 0.5Si0.5, in the temperature range of 13 – 300 K. The room temperature
value of was found to be 0.0025, which was approximately 6 times lower than that at 13 K.
We have also investigated vg and in CFAS, obtaining values as high as 26 km/s and 23.3 m
respectively, at room temperature.
This work was supported by the Singa pore NRF CRP Award No. NRF-CRP 4-2008-06.
8
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10
Figure captions
FIG. 1. (a) Normalized magnetic hysteresis data along the crystallographic hard [100] and easy
[110] axes of MBE-grown CFAS. Ms is the saturation magnetization. (b) -2 XRD data of the
MBE-grown CFAS sample. (c) Optical microscopy image of the CFAS me sa integrated with
asymmetric coplanar waveguides (ACPW). The or ientation of the in-plane magnetic field ( H) is
indicated.
FIG. 2. (a) FMR frequency at different magnetic fields. Inset: FMR frequency for a fixed field
(1040 Oe) at different temperat ures. (b) Resonance linewidth as a function of frequency at
different temperatures (symbols), with correspo nding fit lines. (c) Gilbert damping parameter ( )
at different temperatures. The spin pumping contribution to damping ( sp) for a Fe/Au system
has been included, where all sp values were obtained from literature, except those at 13 K and
room temperature, which were obtained by extra polating the literature valu es. (d) Extrinsic field
linewidth (H0) at different temperatures.
FIG. 3. (a) PIMM data from two differ ent signal line-signal line spacings at H = 50 Oe for 300 K.
(b) Fast Fourier transform (FFT) of room temp erature PIMM data. (c) Room temperature group
velocity ( vg, axis: left and bottom) and attenuation length ( , axis: top and right) at different
magnetic fields. (d) Room temperature Gilbert damping parameter ( ) at different magnetic
fields.
11
FIG. 1
12
FIG. 2 800 1000 1200 140091011121314
(d) (c)(b) (a)
13 K
90 K
210 K
294 KResonance frequency (GHz)
Magnetic field (Oe)10 11 12 13 1450100150200250300
90 K
210 K
294 K H (Oe)
Resonance frequency (GHz)
0 100 200 3000.0000.0050.0100.0150.020
Temperature (K) MBE
Sputtered
sp Au-Fe
0 100 200 300050100150200 H0 (Oe)
Temperature (K) MBE
Sputtered0 200 800 100012000412 fR (GHz)
Temperature (K)13
FIG. 3
|
2201.06060v2.Ferromagnetic_resonance_modulation_in__d__wave_superconductor_ferromagnetic_insulator_bilayer_systems.pdf | Ferromagnetic resonance modulation in d-wave superconductor/ferromagnetic
insulator bilayer systems
Yuya Ominato,1Ai Yamakage,2Takeo Kato,3and Mamoru Matsuo1, 4, 5, 6
1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China.
2Department of Physics, Nagoya University, Nagoya 464-8602, Japan
3Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan
4CAS Center for Excellence in Topological Quantum Computation,
University of Chinese Academy of Sciences, Beijing 100190, China
5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan
6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
(Dated: May 6, 2022)
We investigate ferromagnetic resonance (FMR) modulation in d-wave superconductor
(SC)/ferromagnetic insulator (FI) bilayer systems theoretically. The modulation of the Gilbert
damping in these systems re
ects the existence of nodes in the d-wave SC and shows power-law
decay characteristics within the low-temperature and low-frequency limit. Our results indicate the
e ectiveness of use of spin pumping as a probe technique to determine the symmetry of unconven-
tional SCs with high sensitivity for nanoscale thin lms.
I. INTRODUCTION
Spin pumping (SP)1,2is a versatile method that can
be used to generate spin currents at magnetic junctions.
While SP has been used for spin accumulation in vari-
ous materials in the eld of spintronics3,4, it has recently
been recognized that SP can also be used to detect spin
excitation in nanostructured materials5, including mag-
netic thin lms6, two-dimensional electron systems7{9,
and magnetic impurities on metal surfaces10. Notably,
spin excitation detection using SP is sensitive even for
such nanoscale thin lms for which detection by con-
ventional bulk measurement techniques such as nuclear
magnetic resonance and neutron scattering experiment is
dicult.
Recently, spin injection into s-wave superconductors
(SCs) has been a subject of intensive study both theoret-
ically11{20and experimentally21{34. While the research
into spin transport in s-wave SC/magnet junctions is ex-
pected to see rapid development, expansion of the devel-
opment targets toward unconventional SCs represents a
fascinating research direction. Nevertheless, SP into un-
conventional SCs has only been considered in a few recent
works35,36. In particular, SP into a d-wave SC, which is
one of the simplest unconventional SCs that can be real-
ized in cuprate SCs37, has not been studied theoretically
to the best of our knowledge, although experimental SP
in ad-wave SC has been reported recently38.
In this work, we investigate SP theoretically in a bi-
layer magnetic junction composed of a d-wave SC and
a ferromagnetic insulator (FI), as shown in Fig. 1. We
apply a static magnetic eld along the xdirection and
consider the ferromagnetic resonance (FMR) experiment
of the FI induced by microwave irradiation. In this setup,
the FMR linewidth is determined by the sum of the in-
trinsic contribution made by the Gilbert damping of the
bulk FI and the interface contribution, which originates
from the spin transfer caused by exchange coupling be-
Microwavex yz
Spin current
Ferromagnetic resonanceInteractionFIG. 1. Schematic of the d-wave SC/FI bilayer system. The
two-dimensional d-wave SC is placed on the FI. Precessional
motion of the magnetization is induced by microwave irradia-
tion. The spins are injected and the magnetization dynamics
are modulated because of the interface magnetic interaction.
tween thed-wave SC and the FI. We then calculate the
interface contribution to the FMR linewidth, which is
called the modulation of the Gilbert damping hereafter,
using microscopic theory based on the second-order per-
turbation39{41. We show that the temperature depen-
dence of the modulation of the Gilbert damping exhibits
a coherent peak below the transition temperature that
is weaker than that of s-wave SCs11,13{15. We also show
that because of the existence of nodes in the d-wave SCs,
the FMR linewidth enhancement due to SP remains even
at zero temperature.
The paper is organized as follows. In Sec. II, we in-
troduce the model Hamiltonian of the SC/FI bilayer sys-
tem. In Sec. III, we present the formalism to calculate
the modulation of the Gilbert damping. In Sec. IV, we
present the numerical results and explain the detailed
behavior of the modulation of the Gilbert damping. In
Sec. V, we brie
y discuss the relation to other SC sym-
metries, the proximity e ect, and the di erence between
d-wave SC/FI junctions and d-wave SC/ferromagnetic
metal junctions. We also discuss the e ect of an e ectivearXiv:2201.06060v2 [cond-mat.mes-hall] 5 May 20222
Zeeman eld due to the exchange coupling. In Sec. VI,
we present our conclusion and future perspectives.
II. MODEL
The model Hamiltonian of the SC/FI bilayer system
His given by
H=HFI+HdSC+HT: (1)
The rst term HFIis the ferromagnetic Heisenberg
model, which is given by
HFI= JX
hi;jiSiSj ~
hdcX
jSx
j; (2)
whereJ>0 is the exchange coupling constant, hi;ji
represents summation over all the nearest-neighbor sites,
Sjis the localized spin at site jin the FI,
is the gy-
romagnetic ratio, and hdcis the static magnetic eld.
The localized spin Sjis described as shown using the
bosonic operators bjandby
jof the Holstein-Primako
transformation42
S+
j=Sy
j+iSz
j=
2S by
jbj1=2
bj; (3)
S
j=Sy
j iSz
j=by
j
2S by
jbj1=2
; (4)
Sx
j=S by
jbj; (5)
where we require [ bi;by
j] =i;jto ensure that S+
j,S
j,
andSx
jsatisfy the commutation relation of angular mo-
mentum. The deviation of Sx
jfrom its maximum value S
is quanti ed using the boson particle number. It is conve-
nient to represent the bosonic operators in the reciprocal
space as follows
bk=1p
NX
je ikrjbj; by
k=1p
NX
jeikrjby
j;(6)
whereNis the number of sites. The magnon opera-
tors with wave vector k= (kx;ky;kz) satisfy [bk;by
k0] =
k;k0. Assuming that the deviation is small, i.e., that
hby
jbji=S 1, the ladder operators S
jcan be approx-
imated asS+
j(2S)1=2bjandS
j(2S)1=2by
j, which
is called the spin-wave approximation. The Hamiltonian
HFIis then written as
HFIX
k~!kby
kbk; (7)
where we assume a parabolic dispersion ~!k=Dk2+
~
hdcwith a spin sti ness constant Dand the constant
terms are omitted.
The second term HdSCis the mean- eld Hamiltonian
for the two-dimensional d-wave SC, and is given by
HdSC=X
k(cy
k";c k#)
k k
k kck"
cy
k#
;(8)wherecy
kandckdenote the creation and annihilation
operators, respectively, of the electrons with the wave
vectork= (kx;ky) and thexcomponent of the spin
=";#, andk=~2k2=2m is the energy of conduc-
tion electrons measured from the chemical potential .
We assume that the d-wave pair potential has the form
k= cos 2 kwith the phenomenological temperature
dependence
= 1:76kBTctanh
1:74r
Tc
T 1!
; (9)
where k= arctan(ky=kx) denotes the azimuth angle of
k. Using the Bogoliubov transformation given by
ck"
cy
k#
=
uk vk
vkuk
k"
y
k#
; (10)
where
y
kand
kdenote the creation and annihilation
operators of the Bogoliubov quasiparticles, respectively,
andukandvkare given by
uk=r
Ek+k
2Ek; vk=r
Ek k
2Ek; (11)
with the quasiparticle energy Ek=p
2
k+ 2
k, the mean-
eld Hamiltonian can be diagonalized as
HdSC=X
k(
y
k";
k#)
Ek 0
0 Ek
k"
y
k#
:(12)
The density of states of the d-wave SC is given by43
D(E)=Dn= Re2
K2
E2
; (13)
whereDn=Am= 2~2is the density of states per spin of
the normal state, Ais the system area, and K(x) is the
complete elliptic integral of the rst kind in terms of the
parameterx, where
K(x) =Z=2
0d p
1 xcos2 : (14)
D(E) diverges at E= = 1 and decreases linearly when
E= 1 because of the nodal structure of k. The
density of states for an s-wave SC, in contrast, has a
gap forjEj<. This di erence leads to distinct FMR
modulation behaviors, as shown below.
The third term HTdescribes the spin transfer between
the SC and the FI at the interface
HT=X
q;k
Jq;k+
qS
k+J
q;k
qS+
k
; (15)
whereJq;kis the matrix element of the spin transfer pro-
cesses, and
q= (y
qiz
q)=2 andS
k=Sy
kiSz
kare3
(a) Spin transfer process (b) Self-energy
Jq,kJ*q,k
p/uni2191p+q/uni2193
p/uni2191p+q/uni2193
−k −k
/uni03A3R
k(/uni03C9)=
FIG. 2. (a) Diagrams of the bare vertices of the spin transfer
processes at the interface. (b) Self-energy within the second-
order perturbation.
the Fourier components of the ladder operators and are
given by
+
q=X
pcy
p"cp+q#;
q=X
pcy
p+q#cp"; (16)
S
k(2S)1=2by
k; S+
k(2S)1=2bk: (17)
Using the expressions above, HTcan be written as
HTp
2SX
p;q;k
Jq;kcy
p"cp+q#by
k+J
q;kcy
p+q#cp"b k
:
(18)
The rst (second) term describes a magnon emission
(absorption) process accompanying an electron spin-
ip
from down to up (from up to down). A diagrammatic
representation of the interface interactions is shown in
Fig. 2 (a).
In this work, we drop a diagonal exchange coupling at
the interface, whose Hamiltonian is given as
HZ=X
q;kJq;kx
qSx
k: (19)
This term does not change the number of magnons in
the FI and induces an e ective Zeeman eld on electrons
in the two-dimensional d-wave SC. We expect that this
term does not a ect our main result because the coupling
strength is expected to be much smaller than the super-
conducting gap and the microwave photon energy. We
will discuss this e ect in Sec. V brie
y.
III. FORMULATION
The coupling between the localized spin and the mi-
crowave is given by
V(t) = ~
hacX
i(Sy
icos!t Sz
isin!t); (20)wherehacis the amplitude of the transverse oscillating
magnetic eld with frequency !. The microwave irra-
diation induces the precessional motion of the localized
spin. The Gilbert damping constant can be read from
the retarded magnon propagator de ned by
GR
k(t) =1
i~(t)h[S+
k(t);S
k(0)]i; (21)
where(t) is a step function. Second-order perturbation
calculation of the magnon propagator with respect to the
interface interaction was performed and the expression of
the self-energy was derived in the study of SP39{41. Fol-
lowing calculation of the second-order perturbation with
respect to Jq;k, the Fourier transform of the retarded
magnon propagator is given by
GR
k(!) =2S=~
! !k+i ! (2S=~)R
k(!); (22)
where is the intrinsic Gilbert damping constant that
was introduced phenomenologically44{46. The diagram
of the self-energy R
k(!) is shown in Fig. 2 (b). From the
expressions given above, the modulation of the Gilbert
damping constant is given by
= 2SIm R
k=0(!)
~!: (23)
Within the second-order perturbation, the self-energy
is given by
R
k(!) = X
qjJq;kj2 R
q(!); (24)
where R
q(!) represents the dynamic spin susceptibility
of thed-wave SC de ned by
R
q(!) = 1
i~Z
dtei(!+i0)t(t)h[+
q(t);
q(0)]i:(25)
Substituting the ladder operators in terms of the Bogoli-
ubov quasiparticle operators into the above expression
and performing a straightforward calculation, we then
obtain434
R
q(!) = X
pX
=1X
0=1(p+Ep)(p+q+0Ep+q) + pp+q
4Ep0Ep+qf(Ep) f(0Ep+q)
Ep 0Ep+q+~!+i0; (26)
wheref(E) = 1=(eE=kBT+ 1) is the Fermi distribution
function.
In this paper, we focus on a rough interface modeled in
terms of the mean J1and variance J22of the distribution
ofJq;k(see Appendix A for detail). The con gurationally
averaged coupling constant is given by
jJq;k=0j2=J12q;0+J22: (27)
In this case, is written as
=2SJ12
~!Im R
q=0(!) +2SJ22
~!X
qIm R
q(!):(28)
The rst term represents the momentum-conserved spin-
transfer processes, which vanish as directly veri ed from
Eq. (26). This vanishment always occurs in spin-singlet
SCs, including sandd-wave SCs, since the spin is
conserved43. Consequently, the enhanced Gilbert damp-
ing is contributed from spin-transfer processes induced
by the roughness proportional to the variance J22
=2SJ22
~!X
qIm R
q(!): (29)
The wave number summation can be replaced as
X
q()!Dn
2Z1
1dZ2
0d (): (30)
Changing the integral variable from toEand substi-
tuting Eq. (26) into Eq. (29), we nally obtain
=2SJ 22D2
n
~!Z1
1dE[f(E) f(E+~!)]
Re2
K2
E2
Re2
K2
(E+~!)2
:
(31)
Note that the coherence factor vanishes in the above ex-
pression by performing the angular integral. The en-
hanced Gilbert damping in the normal state is given by
n= 2SJ 22D2
n; (32)
for the lowest order of !. This expression means that
is proportional to the product of the spin-up and spin-
down densities of states at the Fermi level7.IV. GILBERT DAMPING MODULATION
Figure 3 shows the enhanced Gilbert damping constant
as a function of temperature for several FMR frequen-
cies, where is normalized with respect to its value in
the normal state. We compare in thed-wave SC shown
in Figs. 3 (a) and (c) to that in the s-wave SC shown in
Figs. 3 (b) and (d). The enhanced Gilbert damping for
thes-wave SC is given by13
=2SJ 22D2
n
~!Z1
1dE[f(E) f(E+~!)]
1 +2
E(E+~!)
RejEjp
E2 2
Re"
jE+~!jp
(E+~!)2 2#
;
(33)
where the temperature dependence of is the same as
that for the d-wave SC, given by Eq. (9). Note that
the BCS theory we are based on, which is valid when
the Fermi energy is much larger than , is described by
only some universal parameters, including Tc, and inde-
pendent of the detail of the system in the normal state.
When ~!=k BTc= 0:1, shows a coherence peak just
below the transition temperature Tc. However, the co-
herence peak of the d-wave SC is smaller than that of
thes-wave SC. Within the low temperature limit, in
thed-wave SC shows power-law decay behavior described
by /T2. This is in contrast to in thes-wave SC,
which shows exponential decay. The di erence in the low
temperature region originates from the densities of states
in thed-wave ands-wave SCs, which have gapless and full
gap structures, respectively. When the FMR frequency
increases, the coherence peak is suppressed, and de-
cays monotonically with decreasing temperature. has
a kink structure at ~!= 2, where the FMR frequency
corresponds to the superconducting gap.
Figure 4 shows atT= 0 as a function of !. In
thed-wave SC, grows from zero with increasing !as
/!2. When the value of becomes comparable to
the normal state value, the increase in is suppressed,
and then approaches the value in the normal state.
In contrast, in thes-wave SC vanishes as long as the
condition that ~! < 2 is satis ed. When ~!exceeds
2, then increases with increasing !and approaches
the normal state value. This di erence also originates
from the distinct spectral functions of the d-wave and
s-wave SCs. Under the low temperature condition that
T= 0:1Tc, the frequency dependence of does not5
0.1 5.0
T/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.01.2
0.20.40.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1n0.1
0.5
1.0
1.5
2.03.04.05.0/uni210F/uni03C9/kBTc
T/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.02.0
0.51.01.5/uni03B4/uni03B1//uni03B4/uni03B1n(a) d-wave (b) s-wave
(c) d-wave (d) s-wave
T/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.01.2
0.20.40.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.02.0
0.51.01.5/uni03B4/uni03B1//uni03B4/uni03B1n0.1
0.5
1.0
1.5
2.03.04.05.0
FIG. 3. Enhanced Gilbert damping as a function of tem-
peratureT. The left panels (a) and (c) show in thed-
wave SC in the low and high frequency cases, respectively.
The right panels (b) and (d) show in thes-wave SC in the
low and high frequency cases, respectively. nis the normal
state value.
change for the s-wave SC, and it only changes in the
low-frequency region where ~!.kBTfor thed-wave SC
(see the inset in Fig. 4).
V. DISCUSSION
We discuss the modulation of the Gilbert damping
in SCs with nodes other than the d-wave SC consid-
ered in this work. Other SCs with nodes are expected
to exhibit the power-law decay behavior within the low-
temperature and low-frequency limit as the d-wave SCs.
However, the exponent of the power can di er due to
the di erence of the quasiparticle density of states. Fur-
thermore, in the p-wave states, two signi cant di erences
arise due to spin-triplet Cooper pairs. First, the uni-
form spin susceptibility R
q=0(!) can be nite in the spin-
triplet SCs because the spin is not conserved. Second,
the enhanced Gilbert damping exhibits anisotropy and
the value changes by changing the relative angle between
the Cooper pair spin and localized spin35.
In our work, proximity e ect between FIs and SCs
was not taken into account because the FMR modula-
tion was calculated by second-order perturbation based
on the tunnel Hamiltonian. Reduction of superconduct-
/uni210F/uni03C9=2/uni0394(T =0)
s-waved-wave
00.2
2 4 6 81.2
0.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1n
/uni210F/uni03C9/kBTc0.4
0.0
100 10.05
0.000.1T/Tc=0.0FIG. 4. Enhanced Gilbert damping as a function of fre-
quency!. The vertical dotted line indicates the resonance
frequency ~!= 2(T= 0). The inset shows an enlarged
view in the low-frequency region.
ing gap due to the proximity e ect15and e ect of the
subgap Andreev bound states that appear in the ab-axis
junction47would also be an important problem left for
future works.
Physics of the FMR modulation for d-wave
SC/ferromagnetic metal junctions is rather di er-
ent from that for d-wave SC/FI junctions. For d-wave
SC/ferromagnetic metal junctions, spin transport is
described by electron hopping across a junction and
the FMR modulation is determined by the product
of the density of states of electrons for a d-wave SC
and a ferromagnetic metal. (We note that the FMR
modulation is determined by a spin susceptibility of
d-wave SC, which in general includes di erent informa-
tion from the density of states of electrons.) While the
FMR modulation is expected to be reduced below a SC
transition temperature due to opening an energy gap, its
temperature dependence would be di erent from results
obtained in our work.
Finally, let us discuss e ect of the diagonal exchange
coupling given in Eq. (19) (see also the last part of
Sec. II). This term causes an exchange bias, i.e., an e ec-
tive Zeeman eld on conduction electrons in the d-wave
SC, which is derived as follows. First, the x-component
of the localized spin is approximated as hSx
ji S,
which gives Sx
kSp
Nk;0. Next, the matrix element
Jq;k=0is replaced by the con gurationally averaged value
Jq;k=0=J1q;0. Consequently, the e ective Zeeman
eld term is given by
HZEZX
p(cy
p"cp" cy
p#cp#); (34)
where we introduced a Zeeman energy as EZ=J1Sp
N.
This term induces spin splitting of conduction electrons6
in thed-wave SC and changes the spin susceptibility of
the SC. The spin-splitting e ect causes a spin excitation
gap and modi es the frequency dependence in Fig. 4, that
will provide additional information on the exchange cou-
pling at the interface. In actual experimental setup for
thed-wave SC, however, the Zeeman energy, that is less
than the exchange bias between a magnetic insulator and
a metal, is estimated to be of the order of 0 :1 erg=cm2.
This leads to the exchange coupling that is much less
thanJ0:1 meV for YIG48. Therefore, we expect that
the interfacial exchange coupling is much smaller than
the superconducting gap and the microwave photon en-
ergy though it has not been measured so far. A detailed
analysis for this spin-splitting e ect is left for a future
problem.
VI. CONCLUSION
In this work, we have investigated Gilbert damping
modulation in the d-wave SC/FI bilayer system. The
enhanced Gilbert damping constant in this case is pro-
portional to the imaginary part of the dynamic spin sus-
ceptibility of the d-wave SC. We found that the Gilbert
damping modulation re
ects the gapless excitation that
is inherent in d-wave SCs. The coherence peak is sup-
pressed in the d-wave SC when compared with that in
thes-wave SC. In addition, the di erences in the spec-
tral functions for the d-wave ands-wave SCs with gap-
less and full-gap structures lead to power-law and ex-
ponential decays within the low-temperature limit, re-
spectively. Within the low-temperature limit, in the
d-wave SC increases with increasing !, while in the
s-wave SC remains almost zero as long as the excitation
energy ~!remains smaller than the superconducting gap
2.
Our results illustrate the usefulness of measurement of
the FMR modulation of unconventional SCs for determi-
nation of their symmetry through spin excitation. We
hope that this fascinating feature will be veri ed exper-
imentally in d-wave SC/FI junctions in the near future.
To date, one interesting result of FMR modulation in
d-wave SC/ferromagnetic metal structures has been re-
ported38. This modulation can be dependent on metallic
states, which are outside the scope of the theory pre-
sented here. The FMR modulation caused by ferromag-
netic metals is another subject that will have to be clar-
i ed theoretically in future work.
Furthermore, our work provides the most fundamental
basis for application to analysis of junctions with vari-
ous anisotropic SCs. For example, some anisotropic SCs
are topological and have an intrinsic gapless surface state.
SP can be accessible and can control the spin excitation of
the surface states because of its interface sensitivity. The
extension of SP to anisotropic and topological supercon-
ductivity represents one of the most attractive directions
for further development of superconducting spintronics.
Acknowledgments.| This work is partially supportedby the Priority Program of Chinese Academy of Sciences,
Grant No. XDB28000000. We acknowledge JSPS KAK-
ENHI for Grants (No. JP20H01863, No. JP20K03835,
No. JP20K03831, No. JP20H04635, and No.21H04565).
Appendix A: Magnon self-energy induced by a
rough interface
The roughness of the interface is taken into account
as an uncorrelated (white noise) distribution of the ex-
change couplings35, as shown below. We start with an
exchange model in the real space
Hex=X
jZ
d2rJ(r;rj)(r)Sj
=X
q;kJq;kqSk: (A1)
The spin density (r) in the SC and the spin Sjin the
FI are represented in the momentum space as
(r) =1
AX
qeiqrq; (A2)
Sj=1p
NX
keikrjSk; (A3)
whereAdenotes the area of the system and Nis the
number of sites. The exchange coupling constant is also
obtained to be
Jq;k=1
Ap
NX
jZ
d2rei(qr+krj)J(r;rj): (A4)
The exchange model Hexis decomposed into the spin
transfer term HTand the e ective Zeeman eld term HZ
asHex=HT+HZ.
Now we consider the roughness e ect of the interface.
Uncorrelated roughness is expressed by the mean J1and
varianceJ22as
1p
NX
jJ(r;rj) =J1; (A5)
1
NX
jj0J(r;rj)J(r0;rj0) J12=J22A2(r r0);(A6)
whereOis the con gurational average of Oover the
roughness. The above expressions lead to the con gu-
rationally averaged self-energy
R
k=0(!) = X
qjJq;k=0j2 R
q(!)
= J12 R
q=0(!) J22X
q R
q(!); (A7)
which coincides with the model Eq. (27) in the main text.
This model provides a smooth connection between the7
specular (J12 R
q=0) and di use ( J22P
q R
q) limits. The
uncorrelated roughness case introduced above is a simplelinear interpolation of the two. Extensions to correlated
roughness can be made straightforwardly.
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2111.15142v1.First_and_second_order_magnetic_anisotropy_and_damping_of_europium_iron_garnet_under_high_strain.pdf | 1
First and second order magnetic anisotropy and damping of
europium iron garnet under high strain
Víctor H. Ortiz1, Bassim Arkook1, Junxue Li1, Mohammed Aldosary1, Mason Biggerstaff1, Wei
Yuan1, Chad Warren2, Yasuhiro Kodera2, Javier E. Garay2, Igor Barsukov1*, and Jing Shi1*
1Department of Physics and Astronomy, University of California, Riverside, CA 92521,
USA
2 Department of Mechanical and Aerospace Engineering, University of California, San
Diego, CA 92093, USA
Understanding and tailoring static and dynamic properties of magnetic insulator thin films
is important for spintronic device applications. Here, we grow atomically flat epitaxial europium
iron garnet (EuIG) thin films by pulsed laser deposition on (111) -oriented garnet sub strates with a
range of lattice parameters. By controlling the lattice mismatch between EuIG and the substrates,
we tune the strain in EuIG films from compressive to tensile regime, which is characterized by X -
ray diffraction. Using ferromagnetic resonance , we find that in addition to the first -order
perpendicular magnetic anisotropy which depends linearly on the strain, there is a significant
second -order one that has a quadratic strain dependence. Inhomogeneous linewidth of the
ferromagnetic resonance inc reases notably with increasing strain, while the Gilbert damping
parameter remains nearly constant (≈ 2× 10-2). The se results provide valuable insight into the spin
dynamics in ferrimagnetic insulators and useful guidance for material synthesis and engineer ing
of next -generation spintronics applications.
*: Corresponding authors: Igor Barsukov ( igorb@ucr.edu ) and Jing Shi ( jing.shi@ucr.edu ) 2
Ferrimagnetic insulators (FMIs) have played an important role in uncovering a series of
novel spintronic effects such as spin Seebeck effect (SSE) and spin Hall magnetoresistance (SMR).
In addition, FMI thin films have proved to be an excellent source of proximity -induced
ferromagnetism in adjacent layers (e.g., heavy metals [1], graphene [1] and topological
insulators [2]) and of pure spin currents [3–6]. FMIs have also been shown to be a superb medium
for magnon spin currents with a long decay length [7,8]. Among FMIs, rare earth iron garnets
(REIGs) in particular have a plethora of desirable properties for practical applications: high Curie
temperature (T c > 550 K), strong chemical stability, and relatively large band gaps (~ 2.8 eV).
Compared to other magnetic materials, REIGs are distinct owing to their magnetoelastic
effect with the magnetostriction coefficient ranging from -8.5×106 to +21 ×106 at room
temperature [9] and up to two orders of magnitude increases at low temperatures [10]. This unique
feature allows for tailoring ma gnetic anisotropy in REIG thin films via growth, for example, by
means of controlling lattice mismatch with substrates, film thickness, oxygen pressure, and
chemical substitution. In thin films, the magnetization usually prefers to be in the film plane due
to magnetic shape anisotropy; however, the competing perpendicular magnetic anisotropy (PMA)
can be introduced by utilizing magneto -crystalline anisotropy or interfacial strain, both of which
have been demonstrated through epitaxial growth [11–14]. In the study of Tb 3Fe5O12 (TbIG) and
Eu3Fe5O12 (EuIG) thin films, the PMA field H2ꓕ was found to be as high as 7 T under interfacial
strain [11], much stronger than the demagnetizing field. While using strain is proven to be an
effective way of manipulating magn etic anisotropy, it often comes at a cost of increasing magnetic
inhomogeneity and damping of thin films [15,16].
In this work, we investigate the effect of strain on magnetic properties of (111) -oriented
EuIG thin films for the following reasons: (1) The spin dynamics in EuIG bulk crystals is
particularly interesting but has not been studied thoroughly in the thin film form. Compared to
other REIGs, the Eu3+ ions occupying the dodecahedral sites (c -site) should have the J = 0 ground
state according to the Hund’s rules, which do not contribute to the total magnetic moment;
therefore, EuIG thin films can potentially have a ferromagnetic resonance (FMR) linewidt h as
narrow as that of Y 3Fe5O12 (YIG) [17,18] or Lu 3Fe5O12 (LuIG) [19]. In EuIG crystals, a very
narrow linewidth (< 1 Oe) [20] was indeed observed at low temperatures, but it showed a nearly
two orders of magnitude increase at high temperatures, which ra ises fundamental questions
regarding the damping mechanism responsible for this precipitous change. (2) Although it has 3
been shown that the uniaxial anisotropy can be controlled by moderate strain for different substrate
orientations and even in polycrysta lline form [21], the emergence of the higher -order anisotropy
at larger strain, despite its technological significance, has remained elusive.
We grow EuIG films by pulsed laser deposition (PLD) from a target densified by powders
synthesized using the meth od described previously [22]. The films are deposited on (111) -oriented
Gd3Sc2Ga3O12 (GSGG), Nd 3Ga5O12 (NGG), Gd 2.6Ca0.4Ga4.1Mg 0.25Zr0.65O12 (SGGG),
Y3Sc2Ga3O12 (YSGG), Gd3Ga5O12 (GGG), Tb 3Ga5O12 (TGG) and Y 3Al5O12 (YAG) single crystal
substrates, with the lattice mismatch 𝜂=𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 −𝑎𝐸𝑢𝐼𝐺
𝑎𝐸𝑢𝐼𝐺 (where 𝑎 represents the lattice parameter
of the referred material) ranging from +0.45% (GSGG) to -3.95% (YAG) in the decreasing order
(see Table I). After the standard solvent cleaning process, the substrates are annealed at 220 °C
inside the PLD chamber with the base pressure lower than 10-6 Torr for 5 hours prior to deposition .
Then the temperature is increased to ~ 600 °C in the atmosphere of 1.5 mT orr oxygen mixed with
12% (wt.) ozone for 30 minutes. A 248 nm KrF excimer pulsed laser is used to ablate the target
with a power of 156 mJ and a repetition rate of 1 Hz. We crystalize the films by ex situ annealing
at 800 °C for 200 s in a steady flow of oxygen using rapid thermal annealing (RTA) .
Reflection high energy electron diffraction (RHEED) is used to evaluate the crystalline
structural properties of the EuIG films grown on various substrates (Fig. 1a). Immediately after
the deposition, RHEED dis plays the absence of any crystalline order. After ex situ rapid thermal
annealing, all EuIG films turn into single crystals. We carry out atomic force microscopy (AFM)
on all samples and find that they show atomic flatness and good uniformity with root -mean-square
(RMS) roughness < 2 Å (Fig. 1b). In addition, we perform X -ray diffraction (XRD) on all samples
using a Rigaku SmartLab with Cu K α radiation with a Ni filter and Ge(220) mirror as
monochromators, at room temperature in 0.002° steps over the 2 range from 10° to 90° [23]. In
a representative XRD spectrum (Fig. 1c), two (444) Bragg peaks are present, one from the 50 nm
thick EuIG film and the other from the YSGG substrate, which confirms the epitaxial growth and
single crystal structure of the fi lm without evidence of any secondary phases. Other REIG films
grown under similar conditions , i.e., by PLD in oxygen mixed with ozone at ~600 °C during
followed by RTA, have shown no observable interdiffusion across the interface from high
resolution trans mission electron microscopy and energy dispersive X -ray spectroscopy (Fig. S1 ,
[24]). The EuIG Bragg peak ( a0 = 12.497 Å) is shifted with respect to the expected peak position
of unstrained bulk crystal, indicating a change in the EuIG lattice parameter pe rpendicular to the 4
surface ( aꓕ). For the example shown in Fig. 1c, the EuIG (444) peak shifts to left with respect to
its bulk value, indicating an out -of-plane tensile strain and therefore an in -plane compressive strain
in the EuIG lattice.
A common app roach for inferring the in -plane strain ε|| of thin films from the standard −2
XRD measurements involves the following equation [23],
𝜀∥= −𝑐11+2 𝑐12+4 𝑐44
2𝑐11+4 𝑐12−4 𝑐44 𝜀⊥, with 𝜀⊥=𝑎⊥−𝑎𝑜
𝑎𝑜, (1)
where a0 is the lattice parameter of the bulk material, and aꓕ can be calculated using 𝑎⊥=
𝑑ℎ𝑘𝑙√ℎ2+𝑘2+𝑙2 from the interplanar distance 𝑑ℎ𝑘𝑙 obtained from the XRD data (Fig. S 2, [25]),
and cij are the elastic stiffness constants of the crystal which in most cases can be found in the
literature [9]. However, due to the wide range of strain values studied in this work and the
possibility that the films may contain different amounts of crystalline defects, we perform
reciprocal space mapping (RSM) measurements on a subset of our EuIG samples (Fig. S 3, [26])
and compared the measured in -plane lattice parameters with the calculated ones using Eq. 1. We
observe that the average in -plane strain s measured by RSM has a systematic difference of 40%
from the calculated values based on the elastic properties (Fig. S 4, [26]). Given this nearly constant
factor for all measured films, we find that the elastic stiffness constants of our EuIG films may
deviate from the literature reported bulk values , possibly due to stochiometric deviations or slight
unit cell distortion in thin films . Here we adopt the reported lattice parameter value ( a0 = 12.497
Å) as the reference due to the difficulty of grow ing sufficiently thick, unstrained EuIG films usin g
PLD .
In the thickness -tuned magnetic anisotropy study [11], the anisotropy field in REIG films is
found to be proportional to η/(t+t o), which was attributed to the relaxation of strain as the film
thickness t increases. Here in EuIG samples with small lattice mismatch η (e.g., NGG/EuIG), the
strain is mostly preserved in 50 nm thick films (pseudomorphic regime), whereas for larger η (e.g.,
YAG/EuIG ), the lattice parameter of EuIG films shows nearly complete structural relaxation to
the bulk value. For this reason, in the samples with larger η (YAG = -3.95 %, GSGG = 0.45%),
we grow thinner EuIG films (20 nm) in order to retain a larger in -plane strain (compressive for 5
YAG, tensile for GSGG). For EuIG films gr own on TGG and GGG substrates, the paramagnetic
background of the substrates is too large to obtain a reliable magnetic moment measurement of the
EuIG films; therefore, the results of thinner films on these two substrates are not included in this
study.
Room-temperature magnetic hysteresis curves for YSGG/EuIG sample are shown in Fig. 1d
with the magnetic field applied parallel and perpendicular to the film [26]. The saturation field for
the out -of-plane loop (~1100 Oe) is clearly larger than that for the i n-plane loop, indicating that
the magnetization prefers to lie in the film plane. Moreover, since the demagnetizing field 4π Ms
(≈ 920 Oe) is less than the saturation field in the out -of-plane loop (Fig. S 5, [27]), it suggests the
presence of additional easy -plane anisotropy result ing from the magnetoelastic effect due to
interfacial strain. As shown in this example, we can qualitatively track the evolution of the
magnetic anisotropy in samples with different strains. However, this approach cannot provide a
quantitative description when high -order anisotropy contributions are involved.
To quantitatively determine magnetic anisotropy in all EuIG films, we perform polar angle
(H)-dependent FMR measurements using an X -band microwave cavity with f requency f = 9.32
GHz and field modulation. The samples are rotated from H = 0° to H = 180° in 10° steps, where
H = 90° corresponds to the field parallel to the sample plane (Fig. 2a). The spectra at Η = 0° for
all samples are displayed in Fig. 2b and show a single resonance peak which can be well fitted by
a Lorentzian derivative. Despite different strains in all s amples, the resonance field Hres is lower
for the in -plane direction ( H = 90°) than for the out -of-plane direction ( H = 0°). A quick
inspection reveals that the out -of-plane Hres shifts to larger values as η increases in the positive
direction (e.g., fro m YAG/EuIG to GSGG/EuIG), corresponding to stronger easy -plane
anisotropy. Furthermore, the Hres values at θΗ = 0° show a large spread among the samples. Fig. 2c
shows a comparison of FMR spectra at different polar angles between two representative samples:
NGG/EuIG (small η) and YAG/EuIG (large η).
Figs. 3a -c show Hres vs. θH for three representative EuIG films . To evaluate magnetic
anisotropy, we fit the data using the Smit -Beljers formalism by considering the first -order
−𝐾1cos2𝜃 and the second -order −1
2𝐾2cos4𝜃 uniaxial anisotropy energy terms [28]. From this
fitting, we extract the parameters 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠− 2𝐾1
𝑀𝑠= 4𝜋𝑀𝑠- 𝐻2⊥ and 𝐻4⊥= 2𝐾2
𝑀𝑠 (see Table
I), her e 𝐻2⊥ and 𝐻4⊥ being the first- and second - order anisotropy fields, respectively , and favoring 6
out-of-plane (in -plane) orientation of magnetization when they are positive (negative). The
spectroscopic g-factor is treated as a fitted parameter which is found as a nearly constant , g =
1.40 (Fig. S 6, [28]), in accordance to the previous results obtained by Miyadai [31]. In Figs. 3d
and 3e, we present 𝐻2⊥ and 𝐻4⊥ as functions of the measured out -of-plane strain 𝜀⊥ and in -plane
strain 𝜀∥. Clear ly, the magnitude of 4𝜋𝑀𝑒𝑓𝑓 is greater than the demagnetizing field for EuIG
4𝜋𝑀𝑠=920 𝑂𝑒; therefore, 𝐻2⊥ is negative for all samples, i.e., favoring the in -plane orientation.
As shown in Fig. 3d, |𝐻2⊥| increases linearly with increasing in -plane strain η. This is consistent
with the magnetoelastic effect in (111) -oriented EuIG films [9]. As briefly discussed earlier, due
to the constant scaling factor between the calculated and measured 𝜀∥, we rewrite t he
magnetoelastic contribution to the first -order perpendicular anisotropy as −9𝛯
3𝑀𝑠𝜀⊥, with the
parameter 𝛯 containing the information related to the magnetoelastic constant λ111 and elastic
stiffness cii. We fit the magnetoelastic equation in Ref. [11] using the parameter 𝛯 and obtain 𝛯=
−(7.06±0.95)×104 𝑑𝑦𝑛𝑒
𝑐𝑚2 from the slope. On the other hand, based on the reported literature
values ( 𝜆111=+1.8×10−6, c11 = 25.10 ×1011 dyne/cm2, c12 = 10.70 ×1011 𝑑𝑦𝑛𝑒
𝑐𝑚2, c44 = 7.62 ×1011
𝑑𝑦𝑛𝑒
𝑐𝑚2) [10], we obtain 𝛯𝑙𝑖𝑡=−6.12×104 𝑑𝑦𝑛𝑒
𝑐𝑚2. This result suggests that even though the actual
elastic properties of our EuIG films may be different from the ones reported in for EuIG crystals
due to the thin film unit cell distortion (Table S1 , [32]), the pertaining parameter 𝛯 appears to be
relatively insensitive to variations of stoichiometry . The intercept of the straight -line fit should
give the magneto -crystalline anisotropy coefficient of EuIG Kc. We find Kc = (+62.76 ± 0.18 ) ×
103 erg/cm3, which is differ ent from the previously reported values for EuIG bulk crystals in both
the magnitude and sign ( Kc = -38 × 103 erg/cm3) [31]. Similar growth -modified magneto -
crystalline anisotropy was observed in EuIG films grown with relatively lo w temperatures
(requiring post -deposition annealing to crystalize) [10]. In the absence of interfacial interdiffusion,
the anomalous anisotropy may be related to partial deviation from the chemical ordering of the
garnet structure [31].
By comparing the first - and second -order anisotropy fields 𝐻2⊥ and 𝐻4⊥ vs. 𝜀∥ plotted in
Figs. 3d and 3e, we find that the former dominates over the entire range of 𝜀∥ (except for
YAG/EuIG). In contrast to the linear dependence for 𝐻2⊥, 𝐻4⊥ can be fitted well with a quadratic
𝜀∥ dependence , which is not surprising for materials with large magnetostriction constants (such 7
as EuIG) under large strains. For relatively small 𝜀∥, the linear strain term in the magnetic
anisotropy energy dictates . For large 𝜀∥, higher -order strain terms may not be neglected. By
including the ( 𝜀∥cos2θ)2 term, we obtain excellent fitting to the FMR data, indicating that the
second -order expansion in 𝜀∥ is adequate. In contrast to 𝐻2⊥, 𝐻4⊥ is always positive, thus favoring
out-of-plane magnetization orientation. It is worth pointing out that for YAG and TGG, the
magnitude of the 𝐻2⊥ becomes comparable with that of the 𝐻4⊥, but the sign differ s. Comparison
of 𝐻4⊥ with 4𝜋𝑀𝑒𝑓𝑓 reveals that a coexistence (bi -stable) magnetic state can be realized when
𝐻4⊥>4𝜋𝑀𝑒𝑓𝑓 [31, 33 -35]. The results are summarized in Table I.
The above magnetic anisotropy energy analysis only deals with the polar angle dependence ,
but in principle, it can also vary in the film plane and therefore depend on the azimuthal angle. To
understand the latter, w e perform azimuthal angle dependent FMR measurements on all samples.
We indeed observe a six -fold in -plane anisotropy in Hres due to the crystalline symmetry of EuIG
(111). However, the amplitude of the six -fold Hres variation is less than 15 Oe, about two orders
of magnitude smaller than the average value of Hres for most samples, thus we omit the in-plane
anisotropy in our analysis.
Besides the Hres information, t he FMR spectra in Fig. 2 c reveal s significant variations in
FMR linewidth, which contains information of magnetic inhomogeneity and Gilbert damping. To
investigate these properties systematically, we perform broad -band (up to 15 GHz) FMR
measurements with m agnetic field applied in the film plane, using a coplanar waveguide setup.
From the frequency dependence of Hres, we obtain 4𝜋𝑀𝑒𝑓𝑓 and g independently via fitting the data
with the Kittel equation. These values agree very well with those previously found from the polar
angle dependence. We plot the half width at half maximum, ∆𝐻, as a function of frequency f in
Fig. 4a. While ∆𝐻 varies significantly across the samples, the data for each sample fall
approximately on a straight line and the slope of ∆𝐻 vs. 𝑓 appears to be visibly close to each other.
For a quantitative evaluation of ∆𝐻, we consider the following contributions: the Gilbert damping
∆𝐻𝐺𝑖𝑙𝑏𝑒𝑟𝑡 , two -magnon scatt ering ∆𝐻𝑇𝑀𝑆, and the inhomogeneous linewidth ∆𝐻0 [36],
∆𝐻=∆𝐻𝐺𝑖𝑙𝑏𝑒𝑟𝑡 +∆𝐻𝑇𝑀𝑆 +∆𝐻0 . (3)
8
The Gilbert term, ∆𝐻𝐺𝑖𝑙𝑏𝑒𝑟𝑡 =2𝜋𝛼𝑓
|𝛾|, depends linearly on f, where α is the Gilbert damping
parameter; the two -magnon term is described through ∆𝐻𝑇𝑀𝑆 =𝛤0𝑎𝑟𝑐𝑠𝑖𝑛 √√𝑓2+(𝑓𝑜
2)2
−𝑓𝑜
2
√𝑓2+(𝑓𝑜
2)2
+𝑓𝑜
2 [37],
where 𝛤0 denotes the magnitude of the two -magnon scattering, f0 = 2γMeff; and ∆𝐻0, the
inhomogeneous linewidth w hich is frequency independent.
By fitting Eq. (3) to the linewidth data, we obtain quantitative information on magnetic
damping through the Gilbert parameter and two -magnon scattering magnitude as well as the
magnetic inhomogeneity [39–40]. In Fig. 4a, the overall linear behavior for all samples is an
indication of a relatively small two -magnon scattering contribution ∆𝐻𝑇𝑀𝑆 which therefore may
be disregarded in the fitting process. Figs. 4b and 4c show both ∆𝐻0 and α vs. 𝜀∥. It is cl ear that
four of the samples with the smallest ∆𝐻0 (~ 10 Oe) are those with relatively low in -plane strain
(|𝜀∥|<0.30% ). In the meantime, the XRD spectra of these samples show fringes characteristic of
well conformed crystal planes (Fig. S 2), and moreover, the RSM plots (Fig. S 3) reveal a uniform
strain distribution in the films [41]. On the compressive strain side, ∆𝐻0 increases steeply to 400
Oe at 𝜀∥ ~ -0.40 %, and their XRD spectra show no fringes and the RSM graphs indicate non-
uniform strain relaxation in the samples (Figs. S2 and S3 ). In sharp contrast to the ∆𝐻0 trend, the
Gilbert damping α remains about 2 ×10-2 over the entire range of 𝜀∥, sugges ting that the intrinsic
magneti c damping of EuIG films is nearly unaffected by the inhomogeneity. In fact, the magnitude
of α is significantly larger than that of YIG [17,18] or LuIG films [19], which is somewhat
unexpected for Eu3+ in EuIG with J = 0. A possible reason for this enhance d damping is that other
valence states of Eu such as Eu2+ (J =7/2) may be present, which leads to non -zero magnetic
moments of Eu ions in the EuIG lattice and thus results in a larger damping constant, common to
other REIG with non -zero 4f -moments [42]. The X -ray photoelectron spectroscopy data taken on
YSGG(111)/EuIG(50 nm) (Fig. S7 , [43]) indicates such a possibility. While the FMR linewidth
presents large variations across the sample set, we have identified that the non-uniform strain
relaxation process caused by large lattice mismatch with the substrate is a main source of the
inhomogeneity linewidth ∆𝐻0, but it does not affect the Gilbert damp ing α. The results raise
interesting questions on the mechanisms of intrinsic damping and the origin of magnetic
inhomogeneity in EuIG thin films , both of which warrant further investigations. 9
In summary, we find that uniaxial magnetic anisotropy in PLD -grown EuIG(111) thin films
can be tuned over a wide range via magnetostriction and lattice -mismatch induced strain. The first -
order anisotropy field depends linearly on the strain and the second order anisotropy field has a
quadratic dependence. While non -uniform strain relaxation significantly increases the magnetic
inhomogeneity, the Gilbert damping remains nearly constant over a wide range of in -plane strain.
The results demonstrate broad tunab ility of magnetic properties in REIG films and provide
guidance for implementation of EuIG for spintronic applications. Further studies to elucidate the
role of Eu2+ sites in magnetic damping are called upon.
We thank Dong Yan and Daniel Borchardt for the ir technical assistance. This work was supported
as part of the SHINES, an Energy Frontier Research Center funded by the US Department of
Energy, Office of Science, Basic Energy Sciences under Award No. SC0012670. J.S.
acknowledges support by DOE BES Award No. DE -FG02 -07ER46351 and I.B. acknowledges
support by the National Science Foundation under grant number NSF -ECCS -1810541.
10
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Y. Tserk ovnyak, and K. L. Wang, Control of Spin -Wave Damping in YIG Using Spin
Currents from Topological Insulators , Phys. Rev. Appl. 11, 1 (2019).
[39] A. Etesamirad, R. Rodriguez, J. Bocanegra, R. Verba, J. Katine, I. N. Krivorotov, V.
Tyberkevych, B. Ivanov, an d I. Barsukov, Controlling Magnon Interaction by a
Nanoscale Switch , ACS Appl. Mater. Interfaces 13, 20288 (2021).
[40] I. Barsukov, H. K. Lee, A. A. Jara, Y. J. Chen, A. M. Gonçalves, C. Sha, J. A. Katine, R.
E. Arias, B. A. Ivanov, and I. N. Krivorotov, Giant Nonlinear Damping in Nanoscale
Ferromagnets , Sci. Adv. 5, 1 (2019).
[41] X. Guo, A. H. Tavakoli, S. Sutton, R. K. Kukkadapu, L. Qi, A. Lanzirotti, M. Newville,
M. Asta, and A. Navrotsky, Cerium Substitution in Yttrium Iron Garnet: Valence State,
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[43] See Supplemental Information figure S7 at http://placeholder.html for the XPS spectra and
analysis.
14
Figures
Figure SEQ Figure \* ARABIC 1 : Structural and magnetic property characterization of
EuIG 50 nm film grown on YSGG(111) substrate. (a) Reflection high energy electron
diffraction (RHEED) pattern along the direction, displaying single crystal structure after
rapid thermal anneal ing process. (b) 2 mm 2 mm atomic force microscope (AFM) surface
morphology scan, demonstrating a root -mean -square (RMS) roughness of 1.7 Å. (c)
Intensity semi -log plot of
- 2
XRD scan. The dashed line corresponds to the XRD peak
for bulk EuIG. (d) Mag netization hysteresis loops for field out -of-plane and in -plane
directions. Figure 1: Structural and magnetic property characterization of EuIG 50 nm film grown on
TGG(111) substrate . (a) Reflection high energy electron diffraction (RHEED ) pattern along the
⟨112⟩ direction, displaying single crystal structure after rapid thermal annealing process. (b) 5 mm
5 mm atomic force microscope ( AFM ) surface morphology scan, demonstrating a root-mean -
square (RMS) roughness of 1.8 Å. (c) Intensity semi -log plot of - 2 XRD scan. The dashed line
corresponds to the XRD peak for bulk EuIG. (d) Magnetization hysteresis loops for field out -of-
plane and in -plane directions. 15
Figure 2 Polar angle dependent ferromagnetic resonance (FMR). (a) Coordinate system used for
the FMR measurement. (b) Room temperature FMR derivative absorption spectra for θH = 0° (out -
of-plane configuration) for EuIG on different (111) substrates. (c) FMR derivative absorption
spectra for 50 nm EuIG grown on NGG(111) ( 𝜀∥ ≈ 0) and 20 nm EuIG on YAG(111) (𝜀∥< 0) with
polar angle θH ranging from 0° (out -of-plane) to 90° ( in-plane) at 300 K, where 𝜀∥ is in-plane strain
between the EuIG film and substrate.
16
Figure 3 Polar angle dependent ferromagnetic resonance field Hres for (a) tensile in -plane strain
(𝜀∥ > 0), (b) in -plane strain close to zero ( 𝜀∥ ≈ 0), and (c) compressive in -plane strain ( 𝜀∥ < 0). Solid
curves represent the best fitting results. In -plane strain dependence of the anisotropy fields H 2ꓕ (d)
and H 4ꓕ (e).
17
Figure 4 FMR linewidth and magnetic damping of EuIG films as a function of in -plane strain. (a)
Half width at half maximum ∆𝐻 vs. frequency f for EuIG films grown on different substrates, with
the corresponding fitting according to Eq. (3). In -plane strain depen dence of inhomogeneous
linewidth ΔH0 (b) and Gilbert parameter α (c).
18
Substrate asubstrate
(Å) η
(%) t
(nm) 𝜀∥ (%) 𝜀⊥ (%) g H2ꓕ
(Oe) H4ꓕ
(Oe) α (×10-2) ΔHo
(Oe) Γo (Oe)
GSGG 12.554 0.45 50 0.34 -0.16 1.40 -1394.2
± 44.9 339.79
± 6.59 2.46 ±
0.03 21.4 ±
1.3 2.61
25 0.46 -0.21 1.41 -1543.6
± 39.7 709.47
± 27.5 1.58 ±
0.06 10.2 ±
1.7 6.05
NGG 12.508 0.06 50 0.12 -0.06 1.38 -1224.4
± 5.7 18.34 ±
0.05 2.41
±0.01 8.9 ±
0.7 0.20
SGGG 12.480 -
0.14 50 -0.13 0.06 1.40 -909.6
± 15.2 164.8 ±
1.36 2.13 ±
0.01 5.6 ±
0.4 0.50
YSGG 12.426 -
0.57 50 -0.27 0.12 1.37 -709.4
± 22.0 377.3 ±
5.09 2.47 ±
0.03 9.9 ±
1.8 2.47
GGG 12.383 -
0.92 50 -0.45 0.21 1.38 -1015.0
± 81.3 887.2 ±
37.27 2.20 ±
0.14 412.2
± 8.4 3.35
TGG 12.355 -
1.14 50 -0.38 0.18 1.38 -393.4
± 53.6 245.0 ±
10.00 2.29 ±
0.20 253.4
± 11.8 0.20
YAG 12.004 -
3.95 20 -0.42 0.20 1.37 -36.8 ±
47.1 424.8 ±
20.91 1.86 ±
0.20 217.0
± 22.6 0.20
Table 1 Structural and magnetic parameters for the EuIG thin films grown on different substrates. |
1601.06213v1.Nonlinear_magnetization_dynamics_of_antiferromagnetic_spin_resonance_induced_by_intense_terahertz_magnetic_field.pdf | 1Nonlinear magnetization dyna mics of antiferromagnetic
spin resonance induced by inte nse terahertz magnetic field
Y Mukai1,2,4,6, H Hirori2,3,4,7, T Yamamoto5, H Kageyama2,5, and K Tanaka1,2,4,8
1 Department of Physics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto
606-8502, Japan
2 Institute for Integrated Cell-Material Scien ces (WPI-iCeMS), Kyoto University, Sakyo-ku,
Kyoto 606-8501, Japan
3 PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan
4 CREST, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan
5 Department of Energy and Hydrocarbon Chemistr y, Graduate School of Engineering, Kyoto
University, Nishikyo-ku, Kyoto 615-8510, Japan
E-mail:
6 mukai@scphys.kyoto-u.ac.jp
7 hirori@icems.kyoto-u.ac.jp
8 kochan@scphys.kyoto-u.ac.jp
We report on the nonlinear magnetization dynamics of a HoFeO
3 crystal induced by a strong
terahertz magnetic field resonantly enhanced with a split ring resonator and measured with
magneto-optical Kerr effect microscopy. The terahertz magnetic field induces a large change (~40%) in the spontaneous magnetization. The frequency of the antife rromagnetic resonance
decreases in proportion to the square of the magnetization change. A modified
Landau-Lifshitz-Gilbert equation with a phenomenological nonlinear damping term
quantitatively reproduced the nonlinear dynamics.
PACS: 75.78.Jp, 76.50.+g, 78.47.-p, 78.67.Pt
21. Introduction
Ultrafast control of magnetization dynamics by a femtosecond optical laser pulse has
attracted considerable attention from the persp ective of fundamental physics and technological
applications of magnetic recording and inform ation processing [1]. The first observation of
subpicosecond demagnetization of a fe rromagnetic nickel film demonstrated that a femtosecond
laser pulse is a powerful stimulus of ultrafast magnetization dynamics [2], and it has led to numerous theoretical and experimental inves tigations on metallic and semiconducting magnets
[3-8]. The electronic state created by the laser pulse has a strongly nonequilibrium distribution
of free electrons, which consequently leads to demagnetization or even magnetic reversal
[1,2,9-11]. However, the speed of the magnetizat ion change is limited by the slow thermal
relaxation and diffusion, and an alternative t echnique without the limits of such a thermal
control and without excessive thermal energy would be desirable.
In dielectric magnetic media, carrier heating hardly occurs, since no free electrons are present
[12]. Consequently, great effort has been devoted to clarifying the spin dynamics in magnetic
dielectrics by means of femtosecond laser pulses. A typical method for nonthermal optical
control of magnetism is the inverse Faraday effect, where circularly polarized intense laser
irradiation induces an effective magnetic field in the medium. Recently, new optical excitation
methods avoiding the thermal effect such as the ma gneto-acoustic effect is also reported [13,14].
In particular, these techniques have been used in many studies on antiferromagnetic dielectrics
because compared with ferromagne ts, antiferromagnets have inhe rently higher spin precessional
frequencies that extend into the terahertz (THz) regime [12,15]. Additionally, ultrafast
manipulation of the antiferromagnetic order parame ter may be exploited in order to control the
magnetization of an adjacent ferromagnet through the exchange interaction [16]. The THz wave
generation technique is possibly a new way of optical spin control through direct magnetic
excitation without undesirable thermal effects [17-19]. As yet however, no technique has been
successful in driving magnetic motion excited directly by a magnetic field into a nonlinear
dynamics regime that would presumably be fo llowed by a magnetization reversal [20-22].
In our previous work [23], we demonstrated that the THz magnetic field can be resonantly
enhanced with a split ring resonator (SRR) and may become a tool for the efficient excitation of
a magnetic resonance mode of antiferromagnetic dielectric HoFeO
3. We applied a Faraday
rotation technique to detect the magnetization ch ange but the observed Faraday signal averaged 3the information about inhomogeneous magnetiza tion induced by localized THz magnetic field
of the SRR over the sample thickness [23]. In th is Letter, we have developed a time-resolved
magneto-optical Kerr effect (MOKE) micr oscopy in order to access the extremely
field-enhanced region, sample surface near th e SRR structure. As a result, the magnetic
response deviates from the linear response in the strong THz magnetic field regime, remarkably
showing a redshift of the antiferromagnetic r esonance frequency that is proportional to the
square of the magnetization change. The observe d nonlinear dynamics could be reproduced with
a modified Landau-Lifshitz-Gilbert (LLG) e quation having an additional phenomenological
nonlinear damping term.
2. Experimental setup
Figure 1 shows the experimental setup of MOKE microscopy with a THz pump pulse
excitation. Intense single-cycle THz pulses were generated by optical rectification of
near-infrared (NIR) pulses in a LiNbO
3 crystal [24-26]; the maximum peak electric field was
610 kV/cm at focus. The sample was a HoFeO 3 single crystal polished to a thickness of 145 µm,
with a c-cut surface in the Pbnm setting [27]. (The x-, y-, and z-axes are parallel to the
crystallographic a-, b-, and c-axes, respectively. ) Before the THz pump excitation, we applied a
DC magnetic field to the sample to saturate its magnetization along the crystallographic c-axis. We fabricated an array of SRRs on the crystal surface by using gold with a thickness of 250 nm.
The incident THz electric field, parallel to the metallic arm with the SRR gap (the x-axis), drove
a circulating current that resulted in a strong magnetic near-field normal to the crystal surface
[23,28,29]. The SRR is essentially subwavelength LC circuit, and the current induces magnetic
field B
nr oscillating with the LC resonance frequency (the Q-factor is around 4). The right side
of the inset in figure 1 shows the spatial distribut ion of the magnetic field of the SRR at the LC
resonance frequency as calculated by the fin ite-difference time-domain (FDTD) method.
Around the corner the current density in the metal is very high, inducing the extremely
enhanced magnetic field in the HoFeO 3 [29].
At room temperature, the two magnetizations mi (i=1,2) of the different iron sublattices in
HoFeO 3 are almost antiferromagnetically aligned along the x-axis with a slight canting angle
0(=0.63°) owing to the Dzyaloshinskii fiel d and form a spontaneous magnetization MS along
the z-axis [30]. In the THz region, ther e are two antiferromagnetic resonance modes
(quasiantiferromagnetic (AF) and quasiferroma gnetic (F) mode [31]). The magnetic field Bnr 4generated along the z-axis in our setup causes AF-mode motion; as illustrated in figure 2(a), the
Zeeman torque pulls the spins along the y-ax is, thereby triggering precessional motions of mi
about the equilibrium directions. The precessional motions cause the macroscopic magnetization M=m
1+m2 to oscillate in the z-direction [32,33]. The resultant magnetization
change Mz(t) modulates the anti-symmetric off-diagonal element of the dielectric tensor
εxyaሺൌ െεyxaሻ and induces a MOKE signal (Kerr ellipticity change [34,35] (see Appendix A
for the detection scheme of the MOKE measuremen t). The F-mode oscillation is also excited by
THz magnetic field along the x or y-axis. Howe ver, the magnetization deviations associated
with the F-mode, Mx and My, do not contribute to the MOKE in our experimental geometry,
where the probe light was incident no rmal to the c-cut surface of HoFeO 3 (the xy-plane) [34,35].
In addition, the amplitude of the F-mode is much smaller than AF-mode because the F-mode
resonance frequency ( F~0.37 THz) differs from the LC resonance frequency ( LC~0.56 THz).
-10010x position (µm)
-10 0 10
y position (µm)
THz pump HoFeO3
z y
x
SRR
Objective lens
Nonpolarized
beam splitter Quarter
wave plateWollaston
prism
Lens Balanced photodiodes
Visible probe Bin Bnr
Ein
-10 0 10
y position (µm)
120
80
40
0
Figure 1. Schematic setup of THz pump-visible MOKE measurement. The left side
of the inset shows the photograph of SRR fabricated on the c-cut surface of the
HoFeO 3 crystal and the white solid line indicat es the edge of the SRR. The red soli d
and blue dashed circles indicate the probe spots for the MOKE measurement. The
right side of the inset shows the spatia l distribution of the enhancement facto r
calculated by the FDTD method, i.e., the ratio between the Fourier amplitude at LC
of the z-component of Bnr (at z=0) and the incident THz pulse Bin. 5To detect the magnetization change induced onl y by the enhanced magnetic field, the MOKE
signal just around the corner of the SRR (indicated by the red circle in figure 1’s inset), where
the magnetic field is enhanced 50-fold at the LC resonance frequency, was measured with a 400
nm probe pulse focused by an objective lens (spot diameter of ~1.5 µm). Furthermore, although
the magnetic field reaches a maximum at the surface and decreases along the z-axis with a
decay length of lTHz~5 µm, the MOKE measurement in refl ection geometry, in contrast to the
Faraday measurement in transmission [23], can evaluate the magnetization change induced only
by the enhanced magnetic field around the sample surface since the penetration depth of 400 nm
probe light for typical orthoferrites is on the orde r of tens of nm [35]. (The optical refractive
indices of rare-earth orthoferrites in th e near ultraviolet region including HoFeO 3 are similar to
each other, regardless of the rare-earth ion speci es, because it is mostly determined by the strong
optical absorption due to charge transfer and orbital promotion transitions inside the FeO 6
tetragonal cluster [35].) All experiments in this study were performed at room temperature.
3. Results and discussions
Figure 2(a) (upper panel) shows the calculated temporal magnetic waveform together with
the incident magnetic field. The maximum peak am plitude is four times that of the incident THz
pulse in the time domain and reaches 0.91 T. Th e magnetic field continues to ring until around
25 ps after the incident pulse has decayed away. The spectrum of the pulse shown in figure 2(c)
has a peak at the LC resonance frequency ( LC=0.56 THz) of the SRR, which is designed to
coincide with the resonance frequency of the AF-mode ( νAF0=0.575 THz). Figure 2(a) (lower
panel) shows the time development of the MOKE signal for the highest THz excitation
intensity (pump fluence I of 292 µJ/cm2 and maximum peak magnetic field Bmax of 0.91 T). The
temporal evolution of is similar to that of the Faraday rotation measured in the previous
study and the magnetization oscillates harmonically with a period of ~2 ps [23], implying that
the THz magnetic field coherently drives the AF-mode motion.
As shown in figure 2(b), as th e incident pump pulse intensity increases, the oscillation period
becomes longer. The Fourier transform spectra of the MOKE signals for different pump
intensities are plotted in figure 2(c). As the ex citation intensity increases, the spectrum becomes
asymmetrically broadened on the lower freque ncy side and its peak frequency becomes
redshifted. Figure 2(d) plots the center-of-mass fre quency (open circles) and the integral (closed
circles) of the power spectrum P(
) as a function of incident pulse fluence. The center 6frequency monotonically redshifts and P() begins to saturate. As shown in figure 2(c), the
MOKE spectra obtained at the center of the SRR (indicated in the inset of figure 1) does not
show a redshift even for the highest intensity excitation, suggesting that the observed redshift
originates from the nonlinearity of the precessional spin motion rather than that of the SRR
response. We took the analytic signal approach (ASA) to obtain the time development of the
instantaneous frequency (t) (figure 3(c)) and the envelope amplitude 0(t) (figure 3(d)) from
the measured magnetization change (t)=Mz(t)/|MS| (figure 3(b)) (see Appendix B for the
details of the analytic signal approach). As is described in the Appendix C, the MOKE signal
6
4
2
0
Integral of P( )
(arb. units)
300 200 100 0
Fluence (µJ/cm2)0.575
0.570
0.565Frequency (THz)1.0
0.8
0.6
0.4
0.2
0.0Intensity P( ) (arb. units)
0.60 0.58 0.56 0.54
Frequency (THz) 50%
100%
100% (x 3.7)
(center)
10%
|Bnr|2
z
m2 m1
M
x Bnr
y (a) (c)
(d) (b) -0.020.000.02∆(degrees)
40 30 20 10 0
Time (ps)1.0
0.5
0.0
-0.5B (T) Bnr
Bin (x 3)
0.08
0.06
0.04
0.02
0.00∆(degrees)
24 20 16 12 8
Time (ps)10%100%
Figure 2. (a) Upper panel: Incident magnetic field of the THz pump pulse Bin estimated by
electro-optic sampling (dashed line) and the THz magnetic near-field Bnr calculated by the
FDTD method (solid line). The illustration s hows the magnetization motion for the AF-mode.
Lower panel: The MOKE signal for a pump fluence of 292 µJ/cm2 (100%). (b) Comparison o f
two MOKE signals for different pump fluences, vertically offset for clarity. (c) The FFT powe r
spectrum of the magnetic near-field Bnr (black solid line). The spectra P() of the MOKE
signals for a series of pump fluences obtained at th e corner (solid lines) and at the center (blue
dashed circle in the inset of figure 1) for a pump fluence of 100% (dashed line). Each spectru m
of the MOKE signal is normalized by the peak amplitude at the corner for a pump fluence o f
100%. (d) Intensity dependence of the center-of-m ass frequency (open circles) and the integral
(closed circles) of the P(). 7(t) is calibrated to the magnetization change (t) by using a linear relation, i.e., (t)=g(t),
where g (=17.8 degrees−1) is a conversion coefficient. The tim e resolved experiment enables us
to separate the contributions of the applied magnetic field and magnetization change to the
frequency shift in the time domain. A comparison of the temporal profiles between the driving
magnetic field (figure 3(a)) and the frequency e volution (figure 3(c)) shows that for the low
pump fluence (10%, closed blue circles), the frequency is redshifted only when the magnetic
field persists ( t < 25 ps), and after that, it recovers to the constant AF mode frequency
(νAF0=0.575 THz). This result indicates that the signals below t = 25 ps are affected by the
persisting driving field and the redshift may orig inate from the forced oscillation. As long as the
0.575
0.570
0.565
0.560
0.555Frequency (THz)
50 40 30 20 10 0
Time (ps)-0.4-0.20.00.20.4Magnetization
change 0.5
0.0
-0.5Bnr (T)
0.4
0.3
0.2
0.1
0.0Amplitude 0 Experiment
100%
10%
Experiment
100%
Simulation
100% (1=0)
100%
10% Simulation(a)
(b)
(c)
(d)
Figure 3. (a) FDTD calculated magnetic field Bnr for pump fluence of 100%. (b) Temporal
evolution of the magnetization change obtained from the experimental data (gray circles) an d
the LLG model (red line). (c) Instantaneous frequencies and (d) envelope amplitudes fo r
pump fluences of 100% and 10% obtained by the analytic signals calculated from the
experimental data (circles) and the LLG simulation with nonlinear damping paramete r
(1=1×10−3, solid lines) and without one ( 1=0, dashed line). 8magnetic response is under the linear regime, the instantaneous frequency is independent on the
pump fluence. However, for the high pump fluenc e (100%) a redshift (a maximum redshift of
~15 GHz relative to the constant frequency νAF0) appears in the delay time ( t < 25 ps) and even
after the driving field decays away ( t > 25 ps) the frequency continues to be redshifted as long
as the amplitude of the magnetization change is large. These results suggest that the frequency
redshift in the high intensity case depends on the magnitude of the magnetization change,
implying that its origin is a nonlinear precessional spin motion with a large amplitude.
The temperature increase due to the THz absorption (for HoFeO 3 T=1.7×10−3 K, for gold
SRR T=1 K) is very small (see Appendix D). In ad dition, the thermal relaxation of the spin
system, which takes more than a nanosecond [36], is much longer than the frequency
modulation decay (~50 ps) in figure 3(c). Therefore, laser heating can be ignored as the origin of the redshift.
Figure 4 shows a parametric plot of the instantaneous frequency
(t) and envelope amplitude
0(t) for the high pump fluence (100%). The instantaneous frequency shift for t > 25 ps has a
square dependence on the amplitude, i.e., νAF=νAF0(1െCζ02). To quantify the relationship
between the redshift and magnetization change, it would be helpful to have an analytical
expression of the AF mode frequency AF as a function of the magnetization change, which is
derived from the LLG equation based on the two- lattice model [32,33]. The dynamics of the
sublattice magnetizations mi (i=1,2), as shown in the inset of figure 2(a), are described by
dRi
dt=െγ
(1+α2)ቀRi×[B(t)+Beff,i]െαRi×൫Ri×[B(t)+Beff,i]൯ቁ, (1)
where Ri=mi/m0 (m0=|mi|) is the unit directional vector of the sublattice magnetizations,
=1.76×1011 s−1T−1 is the gyromagnetic ratio, V(Ri) is the free energy of the iron spin system
normalized with m0, and Beff,i is the effective magnetic field given by െ∂V/∂Ri (i=1,2) (see
Appendix E). The second term represents the ma gnetization damping with the Gilbert damping
constant
Since Beff,i depends on the sublattice magnetizations mi and the product of these quantities
appears on the right side of Eq. (1), the LLG e quation is intrinsically nonlinear. If the angle of 9the sublattice magnetization precession is sufficien tly small, Eq. (1) can be linearized and the
two fixed AF- and F-modes for the weak excitation can be derived. However, as shown in figure
3(b), the deduced maximum magnetization change reaches ~0.4, corresponding to precession
angles of 0.25° in the xz-plane and 15° in th e xy-plane. Thus, the magnetization change might
be too large to use the linear approximation. For such a large magnetization motion, assuming
the amplitude of the F-mode is zero and =0 in Eq. (1), the AF mode frequency AF in the
nonlinear regime can be deduced as
νAF =νAF0ට1ିζ02tan2β0
K(D), ( 2 )
D =ඨ ζ02(rAF2ି1) tan2β0
1ିζ02tan2β0, ( 3 )
0.575
0.570
0.565Frequency (THz)
0.4 0.3 0.2 0.1 0.0
Amplitude 0Experiment
t > 25 ps
t < 25 ps
Analytic Solution
2nd order expansion
Figure 4. Relation between instantaneous frequency and envelope amplitude 0 obtaine d
from the magnetization change; for t < 25 ps (open circles) and for t > 25 ps (closed circles),
the analytic solution (blue line) and second orde r expansion of the analytic solution (gree n
dashed line). Errors are estim ated from the spatial inhomogeneity of the driving magnetic
field (see Appendix H). 10where K(D) is the complete elliptic integral of the first kind, rAF(≈60) is the ellipticity of the
sublattice magnetization precession trajectory of the AF-mode (see Appendix F), and 0 is the
amplitude of the (t). As shown in figure 4, the analytic solution can be approximated by the
second order expansion νAF≈νAF0(1െtan2β0(rAF2െ1)ζ024⁄) and matches the observed redshift
for t > 25 ps, showing that the frequency appr oximately decreases with the square of (t). The
discrepancy of the experimental data from the theoretical curve ( t < 25 ps) may be due to the
forced oscillation of the AF-mode caused by the driving field.
To elaborate the nonlinear damping effects, we compared the measured (t) with that
calculated from the LLG equation with the damping term. As shown in figures 3(c) and 3(d), the
experiment for the high intensity excitation devi ates from the simulation with a constant Gilbert
damping (dashed lines) even in the t > 25 ps time region, suggesting nonlinear damping
becomes significant in the large amplitude region. To describe the nonlinear damping
phenomenologically, we modified the LLG equa tion so as to make the Gilbert damping
parameter depend on the displacement of th e sublattice magnetization from its equilibrium
position, (Ri)=0+1Ri. As shown in figures 3(b)-(d), the magnetization change (t) derived
with Eq. (1) (solid line) with the damping parameters ( 0=2.27×10−4 and 1=1×10−3) nicely
reproduces the experiments for both the high (100%) and low (10%) excitations.1 These results
suggest that the nonlinear damping plays a signifi cant role in the large amplitude magnetization
dynamics. Most plausible mechanism for the nonlinear damping is four magnon scattering
process, which has been introdu ced to quantitatively evaluate the magnon mode instability of
ferromagnet in the nonlinear response regime [37].
4. Conclusions
In conclusion, we studied the nonlinear magnetization dynamics of a HoFeO 3 crystal excited
by a THz magnetic field and measured by MOKE microscopy. The intense THz field can induce
the large magnetization change (~40%), and the ma gnetization change can be kept large enough
1 The damping parameter 0 (=2.27×10−4) and conversion coefficient g (=17.8 degrees−1) are
determined from the least-squares fit of the calculated result without the nonlinear damping
parameter 1 to the experimental MOKE signal for the low pump fluence of 29.2 µJ/cm2. The
nonlinear damping parameter 1 (=1×10−3) is obtained by fitting the experimental result for the
high intensity case ( I=292 µJ/cm2) with the values of 0 and g obtained for the low excitation
experiment. The estimated value of g is consistent with the stat ic MOKE measurement; the Kerr
ellipticity induced by the spontaneous magnetization MS is ~0.05 degrees ( g~20 degrees−1). See
Appendix G for details on the static Kerr measurement. 11to induce the redshift even after the field has gone , enabling us to separate the contributions of
the applied magnetic field and ma gnetization change to the frequency shift in the time domain.
The resonance frequency decreases in proportion to the square of the magnetization change. A
modified LLG equation with a phenomenologi cal nonlinear damping term quantitatively
reproduced the nonlinear dynamics. This suggest s that a nonlinear spin relaxation process
should take place in a strongly driven regime. Th is study opens the way to the study of the
practical limits of the speed and efficiency of magnetization reversal, which is of vital
importance for magnetic recording and information processing technologies.
12Acknowledgments
We are grateful to Shintaro Takayoshi, Masah iro Sato, and Takashi Oka for their discussions
with us. This study was supported by a J SPS grants (KAKENHI 26286061 and 26247052) and
Industry-Academia Collaborative R&D grant fro m the Japan Science and Technology Agency
(JST).
13Appendix A. Detection sche me of MOKE measurement
We show the details of the detection scheme of the MOKE measurement. A probe pulse for
the MOKE measurement propagates along the z direction. By using the Jones vector [38], an electric field E
0 of the probe pulse polarized linearly along the x-axis is described as
E0 =ቀ1
0ቁ. ( A . 1 )
The probe pulse E1 reflected from the HoFeO 3 surface becomes elliptically polarized with a
polarization rotation angle and a ellipticity angle . It can be written as
E1 =R(െ߶)MR(θ)E0ൌ൬cos θ cos ߶െ ݅ sin θ sin ߶
cos θ sin ߶ ݅ sin θ cos ߶൰, (A.2)
where M is the Jones matrix describing phase retardation of the y component with
respect to the x component
M=ቀ10
0െiቁ, ( A . 3 )
and R(ψ) is the rotation matrix
R(ψ)=൬cosψ sinψ
െsinψcosψ൰. (A.4)
The reflected light passes through the quarter wave plate, which is arranged such that its fast
axis is tilted by an angle of 45° to the x-axis. The Jones matrix of the wave plate is given by
Rቀെπ
4ቁMRቀπ
4ቁ. ( A . 5 )
Thus, the probe light E
2 after the quarter wave plate is described as follows,
14E2 = ൬E2,x
E2,y൰=Rቀെπ
4ቁMRቀπ
4ቁE1
=1
2൬cosሺθ߶ሻെsin (θെ߶+)i(െcosሺθെ߶ሻsin (θ߶))
cosሺθെ߶ሻsin (θϕ)+i(c o sሺθ߶ሻsin (θെ߶))൰. (A.6)
The Wollaston prism after the quarter wave plat e splits the x and y-polarization components of
the probe light E2. The spatially separated two pulses are incident to the balanced detector and
the detected probe pulse intensity ratio of the di fferential signal to the total corresponds to the
Kerr ellipticity angle as follows,
〈หாమ,ೣหమ〉ି〈หாమ,หమ〉
〈หாమ,ೣหమ〉ା〈หாమ,หమ〉ൌെsin2θ. ( A . 7 )
In the main text, we show the Kerr ellipticity change =w−wo, where the ellipticity angles
(w and wo) are respectively obtained with and without the THz pump excitation.
Appendix B. Analytic signal approach and short time Fourier transform
The Analytic signal approach (ASA) allows the extraction of the time evolution of the
frequency and amplitude by a simple procedure and assumes that the signal contains a single
oscillator component. In our study, we measure only the MOKE signal originating from the
AF-mode and it can be expected that the single oscillator assumption is valid. In the ASA, the
time profile of the magnetization change (t) is converted into an analytic signal (t), which is a
complex function defined by using the Hilbert transform [39];
ψ(t)=ζ0(t)exp( i߶(t))=ζ(t)+i ζ෨(t), (B.1)
ζ෨(t)ൌ1
π pζ(t)
tିτ∞
-∞ dτ. ( B . 2 )
where the p is the Cauthy principal value. The real part of (t) corresponds to (t). The real
function 0(t) and (t) represent the envelope amplitude and instantaneous phase of the
magnetization change. The instantaneous frequency (t)(=2(t)) is given by (t)=d(t)/dt. In
the analysis, we averaged 0(t) and (t) over a ten picosecond time range.
To confirm whether the ASA gives appropriate results, as shown in figure B.1 we compare 15them with those obtained by the short time Fourie r transform (STFT). As shown in figure B.1(a),
the time-frequency plot shows only one oscillato ry component of the AF-mode. As shown in
figures B.1(b) and (c), the instantaneous freque ncies and amplitudes obtained by the ASA and
the STFT are very similar. Because the ASA provides us the instantaneous amplitude with a
simple procedure, we showed the time evolu tions of frequency and amplitude derived by the
ASA in the main text.
Appendix C. Determination of conversion coefficient g and linear damping parameter 0
The conversion coefficient g and the linear damping parameter 0(=) in Eq. (1) are
determined by fitting the experimental MOKE signal (t) for the low pump fluence of 29.2
µJ/cm2 with the LLG calculation of the magnetization change (t). Figure C.1 shows the MOKE
signal (t) (circle) and the calculated magnetization change (t) (solid line). From the
least-squares fit of the calculated result to th e experiment by using a linear relation, i.e.,
(t)=g(t), we obtained the parameters g(=17.8 degrees−1) and 0(=2.27×10−4). 0.575
0.570
0.565
0.560
0.555
0.550Frequency (THz)
50 40 30 20 100
Time (ps)ASA
100%
10%
STFT
100%
10%
1.0
0.8
0.6
0.4
0.2
0.0
Fourier am plitude (arb. units)
50 40 30 20 100
Time (ps)0.4
0.3
0.2
0.1
0.0Amplitude 0ASA
100%
10%
STFT
100%
10%
(a) (b) (c)
1.2
1.0
0.8
0.6
0.4
0.2
0.0Frequency (THz)
5040302010
Time (ps)(arb. units)
1.0 0.0
Figure B.1. (a) Time-dependence of the power spectrum of the magnetization
oscillation for the highest THz excitation ( I=292 µJ/cm2) obtained by the STFT.
Comparison of (b) instantaneous frequencies and (c) amplitudes obtained by the ASA
and STFT with a time window with FWHM of 10 ps. 16
Appendix D. Laser heating effect
The details of the calculation of the temperature change are as follows:
For HoFeO 3:
The absorption coefficient abs of HoFeO 3 at 0.5 THz is ~4.4 cm−1 [40]; the fluence IHFO
absorbed by HoFeO 3 can be calculated as IHFO=I(1−exp(−absd)), where d (=145 µm) is the
sample thickness and I is the THz pump fluence. For the highest pump fluence, I=292 µJ/cm2,
IHFO is 18.1 µJ/cm2. Since the sample thickness is much smaller than the penetration depth,
d≪abs−1, we assume that the heating of the sample due to the THz absorption is homogeneous.
By using the heat capacity Cp of 100 J mol−1 K−1 [27], and the molar volume v of ~1.4×102
cm3/mol [27], the temperature change T can be estimated as T=IHFOv/Cpd ~1.7×10−3 K.
For gold resonator (SRR):
The split ring resonator has an absorption band (center frequency ~0.56 THz, band width ~50
GHz) originated from the LC resonance (figure 2( c)). Assuming the SRR absorbs all incident
THz light in this frequency band, the absorbed energy accounts for 3 % of the total pulse energy.
Hence, for the highest THz pump fluence, I=292 µJ/cm2, the fluence absorbed by the SRR is
Igold=8.76 µJ/cm2. By using the heat capacity Cp of 0.13 J g−1 K−1 [41], the number of the SRRs
per unit area N of 4×104 cm−2, and the mass of the SRR m of 1.6×10−9 g, the temperature change -10x10-3-50510degrees)
50 40 30 20 10 0
Time (ps)-0.10.00.1
Magnetization
change Experiment
Simulation
Figure C.1. Experimentally observed MOKE signal (circle) and LLG simulatio n
result of the magnetization change (solid line) for the pump fluence of 29.2
µJ/cm2. 17T can be estimated as T=Igold/CpNm ~ 1 K
Appendix E. Free energy of HoFeO 3
The free energy F of the iron spin (Fe3+) system based on the two-lattice model is a function
of two different iron sublattice magnetizations mi, and composed of the exchange energy and
one-site anisotropy energy [32,33]. The free en ergy normalized by the sublattice magnetization
magnitude, V=F/m0 (m0=|mi|), can be expanded as a power series in the unit directional vector of
the sublattice magnetizations, Ri=mi/m0=(Xi,Yi,Zi). In the magnetic phase 4 (T > 58K), the
normalized free energy is given as follows [32,33]:
V=ER1·R2+D(X1Z2െX2Z1)െAxx(X12+X22)െAzz(Z12+Z22), (E.1)
where E(=6.4×102 T) and D(=1.5×10 T) for HoFeO 3 are respectively the symmetric and
antisymmetric exchange field [42]. Axx and Azz are the anisotropy constants. As mentioned in
Appendix F, the temperature dependent values of the anisotropy constants can be determined
from the antiferromagnetic resonance frequencies. The canting angle of Ri to the x-axis β0
under no magnetic field is given by
tan 2β0=D
E+AxxିAzz. ( E . 2 )
Appendix F. Linearized resonance modes and anisotropy constants ( Axx and Azz)
The nonlinear LLG equation of Eq. (1) can be linearized and the two derived eigenmodes
correspond to the AF and F-mode. The sublatti ce magnetization motion for each mode is given
by the harmonic oscillation of mode coordinates; for the AF-mode ( QAF,
PAF)=((X1−X2)s i nβ0+(Z1+Z2)c o sβ0, Y1−Y2), and for the F-mode ( QF,
PF)=((X1+X2)sinβ0−(Z1−Z2)cosβ0, Y1+Y2),
QAF=AAFcosωAFt, ( F . 1 )
PAF=AAFrAFsinωAFt, ( F . 2 )
QF=AFcosωFt, ( F . 3 )
PF=AFrAFsinωFt, ( F . 4 ) 18
where AAF,F represents the amplitude of each mode. AF,F, and rAF,F are the resonance frequencies
and ellipticities, which are given by
ωAF=γට(b+a)(d-c), ( F . 5 )
ωF=γට(b-a)(d+c), ( F . 6 )
rAF=γටሺௗିሻ
(b+a), ( F . 7 )
rF=γටሺௗାሻ
(b-a), ( F . 8 )
where =1.76×1011 s−1T−1 is the gyromagnetic ratio, and
a=െ2Axxcos2β0െ2Azzsin2β0െEcos 2β0െDsin 2β0, (F.9)
b=E, ( F . 1 0 )
c=2Axxcos2β0െ2Azzcos2β0+Ecos 2β0+Dsin 2β0, (F.11)
d=െEcos 2β0െDsin 2β0. ( F . 1 2 )
Substituting the literature values of the exchange fields ( E=6.4×102 T and D=1.5×10 T [42]) and
the resonance frequencies at room temperature ( AF/2=0.575 THz and F/2=0.37 THz) to
Eqs. (F.5) and (F.6), Axx and Azz can be determined to 8.8×10−2 T and 1.9×10−2 T.
Appendix G. MOKE measurement for the spontaneous magnetization
Figure G.1 shows time-development of the MOKE signals for the different initial condition
with oppositely directed magnetization. We applied the static magnetic field (~0.3 T) to saturate
the magnetization along the z-axis before the TH z excitation. The spontaneous magnetization of
single crystal HoFeO a can be reversed by the much smaller magnetic field (~0.01 T) because of
the domain wall motion [27]. Then, we separately measured the static Kerr ellipticity angle
and THz induced ellipticity change for different initial magnetization Mz=±Ms
without the static magnetic field In figure G.1 we plot the summation of the time resolved
MOKE signal and the static Kerr ellipticity The sings of the ellipticity offset angle 19 for the different spontaneous magnetization (±M S) are different and their magnitudes
are ~0.05 degrees. The conversion coefficient g(=1/~ /0.05 degrees) is estimated to be ~20
degrees−1, which is similar to the value dete rmined by the LLG fitting (~17.8 degrees−1). In the
case of the AF-mode excitation, the phases of the magnetization oscillations are in-phase
regardless of the direction of the spontaneous magnetization M=±Ms, whereas they are
out-of-phase in the case of the F-mode excitation. We can explain this claim as follows: In the
case of AF-mode excitation, the external THz magne tic field is directed along the z-direction as
shown in the inset of figure 2(a), the signs of the torques acting on the sublattice magnetization
mi (i=1,2) depends on the direction of mi, however, the resultant oscillation of the macroscopic
magnetization M= m1+m2 along the z-direction has same phase for the different initial condition
M=±Ms. In the case of the F-mode excitation with the external THz magnetic field along the x
or y-direction, the direction of the torques acting on the magnetization M depends on the initial
direction and the phase of the F-mode osc illation changes depending on the sign of the
spontaneous magnetization ±Ms.
Appendix H. Influence of the spatial distri bution of magnetic field on magnetization
change
As shown in the inset of figure 1, the pump magnetic field strongly localizes near the metallic
arm of the SRR and the magnetic field strength significantly depends on the spatial position r
within the probe pulse spot area. The intensity distribution of the probe pulse Iprobe(r) has an 0.05
0.00
-0.05Kerr ellipticity (degrees)
25 20 15 10 5
Time (ps) +MS
-MS
Figure G.1. The MOKE signals, the temporal change of the Kerr ellipticity , measured
for different initial conditions with oppositely directed magnetizations. 20elongated Gaussian distribution with spatial widt hs of 1.1 µm along the x-axis and 1.4 µm along
the y-axis [full width at half maximum (FWHM) intensity]. The maximum magnetic field is 1.2
times larger than the minimum one in the spot diameter, causing the different magnetization
change dynamics at different positions. To take into account this spatial inhomogeneity to the
simulation, the spatially weighted average of magnetization change ζ̅(t) has to be calculated as
follows:
ζ̅(t)=ζ(r,t)Iprobe(r)dr
Iprobe(r)ௗr , ( H . 1 )
where (r,t) is a magnetization change at a position r and time t.
Figure H.1(a) shows the simulation result of the spatially averaged magnetization change ζ̅(t)
and the non-averaged (r0,t) without the nonlinear damping term ( 1=0), where r0 denotes the
peak position of Iprobe(r). For the low excitation intensity (10%), ζ̅(t) is almost the same as
(r0,t) as shown in figure H.1(a). On the other hand, for the high excitation intensity, the spatial
inhomogeneity of magnetization change dyna mics induces a discrepancy between the ζ̅(t) and (a) (b) (c)
-0.100.000.10Magnetization change
50 40 30 20 100
Time (ps)-0.6-0.4-0.20.00.20.4 Averaged
Non-averaged
Averaged
Non-averaged100% 10%
0.575
0.570
0.565
0.560
0.55550 40 30 20 100
Time (ps)0.575
0.570
0.565
0.560
0.555Frequency (THz) Averaged
Non-averaged
Experiment
Averaged
Non-averaged
Experiment
100% 10%
0.4
0.3
0.2
0.1
0.0
50 40 3020 100
Time (ps)0.12
0.08
0.04
0.00Amplitude Averaged
Non-averaged
Experiment
Averaged
Non-averaged
Experiment100% 10%
Figure H.1. Comparison of the spatially averag ed and non-averaged magnetization
change for the different pump fluences of 10% and 100%. (a) Temporal evolutions of
the magnetization change, (b) instantaneous frequencies and (c) normalized envelope
amplitudes. Open circles show the experimental results. 21(r0,t). This discrepancy is caused by the quasi-i nterference effect between the magnetization
dynamics with different frequencies and amplit udes at different positions. Figures H.1(b) and
(c) show the instantaneous freque ncy and envelope amplitude obt ained from the data shown in
figure H.1(a) by using analytic signal approach with the experimental result. For the averaged
magnetization change, the frequency redshift is more emphasized (figure H.1(b)) and the decay
time becomes shorter (figure H.1(c)). Nonetheless, neither spatially averaged nor non-averaged
simulation reproduces the experimental result of the instantaneous frequency (figure H.1(b))
without nonlinear damping term.
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2106.08528v2.Spin_Torque_driven_Terahertz_Auto_Oscillations_in_Non_Collinear_Coplanar_Antiferromagnets.pdf | Spin-Torque-driven Terahertz Auto Oscillations in Non-Collinear Coplanar
Antiferromagnets
Ankit Shuklaand Shaloo Rakhejay
Holonyak Micro and Nanotechnology Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801
(Dated: January 19, 2022)
We theoretically and numerically study the terahertz auto oscillations, or self oscillations, in thin-
lm metallic non-collinear coplanar antiferromagnets (AFMs), such as Mn 3Sn and Mn 3Ir, under the
e ect of antidamping spin torque with spin polarization perpendicular to the plane of the lm. To
obtain the order parameter dynamics in these AFMs, we solve three Landau-Lifshitz-Gilbert equa-
tions coupled by exchange interactions assuming both single- and multi-domain (micromagnetics)
dynamical processes. In the limit of a strong exchange interaction, the oscillatory dynamics of the
order parameter in these AFMs, which have opposite chiralities, could be mapped to that of two
damped-driven pendulums with signi cant di erences in the magnitude of the threshold currents
and the range of frequency of operation. The theoretical framework allows us to identify the in-
put current requirements as a function of the material and geometry parameters for exciting an
oscillatory response. We also obtain a closed-form approximate solution of the oscillation frequency
for large input currents in case of both Mn 3Ir and Mn 3Sn. Our analytical predictions of threshold
current and oscillation frequency agree well with the numerical results and thus can be used as com-
pact models to design and optimize the auto oscillator. Employing a circuit model, based on the
principle of tunnel anisotropy magnetoresistance, we present detailed models of the output power
and eciency versus oscillation frequency of the auto oscillator. Finally, we explore the spiking
dynamics of two unidirectional as well as bidirectional coupled AFM oscillators using non-linear
damped-driven pendulum equations. Our results could be a starting point for building experimen-
tal setups to demonstrate auto oscillations in metallic AFMs, which have potential applications in
terahertz sensing, imaging, and neuromorphic computing based on oscillatory or spiking neurons.
I. INTRODUCTION
Terahertz (THz) radiation, spanning from 100 Giga-
hertz (GHz) to 10 THz, are non-ionizing, have short
wavelength, o er large bandwidth, scatter less, and are
absorbed or re
ected di erently by di erent materials.
As a result, THz electronics can be employed for safe
biomedical applications, sensing, imaging, security, qual-
ity monitoring, spectroscopy, as well as for high-speed
and energy-ecient non-von Neumann computing (e.g.,
neuromorphic computing). THz electronics also has po-
tential applications in beyond-5G communication sys-
tems and Internet of Things. Particularly, the size of
the antennae for transmitting the electromagnetic signal
could be signi cantly miniaturized in THz communica-
tion networks [1{8]. These aforementioned advantages
and applications have led to an intense research and de-
velopment in the eld of THz technology with an aim
to generate, manipulate, transmit, and detect THz sig-
nals [3, 9]. Therefore, the development of ecient and
low power signal sources and sensitive detectors that op-
erate in the THz regime is an important goal [3].
Most coherent THz signal sources can be categorized
into three types | particle accelerator based sources,
solid state electronics based sources, and photonics based
sources [3, 9]. Particle accelerator based signal genera-
tors include free electron lasers [10], synchrotrons [11],
ankits4@illinois.edu
yrakheja@illinois.eduand gyrotrons [12]. While particle accelerator sources
have the highest power output, they require a large
and complex set-up [13]. Solid state generators include
diodes [14{16], transistors [17, 18], frequency multipli-
ers [19], and Josephson junctions [20], whereas photonics
based signal sources include quantum cascade lasers [21],
gas lasers [22], and semiconductor lasers [23]. Solid-state
generators are ecient at microwave frequencies whereas
their output power and eciency drop signi cantly above
100 GHz [13]. THz lasers, on the other hand, provide
higher output power for frequencies above 30 THz [24],
however, their performance for lower THz frequencies
is plagued by noise and poor eciency [13]. Here, we
present the physics, operation, and performance bench-
marks of a new type of nanoscale THz generator based
on the ultra-fast dynamics of the order parameter of an-
tiferromagnets (AFMs) when driven by spin torque.
Spin-transfer torque (STT) [25, 26] and spin-orbit
torque (SOT) [27] enable electrical manipulation of ferro-
magnetic order in emerging low-power spintronic radio-
frequency nano-oscillators [28]. When a spin current
greater than a certain threshold (typically around 108
109A=cm2[28, 29]) is injected into a ferromagnet (FM)
at equilibrium, the resulting torque due to this current
pumps in energy which competes against the intrinsic
Gilbert damping of the material. When the spin torque
balances the Gilbert damping, the FM magnetization un-
dergoes a constant-energy steady-state oscillation around
the spin polarization of the injected spin current. Such
oscillators are nonlinear, current tunable with frequencies
in the range of hundreds of MHz to a few GHz with out-
put power in the range of nano-Watt (nW). They are alsoarXiv:2106.08528v2 [cond-mat.mes-hall] 15 Jan 20222
compatible with the CMOS technology [28]; however, the
generation of the THz signal using FMs would require
prohibitively large amount of current, which would lead
to Joule heating and degrade the reliability of the elec-
tronics. It would also lead to electromigration and hence
irreversible damage to the device set-up [30].
AFM materials, which are typically used to exchange
bias [31] an adjacent FM layer in spin-valves or magnetic
tunnel junctions for FM memories and oscillators, have
resonant frequencies in the THz regime [32{35] due to
their strong exchange interactions. It was suggested that
STT could, in principle, be used to manipulate the mag-
netic order in conducting AFMs [36], leading to either
stable precessions for their use as high-frequency oscilla-
tors [37, 38] or switching of the AFM order [39] for their
use as magnetic memories in spin-valve structures. The
SOT-based spin Hall e ect (SHE), on the other hand,
could enable the use of both conducting [40{44] and in-
sulating [29, 43, 45, 46] AFMs in a bilayer comprising an
AFM and a non-magnetic (NM) layer, for its use as a
high frequency auto-oscillator [47].
Table I lists the salient results from some of the re-
cently proposed AFM oscillators. These results, how-
ever, are reported mainly for collinear AFMs, while de-
tailed analyses of the dynamics of the order parameter in
the case of non-collinear AFMs is lacking. In this paper,
rstly, we ll this existing knowledge gap in the model-
ing of auto oscillations in thin- lm non-collinear copla-
nar AFMs like Mn 3Ir;Mn3Sn, or Mn 3GaN under the ac-
tion of a dc spin current. Secondly, we compare their
performance (generation and detection) against that of
collinear AFMs such as NiO for use as a THz signal
source. In the case of NiO, inverse spin Hall e ect (iSHE)
is employed for signal detection, whereas in this work we
utilize the large magnetoresistance of metallic AFMs. Fi-
nally, we investigate these auto oscillators as possible can-
didates for neuron emulators. Considering that the spin
polarization is perpendicular to the plane of the AFM
thin- lm, three possible device geometries are identi ed
and presented in Fig. 1 for the generation and detection
of auto oscillations in metallic AFMs.
Figure 1(a) is based on the phenomena of spin injec-
tion and accumulation in a local lateral spin valve struc-
ture. Charge current, I write, injected into the structure
is spin-polarized along the magnetization of FM 1(FM 2)
and gets accumulated in the NM. It then tunnels into the
AFM with the required perpendicular spin-polarization
(adapted from Ref. [50]). The bottom MgO layer used
here would reduce the leakage of charge current into
the metallic AFMs considered in this work. This would
reduce chances of Joule heating in the AFM thin- lm
layer. On the other hand, in Fig. 1(b), spin lter-
ing [51] technique is adopted wherein, a conducting AFM
is sandwiched between two conducting FMs. Di er-
ent scattering rates of the up-spins and down-spins of
the injected electron ensemble at the two FM interfaces
results in a perpendicularly polarized spin current as
shown. The structure in Fig. 1(c) generates spin polar-
FM1
Iwrite
FM2NMJs
npAFMMgOPtLoadIdeal Bias Tee
FM1FM2
IwriteAFM
MgOIdeal Bias Tee
Load
JsnpIread
FM1
NMLoadIdeal Bias Tee(a)
(b) (c)xyzIread
IwriteIread
Pt
MgO
JsnpAFMMgOLbtCbt
Cbt Lbt LbtCbtFIG. 1. Device geometries to inject perpendicularly polar-
ized spin current in thin- lm metallic AFMs. In all the cases,
Iwrite is the charge current injected to generate spin current,
whereas, I readis the charge current injected to extract the
oscillations as a transduced voltage signal using the princi-
ples of tunnel anisotropy magnetoresistance (TAMR). (a) Lat-
eral spin valve structure leads to spin accumulation in NM
followed by injection into the AFM. (b) Perpendicular spin
valve structure spin lters the injected charge current. (c)
FM/NM/AFM trilayer structure generates spin current due
to interfacial spin-orbit torque.
ization perpendicular to the interface due to the interfa-
cial SOTs generated at the FM/NM interface (adapted
from Ref. [52, 53]). In this case, the spin current injected
into the AFM has polarization along both yandzdirec-
tion; however, the interface properties could be tailored
to suppress the spin polarization along y[52]. In order
to extract the THz oscillations of the order parameter as
a measurable voltage signal, the tunnel anisotropy mag-
netoresistance (TAMR) measurements are utilized [54].
In this work, we establish the micromagnetic model
for non-collinear coplanar AFMs with three sublattices
along with the boundary conditions in terms of both the
sublattice magnetizations (Section II A), as well as the
N eel order parameter (Section II B). We show that in the
macrospin limit the oscillation dynamics correspond to
that of a damped-driven pendulum (Section III A and
Section III B). The oscillation dynamics of AFM mate-
rials with two di erent chiralities in then compared in
Section III C. We use the TAMR detection scheme to
extract the oscillations as a voltage signal and present
models of the output power and eciency as a function
of the oscillator's frequency (Section IV). This is followed
by a brief investigation of the e ect of inhomogeneity due
to the exchange interaction on the dynamics of the AFM
order (Section V). Finally, we discuss the implication of
our work towards building coherent THz sources in Sec-3
TABLE I. Recent numerical studies on electrically controlled AFM THz oscillators. The investigated AFM materials, the
direction of their uniaxial anisotropy axis ueand that of the spin polarization of the injected spin current npare listed. Salient
results along with the schemes to extract the oscillation as a voltage signal are also brie
y stated. Ref. [44] does not provide
the name of a speci c AFM, however, an AFM with uniaxial anisotropy is considered.
Ref. AFM material ue np Salient Features Detection Schemes
[45] NiO x x a) THz oscillations for current above a threshold iSHE
b) Feedback in AFM/Pt bilayer sustains oscillation
[29] NiO x -z a) Hysteretic THz oscillation in a biaxial AFM iSHE
b) Threshold current dependence on uniaxial anisotropy
[40] Fe2O3 a)z y a) Monodomain analysis of current driven oscillations in -
AFM insulators with DMI
b)y y Similar to [45] -
[41] Fe2O3 x y a) Canted net magnetization due to DMI Dipolar radiation
b) Small uniaxial anisotropy leads to low power THz frequency
[42] CuMnAs ;Mn2Auy z a) Low dc current THz signal generation due to N eel SOT -
b) Phase locked detector for external THz signal
[43] NiO x Varied a) Comparison of analytical solutions to micromagnetic results AMR/SMR
b) E ect of DMI on hysteretic nature of dynamics
[48] Mn 2RuxGa z y a) Generation of spin current in single AFM layer AMR
b) Oscillation dependence on reactive and dissipative torques
[44] Uniaxial Ani. z Varied a) Non-monotonic threshold current variation with np -
b) E ects of anisotropy and exchange imperfections
[49] Mn 3Sn x-y plane z a) E ective pendulum model based on multipole theory AHE
[46] NiO;Cr2O3 x Varied a) General e ective equation of a damped-driven pendulum iSHE
b) Analytic expression of threshold current and frequency
Our Mn 3Sn;Mn3Ir x-y plane z a) Di erent numerical and analytic models TAMR
Work b) Inclusion of generation current for TAMR eciency
c) Non-linear dynamics of bidirectional coupled oscillators
tion VI, and towards hardware neuron emulators for neu-
romorphic computing architecture in Section VII. Some
of the salient results from this work are listed in Table I.
II. THEORY
A. Magnetization Dynamics
We consider a micromagnetic formalism in the con-
tinuum domain [43, 55] under which a planar non-
collinear AFM is considered to be composed of three
equivalent interpenetrating sublattices, each with a con-
stant saturation magnetization Ms[56]. Each sublattice,
i(= 1, 2 or 3), is represented as a vector eld mi(r;t)
such that for an arbitrary r=r0,kmi(r0;t)k= 1. The
dynamics of the AFM under the in
uence of magnetic
elds, damping, and spin torque is assumed to be gov-
erned by three Landau-Lifshitz-Gilbert (LLG) equations
coupled by exchange interactions. For sublattice i, the
LLG is given as [57]
@mi
@t=
0
miHe
i
+ i
mi@mi
@t
!smi(minp) !s(minp);(1)
wheretis time in seconds, He
iis the position dependent
e ective magnetic eld on i, iis the Gilbert dampingparameter for i, and
!s=~
2e
Js
Msda(2)
is the frequency associated with the input spin current
density,Js, with spin polarization along np. Here,dais
the thickness of the AFM layer, ~is the reduced Planck's
constant,0is the permeability of free space, eis the el-
ementary charge, and
= 17:61010T 1s 1is the gyro-
magnetic ratio. For all sublattices, the spin polarization,
np, is assumed to be along the zaxis. Finally, is a mea-
sure of the strength of the eld-like torque as compared
to the antidamping-like torque. The e ect of eld-like
torque on the sublattice vectors here is the same as that
of an externally applied magnetic eld|canting towards
the spin polarization direction. Results presented in the
main part of this work do not include the e ect of the
eld-like torque; however, a small discussion on the same
is presented in the supplementary material [58].
The e ective magnetic eld, He
iat each sublattice,
includes contributions from internal elds as well as ex-
ternally applied magnetic elds and is obtained as
He
i(r;t) = 1
0MsF
mi(r;t); (3)
where
mi=@
@mi r@
@(rmi), andFis the energy
density of the AFM, considered in our work. It is given
as4
F=X
hi;ji
Jmimj+Aijrmirmj
+Aii3X
i=1(rmi)2+3X
i=1Khm2
i;z Ke(miue;i)2
+DX
hi;jiz(mimj) +Dii3X
i=1(mi;zrmi
(mir)mi;z) +DijX
hi;ji((mi;zrmj
(mir)mj;z) (mj;zrmi (mjr)mi;z))
3X
i=10MsHami;(4)
wherehi;jirepresents the sublattice ordered pairs (1 ;2),
(2;3) and (3;1).
The rst three terms in Eq. (4) represent exchange en-
ergies. HereJ(>0) is the homogeneous inter-sublattice
exchange energy density whereas Aii(>0) andAij(<
0) are the isotropic inhomogeneous intra- and inter-
sublattice exchange spring constants, respectively. The
next two terms in Eq. (4) represent magnetocrystalline
anisotropy energy for biaxial symmetery upto the low-
est order withKe(>0) andKh(>0) being the easy and
hard axes anisotropy constants, respectively. We assume
that the easy axes of sublattices 1, 2 and 3 are along
ue;1= (1=2)x+ (p
3=2)y,ue;2= (1=2)x (p
3=2)y
andue;3=x, respectively, and an equivalent out of plane
hard axis exists along the zaxis. The next three terms
represent the structural symmetry breaking interfacial
Dzyaloshinskii-Moriya Interaction (iDMI) energy density
in the continuum domain. Its origin lies in the interaction
of the antiferromagnetic spins with an adjacent heavy
metal with a large spin-orbit coupling [59, 60]. Here, we
assume the AFM crystal to have Cnvsymmetry [61] such
that the thin- lm AFM is isotropic in its plane, and D,
Dii, andDijrepresent the e ective strength of homoge-
neous and inhomogeneous iDMI, respectively, along the
zdirection. Finally, the last term in Eq. (4) represents
the Zeeman energy due to an externally applied mag-
netic eld Ha. Now, using Eq. (4) in Eq. (3) we get the
e ective eld for sublattice ias
He
i=X
j
j6=i
J
0Msmj+Aij
0Msr2mj
+2Aii
0Msr2mi
2Kh
0Msmi;zz+2Ke
0Ms(miue;i)ue;i
+Dz(mj mk)
0Ms 2Dii
0Ms((rmi)z rmi;z)
Dij
0Ms((r(mj mk))z r(mj;z mk;z))
+Ha;
(5)where (i;j;k ) = (1;2;3);(2;3;1);or (3;1;2), respectively.
In order to explore the dynamics of the AFM, we adopt
a nite di erence discretization scheme and discretize the
thin- lm of dimension LWdainto smaller cells of
sizesLsWsd. Each of these cells is centered around
position rsuch that mi(r;t) denotes the average mag-
netization of the spins within that particular cell [43].
Finally, we substitute Eq. (5) in Eq. (1) and use fourth-
order Runge-Kutta rule along with the following bound-
ary conditions for sublattice iof the thin- lm considered
(see supplementary material [58]):
2Aii@mi
@+AijX
j
j6=imi@mj
@mi
+Diimi(z)
+Dijmi(mi((z)(mk mj))) = 0;(6)
whereis the normal vector perpendicular to a surface
parallel to xory. The above equation ensures that the
net torque due to the internal elds on the boundary
magnetizations of each sublattice is zero in equilibrium
as well as under current injection [62]. For the energy
density presented in Eq. (4), the elds at the boundary
are non-zero only for inhomogeneous inter- and intra-
sublattice exchange, and Dzyaloshinskii-Moriya interac-
tions. Finally, in the absence of DMI, we have the Neu-
mann boundary condition@mi
@= 0, which implies that
the boundary magnetization does not change along the
surface normal .
For all the numerical results presented in this work
we solve the system of Eqs. (1), (5), and (6) with the
equilibrium state as the starting point. The equilibrium
solution in each case was arrived at by solving these three
equations for zero external eld and zero current with a
large Gilbert damping of 0 :5.
B. N eel Order Dynamics
The aforementioned micromagnetic modeling ap-
proach assuming three sublattices is extremely useful in
exploring the physics of the considered AFM systems. It
is, however, highly desirable to study an e ective dynam-
ics of the AFMs under the e ect of internal and external
stimuli in order to gain fundamental insight. Therefore,
we consider an average magnetization vector mand two
staggered order parameters n1andn2to represent an
equivalent picture of the considered AFMs. These vec-
tors are de ned as [37, 56, 63]
m1
3(m1+m2+m3); (7a)
n11
3p
2(m1+m2 2m3); (7b)
n21p
6( m1+m2); (7c)5
such thatkmk2+kn1k2+kn2k2= 1. The energy land-
scape (Eq. (4)) can then be represented as
F
3=3J
2m2+Am(rm)2+An
2
(rn1)2+ (rn2)2
+Kh
m2
z+n2
1;z+n2
2;z
Ke
23
2(n1;x n2;y)2
+1
2(n1;y+n2;x)2+mx
mx p
2(n1;x n2;y)
+my
my+p
2(n1;y+n2;x)
+ 4n1;xn2;y
+p
3Dz(n1n2) +Dii(mzrm (mr)mz
+n1;zrn1 (n1r)n1;z+n2;zrn2
(n2r)n2;z) +p
3Dij(n1;zrn2 (n1r)n2;z
n2;zrn1+ (n2r)n1;z) 0MsHam;
(8)
whereAm=
Aii+Aij
andAn=
2Aii Aij
.
An equation of motion involving the staggered order
parameters can be obtained by substituting Eq. (1) in the
rst-order time derivatives of Eq. (7) and evaluating each
term carefully (see supplementary material [58]). How-
ever, an analytical study of such an equation of motion
that consists of contributions from all the energy terms
of Eqs. (4) or (8) would be as intractable as the dynam-
ics of individual sublattices itself. Therefore, we consider
the case of AFMs with strong inter-sublattice exchange
interaction such that J jDj K e. This corresponds
to systems with ground state con ned to the easy-plane
(x yplane) and those that host kmk 1 (weak ferro-
magnetism), n1?n2, andkn1kkn2k1=p
2 [56, 63].
However, when an input current is injected in the system,
the sublattice vectors cant towards the spin polarization
direction leading to an increase in the magnitude of m
while decreasing that of n1andn2. Spin polarization
along the zdirection and an equal spin torque on each
sublattice vector ensures that n1andn2have negligible
zcomponents at all times (Eqs. (7b), (7c)). Therefore,
we consider
n1(r;t) =0
@p
1 n2
1zcos'(r;t)p
1 n2
1zsin'(r;t)
n1z(r;t)1
A; (9a)
n2(r;t) =0
@p
1 n2
2zcos('(r;t)=2)p
1 n2
2zsin('(r;t)=2)
n2z(r;t)1
A;(9b)
where'is the azimuthal angle from the xaxis and
jn1zj;jn2zj 1.
The two choices for n2correspond to two di erent
classes of materials|one with a positive (+ =2) chiral-
ity and the other with a negative ( =2) chirality [56].
Materials that have a negative (positive) value of Dcor-
respond to + =2( =2) chirality because the respective
con guration reduces the overall energy of the system.
L12phase of AFMs like Mn 3Ir;Mn3Rh, or Mn 3Pt is ex-
pected to host + =2 chirality whereas the hexagonalphase of AFMs like Mn 3Sn;Mn3Ge, or Mn 3Ga is ex-
pected to host =2 chirality [56, 64].
III. SINGLE DOMAIN ANALYSIS
A. Positive Chirality
De ning n3= (n1n2)=, and considering the case
of +=2 chirality, it can be shown that mis just a de-
pendent variable of the N eel order dynamics. To a rst
order, mcould be expressed as [56, 63]
m= 1
!E(n1_ n1+n2_ n2+n3_ n3
0Ha
!snp);
(10)
where!E= 3
J=Ms. One can then arrive at the equa-
tion of motion for the N eel vectors as
n1
n1 c2r2n1 !E!Kn1+!E!Kh(n1z)z
+!E!D(n2z) +!E!ii
Dn1+!E!ij
Dn2+ !E_n1
!E!s(n1np)] +n2
n2 c2r2n2 !E!Kn2
+!E!Kh(n2z)z !E!D(n1z) +!E!ii
Dn2
!E!ij
Dn1+ !E_n2 !E!s(n2np)i
+n3n3
+
0(n1_n1+n2_n2+n3_n3)Ha
0_Ha= 0;
(11)
wherec=p
!E
An=Ms,!Kh= 2
Kh=Ms,!K;n1=
!K
4
n1;y+n2;x+p
2my
^ x+ (n1;x+ 3n2;y
+p
2mx
^ y
,!D=p
3
D
Ms,!K;n2=!K
4((n1;y+n2;x
+p
2my
^ x+
n1;x+ 3n2;y+p
2mx
^ y
,
!ii
D;ni=2
Dii
Ms((rni)z rni;z),
and!ij
D;ni=p
3
Dij
Ms((rni)z rni;z).
The equations of motion (Eqs. (10) and (11)) derived
here are useful in the numerical study of textures like
domain walls, skyrmions, and spin-waves in AFMs with
biaxial anisotropy under the e ect of external magnetic
eld and spin current. However, here we are interested
in analytically studying oscillatory dynamics of the order
parameter in thin- lm AFMs, therefore, we neglect in-
homogeneous interactions compared to the homogeneous
elds. Using Eq. (9) in Eq. (11) and neglecting the time
derivative of n1zandn2z, we have
'+ !E_'+!E!K
2sin 2'+!E!s= 0; (12)
where!K= 2
Ke=Ms. This indicates that in the limit
of strong exchange interaction, the dynamics of the stag-
gered order parameters is identical to that of a damped-
driven non-linear pendulum [65]. This equation is iden-
tical to the case of collinear AFMs such as NiO when the6
(c) (d)
(a) (b)Negative Chirality Positive Chirality
FIG. 2. Stationary solution for AFMs with positive (a, b) and negative chirality (c, d). (a) Sublattice magnetization for currents
below the threshold current Jth1
s. WhenJs= 0 (equilibrium state), the sublattice vectors micoincide with the easy axes ue;i,
whereas for a non-zero current smaller than Jth1
s, the macrospins have stationary solutions other than the equilibrium solution,
as depicted by dashed and dotted line. The zcomponent of these vectors is zero. (b) An equivalent representation of (a) through
the staggered order parameters n1andn2. They are perpendicular to each other and have zero out-of-plane component. The
thinner dash-dotted gold arrows through the thicker arrows of n1represent the analytic expression of the stationary solution
'= 1
2sin 1
2!s
!K
. The average magnetization mis vanishingly small, as can also be noticed from Eq. (10). (c) Sublattice
magnetization at equilibrium ( Js= 0). Thezcomponent of these vectors is zero. Here, only the sublattice vector m3coincides
with its corresponding easy axis. On the other hand, m1andm2are oriented such that the energy due to DMI is dominant
over anisotropy. (d) An equivalent representation of (c) through the staggered order parameters n1andn2, and average
magnetization m.n1andn2are almost perpendicular to each other with a negative chirality, as assumed in Eq.(9). A small
in-plane net magnetization (shown by the magni ed green arrow) also exists in this case [56].
(a) (b) (c) (d)
(e) (g) (f) (h)Positive Chirality Negative Chirality
α = 0.01α = 0.01 α = 0.01
α = 0.01da = 4 nm da = 4 nm
da = 4 nm da = 4 nm
FIG. 3. Upper panel (a-d) shows the time averaged frequency as a function of input spin current, whereas the lower panel (e-h)
shows the FFT of the oscillations corresponding to the cases marked by the dashed red boxes above. In the time averaged
frequency response in the upper panel, the dashed black lines denote the analytic expression of frequency (Eq. (17)). (a), (c)
Frequency response for di erent lm thicknesses for = 0:01. (e), (g) FFT of the signal corresponding to Js=Jth2
s. (b),
(d) Frequency response for di erent damping constants for da= 4 nm. (f), (h) FFT of the signal corresponding to = 0:1
andJth1
s, respectively. Positive chirality: The numerical values of the average frequency match very well against the analytic
expression for lower damping and large current. On the other hand, non-linearity and, hence, higher harmonics are observed
for small current and large damping. Negative chirality: The numerical values of the average frequency exactly match against
the analytic expression for all values of damping and input current considered here.
direction of spin polarization is perpendicular to the easy-
plane [29, 44, 66]. However, the dynamics of the non-
collinear coplanar AFMs discussed here is signi cantlydi erent in the direction of the spin torques, magnitude
of threshold currents as well as the range of possible fre-
quencies. Here, the sin 2 'dependence signi es a two-fold7
anisotropy symmetric system.
B. Negative Chirality
For the case of =2 chirality, it can be shown that m
is a dependent variable of the N eel order; however, in this
case there are additional in-plane terms that arise due to
a competition between the DMI, exchange coupling and
magnetocrystalline anisotropy. To a rst order, mis ex-
pressed as [56, 67] (also see supplementary material [58])
m= 1
!E(n1_ n1+n2_ n2+n3_ n3
0Ha
!snp) !K
2!E(cos'x sin'y);
(13)
which is used to arrive at the equation of motion for the
N eel vectors as
n1
n1 c2r2n1 !E!Kn1+!E!Kh(^ n1z)z
+!E!D(n2z) +!E!ii
Dn1+!E!ij
Dn2+ !E_n1
!E!s(n1np)] +n2
n2 c2r2n2 !E!Kn2
+!E!Kh(^ n2z)z !E!D(n1z) +!E!ii
Dn2
!E!ij
Dn1+ !E_n2 !E!s(n2np)i
+n3n3
+
0(n1_n1+n2_n2+n3_n3)Ha
0_Ha
!K
2(sin'x+ cos'y) _'
0!K
2(Ha;zsin'x
+Ha;zcos'y (Ha;xsin'+Ha;ycos')z) = 0:
(14)
Similar to the previous case, we are interested in a the-
oretical analysis of the oscillation dynamics in thin lm
AFMs with negative chirality. Therefore, we use Eq. (9)
in Eq. (14) and neglect all the inhomegeneous interac-
tions to arrive at a damped-driven linear pendulum equa-
tion given as
'+ !E_'+!E!s= 0: (15)
Here the dependence of the dynamics on anisotropy is
not zero but very small, and it scales proportional to
!3
K
!2
Ecos 6'[67]. However, for a rst-order approxima-
tion in mand dynamics in the THz regime, it can be
safely ignored. The cos 6 'dependence implies that these
materials host a six-fold anisotropic symmetry. Though
this equation is similar to that obtained for the case of
a collinear AFM with spin polarization along the easy
axis [44], the dynamics is signi cantly di erent from that
of the collinear AFM.
C. Comparison of Dynamics for Positive and
Negative Chiralities
Here, we contrast the dynamics of AFM order param-
eter for positive and negative chiralities. The numericalresults presented in this section are obtained in the single-
domain limit assuming thickness da= 4 nm, = 0:01,
Ms= 1:63 T,Ke= 3 MJ=m3,J= 2:4108J=m3,
D= 20 MJ=m3for positive chirality or 20 MJ =m3for
negative chirality [56], unless speci ed otherwise.
Figure 2 shows the stationary solutions of the thin-
lm AFM system with di erent chiralities. For the case
of positive chirality, it can be observed from Fig. 2(a)
that in equilibrium the sublattice vectors micoincide
with the easy axes ue;i. When a non-zero spin current is
applied, the equilibrium state is disturbed; however, be-
low a certain threshold, Jth1
s, the system dynamics con-
verge to a stationary solution in the easy-plane of the
AFM, indicated by dashed blue, and dotted red set of
arrows. An equivalent representation of the stationary
solutions in terms of the staggered order parameters is
presented in Fig. 2(b). n1andn2are perpendicular to
each other with zero out-of-plane component for all val-
ues of the input currents. The gold dash-dotted arrows
passing through n1correspond to the stationary solu-
tions given as '= 1
2sin 1
2!s
!K
, obtained analytically
by setting both _ 'and 'as zero in Eq. (12). In positive
chirality material, the average magnetization mis van-
ishingly small in the stationary state. This can also be
perceived from Eq. (10) as we do not consider any exter-
nal eld. Since these materials have a two-fold symmetry,
they also host '= 1
2sin 1
2!s
!K
stationary states.
For the case of negative chirality, it can be observed
from Fig. 2(c) that in equilibrium only the sublattice
vector m3coincides with the its corresponding easy axis,
whereas both m1andm2are oriented such that the en-
ergy due to DMI is dominant over anisotropy, which in
turn lowers the overall energy of the system. It can be
observed from Fig. 2(d) that n1andn2are almost per-
pendicular to each other. A small in-plane net magneti-
zation exists in this case and is shown here as a zoomed
in value (zoom factor = 100x) for the sake of compari-
son to staggered order parameters. Due to the six-fold
anisotropy dependence other equilibrium states, wherein
either of m1orm2coincide with their easy axis while the
other two sublattice vectors do not, also exist. Finally,
due to the small anisotropy dependence, non-equilibrium
stationary states exist for much lower currents [68] than
those considered here, and therefore are not shown.
For materials with positive chirality, the system be-
comes unstable when the input spin current exceeds the
threshold,Jth1
s. The resultant spin torque pushes the
N eel vectors out of the easy-plane, and they oscillate
around the spin polarization axis, np=z;with THz fre-
quency due to strong exchange. This threshold current
is given as [29, 44, 46]
Jth1
s=da2e
~Ms
!K
2=da2e
~Ke; (16)
while the frequency of oscillation in the limit of large
input current (neglecting the sin 2 'term) from Eq. (12)8
(a) (b)
(c) (d)Positive Chirality Negative Chirality
FIG. 4. The out-of-plane (z) component of the average magnetization, m, and n3for di erent values of input currents for both
positive and negative chirality. Positive chirality: (a) When current is increased from zero but to a value below the threshold
(0:95Jth1
s),mzis zero. However, it increases to a large value when Js=Jth1
s.mzdecreases again to a smaller value when
the current is decreased to Js= 0:86Jth1
s. Finally, when the current is further reduced below the lower threshold to 0 :9Jth2
s,
mzbecomes zero again. (b) n3is initially equal to 1 =p
2, but decreases in magnitude during the AFM dynamics since the
magnitude of mincreases when the sublattice vectors move out of the plane. As soon as the current is lowered below Jth2
s,
the system goes to a stationary state and n3= 1=p
2. Negative chirality: (c) Since the threshold current in this case is small,
non-zeromzis observed for all values of current considered here. (d) n3decreases in magnitude when current increases but
approaches 1=p
2 for lower values of current. Here = 0:01, andda= 4 nm.
is given as[44, 46]
f=1
2!s
=1
2~
2e
Js
Msda1
: (17)
Additionally, for small damping, there exists a lower
threshold,Jth2
s< Jth1
s, which is equal to the current
that pumps in the same amount of energy that is lost in
one time period due to damping [29, 44, 46]. The lower
threshold current is given as
Jth2
s=da2e
~Ms
2
p!E!K=da2e
~2
p
6JKe:(18)
The presence of two threshold currents enables energy-
ecient operation of the THz oscillator in the hysteretic
region [29].
The average frequency response as a function of the
input spin current and the Fourier transform of the os-
cillation dynamics is plotted in the left panel of Fig. 3
for materials with positive chirality. It can be observed
from Fig. 3(a) that the fundamental frequency for dif-
ferent lm thicknesses scales as predicted by Eq. (17)
except for low currents near Jth2
swhere non-linearity in
the form of higher harmonics appears as seen from the
FFT response in Fig. 3(e). Next, Figs. 3(b), (f) show
that the non-linearity in the frequency response for low
input current increases as the value of the damping co-
ecient increases. This is expected as the contribution
from the uniaxial anisotropy (sin 2 'term) becomes sig-
ni cant owing to large damping and low current making
the motion non-uniform ( '6= 0) [29].
In the case of materials with negative chirality, small
equivalent anisotropy suggests that the threshold current
for the onset of oscillations is very small, while the fre-
quency of oscillations increases linearly with the spin cur-
rent considered here and is given by Eq. (17). Indeed the
same can be observed from Figs. 3(c), (d) where the re-
sults of numerical simulations exactly match the analyticexpression. The FFT signal in Figs. 3(g), (h) contains
only one frequency corresponding to uniform rotation of
the order ( '= 0). This coherent rotation of the order
parameter with such small threshold current [68] in AFM
materials with negative chirality opens up the possibil-
ity of operating such AFM oscillators at very low en-
ergy for frequencies ranging from MHz-THz. It can also
be observed from Fig. 3(b), (d) that for lower values of
damping, such as = 0:005, the frequency of oscillations
saturates for input current slightly above Jth1
s. This is
because the energy pumped into the system is larger than
that dissipated by damping. As a result the sublattice
vectors move out of the easy-plane and get oriented along
the spin polarization direction (slip-
op). For larger val-
ues of damping, the same would be observed for larger
values of current. Finally, we would like to point out that
the values of both Jth1
sandJth2
sobserved from numerical
simulations were slightly di erent from their analytical
values for di erent damping constants, similar to that
reported in Ref. [43] for collinear AFMs.
Figure 4 shows the out-of-plane (z) components of m
andn3for non-zero currents for AFMs with di erent chi-
ralities. For negative chirality materials, the steady-state
zcomponent of both mandn3does not oscillate with
time ( '0), whereas, for the case of positive chirality,
the steady-state zcomponents of both mandn3show
small oscillations with time ( '6= 0) similar to the case
of NiO with spin polarization along the hard axis [29].
It can be observed from Fig. 4(a) that for positive chi-
rality, as current increases from below the upper thresh-
old current (0.95 Jth1
s) toJth1
s, the out-of-plane compo-
nent of magnetization vectors and hence the average mag-
netization mincreases from zero to a larger value. Due
to the hysteretic nature of the AFM oscillator, the mag-
nitude of mreduces when current is lowered but is non-
zero as long as the input current is above Jth2
s. Similarly,
it can be observed from Fig. 4(b) that the out-of-plane9
component of n3, which was initially 1 =p
2, decreases
as the current increases above Jth1
s. When the current
is lowered to a value below Jth1
s(0:86Jth1
shere) , the
magnitude of the out-of-plane component of n3increases
again and eventually saturates to 1 =p
2 when the current
is lowered further below Jth2
s(0:9Jth2
s, here).
It can also be observed from Fig. 4(c) that for negative
chirality AFMs, the out-of-plane component of the av-
erage magnetization although small is non-zero even for
small currents due to the lower value of threshold current.
On the other hand, nz
3in Fig. 4(d) decreases in magnitude
from an initial value of = 1=p
2 to<1=p
2 as current
increases sincekmkincreases. The values of current are
assumed to be the same for both positive and negative
chirality AFMs for the sake of comparison. Next, using
(a)
(b)
(c)
(d)
FIG. 5.mzfor four di erent values of input current Js.mz
increases with current and so does the frequency of oscillation.
Here (a)Js= 1:6Jth2
s, (b)Js= 0:9Jth1
s. They are both inside
the hysteretic region bounded by Jth2
sandJth1
s. (c)Js=Jth1
s,
and (d)Js= 1:5Jth1
slies outside the hysteretic region. These
results correspond to = 0:01, andda= 4 nm.
Eq. (9) in Eq. (10), it can be shown that _ 'is directly pro-
portional to mz[56]. Therefore, to present the features
of angular velocity with input current, we show mzfor
four di erent values of input current Jsfor positive chi-
rality material in Fig. 5 . Here, Figs. 5(a)-(c) correspond
to the hysteretic region, whereas Fig. 5(d) is for current
outside the hysteretic region. As mentioned previously,
an increase in current increases the spin torque on the
sublattice vectors which leads to an increase in mzand
hence _'.
IV. SIGNAL EXTRACTION
An important requirement for the realization of an
AFM-based auto-oscillator is the extraction of the gen-
erated THz oscillations as measurable electrical quanti-ties viz. voltage and current. It is expected for the ex-
tracted voltage signal to oscillate at the same frequency
as that of the N eel vector and contain substantial out-
put power ( >1W) [69]. In this regard, the landmark
theoretical work on NiO based oscillator [29] suggested
the measurement of spin pumped [70] time varying in-
verse spin Hall voltage [71] across the heavy metal (Pt)
of a NiO=Pt heterostructure. However, the time varying
voltage at THz frequency requires an AFM with signif-
icant in-plane biaxial anisotropy [29, 69], thus limiting
the applicability of this scheme to only select AFM ma-
terials. In addition, the output power of the generated
signal is sizeable (above 1 W) only for frequencies below
0:5 THz [69]. A potential route to overcoming the afore-
mentioned limitations is coupling the AFM signal genera-
tor to a high-Q dielectric resonator, which would enhance
the output power even for frequencies above 0 :5 THz [41].
This method, however, requires devices with sizes in the
10's micrometers range for frequencies above 2 THz and
for the AFMs to possess a tilted net magnetization in
their ground state [69]. A more recent theoretical work
on collinear AFM THz oscillators [43] suggested employ-
ing Anisotropy Magnetoresistance (AMR) or Spin Mag-
netoresistance (SMR) measurements in a four terminal
AFM/HM spin Hall heterostructure. This would enable
the extraction of the THz oscillations as longitudinal or
transverse voltage signals. However, the reported values
of both AMR and SMR at room temperatures in most
AFMs is low and would, in general, require modulating
the band structure for higher values [72].
Ideal Bias
RLC
LR(t)
A TJ
CbtLbt
Pac(a)
RLPac (b)
Zth
UthIread
FIG. 6. (a) An equivalent circuit representation of Fig. 1
(adapted from [69]). The generation (write) current is not
shown in the circuit, although its e ect is included as a varia-
tion in the resistance R(t) through its frequency dependence.
(b) Thevenin equivalent of (a).
A recent theoretical work [69] proposed employing a
four terminal AFM tunnel junction (ATJ) in a spin Hall
bilayer structure with a conducting AFM to e ectively
generate and detect THz frequency oscillations as vari-
ations in the tunnel anisotropy magnetoresistance [54].
A DC current passed perpendicularly to the plane of
the ATJ generates an AC voltage, which is measured
across an externally connected load. It was shown that
both the output power and its eciency decrease as fre-
quency increases, nevertheless, it was suggested that this
scheme could be used for signal extraction in the fre-
quency range of 0 :1 10 THz, although the lateral size10
of the tunnel barrier required for an optimal performance
depends on the frequency of oscillations (size decreases as
the frequency increases) [69]. The analysis presented in
Ref. [69], however, neglects the generation current com-
pared to the read current while evaluating the eciency
of power extraction. But it can be observed from the
results in Section III C that the threshold current, and,
therefore, the generation current, depend on AFM ma-
terial properties, such as damping, anisotropy, and ex-
change constants, and could be quite large. Therefore, in
our work we include the e ect of the generation current
to accurately model the power eciency of the TAMR
scheme.
(a) (b)
FIG. 7. (a) Output power and (b) eciency dependence on
the area of cross-section of the tunnel barrier for di erent fre-
quencies. The thickness of the barrier is xed to db= 1 nm.
The e ect of write current and the input power associated
with it is not considered here, therefore, these results are in-
dependent of the choice of the AFM material.
In order to evaluate the performance of the TAMR
scheme, an equivalent circuit representation (adapted
from Ref. [69]) of the device setup of Fig. 1 is shown
in Fig. 6(a), while its Thevenin equivalent representation
is shown in Fig. 6(b). The circuits in Fig. 6 only repre-
sent the read component, while the THz generation com-
ponent is omitted for the sake of clarity. In Fig. 6(a),
the dashed red box encloses a circuit representation of
the ATJ, comprising a series combination of an oscillat-
ing resistance R(t) =R0+ Rcos!tand inductance
L=0db, connected in parallel to a junction capacitor
C=Ac0=db(assumed parallel plate). The constant
component, R0, in the oscillating resistance, R(t), is the
equilibrium resistance of the MgO barrier and is given
asR0=RA(0) exp(db)
Ac. Here,RA(0) is the resistance-
area product of a zero-thickness tunnel barrier, is the
tunneling parameter, dbis the barrier thickness, and Ac
is the cross-sectional area. The pre-factor, R, of the
time varying component of R(t) is the resistance variation
due to the oscillation of the magnetization vectors with
respect to the polarization axis. R= (=(2 +))R0,
whereis the TAMR ratio of the barrier and depends
on the temperature and material properties.
Due to the
ow of the DC current, Iread, an alter-
nating voltage develops across the ATJ, which is mea-
sured across an externally connected load RL, separated
from the ATJ via an ideal bias tee (enclosed in the green
dashed box). The bias-tee, characterized by an induc-
tanceLbtand a capacitance Cbt, and assumed to have noTABLE II. List of common antiferromagnetic materials Mn 3X
and their associated parameters. Here Msis in Tesla,Keis
in kJ=m3, andJis in MJ=m3. Sign ofDwhich decides the
chirality is also mentioned.
X M sKeJ D Ref.
Ir 0:01 1:63 3000 240 - [56]
Pt 0:013 1:37 10 280 - [73]
Rh 0:013 2:00 10 230 - [73{75]
Ga 0:008 0:54 100 110 + [75{78]
Sn 0:003 0:50 110 59 + [67, 75, 77, 79]
Ge 0:0009 0:28 1320 77 + [75, 77, 78, 80]
GaN 0:1 0:69 10 280 - [81, 82]
NiN 0:1 1:54 10 177 - [81, 82]
voltage drop across it, blocks any DC current from
ow-
ing into the external load. Therefore, the AC voltage of
the ATJ is divided only into its impedance (a combina-
tion ofR0,L, andC) and that of the load RL[69].
Next, we simplify the ATJ circuit into a Thevenin
impedance Zthand voltage Uthas shown in Fig. 6(b).
They are evaluated as
Zth=R0+j!L
(1 ) +j ; (19)
and
Uth=Uac
(1 ) +j ; (20)
wherej=p 1,!= 2f,=!2LC, =!R0C, and
Uac=IreadR. The output voltage and average power
across the load can then be obtained as
UL=UthRL
Zth+RL=Uacr
1 +jp+r(1 +j );(21)
and
PL=1
2jULj2
RL=U2
ac
2RLr2
1 +qr2+ 2r+p2; (22)
wherer=RL=R0,q= (1 )2+ 2, andp=!L
R0. Finally,
the eciency of the power extraction can be obtained as
=PL
Pin
=0:5r
1 +qr2+ 2r+p21
I2
writeRGenR0=U2ac+ 1;(23)
whereRGenis the resistance faced by the generation cur-
rent. It can be observed from Eqs. (21)-(23) that the
output voltage, output power, and the eciency of power
extraction decrease with an increase in frequency since ,
,q, andpincrease with ![69, 83].
Considering that the load impedance is xed to 50
by the external circuit, one can only optimize the source
impedance to achieve PL>1W andUL>1 mV. In
this regard, the resistance of the source tunnel barrier can11
TABLE III. Material Parameters of the NM, and at the
NM/AFM interface.
Parameters Values Ref.
gM 3:81010S/m2[50]
gm 3:8109S/m2[50]
Cu 610 9
m2[50]
tCu 5 nm [50]
be altered by either varying the thickness of the tunnel
barrier,db, or its cross-sectional area, Ac. However, the
optimum values of dbandAcfor the desired output sig-
nals is frequency dependent, and, therefore, tunnel barri-
ers of di erent sizes would be required for di erent oper-
ating frequencies [69, 83]. For all estimates, we consider
db= 1 nm,= 1:3,= 5:6 nm 1,RA(0) = 0:14
m2,
and= 9:8 [69]. For reliable operation of the tunnel bar-
rier, we consider the electric eld across the barrier to be
E= 0:3 V=nm [69], which is below the barrier break-
down eld. Ignoring the e ect of the generation current
in Eq. (23), as suggested in Ref. [69], we deduce from
Fig. 7 that the optimal cross-sectional area Ac0:36
m2forf= 0:1 THz,Ac0:25m2forf= 1 THz,
Ac0:16m2forf= 10 THz.
IrSnPtRhGeGaGaNNiN
X10-710-510-3ζ(%)
f=2.0THz
FIG. 8. Power eciency for di erent materials Mn 3X listed in
Table II. The dashed horizontal line shows the expected e-
ciency of= 0:011% for the optimized geometry ( db= 1 nm,
andAc= 0:24m2) if write current is neglected. The e-
ciency, however, decreases signi cantly due to the inclusion
of write current. Here the AFM thin- lm thickness dais as-
sumed to be 4 nm.
Table II lists the material properties of various con-
ducting AFMs. Depending on the sign of their DMI
constant, these AFMs could host moments with either
a positive or a negative chirality. The closed-form model
presented in Eq. (17) can be used to evaluate the re-
quired spin current for frequency f= 2 THz, regardless
of the chirality since frequency scales linearly with theinput current in this region (see Fig. 3). For a given spin
current density ( Js), the charge current density ( Jwrite)
for the lateral spin-valve structure of Fig. 1(a) is given as
Jwrite =gM+gm
gM gmJs: (24)
wheregMandgmare the conductance of the majority-
and minority-spin electrons at the NM (Cu)/FM inter-
face. The input power required to start the oscilla-
tions is given as ( JwriteAc)2RCu, whereRGen=RCu=
CuLCu
ACu=CupAc
tCupAcis the resistance of the copper
(NM) underneath the bottom MgO. In order to evaluate
the resistance of the copper layer, we have assumed its
length and width to be the same as MgO and the AFM
thin- lm.
The eciency of power extraction for the listed AFM
materials is presented in Fig. 8. The dashed horizon-
tal line denotes the expected eciency of = 0:011% if
the e ect of generation current is neglected and the area
of cross-section of MgO is optimized for f= 2:0 THz.
However, it can be observed that the eciency decreases
signi cantly i.e. by a few orders when the input power
due to the generation current in included in the analy-
sis. For materials with large damping and large uniaxial
anisotropy constants, the required generation current is
higher leading to lower eciency. This result shows that
further optimization of the device geometry for di erent
materials is required to increase the eciency.
This method of power extraction could be more suit-
able for materials with negative chirality. We can observe
from Fig. 9 that the output power as well as the eciency
for both Mn 3Sn and Mn 3Ge for frequencies between 0.1
THz and 2.0 THz are signi cant. The required genera-
tion current for Mn 3Ge is smaller than that for Mn 3Sn,
therefore, the eciency is higher for the former. Also,
the eciency of power extraction increases with decrease
in area of cross-section in both the cases but this is ac-
companied by a decrease in output power.
It might be possible to increase the output power and
overall eciency of the system if the material properties
of the tunnel barrier such as ;, andRA(0) could be
altered. Large room temperature tunneling magnetore-
sistance in an ATJ is feasible either by using a tunnel
barrier other than MgO [84] or inserting a di usion bar-
rier to enhance magneto-transport [85]. Here we adopted
the TAMR extraction scheme because we have consid-
ered metallic AFMs so a DC current through the ATJ
structure can be easily applied. In addition, the three-
or four-terminal compact ATJ structure along with its
small lateral size enables dense packing of several such
THz oscillators on a chip accompanied with a net in-
crease in the output power and eciency of the oscillator
array [72]. For example, with an array of 10 10 such
AFM oscillators excited in parallel, the output power and
eciency could be scaled up by 100 compared to the
results presented in Figs. 7 and 8.12
(a) (b)
(c) (d)
FIG. 9. Upper panel: (a) Output power and (b) eciency for
Mn3Sn. Lower panel: (c) Output power and (d) eciency for
Mn3Ge. Two di erent cross-section size of the MgO barrier is
considered. The output power depends only on the frequency
of oscillation and therefore is same for both the materials.
The eciency of power extraction depends on the generation
current, which is lower for Mn 3Ge, leading to a higher value
of eciency in that case.
V. EFFECTS OF INHOMOGENEITY DUE TO
EXCHANGE INTERACTION
The results presented in Section III C correspond to
the case of a single-domain AFM particle and are, there-
fore, independent of the lateral dimensions of the thin-
lm. This can also be deduced from the equations of the
threshold current and the average oscillation frequency.
However, when the lateral dimensions of the AFM thin-
lm exceed several 10's of nm, micromagnetic analysis
must be carried out. In this section, we analyze the dy-
namics in thin- lm AFMs of varying dimensions within a
micromagnetic simulation framework. We consider AFM
thin- lms of dimensions 50 nm 50 nm and investigate
the e ect of the inhomogeneity due to exchange interac-
tions. In each case, the thin- lm was divided into smaller
cubes, each of size 1 nm 1 nmdanm, since the do-
main wall width 0=p
(2Aii Aij)=(2Ke)>1 nm for
Kecorresponding to Mn 3Ir as listed in Table II. It can
be observed from Fig. 10 that for materials with positive
chirality the e ects of inhomogeneity becomes important
for low currents. On the other hand, for materials with
negative chirality, inhomogeneities do not appear to have
any e ect. For positive chirality materials, the numerical
values of frequency for di erent spring constants deviates
signi cantly from that obtained from the single domain
solution, as well as analytic results. In this case, the hys-
teretic region reduces in size since the lower threshold
current increases in magnitude as compared to the the-
oretical prediction as can be observed from Fig. 10(a).
Positive Chirality Negative Chirality
(a) (b)
(c) (d)FIG. 10. Frequency vs. input current for di erent values
of inhomogeneous exchange constants (intra-sublattice (a, c),
inter-sublattice (b, d)) for both positive and negative chirality.
In all cases = 0:01, andda= 4 nm. Other parameters
correspond to those of Mn 3Ir as listed in Table II for both
positive and negative chirality materials with the exception
of the sign ofDfor the latter.
While we have not included the e ect of inhomogeneous
DMI in our work, we expect such interactions to lead to
the formation of domain walls in the thin- lm similar to
the case of collinear AFMs [43]. A more detailed analy-
sis of the dynamics of the positive chirality materials due
to variation in exchange interaction as well as inhomoge-
neous DMI would be carried out in a future publication.
VI. DISCUSSION
We focused on the dynamics of the order parameters
in exchange dominant non-collinear coplanar AFMs with
both positive (+ =2) and negative ( =2) chiralities as-
sociated to the orientation of equilibrium magnetization
vectors. In both these classes of AFMs, the exchange en-
ergy is minimized for a 2 =3 relative orientation between
the sublattice vectors. Next, the negative (positive) sign
of the iDMI coecient minimizes the system energy for
counterclockwise (clockwise) ordering of m1;m2, andm3
in the x yplane leading to positive (negative) chiral-
ity. Finally, all the sublattice vectors coincide with their
respective easy axis only in the case of the positive chiral-
ity materials due to the relative anticlockwise orientation
of the easy axes. On the other hand, the negative chi-
rality materials have a six-fold symmetry wherein only
one of the sublattice vectors can coincide with its respec-
tive easy axis. As a result, these AFM materials with
di erent chiralities have signi cantly distinct dynamics
in the presence of an input spin current. For AFM ma-
terials with + =2 chirality, oscillatory dynamics are ex-13
cited only when the injected spin current overcomes the
anisotropy, thus indicating the presence of a larger cur-
rent threshold. Moreover, the dynamics in such AFMs is
hysteretic in nature. Therefore, it is possible to sustain
oscillations by lowering the current below that required
to initiate the dynamics as long the energy pumped in by
the current overcomes that dissipated by damping. On
the other hand, in the case of =2 chirality AFMs, os-
cillations can be excited when signi cantly smaller spin
current with appropriate spin polarization is injected into
the AFM. Hence, =2 chirality AFMs may be more
amenable to tuning the frequency response over a broad
frequency range, from the MHz to the THz range [68].
The oscillation of the AFM N eel vectors can be mea-
sured as a coherent AC voltage with THz frequencies
across an externally connected resistive load through the
tunnel anisotropic magnetoresistance measurements for
both +=2 and =2 chirality materials. In general, as
the frequency increases, the magnitude of both the out-
put power and the eciency of power extraction decrease,
however, it is possible to enhance both these quantities
by optimizing the cross-sectional area of the tunnel junc-
tion. This, however, is limited due to larger threshold
current requirement for materials with large damping.
Therefore, a hybrid scheme of electrically synchronized
AFM oscillators on a chip could be used to further en-
hance the power and eciency [86, 87].
(a) (b)
FIG. 11.mzfor larger damping, = 0:1, andda= 4 nm. (a)
Non-coherent (spike-like) signals near the threshold current
Js= 1:1Jth1
s. (b) Coherent signal for larger current Js=
1:5Jth1
s. The angular frequency is directly proportional to
mz, and therefore it would show the exact same features (in
the absence of any external eld) for the chosen values of
current.
Metallic AFMs such as Mn 3Ir and Mn 3Sn could be
considered as examples of + =2 and =2 chiralities,
respectively. Recently, thin- lms with di erent thickness
ranging from 1 nm to 5 nm of both these materials have
been grown using UHV magnetron sputtering [88{91].
In addition, di erent values of damping constants have
been reported for Mn 3Sn [56, 77]. Therefore, we expect
the results presented in Sections III C, IV and V to be
useful for benchmarking THz dynamics in experimental
set-ups with such thin lms metallic antiferromagnets.
(a) (b)
(c) (d)FIG. 12. Time dynamics (single and train of spikes) of a single
\neuron" for di erent input currents and frequencies. The net
input current should be greater than the threshold current
( > 0:2) for a non-zero dynamics. For an input current
above the threshold, as the external frequency increases the
dynamics changes from (a) bursts of spikes to (c) single spikes
to (d) no spikes ( = 0:3). As the input current increases to
= 0:4 the range of external frequency where the spiking
behaviour is observed increases.
VII. POTENTIAL APPLICATIONS
Neurons in the human brain could be thought of as a
network of coupled non-linear oscillators, while the stim-
uli to excite neuronal dynamics is derived from the neigh-
boring neurons in the network [8, 92{94]. For materials
with +=2 chirality, a non-linear behaviour was observed
for large damping, and input currents near the threshold
current,Jth1
s, in Fig. 3(b), (f). This non-linearity cor-
responds to Dirac-comb-like magnetization dynamics, as
shown in Fig. 11(a), and is similar to the dynamics of
biological neurons in their spiking behaviour as well as
a dependence on the input threshold. However, unlike a
biological neuron which shows various dynamical modes
such as spiking, bursting, and chattering [95], the dy-
namics here shows only spikes and does not show any
refractory (\resting") period. Recent works [96, 97] have
shown that it is possible to generate single spiking as
well as bursting behaviours using NiO-based AFM oscil-
lators by considering an input DC current below Jth1
s,
and superimposing it with an AC current. As the AC
current changes with time, the total current could either
go above the threshold, thereby triggering a non-linear
response, or below the threshold current resulting in a
\resting" period. Here we explore the possibility of spik-
ing behaviours in + =2 chirality materials such as Mn 3Ir
under the e ect of an input spin current. We use the non-
linear pendulum model of Eq. (12) and study the possible
dynamics in case of a single oscillator, two unidirectional
coupled oscillators, and two bidirectional coupled oscil-14
lators.
A. Ultra-fast Hardware Emulator of Neurons
We consider a large damping of = 0:1 while the other
material parameters correspond to that of Mn 3Ir as listed
in Table II. Next we choose an input current Js(t) =
Jdc
s+Jac
s(t), whereJdc
s= 0:8Jth1
sis the dc component of
the input current, superimposed with a smaller ac signal
Jac
s(t) =Jth1
scos(2fact). The time dynamics of this
non-linear oscillator is governed by
'+ !E_'+!E!K
2sin 2'+!E!s(t) = 0; (25)
where!s(t)(/Js(t)) is the time varying input current.
(a) (b)
FIG. 13. The dynamics of two neuron system with unidirec-
tional coupling at fac= 60 GHz, and = 0:3. The dotted
blue curve corresponds to the rst neuron. (a) Second neuron
shows no spike for = 0:028 but a single spike for = 0:032.
(b) The single spiking behaviour changes to bursts with three
spikes asincreases and coupling strengthens.
Figure 12 presents the dynamics of Eq. (25) for dif-
ferent input current and frequencies. Firstly, it can be
observed that the input current must be greater than
the threshold current to excite any dynamics viz.
must be greater than 0.2 (dotted line corresponding to
= 0:2 shows no spikes for any value of external fre-
quency). Secondly, for input currents above the thresh-
old viz.=f0:3;0:4g, a train of spikes is observed for
lower frequency of 20 GHz in Fig. 12(a). However, as
frequency of the input excitation increases the number of
observed spikes decreases for both values of current con-
sidered here (Fig. 12(b, c)). Finally, it can be observed
from Fig. 12(d) that for very large frequency the spiking
behaviour vanishes for lower current ( = 0:3) but per-
sists for higher current ( = 0:4). For higher values of
current, the cut-o frequency is higher. This observed
spiking behaviour is indeed similar to that of biological
neurons [95]. Here, however, the observed dynamics is
very fast in the THz regime and thus the AFM oscilla-
tors could be used as the building blocks of an ultra-high
throughput brain-inspired computing architecture.
(a)
(b)
(c)
(d)FIG. 14. The dynamics of two neuron system with unidirec-
tional coupling at fac= 180 GHz. The dashed blue curve
corresponds to the rst neuron. = 0:3: (a) Single spike
for= 0:04 but not for = 0:036. (b) The single spik-
ing behaviour become prominent as the coupling strengthens.
= 0:4: (c) Single spike for = 0:032 in response to a double
spiking behaviour of the rst neuron. (d) For larger second
neuron shows bursting dynamics with two spikes.
B. Two unidirectional coupled arti cial neurons
A network composed of interacting oscillators forms
the basis of the oscillatory neurocomputing model pro-
posed by Hoppensteadt and Izhikevich [98]. In such a
network, the dynamics of an oscillating neuron (or a
\node") is controlled by the incoming input signal as
well as its coupling to neighboring neurons. To inves-
tigate this coupling behaviour we consider a system of
two unidirectional coupled neurons. The rst neuron is
driven by an external signal and its dynamics is governed
by Eq. (25). The dynamics of the second neuron, on the
other hand, depends on the output signal of the rst neu-
ron as well as the coupling between the two neurons. It
is governed by
'j+ !E_'j+!E!K
2sin 2'j+!E!s
ij!E_'isgn(!s) = 0;
(26)
whereij=is the unidirectional coupling coecient
from neuron i= 1 toj= 2. There is no feedback from
the second neuron to the rst and therefore ji= 0. In
addition to the input from the rst neuron, the second
neuron is also driven by a constant DC current Jdc
s2(/!s
in Eq. (26)). We choose this DC current to be the same
as that for the rst neuron viz. Jdc
s2= 0:8Jth1
s. The
dynamics of the second neuron for two di erent external
input currents ( =f0:3;0:4g) and frequencies ( fac=
f60;180gGHz) is presented in Figs. 13 and 14.15
Firstly, it can be observed that in all the cases the sec-
ond neuron shows a spiking behaviour only for above
a certain value. Secondly, for = 0:3 andfac= 60 GHz,
wherein the rst neuron shows bursting behaviour con-
sisting of three spikes, the second neuron shows a single
spike (Fig. 13(a)) for lower value of , and three spikes
for stronger coupling (Fig. 13(b)). This behaviour is due
to the threshold dependence of the second neuron as well
as due to its inertial dynamics. Similar behaviour is also
observed for = 0:4, andfac= 180 GHz in Figs. 14(c),
(d). Thirdly, for = 0:3 andfac= 180 GHz, wherein the
rst neuron shows a single spike, Fig. 14(a) shows that
compared to the case of fac= 60 GHz a slightly higher
value ofis now required to excite the second neuron.
The single spiking behaviour of the second neuron be-
comes more prominent as the coupling strength increases
because of a stronger input as shown in Fig. 14(b). Re-
cently, it was suggested that this coupled behaviour of
THz arti cial neurons could be used to build ultra-fast
multi-input AND, OR, and majority logic gates [96].
(a)
(b)
(c)
(d)
FIG. 15. The dynamics of two neuron system with bidirec-
tional coupling at fac= 180 GHz, and = 0:3. First neu-
ron shows bursting behaviour in this system while the second
neuron follows the rst neuron for all values of . As the cou-
pling between the two neurons increase the number of spikes
for both the neurons increases.
C. Two bidirectional coupled arti cial neurons
In some circuits it is possible that the coupling be-
tween any two neurons is bidirectional. In such cases, in
addition to a forward coupling from the rst neuron to
the second, a feedback exists from the second neuron to
the rst. The dynamics of each neuron of this coupled
system is governed by Eq. (26), however, !s=!s(t) for
the rst neuron, as discussed previously. We consider
12=21=. Figures 15 and 16 show the dynamics of
the two neurons of this coupled system with the couplingatfac= 180 GHz for = 0:3 and 0.4, respectively.
(a) (b)
(c) (d)
FIG. 16. The dynamics of two neuron system with bidirec-
tional coupling at fac= 180 GHz, and = 0:4. Second
neuron res when the coupling is above a certain threshold
which in turn leads to another spike for the rst neuron. As
the coupling between the two neurons increase the number of
spikes for both the neurons increases.
Firstly, Fig. 15(a) shows that for = 0:04 the dynam-
ics of both _'1and _'2are almost similar to that presented
in Fig. 14(a), viz. the e ects of coupling is very small.
However, as the coupling between the two neurons in-
creases (Fig. 15(b)-(d)), a positive feedback is established
between the two neuron leading to dynamics with two or
more spikes, in general. This is observed after the sec-
ond neuron has red, at least once, because the positive
feedback leads to a net input greater than the threshold
current to the rst neuron, even though the external in-
put has reduced below the threshold. Similar behavior is
also observed in the case of = 0:4, although at lower
values of coupling, as presented in Fig. 16. The results
allude to the threshold behaviour of the neurons, inertial
nature of the dynamics, and a dependence of the dynam-
ics on the phase di erence between the two neurons. The
dynamics of two bidirectional coupled arti cial neurons
presented here could be the rst step towards building
AFM-based recurrent neural networks or reservoir com-
puting [99], instead of the slower FM-based coupled os-
cillator systems [100, 101].
VIII. CONCLUSION
In this work, we numerically and theoretically explore
the THz dynamics of thin- lm metallic non-collinear
coplanar AFMs such as Mn 3Ir and Mn 3Sn, under the
action of an injected spin current with spin polarization
perpendicular to the plane of the lm. Physically, these
two AFM materials di er in their spin con guration viz.
positive chirality for Mn 3Ir, and negative chirality for16
Mn3Sn. In order to explore the dynamics numerically,
we solve three LLG equations coupled to each other via
inter-sublattice exchange interactions. We also analyze
the dynamics theoretically in the limit of strong exchange
and show that it can be mapped to that of a damped-
driven pendulum if the e ects of inhomogeneity in the
material are ignored. We nd that the dynamics of Mn 3Ir
is best described by a non-linear pendulum equation and
has a hysteretic behaviour, while that of Mn 3Sn in the
THz regime is best described by a linear pendulum equa-
tion and has a signi cantly small threshold for oscillation.
The hysteretic dynamics in the case of Mn 3Ir allows for
possibility of energy ecient THz coherent sources. On
the other hand, a small threshold current requirement
in the case of Mn 3Sn indicates the possibility of e-
cient coherent signal sources from MHz to THz regime.
We employ the TAMR detection scheme to extract the
THz oscillations as time-varying voltage signals across
an external resistive load. Including inhomogeneous ef-
fects leads to a variation in the dynamics | the lowerthreshold current for sustaining the dynamics increases,
the hysteretic region reduces, and the frequency of oscil-
lation decreases for lower current levels. Finally, we also
show that the non-linear behaviour of positive chirality
materials with large damping could be used to emulate
arti cial neurons. An interacting network of such oscil-
lators could enable the development of neurocomputing
circuits for various cognitive tasks. The device setup and
the results presented in this paper should be useful in
designing experiments to further study and explore THz
oscillations in thin- lm metallic AFMs.
ACKNOWLEDGEMENTS
This research is funded by AFRL/AFOSR, under
AFRL Contract No. FA8750-21-1-0002. The authors
also acknowledge the support of National Science Foun-
dation through the grant no. CCF-2021230. Ankit
Shukla is also grateful to Siyuan Qian for fruitful dis-
cussions.
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1803.07280v1.Stability_of_the_wave_equations_on_a_tree_with_local_Kelvin_Voigt_damping.pdf | STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL
KELVIN-VOIGT DAMPING
KAIS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL
Abstract. In this paper we study the stability problem of a tree of elastic strings with
local Kelvin-Voigt damping on some of the edges. Under the compatibility condition of
displacement and strain and continuity condition of damping coecients at the vertices of
the tree, exponential/polynomial stability are proved. Our results generalizes the cases of
single elastic string with local Kelvin-Voigt damping in [21, 24, 5].
Contents
1. Introduction 1
2. Well-posedness of the system 4
3. Asymptotic behaviour 6
4. Further comments: graph case 14
References 16
1.Introduction
In this paper, we investigate the asymptotic stability of a tree of elastic strings with local
Kelvin-Voigt damping. We rst introduce some notations needed to formulate the problem
under consideration. Let Tbe a tree ( i.e.Tis a planar connected graph without closed paths).
degree of a vertex { number of incident edges at that vertex
R{ root ofT, a designated vertex with degree 1
exterior vertex { vertex with degree 1
interior vertex { vertex with degree greater than 1
e{ the edge incident the root R
O{ the vertex ofTother thanR
{ multi-index of length k, = ( 1;; k)
2010 Mathematics Subject Classi cation. 35B35, 35B40, 93D20.
Key words and phrases. Tree, dissipative wave operator, Kelvin-Voigt damping, Frequency approach.
1arXiv:1803.07280v1 [math.AP] 20 Mar 20182 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL
O { vertex ofTwith index
e { edge ofTwith index
M{ set of the interior vertices of T
S{ set of the exterior vertices of T, excludingR
IM{ index set ofM
IS{ index set ofS
We choose empty index for edge eand vertexO. Assume there are m edges, di erent from
e ;that branch out from O ;we denote these edges by e o ; = 1;:::;m and the other vertex
of the edge e o byO o , i.e. the interior vertex O , contained in the edge e ;has multiplicity
equal tom + 1.
Furthermore, length of the edge e is denoted by ` . Then,e may be parametrized by its arc
length by means of the functions , de ned in [0 ;` ] such that (` ) =O and (0) is the
other vertex of this edge.
•R •O•
O1
•O2•O1,1
•
O2,2e•O1,2
•
O2,1Dirichlet boundary condition
Dirichlet boundary conditions
Figure 1. A Tree-Shaped network
Now, we are ready to introduce a planar tree-shaped network of Nelastic strings, where N2,
see [20, 22, 26, 17, 19] and [16] concerning the model. More precisely, we consider the following
initial and boundary value problem :
(1.1)@2u
@t2(x;t) @
@x@u
@x+a (x)@2u
@x@t
(x;t) = 0;0<x<` ; t> 0; 2I:=IM[IS;
(1.2) u(0;t) = 0; u (` ;t) = 0; 2IS; t> 0;
(1.3) u (0;t) =u (` ;t); t> 0; = 1;2;:::;m ; 2IM;
(1.4)
m X
=1@u
@x(0;t) +a (0)@2u
@x@t(0;t) =@u
@x(` ;t) +a (` )@2u
@x@t(` ;t); t> 0; 2IM;STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 3
(1.5) u (x;0) =u0
(x);@u
@t(x;0) =u1
(x);0<x<` ; 2I;
whereu : [0;` ](0;+1)!R; 2I;be the transverse displacement with index ,a 2
L1(0;` ) and, either a is zero, that is, e is a purely elastic edge, or a (x)>0, on (0;` ).
Such edge will be called a K-V edge. This setting also includes the case that a (x)>0 only on
subintervals of (0 ;` ), since we then can consider this edge as the union of pure elastic edges
and K-V edges.
We assume thatTcontains at least one K-V edge. Furthermore, we suppose that every maximal
subgraph of purely elastic edges is a tree, whose leaves are attached to K-V edges.
Models of the transient behavior of some or all of the state variables describing the motion
of
exible structures have been of great interest in recent years, for details about physical
motivation for the models, see [16], [20] and the references therein. Mathematical analysis
of transmission partial di erential equations is detailed in [20]. For the feedback stabilization
problem the wave or Schr odinger equations in networks, we refer the readers to references [7]-[11],
[20].
Our aim is to prove, under some assumptions on damping coecients a , 2I, exponential
and polynomial stability results for the system (1.1)-(1.5).
We de ne the natural energy E(t) of a solution u= (u ) 2Iof (1.1)-(1.5) by
(1.6) E(t) =1
2X
2IZ`
0 @u
@t(x;t) 2
+ @u
@x(x;t) 2!
dx:
It is straightforward to check that every suciently smooth solution of (1.1)-(1.5) satis es the
following dissipation law
(1.7)d
dtE(t) = X
2IZ`
0a (x) @2u
@x@t(x;t) 2
dx0;
and therefore, the energy is a nonincreasing function of the time variable t.
The main results of this paper then concern the precise asymptotic behavior of the solutions of
(1.1)-(1.5). Our technique is a special frequency domain analysis of the corresponding operator.
This paper is organized as follows: In Section 2, we give the proper functional setting for system
(1.1)-(1.5) and prove that the system is well-posed. In Section 3, we analyze the resolvent of the
wave operator associated to the dissipative system (1.1)-(1.5) and prove the asymptotic behavior
of the corresponding semigroup. In the last section we give some comments on the cases of more
general graph for the network.4 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL
2.Well-posedness of the system
In order to study system (1.1)-(1.5) we need a proper functional setting. We de ne the following
space
H=VH
whereH=Y
2IL2(0;` ) andV=(
u2Y
2IH1(0;` ) :u(0) = 0; u (` ) = 0; 2IS;satis es (2.8))
(2.8) u (0) =u (` ); = 1;:::;m ; 2IM;
and equipped with the inner products
(2.9) <u;~u>H=X
2IZ`
0(u (x)~u (x) +u0
(x)~u0 (x))dx:
System (1.1)-(1.5) can be rewritten as the rst order evolution equation
(2.10)8
>>><
>>>:@
@t0
@u
@u
@t1
A=Ad0
@u
@u
@t1
A;
u(0) =u0;@u
@t=u1
where the operator Ad:D(Ad)H!H is de ned by
Ad0
@u
v1
A:=0
@v
(u0+av0)01
A;
with
a:= (a ) 2Iandav0:= (a v0
) 2I;
and
D(Ad) :=(
(u;v)2H; v2V;(u0+av0)2Y
2IH1(0;` ) : (u;v) satis es (2 :11))
;
(2.11)m X
=1u (0) +a (0)v0
(0) =u0
(` ) +a (` )v0
(` ); 2IM:
Lemma 2.1. The operatorAdis dissipative,f0;1g(Ad) :the resolvent set of Ad:
Proof. For (u;v)2D(Ad);we have
Re(hAd(u;v);(u;v)iH) =ReX
2I Z`
0v0
u0
dx+Z`
0(u0
+a v0
)0v dx!
:
Performing integration by parts and using transmission and boundary conditions, a straightfor-
ward calculations leads to
Re(hAd(u;v);(u;v)iH) = X
2IZ`
0a (x)jv0
(x)j2dx0STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 5
which proves the dissipativeness of the operator AdinH:
Next, using Lax-Milgram's lemma, we prove that 1 2(Ad):For this, let ( f;g)2H and we look
for (u;v)2D(Ad) such that
(I Ad)(u;v) = (f;g)
which can be written as
u v =f ; 2I; (2.12)
v (u0
+a v0
)0=g ; 2I: (2.13)
Letw2V; multiplying (2.13) by w , then summing over 2I, we obtain
X
2IZ`
0v w dx X
2I[(u0
+a v0
) (x)w (x)]`
0+
(2.14)X
2IZ`
0(u0
+a v0
)w0
dx=X
2IZ`
0g w dx:
Replacingv in the last equality by (2.12), we get
(2.15) '(u;w) = (w)
where
'(u;w) =X
2I Z`
0u w dx+Z`
0(1 +a )u0
w0
!
and
(w) =X
2I Z`
0(f +g )w dx+Z`
0a f0
w0
dx!
:
The fuction 'is a continuous sesquilinear form on VVand is a continuous anti-linear form
onV; hereVis equipped with the inner product
f;g
=X
2I Z`
0u w dx+Z`
0u0
w0
!
:
Since'is coercive on V;the conclusion is deduced by the Lax-Milgram lemma :
By the same why we prove that 0 2(Ad).
By the Lumer-Phillip's theorem (see [27, 29]), we have the following proposition.
Proposition 2.2. The operatorAdgenerates aC0-semigroup of contraction (Sd(t))t0on the
Hilbert spaceH.
Hence, for an initial datum (u0;u1)2H, there exists a unique solution
u;@u
@t
2C([0;+1);H)
to problem (2.10). Moreover, if (u0;u1)2D(Ad), then
u;@u
@t
2C([0;+1);D(Ad)):6 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL
Furthermore, the solution ( u;@u
@t) of (1.1)-(1.5) with initial datum in D(Ad) satis es (1.7). There-
fore the energy is decreasing.
3.Asymptotic behaviour
In order to analyze the asymptotic behavior of system (1.1)-(1.5), we shall use the following
characterizations for exponential and polynomial stability of a C0-semigroup of contractions:
Lemma 3.1. [18, 28] AC0-semigroup of contractions (etA)t0de ned on the Hilbert space H
is exponentially stable if and only if
(3.16) iR(A)
and
(3.17) lim sup
j j!+1
(i I A ) 1
L(H)<1
Lemma 3.2. [14]AC0-semigroup of contractions (etA)t0on the Hilbert space Hsatis es
etAy0
C
t1
ky0kD(A)
for some constant C > 0and for >0if and only if (3.16) holds and
(3.18) lim sup
j j!+11
(i I A ) 1
L(H)<1:
Lemma 3.3 (Asymptotic stability) .The operatorAdveri es (3.16) and then the associated
semigroup (Sd(t))t0is asymptotically stable on H.
Proof. Since 02(Ad) we only need here to prove that ( i I Ad) is a one-to-one correspondence
in the energy space Hfor all 2R. The proof will be done in two steps: in the rst step we
will prove the injective property of ( i I Ad) and in the second step we will prove the surjective
property of the same operator.
Suppose that there exists 2Rsuch thatKer(i I Ad)6=f0g. So=i is an
eigenvalue ofAd;then let (u;v) an eigenvector of D(Ad) associated to :For every in
Iwe have
v =i u ; (3.19)
(u0
+a v0
)0=i v : (3.20)
We have
hAd(u;v);(u;v)iH=X
2IZ`
0a jv0
j2dx= 0:
Thena v0
= 0 a.e. on (0 ;` ).STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 7
Lete a K-V edge. According to (3.19) and the fact that a v0
= 0 a.e. on (0 ;` );
we haveu0
= 0 a.e. on ! :Using (3.20), we deduce that v = 0 on! :Return back to
(3.19), we conclude that u = 0 on! :
Puttingy=u0
+a v0
= (1 + i a )u0
;we havey2H2(0;` ) andy0= 2u :
Henceysatis es the Cauchy problem
y00+ 2
1 +i a y= 0; y(z0) = 0; y0(z0) = 0
for somez0in! :Thenyis zero on (0 ;` ) and hence u0
andu are zero on (0 ;` ).
Moreoveru andu0
+a v0
vanish at 0 and at ` .
Ife is a purely elastic edge attached to a K-V edge at one of its ends, denoted by
x ;thenu (x ) = 0; u0
(x ) = 0:Again, by the same way we can deduce that u0
and
u are zero in L2(0;` ) and at both ends of e . We iterate such procedure on every
maximal subgraph of purely elastic edges of T(from leaves to the root), to obtain nally
that (u;v) = 0 inD(Ad);which is in contradiction with the choice of ( u;v):
Now given ( f;g)2H, we solve the equation
(i I Ad)(u;v) = (f;g)
or equivalently,
(3.21)8
<
:v=i u f
2u+u00+i (au0)0= (af0)0 i f g:
Let's de ne the operator
Au= u00 i (au0)0;8u2V:
It is easy to show that Ais an isomorphism from VontoV0(whereV0is the dual space
ofVobtained by means of the inner product in H). Then the second line of (3.21) can
be written as follow
(3.22) u 2A 1u=A 1
g+i f (af0)0
:
Ifu2Ker(I 2A 1), then 2u Au= 0. It follows that
(3.23) 2u+u00+i (au0)0= 0:
Multiplying (3.23) by uand integrating over T, then by Green's formula we obtain
2X
2IZ`
0ju (x)j2dx X
2IZ`
0ju0
(x)j2dx i X
2IZ`
0a (x)ju0
(x)j2dx= 0:
This shows that
X
2IZ`
0a (x)ju0
(x)j2dx= 0;8 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL
which imply that au0= 0 inT.
Inserting this last equation into (3.23) we get
2u+u00= 0; inT:
According to the rst step, we have that Ker( I 2A 1) =f0g. On the other hand
thanks to the compact embeddings V ,!HandH ,!V0we see that A 1is a compact
operator in V. Now thanks to Fredholm's alternative, the operator ( I 2A 1) is
bijective in V, hence the equation (3.22) have a unique solution in V, which yields that
the operator ( i I Ad) is surjective in the energy space H. The proof is thus complete.
Before stating the main results, we de ne a property (P) on aas follows
(P)8 2I; a0
;a00
2L1(0;` ) and8 2IM; a0
(` ) m X
=1a0
(0)0:
Theorem 3.4. Suppose that the function asatis es property (P), then
(i)Ifais continuous at every inner node of Tthen (Sd(t))t0is exponentially stable on H.
(ii)Ifais not continuous at least at an inner node of Tthen (Sd(t))t0is polynomially
stable onH, in particular there exists C > 0such that for all t>0we have
eAt(u0;u1)
HC
t2
(u0;u1)
D(A);8(u0;u1)2D(A):
Proof. According to Lemma 3.1, Lemma 3.2, and Lemma 3.3, it suces to prove that for
= 0,
whenais continous at every inner node, or
= 1=2, whenais not continuous at an inner node,
there exists r>0 such that
(3.24) inf
k(u;v)kH; 2R
k(i I Ad)(u;v)kHr:
Suppose that (3.24) fails. Then there exists a sequence of real numbers n, with n!1
(without loss of generality, we suppose that n>0 ), and a sequence of vectors ( un;vn) in
D(Ad) withk(un;vn)kH= 1 such that
(3.25)
nk(i nI Ad)(un;vn)kH!0:
We shall prove that k(un;vn)kH=o(1);which contradict the hypotheses on ( un;vn):
Writing (3.25) in terms of its components, we get for every 2I;
n(i nu ;n v ;n) =:f ;n=o(1) inH1(0;` ); (3.26)
n(i nv ;n (u0
;n+a v0
;n)0) =:g ;n=o(1) inL2(0;` ): (3.27)STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 9
Note that
nX
2IZ`
0a (x)jv0
(x)j2dx=Re(h
n(i nI Ad)(un;vn);(un;vn)iH) =o(1):
Hence, for every 2I
(3.28)
2n
a1
2
v0
;n
L2(0;` )=o(1):
Then from (3.26), we get that
(3.29)
2n
a1
2
nu0
;n
L2(0;` )=o(1):
De neT ;n= (u0
;n+a v0
;n) and multiplying (3.27) by
nqT ;nwhereqis any real function
inH2(0;` ), we get
(3.30) ReZ`
0i nv ;nqT ;ndx ReZ`
0T0
;nqT ;ndx=o(1):
Using (3.26) we have
ReZ`
0i nv ;nqT ;ndx
= ReZ`
0v ;nq(v0
;n+
nf0
;n)dx+ReZ`
0i nv ;nqa v0
;ndx
= 1
2h
q(x)jv ;n(x)j2i`
0+1
2Z`
0q0jv ;nj2dx ImZ`
0qa nv ;nv0
;ndx+o(1): (3.31)
On the other hand, integrating the second term in (3.30) by parts, yields
(3.32) ReZ`
0T0
;nqT ;ndx=1
2h
q(x)jT ;n(x)j2i`
0 1
2Z`
0q0jT ;nj2dx:
Hence, by substituing (3.31) and (3.32) into (3.30), we obtain
1
2Z`
0q0jv ;nj2dx+1
2Z`
0q0jT ;nj2dx ImZ`
0qa nv ;nv0
;ndx
1
2h
q(x)jv ;n(x)j2i`
0+h
q(x)jT ;n(x)j2i`
0
=o(1): (3.33)
Lemma 3.5. The following property holds
(3.34) ImZ`
0qa nv ;nv0
;ndx=o(1):
Proof. Since
2na1
2
v0
;n!0 inL2(0;` ) andqa1
2
2L1(0;` );it suces to prove that
(3.35) 1
2n
a1
2
v ;n
L2(0;` )=O(1):
For this, taking the inner product of (3.27) by i 1 2
na v ;nleads to
(3.36) 2
n
a1
2
v ;n
2
L2(0;` )= i 1
nZ`
0T0
;na v ;ndx i 1 2
nZ`
0g ;na v ;ndx:10 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL
Sincea 2L1(0;` ) andg ;n!0 inL2(0;` ) we can deduce the inequality
(3.37) Re(i 1 2
nZ`
0g ;na v ;ndx)1
4 2
n
a1
2
v ;n
2
L2(! )+o(1):
On the other hand, we have
Re
i 1
nZ`
0T0
;na v ;ndx!
= Re
i 1
nT ;n(x)a (x)v ;n(x)`
0
+Re"
i 1
nZ`
0
a0
u0
;nv ;n+a a0
v0
;nv ;n+a u0
;nv0
;n
dx#
: (3.38)
Using (3.28) and (3.29) we have
(3.39) Re
i 1
nZ`
0a u0
;nv0
;ndx!
=o(1):
Using again (3.28) and the fact that a0
2L1(0;` );we conclude that
(3.40) Re
i 1
nZ`
0a a0
v0
;nv ;ndx!
1
4 2
n
a1
2
v ;n
2
L2(0;` )+o(1):
Now by (3.27), we obtain after integrating by parts that
Re"
i 1
nZ`
0a0
u0
;nv ;ndx#
=Re"
nZ`
0a0
(v0
;n+
nf0
;n)v ;ndx#
=1
2h
na0
(x)jv ;n(x)j2i`
0 1
2
nZ`
0a00
jv ;nj2dx+o(1):
Furthermore, using that a00
2L1(0;` ) and thatv ;nis bounded, we deduce
(3.41) Re"
i 1
nZ`
0a0
u0
;nv ;ndx#
1
2h
na0
(x)jv ;n(x)j2i`
0+O(1):
Combining (3.39), (3.40), (3.41) with (3.38), we get
Re(i 1
nZ`
0T0
;nav ;ndx) Re
i 1
nT ;n(x)a (x)v ;n(x)`
0
+1
2h
na0
(x)jv ;n(x)j2i`
0+1
4 2
n
a1
2
v ;n
2
L2(0;` )+O(1): (3.42)
Thus, substituting (3.37) and (3.42) into (3.36) leads to
1
2 2
n
a1
2
v ;n
2
L2(0;` ) Re
i 1
nT ;n(x)a (x)v ;n(x)`
0
+X
2I
a0
(` )jv ;n(` )j2 a0
(0)jv ;n(0)j2
+O(1) (3.43)
Case (i): Here ais continuous in all nodes. For
= 0, it follows from (3.43) that
(3.44)X
2I 2
n
a1
2
v ;n
2
L2(0;` )2X
2I
a0
(` )jv ;n(` )j2 a0
(0)jv ;n(0)j2
+O(1):STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 11
We have used the continuity condition of vandaand the compatibilty condition (1.5) at inner
nodes and the Dirichlet condition of uandvat externel nodes.
To conclude, notice that from the property (P) we deduce that
X
2I
a0
(` )jv ;n(` )j2 a0
(0)jv ;n(0)j2
0:
Then, (3.44), yields
2
n
a1
2
v ;n
2
L2(0;` )=O(1)
for every 2I;and the proof of Lemma 3.5 is complete for case (i).
Case (ii): Recall that here the function ais not continuous at some internal nodes. For
=1
2,
we want estimate the rst term in the right hand side of (3.43). To do this it suces to estimate
Re(i 1
nT ;n(x)a (x )v ;n(x)) at an inner node x=x whena (x )6= 0. For simplicity and
without loss of generality we suppose that x is the end of e ;nidenti ed to 0 via .
Sincea is continuous on [0 ;` ], there exists a positive number k <` such thata (x)6= 0 on
[0;` ]. We rst prove
(3.45) nkv ;nk2
L2(0;k )=o(1):
We need the following Gagliardo-Nirenberg inequality [25] in estimation:
There exists two positives constants C1andC2such that, for any winH1(0;k ),
(3.46) kwkL1(0;k )C1kwk1
2
L2(0;k )kw0k1
2
L2(0;k )+C2kwkL2(0;k ):
Multiplying (3.27) by i 1
nv ;ninL2(0;` ) and integrating by parts, we obtain
1
2nkv ;nk2
L2(0;` )= i 1
2n[T ;n(x)v ;n(x)]k
0+i 1
2nZk
0T ;nv0
;ndx+o(1):
By (3.28) and (3.29) we have i 1
2nRk
0T ;nv0
;ndx=o(1):
Using Gagliardo-Nerenberg inequality (3.46), (3.28), (3.29) and the boundedness of v ;n,
kv ;nkL1(0;k )C1kv ;nk1
2
L2(0;k )
v0
;n
1
2
L2(0;k )+C2kv ;nkL2(0;k )=O(1);
3
8nkT ;nkL1(0;k )C1
1
4nT ;n
1
2
L2(0;k )
1
nT0
;n
1
2
L2(0;k )+C2 3
8nkT ;nkL2(0;k )=o(1):
It follows that i 1
2n[T ;n(x)v ;n(x)]k
0=o(1) and then 1
2nkv ;nk2
L2(0;k )=o(1):
Then, we multiply (3.27) by i 1
2nv ;nand we repeat exactly the same strategy as before, using
(3.27) and 1
2nkv ;nk2=o(1), we obtain (3.45).
We are now ready to estimate Re(i 1
2nT ;n(0)v ;n(0)):12 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL
Applying Gagliardo-Nerenberg inequality (3.46) to w=v ;nwe obtain, using (3.28),
1
2nkv ;nkL1(0;k )C1
3
4nv ;n
1
2
L2(0;k )
1
4nv0
;n
1
2
L2(0;k )+C2 1
2nkv ;nkL2(0;k )
o(1) +
3
4nv ;n
o(1):
Using again the Gagliardo-Nerenberg inequality (3.46) with w=T ;n,
kT ;nkL1(0;k )C1
1
4nT ;n
1
2
L2(0;k )
1
4nT0
;n
1
2+C2kT ;nkL2(0;k )
o(1)
1
4nT0
;n
1
2
L2(0;k )+o(1)
o(1) +o(1)
3
4nv ;n
L2(0;k ):
Here, we have used (3.27),(3.28) and (3.29). Then
(3.47)jRe(i 1
2nT ;n(0)v ;n(0))j 1
2nkv ;nkL1(0;` )kT ;nkL1(0;` )1
4 3
2nkv ;nk2
L2(0;k )+o(1)
and
(3.48)
Reh
i 1
2nT ;n(x)v ;n(x)ik
02 1
2nkv ;nkL1(0;` )kT ;nkL1(0;` )1
2 3
2nkv ;nk2
L2(0;k )+o(1)
Multiplying (3.27) by iv ;ninL2(0;k ) and integrating by parts, we obtain
(3.49) 3
2nkv ;nk2
L2(0;k )= i 1
2n[T ;n(x)v ;n(x)]k
0+i 1
2nZk
0T ;nv0
;ndx+o(1):
Using (3.28) and (3.29), the second term on the left hand side of (3.49) converge to zero. We
conclude, using (3.48) that
3
2nkv ;nk2
L2(0;k )=o(1):
Return back to (3.47) which yields
Re(i 1
2nT ;n(0)v ;n(0)) =o(1):
We obtain the same result if we suppose that x is the end of e ;nidenti ed to ` via , that
is
Re(i 1
2nT ;n(` )v ;n(` )) =o(1);
and we then conclude that the rst term on the right hand side of (3.43) converge to zero.
Now, summing over 2Iin (3.43) by taking into account the estimate of Reh
i 1
2nT ;n(x)v ;n(x)i`
0
and the inequality in (P), we obtain that
X
2I 1
2n
a1
2
nv ;n
2
L2(0;` )=O(1);
then
3
2n
a1
2
v ;n
2
L2(0;` )=O(1)
for every 2I;and the proof of Lemma 3.5 is complete for case (ii). STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 13
Return back to the general case. Substituting (3.34) in (3.33) leads to
(3.50)1
2Z`
0q0jv ;nj2dx+1
2Z`
0q0jT ;nj2dx 1
2h
q(x)
jv ;n(x)j2+jT ;n(x)j2i`
0=o(1)
for every 2I:
Let 2Isuch thate is a K-V string. First, note that from (3.35), we deduce that
a1
2
v ;n
2
L2(0;` )=o(1):
Then, we take q(x) =Rx
0a (s)dsin (3.50) to obtain
(3.51)1
2Z`
0a jT ;nj2dx 1
2 Z`
0a (s)ds!
jv ;n(` )j2+jT ;n(` )j2
=o(1):
Since1
2R`
0a jT ;nj2dx=o(1) andR`
0a (s)ds> 0, then (3.51) implies
(3.52) jT ;n(` )j2+jv ;n(` )j2=o(1):
Therefore (3.50) can be rewritten as
1
2Z`
0q0jv ;nj2dx+1
2Z`
0q0jT ;nj2dx
+1
2
q(0)jv ;n(0)j2+q(0)jT ;n(0)j2
=o(1): (3.53)
Takingq=xin (3.53) implies that kv ;nkL2(0;` )=o(1) andkT ;nkL2(0;` )=o(1):Moreover,
u0
;n
L2(0;` )=kT ;n a v ;nkL2(0;` )=o(1):
By lettingq=` xin (3.53) and by taking into account the convergence of v ;nandT ;nin
L2(0;` );we get
(3.54) v ;n(0) =o(1) andT ;n(0) =o(1):
Finally, notice that (3.52) signi es that
(3.55) v ;n(` ) =o(1) andT ;n(` ) =o(1):
To conclude, it suces to prove that
(3.56)kv ;nkL2(0;` )=o(1) and
u0
;n
L2(0;` )=kT ;nkL2(0;` )=o(1)
for every 2Isuch thate is purely elastic.
To do this, starting by a string e attached at one end to only K-V strings. Using continuity
condition of vand the compatibility condition at inner nodes, implies that e satis es (3.54)
or (3.55). Moreover, by taking q= 1 in (3.50), we conclude that e satis es (3.54) and (3.55).
Then using again (3.50) with q=x;we deduce that (3.56) is satis ed by e :We iterate such
procedure on each maximally connected subgraph of purely elastic strings (from leaves to the
root).
Thusk(un;vn)kH=o(1);which contradicts the hypoyhesis k(un;vn)kH= 1: 14 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL
Remark 3.6. (1)If for every 2I,a is continuous on [0;` ]and not vanish in such
interval then we don't need the property (P) in the Theorem 3.4.
Indeed (P) is used only to estimate
Re
i 1
nZ`
0T0
;na v ;ndx!
in(3.36) , according to 1
2n
a1
2
v ;n
L2(0;` ).
This is equivalent to estimate
Re
i 1
nZ`
0T0
;nv ;ndx!
according to 1
2nkv ;nkL2(0;` ):
Re
i 1
nZ`
0T0
;nv ;ndx!
= Re
i 1
nT ;nv ;n`
0+Re
i 1
nZ`
0T ;nv0
;ndx!
=
Re
i 1
nT ;n(x)v ;n(x)`
0+o(1)
as in case (ii) (proof of Theorem 3.4) we prove without using (P) that
Re
i 1
nT ;n(x)v ;n(x)`
0 2
n
4kv ;nk2
L2(0;` )+o(1):
(2)We nd here the particular cases studied in [23, 24, 1, 17, 22] . Note that concerning
the result of polynomial stability in [1, 17] the authors proved that the1
t2decay rate of
solution is optimal when the damping coecient is a characteristic function.
4.Further comments: graph case
In this section we want generalize the previous results to a general graph. Then we suppose that
Tis a connected graph G(thenGcan contains some circuits).STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 15
••
••
•••
••
•
Figure 2. A Graph
We conserve the same notations as in T, we just replace SbyS0which is the set of all the
external nodes, and IS0denote the set of such nodes. Then I=IM[IS0. We denote by Jthe
setf1;:::;Ngand fork2Iwe will denote by Jkthe set of indices of edges adjacent to sk. If
k2IS0;then the index of the unique element of Jkwill be denoted by jk:
We suppose that the graph is directed, then we need to de ne the incidence matrix D=
(dkj)pN;p=jIj;as follows,
dkj=8
>><
>>:1 ifj(`j) =sk;
1 ifj(0) =sk;
0 otherwise,
The system (1.1)-(1.5) is rewritten as follows
(4.57)@2uj
@t2(x;t) @
@x@uj
@x+aj(x)@2uj
@x@t
(x;t) = 0;0<x<`j; t> 0; j2J;
(4.58) ujk(sk;t) = 0; k2IS0; t> 0;
(4.59) uj(sk;t) =ul(sk;t); t> 0; j;l2Jk; k2IM;
(4.60)X
j2Jkdkj@uj
@x(sk;t) +aj(sk)@2uj
@x@t(sk;t)
= 0; t> 0; k2IM;
(4.61) uj(x;0) =u0
j(x);@uj
@t(x;0) =u1
j(x);0<x<`j; j2J:
As in the case of a tree, we suppose that Gcontains at least a K-V edge and that every max-
imal subgraph of purely elastic edges is (a tree), the leaves of which K-V edges are attached.
Furthermore, we suppose that S06=;:16 KA IS AMMARI, ZHUANGYI LIU, AND FARHAT SHEL
Finally the property (P) is rewritten as follows,
(P0)8j2J; a0
j;a00
j2L1(0;`j) and8k2IM;X
j2Jkdkja0
j(sk)0:
Under the property (P') the system (4.57)-(4.61) is polynomially stable, and it is exponentially
stable if and only if ais continuous at each inner nodes, i.e., uj(sk) =ul(sk); t > 0;;j;l2
Jk; k2IM.
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(2005), 177-193.
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Math. Soc., 305(1988), 837{852.
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50 of Math ematiques & Applications (Berlin), Springer-Verlag, 2006.
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Ann. Di erential Equations ,1(1985), 43-56.STABILITY OF THE WAVE EQUATIONS ON A TREE WITH LOCAL KELVIN-VOIGT DAMPING 17
[19] S. Chen, K. Liu and Z. Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt
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Basler Lehrb ucher , Birkh auser Verlag, Basel, 2009.
UR Analysis and Control of PDEs, UR 13ES64, Department of Mathematics, Faculty of Sciences of
Monastir, University of Monastir, Tunisia
E-mail address :kais.ammari@fsm.rnu.tn
Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812-3000, United
States
E-mail address :zliu@d.umn.edu
UR Analysis and Control of PDEs, UR 13ES64, Department of Mathematics, Faculty of Sciences of
Monastir, University of Monastir, Tunisia
E-mail address :farhat.shel@ipeit.rnu.tn |
1405.4677v1.Comparison_of_micromagnetic_parameters_of_ferromagnetic_semiconductors__Ga_Mn__As_P__and__Ga_Mn_As.pdf | 1
Comparison of micromagnetic parameters of ferromagnetic
semiconductors (Ga,Mn)(As,P) and (Ga,Mn)As
N. Tesařová1, D. Butkovi čová1, R. P. Campion2, A.W. Rushforth2, K. W. Edmonds,
P. Wadley2, B. L. Gallagher2, E. Schmoranzerová,1 F. Trojánek1, P. Malý1, P. Motloch4,
V. Novák3, T. Jungwirth3, 2, and P. N ěmec1,*
1 Faculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 3, 121 16 Prague 2,
Czech Republic
2School of Physics and Astronomy, University of Nottingham, Nottingham NG72RD, United Kingdom
3 Institute of Physics ASCR, v.v.i., Cukrovar nická 10, 16253 Prague 6, Czech Republic
4 University of Chicago, Chicago, IL 60637, USA
We report on the determination of microm agnetic parameters of epilayers of the
ferromagnetic semiconductor (Ga,Mn)As, which has easy axis in the sample plane, and
(Ga,Mn)(As,P) which has easy axis perpendicula r to the sample plane. We use an optical
analog of ferromagnetic resonance where the laser-pulse-induced precession of
magnetization is measured directly in the time domain. By the analysis of a single set of pump-and-probe magneto-optical data we determined the magnetic anisotropy fields, the
spin stiffness and the Gilbert damping consta nt in these two materials. We show that
incorporation of 10% of phosphorus in (Ga, Mn)As with 6% of manganese leads not only
to the expected sign change of the perpendicu lar-to-plane anisotropy field but also to an
increase of the Gilbert damping and to a reduction of the spin stiffness. The observed changes in the micromagnetic parameters upon incorporating P in (Ga,Mn)As are
consistent with the reduced hole density, conductivity, and Curie temperature of the (Ga,Mn)(As,P) material. We report that th e magnetization precession damping is stronger
for the n = 1 spin wave resonance mode than for the n = 0 uniform magnetization
precession mode.
PACS numbers: 75.50.Pp, 75.30.Gw, 75.70.-i, 78.20.Ls, 78.47.D-
I. INTRODUCTION
(Ga,Mn)As is the most widely studied dilute d magnetic semiconductor (DMS) with a carrier-
mediated ferromagnetism.1 Investigation of this material system can provide fundamental
insight into new physical phenomen a that are present also in ot her types of magnetic materials
– like ferromagnetic metals – where they can be exploited in spintronic applications.2-5
Moreover, the carrier concentration in DMSs is several orders of ma gnitude lower than in
conventional FM metals which enables manipul ation of magnetization by external stimuli –
e.g. by electric6,7 and optical8,9 fields. Another remarkable propert y of this material is a strong
sensitivity of the magnetic anisotropy to the ep itaxial strain. (Ga,Mn)A s epilayers are usually
prepared on a GaAs substrate where the growth-i nduced compressive strain leads to in-plane
orientation of the easy axis (EA) for Mn concentrations ≥2%.10 However, for certain
experiments – e.g., for a visualization of magn etization orientation by the magneto-optical
polar Kerr effect11-17 or the anomalous Hall effect12,18 – the EA orientation in the direction
perpendicular to the sample plane is more suitable. To achieve this, (Ga,Mn)As layers have been grown on relaxed (In,Ga)As buffer laye rs that introduce a tensile strain in
(Ga,Mn)As.
11,12,14,16-18 However, the growth on (In,Ga)As la yers can result in a high density
of line defects that can lead to high coerci vities and a strong pinning of domain walls 2
(DW).16,17 Alternatively, tensile strain and perpendicular-to-plane or ientation of the EA can be
achieved by incorporation of small amount s of phosphorus in (Ga,Mn)(As,P) layers.19,20 In
these epilayers, the EA can be in the sample plane for the as-grown material and perpendicular to the plane for fully annealed (Ga,Mn)(As,P).
21 The possibility of magnetic
anisotropy fine tuning by the thermal annealing tu rns out to be a very favorable property of
(Ga,Mn)(As,P) because it enables the preparation of materials with extr emely low barriers for
magnetization switching.22,23 Compared to tensile-stained (Ga,Mn)As/(In,Ga)As films,
(Ga,Mn)(As,P)/GaAs epilayers show weaker DW pinning, which allows observation of the
intrinsic flow regimes of DW propagation.13,15,24
Preparation of uniform (Ga,Mn)As epilayers with minimized dens ity of unintentional
extrinsic defects is a rather challenging task which requires optimized growth and post-
growth annealing conditions.25 Moreover, the subsequent determination of material
micromagnetic parameters by the standard char acterization techniques, such as ferromagnetic
resonance (FMR), is complicated by the fact th at these techniques require rather thick films,
which may be magnetically inhomogeneous.25,26 Recently, we have reported the preparation
of high-quality (Ga,Mn)As epila yers where the individually optimized synthesis protocols
yielded systematic doping trends, whic h are microscopically well understood.25
Simultaneously with the optimization of the ma terial synthesis, we developed an optical
analog of FMR (optical-FMR)25, where all micromagnetic pa rameters of the in-plane
(Ga,Mn)As were deduced from a single magneto -optical (MO) pump-and-probe experiment
where a laser pulse induces precession of magnetization.27,28 In this method the anisotropy
fields are determined from the dependence of the precession frequency on the magnitude and
the orientation of the external magnetic field, the Gilbert damping cons tant is deduced from
the damping of the precession signal, and the sp in stiffness is obtained from the mutual
spacing of the spin wave resonance modes observe d in the measured MO signal. In this paper
we apply this all optical-FMR to (Ga,Mn)(As,P) . We demonstrate the applicability of this
method also for the determination of microma gnetic parameters in DMS materials with a
perpendicular-to-plane orientation of the EA. By this method we show that the incorporation
of P in (Ga,Mn)As leads not only to the expect ed sign change of the perpendicular-to-plane
anisotropy field but also to a considerable in crease of the Gilbert damping and to a reduction
of the spin stiffness. Moreover, we illustrate that the all optical-FMR can be very effectively
used not only for an investig ation of the uniform magnetizati on precession but also for a study
of spin wave resonances.
II. EXPERIMENTAL
In our previous work we reported in de tail on the preparation and micromagnetic
characterization of (Ga,Mn)A s epilayers prepared in MBE laboratory in Prague.
25 We also
pointed out that the preparati on of (Ga,Mn)As by this highly non-equilibrium synthesis in two
distinct MBE laboratories in Prague and in No ttingham led to a growth of epilayers with
micromagnetic parameters that showed the same doping trends.25 Nevertheless, the
preparation of epilayers with identical paramete rs (e.g., thickness, nominal Mn content, etc.)
in two distinct MBE machines is still a nontrivial task. Therefore, in this study of the role of
the phosphorus incorporation to (Ga,Mn)As we opted for a dire ct comparison of materials
prepared in one MBE mach ine. The investigated Ga 1-xMn xAs and Ga 1-xMn xAs1-yPy epilayers
were prepared in Nottingham20 with the same nominal amount of Mn (x = 6%) and the same
growth time on a GaAs substrate (with 50 nm thick GaAsP buffer layer in the case of
(Ga,Mn)(As,P)]. They differ only in the incorpor ation of P (y = 10%) in the latter epilayer. 3
The inferred epilayer thicknesses are (24.5 േ 1.0) nm for both (Ga,Mn)As and
(Ga,Mn)(As,P).29 The as-grown layers, wh ich both had the EA in th e epilayer plane, were
thermally annealed (for 48 hours at 180°C). This led to an increase in Curie temperature and
to a rotation of the EA to the perpendicular-to-plane orientation for (Ga,Mn)(As,P).20,21
The magnetic anisotropy of the samples was studied using a superconducting quantum
interference device (SQUID) magneto meter and by the all-optical FMR.25 The hole
concentration was determined by fitting to Hall effect measurements at low temperatures
(1.8 K) for external magnetic fields from 2 T to 6 T. In this range the magnetization is
saturated and one can obtain th e normal Hall coefficient af ter correction for the field
dependence of the anomalous Hall du e to the weak magnetoresistance.30 The time-resolved
pump-and-probe MO experiments were performe d using a titanium sapphire pulsed laser
(pulse width 200 fs) with a repetition rate of 82 MHz, which was tuned ( hυ = 1.64 eV)
above the GaAs band gap. The energy fl uence of the pump pulses was around 30 μJcm-2 and
the probe pulses were at least ten times weak er. The pump pulses were circularly polarized
(with a helicity controlled by a quarter wave plate) and the probe pulses were linearly
polarized (in a direction perpendicular to the external magnetic field). The time-resolved MO data reported here correspond to the polariz ation-independent part of the pump-induced
rotation of probe polarization plane, which was computed from the measured data by
averaging the signals obtained for the opposite helicities of circularly polarized pump
pulses.
27, 28 The experiment was performed close to the normal-incidence geometry, where the
angles of incidence were 9° and 3° (measured from the sample normal) for the probe and the pump pulses, respectively. The rotation of the probe polarization plane is caused by two MO effects – the polar
Kerr effect and the magnetic linear dichroism, which are sensitiv e to perpendicular-to-plane
and in-plane components of magnetization, respectively.
31-33 For all MO experiments, samples
were mounted in a cryostat and cooled down to ≈ 15 K. The cryostat was placed between the
poles of an electromagnet and the external magnetic field Hext ranging from ≈ 0 to 585 mT
was applied in the sample plane, either in the [010] or [110] crystallographic di rection of the
sample (see inset in Fig. 1 for a definition of the coordinate system). Prior to all
measurements, we always prepared the magnetiza tion in a well-defined state by first applying
a strong saturating magnetic field and then reducing it to the desired magnitude of Hext.
III. RESULTS AND DISCUSSION
A. Sample characterization
The hysteresis loops measured by SQUID magnetometry for external magnetic field
applied along the in-plane [-110] and perpendicu lar-to-plane [001] crystallographic directions
in (Ga,Mn)As and (Ga,Mn)(As,P) samples are s hown in Fig. 1(a) and Fig. 1(c), respectively.
These data confirm the expected in-plane and perp endicular-to-plane orient ations of the EA in
(Ga,Mn)As and (Ga,Mn)(As,P), respectively. Moreover, they reveal that for the
(Ga,Mn)(As,P) sample, an external magnetic field of 250 mT is needed to rotate the
magnetization into the sample plane. In Fig. 1(b) and Fig. 1(d) we show the temperature
dependences of the remanent magnetization of the samples from which the Curie temperature
T
c of 130 K and 110 K can be deduced. The measur ed saturation ma gnetization also
indicates very similar density of Mn moments contributing to the ferromagnetic state in the
two samples. 4
Fig. 1 (Color online): Magnetic characterization of samples: (a), (b) (Ga,Mn)As and (c), (d) (Ga,Mn)(As,P). (a),
(c) Hysteresis loops measured in at 2 K for the external magnetic field applied in the sample plane (along the
crystallographic direction [-110]) and perpendicular to sample plane (along the crystallographic direction [001]). (b), (d) Temperature dependence of the remanent magnetization. Inset: Definition of the coordinate system.
The electrical characterization of the samp les is shown in Fig. 2. The measured data
show a sharp Curie point singula rity in the temperature derivative of the resistivity which
confirms the high quality of the samples.25 The hole densities inferred from Hall
measurements are (1.3 0.2) 1021 cm-3 and (0.8 0.2) 1021 cm-3 for (Ga,Mn)As and
(Ga,Mn)(As,P), respectively. The hole density obtained for (Ga, Mn)As is in agreement with
our previous measurements for simila r films in magnetic fields up 14 T.30 The reduction of
the density of itinerant holes quantitatively correlates with the observed increase of the resistivity of the (Ga,Mn)(As,P) film as compared to the (Ga,Mn)As sample.
5
Fig. 2 (Color online): Electrical char acterization of samples. Temperature dependence of the resistivity (a) and
its temperature derivative (b).
B. Time-resolved magnet o-optical experiment
In Fig. 3(a) and 3(b) we show the measur ed MO signals that reflect the magnetization
dynamics in (Ga,Mn)As and (Ga,Mn)(As,P) sa mples, respectively. Th ese signals can be
decomposed into the oscillatory parts [Figs. 3(c) and 3(d)] and the non-oscillatory pulse-like
background [Fig. 3(e) and 3(f)].27, 28 The oscillatory part arises from the precessional motion
of magnetization around the quasi-equilibrium EA and the pulse-like function reflects the
laser-induced tilt of the EA and the laser-induced demagnetization.25,31 The pump
polarization-independent MO data reported here, which were measured at a relatively low
excitation intensity of 30 μJcm-2, can be attributed to the ma gnetization precession induced by
a transient heating of the sample due to the absorption of the laser pulse.8,9 Before absorption
of the pump pulse the magnetization is along th e EA direction. Absorptio n of the laser pulse
leads to a photo-injection of electron-hole pa irs. The subsequent fast non-radiative
recombination of photo-injected electrons induces a transi ent increase of the lattice
temperature (within tens of picoseconds afte r the impact of the pu mp pulse). The laser-
induced change of the lattice temperature then leads to a change of the EA position.34 As a
result, magnetization starts to follow th e EA shift by the precessional motion. Finally,
dissipation of the heat leads to a return of the EA to the equilibrium position and the
precession of magnetization is stopped by a Gilbert damping.25 It is apparent from Fig. 3
that the measured MO signals are strongly dependent on a magnit ude of the external magnetic
field, which was applied in the epilayer plan e along the [010] crystall ographic direction in
both samples. In particular, absorption of the laser pulse does not induce precession of
magnetization in (Ga,Mn)(As,P) unless magnetic field stronger than 20 mT is applied [see
Fig. 3(d)].
6
Fig. 3 (Color online): Time-resolved magneto-optical (MO) signals measured in (Ga,Mn)As (a) and
(Ga,Mn)(As,P) (b) for two magnitudes of the external magnetic field applied along the [010] crystallographic
direction. The measured MO signals were decomposed in to oscillatory parts [(c) and (d]), which correspond to
the magnetization precession, and to non-oscillatory part s [(e) and (f)], which are connected with the quasi-
equilibrium tilt of the easy axis and with the demagnetization. Note different x-scales in the left and in the right
columns.
The magnetization dynamics is describe d by the Landau-Lifshitz-Gilbert (LLG)
equation that is usually expressed in the form35,36:
ௗሺ௧ሻ
ௗ௧ൌെ ߛൣ ሺݐሻൈሺݐሻ൧ఈ
ெೞቂሺݐሻൈௗሺ௧ሻ
ௗ௧ቃ, ( 1 )
where = (gμB)/ћ is the gyromagnetic ratio, g is the Landé g-factor, μB is the Bohr magneton,
ħ is the reduced Planck constant, is the Gilbert damping constant, and Heff is the effective
magnetic field. Nevertheless, it is more conve nient to express this equation in spherical
coordinates where the directi on of the magnetization vector M is given by the polar angle θ
and azimuthal angle φ and where Heff can be directly connected w ith angular derivatives of the
free energy density functional F (see the Appendix).37 For small deviations δ and δ of
magnetization from its equilibrium position (given by 0 and 0), the solution of LLG
equation can be written in the form (t) = 0 + δ(t) and (t) = 0 + δ(t) as
ߠሺݐሻൌߠܣఏ݁ି௧ݏܿሺ2ݐ݂ߨΦ ఏሻ, ( 2 )
߮ሺݐሻൌ߮ܣఝ݁ି௧ݏܿ൫2ݐ݂ߨΦ ఝ൯, ( 3 )
where the constants A (A) and () represent the initial amplitude and phase of (),
respectively, f is the magnetization precession frequency, and kd is the precession damping
rate (see the Appendix). The pr ecession frequency reflects the in ternal magnetic anisotropy of
the sample that can be characterized by the cubic ( KC), in-plane uniaxial ( Ku) and out-of-plane
uniaxial ( Kout) anisotropy fields (see Eq. (A4) in the Appendix).10 Moreover, f depends also on
the magnitude and on the orientation of Hext (see the Appendix) and, therefore, the magnetic
7
field dependence of f can be used to evalua te the magnetic anisotropy fields in the sample. If
the applied in-plane magnetic field is strong e nough to align the magnetiz ation parallel with
Hext (i.e., for Hext exceeding the saturation field in the sa mple for a particular orientation of
Hext), = H = π/2 and = H and if the precession damping is relatively slow , i.e. α2 ≈ 0 f
can be expressed as
݂ൌఓಳ
ඩ൬ܪ௫௧െ2ܭ௨௧ሺଷା௦ସఝ ሻ
ଶ2ܭ௨݊݅ݏଶቀ߮ுെగ
ସቁ൰
ൈሺܪ௫௧2ܭݏܿ4߮ ுെ2ܭ௨݊݅ݏ2߮ுሻ, (4)
Fig. 4 (Color online): Fourier spectrum of the oscillatory part of the MO signal measured in (Ga,Mn)As for
external magnetic fields applied alon g the [010] crystallo graphic direction. f0 and f1 indicate the frequencies of
the uniform magnetization precession and the fi rst spin wave resona nce, respectively.
In Fig. 4 we show the fast Fourier transfor m (FFT) spectra of the oscillatory parts of
the MO signals measured in the (Ga,Mn )As sample for different values of Hext. This figure
clearly reveals that for all external magnetic fields there are two distinct oscillatory
frequencies present in the measured data . These precession modes are the spin wave
resonances (SWRs) – i.e., spin waves (or magno ns) that are selectively amplified by fulfilling
the boundary conditions: In a homogeneous thin magnetic film with a thickness L, only the
perpendicular standing waves with a wave vector k fulfilling the resonant condition kL = n
(where n is the mode number) are amplified.25,38-41 In our case – using the ferromagnetic films
with a thickness around 25 nm – we detect only42 the uniform magnetiza tion precession with
zero k vector (i.e. the precession where at any instant of time all magnetic moments are
parallel over the entire sample; n = 0 at frequency f0) and the first SWR (i.e. n = 1 at
frequency f1). See the inset in Fig. 8 for a schematic de piction of the modes. In Fig. 5 we plot
the amplitudes of the uniform magnetization precession ( A0) and of the first SWR ( A1) as a
function of the exte rnal magnetic field Hext. In the (Ga,Mn)As sample, the oscillations are
present even when no magnetic field is applied and the precession amplitude increases
slightly with an increasing Hext (up to 20 mT for A0 and up to 60 mT for A1). Above this
value, a further increase of Hext leads to a suppression of the oscillations, but the suppression
of the first SWR is slower than that of the uniform magnetization precession [see Fig. 5(c)]. In
8
(Ga,Mn)(As,P), the oscillatory signal starts to appear at 50 mT, reaches its maximum for
Hext 175 mT, and a further increase of Hext leads to its monotonic d ecrease, like in the case
of (Ga,Mn)As. The observed field dependence of the precession amplitude, which expresses
the sensitivity of the EA position on the laser- induced sample temperature change, can be
qualitatively understood as follows. In (Ga,Mn)As, the position of the EA in the sample plane
is given by a competition between the cubic and the in-plane uniaxial magnetic
anisotropies.10,25 The laser-induced heating of the sa mple leads to a reduction of the
magnetization magnitude M and, consequently, it enhances th e uniaxial anisotropy relative to
the cubic anisotropy.9 This is because the uniaxial anisotropy component scales with
magnetization as ~ M2 while the cubic component scales as ~ M4. The application of Hext
along the [010] crystallographic di rection deepens the minimum in the [010] direction in the
free energy density functional F (due to the Zeeman term in F, see Eq. (A4) in the Appendix).
Measured data shown in Fig. 5 reve al that in the (Ga,Mn)As sample, Hext initially (for Hext up
to 20 mT) destabilizes the posit ion of EA but stabilizes it for large values of Hext (where the
position of the energy minimum in F is dominated by the Zeeman term, which is not
temperature dependent). In the case of (Ga,Mn)( As,P), the position of the EA is determined
by the strong perpendicular-to-p lane anisotropy. Therefore, w ithout an external magnetic
field, the laser-induced heating of the sample doe s not change significantl y the position of EA
and, consequently, does not initiate the pr ecession of magnetization [see Fig. 5(b)]. The
application of an in-plane fi eld moves the energy minimum in F towards the sample plane
[see Fig. 1(c)] which makes the EA position more sensitive to the laser-induced temperature
change. Finally, for a sufficiently strong Hext, the sample magnetic anisotropy is dominated by
the temperature-independent Zeeman term, wh ich again suppresses the precession amplitude.
The markedly different ratio A1/A0 in the (Ga,Mn)As and (Ga,Mn )(As,P) samples is probably
connected with a different surface magnetic anis otropy and/or a slight difference in magnetic
homogeneity in these two samples.43,44
Fig. 5 (Color online): Dependence of the amplitude of the uniform magnetization precession ( A0) and the first
spin wave resonance ( A1) on the magnitude of the external magnetic field ( Hext) applied along the [010]
crystallographic direction in (Ga,Mn )As (a) and (Ga,Mn)(As,P) (b). (c) and (d) Dependence of the ratio A1 / A0
on Hext.
9
C. Determination of magnetic anisotropy
In Fig. 6 we plot the ma gnetic field dependences of f0 and f1 for two different
orientations of Hext. The frequency f0 of the spatially uniform precession of magnetization is
given by Eq. (4). For the SWRs, where the local moments are no longer para llel (see the inset
in Fig. 8), restoring torques due to exchange interaction and internal magnetic dipolar
interaction have to be included in the analysis.39-41,45 For Hext along the [010] crystallographic
direction (i.e., for φH = /2) Eq. (4) can be written as
݂ൌఓಳ
ඥሺܪ௫௧െ2ܭ௨௧ܭ∆ܪሻሺܪ௫௧െ2ܭെ2ܭ௨∆ܪሻ , (5)
where Hn is the shift of the resonant field for the nth spin-wave mode with respect to the
n = 0 uniform precession mode. Analogically, for Hext applied in the [110] crystallographic
direction (i.e., for φH = /4)
݂ൌఓಳ
ඥሺܪ௫௧െ2ܭ௨௧2ܭܭ௨∆ܪሻሺܪ௫௧2ܭ∆ܪሻ. (6)
The lines in Fig. 6 represent the fits of all four measured dependencies fn = fn (Hext, H) [where
n = 0; 1 and H = /4; /2] with a single set of anisotropy constants for each of the samples,
which confirms the credibility of the fitting pr ocedure. The obtained an isotropy constants at
≈ 15 K are: KC = (17 ± 3) mT, Ku = (11 ± 5) mT, Kout = (-200 ± 20) mT for (Ga,Mn)As and KC
= (14 ± 3) mT, Ku = (11 ± 5) mT, Kout = (90 ± 10) mT for (Ga,Mn)(As,P), respectively (in
both cases we considered the Mn g-factor of 2). For (Ga,Mn)As, we can now compare these
anisotropy constants with those obtained by the same fitting procedure for samples prepared
in a different MBE laboratory (in Prague) – see Fig. 4 in Ref. 25. We see that the previously
reported25 doping trends of KC and Kout predict for a sample with nominal Mn doping x = 6%
the anisotropy fields which are the same as thos e reported in this pape r for the sample grown
in Nottingham. This observation is in accord with the current microscopic understanding of
their origin – KC reflects the zinc-blende crystal st ructure of the host semiconductor and Kout
Fig. 6 (Color online): Magnetic field dependence of the precession frequencies f0 and f1 for two different
orientations of the external magnetic field (points) measured in (Ga,Mn)As (a) and (Ga,Mn)(As,P) (b). Lines are the fits by Eqs. (5) and (6). ΔH
1 indicates the shift of the resonant field for the first spin-wave mode with respect
to the uniform precession mode.
10
is a sum of the anisotropy due to the growth-induced lattice-ma tching strain and of the thin-
film shape anisotropy, which should be the sa me for equally doped and optimally synthesized
samples, independent of the growth chamber. On the other hand, the micr oscopic origin of in-
plane uniaxial anisotropy field K
u is still not established10,25 and our data reveal that it is
considerably smaller in the sample grown in Nottingham. Th e incorporation of phosphorus
does not change significa ntly the values of KC and Ku but it strongly modi fies the magnitude
and changes the sign of Kout, which is in agreement with the previous results obtained by
FMR experiment.22
D. Determination of spin stiffness
The observation of a higher-o rder SWR enables us to also determine the exchange
spin stiffness constant D, which is a parameter that is rather difficult to extract from other
experiments in (Ga,Mn)As.25,46 In homogeneous thin films, Hn is given by the Kittel
formula43
Δܪ ܪെܪൌ݊ଶ
ఓಳగమ
మ, ( 7 )
where L is the thickness of the magnetic film. By fitting the data in Fig. 6, we obtained H1 =
(363 ± 2) mT for (Ga,Mn)As and (271 ± 2) mT for (Ga,Mn)(As,P) which correspond to D =
(2.5 ± 0.2) meVnm2 and (1.9 ± 0.2) meVnm2 for (Ga,Mn)As and (Ga,Mn)(As,P), respectively
(note that the relatively large experimental error in D is given mainly by the uncertainty of the
epilayer thickness).29 The value obtained for (Ga,Mn)As is again in agreement with that
reported previously for samples grown in Prague,25 which also confirms the consistent
determination of the epilayer thicknesses in both MBE laboratories.29 The incorporation of
phosphorus leads to a reduction of D which correlates with the decrease of the hole density,47
and the reduced Tc in (Ga,Mn)(As,P), as compared to its (Ga,Mn)As counterpart.
E. Determination of Gilbert damping
The Gilbert damping constant α can be determined by fitting the measured dynamical
MO signals by the LLG equation.
35,36,48 For a relatively slow precession damping and a
sufficiently strong external magnetic field, the analytical solution of the LLG equation gives
(see the Appendix)
݇
ௗൌߙఓಳ
ଶ൬2ܪ௫௧െ2ܭ௨௧ሺଷାହ௦ସఝ ಹሻ
ଶܭ௨ሺ1െ3݊݅ݏ2߮ ுሻ൰. (8)
Eq. (8) shows not only that kd is proportional to but also that for obtaining a correct value of
from the measured MO precession signal damp ing it is necessary to take into account a
realistic magnetic anisotropy of the investigated sample. Nevert heless, the correct dependence
of kd on magnetic anisotropy was not cons idered in the previous studies35,36,48 where only one
effective magnetic field was used, which is probably one of the reasons why mutually
inconsistent results were obtained for Ga 1-xMn xAs with a different Mn content x. An increase
of from 0.02 to 0.08 for an increase of x from 3.6% to 7.5% was reported in Ref. 36. On 11
the contrary, in Ref. 48 values of from 0.06 to 0.19 – without any apparent doping trend –
were observed for x from 2% to 11%.
For numerical modeling of the measured MO data, we first computed from the LLG
equation (Eqs. (A1) and (A2) in the Appendix with th e measured magnetic anisotropy fields)
the time-dependent deviations of the spherical angles [ (t) and (t)] from the corresponding
equilibrium values ( 0, 0). Then we calculated how such changes of and modify the
static magneto-optical response of the samp le, which is the signal that we detect
experimentally31
0
00 2sin2 2cos2 ,MLD s MLD PKEPMt MPt Pt t MO . (9)
The first two terms in Eq. (9) are connected wi th the out-of-plane and in-plane movement of
magnetization, and the last term describes a change of the sta tic magneto-optical response of
the sample due to the laser-induced demagnetization.31 PPKE and PMLD are MO coefficients
that describe the MO response of the sample which we measured independently in a static
MO experiment,32,33 and β is the probe polarization orientation with respect to the
crystallographic direction [100].31 To further simplify the fitting procedure, we can extract the
oscillatory parts from the measured MO data (cf. Fig. 3), which effectively removes the MO
signals due to the laser-induced demagnetization [i .e., the last term in Eq. (9)] and due to the
in-plane movement of the easy axis [i.e., a part of the MO signal desc ribed by the second term
in Eq. (9)].31 Examples of the fitting of the precessional MO optical data are shown in Fig.
7(a) and (b) for (Ga,Mn)As and (Ga,Mn)(As,P), respectively. We stress that in our case the
only fitting parameters in the modeling are the damping coefficient and the initial
deviations of the spherical angles from the corresponding equilibrium values. By this
numerical modeling we deduced a de pendence of the damping factor on the external
magnetic field for two different orientations of Hext. At smaller fields, the dependences
obtained show a strong anisotropy w ith respect to the field angle th at can be fully ascribed to
the field-angle dependence of the precession frequency.25 However, when plotted as a
function of the precession frequency, the de pendence on the field-an gle disappears – see
Fig. 7(c) and (d) for (Ga,Mn)As and (Ga,Mn )(As,P), respectively. For both materials,
initially decreases monotonously with f and finally it saturates at a certain value for f ≥
10 GHz. A frequency-dependent (or magnetic field-dependent) damping parameter was
reported in various magnetic materials and a va riety of underlying mechanisms responsible
for it were suggested as an explanation.49-51 In our case, the most probable explanation seems
to be the one that was used by Walowski et al. to explain the experimental results obtained in
thin films of nickel.49 They argued that in the low field range small magnetization
inhomogeneities can be formed – the magnetizati on does not align parallel in an externally
applied field, but forms ripples.49 Consequently, the measured MO signal which detects
sample properties averaged over the laser spot size, which is in our case about 30 m wide
(FWHM), experiences an apparent oscillation damping because the magnetic properties
(i.e., the precession frequencies) are slightly differing within the spot size (see Fig. 6 and 7 in
Ref. 49). On the other hand, for stronger external fields the sample is fully homogeneous and,
therefore, the precession damping is not de pendent on the applied field (the precession
frequency), as expected for the in trinsic Gilbert damping coefficient.52,53 We note that the
observed monotonous frequency decrease of α is in fact a signature of a magnetic
homogeneity of the studied epilayers.25 The obtained frequency-independent values of α are
(0.9 ± 0.2) 10-2 for (Ga,Mn)As and (1.9 ± 0.5) 10-2 for (Ga,Mn)(As,P), respectively. The 12
observed enhancement of the magnetization pre cession damping due to the incorporation of
phosphorus is also clearly apparent directly from Figs. 7(a) and 7(b) where the MO data with
similar precession frequencies are shown for (G a,Mn)As and (Ga,Mn)(As,P), respectively. In
(Ga,Mn)As the value of α obtained is again fully in accord with the reported Mn doping trend
in α in this material.25 In (Ga,Mn)(As,P), the determined α is similar to the value 1.2 10-2
which was reported by Cubukcu et al. for (Ga,Mn)(As,P) with a si milar concentration of Mn
and P.22 Comparing to the doping trends in the se ries of optimized (Ga,Mn)As materials,25 the
value of α i n o u r ( G a , M n ) ( A s , P ) s a m p l e i s c o n s i s tent with the measured Gilbert damping
constant in lower Mn-doped (Ga,Mn)As epilayers with similar hole densities and resistivities
to those of the (Ga,Mn)(As,P) film.
Fig. 7 (Color online): Determination of the Gilbert da mping. (a) and (b) Oscillatory part of the MO signal
(points) measured in (Ga,Mn)As for the external magnetic field 100 mT (a) and in (Ga,Mn)(As,P) for 350 mT
(b); magnetic field applied along the [010] crystallographic direction leads to a similar frequency ( f0 7.5 GHz)
in both cases. Lines are fits by the Landau-Lifshitz-G ilbert equation. (c) and (d) Dependence of the damping
factor () on the precession frequency for two different orienta tions of the external magnetic field in (Ga,Mn)As
(c) and (Ga,Mn)(As,P) (d).
The high quality of our MO data enables us to evaluate not only the damping of the
uniform magnetization precession, which is addresse d above, but also the damping of the first
SWR. To illustrate this procedure, we show in Fig. 8(a) the MO data measured for Hext =
13
250 mT applied along the [010] crystallographic direction in (Ga,Mn)As. The experimental
data (points) obtained can be fitted by a sum of two expone ntially damped cosine functions
(line) which enables us to separate, directly in a time domain, the contributions of the individual precession modes to the measured MO signal. In this particular case, the uniform
magnetization precession occurs at a frequency f
0 = 12.2 GHz and this precession mode is
damped with a rate constant kd0 = 0.79 ns-1. Remarkably, the first SWR, which has a
frequency f1 = 23.0 GHz, has a considerably la rger damping rate constant kd1 = 1.7 ns-1 – see
Fig. 8(b) where the contribution of individual modes ar e directly compared and also Fig. 8(c)
where Fourier spectra computed from the measured MO data for two diffe rent ranges of time
delays are shown. To convert the damping rate constant kdn obtained to the damping constant
Fig. 8 (Color online): Comparison of the Gilbert damping of the uniform magnetization precession and of the
first spin wave resonance. (a) Oscillatory part of the MO signal (points) measured in (Ga,Mn)As for the external
magnetic field 250 mT applied along the [010] crystallographic direction. The solid line is a fit by a sum of two exponentially damped cosine functions that are shown in (b). Inset: Schematic illustration
39 of the spin wave
resonances with n = 0 (uniform magnetization precession with zero k vector) and n = 1 (perpendicular standing
wave with a wave vector k fulfilling the resonant condition kL = ) in a magnetic film with a thickness L. (c)
Normalized Fourier spectra computed fo r the depicted ranges of time delays from the measured MO data, which
are shown in (a). (d) Dependence of the damping factor ( n) on the precession frequency for the uniform
magnetization precession ( n = 0) and the first spin wave resonance ( n = 1).
14
n for the n-th mode, we can use the ge neralized analytic al solution of the LLG equation. For
a sufficiently strong Hext along the [010] crystallographic direction (i.e., when φ φH = /2),
Eq. (8) can be written as
݇ௗൌߙఓಳ
ଶሺ2ܪ௫௧2∆ܪ െ2ܭ௨௧2ܭܭ௨ሻ. (10)
For the case of MO data measured at Hext = 250 mT, the damping constants obtained for
modes with n = 0 and 1 are 0 = 0.009 and 1 = 0.011, respectively. [We note that the value
of 0 obtained from the analytical solution of LLG equation is identical to that determined by
the numerical fitting and shown in Fig. 7(c), which confirms the consistency of this
procedure.] In Fig. 8(d) we show the dependence of 0 and 1 on the precession frequency.
These data clearly show that even if the modes with n = 0 and 1 were oscillating with the
same frequency, the SWR mode with n = 1 would have a larger damping coefficient.
However, for sufficiently high fr equencies (i.e., external magnetic fields) the damping of the
two modes is nearly equal [see Fig. 8(d)]. This feature can be ascribed to the presence of an
extrinsic contribution to the damping coeffici ent for the SWR modes. The extrinsic damping
probably originates from small variations of the sample thickness (< 1 nm) within the laser
spot size54 and/or from the presence of a weak bulk inhomogeneity,43 which is apparent as
small variations of ΔHn. The frequency spacing and the (Ki ttel) character of the SWR modes
is insensitive to such small variations of ΔHn but the resulting frequency variations (see Eq. 5)
can still strongly affect the observed damping of the oscillations. For high enough external
magnetic fields, the variations of ΔHn have a negligible role a nd the damping of the SWR
modes is governed solely by the intrinsic Gilbert damping parameter.
IV. CONCLUSIONS
We used the optical analog of FMR, wh ich is based on a pump-and-probe magneto-
optical technique, for the determination of micromagnetic parameters of (Ga,Mn)As and
(Ga,Mn)(As,P) DMS materials. The main advantage of this technique is that it enables us to
determine the anisotropy constants, the spin s tiffness and the Gilbert damping parameter from
a single set of the experimental magneto-optical data measured in films with a thickness of
only several tens of nanometers. To addres s the role of phosphorus incorporation in
(Ga,Mn)As, we measured simultaneously proper ties of (Ga,Mn)As and (Ga,Mn)(As,P) with
6% Mn-doping which were grown under identical conditions in the sa me MBE laboratory.
We have shown that the laser-i nduced precession of magnetization is closely connected with a
magnetic anisotropy of the samples. In partic ular, in (Ga,Mn)As with in-plane magnetic
anisotropy the laser-pulse-induced precession of magnetization was observed even when no
external magnetic field was applied. On the cont rary, in (Ga,Mn)(As,P) with perpendicular-to-
plane magnetic EA the precession of magnetizat ion was observed only when the EA position
was destabilized by an external in-plane ma gnetic field. From the measured magneto-optical
data we deduced the anisotropy constants, spin stiffness, and Gilber t damping parameter in
both materials. We have shown that the incorp oration of 10% of P in (Ga,Mn)As leads not
only to the expected sign change of the perpendi cular-to-plane anisotropy field but also to a
considerable increase of the G ilbert damping which correlates with the increased resistivity
and reduced itinerant hole density in the (Ga,Mn)(As,P) material. We also observed a reduction of the spin stiffness consistent with the suppression of T
c upon incorporating P in 15
(Ga,Mn)As. Finally, we found that in small exte rnal magnetic fields the damping of the first
spin wave resonance is sizably stronger than that of the uniform magnetization precession.
ACKNOWLEDGEMENTS
This work was supported by the Grant Agency of the Czech Republic grant no.
P204/12/0853 and 202/09/H041, by the Grant Agency of Charles University in Prague grant
no. 1360313 and SVV-2013-267306, by EU grant ERC Advanced Grant 268066 - 0MSPIN,
and by Praemium Academiae of the Academy of Sciences of the Czech Republic, from the
Ministry of Education of the Czech Re public Grant No. LM2011026, and from the Czech
Science Foundation Grant No. 14-37427G.
APPENDIX
Due to symmetry reasons, it is conveni ent to rewrite the LLG equation given by
Eq. (1) in spherical coordinates where M
S describes the magnetiza tion magnitude and polar θ
and azimuthal φ angles characterize its orientation. We define the perpendicular-to-plane
angle θ (in-plane angle φ) in such a way that it is counted from the [001] ([100])
crystallographic direction and it is positive wh en magnetization is tilted towards the [100]
([010]) direction (see inset of Fig. 1 for the co ordinate system definition). The time evolution
of magnetization is given by37
ௗெೞ
ௗ௧ൌ0, ( A 1 )
ௗఏ
ௗ௧ൌെఊ
ሺଵାఈమሻெೞቀߙ ∙ܣ
௦ఏቁൌΓఏሺߠ,߮ሻ , ( A 2 )
ௗఝ
ௗ௧ൌఊ
ሺଵାఈమሻெೞ௦ఏቀܣെఈ∙
௦ఏቁൌΓఝሺߠ,߮ሻ , ( A 3 )
where A = dF/d and B = dF/d are the derivatives of the free energy density functional F
with respect to and , respectively. We express F in a form10
ܨൌܯ ௌܭ݊݅ݏଶߠቀଵ
ସ݊݅ݏଶ2݊݅ݏ߮ଶߠ ݏܿଶߠቁെܭ ௨௧ݏܿଶߠെೠ
ଶ݊݅ݏଶߠሺ1െ݊݅ݏ2߮ ሻെ
െܪ௫௧൫ߠݏܿߠݏܿ ுߠ݊݅ݏߠ݊݅ݏ ுݏܿሺ߮െ߮ுሻ൯൩, (A4)
where KC, Ku and Kout are the constants that characterize the cubic, uniaxial and out-of-plane
magnetic anisotropy fields in (Ga,Mn)As, respectively. Hext is the magnitude of the external
magnetic field whose orientati on is described by the angles θH and φH, which are again
counted from the [001] and [100] crystallographic direc tions, respectively. For small
deviations δθ and δφ from the equilibrium values θ0 and φ0, the Eqs. (A2) and (A3) can be
written in a linear form as
ௗఏ
ௗ௧ൌܦଵሺߠെߠሻܦଶሺ߮െ߮ሻ, ( A 5 )
ௗఝ
ௗ௧ൌܦଷሺߠെߠሻܦସሺ߮െ߮ሻ, ( A 6 )
where 16
ܦଵൌௗഇ
ௗୀబ,ୀబ , ( A 7 a )
ܦଶൌௗഇ
ௗୀబ,ୀబ , ( A 7 b )
and analogically for D3, D4. The solution of Eqs. (A5) and (A6) is expressed by Eqs. (2) and
(3) where the magnetizati on precession frequency f and the damping rate kd are given by
݂ൌඥସሺభరିమయሻିሺభାరሻమ
ସగ, ( A 8 )
݇ௗൌെభାర
ଶ. ( A 9 )
Eqs. (A8) and (A9) for F in the form (A4) can be simplified when the geometry of our
experiment – i.e., the in-plane orientation of the external magnetic field ( θH = π/2) – is taken
into account. The equilibrium orientation of magnetization is in the sample plane for
(Ga,Mn)As ( θ0 = π/2) and the same applies for (Ga,Mn)(As, P) if sufficiently strong external
magnetic field (see Fig. 1) is applied ( θ0 ≈ θH = π/2). In such conditi ons, the precession
frequency f and the damping rate kd are given by the following equations
݂ൌఓಳ
ଶగሺଵାఈమሻ
ۣളളളളളളളളളളളളളളളለ
൬ܪ௫௧ݏܿሺ߮െ߮ுሻെ2ܭ௨௧ሺଷା௦ସఝ ሻ
ଶ2ܭ௨݊݅ݏଶቀ߮െగ
ସቁ൰ൈ
ൈሺܪ௫௧ݏܿሺ߮െ߮ுሻ2ܭݏܿ4߮െ2ܭ ௨݊݅ݏ2߮ሻ
ߙଶ
ەۖ۔ۖۓ൬ܪ௫௧ݏܿሺ߮െ߮ுሻെ2ܭ௨௧ሺଷା௦ସఝ ሻ
ଶ2ܭ௨݊݅ݏଶቀ߮െగ
ସቁ൰ൈ
ൈሺܪ௫௧ݏܿሺ߮െ߮ுሻ2ܭݏܿ4߮െ2ܭ ௨݊݅ݏ2߮ሻെ
െቀܪ௫௧ݏܿሺ߮െ߮ுሻെܭ௨௧ሺଷାହ௦ସఝ ሻ
ସೠሺଵିଷ௦ଶఝ ሻ
ଶቁଶ
ۙۖۘۖۗ( A10)
݇ௗൌߙఓಳ
ଶሺଵାఈమሻ൬2ܪ௫௧ݏܿሺ߮െ߮ுሻെ2ܭ௨௧ሺଷାହ௦ସఝ ሻ
ଶܭ௨ሺ1െ3݊݅ݏ2߮ ሻ൰. (A11)
17
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2107.14254v2.Microscopic_analysis_of_sound_attenuation_in_low_temperature_amorphous_solids_reveals_quantitative_importance_of_non_affine_effects.pdf | Microscopic analysis of sound attenuation in low-temperature amorphous
solids reveals quantitative importance of non-ane e ects
Grzegorz Szamel1,a)and Elijah Flenner1
Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523,
USA
(Dated: 4 April 2022)
Sound attenuation in low temperature amorphous solids originates from their disordered structure. However,
its detailed mechanism is still being debated. Here we analyze sound attenuation starting directly from the
microscopic equations of motion. We derive an exact expression for the zero-temperature sound damping
coecient. We verify that the sound damping coecients calculated from our expression agree very well with
results from independent simulations of sound attenuation. The small wavevector analysis of our expression
shows that sound attenuation is primarily determined by the non-ane displacements' contribution to the
sound wave propagation coecient coming from the frequency shell of the sound wave. Our expression involves
only quantities that pertain to solids' static con gurations. It can be used to evaluate the low temperature
sound damping coecients without directly simulating sound attenuation.
I. INTRODUCTION
The physics of sound attenuation in amorphous solids
is drastically di erent than in crystalline solids. At low
temperatures, when thermal e ects can be neglected,
sound is attenuated due to the inherent disorder of amor-
phous solids, whereas the attenuation is absent in crys-
talline solids. To understand the physical mechanism be-
hind sound attenuation one can examine its wavevector
kdependence. Sound attenuation in amorphous solids
has a complicated dependence on the wavevector1, but
small wavevector k4scaling of sound damping coe-
cients has long been conjectured on an experimental
basis2,3. An initial interpretation was that this small
wavevector behavior originates from Rayleigh scattering
of sound waves from the solid's inhomogeneities. Recent
computer simulations4{6veri ed that in classical three-
dimensional zero-temperature amorphous solids at the
smallest wavevectors sound damping coecients scale as
k4, although a logarithmic correction to this scaling was
also claimed7.
The speci c physical mechanism of sound attenuation
in low temperature amorphous solids is still debated.
Zeller and Pohl2obtained the Rayleigh scattering law us-
ing an \isotopic scattering"3model in which every atom
of the glass is an independent source of scattering. Sev-
eral recent experimental and simulational results were
analyzed within the framework of the
uctuating elastic-
ity theory of Schirmacher8{10. This theory posits that an
amorphous solid can be modeled as a continuous medium
with spatially varying elastic constants. The inhomo-
geneity of the elastic constants causes sound scattering
and attenuation. In the limit of the wavelength being
much larger than the characteristic spatial scale of the
inhomogeneity this mechanism is equivalent to Rayleigh
scattering and the theory predicts that sound damping
a)Email: grzegorz.szamel@colostate.educoecients scale with the wavevector as k4. If the elastic
constant variations have slowly decaying, power-law-like
correlations, the theory predicts a logarithmic correction
to Rayleigh scattering7,15. Other physical approaches,
e.g.local oscillator9,11{14and random matrix16{18mod-
els, can also be used to derive the Rayleigh scattering
law. For this reason, Rayleigh scaling cannot serve to
distinguish between di erent models9, and other model
predictions must be used to determine the mechanism
behind sound attenuation.
Three recent studies came to very di erent conclu-
sions regarding the applicability of the
uctuating elas-
ticity theory for sound attenuation. First, Caroli and
Lema^ tre19analyzed a version of the theory derived from
microscopic equations of motion. They obtained allthe
parameters needed to calculate sound attenuation from
the theory from the same simulations that were used
to test the theoretical predictions. Caroli and Lema^ tre
showed that this version of the theory underestimates
sound damping coecients by about two orders of mag-
nitude.
Second, Kapteijns et al.20analyzed the dependence of
sound attenuation in a two-dimensional glass on a param-
eter, which \resembles" changing the stability of the
amorphous solid. To calculate the disorder parameter8
of the
uctuating elasticity theory they replaced
uctu-
ations of local elastic constants (which are used in the
theoretical description) by the sample-to-sample
uctu-
ations of bulk elastic constants. In this way they were
able to sidestep the issue of the de nition of local elastic
constants21and of the correlation volume. While Kaptei-
jnset al. showed that the disorder parameter and the
sound damping coecient have the same dependence on
, they left the calculation of the pre-factor for the scaling
for further research.
Finally, Mahajan and Pica Ciamarra22argued that
sound attenuation is proportional to the square of the
disorder parameter
according to a version of
uctuating
elasticity theory that incorporates an elastic correlation
length9,23. They relied upon a relation between the bo-arXiv:2107.14254v2 [cond-mat.dis-nn] 31 Mar 20222
son peak, the speed of sound, and an elastic correlation
length to show that the speed of sound and the boson
peak frequency can be used to infer the change of the
sound damping coecient. Again, the magnitude of the
sound damping coecient was not addressed.
The results described above show that it is dicult
to distinguish between and to validate di erent semi-
phenomenological models invoked to describe sound at-
tenuation in zero-temperature amorphous solids. One
of the reasons is that most of these approaches involve
an adjustable parameter (or parameters) and therefore
are able to predicts trends rather than absolute values of
sound damping coecients. For example, neither Kaptei-
jnset al. nor Mahajan and Pica Ciamarra calculated the
values of sound damping coecients (in contrast to Car-
oli and Lema^ tre), but rather investigated the variation of
the sound attenuation between di erent glasses. Limited
range of glasses that can be created in silico makes it dif-
cult to distinguish between trends predicted by di erent
models or di erent versions of a model.
Our goal is to understand the microscopic origin of
the sound attenuation. We derive an exact expression
for the sound damping coecient in terms of quanti-
ties that can be calculated from static con gurations of
amorphous solids, without the need to directly simulate
sound attenuation. Our expression is analogous to well-
known Green-Kubo formulae for transport coecients24.
The latter expressions allow one to calculate transport
coecients without explicitly simulating transport pro-
cesses. While both our expression and Green-Kubo for-
mulae need to be evaluated numerically, they can also
serve as starting points for approximate analyses and
treatments that can shed light at the validity of semi-
phenomenological models. We hope that the results of
one such analysis, which we present at the end of the pa-
per, can inspire new models or be incorporated into the
existing ones.
In Sec. II we start from the microscopic equations
of motion for harmonic vibrations. We derive an ex-
act equation of motion for an auto-correlation function
that has been used to determine sound attenuation. We
identify the self-energy and show that its real part repro-
duces the non-Born contribution to the zero-temperature
wave propagation coecients. The imaginary part of the
self-energy is the origin of sound attenuation. We show
that sound damping coecients calculated this way agree
very well with those obtained from direct simulations of
sound attenuation in zero-temperature glasses with dif-
ferent stability. In Sec. III we present the small wavevec-
tor expansion of our expression for the sound damping co-
ecient. It shows that the limiting k4sound attenuation
originates from the same physics as the non-Born con-
tribution to the elastic constants and wave propagation
coecients, i.e.from the forces inducing non-ane dis-
placements, which appear due to the amorphous solids'
disordered structure. More precisely, attenuation of the
sound wave is primarily determined by the contribution
to the non-Born part of the wave propagation coecientfrom a shell of frequencies around the frequency of the
sound wave. We thus show the common origin of the
renormalization of the elastic constants and of sound at-
tenuation. In Sec. IV we discuss the results of an approx-
imate evaluation of our expression for the sound damp-
ing coecient which assumes that the exact eigenvectors
of the Hessian matrix can be replaced by plane waves.
These results allow us to critically evaluate the relation
between our exact expression and the
uctuating elas-
ticity theory. We end the paper with a discussion of our
results and related descriptions of the sound attenuation.
II. MICROSCOPIC ANALYSIS OF SOUND
ATTENUATION
We start from microscopic equations of motion for
small displacements of Nspherically symmetric particles
of unit mass comprising our model amorphous solid,
@2
tui= X
jHijuj: (1)
Hereuiis the displacement of the ith particle from its in-
herent structure (potential energy minimum) position Ri
andHis the Hessian calculated at the inherent structure,
Hij=X
l6=i@2V(Ril)
@Ri@Rj(2)
whereV(r) is the pair potential and Hijis a 3x3 tensor.
To derive an expression for the sound damping coe-
cient we use a slightly modi ed procedure proposed by
Gelin et al.7. We assume that at t= 0 the particles are
displaced from their equilibrium positions according to
ui(t= 0) = ^eexp( ikRi),_ui(t= 0) = 0, where ^eis
a unit vector and wavevector kis one of the wavevectors
allowed by periodic boundary conditions. We then ana-
lyze the time dependence of the auto-correlation func-
tion of the single-particle displacement averaged over
the whole system, C(t) =N 1P
iu
i(t= 0)ui(t).
We anticipate that in the limit of small wavevectors k
the auto-correlation function will exhibit damped oscilla-
tions,C(t)/cos(vkt) exp( (k)t=2), and we will iden-
tifyvas the speed of sound and ( k) as the damping
coecient.
Solving Eqs. (1) with our initial conditions is equiva-
lent to solving the following equations
@2
tai= X
jHij(k)aj; (3)
whereH(k) is the wavevector-dependent Hessian,
Hil(k) =Hilexp[ik(Ri Rl)], with initial conditions
ai(t= 0) = ^e,_ai(t= 0) = 0. In terms of the new
variables,C(t) =N 1P
iai(t= 0)ai(t).
To analyze C(t) we use the standard projection op-
erator approach25. First, we de ne a scalar product
of two displacement vectors, aiandbj,i;j= 1;:::;N ,
hajbi=P
ia
ibi. Next, we de ne a unit vector j1iwith3
components 1i=N 1=2^e, and projection operator Pon
the unit vector,P=j1ih1jand orthogonal projection
Q,Q=I j 1ih1jwhereIis the identity matrix.
Using the projection operator approach we obtain the
following expression for the Fourier transform C(!) =1
0C(t) exp(i(!+i))dtof the displacement auto-
correlation function,
C(!) =i(!+i)
(!+i)2 h1jH(k)j1i+(k;!); (4)
where the self-energy (k;!) reads
(k;!) =
1 H(k)Q1
(!+i)2+QH(k)QQH(k) 1
:
(5)
Equations (4-5) are exact. While it is straightforward
to calculateh1jH(k)j1i, evaluation of the self-energy re-
quires inversion of a large-dimensional matrix for each
allowed wavevector. To make the numerical e ort man-
ageable, in the denominator in Eq. (5) we approximate
H(k) byH. As argued in Appendix A, this approxima-
tion does not in
uence the small wavevector dependence
of the sound damping coecients.
In the small wavevector limit, the rst non-trivial term
in the denominator in Eq. (4), h1jH(k)j1i, can be ex-
pressed in terms of the Born contributions to the zero-
temperature wave propagation coecients26,
h1jH(k)j1i= 1^e ABorn
k ^e
k+o(k2) (6)
where=N=V is the number density, Greek indices
denote Cartesian components, the Einstein summation
convention for Greek indices is hereafter adopted, and
ABorn
is the Born wave propagation coecient, which
can be expressed as the average of the local Born wave
propagation coecients ABorn
j;
,
ABorn
j;
=
2X
l6=j@2V(Rjl)
@Rj; @Rj;
Rjl; Rjl; (7)
over the whole system,
ABorn
=N 1X
jABorn
j;
: (8)
For example, if the coordinate system is chosen such that
^eis in the x direction, and we are interested in a trans-
verse wave and choose kin theydirection, then the right
hand side of (6) becomes 1ABorn
xyxyk2.
In the absence of the self-energy term, (4) predicts the
Born value of the speed of sound and no sound damp-
ing. Both the renormalization of the sound speed and
the sound attenuation originate from the self-energy.
The self-energy can be calculated using the eigenvalues
and eigenvectors of the Hessian. In the thermodynamic
limit27, when the spectrum of the Hessian becomes con-
tinuous, we can use the Plemelj identity to identify the
Born Term
Deform
Full Theory
(a) (b)vT
2.02.53.0vL
5.05.5
Tp0.1 0.2Tp0.1 0.2
Tp = 0.200
Tp = 0.085
Tp = 0.062
Theory; T p = 0.200
Theory; T p = 0.085
Theory; T p = 0.062(c) (d)
e ||k e 㾉 k
k4k4ΓT
10−410−310−210−11ΓL
10−410−310−210−11
k0.1 1
k0.1 1FIG. 1. Upper panels: the transverse (a) and longitudi-
nal (b) speed of sound obtained from the theory, Eq. (13),
(red squares) and calculated from the elastic constants (black
circles) as a function of parent temperature Tp. The black
triangles are the Born values of the speed of sound. Lower
panels: the transverse (c) and longitudinal (d) damping co-
ecients obtained from the theory, Eq. (14), (squares) and
obtained from sound attenuation simulations5(circles) for dif-
ferent parent temperatures. Rayleigh scaling /k4is recov-
ered at small wavevectors. The error bars for the theoretical
calculation represent the uncertainty due to using di erent
bin sizes.
imaginary component of the self-energy, which is respon-
sible for sound attenuation. The real 0(k;!) and imag-
inary 00(k;!) parts of the self-energy read,
0(k;!) =
d
(k;
)
2 !2 1; (9)
00(k;!) =
2!(k;j!j); (10)
where
denotes the Cauchy principal value. The func-
tion ( k;
) is de ned through the sum over eigenvectors
Epof the Hessian matrix with non-zero28eigenvalues
2
p
such that
p2[
;
+ d
], where d
is the bin size,
(k;
) = (1=d
)X
p2[
;
+d
]jh1jH(k)QjEpij2:(11)
The key conceptual issue in writing Eq. (11) (and closely
related equations (18,21)) is that the thermodynamic4
limit is implied for the expression at the right-hand-side.
In this limit the spectrum becomes dense and phonon
bands are not distinguishable. Thus, to calculate ( k;
)
from the analysis of nite-size simulations we need to
choose bin size d
such that phonon bands are not re-
solved. In the numerical calculations described below we
tried a few bin sizes between 0.1 and 0.2 and found that
within this range the results were not very sensitive to
the bin size.
To evaluate the displacement auto-correlation function
we need to nd complex poles of the denominator at the
right-hand-side of Eq. (4). In the small wavevector limit
this can be done perturbatively, using kas the small pa-
rameter. This leads to the following pair of poles,
!=vk i00(k;vk)=(2vk) (12)
where the renormalized speed of sound vis given by
v2= lim
k!0k 2
h1jH(k)j1i 0(k; 0)
: (13)
The last term in Eq. (12) is our main result. It says that
the sound damping in zero-temperature amorphous solids
is determined by ( k;
)=
2calculated at the wave's fre-
quency,
= vk,
(k) =00(k;vk)
vk=
2(k;vk)
(vk)2: (14)
We emphasize that ( k;
)=
2is the same function that,
after integration over the whole frequency spectrum, de-
termines the renormalization of the wave propagation
coecients. Note that v, (k), and related quantities
de ned below depend on the angle between the polariza-
tion of the initial condition ^eand the direction of the
wavevector ^k.
To verify Eqs. (13-14) we calculated vand (k) for
model zero-temperature glasses analyzed in Ref.5. These
glasses were obtained by instantaneously quenching su-
percooled liquids equilibrated using the swap Monte
Carlo algorithm29at di erent parent temperatures Tpto
their inherent structures using the fast inertial relaxation
engine minimization30. The glasses consist of spherically
symmetric, polydisperse particles which interact via a po-
tential/r 12, with a smooth cuto , see Appendix B and
Refs.5,29for details. The parent temperature controls the
glass's stability and thus its properties5,31,32.
We calculated eigenvalues and eigenvectors of the Hes-
sian using ARPACK33and Intel Math Kernel Library34.
Then, using Eqs. (13-14) we evaluated vand (k) for
the longitudinal, ^ek^k, and the transverse, ^e?^k,
sound. To calculate ( k;
) one chooses a kncompatible
with periodic boundary conditions. Then one calculates
jh1jH(kn)QjEp(
p)ijand bins the results according to
the square root of the eigenvalue of jEpito determine
(kn;
). Note that
2
pis the eigenvalue corresponding
tojEpi. The damping is given by (14) where ( kn;
) is
evaluated at
= jknjv.
Fig. 1 shows results for vand for three parent tem-
peratures. Tp= 0:2; glasses obtained by quenching liq-
uid samples equilibrated at 0 :2 are much less stable thantypical laboratory glasses. Tp= 0:085, which is be-
tween the mode-coupling temperature Tc:108 and
the estimated laboratory glass transition temperature
Tg0:072; glasses obtained by quenching samples equili-
brated at 0:085 are about as stable as typical laboratory
glasses.Tp= 0:062, which is well below estimated Tg;
glasses obtained by quenching liquid samples equilibrated
at 0:062 are as stable as laboratory ultrastable glasses ob-
tained by the vapor deposition method35,36. We previ-
ously showed5that sound damping coecients decrease
by more than an order of magnitude over this range of
stability.
For all three parent temperatures there is excellent
agreement between results of Eqs. (13-14) and transverse
and longitudinal sound speeds, vTandvL, and transverse
and longitudinal sound damping coecients, Tand L
obtained previously5from direct simulations of sound
attenuation. At small wavevectors we recover Rayleigh
scaling, /k4, but the theory also accurately predicts
sound damping for wavevectors outside the Rayleigh scal-
ing regime. The predicted damping coecients depart
from the simulation results for larger wavevectors, but
at larger wavevectors the assumptions used to nd the
poles, Eq. (12), become invalid.
III. THE ORIGIN OF SOUND ATTENUATION:
NON-AFFINE EFFECTS
To get some physical insight into the origin of sound
attenuation in zero-temperature amorphous solids we ex-
amine the small wavevector expansion of h1jH(k)QjEpi,
h1jH(k)QjEpi= iN 1=2X
jj;
^e
kEp
j (15)
+ 1N 1=2X
j
ABorn
j;
^e ^eABorn
^e
k kEp
j; +o(k2):
In Eq. (15) Ep
jdenotes the component of the pth eigen-
vector of the Hessian corresponding to particle jandEp
j;
denotes its Cartesian component . Furthermore, j;
denotes the vector eld describing forces due ane de-
formations,
j;
= X
l6=j@2V(Rjl)
@Rj;
@RjRjl;: (16)
Speci cally, j;
is proportional to the force on parti-
clejresulting from a deformation along the
direction
that linearly depends on the coordinates. Finally, the
2nd term at the right-hand-side of Eq. (15) accounts for
the spatial variation of the local Born wave propagation
coecients.
As discussed in the literature37,38, forces encoded in
vector eld j;
do not seem to posses any longer-
range correlations. In contrast, non-ane displace-
ments given byH 1exhibit characteristic vortex-like
structures and correlations extending over many particle5
Tp = 0.200
vTk1Γ(vTk1)(a)ΓT terms
10−410−21
Ω 0 0.5
πΥ/[2(vTk1)2]
πΘ/(2v2T)
k21 πΔ/(2v2T)Tp = 0.062
vTk1ΓT(vTk1)(b)ΓT terms
10−410−21
Ω0.5
FIG. 2. The terms that contribute to the transverse sound
attenuation for k1= 2=L = 0:13722, where Lis the length
of the simulation box, given by Eq. 20 for Tp= 0:200 (a)
andTp= 0:062 (b). The vertical lines marks the frequency
=vTk1, wherevTis the transverse speed of sound, and the
horizontal line is the sound attenuation calculated in simula-
tions from Ref.5. The value of (k1;
=vTk1)=[2(vTk1)2]
gives the damping for k1.
diameters37{40. The characteristic length of these cor-
relations determines the minimal length scale on which
a macroscopic elastic approach can be used to describe
the response of amorphous solids39. It follows from the
combination of Eqs. (9), (13) and (15) that the renor-
malization of the wave propagation coecients originates
from the rst term in Eq. (15),
lim
k!0k 20(k; 0) =N 1
d
(
)
2(17)
where (
) is de ned analogously to ( k;
),
(
) = (1=d
)X
p2[
;
+d
] p
^e
^k 2
(18)
with
p
=N 1=2X
jj;
Ep
j: (19)
Equations (17-18) reproduce the exact expression for
the non-Born contribution to the wave propagation co-
ecients derived from the analysis of the non-ane
displacements37,38. We note that function (
) is closely
related to function (!) introduced and evaluated
by Lema^ tre and Maloney, see Eq. (32) of Ref.38.While only the rst term in Eq. (15) determines
the renormalization of the wave propagation coecients,
both terms contribute to sound attenuation,
(k) =
2v2
(vk) +k2(vk)
(20)
where (
) is de ned analogously to ( k;
),
(
) = (1=d
)X
p2[
;
+d
] Ap
^k ^e
^k 2
(21)
with
Ap
= 1N 1=2X
j
ABorn
j;
^e ^eABorn
Ep
j; :
(22)
We note that the second term in Eq. (20) is expressed in
term of the
uctuations of the local Born wave propaga-
tion coecients, see Eq. (22). Thus, the physical content
of the second term resembles that of the
uctuating elas-
ticity theory. We will discuss this correspondence further
in the next section.
It is the rst term in Eq. (20) that makes the dominant
contribution to the damping coecient, see Fig. 2. This
implies that the sound damping is primarily determined
by function (
), which is the same function that also
determines the renormalization of the wave propagation
coecients, Eq. (17). While previous studies suggested41
and analyzed approximately42the importance the non-
ane e ects for the sound attenuation, we have presented
the rst approach that accounts for these e ects exactly.
IV. SOUND DAMPING IN THE PLANE-WAVE
APPROXIMATION
The most recent version of the
uctuating elastic-
ity theory discussed by Mahajan and Pica Ciammarra22
posits that \amorphous materials can be described as ho-
mogeneous isotropic elastic media punctuated by quasilo-
calized modes acting as elastic heterogeneities." This
suggests that plane waves should be a reasonable zeroth
order approximation for the eigenvectors of the Hessian
matrix describing an amorphous solid. To check this sup-
position we calculated and contributions in Eq. (20)
approximating the exact eigenvectors by plane waves,
Ep
j/^eqe iqRj, see Appendix C for details. For the
contributions to transverse wave damping coecient we
obtained the following expressions
2v2(vTk)1
60k4
3v2
T
v5
L+4
v3
T
ABorn
xyABorn
xy
+v2
T
v5
L 1
v3
T
ABorn
xyABorn
xy+ABorn
xyABorn
xy
;(23)
k2
2v2(vTk)1
12k4
31
v3
L+2
v3
ThD
ABorn
xyxy2E
+D
ABorn
yyxy2E
+D
ABorn
zyxy2Ei
; (24)6
where we implicitly assumed analyticity of the correlation
functions of local wave propagation coecients at the
vanishing wavevector. For example, we assumed that at
q!0,
D
ABorn
xyxy2E
= lim
q!0N 1 X
jeiqRj
ABorn
j;xyxy ABorn
xyxy 2
(25)
and other similar equalities, as discussed in Appendix C.
We note that while the exact formula (18) for contri-
bution involves non-ane forces , approximate formula
(23) is expressed in terms of correlations of local wave
propagation coecients. This follows from the fact that,
as shown in Appendix C, for small wavevectors q
X
jj;
^eqe iqRj=i
X
jABorn
j;
^eq q e iqRj+o(q):
(26)
Furthermore, we note that formulae (23-24) are reminis-
cent of Zeller and Pohl's \isotopic scattering" model in
that every atom jis a source of scattering of a plane wave,
with the amplitude depending on its local wave propa-
gation coecient ABorn
j;
. Importantly, our approximate
formulae involve correlations of local wave propagation
coecients that vanish at the macroscopic level and thus
do not appear in the semi-phenomenological
uctuating
elasticity theory, e.g.ABorn
j;yyxy .
The plane wave approximation recovers analytically
the Rayleigh scattering k4scaling of the sound damp-
ing coecient. However, it is quantitatively quite inac-
curate, see Fig. 3. This implies that at least for the pur-
pose of calculating sound damping, eigenvectors of the
Hessian are not well approximated by plane waves. We
note that the plane-wave approximation becomes more
accurate with decreasing parent temperature or increas-
ing glass stability.
Finally, we note that the rst term in square brackets
in Eq. (24), which involves correlations of the
uctua-
tions of the local shear modulus, ABorn
j;xyxy , represents the
result of the microscopic, isotopic scattering-like, version
of the
uctuating elasticity theory. As shown in Fig. 3,
this term is about 2.5-4 times smaller than the complete
plane-wave result, and thus it severely underestimates
sound attenuation.
In Fig. 3 we also show the result of a semi-
phenomenological
uctuating elasticity theory. To cal-
culate this result we started from the celebrated formula
of Rayleigh43that predicts the attenuation of a trans-
verse wave due to inclusions of volume Vdand number
densityn, R(k) =nVdvT
k4=(6), where
is the dis-
order parameter. In Rayleigh's calculation
character-
ized the variation of \optical density". To adopt his
calculation to the present problem we expressed
in
terms of the variation of the square of the transverse
speed of sound,
= (v2
T)2Vd=v4
T. Next, we added to
Tp = 0.062 Tp = 0.2Simulation
TheoryPlane WaveFET(ΔABorn
xyxy)2ΓT
10−410−310−210−11ΓT
10−410−310−210−11
k0.1 1
k0.1 1FIG. 3. Comparison of the sound damping coecients ob-
tained from the theory (squares) and evaluated from sound
attenuation simulations (circles) with predictions of the plane-
wave approximation, Eqs. (23-24) (red symbols) and micro-
scopic version of the
uctuating elasticity theory, i.e. the
contribution due toD
ABorn
xyxy2E
term in Eq. (24) (black
line). Also shown is the result of a semi-phenomenological
uctuating elasticity theory (FET, green line).
Rayleigh's expression the the contribution of the longitu-
dinal waves excited due to the presence of the inclusions,
(k) =nVdvT
k4(v3
T=(2v3
L))=(6). The complete for-
mula of the semi-phenomenological
uctuating elasticity
theory thus reads
FET(k) =k4vT
6
1 +1
2v3
T
v3
L
nVd
: (27)
We note that if one makes the identi cationD
ABorn
xyxy2E
=(3v4
T) =nVd
, the contribution to
sound attenuation due to the rst term in square brack-
ets in Eq. (24) becomes identical to expression (27).
To calculate the value of FETwe need the disorder
parameter
and the volume fraction of the inclusions
nVd. For
we used previously obtained results for the
uctuations of local elastic constants44. We recall that
disorder parameters calculated this way increase slightly
with increasing box size used to de ne local elastic
constants, thus we used the largest box size considered
in Ref.44. Furthermore, we note that Mahajan and Pica
Ciamarra's formulation of
uctuating elasticity theory
assumesnVd 1, see the SI of Ref.22. To calculate the
upper bound for FETwe substituted nVd= 1. Figure
3 shows that the result of this procedure signi cantly
underestimates sound attenuation.
We note that in addition to the microscopic ver-
sion of the
uctuating elasticity theory, originally de-
rived by Caroli and Lema^ tre and embodied in the rst
term in square brackets in Eq. (24), and the semi-
phenomemonogical approach resulting in expression (27)
one could compare our results to predictions of more so-
phisticated versions of the
uctuating elasticity theory,
e.g. the version relying upon the self consistent Born
approximation8. This comparison is left for future work.7
V. DISCUSSION
According to our microscopic analysis, sound attenu-
ation in zero-temperature amorphous solids is primarily
determined by internal forces induced by initial ane
displacements of the particles, i.e.by the physics of non-
ane displacement elds. Quantitatively, the damping
coecient is proportional to the non-ane contribution
to the wave propagation coecients from the frequency
equal to the frequency of the sound wave. It is not trivial
that our exact calculation (as opposed to the plane-wave
approximation discussed in the previous section) repro-
duces the Rayleigh scaling of sound damping coecients.
This fact results from the frequency dependence of and
, which deserves further theoretical study.
The mechanism of the attenuation revealed by our mi-
croscopic analysis was mentioned by Caroli and Lema^ tre
in Ref.19. It was investigated in Ref.41, where Caroli and
Lema^ tre considered separately the e ects of the long-
wavelength, elastic continuum-like, and small-scale, pri-
marily non-ane, motions with the small-scale motions
being the scatterers for the long-wavelength ones.
An earlier study by Wang, Szamel and Flenner5found
a strong correlation between the sound attenuation co-
ecient and the amplitude of the vibrational density of
states of quasilocalized modes. The latter modes were
de ned using a cuto in the participation ratio, folow-
ing Mizuno et al.45and Wang et al.31. We attempted to
quantify the relative contributions of the extended and
quasi-localized modes by separating the contributions of
modes with small and large participation ratio. We did
not nd convincing evidence for the dominance of small
participation ratio modes versus larger participation ra-
tio modes.
We note in this context that local oscillator
models11{14express the sound attenuation coecient in
terms of the contributions from localized \defects"46,47
referred to as \soft modes". The formulas derived in
these approaches are similar to our Eqs. (14) and (20).
The details of expressions of Refs.11{14and our Eqs. (14)
and (20) di er; in particular we express the self-energy in
terms of all the exact eigenvectors and eigenvalues of the
Hessian matrix. In order to evaluate the local oscillator
based sound attenuation coecient formulas one needs to
characterize the properties of the soft modes. In practical
applications one may parametrize the soft modes' prop-
erties and t the parameters to the experimental results.
Such a procedure was used by Schober14and resulted in
a good agreement between the theory and experiment.
In view of both the previously found5correlation be-
tween the sound attenuation coecient and the ampli-
tude of the vibrational density of states of quasilocalized
modes and the success of local oscillator approach14we
believe that future work should investigate whether dom-
inant contributions to the sound atteanuation coecient
formulas (14) and (20) originate from well de ned regions
that can be identi ed as \defects".
Damart et al.40demonstrated that the non-ane dis-placement eld was responsible for high-frequency har-
monic dissipation in a simulated amorphous SiO 2. There-
fore, it appears that non-ane displacements are respon-
sible for dissipation over the full frequency range. Fur-
ther theoretical development is needed to connect the
low-frequency and high-frequency theories.
Recently, Baggioli and Zaccone developed an approxi-
mate microscopic theory for the sound attenuation that
takes into account non-ane displacements42. This the-
ory shares physical insight with our approach but it is
quantitatively as inaccurate as the plane-wave version of
our exact formula.
As we mentioned in the introduction, Gelin et al.7
found a logarithmic correction to the Rayleigh scattering
scaling of the sound damping coecients, which within
the
uctuating elasticity theory could originate from the
slowly decaying correlations between local values of the
elastic constants7,15. Within our approach, a logarithmic
correction could originate from a logarithmic dependence
of (
) or (
) on frequency
. Our present numerical
data are consistent with the absence of such a logarith-
mic dependence but it would be interesting to investigate
this issue farther.
Within the plane-wave approximation a logarithmic
correction could result from a logarithmic small wavevec-
tor divergence of the correlation functions of local Born
wave propagation coecients. We did not observe such
a divergence but we note that our systems were signif-
icantly smaller that those discussed in Ref.7. We note
that if the correlation functions of local wave propagation
coecients are singular, additional terms in the plane-
wave approximation will appear. These terms will origi-
nate from the anisotropic small wavevector character of
the correlation functions of local wave propagation coef-
cients.
Our approach arrives at the physical picture of sound
attenuation di erent from that postulated in the
uctuat-
ing elasticity theory. While the latter theory can predict
trends20,22, it is quantitatively very inaccurate, as noted
earlier by Caroli and Lema^ tre19. Our analysis revealed
that the
uctuating elasticity theory misses the domi-
nant non-ane e ects. In addition, it does not include
the contributions due to
uctuations of local microscopic
wave propagation coecients that vanish at the macro-
scopic level. Most importantly, the
uctuating elastic-
ity theory uses plane-wave-like picture of sound in low-
temperature amorphous solids. The comparison of the
results obtained using the full theoretical expression and
adopting the plane-wave approximation, shown in Fig. 3,
suggests that this leads to large quantitative discrepan-
cies.
Finally, we note that calculating sound attenuation us-
ing Eq. (14) or (20) is somewhat numerically demanding
but more straightforward than analyzing the time de-
pendence of the velocity or displacement auto-correlation
functions. The latter analysis su ers from large nite-size
e ects5,48that make the evaluation of the sound damping
coecients at the smallest wavevectors allowed by the pe-8
riodic boundary conditions dicult. Our approach o ers
an attractive alternative way to evaluate sound damping
coecients of low temperature elastic solids that does not
su er from nite size e ects.
ACKNOWLEDGMENTS
We thank A. Ninarello for generously providing equi-
librated con gurations at very low parent temperatures
and E. Bouchbinder for comments on the manuscript.
We gratefully acknowledge the support of NSF Grant
No. CHE 1800282.AUTHOR DECLARATIONS
Con
ict of interest
The authors have no con
icts to disclose.
DATA AVAILABILITY
The data that support the ndings of this study are
available from the corresponding author upon reasonable
request.
Appendix A: Approximation H(k)H in Eq. (5) of the
main text
First, we examine the small wavevector expansion of
QH(k)Q. Thei;jelement, which is a 3x3 tensor, reads
[QH(k)Q]ij=Hije ik(Ri Rj) N 1X
lHile ik(Ri Rl)^e^e N 1^eX
l^eHlje ik(Rl Rj)
+N 2^eX
l;m^eHlm^ee ik(Rl Rm)^e= [QH(k= 0)Q]ij+
HQ1(k)
ij+
HQ2(k)
ij+o(k2); (A.1)
where the matrix elements of the terms of the rst and second order in k,HQ1(k) andHQ2(k), read
HQ1(k)
ij=i(1 ij)(
@2V(Rij)
@R2
ik(Ri Rj) N 1X
l^e"
@2V(Ril)
@R2
ik(Ri Rl) @2V(Rlj)
@R2
jk(Rj Rl)#
^e)
;
(A.2)
HQ2(k)
ij=1
2(1 ij)@2V(Rij)
@R2
i(k(Ri Rj))2 1
2N 1X
l6=i@2V(Ril)
@R2
i^e(k(Ri Rl))2^e
1
2N 1^eX
l6=j^e@2V(Rlj)
@R2
l(k(Rl Rj))2+1
2N 2^eX
l6=m^e@2V(Rlm)
@R2
l^e(k(Rl Rm))2^e:
(A.3)
Next, we assume that for small wavevectors kwe can
treat terms HQ1(k) andHQ2(k) in the denominator
of Eq. (5) of the main text perturbatively. Due to the
symmetry, the term of the rst order in k,HQ1(k), willcontribute in the second order of the perturbation ex-
pansion. In contrast, the term of the second order in k,
HQ2(k), will contribute in the rst order. Here we will
show the contribution of HQ2(k) term. It reads
Q2(k;!) (A.4)
= X
eigenvec.
p;qh1jH(k)QjEpiD
Ep
(!+i)2+p 1 EpE
Ep HQ2(k) Eq D
Eq
(!+i)2+q 1 EqE
hEqjQH(k)j1i:
Counting powers of kin the expression above shows that, at least perturbatively, term HQ2(k) results in a correc-9
tion that is higher order in kthan the dominant small
wavevector result of approximation H(k)H in the de-
nominator of Eq. (5).
Appendix B: Simulation details
We obtained zero-temperature glasses by instanta-
neously quenching supercooled liquids of unit number
density,= 1:0, equilibrated through the swap Monte
Carlo algorithm29. The constituent particles of these
liquids have unit mass and diameters selected using
distribution P() =A
3, where2[0:73;1:63] and
Ais a normalization factor. The cross-diameter ij
is determined according to a non-additive mixing rule,
ij=i+j
2(1 ji jj) with= 0:2. The interac-
tion between two particles iandjis given by the inverse
power law potential, V(rij) = (ij=rij)12+Vcut(rij),
when the separation rijis smaller than the potential cut-
o rc
ij= 1:25ij, and zero otherwise. Here, Vcut(rij) =
c0+c2(rij=ij)2+c4(rij=ij)4, and the coecients c0,c2
andc4are chosen to guarantee the continuity of V(rij)
atrc
ijup to the second derivative.
The number of particles Nvaried between 48000 and
192000. The largest systems had to be analyzed to de-
termine sound attenuation at the lowest wavevectors re-ported.
Appendix C: Plane-wave approximation
We assume that for small wavevectors we can approxi-
mate eigenvectors of the Hessian matrix by plane waves.
We note that strictly speaking, for our amorphous solids
the normalization factor is con guration-dependent. We
checked that this dependence is weak and for this reason
we use the following approximation,
Ep
jN 1=2^eqe iqRj: (C.1)
Approximate plane-wave eigenvectors are labeled by
their wavevector qand their polarization ^eq. For each
wavevector qwe have one longitudinal and two trans-
verse modes. We assume that the associated eigenvalues
are given by ( vLq)2and (vTq)2for the longitudinal and
transverse modes, respectively.
Here we will present the derivation of approximate for-
mula for the contribution to the transverse sound damp-
ing coecient originating from , Eq. (23) of the main
text. The contribution originating from , Eq. (24) of
the main text and the approximate expression for the
longitudinal sound damping can be derived in a similar
way.
First, we need to calculate
iN 1=2X
jj;
^e
kEp
j iN 1X
jj;
^e
k^eqe iqRj=iN 1X
jX
l6=j@2V(Rjl)
@Rj;
@RjRjl;^e
k^eqe iqRj:(C.2)
Using thei$jsymmetry we get
iN 1X
jX
l6=j@2V(Rjl)
@Rj;
@RjRjl;^e
k^eqe iqRj=i
2NX
jX
l6=j@2V(Rjl)
@Rj; @Rj;
Rjl;^eq ^e
kh
1 eiq(Rj Rl)i
e iqRj
=1
2NX
jX
l6=j@2V(Rjl)
@Rj; @Rj;
Rjl;Rjl; ^eq q ^e
ke iqRj+o(q) = (N) 1X
jABorn
j;
^eq q ^e
ke iqRj+o(q):(C.3)
Next, we need to take the square of the absolute value
of expression (C.3) for a given wavevector qand polar-
ization ^eqand then integrate over spherical shell with
frequencyqvL=kvTfor longitudinal and qvT=kvTfor
transverse modes. We shall note that since the spher-
ical shell is speci ed in the frequency space, there will
be additional factors, 1 =vLfor longitudinal and 1 =vTfor
transverse modes. Finally, we need to multiply the result
by=(2v2
Tk2) to get the contribution to the transverse
sound damping coecient.
To perform these calculations we assume that wavevec-
torkis parallel to the yaxis and the sound polarization ^e
is along the xaxis. Furthermore, we specify the polariza-
tion vector for the approximate plane-wave eigenvectors
as^eL
q=^q(sincos ;sinsin ;cos) for the longi-tudinal modes and ^eT1
q= (coscos ;cossin ; sin)
and ^eT2
q= ( sin ;cos ;0) for the two transverse
modes.
The contribution of the longitudinal modes reads
2 (vTk)2v4
T
v5
LVk6
(2)3(C.4)
d^q (N) 1X
jABorn
j; xy ^eL
q ^q e i^q(kvT=vL)Rj 2
:
Guided by our numerical calculations we assume that
the following small wavevector limit is nite and does not10
depend on the direction
lim
q!0N 1X
jABorn
j; xyeiqRjX
lABorn
l;
xye iqRl
ABorn
xyABorn
xy
: (C.5)
Expression (C.4) becomes
2 (vTk)2v4
T
v5
LVk6
(2)31
2N
ABorn
xyABorn
xy
d^q^eL
q ^q ^eL
q
^q
=
2 (vTk)2v4
T
v5
LVk6
(2)31
2N
ABorn
xyABorn
xy
4
15(
+
+
)
=1
60k4
3v2
T
v5
L
ABorn
xyABorn
xy
+
ABorn
xyABorn
xy
+
ABorn
xyABorn
xy
: (C.6)
Assuming again that the small wavevector limit of the
correlation functions of local wave propagation coe-
cients is nite and does not depend on the direction, the
contribution of the two transverse modes reads
2 (vTk)21
vTVk6
(2)31
2N
ABorn
xyABorn
xy
d^q
^eT1
q ^q ^eT1
q
^q+ ^eT2
q ^q ^eT2
q
^q
=
2 (vTk)21
vTVk6
(2)31
2N
ABorn
xyABorn
xy
4
15(4
)
=1
60k4
31
v3
T
4
ABorn
xyABorn
xy
ABorn
xyABorn
xy
ABorn
xyABorn
xy
(C.7)
Adding expressions (C.6) and (C.7) we get Eq. (23) of
the main text.
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2205.06399v1.Precession_dynamics_of_a_small_magnet_with_non_Markovian_damping__Theoretical_proposal_for_an_experiment_to_determine_the_correlation_time.pdf | arXiv:2205.06399v1 [cond-mat.mes-hall] 13 May 2022Precession dynamics of a small magnet with non-Markovian da mping: Theoretical
proposal for an experiment to determine the correlation tim e,✩✩
Hiroshi Imamura, Hiroko Arai, Rie Matsumoto, Toshiki Yamaj i, Hiroshi Tsukahara✩
National Institute of Advanced Industrial Science and Tech nology (AIST), Tsukuba, Ibaraki 305-8568, Japan
Abstract
Recent advances in experimental techniques have made it pos sible to manipulate and measure the magnetization dynamics on
the femtosecond time scale which is the same order as the corr elation time of the bath degrees of freedom. In the equations of
motion of magnetization, the correlation of the bath is repr esented by the non-Markovian damping. For development of th e science
and technologies based on the ultrafast magnetization dyna mics it is important to understand how the magnetization dyn amics
depend on the correlation time. It is also important to deter mine the correlation time experimentally. Here we study the precession
dynamics of a small magnet with the non-Markovian damping. E xtending the theoretical analysis of Miyazaki and Seki [J. C hem.
Phys. 108, 7052 (1998)] we obtain analytical expressions of the prece ssion angular velocity and the e ffective damping constant for
any values of the correlation time under assumption of small Gilbert damping constant. We also propose a possible experi ment for
determination of the correlation time.
Keywords: non-Markovian damping, generalized Langevin equation, LL G equation, ultrafast spin dynamics, correlation time
1. Introduction
Dynamics of magnetization is the combination of precession
and damping. The precession is caused by the torque due to
the internal and external magnetic fields. Typical time scal e
of the precession around the external field and the anisotrop y
field is nanosecond. The damping is caused by the coupling
with the bath degrees of freedom such as conduction electron s
and phonons. The typical time scale of the relaxation of con-
duction electrons and phonons is picosecond or sub-picosec ond
which is much faster than precession. In typical experiment al
situations such as ferromagnetic resonance and magnetizat ion
process, the time correlation of the bath degrees of freedom
can be neglected and the magnetization dynamics is well repr e-
sented by the Landau-Lifshitz-Gilbert (LLG) equation with the
Markovian damping term[1–3].
Recent advances in experimental techniques such as fem-
tosecond laser pulse and time-resolved magneto-optical Ke rr
effect measurement have made it possible to manipulate and
measure magnetization dynamics on the femtosecond time
scale[4–11]. In 1996, Beaurepaire et al. observed the femto sec-
ond laser pulse induced sub-picosecond demagnetization of a
Ni thin film[4], which opens the field of ultrafast magnetiza-
tion dynamics. The all-optical switching of magnetization in a
✩Permanent address: High Energy Accelerator Research Organ ization
(KEK), Institute of Materials Structure Science (IMSS), Ts ukuba, Ibaraki 305-
0801, Japan
✩✩This work is partly supported by JSPS KAKENHI Grant Numbers
JP19H01108 and JP18H03787.
Email addresses: h-imamura@aist.go.jp (Hiroshi Imamura),
arai-h@aist.go.jp (Hiroko Arai)ferrimagnetic GdFeCo alloy was demonstrated by Stanciu et a l.
using a 40 femtosecond circularly polarized laser pulse[5] . The
helicity-dependent laser-induced domain wall motion in Co /Pt
multilayer thin films was reported by Quessab et al.[11].
To understand the physics behind the ultrafast magnetizati on
dynamics it is necessary to take into account the time correl a-
tion of bath in the equations of motion of magnetization. The
first attempt was done by Kawabata in 1972[12]. He derived the
Bloch equation and the Fokker-Planck equation for a classic al
spin interacting with the surroundings based on the Nakajim a-
Zwanzig-Mori formalism[13–15]. In 1998, Miyazaki and Seki
constructed a theory for the Brownian motion of a classical
spin and derived the integro-di fferential form of the generalized
Langevin equation with non-Markovian damping[16]. They
also showed that the generalized Langevin equation reduces to
the LLG equation with modified parameters in a certain limit.
Atxitia et al. applied the theory of Miyazaki and Seki to the
atomistic model simulations and showed that materials with
smaller correlation time demagnetized faster[17].
Despite the experimental and theoretical progresses to dat e
little attention has been paid to how to determine the correl a-
tion time experimentally. For development of the science an d
technologies based on the ultrafast magnetization dynamic s it
is important to determine the correlation time experimenta lly
as well as to understand how the magnetization dynamics de-
pend on the correlation time.
In this paper the precession dynamics of a small magnet with
non-Markovian damping is theoretically studied based on th e
macrospin model. The magnet is assumed to have a uniaxial
anisotropy and to be subjected to an external magnetic field
parallel to the magnetization easy axis. The non-Markovian ity
Preprint submitted to Journal of Magnetism and Magnetic Mat erials May 16, 2022enhances the precession angular velocity and reduces the da mp-
ing. Assuming that the Gilbert damping constant is much
smaller than unity, the analytical expressions of the prece ssion
angular velocity and the e ffective damping constant are derived
for any values of the correlation time by extending the analy sis
of Miyazaki and Seki[16]. We also propose a possible exper-
iment for determination of the correlation time. The correl a-
tion time can be determined by analyzing the external field at
which the enhancement of the precession angular velocity is
maximized.
The paper is organized as follows. Section 2 explains the
theoretical model and the equations of motion. Section 3 giv es
the numerical and theoretical analysis of the precession dy nam-
ics in the absence of an anisotropy field. The e ffect of the
anisotropy field is discussed in Sec. 4. A possible experimen t
for determination of the correlation time is proposed in Sec . 5.
The results are summarized in Sec. 6.
2. Theoretical model
We calculate the magnetization dynamics in a small mag-
net with a uniaxial anisotropy under an external magnetic fie ld
based on the macrospin model. The magnetization easy axis
is taken to be z-axis and the magnetic field is applied in the
positive z-direction. In terms of the magnetization unit vector,
m=(mx,my,mz), the energy density is given by
E=K(1−m2
z)−µ0MsH m z, (1)
where Kis the effective anisotropy constant including the crys-
talline, interfacial, and shape anisotropies. µ0is the vacuum
permeability, Msis the saturation magnetization, His the exter-
nal magnetic field. The e ffective field is obtained as
Heff=(Hkmz+H)ez, (2)
where ezis the unit vector in the positive zdirection and Hk=
2K/(µ0Ms) is the effective anisotropy field.
The magnetization precesses around the e ffective field with
damping. The energy and angular momentum are absorbed by
the bath degrees of freedom such as conduction electrons and
phonons until the magnetization becomes parallel to the e ffec-
tive field. The equations of motion of mcoupled with the bath
is given by the Langevin equation with the stochastic field re p-
resenting the bath degrees of freedom. If the time scale of th e
bath is much smaller than the precession frequency the stoch as-
tic field can be treated as the Wiener process[18] as shown by
Brown[3].
Since we are interested in the ultrafast magnetization dyna m-
ics of which time scale is the same order as the correlation ti me
of the bath degrees of freedom, the stochastic field should be
treated as the Ornstein-Uhlenbeck process[18, 19]. As show n
by Miyazaki and Seki [16] the equations of motion of mtakes
the following integro-di fferential form:
˙m=−γm×(Heff+r)+αm×/integraldisplayt
−∞ν(t−t′) ˙m(t′)dt′,(3)whereγis the gyromagnetic ratio, αis the Gilbert damping con-
stant, and ris the stochastic field. The first term represents the
precession around the sum of the e ffective field and the stochas-
tic field, and the second term represents the non-Markovian
damping. The memory function in the non-Markovian damping
term is defined as
ν(t−t′)=1
τcexp/parenleftigg
−|t−t′|
τc/parenrightigg
, (4)
whereτcis the correlation time of the bath degrees of freedom.
The stochastic field, r, satisfies/angbracketleftri(t)/angbracketright=0 and
/angbracketleftrj(t)rk(t′)/angbracketright=µ
2δj,kν(t−t′), (5)
where/angbracketleft/angbracketrightrepresents the statistical mean, and
µ=2αkBT
γMsV. (6)
The subscripts jandkstand for x,y, orz,kBis the Boltzmann
constant, Tis the temperature, Vis the volume of the mag-
net, andδj,kis Kronecker’s delta. The LLG equation with the
Markovian damping derived by Brown [3] is reproduced in the
limit ofτc→0 because lim τc→0ν(t−t′)=2δ(t−t′), where
δ(t−t′) is Dirac’s delta function. Equation (3) is equivalent to
the following set of the first order di fferential equations,
˙m=−γm×[Heff+δH] (7)
˙δH=−1
τcδH−α
τ2cm−γ
τcR, (8)
where Rrepresents the stochastic field due to thermal agita-
tion. Equations (7), (8) are used for numerical simulations . The
stochastic field, R, satisfies/angbracketleftRj(t)/angbracketright=0 and
/angbracketleftRj(t)Rk(t′)/angbracketright=µδj,kδ(t−t′). (9)
3. Precession dynamics in the absence of an anisotropy field
In this section the precession dynamics in the absence of an
anisotropy field, i.e. Hk=0, is considered. The initial di-
rection of magnetization is assumed to be m=(1,0,0). The
numerical simulation shows that the non-Markovian damping
enhances the precession angular velocity and reduces the da mp-
ing. The numerical results are theoretically analyzed assu ming
thatα≪1. The analytical expressions of the precession an-
gular velocity and the e ffective damping constant are obtained.
The case with Hk/nequal0 will be discussed in Sec. 4.
3.1. numerical simulation results
We numerically solve Eqs. (7), (8) for H=10 T,α=0.05,
andτc=1 ps. The temperature is assumed to be low enough
to set R=0 in Eq. (8). Figure 1(a) shows the trajectory of m
on a unit sphere. The initial direction is indicated by the fil led
circle. The plot of the temporal evolutions of mx,my, and mzare
shown in Fig. 1(b). The magnetization relaxes to the positiv ez
direction with precessing around the external field. The res ults
2t [ps] 0.00 0.01 0.03
0.02 0.04
100 200 0
t [ps]
a) b)
c) d) z
x yHφ [rad / ps]
1.76 1.77 1.79
1.78 1.80 100 200 0
100 200 0-1 1
0mx, m y, m z
t [ps]
φ00.05
αmzmymx
yαeff
Figure 1: (a) Trajectory of mon a unit sphere. The external field of H=
10 T is applied in the positive zdirection. The initial direction is assumed to
bem=(1,0,0) as indicated by the filled circle. The other parameters are
τc=1 ps, andα=0.05. (b) Temporal evolution of mx,my,mz. (c) Temporal
evolution of the precession angular velocity, ˙φ. The solid red curve shows the
simulation result. The dotted black line indicates the resu lt of the Markovian
LLG equation, i.e. ˙φ0=γH/(1+α2). (d) Temporal evolution of the e ffective
damping constant, αeff. The solid red curve shows the simulation result. The
dotted black line indicates α=0.05.
are quite similar to that of the Markovian LLG equation, whic h
implies that the non-Markovianity in damping causes renorm al-
ization of the gyromagnetic ratio and the Gilbert damping co n-
stant in the Markovian LLG equation.
The renormalized value of the gyromagnetic ratio can be
observed as a variation of the precession angular velocity, ˙φ,
where the polar and azimuthal angles are defined as m=
(sinθcosφ,sinθsinφ,cosθ). Figure 1(c) shows that temporal
evolution of ˙φ(solid red) together with the precession angular
velocity without non-Makovianity, ˙φ0=γH/(1+α2), (dotted
black). The precession angular velocity increases with inc rease
of time and saturates to a certain value around 1.798. The sha pe
of the time dependence of ˙φis quite similar to that of mzshown
in Fig. 1(b), which suggests that the non-Markovian damping
acts as an effective anisotropy field in the precession dynamics.
The renormalization of the Gilbert damping constant can be
observed as a variation of the temporal evolution of the pola r
angle, ˙θ. Rearranging the LLG equation for ˙θ, the effective
damping constant can be defined as
αeff=−˙θ/(γHsinθ). (10)
In Fig. 1(d)αeffis shown by the red solid curve as a function of
time. The effective damping constant is reduced to about one-
fifth of the original value of α=0.05 (dotted black). Contrary
to˙φ,αeffdoes not show clear correlation with the dynamics of
m. During the precession, αeffis kept almost constant.
The enhancement of the precession angular velocity and the
reduction of the Gilbert damping constant due to the non-
Markovian damping will be explained by deriving the e ffectiveLLG equation that is valid up to the first order of αin the next
subsection.
3.2. Theoretical analysis
Since the Gilbert damping constant, α, of a conventional
magnet is of the order of 0 .01∼0.1, it is natural to take the
first order ofαto derive the effective equations of motion for
m. The other parameters related to the motion of mareγ,H,
andτc. Multiplying these parameters we can obtain the dimen-
sionless parameter, ξ=γHτc, which represents the increment
of the precession angle during the correlation time.
In the case ofξ<1 Miyazaki and Seki dereived the e ffective
LLG equation using time derivative series expansion[16]. W e
first briefly review their analysis. Then we derive the e ffective
LLG equation forξ>1 using the time-integral series expansion
and show that the e ffective LLG equation has the same form for
bothξ< 1 andξ> 1. Therefore, it is natural to assume that
the derived effective LLG equation is valid for any values of ξ
includingξ=1.
3.2.1. Brief review of Miyazaki and Seki’s derivation of the ef-
fective LLG equation for ξ<1
In Ref. 16, Miyazaki and Seki derived the e ffective LLG
equation with renormalized parameters using the time deriv a-
tive series expansion. Similar analysis of the LLG equation
was also done by Shul in the study of the damping due to
strain[20, 21]. The following is the brief summary of the der iva-
tion.
Successive application of the integration by parts using ν(t−
t′)=τc[dν(t−t′)/dt′] gives the following time derivative se-
ries expansion:
/integraldisplayt
−∞ν(t−t′) ˙m(t′)dt′=∞/summationdisplay
n=1(−τc)n−1dnm
dtn. (11)
Then the non-Markovian damping term in Eq. (3) is expressed
as
α∞/summationdisplay
n=1(−τc)n−1/parenleftigg
m×dnm
dtn/parenrightigg
. (12)
The first derivative, n=1, is given by
˙m=−γHm×ez+O(α), (13)
where Ois the Bachmann–Landau symbol. For n=2, substi-
tution of Eq. (13) into the time derivative of Eq. (13) gives
¨m=(−γH)2(m×ez)×ez+O(α). (14)
The n-th order time derivative is obtained by using the same
algebra as
dn
dtnm=(−γH)n/bracketleftbig(m×ez)×ez.../bracketrightbig+O(α), (15)
where ezappears ntimes. Expanding the vector products we
obtain for even order time derivatives
d2nm
dt2n=(−1)n(γH)2n/bracketleftbigm−mzez/bracketrightbig+O(α), (16)
3and for odd order time derivatives
d2n+1m
dt2n+1=(−1)n(γH)2n˙m+O(α). (17)
Substituting Eqs. (16) and (17) into Eq. (12) the non-
Markovian damping term is expressed as
−∞/summationdisplay
n=1γ2nm×ez+∞/summationdisplay
n=0α2n+1m×˙m, (18)
where
γ2n=αγH m z(−1)n−1ξ2n−1(19)
α2n+1=α(−1)nξ2n. (20)
The sums in Eq. (18) converge for ξ<1. Introducing
˜γ=γ/parenleftigg
1+αmzξ
1+ξ2/parenrightigg
(21)
˜α=α
1+ξ2, (22)
Eq. (3) can be expressed as the following e ffective LLG equa-
tion with renormalized gyromagnetic ratio, ˜ γ, and damping
constant, ˜α:
˙m=−˜γm×(H+r)+˜αm×˙m+O(α2). (23)
3.2.2. Derivation of the e ffective LLG equation for ξ>1
Forξ>1 we expand Eq. (3) in power series of 1 /ξusing the
time integral series expansion approach. Using the integra tion
by parts with dν(t−t′)/dt′=ν(t−t′)/τcthe integral part of the
non-Markovian damping can be written as
/integraldisplayt
−∞ν(t−t′) ˙m(t′)dt′=1
τc/integraldisplayt
−∞˙m(t′)dt′
−1
τc/integraldisplayt
−∞ν(t−t′)/bracketleftigg/integraldisplayt′
−∞˙m(t′′)dt′′/bracketrightigg
dt′. (24)
Successive application of the integration by parts gives
/integraldisplayt
−∞ν(t−t′) ˙m(t′)dt′=−∞/summationdisplay
n=1/parenleftigg
−1
τc/parenrightiggn
Jn, (25)
where Jnis the nth order multiple integral defined as
Jn=/integraldisplayt
−∞/integraldisplayt1
−∞···/integraldisplaytn−1
−∞˙m(tn)dtn···dt2dt1. (26)
From Eq. (17), on the other hand, ˙ mis expressed as
˙m=1
(−1)n(γH)2nd2n
dt2n˙m+O(α). (27)
Substituting Eq. (27) into Eq. (26) the multiple integrals a re
calculated as
J2n=1
(−1)n(γH)2n˙m (28)
J2n−1=1
(−1)n(γH)2n¨m. (29)Then Eq. (25) becomes
/integraldisplayt
−∞ν(t−t′) ˙m(t′)dt′=∞/summationdisplay
n=11
(−1)n−1ξ2n˙m
+∞/summationdisplay
n=1τc
(−1)nξ2n¨m. (30)
Substituting Eq. (30) into the second term of Eq. (3) the non-
Markovian damping term is expressed as
α∞/summationdisplay
n=11
(−1)n−1ξ2nm×[ ˙m+τc¨m]+O(α2). (31)
From Eq. (16) ¨ mis expressed as
¨m=(−1)(γH)2/bracketleftbigm−mzez/bracketrightbig. (32)
Substituting Eqs. (31) and (32) into Eq. (3) we obtain
˙m=−γH∞/summationdisplay
n=1/bracketleftigg
1+αmz
(−1)n−1ξ2n−1/bracketrightigg
m×ez−γm×r
+α∞/summationdisplay
n=11
(−1)n−1ξ2nm×˙m+O(α2). (33)
The sums converge for ξ>1, and the effective LLG equation
forξ>1 has the same form as ξ<1, i.e. Eq. (23). Since the
effective LLG equation has the same form for both ξ< 1 and
ξ>1, it is natural to Eq. (23) is valid for any values of ξ.
As pointed out by Miyazaki and Seki, and independently by
Suhl the effect of the non-Markovian damping on the precession
can be regarded as the renormalization of the e ffective field [16,
20, 21]. Equation (23) can be expressed as
˙m=−γm×/parenleftigg
H+αHξ
1+ξ2mz/parenrightigg
ez−γm×r
+˜αm×˙m+O(α2). (34)
The second term in the bracket represents the fictitious unia xial
anisotropy field originated from the non-Markovian damping .
The fictitious anisotropy field increases with increase of ξfor
ξ<1 and takes the maximum value of αH m z/2 atξ=1, i.e.
γHτc=1. Forξ>1 the fictitious anisotropy field decreases
with increase ofξand vanishes in the limit of ξ→∞ because
the non-Markovian damping term vanishes in the limit of τc→
∞. The precession angular velocity, ˙φ, is expected to have the
sameξdependence as the fictitious anisotropy field and to have
the same temporal evolution as mzas shown in Figs. 1(b) and
1(c).
3.2.3. The Correlation time dependence of the precession an -
gular velocity, and e ffective damping constant
Equation (21) tells us that up to the first order of αthe pre-
cession angular velocity can be approximated as
˙φ≃˜γH=γH/bracketleftigg
1+αmzγHτc
1+(γHτc)2/bracketrightigg
, (35)
4τc’ =1/( γH)
0.1 1 10 0.01
τc [ps] τc [ps] a) b)
φ [rad / ps]
1.76 1.77 1.79
1.78 1.80
0.1 1 10 0.01
α, αeff ~
αeffα0.04
0.00 0.02 0.05
0.01 0.03
ααα
effeffeffαeffαeff~sim. approx.
Figure 2: (a) The correlation time, τc, dependence of the precession angular
velocity,δ˙φ, atθ=5◦forH=10 T. The solid yellow curve shows the ap-
proximation result, ˜ γH. The dotted black curve shows the simulation results
obtained by numerically solving Eqs. (7) and (8). The thin ve rtical dotted line
indicates the critical value of the correlation time, τ′
c=1/(γH). (b)τcdepen-
dence of ˜α(solid yellow) andαeff(dotted black). The parameters and the other
symbols are the same as panel (a).
where the second term in the square bracket represents the en -
hancement due to the fictitious anisotropy field.
In Fig. 2(a) the approximation result of Eq. (35) at θ=5◦
where ˙φis almost saturated is plotted as a function of τcby the
solid yellow curve. The external field and the Gilbert damp-
ing constant are assumed to be H=10 T andα=0.05, re-
spectively. The corresponding simulation results obtaine d by
numerically solving Eqs. (7) and (8) are shown by the dotted
black curve. Both curves agree well with each other because
αis as small as 0.05. The precession angular velocity is maxi-
mized at the critical value of the correlation time τ′
c=1/(γH).
Figure 2(b) shows the τcdependence of ˜α(solid yellow) and
αeff(dotted black) for the same parameters as panel (a). Both
curves agree well with each other and are monotonic decreasi ng
functions ofτc. They vanish in the limit of τc→∞ similar to
the non-Markovian damping term.
4. Effect of an anisotropy field on precession dynamics
The theoretical analysis given in the previous section can
be applied to the case with Hk/nequal0 by replacingξwithξk=
γ(H+Hkmz)τc. Following the same procedure as for Hk=0
Eq. (3) can be expressed as
˙m=−γm×/parenleftig
H+αHξk
1+ξ2
kmz+αHkξk
1+ξ2
km2
z/parenrightig
ez
−γm×r+α
1+ξ2
km×˙m+O(α2). (36)
The second and the third terms in the bracket can be regarded
as the fictitious uniaxial and unidirectional anisotropy fie lds
caused by the non-Markovian damping. Similar to the re-
sults for Hk=0 the precession angular velocity is maximized
atξk=1. The renormalized damping constant is given by
α/(1+ξ2
k) which is a monotonic decreasing function of ξkand
vanishes in the limit of ξk→∞ .c) d) a) b)
24 6 14 12 10 8 0
H [T] δφ /φ0 [%] 2
013
δφ /φ0 [%] 2
01
24 6 14 12 10 8 0
H [T] Hk = 0 H’=1/( γτ c)
τc = 1 ps
θ = 5 oθ = 5 oHk = 1 T
τc = 1 ps H’=1/( γτ c) − HkmzH = 6, 7, 8, 9 T
H = 2, 3, 4, 5 T
= 1 ps
= 5 = 0
= 1 ps τ
θH
τ = 1 ps = 1 ps = 1 ps τc = 1 ps τ = 1 ps τ = 1 ps
θτH
τ
= 5 = 1 ps = 1 T
= 1 ps = 1 ps = 1 T
= 1 ps = 1 ps τc = 1 ps ττ = 1 ps = 1 ps 3δφ /φ0 [%] 2
013
δφ /φ0 [%] 2
013
t [ps] 100 200 0
t [ps] 100 200 0Hk = 0
τc = 1 ps Hk = 0
τc = 1 ps
Figure 3: (a)τcdependence ofδ˙φ/˙φ0atθ=5◦. From top to bottom the
external field is H=2,3,4,5 T. The parameters are Hk=0, andτc=1 ps. (b)
The same plot as panel (a) for H≥5 T. From top to bottom the external field is
H=6,7,8,9 T. (c) Hdependence ofδ˙φ/˙φ0atθ=5◦obtained by solving Eqs.
(7) and (8). The parameters are Hk=0, andτc=1 ps. The critical value of
the external field, H′=1/(γτc), is indicated by the thin vertical dotted line. (d)
The same plot as panel (c) for Hk=1 T. The thin vertical dotted line indicates
the critical value of the external field, H′=1/(γτc)−Hkmz.
5. A possible experiment to determine the correlation time
Based on the results shown in Secs. 3 and 4 we propose a
possible experiment to determine the correlation time, τc. Sim-
ilar to the previous sections we first discuss the case withou t
anisotropy field, i.e. Hk=0, and then extend the discussion to
the case with Hk/nequal0.
In Figs. 3(a) and 3(b) we show the temporal evolution of the
enhancement of angular velocity, δ˙φ/˙φ0, obtained by the solv-
ing Eqs. (7) and (8) for various values of H. The increment
of the precession angular velocity is defined as δ˙φ=˙φ−˙φ0.
The initial state and the correlation time are assumed to be
m=(1,0,0) andτc=1 ps, respectively. As shown in Fig.
3(a),δ˙φ/˙φ0increases with increase of HforH≤5T. Once
the external field exceeds the critical value of 1 /(γτc)=5.7 T,
δ˙φ/˙φ0decreases with increase of Has shown in Fig. 3 (b). The
results suggest that correlation time can be determined by a na-
lyzing the external field that maximizes the enhancement of t he
precession angular velocity.
Figure 3(c) shows the Hdependence ofδ˙φ/˙φ0atθ=5◦
whereδ˙φ/˙φ0is almost saturated. The enhancement is maxi-
mized at the critical value of the external field, H′=5.7 T. The
correlation time is calculated as τc=1/(γH′)=1 ps.
If the system has a uniaxial anisotropy field, Hk, the en-
hancement of the precession angular velocity is maximized a t
H′=1/(γτc)−Hkmzas shown in Fig. 3(d). The correlation
time is obtained as τc=1/γ(H′+Hkmz).
The above analysis is expected to be performed experimen-
5tally using the time resolved magneto optical Kerr e ffect mea-
surement technique. In the practical experiments the analy sis
can be simplified as follows. The polar angle of the initial st ate
is not necessarily large. It can be small as far as the preces-
sion angular velocity can be measured. Instead of analyzing
δ˙φ/˙φ0, one can analyze ˙φ/Hor˙φ/(H+Hkmz) because they are
maximized at the same value of Hasδ˙φ/˙φ0. Since the required
magnetic field is as high as 10 T, a superconducting magnet [22 ]
is required.
6. Summary
In summary we theoretically analyze the ultrafast precessi on
dynamics of a small magnet with non-Markovian damping. As-
sumingα≪1, we derive the effective LLG equation valid for
any values ofτc, which is a direct extension of Miyazaki and
Seki’s work[16]. The derived e ffective LLG equation reveals
the condition for maximizing ˙φin terms of Handτc. Based on
the results we propose a possible experiment for determinat ion
ofτc, whereτccan be determined from the external field that
maximizesδ˙φ/˙φ0.
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6 |
1211.2900v1.Critical_exponent_for_the_semilinear_wave_equation_with_scale_invariant_damping.pdf | arXiv:1211.2900v1 [math.AP] 13 Nov 2012CRITICAL EXPONENT FOR THE SEMILINEAR WAVE
EQUATION WITH SCALE INVARIANT DAMPING
YUTA WAKASUGI
Abstract. In this paper we consider the critical exponent problem for t he
semilinear damped wave equation with time-dependent coeffic ients. We treat
the scale invariant cases. In this case the asymptotic behav ior of the solution is
very delicate and the size of coefficient plays an essential ro le. We shall prove
that if the power of the nonlinearity is greater than the Fuji ta exponent, then
there exists a unique global solution with small data, provi ded that the size of
the coefficient is sufficiently large. We shall also prove some b low-up results
even in the case that the coefficient is sufficiently small.
1.Introduction
We consider the Cauchy problem for the semilinear damped wave equa tion
(1.1)/braceleftBigg
utt−∆u+µ
1+tut=|u|p,(t,x)∈(0,∞)×Rn,
(u,ut)(0,x) = (u0,u1)(x), x∈Rn,
whereµ >0, (u0,u1)∈H1(Rn)×L2(Rn) have compact support and 1 < p≤
n
n−2(n≥3),1< p <∞(n= 1,2). Our aim is to determine the critical exponent
pc, which is the number defined by the following property:
Ifpc<p, all small data solutions of (1.1) are global; if 1 <p≤pc, the time-local
solution cannot be extended time-globally for some data regardless of smallness.
We note that the linear part of (1.1) is invariant with respect to the h yperbolic
scaling
˜u(t,x) =u(λ(1+t)−1,λx).
In this case the asymptotic behavior of solutions is very delicate. It is known that
the size of the damping term µplays an essential role. The damping term µ/(1+t)
is known as the borderline between the effective andnon-effective dissipation, here
effective means that the solution behaves like that of the corresponding par abolic
equation, and non-effective means that the solution behaves like that of the free
wave equation. Concretely, for linear damped wave equation
(1.2) utt−∆u+(1+t)−βut= 0,
if−1<β <1, then the solution uhas the same Lp-Lqdecay rates as those of the
solution of the corresponding heat equation
(1.3) −∆v+(1+t)−βvt= 0.
Moreover, if −1/3<β <1 then the asymptotic profile of uis given by a solution of
(1.3) (see [14]). This is called the diffusion phenomenon . In particular, the constant
Key words and phrases. damped wave equation; time dependent coefficient; scale inva riant
damping; critical exponent.
12 YUTA WAKASUGI
coefficient case β= 0 has been investigated for a long time. We refer the reader to
[8, 9]. On the other hand, if β >1 then the asymptotic profile of the solution of
(1.2) is given by that of the free wave equation /squarew= 0 (see [13]).
Wirth [12] considered the linear problem
(1.4)/braceleftBigg
utt−∆u+µ
1+tut= 0,
(u,ut)(0,x) = (u0,u1)(x).
He proved several Lp-Lqestimates for the solutions to (1.4). For example, if µ>1
it follows that
/ba∇dblu(t)/ba∇dblLq/lessorsimilar(1+t)max{−n−1
2(1
p−1
q)−µ
2,−n(1
p−1
q)}(/ba∇dblu0/ba∇dblHsp+/ba∇dblu1/ba∇dblHs−1
p),
/ba∇dbl(ut,∇u)(t)/ba∇dblLq/lessorsimilar(1+t)max{−n−1
2(1
p−1
q)−µ
2,−n(1
p−1
q)−1}(/ba∇dblu0/ba∇dblHs+1
p+/ba∇dblu1/ba∇dblHs
p),
where 1< p≤2, 1/p+ 1/q= 1 ands=n(1/p−1/q). This shows that if µ
is sufficiently large then the solution behaves like that of the corresp onding heat
equation
(1.5)µ
1+tvt−∆v= 0
ast→ ∞, and ifµis sufficiently small then the solution behaves like that of the
free wave equation. We mention that for the wave equation with spa ce-dependent
damping /squareu+V0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−1ut= 0, a similar asymptotic behavior is obtained by Ikehata,
Todorova and Yordanov [6].
The Gauss kernel of (1.5) is given by
Gµ(t,x) =/parenleftbiggµ
2π((1+t)2−1)/parenrightbiggn
2
e−µ|x|2
2((1+t)2−1).
We can obtain the Lp-Lqestimates of the solution of (1.5). In fact, it follows that
/ba∇dblv(t)/ba∇dblLq/lessorsimilar(1+t)−n(1
p−1
q)/ba∇dblv(0)/ba∇dblLp
for 1≤p≤q≤ ∞. In particular, taking q= 2 andp= 1, we have
/ba∇dblv(t)/ba∇dblL2/lessorsimilar(1+t)−n
2/ba∇dblv(0)/ba∇dblL1.
From the point of view of the diffusion phenomenon, we expect that t he same type
estimate holds for the solution of (1.4) when µis large. To state our results, we
introduce an auxiliary function
ψ(t,x) :=a|x|2
(1+t)2, a=µ
2(2+δ)
with a positive parameter δ. We have the following linear estimate:
Theorem 1.1. For anyε >0, there exist constants δ >0andµ0>1such that
for anyµ≥µ0the solution of (1.4)satisfies
/integraldisplay
Rne2ψu2dx≤Cµ,ε(1+t)−n+ε(1.6)
/integraldisplay
Rne2ψ(u2
t+|∇u|2)dx≤Cµ,ε(1+t)−n−2+ε(1.7)
fort≥0, whereCµ,εis a positive constant depending on µ,εand/ba∇dbl(u0,u1)/ba∇dblH1×L2.SEMILINEAR DAMPED WAVE EQUATION 3
Remark 1.1. The constant µ0depends on ε. The relation is
µ0∼ε−2.
Therefore, as εsmaller,µ0must be larger. We can expect that εcan be removed
and the same result holds for much smaller µ. However, we have no idea for the
proof.
We also consider the critical exponent problem for (1.1). For the co rresponding
heat equation (1.5) with nonlinear term |u|p, the critical exponent is given by
pF:= 1+2
n,
which is well known as the Fujita critical exponent (see [4]). Thus, w e can expect
that the critical exponent of (1.1) is also given by pFifµis sufficiently large.
For the damped wave equation with constant coefficient
utt−∆u+ut=|u|p,
Todorovaand Yordanov[11] provedthe critical exponent is given b ypF. Lin, Nishi-
hara and Zhai [7] (see also Nishihara[10]) extended this result to time -dependent
coefficient cases
utt−∆u+(1+t)−βut=|u|p
with−1<β <1. Theyprovedthat pFisstillcritical. Recently, D’abbicco, Lucente
and Reissig [3] extended this result to more general effective b(t) by using the linear
decay estimates, which are established by Wirth [14]. For the scale-in variant case
(1.1), very recently, D’abbicco [1] proved the existence of the glob al solution with
small data for (1.1) in the case n= 1,2,µ≥n+2 andpF< p. He also obtained
the decay rates of the solution without any loss ε.
Our main result is following:
Theorem 1.2. Letp > pFand0< ε <2n(p−pF)/(p−1). Then there exist
constantsδ>0andµ0>1having the following property: if µ≥µ0and
I2
0:=/integraldisplay
Rne2ψ(0,x)(u2
0+|∇u0|2+u2
1)dx
is sufficiently small, then there exists a unique solution u∈C([0,∞);H1(Rn))∩
C1([0,∞);L2(Rn))of(1.1)satisfying
/integraldisplay
Rne2ψu2dx≤Cµ,εI2
0(1+t)−n+ε(1.8)
/integraldisplay
Rne2ψ(u2
t+|∇u|2)dx≤Cµ,εI2
0(1+t)−n−2+ε(1.9)
fort≥0, whereCµ,εis a positive constant depending on µandε.
Remark 1.2. Similarly as before, we note that µ0depends on ε. The relation is
µ0∼ε−2∼(p−pF)−2.
Therefore, as pis closer to pF,µ0must be larger. Thus, we can expect that εcan
be removed and the same result holds for much smaller µ. As mentioned above,
D’abbicco [1]obtained an affirmative result for this expectation when n= 1,2.
However, we have no idea for the higher dimensional cases.4 YUTA WAKASUGI
We prove Theorem 1.2 by a multiplier method which is essentially develope d
in [11]. Lin, Nishihara and Zhai [7] refined this method to fit the damping term
b(t) = (1+t)−βwith−1<β <1. They used the property β <1 and so we cannot
apply their method directly to our problem (1.1). Therefore, we nee d a further
modification. Instead of the property β <1, we assume that µis sufficiently large
and modify the parameters used in the calculation.
Remark1.3. We can also treat other nonlinear terms, for example −|u|p,±|u|p−1u.
We also have a blow-up result when µ>1 and 1<p≤pF.
Theorem 1.3. Let1<p≤pFandµ>1. Moreover, we assume that/integraldisplay
Rn(µ−1)u0+u1dx>0.
Then there is no global solution for (1.1).
Remark 1.4. Theorem 1.3 is essentially included in a recent work by D’abb icco
and Lucente [2]. In this paper we shall give a much simpler proof.
One of our novelty is blow-up results for the non-effective damping c ases. We
also obtain blow-up results in the case 0 <µ≤1.
Theorem 1.4. Let0<µ≤1and
1<p≤1+2
n+(µ−1).
We also assume /integraldisplay
Rnu1(x)dx>0.
Then there is no global solution for (1.1).
Remark 1.5. In Theorem 1.4, we do not put any assumption on the data u0, and
the blow-up results hold even for the case p≥pF. We can interpret this phenomena
as that the equation (1.1)loses the parabolic structure and recover the hyperbolic
structure if µis sufficiently small.
We prove this theorem by a test-function method developed by Qi S . Zhang [15].
In the same way of the proof of Theorem 1.4, we can treat the damp ing terms
(1+t)−βwithβ >1 (see Remark 3.1).
In the next section, we give a proof of Theorem 1.2. We can prove Th eorem 1.1
by the almost same way, and so we omit the proof. In Section 3, we sh all prove
Theorem 1.3 and Theorem 1.4.
2.Proof of Theorem 1.2
In this section we prove our main result. First, we prepare some not ation and
terminology. We put
/ba∇dblf/ba∇dblLp(Rn):=/parenleftbigg/integraldisplay
Rn|f(x)|pdx/parenrightbigg1/p
.
ByH1(Rn) we denote the usual Sobolev space. For an interval Iand a Banach
spaceX, we define Cr(I;X) as the Banach space whose element is an r-times
continuously differentiable mapping from ItoXwith respect to the topology in
X. The letter Cindicates the generic constant, which may change from a line toSEMILINEAR DAMPED WAVE EQUATION 5
the next line. We also use the symbols /lessorsimilarand∼. The relation f/lessorsimilargmeansf≤Cg
with some constant C >0 andf∼gmeansf/lessorsimilargandg/lessorsimilarf.
We first describe the local existence:
Proposition 2.1. For anyp >1,µ >0andδ >0, there exists Tm∈(0,+∞]
depending on I2
0such that the Cauchy problem (1.1)has a unique solution u∈
C([0,Tm);H1(Rn))∩C1([0,Tm);L2(Rn)), and ifTm<+∞then we have
liminf
t→Tm/integraldisplay
Rneψ(t,x)(u2
t+|∇u|2+u2)dx= +∞.
We can provethis propositionby standard arguments(see [5]). We p rovea priori
estimate for the following functional:
M(t) := sup
0≤τ≤t/braceleftbigg
(1+τ)n+2−ε/integraldisplay
Rne2ψ(u2
t+|∇u|2)dx+(1+τ)n−ε/integraldisplay
Rne2ψu2dx/bracerightbigg
.
We putb(t) =µ
1+tandf(u) =|u|p. By a simple calculation, we have
−ψt=2
1+tψ,∇ψ=2ax
(1+t)2,|∇ψ|2
−ψt=b(t)
2+δ
and
∆ψ=n
2+δb(t)
1+t=:/parenleftBign
2−δ1/parenrightBigb(t)
1+t.
Here and after, δi(i= 1,2,...) denote a positive constant depending only on δsuch
that
δi→0+asδ→0+.
Multiplying (1.1) by e2ψut, we obtain
∂
∂t/bracketleftbigge2ψ
2(u2
t+|∇u|2)/bracketrightbigg
−∇·(e2ψut∇u) (2.1)
+e2ψ/parenleftbigg
b(t)−|∇ψ|2
−ψt−ψt/parenrightbigg
u2
t+e2ψ
−ψt|ψt∇u−ut∇ψ|2
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
T1
=∂
∂t/bracketleftbig
e2ψF(u)/bracketrightbig
+2e2ψ(−ψt)F(u),
whereFis the primitive of fsatisfyingF(0) = 0. Using the Schwarz inequality,
we can calculate
T1≥e2ψ/parenleftbigg1
5(−ψt)|∇u|2−b(t)
4(2+δ)u2
t/parenrightbigg
.
From this and integrating (2.1), we have
d
dt/integraldisplay
Rne2ψ
2(u2
t+|∇u|2)dx+/integraldisplay
Rne2ψ/braceleftbigg/parenleftbiggb(t)
4−ψt/parenrightbigg
u2
t+−ψt
5|∇u|2/bracerightbigg
dx (2.2)
≤d
dt/integraldisplay
Rne2ψF(u)dx+2e2ψ(−ψt)F(u)dx.6 YUTA WAKASUGI
On the other hand, by multiplying (1.1) by e2ψu, it follows that
∂
∂t/bracketleftbigg
e2ψ/parenleftbigg
uut+b(t)
2u2/parenrightbigg/bracketrightbigg
−∇·(e2ψu∇u)
+e2ψ/braceleftBig
|∇u|2+/parenleftbigg
−ψt+1
2(1+t)/parenrightbigg
b(t)u2+2u∇ψ·∇u/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
T2−2ψtuut−u2
t/bracerightBig
=e2ψuf(u).
We calculate
e2ψT2= 4e2ψu∇ψ·∇u−2e2ψu∇ψ·∇u
= 4e2ψu∇ψ·∇u−∇·(e2ψu2∇ψ)+2e2ψu2|∇ψ|2+e2ψ(∆ψ)u2
and have
∂
∂t/bracketleftbigg
e2ψ/parenleftbigg
uut+b(t)
2u2/parenrightbigg/bracketrightbigg
−∇·(e2ψ(u∇u+u2∇ψ)) (2.3)
+e2ψ/braceleftBig
|∇u|2+4u∇u·∇ψ+((−ψt)b(t)+2|∇ψ|2)u2
/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright
T3
+(n+1−2δ1)b(t)
2(1+t)u2−2ψtuut−u2
t/bracerightBig
=e2ψuf(u).
T3is estimated as
T3=/parenleftbigg
1−4
4+δ/2/parenrightbigg
|∇u|2+δ
2|∇ψ|2u2+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/radicalbig
4+δ/2∇u+/radicalbig
4+δ/2∇ψ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
≥δ2(|∇u|2+b(t)(−ψt)u2).
Thus, we can rewrite (2.3) as
∂
∂t/bracketleftbigg
e2ψ/parenleftbigg
uut+b(t)
2u2/parenrightbigg/bracketrightbigg
−∇·(e2ψ(u∇u+u2∇ψ))
+e2ψ/braceleftbigg
δ2(|∇u|2+b(t)(−ψt)u2)+(n+1−2δ2)b(t)
2(1+t)u2−2ψtuut−u2
t/bracerightbigg
≤e2ψuf(u).
Integrating the above inequality and then multiplying by a large param eterνand
adding (1+ t)×(2.2), we obtain
d
dt/bracketleftbigg/integraldisplay
e2ψ/braceleftbigg1+t
2(u2
t+|∇u|2)+νuut+νb(t)
2u2/bracerightbigg
dx/bracketrightbigg
+/integraldisplay
e2ψ/braceleftBig/parenleftBigµ
4−ν−1
2/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
T4+(−ψt)(1+t)/parenrightBig
u2
t+/parenleftBig
νδ2−1
2/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
T5+(−ψt)(1+t)
5/parenrightBig
|∇u|2
+νδ2b(t)(−ψt)u2+(n+1−2δ1)νb(t)
2(1+t)u2+2ν(−ψt)uut/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
T6/bracerightBig
dx
≤d
dt/bracketleftbigg
(1+t)/integraldisplay
e2ψF(u)dx/bracketrightbigg
+C/integraldisplay
e2ψ(1+(1+t)(−ψt))|u|p+1dx.SEMILINEAR DAMPED WAVE EQUATION 7
We put the condition for µandνas
µ
4−ν−1
2>0 (2.4)
νδ2−1
2>0. (2.5)
Then the terms T4andT5are positive. Using the Schwarz inequality, we can
estimateT6as
|T6| ≤1
2(−ψt)(1+t)u2
t+2ν2
1+t(−ψt)u2.
Now we put an another condition
(2.6) µ≥2ν
δ2.
Then we obtain the following estimate.
d
dtˆE(t)+H(t)+(n+1−2δ1)νb(t)
2(1+t)J(t;u2) (2.7)
≤d
dt[(1+t)J(t;F(u))]+C(J(t;|u|p+1)+(1+t)Jψ(t;|u|p+1)),
where
ˆE(t) :=/integraldisplay
e2ψ/braceleftbigg1+t
2(u2
t+|∇u|2)+νuut+νb(t)
2u2/bracerightbigg
dx,
H(t) =/integraldisplay
e2ψ/braceleftbigg/parenleftbiggµ
4−ν−1
2/parenrightbigg
u2
t+/parenleftbigg
νδ2−1
2/parenrightbigg
|∇u|2/bracerightbigg
dx
J(t;u) =/integraldisplay
e2ψudx, Jψ(t;u) =/integraldisplay
e2ψ(−ψt)udx.
Multiplying (2.7) by (1+ t)n+1−ε, we have
d
dt[(1+t)n+1−εˆE(t)]−(n+1−ε)(1+t)n−εˆE(t)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
T7
+(1+t)n+1−εH(t)+(n+1−2δ1)(1+t)n+1−ενb(t)
2(1+t)J(t;u2)
≤d
dt[(1+t)n+2−εJ(t;F(u))]
+C((1+t)n+1−ε(J(t;|u|p+1)+(1+t)Jψ(t;|u|p+1)).
Now we estimate the bad term T7. First, by the Schwarz inequality, one can obtain
|νuut| ≤ν
4δ3b(t)u2
t+δ3νb(t)u2,
whereδ3determined later. From this, T7is estimated as
T7≤(n+1−ε)(1+t)n−ε
×/integraldisplay
e2ψ/braceleftbigg/parenleftbigg1+t
2+ν(1+t)
4δ3µ/parenrightbigg
u2
t+1+t
2|∇u|2+νb(t)
2(1+2δ3)u2/bracerightbigg
dx.8 YUTA WAKASUGI
We strengthen the conditions (2.4) and (2.5) as
µ
4−ν−1
2−(n+1−ε)/parenleftbigg1
2+ν
4δ3µ/parenrightbigg
>0, (2.8)
νδ2−1
2(n+2−ε)>0. (2.9)
Moreover, we take ε= 3δ1and then choose δ3such that
(n+1−2δ1)−(n+1−3δ1)(1+2δ3)>0.
Under these conditions, we can estimate T7and obtain
d
dt[(1+t)n+1−εˆE(t)]≤d
dt[(1+t)n+2−εJ(t;F(u))]
+C(1+t)n+1−ε(J(t;|u|p+1)+(1+t)Jψ(t;|u|p+1)).
By integrating the above inequality, it follows that
(1+t)n+1−εˆE(t)≤CI2
0+(1+t)n+2−εJ(t;|u|p+1)
+C/integraldisplayt
0(1+τ)n+1−ε(J(τ;|u|p+1)+(1+t)Jψ(τ;|u|p+1))dτ.
By a simple calculation, we have
(1+t)E(t)+1
1+tJ(t;u2)≤CˆE(t),
where
E(t) :=/integraldisplay
e2ψ(u2
t+|∇u|2)dx.
Thus, we obtain
(1+t)n+2−εE(t)+(1+t)n−εJ(t;u2) (2.10)
≤CI2
0+(1+t)n+2−εJ(t;|u|p+1)
+C/integraldisplayt
0(1+τ)n+1−ε(J(τ;|u|p+1)+(1+t)Jψ(τ;|u|p+1))dτ.
Now we turn to estimate the nonlinear terms. We need the following lem ma:
Lemma 2.2 (Gagliardo-Nirenberg) .Letp,q,r∈[1,∞]andσ∈[0,1]satisfy
1
p=σ/parenleftbigg1
r−1
n/parenrightbigg
+(1−σ)1
q
except forp=∞orr=nwhenn≥2. Then for some constant C=C(p,q,r,n)>
0, the inequality
/ba∇dblh/ba∇dblLp≤C/ba∇dblh/ba∇dbl1−σ
Lq/ba∇dbl∇h/ba∇dblσ
Lrfor anyh∈C1
0(Rn)
holds.
Noting that
J(t;|u|p+1) =/integraldisplay/vextendsingle/vextendsingle/vextendsinglee2
p+1ψu/vextendsingle/vextendsingle/vextendsinglep+1
dxSEMILINEAR DAMPED WAVE EQUATION 9
and∇(e2
p+1ψu) =2
p+1e2
p+1ψ(∇ψ)u+e2
p+1ψ∇u, we apply the above lemma to
J(t;|u|p+1) withσ=n(p−1)
2(p+1)and have
J(t;|u|p+1)≤C/parenleftbigg/integraldisplay
e4
p+1ψu2dx/parenrightbigg1−σ
2(p+1)
×/parenleftbigg/integraldisplay
e4
p+1ψ|∇ψ|2u2dx+/integraldisplay
e4
p+1ψ|∇u|2dx/parenrightbiggσ
2(p+1)
.
We note that
e4
p+1ψ|∇ψ|2=4a2|x|2
(1+t)4e4
p+1ψ≤C1
(1+t)2e2ψ
and obtain
J(t;|u|p+1)≤C/parenleftbigg/integraldisplay
e2ψu2dx/parenrightbigg1−σ
2(p+1)
×/parenleftbigg1
(1+t)2/integraldisplay
e2ψu2dx+/integraldisplay
e2ψ|∇u|2dx/parenrightbiggσ
2(p+1)
.
Therefore, we can estimate
(1+t)n+2−εJ(t;|u|p+1)≤(1+t)n+2−ε{(1+t)−(n−ε)M(t)}1−σ
2(p+1)
×{(1+t)−(n+2−ε)M(t)}σ
2(p+1).
By a simple calculation, if
(2.11) ε<2n/parenleftbig
p−/parenleftbig
1+2
n/parenrightbig/parenrightbig
p−1
then we have
(1+t)n+2−εJ(t;|u|p+1)≤CM(t)p+1.
We note that the conditions (2.6), (2.8), (2.9), (2.11) are fulfilled by the determi-
nation of the parameters in the order
p→ε→δ→ν→µ.
In a similar way, we can estimate the other nonlinear terms. Consequ ently, we
obtain the a priori estimate
M(t)≤CI2
0+CM(t)p+1.
This proves Theorem 1.2.
3.Proof of Theorem 1.3 and Theorem 1.4
In this section we first give a proof of Theorem 1.3. We use a method b y Lin,
Nishihara and Zhai [7] to transform (1.1) into divergence form and t hen a test-
function method by Qi S. Zhang [15].
Letµ>1. We multiply (1.1) by a positive function g(t)∈C2([0,∞)) and obtain
(gu)tt−∆(gu)−(g′u)t+(−g′+gb)ut=g|u|p.
We now choose g(t) as the solution of the initial value problem for the ordinary
differential equation
(3.1)/braceleftBigg−g′(t)+g(t)b(t) = 1, t>0,
g(0) =1
µ−1.10 YUTA WAKASUGI
The solution g(t) is explicitly given by
g(t) =1
µ−1(1+t).
Thus, we obtain the equation in divergence form
(3.2) ( gu)tt−∆(gu)−(g′u)t+ut=g|u|p.
Next, we apply a test function method. We first introduce test fun ctions having
the following properties:
η(t)∈C∞
0([0,∞)),0≤η(t)≤1, η(t) =/braceleftbigg1,0≤t≤1
2,
0, t≥1,(t1)
φ(x)∈C∞
0(Rn),0≤φ(x)≤1, φ(x) =/braceleftbigg1,|x| ≤1
2,
0,|x| ≥1,(t2)
η′(t)2
η(t)≤C/parenleftBig1
2≤t≤1/parenrightBig
,|∇φ(x)|2
φ(x)≤C/parenleftBig1
2≤ |x| ≤1/parenrightBig
. (t3)
LetRbe a large parameter in (0 ,∞). We define the test function
ψR(t,x) :=ηR(t)φR(x) :=η/parenleftbiggt
R/parenrightbigg
φ/parenleftBigx
R/parenrightBig
,
Letqbe the dual of p, that isq=p
p−1. Suppose that uis a global solution with
initial data ( u0,u1) satisfying
/integraldisplay
Rn((µ−1)u0+u1)dx>0.
We define
IR:=/integraldisplay
QRg(t)|u(t,x)|pψR(t,x)qdxdt,
whereQR:= [0,R]×BRandBR:={x∈Rn;|x| ≤R}. By the equation (3.2) and
integration by parts one can calculate
IR=−/integraldisplay
BR((µ−1)u0+u1)φq
Rdx
+/integraldisplay
QRgu∂2
t(ψq
R)dxdt+/integraldisplay
QR(g′u−u)∂t(ψq
R)dxdt−/integraldisplay
QRgu∆(ψq
R)dxdt
=:−/integraldisplay
BR((µ−1)u0+u1)φq
Rdx+J1+J2+J3.
By the assumption on the data ( u0,u1) it follows that
IR<J1+J2+J3
for largeR. We shall estimate J1,J2andJ3, respectively. We use the notation
ˆQR:= [R/2,R]×BR,˜QR:= [0,R]×(BR\BR/2).SEMILINEAR DAMPED WAVE EQUATION 11
We first estimate J3. By the conditions (t1)-(t3) and the H¨ older inequality we have
the following estimate:
|J3|/lessorsimilarR−2/integraldisplay
˜QRg(t)|u|ψq−1
Rdxdt
/lessorsimilarR−2/parenleftbigg/integraldisplay
˜QRg(t)|u|pψq
R(t,x)dxdt/parenrightbigg1/p/parenleftbigg/integraldisplay
˜QRg(t)dxdt/parenrightbigg1/q
/lessorsimilar˜I1/p
RRn+2
q−2,
where
˜IR:=/integraldisplay
˜QRg(t)|u|pψq
R(t,x)dxdt.
In a similar way, we can estimate J1andJ2as
|J1|+|J2|/lessorsimilarˆI1/p
RRn+2
q−2,ˆIR:=/integraldisplay
ˆQRg(t)|u|pψq
R(t,x)dxdt.
Hence, we obtain
(3.3) IR/lessorsimilar(˜I1/p
R+ˆI1/p
R)Rn+2
q−2,
in particular I1−1/p
R/lessorsimilarRn+2
q−2. If 1<p<pF, by letting R→ ∞we haveIR→0
and henceu= 0, which contradicts the assumption on the data. If p=pF, we have
onlyIR≤Cwith some constant Cindependent of R. This implies that g(t)|u|pis
integrable on (0 ,∞)×Rnand hence
lim
R→∞(˜IR+ˆIR) = 0.
By (3.3), we obtain lim R→∞IR= 0. Therefore, umust be 0. This also leads a
contradiction.
Proof of Theorem 1.4. The proof is almost same as above. Let 0 <µ≤1. Instead
of (3.1), we consider the ordinary differential equation
(3.4) −g′(t)+g(t)b(t) = 0
withg(0)>0. We can easily solve this and have
g(t) =g(0)(1+t)µ.
Then we have
(3.5) ( gu)tt−∆(gu)−(g′u)t=g|u|p.
Using the same test function ψR(t,x) as above, we can calculate
IR:=/integraldisplay
QRg(t)|u|pψq
Rdxdt
=−/integraldisplay
BRg(0)u1φq
Rdx+3/summationdisplay
k=1Jk,
where
J1=/integraldisplay
QRgu∂2
t(ψq
R)dxdt, J 2=/integraldisplay
QRg′u∂t(ψq
R)dxdt, J 3=−/integraldisplay
QRgu∆(ψq
R)dxdt.12 YUTA WAKASUGI
We note that the term of u0vanishes and so we put the assumption only for u1.
We first estimate J2. Notingg′(t) =µg(0)(1+t)µ−1, we have
|J2|/lessorsimilar1
R/integraldisplay
ˆQR(1+t)µ−1|u|ψq−1
Rdxdt.
By noting that (1+ t)µ−1∼g(t)1/p(1+t)µ/q−1and the H¨ older inequality, it follows
that
|J2|/lessorsimilar1
R/parenleftbigg/integraldisplay
ˆQRg|u|pψq
Rdxdt/parenrightbigg1/p/parenleftBigg/integraldisplayR
R/2/integraldisplay
BR(1+t)µ−qdxdt/parenrightBigg1/q
/lessorsimilarˆI1/p
RR−1+(n+(µ−q+1))/q,
whereˆIRis defined as before. A simple calculation shows
−1+(n+(µ−q+1))/q≤0⇔p≤1+2
n+(µ−1).
In the same way, we can estimate J1andJ3as
|J1|+|J3|/lessorsimilar(ˆI1/p
R+˜I1/p
R)R−2+(n+µ+1)/q,
where˜IRis same as before. It is also easy to see that
−2+(n+µ+1)/q≤0⇔p≤1+2
n+(µ−1).
Finally, we have
IR/lessorsimilar˜I1/p
R+ˆI1/p
R
ifp≤1+2/(n+(µ−1)). The rest of the proof is same as before. /square
Remark 3.1. We can apply the proof of Theorem 1.4 to the wave equation with
non-effective damping terms
/braceleftbigg
utt−∆u+b(t)ut=|u|p,
(u,ut)(0,x) = (u0,u1)(x),
where
b(t) = (1+t)−β
withβ >1. We can easily solve (3.4)and have
g(t) =g(0)exp/parenleftbigg1
−β+1((1+t)−β+1−1)/parenrightbigg
.
We note that g(t)∼1. The same argument implies that if
1<p≤1+2
n−1,/integraldisplay
u1dx>0,
then there is no global solution. We note that the exponent 1+2/(n−1)is greater
than the Fujita exponent. This shows that when β >1, the equation loses the
parabolic structure even in the nonlinear cases. One can exp ect that the critical ex-
ponent is given by the well-known Strauss critical exponent . However, this problem
is completely open as far as the author’s knowledge.SEMILINEAR DAMPED WAVE EQUATION 13
Acknowledgement
The author has generous support from Professors Tatsuo Nishit ani and Kenji
Nishihara. In particular, Prof. Nishihara gave the author an essen tial idea for the
proof of Theorem 1.4.
References
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damping , arXiv:1211.0731v1.
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arXiv:1211.0453v1.
[3]M. D’abbicco, S. Lucente, M. Reissig ,Semi-Linear wave equations with effective damping ,
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tions with critical potential , J. Math. Soc. Japan (in press).
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dependent damping , Discrete Contin. Dyn. Syst., 32(2012), 4307-4320.
[8]A. Matsumura ,On the asymptotic behavior of solutions of semi-linear wave equations , Publ.
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space and their application , Math. Z., 244(2003), 631-649.
[10]K. Nishihara ,Asymptotic behavior of solutions to the semilinear wave equ ation with time-
dependent damping , Tokyo J. Math., 34(2011), 327-343.
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Appl. Sci., 27(2004), 101-124.
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Differential Equations, 222(2006), 487-514.
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E-mail address :y-wakasugi@cr.math.sci.osaka-u.ac.jp
Department of Mathematics, Graduate School of Science, Osa ka University, Toy-
onaka, Osaka, 560-0043, Japan |
1605.04543v1.Propagation_of_Thermally_Induced_Magnonic_Spin_Currents.pdf | arXiv:1605.04543v1 [cond-mat.mtrl-sci] 15 May 2016Propagation of Thermally Induced Magnonic Spin Currents
Ulrike Ritzmann, Denise Hinzke, and Ulrich Nowak
Fachbereich Physik, Universit¨ at Konstanz, D-78457 Konst anz, Germany
(Dated: 19.12.2013)
The propagation of magnons in temperature gradients is inve stigated within the framework of an
atomistic spin model with the stochastic Landau-Lifshitz- Gilbert equation as underlying equation
of motion. We analyze the magnon accumulation, the magnon te mperature profile as well as the
propagation length of the excited magnons. The frequency di stribution of the generated magnons
is investigated in order to derive an expression for the influ ence of the anisotropy and the damping
parameter on the magnon propagation length. For soft ferrom agnetic insulators with low damping
a propagation length in the range of some µm can be expected for exchange driven magnons.
PACS numbers: 75.30.Ds, 75.30.Sg, 75.76.+j
I. INTRODUCTION
Spin caloritronics is a new, emerging field in mag-
netism describing the interplay between heat, charge and
spin transport1,2. A stimulation for this field was the dis-
covery of the spin Seebeck effect in Permalloy by Uchida
et al.3. Analog to the Seebeck effect, where in an elec-
tric conductor an electrical voltage is created by apply-
ing a temperature gradient, in a ferromagnet a temper-
ature gradient excites a spin current leading to a spin
accumulation. The generated spin accumulation was de-
tected by measuring the spin current locally injected into
a Platinum-contact using the inverse spin Hall effect3,4.
A first explanation of these effect was based on a spin-
dependent Seebeckeffect, wherethe conductionelectrons
propagate in two different channels and, due to a spin
dependent mobility, create a spin accumulation in the
system5.
Interestingly, it was shown later on that this effect also
appears in ferromagnetic insulators6. This shows that in
addition to conduction-electron spin-currents, chargeless
spin-currents exist as well, where the angular momentum
is transported by the magnetic excitations of the system,
so-called magnons. A first theoretical description of such
a magnonic spin Seebeck effect was developed by Xiao et
al.7. With a two temperature model including the local
magnon (m) and phonon (p) temperatures the measured
spin Seebeck voltage is calculated to be linearly depen-
dent on the local difference between magnon and phonon
temperature, ∆ Tmp=Tm−Tp. This temperature dif-
ference decays with the characteristic lengthscale λ. For
the ferromagnetic material YIG they estimate the length
scale in the range of several millimeters.
The contribution of exchange dominated magnons to
the spin Seebeck effect was investigated in recent experi-
ments by Agrawalet al.8. Using Brillouin lightscattering
the difference between the magnon and the phonon tem-
perature in a system with a linear temperature gradient
was determined. They found no detectable temperature
difference and estimate a maximal characteristic length
scale of the temperature difference of 470 µm. One pos-
sible conclusion from this results might be be that in-
stead of exchangemagnons, magnetostatic modes mainlycontribute to the spin Seebeck effect and are responsible
for the long-range character of this effect. Alternatively,
phononsmightcontributetothemagnonaccumulationas
well viaspin-phonon drag9,10. A complete understanding
of these different contributions to the spin Seebeck effect
is still missing.
In this paper thermally excited magnonic spin currents
and their length scale of propagation are investigated.
Usingatomisticspinmodelsimulationwhichdescribethe
thermodynamics of the magnetic system in the classical
limit including the whole frequency spectra of excited
magnons,wedescribespincurrentsbyexchangemagnons
in the vicinity of a temperature step. After introducing
our model, methods and basic definitions in Section II
we determine the magnon accumulation as well as the
corresponding magnon temperature and investigate the
characteristic lengthscale of the decay of the magnon ac-
cumulation in Section III. In Section IV we introduce
an analytical description which is supported by our sim-
ulations shown in Section V and gives insight into the
material properties dependence of magnon propagation.
II. MAGNETIZATION PROFILE AND
MAGNON TEMPERATURE
For the investigationofmagnonic spin currentsin tem-
perature gradients we use an atomistic spin model with
localized spins Si=µi/µsrepresenting the normalized
magnetic moment µsof a unit cell. The magnitude of
the magnetic moment is assumed to be temperature in-
dependent. Wesimulateathree-dimensionalsystemwith
simple cubic lattice structure and lattice constant a. The
dynamics of the spin system are described in the classical
limit by solving the stochastic Landau-Lifshitz-Gilbert
(LLG) equation,
∂Si
∂t=−γ
µs(1+α2)Si×(Hi+α(Si×Hi)), (1)
numerically with the Heun method11withγbeing the
gyromagnetic ratio. This equation describes a preces-
sion of each spin iaround its effective field Hiand the2
coupling with the lattice by a phenomenological damp-
ing term with damping constant α. The effective field
Hiconsists of the derivative of the Hamiltonian and an
additional white-noise term ζi(t),
Hi=−∂H
∂Si+ζi(t) . (2)
The Hamiltonian Hin our simulation includes exchange
interaction of nearest neighbors with isotropic exchange
constant Jand an uniaxial anisotropy with an easy axis
inz-direction and anisotropy constant dz,
H=−J
2/summationdisplay
<i,j>SiSj−dz/summationdisplay
iS2
i,z. (3)
The additional noise term ζi(t) of the effective field Hi
includes the influence of the temperature and has the
following properties:
/angbracketleftζ(t)/angbracketright= 0 (4)
/angbracketleftBig
ζi
η(0)ζj
θ(t)/angbracketrightBig
=2kBTpαµs
γδijδηθδ(t) . (5)
Herei,jdenote lattice sites and ηandθCartesian com-
ponents of the spin.
We simulate a model with a given phonon temperature
Tpwhich is space dependent and includes a temperature
step inz-direction in the middle of the system at z= 0
fromatemperature T1
pinthehotterareato T2
p= 0K(see
Fig. 1). We assume, that this temperature profile stays
constant during the simulation and that the magnetic
excitationshavenoinfluence onthe phonontemperature.
The system size is 8 ×8×512, large enough to minimize
finite-size effects.
All spins are initialized parallel to the easy-axis in z-
direction. Due tothetemperaturestepanon-equilibrium
in the magnonic density of states is created. Magnons
propagate in every direction of the system, but more
magnons exist in the hotter than in the colder part of
the system. This leads to a constant net magnon current
from the hotter towards the colder area of the system.
Due to the damping of the magnons the net current ap-
pears around the temperature step with a finite length
scale.After an initial relaxation time the system reaches
a steadystate. In this steady state the averagedspin cur-
rent from the hotter towardsthe colderregionis constant
and so the local magnetization is time independent. We
can now calculate the local magnetization m(z) depend-
ing on the space coordinate zas the time average over
all spins in the plane perpendicular to the z-direction.
We use the phonon temperature T1
p= 0.1J/kBin the
heated area, the anisotropy constant dz= 0.1Jand vary
the damping parameter α. The resulting magnetization
versus the space coordinate zfor different damping pa-
rameters in a section around the temperature step is
shown in Fig. 1. For comparison the particular equi-
librium magnetization m0of the two regions is also cal-
culated and shown in the figure.m0α= 1α= 0.1α= 0.06Tp
space coordinate z/a
phonon temperature kBTp/Jmagnetization m0.1
0
403020100-10-20-30-401
0.995
0.99
0.985
0.98
0.975
FIG. 1. Steady state magnetization mand equilibrium mag-
netization m0over space coordinate zfor a given phonon
temperature profile and for different damping parameters α
in a small section around the temperature step.
Far away from the temperature step on both sides the
amplitudes of the local magnetization m(z) converge to
the equilibrium values, only in the vicinity of the tem-
perature step deviations appear. These deviations de-
scribe the magnon accumulation, induced by a surplus
of magnons from the hotter region propagating towards
the colder one. This leads to a less thermal excitation
in the hotter area and the value of the local magneti-
zation increases. In the colder area the surplus of in-
coming magnons decrease the value of the local magneti-
zation. For smaller values of αthe magnons can propa-
gate overlargerdistances before they are finally damped.
This leads to a damping-dependent magnon accumula-
tion which increases with decreasing damping constant
α.
Forafurtheranalysisinthecontextofthespin-Seebeck
effect we define a local magnon temperature Tm(z) via
the magnetization profile m(z). For that the equilib-
rium magnetization m0(T) is calculated for the same
model but homogeneous phonon temperature Tp. In
equilibrium magnon temperature Tmand the phonon
temperature Tpare the same and we can determine the
fitequilibrium data
magnon temperature kBTm/Jmagnetization m0
10.90.80.70.60.50.40.30.20.101
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
FIG. 2. Equilibrium magnetization m0over the magnon
temperature Tm. Red points show the simulated equilibrium
magnetization and the black line shows a fit of the data.3
T0
mα= 1α= 0.1α= 0.06Tp
space coordinate z/a
phonon temperature kBTp/Jmagnon temperature kBTm/J 0.1
0
403020100-10-20-30-400.1
0.08
0.06
0.04
0.02
0
FIG. 3. Magnon temperature Tmover the space coordinate
zfor different damping parameters αcorresponding to the
results in Fig. 1.
(magnon) temperature dependence of the equilibrium
magnetization m0(Tm) of the system. The magnetiza-
tion of the equilibrium system decreases for increasing
magnontemperatureasshownin Fig. 2 andthe behavior
can be described phenomenologically with a function12
m0(T) = (1−Tm/Tc)βwhereTcistheCurietemperature.
Fitting ourdatawefind Tc= (1.3326±0.00015)J/kBand
for the exponent we get β= 0.32984±0.00065. This fit
of the data is also shown in Fig. 2 and it is a good
approximation over the whole temperature range. The
inverse function is used in the following to determine the
magnon temperature for a given local magnetization and
with that a magnon temperature profile Tm(z).
The resulting magnon temperature profiles are shown
in Fig. 3. Far away from the temperature step the
magnon temperature Tm(z) coincides with the given
phonon temperature Tp, and deviations — dependent on
the damping constant α— appear only around the tem-
perature step. These deviations correspond to those of
the local magnetization discussed in connection with Fig.
1.
III. MAGNON PROPAGATION LENGTH
To describe the characteristic lengthscale of the
magnon propagationaround the temperature step we de-
fine the magnon accumulation ∆ m(z) as the difference
between the relative equilibrium magnetization m0(z) at
the given phonon temperature Tp(z) and the calculated
local magnetization m(z):
∆m(z) =m0(z)−m(z) . (6)
Weinvestigatethemagnonpropagationinthecolderpart
of the system, where Tp(z) = 0. For a small magnon
temperature, the temperature dependence of the magne-
tization can be approximated as
m(Tm)≈1−β
TcTm. (7)α= 1.00α= 0.50α= 0.10α= 0.08α= 0.06
space coordinate z/amagnon accumulation ∆m
2502001501005001
10−2
10−4
10−6
10−8
10−10
FIG. 4. Magnon accumulation ∆ mover space coordinate z
in the colder region of the system at Tp= 0K for different
damping constants αshows exponential decay with magnon
propagation length ξ. The points show the data from our
simulation and the lines the results from an exponential fit.
These linear equation is in agreement with an analytical
solution for low temperatures presented by Watson et
al.12. For low phonon temperatures one obtains for the
difference between phonon and magnon temperature
∆T=Tm−Tp=β
Tc∆m. (8)
Note, that the proportionality between magnon accumu-
lation and temperature difference holds for higher tem-
peratures as well as long as magnon and phonon temper-
ature are sufficiently close so that a linear approximation
applies,thoughtheproportionalityfactorincreases. Note
also, that this proportionalitywas determined in theoret-
ical descriptions of a magnonic spin Seebeck effect7. Our
results for the magnon accumulation should hence be rel-
evantfortheunderstandingofthemagnonicspinSeebeck
where the temperature difference between the magnons
in the ferromagnet and the electrons in the non-magnet
plays a key role.
We further investigate our model as before with a tem-
perature in the heated area of T1
p= 0.1J/kB, anisotropy
constant dz= 0.1Jand different damping parameters.
The magnon accumulation ∆ mversus the space coordi-
natezin the colder region of the system at Tp= 0K is
shown in Fig. 4. Apart from a sudden decay close to the
temperature step the magnon accumulation ∆ m(z) then
decays exponentially on a length scale that depends on
the damping constant α. To describe this decay we fit
the data with the function
∆m(z) = ∆m(0)·e−z
ξ. (9)
We define the fitting parameter ξas the propagation
length of the magnons. Here, the deviations from the
exponential decay at the beginning of the system are ne-
glected. The fits for the data are shown in Fig. 4 as
continuous lines.
The propagation length dependence on the damping
parameter αis shown in Fig. 5. The values of the prop-
agation length from our simulations, shown as points,4
dz= 0.01Jdz= 0.05Jdz= 0.10Jdz= 0.50J
damping constant αpropagation length ξ/a
1 0.1100
10
1
FIG. 5. Magnon propagation length ξover the damping con-
stantαfor different anisotropy constant dz. Numerical data
is shown as points and the solid lines are from Eq. (19).
are inversely proportional to the damping constant α
and, furthermore, show also a strong dependence on the
anisotropy constant dz. This behavior will be discussed
in the next two sections with an analytical analysisof the
magnon propagation and an investigation of the frequen-
cies of the propagating magnons. A simple approxima-
tion for the propagation length leads to Eq. (19) which
is also shown as solid lines in Fig. 5.
IV. ANALYTICAL DESCRIPTION WITH
LINEAR SPIN-WAVE THEORY
For the theoretical description of the magnon accu-
mulation, excited by a temperature step in the system,
we solve the LLG equation (Eq. (1)), analytically in
the area with Tp= 0K. We consider a cubical system
with lattice constant awhere all spins are magnetized in
z-direction parallel to the easy-axis of the system. As-
suming only small fluctuations in the x- andy-direction
we have Sz
i≈1 andSx
i,Sy
i≪1. In that case we can
linearize the LLG-equation and the solution of the re-
sulting equation consists of a sum over spin waves with
wavevectors qand the related frequency ωqwhich decay
exponentially in time dependent on their frequency and
the damping constant αof the system,
S±
i(t) =1√
N/summationdisplay
qS±
q(0)e∓iqri±iωqt·e−αωqt. (10)
The frequency ωqof the magnons is described by the
usual dispersion relation
¯hωq=1
(1+α2)/parenleftBig
2dz+2J/summationdisplay
θ(1−cos(qθaθ))/parenrightBig
. (11)
The dispersion relation includes a frequency gap due to
the anisotropy constant and a second wavevector depen-
dent term with a sum over the Cartesian components13.
Considering now the temperature step, magnons from
the hotter area propagate towards the colder one. Weinvestigate the damping process during that propagation
inordertodescribethe propagatingfrequenciesaswellas
to calculate the propagation length ξof the magnons for
comparisonwiththeresultsfromsectionIII.Themagnon
accumulation will depend on the distance to the temper-
ature step and — for small fluctuations of the SxandSy
components — can be expressed as
∆m(z) = 1−/angbracketleftSz(z)/angbracketright ≈1
2/angbracketleftbig
Sx(z)2+Sy(z)2/angbracketrightbig
, (12)
where the brackets denote a time average. We assume
that the local fluctuations of the SxandSycomponents
can be described with a sum over spin waves with differ-
ent frequencies and damped amplitudes aq(z),
Sx(z) =/summationdisplay
qaq(z)cos(ωqt−qr) , (13)
Sy(z) =/summationdisplay
qaq(z)sin(ωqt−qr) . (14)
In that case for the transverse component of the magne-
tization one obtains
/angbracketleftbig
Sx(z)2+Sy(z)2/angbracketrightbig
=/angbracketleftBigg/summationdisplay
qaq(z)2/angbracketrightBigg
, (15)
where mixed terms vanish upon time averaging. The
magnon accumulation can be written as:
∆m(z) =1
2/angbracketleftBigg/summationdisplay
qaq(z)2/angbracketrightBigg
. (16)
The amplitude aq(z) of a magnon decays exponentially
as seen in Eq. (10) dependent on the damping constant
and the frequency of the magnons. In the next step we
describe the damping process during the propagation of
the magnons. In the one-dimensional limit magnons only
propagate in z-direction with velocity vq=∂ωq
∂q. Then
the propagation time can be rewritten as t=z/vqand
we can describe the decay of the amplitude with aq(z) =
aq(0)·f(z) with a damping function
f(z) = exp/parenleftBig
−αωqz
∂ωq
∂qz/parenrightBig
. (17)
The amplitudes are damped exponentially during the
propagation which defines a frequency dependent propa-
gation length
ξωq=/radicalbigg
J2−/parenleftBig
1
2(1+α2)(¯hωq−2dz)−J/parenrightBig2
α(1+α2)¯hωq,(18)
where we used γ=µs/¯h. In the low anisotropy limit
this reduces to ξωq=λ/παwhereλ= 2π/qis the wave
length of the magnons.5
The total propagation length is then the weighted av-
erage over all the excited frequencies. The minimal fre-
quency is defined by the dispersion relation with a fre-
quency gap of ωmin
q= 1/(¯h(1 +α2))2dz. For small fre-
quencies above that minimum the velocity is small, so
the magnons are damped within short distances. Due to
the fact that the damping process is also frequency de-
pendent higher frequencies will also be damped quickly.
In the long wave length limit the minimal damping is
at the frequency ωmax
q≈4dz/(¯h(1 +α2)) which can be
determined by minimizing Eq. (17) .
In a three-dimensional system, besides the z-
component of the wavevector, also transverse compo-
nents of the wavevector have to be included. The
damping of magnons with transverse components of
the wavevector is higher than described in the one-
dimensionalcase,becausethe additionaltransverseprop-
agationincreasethepropagationtime. Inoursimulations
the cross-section is very small, so that transverse compo-
nents of the wave-vectors are very high and get damped
quickly. Thisfact andthe highdamping forhighfrequen-
cies described in Eq. (17) can explain the very strong
damping at the beginning of the propagation shown in
Fig. 4.
V. FREQUENCIES AND DAMPING OF
PROPAGATING MAGNONS
In this section we investigate the frequency distribu-
tionofthemagnonicspincurrentwhilepropagatingaway
from the temperature step. First we determine the fre-
quencies of the propagating magnons in our simulations
with Fourier transformation in time to verify our as-
sumptions from the last section. As before a system of
8×8×512spins with a temperature step in the center of
the system is simulated with an anisotropy of dz= 0.1J.
The temperature of the heated area is T1
p= 0.1J/kBand
the damping constant is α= 0.1. After an initial relax-
ation to a steady-state the frequency distribution of the
propagating magnons in the colder area is determined by
Fourier transformation in time of S±(i) =Sx(i)±iSy(i).
The frequency spectra are averaged over four points in
thex-y-plane and analyzed depending on the distance z
of the plane to the temperature step.
The results for small values of zare shown in Fig. 6(a)
and for higher values of z, far away from the temper-
ature step, for the regime of the exponential decay, in
Fig. 6(b). For small values of z, near the temperature
step, the frequency range of the propagating magnons is
very broad. The minimum frequency is given by ωmin
q=
2dz/(¯h(1+α2)) and far away from the temperature step
the maximum peak is around ωmax
q= 4dz/(¯h(1 +α2)).
These characteristic frequencies are in agreement with
our findings in section IV.
Furthermore, a stronger damping for higher frequen-
cies can be observed. This effect corresponds to the
strong damping of magnons with wavevector compo-10864204·10−3
3·10−3
2·10−3
1·10−3
0z= 20az= 10az= 1a(a)
frequency ¯hωq/Jamplitude |S+(ωq)|
1.41.210.80.60.40.208·10−5
6·10−5
4·10−5
2·10−5
0ωminz= 100az= 90az= 80a(b)
frequency ¯hωq/Jamplitude |S+(ωq)|
FIG.6. Amplitude |S+(ωq)|versusthefrequency ωqfor asys-
tem with 8 ×8×512 spins. (a):after propagation over short
distances form 1 to 20 lattice constants. (b): after propaga -
tion over longer distances from 80 to 100 lattice constants.
nents transverse to the z-direction and it explains the
higher initial damping, which was seen in the magnon
accumulation in Fig. 4. A much narrower distribu-
tion propagates over longer distances and reaches the
area shown in Fig. 6(b). In that area the damping
can be described by one-dimensional propagation of the
magnons in z-direction with a narrow frequency distri-
bution around the frequency with the lowest damping
ωmax
q= 4dz/(¯h(1+α2)).The wavelength and the belong-
ing group velocity of the magnons depending on their
frequency in the one-dimensional analytical model are
shown in Fig.7(a). In the simulated system magnons
with the longest propagation length have a wavelength
ofλ= 14a. Depending on the ratio dz/Jthe wave-
length increases for systems with lower anisotropy. As
discussed in the last chapter, magnons with smaller fre-
quencies are less damped in the time domain, but due to
their smaller velocity the magnons very close to the min-
imum frequency also have a smaller propagation length.
To investigate the frequency-dependent damping-
process during the propagation of the magnons we calcu-
latetheratiooftheamplitudeofthemagnons |S+(ωq,x)|
forz= 80aandz= 80a+∆ with ∆ = 10 a,20a,50aand
normalize it to a damping per propagation of one spin.
The resulting ratios ( |S+(ωq,x)|/|S+(ωq,x−∆)|)1/∆are
shown in Fig. 7 in comparison with the frequency-6
2
1.5
1
0.5
0
43.532.521.510.50100
80
60
40
20
0ωmin
qvqλ(a)
frequency ¯hωq/J
velocityvq¯h/(Ja)wave length λ/a
damping function∆ = 10a∆ = 20a∆ = 50a(b)
frequency ω[J/¯h]damping ratio
10.90.80.70.60.50.40.30.21
0.98
0.96
0.94
0.92
0.9
FIG. 7. (a): Wavelength λand group velocity vqof the
magnons in a one-dimensional model dependent on the fre-
quencyωq. (b):Damping ratio as explained in the text versus
the frequency ωqfor different distances ∆ and compared to
the damping function (Eq. (17)).
dependent damping-function (Eq. (17)). The figure
shows a good agreement between simulation and our an-
alytical calculations.
These results explain the dependence of the magnon
propagation length on the model parameters. The fre-
quency with the maximal amplitude is determined by
the anisotropy constant. Under the assumption that the
frequency with the lowest damping is dominant and the
contribution of other frequencies can be neglected the
propagation length can be calculated as
ξ=a
2α/radicalbigg
J
2dz, (19)
where the square-root term is the domain wall width of
the model. This formula is also plotted in Fig. 5.
The comparison with our simulations shows good
agreement though the equation above gives only the
propagation of those magnons with the smallest damp-
ingduringthe propagation.Inthe consideredsystemwith
α= 0.1 anddz= 0.1Jwe get a propagation length of
aboutξ= 11aat a wavelength of the magnons λ= 14a.
For smaller values of the anisotropy and smaller damp-
ing parameters the frequency distribution of the thermal
magnons is broader and Eq. (19) is an overestimation
of the real propagation length since the magnon accu-mulation is no longer exponentially decaying due to the
broader spectrum of propagating frequencies. However
we would expect for soft ferromagnetic insulators with a
smalldamping constantof10−4−10−3and ananisotropy
constant in the range of 10−3J−10−2Ja propagation
length of 103a−105awhich would be in the micrometer-
range.
VI. SUMMARY AND DISCUSSION
Using the frameworkof an atomistic spin model we de-
scribe thermally induced magnon propagationin a model
containing a temperature step. The results give an im-
pression of the relevant length scale of the propagation
of thermally induced exchange magnons and its depen-
dence on system parameters as the anisotropy, the ex-
change and the damping constant. In the heated area
magnons with a broad frequency distribution are gener-
atedandbecauseoftheverystrongdampingformagnons
with high frequency, especially those with wave-vector
components transverse to the propagation direction in
z-direction, most of the induced magnons are damped
on shorter length scales. Behind this region of strong
damping near the temperature step, the propagation of
magnons is unidirectional and the magnon accumula-
tion decays exponentially with the characteristic prop-
agation length ξ. This propagation length depends on
the damping parameter but also on system properties as
the anisotropy of the system, because of the dependence
on the induced frequencies.
In contrast to long range magnetostatic spin waves,
which can propagate over distances of some mm14,15, we
find that for exchange magnons the propagation length
is considerably shorter and expect from our findings for
soft ferromagnetic insulators with a low damping con-
stant a propagation length in the range of some µm for
those magnons close to the frequency gap and the lowest
damping. These findings will contribute to the under-
standing of length scale dependent investigations of the
spin Seebeck effect8,16–18.
Recent experiments investigate the longitudinal spin
Seebeck effect, where the generated spin current
longitudinal to the applied temperature gradient is
measured19–22. In this configuration Kehlberger et al.
show that the measured spin current is dependent on the
thickness of the YIG layer and they observe a saturation
ofthespincurrentonalengthscaleof100 nm16. Thissat-
uration can be explained by the lengthscale of the prop-
agation of the thermally excited magnons. Only those
magnons reaching the YIG/Pt interface of the sample
contribute to the measured spin current and — as shown
here — exchange magnons thermally excited at larger
distances are damped before they can reach the inter-
face. In this paper, we focus on the propagation length
of those magnons with the lowest damping, however the
lengthscale of the magnon accumulation at the end of
a temperature gradient is dominated by a broad range7
of magnons with higher frequencies which are therefore
damped on shorter length scales.
ACKNOWLEDGMENTS
The authors would like to thank the Deutsche
Forschungsgemeinschaft (DFG) for financial support viaSPP 1538 “Spin Caloric Transport” and the SFB 767
“Controlled Nanosystem: Interaction and Interfacing to
the Macroscale”.
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0809.4644v2.Damping_and_magnetic_anisotropy_of_ferromagnetic_GaMnAs_thin_films.pdf | Anisotropic Magnetization Relaxa tion in Ferromagnetic GaMnAs
Thin Films
Kh.Khazen, H.J.von Bardeleben, M.Cubukcu, J.L.Cantin
Institut des Nanosciences de Paris,
Université Paris 6, UMR 7588 au CNRS
140, rue de Lourmel, 75015 Paris, France
V.Novak, K.Olejnik, M.Cukr
Institut of Physics, Academy of Sciences,
Cukrovarnicka 10, 16253 Praha, Czech Republic
L.Thevenard, A. Lemaître
Laboratoire de Photonique et des Nanostructures, CNRS
Route de Nozay, 91460 Marcoussis, France
Abstract:
The magnetic properties of annealed, epitaxial Ga 0.93Mn 0.07As layers under tensile and
compressive stress have been investigat ed by X-band (9GHz) and Q-band (35GHz)
ferromagnetic resonance (FMR) spectroscopy. From the analysis of the linewidths of the
uniform mode spectra the FMR Gilbert damping factor α has been determined. At T=4K we
obtain a minimum damping factor of α = 0.003 for the compressively stressed layer. Its value
is not isotropic. It has a minimum value for th e easy axes orientations of the magnetic field
and increases with the measuring temperature. It s average value is for both type of films of
the order of 0.01 in spite of strong differences in the inhomogeneous linewidth which vary
between 20 Oe and 600 Oe for the layers grown on GaAs and GaInAs substrates respectively .
PACS numbers: 75.50.Pp, 76.50.+g, 71.55.Eq
Introduction:
The magnetic properties of ferromagnetic Ga 1-xMn xAs thin films with Mn
concentrations between x=0.03 and 0.08 have been studied in great detail in the recent years
both theoretically and experimentally. For recent reviews see references [1, 2]. A
particularity of GaMnAs ferro magnetic thin films as comp ared to conventional metal
ferromagnetic thin films is the predominance of the magnetocrystalline anisotropy fields over
the demagnetization fields. The strong anisotropy fields are not directly related to the crystal
structure of GaMnAs but are induced by the la ttice mismatch between the GaMnAs layers and
the substrate material on which they are grow n. When grown on (100) GaAs substrates the
difference in the lattice constants induces biaxial strains of ≈ 0.2% which give rise to
anisotropy fields of several 103 Oe. The low value of the de magnetization fields (~300Oe) is
the direct consequence of the small spin conc entration in diluted magnetic semiconductors
(DMS) which for a 5% Mn doping leads to a saturation magnetization of only 40 emu/cm3.
As the strain is related to the lattice mismat ch it can be engineer ed by choosing different
substrate materials. The two systems which have been investigated most often are (100) GaAs
substrates and (100)GaInAs pa rtially relaxed buffer layers. These two cases correspond to
compressive and tensile strained Ga MnAs layers respectively [3].
The static micro-magnetic pr operties of GaMnAs layers can be determined by
magnetization, transport, magneto-optical and ferromagnetic resonance techniques. For the
investigation of the ma gnetocrystalline anisotropies the ferromagnetic resonance spectroscopy
(FMR) technique has been shown to be partic ularly well adapted [2, 4]. The dynamics and
relaxation processes of the magnetization of such layers have hardly been investigated up to
now [5-7]. The previous FMR studies on this subject concerned either unusually low doped
GaMnAs layers [5, 7] or employed a single microwave frequency [6] which leads to an
overestimation of the damping factor. The knowle dge and control of the relaxation processes
is in particular important for device applications as they de termine for example the critical
currents necessary for current induced magne tization switching. It is thus important to
determine the damping factor for state of the art samples with high Curie temperatures of T C ≈
150K, such as those used in this work. Anothe r motivation of this work is the search for a
potential anisotropy of the ma gnetization relaxation in a dilu ted ferromagnetic semiconductor in which the m agnetocrystalline anisotro pies are strong and dom inant over the
demagnetization contribution.
The intr insic sm all angle m agnetization re laxation is generally described by one
param eter, the Gilbert d amping coefficient α, which is defined by the Landau Lifshitz Gilbert
(LLG) equation of m otion for the m agnetization:
⎥⎦⎤
⎢⎣⎡× +⎥⎦⎤
⎢⎣⎡× −= ⋅dtsdMeffH MdtMdrr rrr
γα
γ1 eq.1
with M the m agnetization, H eff the effective m agnetic field, α the dam ping fa ctor, γ the
gyrom agnetic ratio and s the uni t vector parallel to M.
The dam ping factor α is generally assum ed to be a scal ar quantity [8, 9] . It is defined
for sm all angle precess ion relaxatio n which is the case of FMR experim ents. This param eter
can be experim entally determ ined by FMR spectr oscopy either from the angular variation of
the linewidth or from the variati on of the uniform mode linewidth ∆Hpp with th e microwave
frequency. In this second case the linewidth is given by:
ω
γω ⋅
⋅⋅ + ∆= ∆+ ∆= ∆
MGH H H Hin inpp
2 hom hom hom32)( eq.2
With ∆Hpp the first derivative p eak-to -peak linewid th of the uniform mode of
Lorentzian lineshape, ω the angular m icrowave frequency an d G the Gil bert dam ping factor
from which the m agnetization independent damping factor α can be deduced as α=G/γM. In
eq. 2 it is assum ed that the m agnetiz ation and th e applied magnetic f ield are collin ear which is
fulfilled for high symm etry direction s in GaMnAs such as [001], [110] and [100]. Ot herwise a
1/cos ( θ-θH) term has to be added to eq.2 [8].
∆Hinhom is the inhom ogeneous, frequency indepe ndent linewidth; it can be further
decom posed in three con tributions, re lated to the crysta lline imperf ection of the f ilm [10]:
int
inthom HHH H HHr
H
Hr
H
Hr
in ∆⋅ + ∆⋅ + ∆⋅∂= ∆δδφδφδθθδ eq.3
These three term s were introduced to take in to account a slight m osaic structure of the
metallic thin f ilms def ined by the polar angles (θ, φ) and their distributions ( ∆θ, ∆φ) -
expressed in the first two term s in eq.3- and a distribution of the internal anisotropy fields H int – the last term of eq.3. In the case of homo epitaxial III-V films obtained by MBE growth like
GaMnAs on GaAs, films of high crystalline qu ality are obtained [LPN] and only the third
component ( ∆Hint) is expected to play an important role.
Practically, the variation of the FMR linewid th with the microwave frequency can be
measured with resonant cavity systems at different frequencies between 9GHz and 35GHz;
the minimum requirement -used also in this work - is the use of two frequencies. We disposed
in this work of 9GHz and 35GHz spectrometers. The linewidth is decomposed in a frequency independent inhomogeneous part and a linear fr equency dependent homogeneously broadened
part. For most materials the inhomogeneous fr action of the linewidth is strongly sample
dependent and depends further on the interface quali ty and the presence of cap layers. It can
be smaller but also much larger than the intrinsic linewidth. In Ga
0.95Mn 0.05As single films
total X-band linewidths be tween 100Oe and 1000Oe have been encountered. These
observations indicate already the impor tance of inhomogeneous broadening. The
homogeneous linewidth will depend on the intrinsic sample properties. This approach supposes that the inhomogeneous linewidth is frequency independent and the homogenous
linewidth linear dependent on the frequency, two assumptions generally valid for high
symmetry orientations of the a pplied field for which the magne tization is parallel to the
magnetic field.
It should be underlined that in diluted magnetic semiconductor (DMS) materials like
GaMnAs the damping parameter is not only determined by the sample composition x Mn [5]. It
is expected to depend as we ll on (i) the magnetic compensati on which will vary with the
growth conditions, (ii) the (hol e) carrier concentration respon sible for the ferromagnetic Mn-
Mn interaction which is influenced by the presence of native donor defects like arsenic
antisite defects or Mn interstitial ions [11] and (iii) the valence bandstructure, sensitive to the
strain in the film. Due to the high out-of –plane and in-plane anisotropy of the magnetic
parameters [12] which further vary with the applied field and the temperature a rather
complex situation with an anisotropic and te mperature dependent da mping factor can be
expected in GaMnAs.
Whereas the FMR Gilbert damping factor has been determined for many metallic
ferromagnetic thin films [8] only three experimental FMR studies have been published for GaMnAs thin films up to now [5-7]. In ref.[ 5,7] low doped GaMnAs laye rs with a critical
temperature of 80K which do not correspond to the high quality, standa rd layers available
today were studied. In the ot her work [6] higher doped layers were investigated but the
experiments were limited to a single microw ave frequency (9GHz) and thus no frequency dependence could be studied. In this work we present the results of FMR studies at 9GHz and
35 GHz on two high quality GaMnAs layers with optimum critical temperatures: one is a
compressively strained layer grown on a GaAs buffer layer and the othe r a tensile strained
layer grown on a (Ga,In)As buffer layer. Due to the opposite sign of the strains the easy axis
of magnetization is in-plane [ 100] in the first case and out-o f-plane [001] in the second. The
GaMnAs layers have been annealed ex-situ after their growth in order to reduce the electrical
and magnetic compensation, to homogenize the laye rs and to increase the Curie temperature
to ≈ 130K. Such annealings have become a st andard procedure for improving the magnetic
properties of low temperature molecular b eam epitaxy (LTMBE) grown GaMnAs films.
Indeed, the low growth temperature required to incorporate the high Mn concentration
without the formation of precipitates gives rise to native defect the conc entration of which can
be strongly reduced by the annealing.
Experimental details
A first sample consisting of a Ga 0.93Mn 0.07As layer of 50nm thickness has been grown
at 250° C by low temperature molecular beam epitaxy on a semi-insulating (100) oriented
GaAs substrate. A thin GaAs buffer layer has been grown before the deposition of the
magnetic layer. The second sample, a 50 nm thick Ga 0.93Mn 0.07As layer have been grown
under very similar conditions on a partially relaxed (100) Ga 0.902In0.098As buffer layer; for
more details see ref. [13]. After the growth the structure was thermally annealed at 250° C for
1h under air or nitrogen gas fl ow. The Curie temperatures were 157K and 130K respectively.
Based on conductivity measuremen ts the hole concentratio n is estimated in the 1020cm-3
range.
The FMR measurements were performed with Bruker X-band and Q-band
spectrometers under standard conditions: mW microwave power and 100 KHz field
modulation. The samples were measured at te mperatures between 4K and 170K. The angular
variation of the FMR spectra was measured in the two rotation planes (110) and (001). The
peak-to peak linewidth of the first derivati ve spectra were obtained from a lineshape
simulation. The value of the st atic magnetization M(T) had been determined by a commercial
superconducting quantum interference device (SQUID) magnetometer. A typical hysteresis
curve is shown in the inset of fig.8.
Experimental results: The saturation magnetizations of the two laye rs and the magneto crystalline anisotropy
constants which had been previously de termined by SQUID and FMR measurements
respectively are given in table I. The anisotropy constants had been determined in the whole
temperature range but for clarity only its values at T=55K and T=80K are given in table I. We
see that the dominant anisotropy constant K 2⊥ are of different sign with -55000 erg/cm3 to
+91070 erg /cm3 and that the other three constants ha ve equally opposite signs in the two
types of layers. The easy axes of magnetization are the in-pla ne [100] and the out-of-plane
[001] direction respectively. Howe ver the absolute values of th e total effective perpendicular
anisotropy constant Ku=K 2⊥ +K 4⊥ are less different for the two samples: -46517erg/cm3 and
+57020erg/cm3 respectively. More detailed inform ation on the measurements of these
micromagnetic parameters will be published elsewhere.
For the GaMnAs/GaAs layers the peak-to-peak linewidth of the first derivative
uniform mode spectra has been strongly re duced by the thermal annealing; in the non
annealed sample the X-band linewidth was highl y anisotropic with va lues between 150Oe and
500Oe at T=4K. After annealing it is reduced to an quasi isotropic average value of 70Oe at
X-band. Quite differently, for the GaMnAs/GaInA s system the annealing process decreases
the linewidth of the GaMnAs layers only marginally. Although full angular dependencies
have been measured by FMR we will analyze only the linewidth of the four high symmetry field orientations H//[001], H //[100], H//[1-10], H//[ 110] corresponding to the hard and easy
axes of magnetization. As will be shown below, in spite of rather similar high critical temperatures (157K/130K) the linewidth are drastically di fferent for the two cases.
1. GaMnAs on GaAs
In fig. 1a and 1b we show typical low te mperature FMR spectra at X-band and Q-band
frequencies for the hard [001] /intermediate [100] axis orientation of the applied magnetic
field. The spectra are characterized by excelle nt signal to noise ra tios and well defined
lineshapes. We see that at both frequencies the lineshape is close to a Lorentzian. In addition
to the main mode one low intensity spin wave resonance is observed at both frequencies at
lower fields (not shown).
The linewidth at X-band (fig.2) is of the order of 50Oe to 75Oe with a weak
orientation and temperature dependence. Above T>130K, close to the criti cal temperature, the
linewidth increase strongly. At Q-band we observe a systematic increase by a factor of two of
the total linewidth (fig.3) with an increase d temperature and orient ation dependence. As generally observed in GaMnAs, the easy axis orie ntation gives rise to th e lowest linewidth. At
Q-band the lineshape is perfectly Lorentzian (f ig.1b). These linewidth are among the smallest
ever reported for GaMnAs thin films, which re flects the high crystal line and magnetic quality
of the film.
To determine the damping factor α we have plotted the frequency dependence of the
linewidth for the different orientations and at various temperatures. An example is given in
fig. 4 for T=80K; this allows us to determin e the inhomogeneous linewidth obtained from a
linear extrapolation to zero frequency and the damping factor from the slope. The
inhomogeneous linewidth at T=80K is of the order of 30 Oe, i.e. 50% of the total linewidth at
X-band. This shows that the approximation ∆Hinhom<< ∆Hhomo which had been previously
used [5] to deduce the damping factor from a single (X-band) frequenc y measurement is not
fulfilled here.
The temperature dependence of the inhomogeneous linewidth is shown in fig.5.
Similar trends as for the total linewidth in the non annealed films are observed: the linewidth
is high at the lowest temperatures, decreases with increasing temperat ures up to 120K and
increases again close to T C.
From the slope of the linewidth variati on with microwave frequency we obtain the
damping factor α (fig.6). Its high temperature value is of the order of 0.010 but we observe a
systematic, linear variation with the temperatur e and a factor two difference between the easy
axis orientation [100] and the hard axis orientation [001].
2. GaMnAs on GaInAs
Similar measurements have been performed on the annealed tensile strained layer. In
tensile strained GaMnAs films the easy axis of magnetization ([001]) coincides with the
strong uniaxial second order anisotropy directio n. For that reason no FMR resonance can be
observed at temperatures below T=80K for the easy axis orientation H// [001] at X-band. For
the other three orientations the resonances can be observed at X-band in the whole
temperature range 4K to T C. Due to the strong temperature dependence of the anisotropy
constants and the parallel decr ease of the internal anisotropy fields the easy axis resonance
becomes observable at X-band for temperatures above 80K. In the films on GaInAs much
higher linewidth are encountered th an in films on GaAs, the values are up to ten times higher
indicating a strong inhomogeneity in this film. A second low fiel d resonance is systematically
observed at X-band and Q-band; it is equally attributed to a spin wave resonance. Figures 7a and 7b show typical FMR spectra at X- and Q-band re spectively. At both
frequencies the lineshape can no longer be simu lated by a Lorentzian but has changed into a
Gaussian lineshape.
Contrary to the first cas e of GaMnAs/GaAs the X-band linewidth varies monotonously
in the whole temperature region (fig.8). We observe a linewidth of ~600Oe at T=4K, which
decreases only slowly with temperature; the linewidth becomes minimal in the 100 K to 140K
range. The Curie temperature “s een” by the FMR spectroscopy is s lightly higher as compared
to the one measured by SQUID due to th e presence of the applied magnetic field.
At low temperature the Q-ba nd linewidth vary strongly w ith the orientation of the
applied field with values be tween 500Oe and 700Oe. The lowest value is observed for the
easy axis orientation. They decrease as at X-band only slowly with increasing temperature
and increase once again when approaching the Curie temperature. At Q-band the easy axis
FMR spectrum, which is also accompanied by a str ong spin wave spectrum at lower fields, is
observable in the whole temperature range.
For this sample we observe especially at Q-band a systematic difference between the cubic axes [100], [001] linewidth and the one for the in-plane [110] and [1-10] field
orientations (fig.8). The most surprising observation is that for temperatures below T<100K
the linewidth for H//[100] and H//[110] are co mparable at X-band and Q-band and thus an
analysis in the simple model discussed above is not possible. We attribute this to much higher
crystallographic/magnetic inhomogeneities, which mask the homogenous linewidth. The
origin of the strong inhomogeneity is still unclear. The only orientation for which in the whole temperature range a systematic increas e in the linewidth between X-and Q-band is
observed is the H//[1-10] orienta tion. We have thus analyzed th is variation (fig.10) according
to eq.1.
In spite of important differences in the lin ewidth the slope varies only weakly which
indicates that the inhomogeneous linewidth is very temperature dependent and decreases
monotonously with increasing temperature from 570Oe to 350Oe.
In the high temperature range (T ≥100K) the easy axis orientation could also be
analyzed (fig.11). The inhomogeneous linewidth are lower than for the hard axis orientation at
the same temperatures and are in the 300Oe range (fig.12). The homogenous linewidth at
9Ghz is in the 50Oe range which is close to the values determined in the first case of
GaMnAs/GaAs.
From the slope (fig.13) we obtain the da mping factor which for the hard axis
orientation is α=0.010 in the whole temperature range. Th is value is comparable to the one measured for the GaMnAs/GaAs film for H//[110]. The damping factor for the easy axis
orientation is lower but increases close to T C as in the previous case.
Discussion:
An estimation of the FMR intrinsic damping factor in a ferromagnetic GaMnAs thin
film has been made within a model of localiz ed Mn spins coupled by p-d kinetic exchange
with the itinerant-spin of holes treated by the 6-band Kohn-Luttinger Hamiltonian [5]. Note, that these authors take for the effective kinetic exchange field the value in the mean-
field approximation, i.e. H
eff=JN<S>, so that their calculation are made within the random
phase approximation (RPA). RPA calculations of α have been made by Heinrich et al. [14]
and have recently been used by Tserkovnyak et al .[15] for numerical app lications to the case
of Ga 0.95Mn 0.05As. Both models however, are phenomenological and include an
adjustable parameter: the quasiparticle lifetime Γ for the holes in [5] and the spin-flip
relaxation T 2 in [15]. These models do not take into account neither multi-magnon
scattering nor any damping beyond the RPA. It has been argued elsewhere [16], that in diluted
magnetic semiconductors such affects are only impor tant at high temperatur e (i.e. at T>Tc). In
particular, the increase of α in the vicinity of Tc may be attributed to such effects that are
beyond the scope of the models of references [5] and [14]. At low temperatures T<<Tc
however, where the corrections to the RPA are expected to be negligib le, the models of [5,
14,15] provide us with a numerical value of α in agreement with our experiments if we
introduce reasonable values of these parameters. For GaMnAs films with metallic
conductivity, Mn concentrations of x= 0.05 and hole concentr ations of 0.5 nm-3 (5x1020cm-3)
Sinova et al [5] predict an isotropi c low temperature damping factor α between 0.02 and 0.03
depending on the quasiparticle life time broa dening. Tserkovnyak et al [15] found a similar
value of α ≈ 0.01 for the isotropic damping factor fo r a typical GaMnAs film with 5% Mn
doping and full hole polarization.
Both predicted values are of the same orde r of magnitude as the experimental values
determined in this study. Our results for GaMn As/GaAs show further that the damping factor
is not isotropic as generally a ssumed but is anisotropic with a lowest value for the in-plane
easy axis orientations of the applied magnetic field H//[100], H//[1-10] and an increase of up
to a factor of two for the hardest axis orientat ion H// [001]. Intrinsic anisotropic damping is
related to the fact that the free energy density depends on the orientation of the magnetization which in the case of GaMnAs is related to th e anisotropy of the p-hole Fermi surface. We have shown (table I) that the anisotropy of the magnetocristalline constants and the related
fields are important in these strained layers and it is thus not surprisi ng to find also anisotropy
of the damping factor. For further discussions on this subject see reference [9]. The system
might also contain extrinsic an isotropies related to the pres ence of lattice defects. Their
influence can be deduced from the value and anisotropy of the inhomogeneous linewidth. In
the case of the compressively strained layers (G aMnAs/GaAs) we see that their value is small
and rather isotropic quite differen tly from the tensile strained film. It is in the first case that
our measurements show a factor of two anisot ropy of the Gilbert damping factors. A further
indication for the intrinsic charac ter of the anisotropy is the fact that the damping factor for
the perpendicular orientation has the highest value. In this case any contribution from two
magnon scattering will be minimized. Anyway, such contributions are generally only
important at low frequency measurements in the 2-6GHz range but even there they were
found to be negligible [7].
Additional material related parameters are expected to further influence the damping
factor. As the spin flip relaxation times will depend on the sample properties and in particular
the presence of scattering centers we will not expect to find a unique damping factor even for GaMnAs/GaAs samples with the same Mn composition x. More likely, different damping
factors are expected to be found in real films and their values might be used to assess the film
quality. In this sense the GaMnAs/GaAs film studi ed here is of course “better” than the one
on GaInAs in line with the strong difference in the sample inhomogeneities.
The inhomogeneous linewidth originates from spatial inhomogeneities in the local
magnetic anisotropy fields and inhomogeneities in the local exchange interactions. Given the particular growth conditions of these films, low temper ature molecular beam epitaxy,
inhomogeneities can not be expected to be neglig ible in these materials. If we had analyzed
our X-band results of the GaMnAs/GaAs film in the spirit of ref. [5], i.e. assuming a
negligible inhomoge neous linewidth - ( ∆H
inhom=0) -, we would have obtained artificially
increased damping factors. A further contribution might be expected from the intrinsic
disorder in these films: as GaMnAs is a diluted magnetic semiconductor with random
distribution of the Mn ions, this disorder will even for crysta llographically perfect crystals
give some importance to this term.
In the previous studies of the FMR damping factor in GaMnAs/GaAs single films
higher values have been reported. Matsuda et al [6] found damping factors between 0.02 and
0.06 in the T=10K to T=20K temperature range. They observed the same tend encies as in this
work concerning the anisotropy and temperature dependence of α: the lowest damping factor is seen for the easy axis orientation and its va lues increases with increasing temperatures. The
films of their study were however significantly different: (i) the Mn doping concentration was
lower, x=0.03 and thus the hole concentratio n was equally lower and (ii) the critical
temperature of the annealed film was only T=80K. The anisotropy in the inhomogeneous
linewidth at T=20K was equally much higher, varying between 30Oe for the easy axis to
250Oe for the hardest axis [110]. In a second study Sinova et al [5] have measured an
annealed GaMnAs/GaAs sample with a similar composition (x= 0.08) and critical temperature
(TC=130K) as the one studied here. They deduced a damping factor of the order of α=0.025
with only a slight temperature dependence betw een 4K and 80K and an increase close to T C.
However, these measurements were done at one (X-band) microwave frequency only and the
numerical value of α was obtained by assuming a neglig ible inhomogeneous linewidth. As
explained above, the value of α can be expected to be overestimated in this case.In the
photovoltage measurements of ref.7 the dampi ng factor of a low (x=0.02) doped GaMnAs
layer has been determined with microwave fr equencies from 2 to 19.6GHz but for one field
orientation H//[001] (h ard axis) and one temperature (T =9K) only. Interestingly, their
measurements show a linear behavior even in the low frequency range down to 4GHz which
demonstrates the negligible contribution from two magnon scattering in this case.
The intrinsic damping factor α plays also an important role in the critical currents
required to switch the magnetization in FM/NM/ FM trilayers [5]. However, in trilayer
structures interface and spin pumping effects wi ll add to the intrinsic damping factor of the
ferromagnetic material and give rise to an incr eased effective damping fa ctor. Sinova et al [5]
have estimated the critical current for real istic GaMnAs layers: based on a value of α= 0.02,
they estimated the critica l current density to J C=105A/cm2. It should be noted that the damping
factor involved in the domain wall motion [ 17, 18] is by definition different form the FMR
damping factor. Both are however linearly related with αFMR<αDW [17, 18]. First observations
of current induced magne tization switching in Ga 0.956Mn 0.044As/GaAs/Ga 0.967Mn 0.033As
trilayers confirm these theoretical predictions [19]. These authors observed a critical current
density of ≈ 105A/cm2 which would have been predicted from Slonczewski’s formula [20] for
a domain wall damping factor of αDW=0.002.
Conclusion:
We have determined the intrinsic FM R Gilbert damping factor for annealed
Ga0.93Mn 0.07As thin films with high critical temperatures. To evaluate the influence of the
strain the two prototype cases of compressive and tensile strained layers were studied. In
both cases we find an average damping factor of the order of 0.01. We thus see that the sign
of the strain does not seem to influence the damping factor strongly. The homogeneity of the
films as judged from the inhomogeneous linewid th is much higher in the case of GaAs
substrates than for GaInAs substrates. This must be attributed to the high dislocation density
in the GaInAs layer [13]. In the case of the GaMnAs/GaAs layers, where the small linewidth
allows a finer analysis of the data, we observe an anisotropy of the da mping factor, which has
the lowest value for the easy axis orientation. This value of α[1−10]=0.003 is an order of
magnitude lower than the previous reported va lues. The few experiment al results available
seem to indicate that the damping factor decr eases with increasing Mn concentration. This
corresponds well to the th eoretical predictions by Sinova et al [5] for the case of small quasi-
particle lifetime broadening. It wi ll be interesting to test this behavior further in more highly
doped layers (x ≈0.15), which become now available.
Acknowledgment: We thank Alain Mauger of the IMPMC laboratory of the University Paris
6 and Bret Heinrich from the Simon Frazer University in Vancouver for many helpful
discussions.
References :
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809 (2006).
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Lemaître, Phys.Rev. B77, 165204 (2008) .
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A.H.MacDonald, Phys.Rev. B69, 085209 (2004).
[6] Y.H.Matsuda, A.Oiwa, K.Ta naka, and H.Munekata, Physica B 376-377,668 (2006)
[7] A.Wirthmann, X.Hui, N.Mecking, Y.S.Gu i, T.Chakraborty, C.M.Hu, M.Reinwald,
C.Schüller, and W.Wegschneider, Appl.Phys.Lett.92, 232106 (2008)
[8] M. Farle, Rep.Prog.Phys. 61, 755 (1998). [9] B.Heinrich, « Spin relaxation in magnetic me tallic layers and multilayers », in Ultrathin
Magnetic Structures III, ed. by J.A.C.Bland, and B.Heinrich, Springe r Verlag (Berlin 2005),
p.143
[10] C.Chappert,K.LeDang,P.Beauvilain,H .Hurdequint, and D.Renard, Phys.Rev.B34 , 3192
(1986)
[11] F. Glas, G. Patriarche, L. Largeau, A. Lemaître, Phys. Rev. Lett., 93, 086107 (2004)
[12] L.Thevenard, L.Largeau,, O.Mauguin, A.Lemaitre, K.Khazen and H.J.von Bardeleben,
Phys. Rev.B75 ,195218 (2006)
[13] L. Thevenard, L. Largeau, O. Mauguin, G. Patriarche, A. Lemaître, N. Vernier and J. Ferré, Phys. Rev. B 73, 195331 (2006)
[14] B.Heinrich, D.Fraitova, V.Kambersky , Phys. Stat. Solidi 23, 501 (1967).
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[16] A.Mauger, D.L. Mills, Phys.Rev. B29, 3815 (1984)
[17] S.C.Chen and HL.Huang, I EEE Transactions on Magnetics 33, 3978 (1997)
[18] H.L.Huang, V.L.Sobolev, and S.C.Chen, J.Appl.Phys. 81, 4066(1997)
[19] D.Chiba, Y.Sato, T.Kita, F. Matsukura, and H.Ohno, Phys.Rev.Lett. 93, 216602(2004)
[20] J.C.Slonczewski, J.Magn.Magn.Mater. 159, L1(1996)
Figure Captions:
Figure 1a: X-band FMR spectrum of the GaMnAs /GaAs film taken at T=20K and for H//
[001]; the peak-to-peak linewidth is 60Oe. The experimental spectrum is shown by circles and
the Lorentzian lineshape simulation by a line.
Figure 1b: Q-band FMR spectrum of the GaMnAs /GaInAs film taken at T=80K and for H//
[100]; the peak to peak linewidth is 120Oe. The experimental spectrum is shown by circles
and the Lorentzian lineshape simulation by a line
Figure 2 (color on line): X-band peak-to-peak li ne widths for the GaMnAs/GaAs film for the
four main field orientations: H//[001] black s quares, H//[1-10] olive lower triangles, H//[110]
red circles, H//[100] blue upper triangles.
Figure 3 color on line): Q-band peak-to-peak line widths for the GaMnAs/GaAs film for the
four main field orientations: H//[001] black s quares, H//[1-10] olive lower triangles, H//[110]
red circles, H//[100] blue upper triangles.
Figure 4 (color on line): Peak-t o-peak linewidth at 9GHz and 35GHz for the GaMnAs/GaAs
film; T=80K and H//[001] black squares, H//[1-10] olive lowe r triangles, H//[110] red circles,
H//[100] blue upper triangles
Figure 5 (color on line): GaMnAs/GaAs inhomogeneous linewidth as a function of temperature for four orientations of the applie d field: H//[001] black squares, H//[1-10] olive
lower triangles, H//[110] red circles, H//[100] blue upper triangles
Figure 6 (color on line): damping factor α as a function of temperature and magnetic field
orientation; H//[001] black s quares, H//[1-10] olive lower triangles, H//[110] blue upper
triangles, H//[100] red circles; the maximum erro r in the determination of the linewidth is
estimated to 10G which co rresponds to an error in α of 0.001
Figure 7a: X-band FMR spectra for the GaMnAs /GaInAs film at 25K and H// [110] (hard
axis); the low field spin wave resonance (SW) is of high intensity in this case; circles:
experimental points, line: simulation with Gaussian lineshape
Figure 7b: Q-band FMR spectra for the GaMnAs /GaInAs film at 25K and H// [110] (hard
axis); the low field spin wave resonance (SW) is of high intensity in this case; circles: experimental points, line: simulation with Gaussian lineshape
Figure 8 (color on line): GaMnAs/GaInAs: X-band FMR linewidth as a function of
temperature for the four orientations of the applied magnetic field: H//[001] black squares,
H//[1-10] olive lower triangles, H//[110] red circ les, H//[100] blue upper triangles; the easy
axis FMR spectrum is not observable below T=100K ; a typical hysteresis curve as measured
by SQUID is shown in the inset.
Figure 9 (color on line): GaMnAs/GaInAs : Q-band FMR linewidth as a function of
temperature for the four orientations of the applied magnetic field : H//[001] black squares,
H//[1-10] olive lower triangl es, H//[110] red circles, H//[100] blue upper triangles
Figure 10 (color on line): GaMnAs/GaInAs : FM R linewidth as a function of microwave
frequency for H//[1-10] at different temper atures; T=10K (black squares), T=25K (red
circles), T=55K(black spades), T=80K( blue stars), T=115K (red triangles)
Figure 11 (color on line): GaMnAs/GaInAs: FMR linewidth as a function of microwave
frequency for H//[001] (easy ax is) at different temperatur es; T=100K (red circles), T=120K
(blue spades),T=130K(black squares)
Figure 12: GaMnAs/GaInAs inhomogeneous linewi dth as a function of temperature for two
orientations of the applied field: H//[001] squares, H//[1-10] lower triangles,
Figure 13 : GaMnAs/GaInAs damping factor α as a function of temperature and magnetic
field orientation; H//[001] squa res, H//[1-10] lower triangles.
Tables
Ga 0.93Mn 0.07As/GaAs G a0.93Mn 0.07As/Ga 0.902In0.098As
Ms(T=4K)= 47e mu/cm3 Ms(T=4K)= 38 e mu/cm3
TC =157K (SQUID) TC=130K (SQUID)
Anisotropy constants (T=80K) Anisotropy constants (T=55K)
K2⊥ = - 55000 e rg/cm3 K 2⊥ = +91070 e rg/cm3
K2// =2617 er g/cm3 K 2// = -2464 erg/cm3
K4⊥ = 8483 erg/cm3 K 4⊥ = -34050 erg/cm3
K2// =2590 er g/cm3 K 2// = -1873er g/cm3
Easy axis of magnetization Easy axis of magnet ization
[100] 4K<T<T C [001] 4K<T< T C
Table I
Table caption
Table I: M icrom agnetic param eters of the two sam ples studied in this work: s aturation
magnetization M s at T=4K, critical tem perature T C, magneto crystalline anisotropy constants
of second and fourth order K 2⊥ , K 2//, K 4⊥, K 4// at T=55K and T=80K respectively and the
orientation of the easy axis for m agnetization.
7500 7600 7700 7800 7900 8000SW
FMR Signal (arb.u.)
Magnetic Field H ( Oe)
Figure 1a
10000 1020 0 10400 1060 0 10800 1100 0
FMR Signal (arb.u.)
Magnetic Field H (Oe)
Figure 1b
0 20 40 60 80 100 120140 160050100150200250300Linew idth ∆H (Oe)
Tempera ture (K) 001
110
100 1-10
Figure 2
0 20 40 60 80 100 120 140160050100150200250300Linew idth ∆H (Oe)
Temperature ( K) 001
110
100 1-10
Figure 3
0 5 10 15 20 25 30 35 4004080120160200Linewi dth ( G)
Freq uency (GHz)
Figure 4
20 40 60 80 100 120 140 160020406080100120∆Hinhom (Oe)
Temperature (K) 001
110
100
1-10
Figure 5
0 20 40 60 80 100 120140 1600.0040.0060.0080.0100.0120.0140.0160.0180.020Damping factor α
Temperature (K ) 001
110
100
1-10
Figure 6
4000 5000 6000 7000 8000SW
FMR S ignal (arb. u.)
Magnetic Field (Oe )
Figure 7a
13000 14000 15000 16000 17000SW
FMR Signal (arb. u.)
Magnet ic Field (Oe)
Figure 7b
0 20 406 08 0 100120140300400500600700800
Linewidth ∆H (Oe)
Temperature (K) [110]
[100]
[1-10]
[001]-500 0 500-40040 M (emu/cm2)
Magnetic F ield (O e)
Figure 8
0 204 0 608 0 100 120 140300400500600700800
Linewidth ∆H (Oe)
Temperat ure (K ) [110]
[100]
[1-10]
[001]
Figure 9
0 5 10 15 20 25 30 35 40350400450500550600650700750
115K80K55K25K10K
Linewidth ∆H (Oe)
Freque ncy (GHz)
Figure 10
0 5 10 15 20 25 30 35 40200250300350400450500
100K120K130K
Linewidth ∆H (Oe)
Freque ncy (GHz)
Figure 11
0 204 0 608 0 100120140200300400500600
∆Hinhom (Oe)
Temperat ure (K ) [001]
[1-10]
Figure 12
0 204 06 0 80 100 120 1400.0040.0060.0080.0100.0120.0140.0160.0180.020Dam ping factor α
Temperature (K)
Figure 13
|
2311.16268v2.Gilbert_damping_in_two_dimensional_metallic_anti_ferromagnets.pdf | Gilbert damping in two-dimensional metallic anti-ferromagnets
R. J. Sokolewicz,1, 2M. Baglai,3I. A. Ado,1M. I. Katsnelson,1and M. Titov1
1Radboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, the Netherlands
2Qblox, Delftechpark 22, 2628 XH Delft, the Netherlands
3Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden
(Dated: March 29, 2024)
A finite spin life-time of conduction electrons may dominate Gilbert damping of two-dimensional
metallic anti-ferromagnets or anti-ferromagnet/metal heterostructures. We investigate the Gilbert
damping tensor for a typical low-energy model of a metallic anti-ferromagnet system with honeycomb
magnetic lattice and Rashba spin-orbit coupling for conduction electrons. We distinguish three
regimes of spin relaxation: exchange-dominated relaxation for weak spin-orbit coupling strength,
Elliot-Yafet relaxation for moderate spin-orbit coupling, and Dyakonov-Perel relaxation for strong
spin-orbit coupling. We show, however, that the latter regime takes place only for the in-plane
Gilbert damping component. We also show that anisotropy of Gilbert damping persists for any
finite spin-orbit interaction strength provided we consider no spatial variation of the N´ eel vector.
Isotropic Gilbert damping is restored only if the electron spin-orbit length is larger than the magnon
wavelength. Our theory applies to MnPS 3monolayer on Pt or to similar systems.
I. INTRODUCTION
Magnetization dynamics in anti-ferromagnets con-
tinue to attract a lot of attention in the context
of possible applications1–4. Various proposals utilize
the possibility of THz frequency switching of anti-
ferromagnetic domains for ultrafast information storage
and computation5,6. The rise of van der Waals magnets
has had a further impact on the field due to the pos-
sibility of creating tunable heterostructures that involve
anti-ferromagnet and semiconducting layers7.
Understanding relaxation of both the N´ eel vector and
non-equilibrium magnetization in anti-ferromagnets is
recognized to be of great importance for the function-
ality of spintronic devices8–13. On one hand, low Gilbert
damping must generally lead to better electric control of
magnetic order via domain wall motion or ultrafast do-
main switching14–16. On the other hand, an efficient con-
trol of magnetic domains must generally require a strong
coupling between charge and spin degrees of freedom due
to a strong spin-orbit interaction, that is widely thought
to be equivalent to strong Gilbert damping.
In this paper, we focus on a microscopic analysis of
Gilbert damping due to Dyakonov-Perel and Elliot-Yafet
mechanisms. We apply the theory to a model of a two-
dimensional N´ eel anti-ferromagnet with a honeycomb
magnetic lattice.
Two-dimensional magnets typically exhibit either
easy-plane or easy-axis anisotropy, and play crucial
roles in stabilizing magnetism at finite temperatures17,18.
Easy-axis anisotropy selects a specific direction for mag-
netization, thereby defining an axis for the magnetic or-
der. In contrast, easy-plane anisotropy does not select a
particular in-plane direction for the N´ eel vector, allowing
it to freely rotate within the plane. This situation is anal-
ogous to the XY model, where the system’s continuous
symmetry leads to the suppression of out-of-plane fluc-
tuations rather than fixing the magnetization in a spe-
cific in-plane direction19,20. Without this anisotropy, themagnonic fluctuations in a two-dimensional crystal can
grow uncontrollably large to destroy any long-range mag-
netic order, according to the Mermin-Wagner theorem21.
Recent density-functional-theory calculations for
single-layer transition metal trichalgenides22, predict the
existence of a large number of metallic anti-ferromagnets
with honeycomb lattice and different types of magnetic
order as shown in Fig. 1. Many of these crystals may
have the N´ eel magnetic order as shown in Fig. 1a and are
metallic: FeSiSe 3, FeSiTe 3, VGeTe 3, MnGeS 3, FeGeSe 3,
FeGeTe 3, NiGeSe 3, MnSnS 3, MnSnS 3, MnSnSe 3,
FeSnSe 3, NiSnS 3. Apart from that it has been predicted
that anti-ferromagnetism can be induced in graphene by
bringing it in proximity to MnPSe 323or by bringing it
in double proximity between a layer of Cr 2Ge2Te6and
WS224.
Partly inspired by these predictions and recent
technological advances in producing single-layer anti-
ferromagnet crystals, we propose an effective model to
study spin relaxation in 2D honeycomb anti-ferromagnet
with N´ eel magnetic order. The same system was studied
by us in Ref. 25, where we found that spin-orbit cou-
pling introduces a weak anisotropy in spin-orbit torque
and electric conductivity. Strong spin-orbit coupling was
shown to lead to a giant anisotropy of Gilbert damping.
Our analysis below is built upon the results of Ref. 25,
and we investigate and identify three separate regimes
of spin-orbit strength. Each regime is characterized by
qualitatively different dependence of Gilbert damping on
spin-orbit interaction and conduction electron transport
time. The regime of weak spin-orbit interaction is dom-
inated by exchange field relaxation of electron spin, and
the regime of moderate spin-orbit strength is dominated
by Elliot-Yafet spin relaxation. These two regimes are
characterized also by a universal factor of 2 anisotropy
of Gilbert damping. The regime of strong spin-orbit
strength, which leads to substantial splitting of electron
Fermi surfaces, is characterized by Dyakonov-Perel relax-
ation of the in-plane spin component and Elliot-Yafet re-arXiv:2311.16268v2 [cond-mat.dis-nn] 28 Mar 20242
FIG. 1. Three anti-ferromagnetic phases commonly found
among van-der-Waals magnets. Left-to-right: N´ eel, zig-zag,
and stripy.
laxation of the perpendicular-to-the-plane Gilbert damp-
ing which leads to a giant damping anisotropy. Isotropic
Gilbert damping is restored only for finite magnon wave
vectors such that the magnon wavelength is smaller than
the spin-orbit length.
Gilbert damping in a metallic anti-ferromagnet can be
qualitatively understood in terms of the Fermi surface
breathing26. A change in the magnetization direction
gives rise to a change in the Fermi surface to which the
conduction electrons have to adjust. This electronic re-
configuration is achieved through the scattering of elec-
trons off impurities, during which angular momentum is
transferred to the lattice. Gilbert damping, then, should
be proportional to both (i) the ratio of the spin life-time
and momentum life-time of conduction electrons, and (ii)
the electric conductivity. Keeping in mind that the con-
ductivity itself is proportional to momentum life-time,
one may conclude that the Gilbert damping is linearly
proportional to the spin life-time of conduction electrons.
At the same time, the spin life-time of localized spins is
inversely proportional to the spin life-time of conduc-
tion electrons. A similar relation between the spin life-
times of conduction and localized electrons also holds
for relaxation mechanisms that involve electron-magnon
scattering27.
Our approach formally decomposes the magnetic sys-
tem into a classical sub-system of localized magnetic mo-
ments and a quasi-classical subsystem of conduction elec-
trons. A local magnetic exchange couples these sub-
systems. Localized magnetic moments in transition-
metal chalcogenides and halides form a hexagonal lat-
tice. Here we focus on the N´ eel type anti-ferromagnet
that is illustrated in Fig. 1a. In this case, one can de-
fine two sub-lattices A and B that host local magnetic
moments SAandSB, respectively. For the discussion of
Gilbert damping, we ignore the weak dependence of both
fields on atomic positions and assume that the modulus
S=|SA(B)|is time-independent.
Under these assumptions, the magnetization dynamics
of localized moments may be described in terms of two
fields
m=1
2S
SA+SB
,n=1
2S
SA−SB
, (1)
which are referred to as the magnetization and staggeredmagnetization (or N´ eel vector), respectively. Within the
mean-field approach, the vector fields yield the equations
of motion
˙n=−Jn×m+n×δs++m×δs−, (2a)
˙m=m×δs++n×δs−, (2b)
where dot stands for the time derivative, while δs+and
δs−stand for the mean staggered and non-staggered non-
equilibrium fields that are proportional to the variation of
the corresponding spin-densities of conduction electrons
caused by the time dynamics of nandmfields. The en-
ergy Jis proportional to the anti-ferromagnet exchange
energy for localized momenta.
In Eqs. (2) we have omitted terms that are propor-
tional to easy axis anisotropy for the sake of compact-
ness. These terms are, however, important and will be
introduced later in the text.
In the framework of Eqs. (2) the Gilbert damping can
be computed as the linear response of the electron spin-
density variation to a time change in both the magneti-
zation and the N´ eel vector (see e. g. Refs.25,28,29).
In this definition, Gilbert damping describes the re-
laxation of localized spins by transferring both total and
staggered angular momenta to the lattice by means of
conduction electron scattering off impurities. Such a
transfer is facilitated by spin-orbit interaction.
The structure of the full Gilbert damping tensor can be
rather complicated as discussed in Ref. 25. However, by
taking into account easy axis or easy plane anisotropy we
may reduce the complexity of relevant spin configurations
to parameterize
δs+=α∥
m˙m∥+α⊥
m˙m⊥+αmn∥×(n∥×˙m∥),(3a)
δs−=α∥
n˙n∥+α⊥
n˙n⊥+αnn∥×(n∥×˙n∥), (3b)
where the superscripts ∥and⊥refer to the in-plane
and perpendicular-to-the-plane projections of the corre-
sponding vectors, respectively. The six coefficients α∥
m,
α⊥
m,αm,α∥
n,α⊥
n, and αnparameterize the Gilbert damp-
ing.
Inserting Eqs. (3) into the equations of motion of
Eqs. (2) produces familiar Gilbert damping terms. The
damping proportional to time-derivatives of the N´ eel vec-
tornis in general many orders of magnitude smaller than
that proportional to the time-derivatives of the magneti-
zation vector m25,30. Due to the same reason, the higher
harmonics term αmn∥×(n∥×∂tm∥) can often be ne-
glected.
Thus, in the discussion below we may focus mostly on
the coefficients α∥
mandα⊥
mthat play the most important
role in the magnetization dynamics of our system. The
terms proportional to the time-derivative of ncorrespond
to the transfer of angular momentum between the sub-
lattices and are usually less relevant. We refer to the
results of Ref. 25 when discussing these terms.
All Gilbert damping coefficients are intimately related
to the electron spin relaxation time. The latter is rel-
atively well understood in non-magnetic semiconductors3
with spin-orbital coupling. When a conducting electron
moves in a steep potential it feels an effective magnetic
field caused by relativistic effects. Thus, in a disordered
system, the electron spin is subject to a random magnetic
field each time it scatters off an impurity. At the same
time, an electron also experiences precession around an
effective spin-orbit field when it moves in between the
collisions. Changes in spin direction between collisions
are referred to as Dyakonov-Perel relaxation31,32, while
changes in spin-direction during collisions are referred to
as Elliot-Yafet relaxation33,34.
The spin-orbit field in semiconductors induces a char-
acteristic frequency of spin precession Ω s, while scalar
disorder leads to a finite transport time τof the con-
ducting electrons. One may, then, distinguish two limits:
(i) Ω sτ≪1 in which case the electron does not have
sufficient time to change its direction between consec-
utive scattering events (Elliot-Yafet relaxation), and (ii)
Ωsτ≫1 in which case the electron spin has multiple pre-
cession cycles in between the collisions (Dyakonov-Perel
relaxation).
The corresponding processes define the so-called spin
relaxation time, τs. In a 2D system the spin life-time
τ∥
s, for the in-plane spin components, appears to be dou-
ble the size of the life-time of the spin component that
is perpendicular to the plane, τ⊥
s32. This geometric ef-
fect has largely been overlooked. For non-magnetic 2D
semiconductor one can estimate35,36
1
τ∥
s∼(
Ω2
sτ,Ωsτ≪1
1/τ, Ωsτ≫1, τ∥
s= 2τ⊥
s. (4)
A pedagogical derivation and discussion of Eq. 4 can
be found in Refs. 35 and 36. Because electrons are con-
fined in two dimensions the random spin-orbit field is
always directed in-plane, which leads to a decrease in the
in-plane spin-relaxation rate by a factor of two compared
to the out-of-plane spin-relaxation rate as demonstrated
first in Ref. 32 (see Refs. 36–40 as well). The reason is
that the perpendicular-to-the-plane component of spin is
influenced by two components of the randomly changing
magnetic field, i. e. xandy, whereas the parallel-to-the-
plane spin components are only influenced by a single
component of the fluctuating fields, i. e. the xspin pro-
jection is influenced only by the ycomponent of the field
and vice-versa. The argument has been further general-
ized in Ref. 25 to the case of strongly separated spin-orbit
split Fermi surfaces. In this limit, the perpendicular-to-
the-plane spin-flip processes on scalar disorder potential
become fully suppressed. As a result, the perpendicular-
to-the-plane spin component becomes nearly conserved,
which results in a giant anisotropy of Gilbert damping in
this regime.
In magnetic systems that are, at the same time, con-
ducting there appears to be at least one additional energy
scale, ∆ sd, that characterizes exchange coupling of con-
duction electron spin to the average magnetic moment of
localized electrons. (In the case of s-d model descriptionit is the magnetic exchange between the spin of conduc-
tionselectron and the localized magnetic moment of d
orfelectron on an atom.) This additional energy scale
complicates the simple picture of Eq. (4) especially in the
case of an anti-ferromagnet. The electron spin precession
is now defined not only by spin-orbit field but also by
∆sd. As the result the conditions Ω sτ≪1 and ∆ sdτ≫1
may easily coexist. This dissolves the distinction between
Elliot-Yafet and Dyakonov-Perel mechanisms of spin re-
laxation. One may, therefore, say that both Elliot-Yafet
and Dyakonov-Perel mechanisms may act simultaneously
in a typical 2D metallic magnet with spin-orbit coupling.
The Gilbert damping computed from the microscopic
model that we formulate below will always contain both
contributions to spin-relaxation.
II. MICROSCOPIC MODEL AND RESULTS
The microscopic model that we employ to calculate
Gilbert damping is the so-called s–dmodel that couples
localized magnetic momenta SAandSBand conducting
electron spins via the local magnetic exchange ∆ sd. Our
effective low-energy Hamiltonian for conduction electrons
reads
H=vfp·Σ+λ
2
σ×Σ
z−∆sdn·σΣzΛz+V(r),(5)
where the vectors Σ,σandΛdenote the vectors of Pauli
matrices acting on sub-lattice, spin and valley space,
respectively. We also introduce the Fermi velocity vf,
Rashba-type spin-orbit interaction λ, and a random im-
purity potential V(r).
The Hamiltonian of Eq. (5) can be viewed as the
graphene electronic model where conduction electrons
have 2D Rashba spin-orbit coupling and are also cou-
pled to anti-ferromagnetically ordered classical spins on
the honeycomb lattice.
The coefficients α∥
mandα⊥
mare obtained using linear
response theory for the response of spin-density δs+to
the time-derivative of magnetization vector ∂tm. Impu-
rity potential V(r) is important for describing momen-
tum relaxation to the lattice. This is related to the an-
gular momentum relaxation due to spin-orbit coupling.
The effect of random impurity potential is treated pertur-
batively in the (diffusive) ladder approximation that in-
volves a summation over diffusion ladder diagrams. The
details of the microscopic calculation can be found in the
Appendices.
Before presenting the disorder-averaged quantities
α∥,⊥
m, it is instructive to consider first the contribution
to Gilbert damping originating from a small number of
electron-impurity collisions. This clarifies how the num-
ber of impurity scattering effects will affect the final re-
sult.
Let us annotate the Gilbert damping coefficients with
an additional superscript ( l) that denotes the number
of scattering events that are taken into account. This4
01234 (i)
?[" ]
(0)
? (1)
? (2)
? (1)
?
10 210 1100101
01234 (i)
k[" ]
(0)
k (1)
k (2)
k (1)
k
FIG. 2. Gilbert damping in the limit ∆ sd= 0. Dotted (green)
lines correspond to the results of the numerical evaluation of
¯α(l)
m,⊥,∥forl= 0,1,2 as a function of the parameter λτ. The
dashed (orange) line corresponds to the diffusive (fully vertex
corrected) results for ¯ α⊥,∥.
m.
means, in the diagrammatic language, that the corre-
sponding quantity is obtained by summing up the ladder
diagrams with ≤ldisorder lines. Each disorder line cor-
responds to a quasi-classical scattering event from a sin-
gle impurity. The corresponding Gilbert damping coeffi-
cient is, therefore, obtained in the approximation where
conduction electrons have scattered at most lnumber
of times before releasing their non-equilibrium magnetic
moment into a lattice.
To make final expressions compact we define the di-
mensionless Gilbert damping coefficients ¯ α∥,⊥
mby extract-
ing the scaling factor
α∥,⊥
m=A∆2
sd
πℏ2v2
fS¯α∥,⊥
m, (6)
where Ais the area of the unit cell, vfis the Fermi ve-
locity of the conducting electrons and ℏ=h/2πis the
Planck’s constant. We also express the momentum scat-
tering time τin inverse energy units, τ→ℏτ.
Let us start by computing the coefficients ¯ α∥,⊥(l)
m in the
formal limit ∆ sd→0. We can start with the “bare bub-
ble” contribution which describes spin relaxation without
a single scattering event. The corresponding results read
¯α(0)
m,⊥=ετ1−λ2/4ε2
1 +λ2τ2, (7a)
¯α(0)
m,∥=ετ1 +λ2τ2/2
1 +λ2τ2−λ2
8ε2
, (7b)
where εdenotes the Fermi energy which we consider pos-
itive (electron-doped system).In all realistic cases, we have to consider λ/ε≪1,
while the parameter λτmay in principle be arbitrary. For
λτ≪1 the disorder-induced broadening of the electron
Fermi surfaces exceeds the spin-orbit induced splitting.
In this case one basically finds no anisotropy of “bare”
damping: ¯ α(0)
m,⊥= ¯α(0)
m,∥. In the opposite limit of substan-
tial spin-orbit splitting one gets an ultimately anisotropic
damping ¯ α(0)
m,⊥≪¯α(0)
m,∥. This asymptotic behavior can be
summarized as
¯α(0)
m,⊥=ετ(
1 λτ≪1,
(λτ)−2λτ≫1,(8a)
¯α(0)
m,∥=ετ(
1 λτ≪1,
1
2
1 + (λτ)−2
λτ≫1,(8b)
where we have used that ε≫λ.
The results of Eq. (8) modify by electron diffusion. By
taking into account up to lscattering events we obtain
¯α(l)
m,⊥=ετ(
l+O(λ2τ2) λτ≪1,
(1 +δl0)/(λτ)2λτ≫1,(9a)
¯α(l)
m,∥=ετ(
l+O(λ2τ2) λτ≪1,
1−(1/2)l+1+O((λτ)−2)λτ≫1,(9b)
where we have used ε≫λagain.
From Eqs. (9) we see that the Gilbert damping for
λτ≪1 gets an additional contribution of ετfrom each
scattering event as illustrated numerically in Fig. 2. This
leads to a formal divergence of Gilbert damping in the
limit λτ≪1. While, at first glance, the divergence looks
like a strong sensitivity of damping to impurity scatter-
ing, in reality, it simply reflects a diverging spin life-time.
Once a non-equilibrium magnetization mis created it
becomes almost impossible to relax it to the lattice in
the limit of weak spin-orbit coupling. The formal diver-
gence of α⊥
m=α∥
msimply reflects the conservation law
for electron spin polarization in the absence of spin-orbit
coupling such that the corresponding spin life-time be-
comes arbitrarily large as compared to the momentum
scattering time τ.
By taking the limit l→ ∞ (i. e. by summing up the
entire diffusion ladder) we obtain compact expressions
¯α⊥
m≡¯α(∞)
m,⊥=ετ1
2λ2τ2, (10a)
¯α∥
m≡¯α(∞)
m,∥=ετ1 +λ2τ2
λ2τ2, (10b)
which assume ¯ α⊥
m≪¯α∥
mforλτ≫1 and ¯ α⊥
m= ¯α∥
m/2
forλτ≪1. The factor of 2 difference that we observe
when λτ≪1, corresponds to a difference in the elec-
tron spin life-times τ⊥
s=τ∥
s/2 that was discussed in the
introduction32.
Strong spin-orbit coupling causes a strong out-of-plane
anisotropy of damping, ¯ α⊥
m≪¯α∥
mwhich corresponds to5
a suppression of the perpendicular-to-the-plane damping
component. As a result, the spin-orbit interaction makes
it much easier to relax the magnitude of the mzcompo-
nent of magnetization than that of in-plane components.
Let us now turn to the dependence of ¯ αmcoefficients on
∆sdthat is illustrated numerically in Fig. 3. We consider
first the case of absent spin-orbit coupling λ= 0. In
this case, the combination of spin-rotational and sub-
lattice symmetry (the equivalence of A and B sub-lattice)
must make Gilbert damping isotropic (see e. g.25,41). The
direct calculation for λ= 0 does, indeed, give rise to the
isotropic result ¯ α⊥
m= ¯α∥
m=ετ(ε2+∆2
sd)/2∆2
sd, which is,
however, in contradiction to the limit λ→0 in Eq. (10).
At first glance, this contradiction suggests the exis-
tence of a certain energy scale for λover which the
anisotropy emerges. The numerical analysis illustrated
in Fig. 4 reveals that this scale does not depend on the
values of 1 /τ, ∆sd, orε. Instead, it is defined solely by
numerical precision. In other words, an isotropic Gilbert
damping is obtained only when the spin-orbit strength
λis set below the numerical precision in our model.
We should, therefore, conclude that the transition from
isotropic to anisotropic (factor of 2) damping occurs ex-
actly at λ= 0. Interestingly, the factor of 2 anisotropy is
absent in Eqs. (8) and emerges only in the diffusive limit.
We will see below that this paradox can only be re-
solved by analyzing the Gilbert damping beyond the in-
finite wave-length limit.
One can see from Fig. 3 that the main effect of finite
∆sdis the regularization of the Gilbert damping diver-
gency ( λτ)−2in the limit λτ≪1. Indeed, the limit of
weak spin-orbit coupling is non-perturbative for ∆ sd/ε≪
λτ≪1, while, in the opposite limit, λτ≪∆sd/ε≪1,
the results of Eqs. (10) are no longer valid. Assuming
∆sd/ε≪1 we obtain the asymptotic expressions for the
results presented in Fig. 3 as
¯α⊥
m=1
2ετ(2
3ε2+∆2
sd
∆2
sdλτ≪∆sd/ε,
1
λ2τ2 λτ≫∆sd/ε,(11a)
¯α∥
m=ετ(2
3ε2+∆2
sd
∆2
sdλτ≪∆sd/ε,
1 +1
λ2τ2λτ≫∆sd/ε,(11b)
which suggest that ¯ α⊥
m/¯α∥
m= 2 for λτ≪1. In the op-
posite limit, λτ≫1, the anisotropy of Gilbert damping
grows as ¯ α∥
m/¯α⊥
m= 2λ2τ2.
The results of Eqs. (11) can also be discussed in terms
of the electron spin life-time, τ⊥(∥)
s = ¯α⊥(∥)
m/ε. For the
inverse in-plane spin life-time we find
1
τ∥
s=
3∆2
sd/2ε2τ λτ ≪∆sd/ε,
λ2τ ∆sd/ε≪λτ≪1,
1/τ 1≪λτ,(12)
that, for ∆ sd= 0, is equivalent to the known result of
Eq. (4). Indeed, for ∆ sd= 0, the magnetic exchange
10 310 210 1100101
10 1101103105 m;k;?[" ]
sd="= 0:1sd="= 0 m;k
m;?FIG. 3. Numerical results for the Gilbert damping compo-
nents in the diffusive limit (vertex corrected)as the function
of the spin-orbit coupling strength λ. The results correspond
toετ= 50 and ∆ sdτ= 0.1 and agree with the asymptotic
expressions of Eq. (11). Three different regimes can be dis-
tinguished for ¯ α∥
m: i) spin-orbit independent damping ¯ α∥
m∝
ε3τ/∆2
sdfor the exchange dominated regime, λτ≪∆sd/ε, ii)
the damping ¯ α∥
m∝ε/λ2τfor Elliot-Yafet relaxation regime,
∆sd/ε≪λτ≪1, and iii) the damping ¯ α∥
m∝ετfor the
Dyakonov-Perel relaxation regime, λτ≫1. The latter regime
is manifestly absent for ¯ α⊥
min accordance with Eqs. (12,13).
plays no role and one observes the cross-over from Elliot-
Yafet ( λτ≪1) to Dyakonov-Perel ( λτ≫1) spin relax-
ation.
This cross-over is, however, absent in the relaxation of
the perpendicular spin component
1
τ⊥s= 2(
3∆2
sd/2ε2τ λτ ≪∆sd/ε,
λ2τ ∆sd/ε≪λτ,(13)
where Elliot-Yafet-like relaxation extends to the regime
λτ≫1.
As mentioned above, the factor of two anisotropy in
spin-relaxation of 2 Dsystems, τ∥
s= 2τ⊥
s, is known in the
literature32(see Refs.36–38as well). Unlimited growth of
spin life-time anisotropy, τ∥
s/τ⊥
s= 2λ2τ2, in the regime
λτ≪1 has been described first in Ref. 25. It can be qual-
itatively explained by a strong suppression of spin-flip
processes for zspin component due to spin-orbit induced
splitting of Fermi surfaces. The mechanism is effective
only for scalar (non-magnetic) disorder. Even though
such a mechanism is general for any magnetic or non-
magnetic 2D material with Rashba-type spin-orbit cou-
pling, the effect of the spin life-time anisotropy on Gilbert
damping is much more relevant for anti-ferromagnets. In-
deed, in an anti-ferromagnetic system the modulus of m
is, by no means, conserved, hence the variations of per-
pendicular and parallel components of the magnetization
vector are no longer related.
In the regime, λτ≪∆sd/εthe spin life-time is de-
fined by exchange interaction and the distinction between
Dyakonov-Perel and Elliot-Yafet mechanisms of spin re-
laxation is no longer relevant. In this regime, the spin-
relaxation time is by a factor ( ε/∆sd)2larger than the
momentum relaxation time.
Let us now return to the problem of emergency of the6
10 6410 5410 4410 3410 2410 14
12 k= ?n= 32
n= 64n= 96
n= 128
FIG. 4. Numerical evaluation of Gilbert damping anisotropy
in the limit λ→0. Isotropic damping tensor is restored only
ifλ= 0 with ultimate numerical precision. The factor of 2
anisotropy emerges at any finite λ, no matter how small it
is, and only depends on the numerical precision n, i.e. the
number of digits contained in each variable during computa-
tion. The crossover from isotropic to anisotropic damping can
be understood only by considering finite, though vanishingly
small, magnon qvectors.
factor of 2 anisotropy of Gilbert damping at λ= 0. We
have seen above (see Fig. 4) that, surprisingly, there ex-
ists no energy scale for the anisotropy to emerge. The
transition from the isotropic limit ( λ= 0) to a finite
anisotropy appeared to take place exactly at λ= 0. We
can, however, generalize the concept of Gilbert damping
by considering the spin density response function at a
finite wave vector q.
To generalize the Gilbert damping, we are seeking a
response of spin density at a point r,δs+(r) to a time
derivative of magnetization vectors ˙m∥and ˙m⊥at the
point r′. The Fourier transform with respect to r−r′
gives the Gilbert damping for a magnon with the wave-
vector q.
The generalization to a finite q-vector shows that the
limits λ→0 and q→0 cannot be interchanged. When
the limit λ→0 is taken before the limit q→0 one
finds an isotropic Gilbert damping, while for the oppo-
site order of limits, it becomes a factor of 2 anisotropic.
In a realistic situation, the value of qis limited from
below by an inverse size of a typical magnetic domain
1/Lm, while the spin-orbit coupling is effective on the
length scale Lλ= 2πℏvf/λ. In this picture, the isotropic
Gilbert damping is characteristic for the case of suffi-
ciently small domain size Lm≪Lλ, while the anisotropic
Gilbert damping corresponds to the case Lλ≪Lm.
In the limit qℓ≪1, where ℓ=vfτis the electron mean
2 0 2
k[a.u.] 2:50:02:5energy [a.u.]=sd= 4
2 0 2
k[a.u.]=sd= 2
2 0 2
k[a.u.]=sd= 1FIG. 5. Band-structure for the effective model of Eq. (5)
in a vicinity of Kvalley assuming nz= 1. Electron bands
touch for λ= 2∆ sd. The regime λ≤2∆sdcorresponds to
spin-orbit band inversion. The band structure in the valley
K′is inverted. Our microscopic analysis is performed in the
electron-doped regime for the Fermi energy above the gap as
illustrated by the top dashed line. The bottom dashed line
denotes zero energy (half-filling).
free path, we can summarize our results as
¯α⊥
m=ετ
ε2+∆2
sd
2∆2
sdλτ≪qℓ≪∆sd/ε,
1
3ε2+∆2
sd
∆2
sdqℓ≪λτ≪∆sd/ε,
1
2λ2τ2 λτ≫∆sd/ε,, (14a)
¯α∥
m=ετ
ε2+∆2
sd
2∆2
sdλτ≪qℓ≪∆sd/ε,
2
3ε2+∆2
sd
∆2
sdqℓ≪λτ≪∆sd/ε,
1 +1
λ2τ2λτ≫∆sd/ε,(14b)
which represent a simple generalization of Eqs. (11).
The results of Eqs. (14) correspond to a simple behav-
ior of Gilbert damping anisotropy,
¯α∥
m/¯α⊥
m=(
1 λτ≪qℓ,
2
1 +λ2τ2
qℓ≪λτ,(15)
where we still assume qℓ≪1.
III. ANTI-FERROMAGNETIC RESONANCE
The broadening of the anti-ferromagnet resonance
peak is one obvious quantity that is sensitive to Gilbert
damping. The broadening is however not solely defined
by a particular Gilbert damping component but depends
also on both magnetic anisotropy and anti-ferromagnetic
exchange.
To be more consistent we can use the model of Eq. (5)
to analyze the contribution of conduction electrons to an
easy axis anisotropy. The latter is obtained by expanding
the free energy for electrons in the value of nz, which has
a form E=−Kn2
z/2. With the conditions ε/λ≫1 and
ε/∆sd≫1 we obtain the anisotropy constant as
K=A
2πℏ2v2(
∆2
sdλ 2∆sd/λ≤1,
∆sdλ2/2 2∆ sd/λ≥1,(16)7
where Ais the area of the unit cell. Here we assume
both λand ∆ sdpositive, therefore, the model natu-
rally gives rise to an easy axis anisotropy with K > 0.
In real materials, there exist other sources of easy axis
or easy plane anisotropy. In-plane magneto-crystalline
anisotropy also plays an important role. For example,
N´ eel-type anti-ferromagnets with easy-axis anisotropy
are FePS 3, FePSe 3or MnPS 3, whereas those with easy
plane and in-plane magneto-crystalline anisotropy are
NiPS 3and MnPSe 3. Many of those materials are, how-
ever, Mott insulators. Our qualitative theory may still
apply to materials like MnPS 3monolayers at strong elec-
tron doping.
The transition from 2∆ sd/λ≥1 to 2∆ sd/λ≤1 in
Eq. (16) corresponds to the touching of two bands in the
model of Eq. (5) as illustrated in Fig. 5.
Anti-ferromagnetic magnon frequency and life-time in
the limit q→0 are readily obtained by linearizing the
equations of motion
˙n=−Jn×m+Km×n⊥+n×(ˆαm˙m), (17a)
˙m=Kn×n⊥+n×(ˆαn˙n), (17b)
where we took into account easy axis anisotropy Kand
disregarded irrelevant terms m×(ˆαn˙n) and m×(ˆαm˙m).
We have also defined Gilbert damping tensors such as
ˆαm˙m=α∥
m˙m∥+α⊥
m˙m⊥, ˆαn˙n=α∥
n˙n∥+α⊥
n˙n⊥.
In the case of easy axis anisotropy we can use the lin-
earized modes n=ˆz+δn∥eiωt,m=δm∥eiωt, hence we
get the energy of q= 0 magnon as
ω=ω0−iΓ/2, (18)
ω0=√
JK, Γ =Jα∥
n+Kα∥
m (19)
where we took into account that K≪J. The expression
forω0is well known due to Kittel and Keffer42,43.
Using Ref. 25 we find out that α∥
n≃α⊥
m(λ/ε)2and
α⊥
n≃α∥
m(λ/ε)2, hence
Γ≃α∥
m
K+J/2
ε2/λ2+ε2τ2
, (20)
where we have simply used Eqs. (10). Thus, one may
often ignore the contribution Jα∥
nas compared to Kα∥
m
despite the fact that K≪J.
In the context of anti-ferromagnets, spin-pumping
terms are usually associated with the coefficients α∥
nin
Eq. (3b) that are not in the focus of the present study.
Those coefficients have been analyzed for example in Ref.
25. In this manuscript we simply use the known results
forαnin Eqs. (17-19), where we illustrate the effect of
both spin-pumping coefficient αnand the direct Gilbert
damping αmon the magnon life time. One can see from
Eqs. (19,20) that the spin-pumping contributions do also
contribute, though indirectly, to the magnon decay. The
spin pumping contributions become more important in
magnetic materials with small magnetic anisotropy. The
processes characterized by the coefficients αnmay also be
10 310 210 1100101
0:000:010:021= k
m="= 0:04
="= 0:02
="= 0:01FIG. 6. Numerical evaluation of the inverse Gilbert damping
1/¯α∥
mas a function of the momentum relaxation time τ. The
inverse damping is peaked at τ∝1/λwhich also corresponds
to the maximum of the anti-ferromagnetic resonance quality
factor in accordance with Eq. (21).
interpreted in terms of angular momentum transfer from
one AFM sub-lattice to another. In that respect, the spin
pumping is specific to AFM, and is qualitatively differ-
ent from the direct Gilbert damping processes ( αm) that
describe the direct momentum relaxation to the lattice.
As illustrated in Fig. 6 the quality factor of the anti-
ferromagnetic resonance (for a metallic anti-ferromagnet
with easy-axis anisotropy) is given by
Q=ω0
Γ≃1
α∥
mr
J
K. (21)
Interestingly, the quality factor defined by Eq. (21) is
maximized for λτ≃1, i. e. for the electron spin-orbit
length being of the order of the scattering mean free path.
The quantities 1 /√
Kand 1 /¯α∥
mare illustrated in
Fig. 6 from the numerical analysis. As one would ex-
pect, the quality factor vanishes in both limits λ→0
andλ→ ∞ . The former limit corresponds to an over-
damped regime hence no resonance can be observed. The
latter limit corresponds to a constant α∥
m, but the reso-
nance width Γ grows faster with λthan ω0does, hence
the vanishing quality factor.
It is straightforward to check that the results of
Eqs. (20,21) remain consistent when considering systems
with either easy-plane or in-plane magneto-crystalline
anisotropy. Thus, the coefficient α⊥
mnormally does not
enter the magnon damping, unless the system is brought
into a vicinity of spin-flop transition by a strong external
field.
IV. CONCLUSION
In conclusion, we have analyzed the Gilbert damping
tensor in a model of a two-dimensional anti-ferromagnet
on a honeycomb lattice. We consider the damping mech-
anism that is dominated by a finite electron spin life-time8
due to a combination of spin-orbit coupling and impu-
rity scattering of conduction electrons. In the case of a
2D electron system with Rashba spin-orbit coupling λ,
the Gilbert damping tensor is characterized by two com-
ponents α∥
mandα⊥
m. We show that the anisotropy of
Gilbert damping depends crucially on the parameter λτ,
where τis the transport scattering time for conduction
electrons. For λτ≪1 the anisotropy is set by a geo-
metric factor of 2, α∥
m= 2α⊥
m, while it becomes infinitely
large in the opposite limit, α∥
m= (λτ)2α⊥
mforλτ≫1.
Gilbert damping becomes isotropic exactly for λ= 0, or,
strictly speaking, for the case λ≪ℏvfq, where qis the
magnon wave vector.
This factor of 2 is essentially universal, and is a geomet-
ric effect: the z-component relaxation results from fluctu-
ations in two in-plane spin components, whereas in-plane
relaxation stems from fluctuations of the z-component
alone. This reflects the subtleties of our microscopic
model, where the mechanism for damping is activated
by the decay of conduction electron momenta, linked to
spin-relaxation through spin-orbit interactions.
We find that Gilbert damping is insensitive to mag-
netic order for λ≫∆sd/ετ, where ∆ sdis an effective
exchange coupling between spins of conduction and local-
ized electrons. In this case, the electron spin relaxation
can be either dominated by scattering (Dyakonov-Perel
relaxation) or by spin-orbit precession (Elliot-Yafet re-
laxation). We find that the Gilbert damping component
α⊥
m≃ε/λ2τis dominated by Elliot-Yafet relaxation irre-
spective of the value of the parameter λτ, while the other
component crosses over from α∥
m≃ε/λ2τ(Elliot-Yafet
relaxation) for λτ≪1, to α∥
m≃ετ(Dyakonov-Perel re-
laxation) for λτ≫1. For the case λ≪∆sd/ετthe spin
relaxation is dominated by interaction with the exchange
field.
Crucially, our results are not confined solely to the N´ eel
order on the honeycomb lattice: we anticipate a broader
applicability across various magnetic orders, including
the zigzag order. This universality stems from our focus
on the large magnon wavelength limit. The choice of the
honeycomb lattice arises from its unique ability to main-
tain isotropic electronic spectra within the plane, coupled
with the ability to suppress anisotropy concerning in-
plane spin rotations. Strong anisotropic electronic spec-
tra would naturally induce strong anisotropic in-plane
Gilbert damping, which are absent in our results.
Finally, we show that the anti-ferromagnetic resonance
width is mostly defined by α∥
mand demonstrate that the
resonance quality factor is maximized for λτ≈1. Our
microscopic theory predictions may be tested for systems
such as MnPS 3monolayer on Pt and similar heterostruc-
tures.ACKNOWLEDGMENTS
We are grateful to O. Gomonay, R. Duine, J. Sinova,
and A. Mauri for helpful discussions. This project has
received funding from the European Union’s Horizon
2020 research and innovation program under the Marie
Sklodowska-Curie grant agreement No 873028.
Appendix A: Microscopic framework
The microscopic model that we employ to calculate
Gilbert damping belongs to a class of so-called s–dmod-
els that describe the physical system in the form of a
Heisenberg model for localized spins and a tight-binding
model for conduction electrons that are weakly coupled
by a local magnetic exchange interaction of the strength
∆sd.
Our effective electron Hamiltonian for a metallic
hexagonal anti-ferromagnet is given by25
H0=vfp·Σ+λ
2[σ×Σ]z−∆sdn·σΣzΛz,(A1)
where the vectors Σ,σandΛdenote the vectors of Pauli-
matrices acting on sub-lattice, spin and valley space re-
spectively. We also introduce the Fermi velocity vf,
Rashba-type spin-orbit interaction λ.
To describe Gilbert damping of the localized field n
we have to add the relaxation mechanism. This is pro-
vided in our model by adding a weak impurity potential
H=H0+V(r). The momentum relaxation due to scat-
tering on impurities leads indirectly to the relaxation of
Heisenberg spins due to the presence of spin-orbit cou-
pling and exchange couplings.
For modeling the impurity potential, we adopt a delta-
correlated random potential that corresponds to the
point scatter approximation, where the range of the im-
purity potential is much shorter than that of the mean
free path (see e.g. section 3.8 of Ref. 44), i.e.
⟨V(r)V(r′)⟩= 2πα(ℏvf)2δ(r−r′), (A2)
where the dimensionless coefficient α≪1 characterizes
the disorder strength. The corresponding scattering time
for electrons is obtained as τ=ℏ/παϵ , which is again
similar to the case of graphene.
The response of symmetric spin-polarization δs+to the
time-derivative of non-staggered magnetization, ∂tm, is
defined by the linear relation
δs+
α=X
βRαβ|ω=0˙mβ, (A3)
where the response tensor is taken at zero frequency25,45.
The linear response is defined generally by the tensor
Rαβ=A∆2
sd
2πSZdp
(2πℏ)2
Tr
GR
ε,pσαGA
ε+ℏω,pσβ
,(A4)9
where GR(A)
ε,pare standing for retarded(advanced) Green
functions and the angular brackets denote averaging over
disorder fluctuations.
The standard recipe for disorder averaging is the diffu-
sive approximation46,47that is realized by replacing the
bare Green functions in Eq. (A4) with disorder-averaged
Green functions and by replacing one of the vertex op-
erators σxorσywith the corresponding vertex-corrected
operator that is formally obtained by summing up ladder
impurity diagrams (diffusons).
In models with spin-orbit coupling, the controllable dif-
fusive approximation for non-dissipative quantities may
become, however, more involved as was noted first in
Ref. 48. For Gilbert damping it is, however, sufficient to
consider the ladder diagram contributions only.
The disorder-averaged Green function is obtained by
including an imaginary part of the self-energy ΣR(not
to be confused here with the Pauli matrix Σ 0,x,y,z) that
is evaluated in the first Born approximation
Im ΣR= 2παv2
fZdp
(2π)2Im1
ε−H0+i0. (A5)
The real part of the self-energy leads to the renormaliza-
tion of the energy scales ε,λand ∆ sd.
In the first Born approximation, the disorder-averaged
Green function is given by
GR
ε,p=1
ε−H0−iIm ΣR. (A6)
The vertex corrections are computed in the diffusive
approximation. The latter involves replacing the vertex
σαwith the vertex-corrected operator,
σvc
α=∞X
l=0σ(l)
α, (A7)
where the index lcorresponds to the number of disorder
lines in the ladder.
The operators σ(l)
αcan be defined recursively as
σ(l)
α=2ℏv2
f
ετZdp
(2π)2GR
ε,pσ(l−1)
αGA
ε+ℏω,p, (A8)
where σ(0)
α=σα.
The summation in Eq. (A7) can be computed in the
full operator basis, Bi={α,β,γ}=σαΣβΛγ, where each
index α,βandγtakes on 4 possible values (with zero
standing for the unity matrix). We may always normalize
TrBiBj= 2δijin an analogy to the Pauli matrices. The
operators Biare, then, forming a finite-dimensional space
for the recursion of Eq. (A8).
The vertex-corrected operators Bvc
iare obtained by
summing up the matrix geometric series
Bvc
i=X
j1
1− F
ijBj, (A9)where the entities of the matrix Fare given by
Fij=ℏv2
f
ετZdp
(2π)2Tr
GR
ε,pBiGA
ε+ℏω,pBj
.(A10)
Our operators of interest σxandσycan always be de-
composed in the operator basis as
σα=1
2X
iBiTr (σαBi), (A11)
hence the vertex-corrected spin operator is given by
σvc
α=1
2X
ijBvc
iTr(σαBi). (A12)
Moreover, the computation of the entire response tensor
of Eq. (A4) in the diffusive approximation can also be
expressed via the matrix Fas
Rαβ=α0ετ
8ℏX
ij[TrσαBi]F
1− F
ij[TrσβBj],(A13)
where α0=A∆2
sd/πℏ2v2
fSis the coefficient used in
Eq. (6) to define the unit of the Gilbert damping.
It appears that one can always choose the basis of
Bioperators such that the computation of Eq. (A13)
is closed in a subspace of just three Bioperators with
i= 1,2,3. This enables us to make analytical computa-
tions of Eq. (A13).
Appendix B: Magnetization dynamics
The representation of the results can be made some-
what simpler by choosing xaxis in the direction of the
in-plane projection n∥of the N´ eel vector, hence ny= 0.
In this case, one can represent the result as
δs+=c1n∥×(n∥×∂tm∥) +c2∂tm∥+c3∂tm⊥+c4n,
where ndependence of the coefficients cimay be param-
eterized as
c1=r11−r22−r31(1−n2
z)/(nxnz)
1−n2z, (B1a)
c2=r11−r31(1−n2
z)/(nxnz), (B1b)
c3=r33, (B1c)
c4= (r31/nz)∂tmz+ζ(∂tm)·n. (B1d)
The analytical results in the paper correspond to the
evaluation of δs±up to the second order in ∆ sdusing
perturbative analysis. Thus, zero approximation corre-
sponds to setting ∆ sd= 0 in Eqs. (A1,A5).
The equations of motion on nandmare given by
Eqs. (2),
∂tn=−Jn×m+n×δs++m×δs−, (B2a)
∂tm=m×δs++n×δs−, (B2b)10
It is easy to see that the following transformation leaves
the above equations invariant,
δs+→δs+−ξn, δ s−→δs−−ξm, (B3)
for an arbitrary value of ξ.
Such a gauge transformation can be used to prove that
the coefficient c4is irrelevant in Eqs. (B2).
In this paper, we compute δs±to the zeroth order in
|m|– the approximation which is justified by the sub-
lattice symmetry in the anti-ferromagnet. A somewhat
more general model has been analyzed also in Ref. 25 to
which we refer the interested reader for more technical
details.
Appendix C: Anisotropy constant
The anisotropy constant is obtained from the grand po-
tential energy Ω for conducting electrons. For the model
of Eq. (A1) the latter can be expressed as
Ω =−X
ς=±1
βZ
dε g(ε)νς(ε), (C1)
where β= 1/kBTis the inverse temperature, ς=±is
the valley index (for the valleys KandK′),GR
ς,pis the
bare retarded Green function with momentum pand in
the valley ς. We have also defined the function
g(ε) = ln (1 + exp[ β(µ−ε)]), (C2)
where µis the electron potential, and the electron density
of states in each of the valleys is given by,
νς(ε) =1
πZdp
(2πℏ)2Im Tr GR
ς,p, (C3)
where the trace is taken only over spin and sub-lattice
space,
In the metal regime considered, the chemical potential
is assumed to be placed in the upper electronic band.
In this case, the energy integration can be taken only for
positive energies. The two valence bands are always filled
and can only add a constant shift to the grand potential
Ω that we disregard.
The evaluation of Eq. (C1) yields the following density
of states
ντ(ε) =1
2πℏ2v2
f
0 0 < ε < ε 2
ε/2 +λ/4ε2< ε < ε 1,
ε ε > ε 1,(C4)where the energies ε1,2correspond to the extremum
points (zero velocity) for the electronic bands. These
energies, for each of the valleys, are given by
ε1,ς=1
2
+λ+p
4∆2+λ2−4ς∆λnz
, (C5a)
ε2,ς=1
2
−λ+p
4∆2+λ2+ 4ς∆λnz
(C5b)
where ς=±is the valley index.
In the limit of zero temperature we can approximate
Eq. (C1) as
Ω =−X
ς=±1
βZ∞
0dε(µ−ε)νς(ε). (C6)
Then, with the help of Eq. (C1) we find,
Ω =−1
24πℏ2v2
fX
ς=±
(ε1,ς−µ)2(4ε1,ς−3λ+ 2µ)
+(ε2,ς−µ)2(4ε2,ς+ 3λ+ 2µ)
. (C7)
By substituting the results of Eqs. (C5) into the above
equation we obtain
Ω =−1
24πℏ2v2
fh
(4∆2−4nz∆λ+λ2)2/3
+(4∆2+ 4nz∆λ+λ2)2/3−24∆µ+ 8µ3i
.(C8)
A careful analysis shows that the minimal energy cor-
responds to nz=±1 so that the conducting electrons
prefer an easy-axis magnetic anisotropy. By expanding
in powers of n2
zaround nz=±1 we obtain Ω = −Kn2
z/2,
where
K=1
2πℏ2v2(
|∆2λ| | λ/2∆| ≥1,
|∆λ2|/2|λ/2∆| ≤1.(C9)
This provides us with the easy axis anisotropy of Eq. (16).
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2203.01632v1.Stability_results_of_locally_coupled_wave_equations_with_local_Kelvin_Voigt_damping__Cases_when_the_supports_of_damping_and_coupling_coefficients_are_disjoint.pdf | arXiv:2203.01632v1 [math.AP] 3 Mar 2022STABILITY RESULTS OF LOCALLY COUPLED WAVE EQUATIONS WITH LO CAL
KELVIN-VOIGT DAMPING: CASES WHEN THE SUPPORTS OF DAMPING AN D
COUPLING COEFFICIENTS ARE DISJOINT
MOHAMMAD AKIL1, HAIDAR BADAWI1, AND SERGE NICAISE1
Abstract. In this paper, we study the direct/indirect stability of loc ally coupled wave equations with local
Kelvin-Voigt dampings/damping and by assuming that the sup ports of the dampings and the coupling coeffi-
cients are disjoint. First, we prove the well-posedness, st rong stability, and polynomial stability for some one
dimensional coupled systems. Moreover, under some geometr ic control condition, we prove the well-posedness
and strong stability in the multi-dimensional case.
Contents
1. Introduction 1
2. Direct and Indirect Stability in the one dimensional case 4
2.1. Well-Posedness 4
2.2. Strong Stability 5
2.3. Polynomial Stability 9
2.3.1. Proof of Theorem 2.6 9
2.3.2. Proof of Theorem 2.7 13
3. Indirect Stability in the multi-dimensional case 16
3.1. Well-posedness 16
3.2. Strong Stability 17
Appendix A. Some notions and stability theorems 20
References 21
1.Introduction
The direct and indirect stability of locally coupled wave equations with lo cal damping arouses many interests in
recent years. The study of coupled systems is also motivated by se veralphysicalconsiderationslike Timoshenko
and Bresse systems (see for instance [ 10,6,3,2,1,15,14]). The exponential or polynomial stability of the wave
equation with a local Kelvin-Voigt damping is considered in [ 20,23,13], for instance. On the other hand, the
direct and indirect stability of locally and coupled wave equations with lo cal viscous dampings are analyzed in
[8,18,16]. In this paper, we are interested in locally coupled wave equations wit h local Kelvin-Voigt dampings.
Before stating our main contributions, let us mention similar results f or such systems. In 2019, Hayek et al.in
[17], studied the stabilization of a multi-dimensional system of weakly cou pled wave equations with one or two
locally Kelvin-Voigt damping and non-smooth coefficient at the interfa ce. They established different stability
1Universit ´e Polytechnique Hauts-de-France, CERAMATHS/DEMAV, Valencien nes, France
E-mail address :Mohammad.Akil@uphf.fr, Haidar.Badawi@uphf.fr, Serge.N icaise@uphf.fr .
Key words and phrases. Coupled wave equations, Kelvin-Voigt damping, strong stab ility, polynomial stability .
1results. In 2021, Akil et al.in [24], studied the stability of an elastic/viscoelastic transmission problem of
locally coupled waves with non-smooth coefficients, by considering:
utt−/parenleftbig
aux+b0χ(α1,α3)utx/parenrightbig
x+c0χ(α2,α4)yt= 0,in (0,L)×(0,∞),
ytt−yxx−c0χ(α2,α4)ut= 0, in (0,L)×(0,∞),
u(0,t) =u(L,t) =y(0,t) =y(L,t) = 0, in (0,∞),
wherea,b0,L >0,c0/\e}atio\slash= 0, and 0 < α1< α2< α3< α4< L. They established a polynomial energy decay
rate of type t−1. In the same year, Akil et al.in [5], studied the stability of a singular local interaction
elastic/viscoelastic coupled wave equations with time delay, by consid ering:
utt−/bracketleftbig
aux+χ(0,β)(κ1utx+κ2utx(t−τ))/bracketrightbig
x+c0χ(α,γ)yt= 0,in (0,L)×(0,∞),
ytt−yxx−c0χ(α,γ)ut= 0, in (0,L)×(0,∞),
u(0,t) =u(L,t) =y(0,t) =y(L,t) = 0, in (0,∞),
wherea,κ1,L >0,κ2,c0/\e}atio\slash= 0, and 0 < α < β < γ < L . They proved that the energy of their system decays
polynomially in t−1. In 2021, Akil et al.in [4], studied the stability of coupled wave models with locally
memory in a past history framework via non-smooth coefficients on t he interface, by considering:
utt−/parenleftbigg
aux+b0χ(0,β)/integraldisplay∞
0g(s)ux(t−s)ds/parenrightbigg
x+c0χ(α,γ)yt= 0,in (0,L)×(0,∞),
ytt−yxx−c0χ(α,γ)ut= 0, in (0,L)×(0,∞),
u(0,t) =u(L,t) =y(0,t) =y(L,t) = 0, in (0,∞),
wherea,b0,L >0,c0/\e}atio\slash= 0, 0< α < β < γ < L , andg: [0,∞)/ma√sto−→(0,∞) is the convolution kernel function.
They established an exponential energy decay rate if the two wave s have the same speed of propagation. In
case of different speed of propagation, they proved that the ene rgy of their system decays polynomially with
ratet−1. In the same year, Akil et al.in [7], studied the stability of a multi-dimensional elastic/viscoelastic
transmission problem with Kelvin-Voigt damping and non-smooth coeffi cient at the interface, they established
some polynomial stability results under some geometric control con dition. In those previous literature, the
authors deal with the locally coupled wave equations with local dampin g and by assuming that there is an
intersection between the damping and coupling regions. The aim of th is paper is to study the direct/indirect
stability of locally coupled wave equations with Kelvin-Voigt dampings/d amping localized via non-smooth
coefficients/coefficient and by assuming that the supports of the d ampings and coupling coefficients aredisjoint.
In the first part of this paper, we consider the following one dimensio nal coupled system:
utt−(aux+butx)x+cyt= 0,(x,t)∈(0,L)×(0,∞), (1.1)
ytt−(yx+dytx)x−cut= 0,(x,t)∈(0,L)×(0,∞), (1.2)
with fully Dirichlet boundary conditions,
(1.3) u(0,t) =u(L,t) =y(0,t) =y(L,t) = 0, t∈(0,∞),
and the following initial conditions
(1.4) u(·,0) =u0(·), ut(·,0) =u1(·), y(·,0) =y0(·) and yt(·,0) =y1(·), x∈(0,L).
In this part, for all b0,d0>0 andc0/\e}atio\slash= 0, we treat the following three cases:
Case 1 (See Figure 1):
(C1)/braceleftiggb(x) =b0χ(b1,b2)(x), c(x) =c0χ(c1,c2)(x), d(x) =d0χ(d1,d2)(x),
where 0< b1< b2< c1< c2< d1< d2< L.
Case 2 (See Figure 2):
(C2)/braceleftiggb(x) =b0χ(b1,b2)(x), c(x) =c0χ(c1,c2)(x), d(x) =d0χ(d1,d2)(x),
where 0< b1< b2< d1< d2< c1< c2< L.
2Case 3 (See Figure 3):
(C3)/braceleftiggb(x) =b0χ(b1,b2)(x), c(x) =c0χ(c1,c2)(x), d(x) = 0,
where 0< b1< b2< c1< c2< L.
While in the second part, we consider the following multi-dimensional co upled system:
b1b2c1c2d1d2Lb0c0
0d0
Figure 1. Geometric description of the functions b,canddin Case 1.
b1b2 0 d1d2c1c2Lb0d0c0
Figure 2. Geometric description of the functions b,canddin Case 2.
0b1b2c1c2Lb0c0
Figure 3. Geometric description of the functions bandcin Case 3.
utt−div(∇u+but)+cyt= 0 in Ω ×(0,∞), (1.5)
ytt−∆y−cyt= 0 in Ω ×(0,∞), (1.6)
with full Dirichlet boundary condition
(1.7) u=y= 0 on Γ ×(0,∞),
and the following initial condition
(1.8) u(·,0) =u0(·), ut(·,0) =u1(·), y(·,0) =y0(·) andyt(·,0) =y1(·) in Ω,
3where Ω ⊂Rd,d≥2 is an open and bounded set with boundary Γ of class C2. Here,b,c∈L∞(Ω) are such
thatb: Ω→R+is the viscoelastic damping coefficient, c: Ω→Ris the coupling function and
(1.9) b(x)≥b0>0 inωb⊂Ω, c(x)≥c0/\e}atio\slash= 0 inωc⊂Ω and c(x) = 0 on Ω \ωc
and
(1.10) meas( ωc∩Γ)>0 and ωb∩ωc=∅.
In the first part of this paper, we study the direct and indirect sta bility of system ( 1.1)-(1.4) by consider-
ing the three cases ( C1), (C2), and (C3). In Subsection 2.1, we prove the well-posedness of our system by using
a semigroup approach. In Subsection 2.2, by using a general criteria of Arendt-Batty, we prove the stron g
stability of our system in the absence of the compactness of the re solvent. Finally, in Subsection 2.3, by using
a frequency domain approach combined with a specific multiplier metho d, we prove that our system decay
polynomially in t−4or int−1.
In the second part of this paper, we study the indirect stability of s ystem (1.5)-(1.8). In Subsection 3.1,
we prove the well-posedness of our system by using a semigroup app roach. Finally, in Subsection 3.2, under
some geometric control condition, we prove the strong stability of this system.
2.Direct and Indirect Stability in the one dimensional case
In this section, we study the well-posedness, strong stability, and polynomial stability of system ( 1.1)-(1.4).
The main result of this section are the following three subsections.
2.1.Well-Posedness. In this subsection, we will establish the well-posedness of system ( 1.1)-(1.4) by using
semigroup approach. The energy of system ( 1.1)-(1.4) is given by
E(t) =1
2/integraldisplayL
0/parenleftbig
|ut|2+a|ux|2+|yt|2+|yx|2/parenrightbig
dx.
Let (u,ut,y,yt) be a regular solution of ( 1.1)-(1.4). Multiplying ( 1.1) and (1.2) byutandytrespectively, then
using the boundary conditions ( 1.3), we get
E′(t) =−/integraldisplayL
0/parenleftbig
b|utx|2+d|ytx|2/parenrightbig
dx.
Thus, if ( C1) or (C2) or (C3) holds, we get E′(t)≤0. Therefore, system ( 1.1)-(1.4) is dissipative in the sense
that its energy is non-increasing with respect to time t. Let us define the energy space Hby
H= (H1
0(0,L)×L2(0,L))2.
The energy space His equipped with the following inner product
(U,U1)H=/integraldisplayL
0vv1dx+a/integraldisplayL
0ux(u1)xdx+/integraldisplayL
0zz1dx+/integraldisplayL
0yx(y1)xdx,
for allU= (u,v,y,z)⊤andU1= (u1,v1,y1,z1)⊤inH. We define the unbounded linear operator A:D(A)⊂
H −→ H by
D(A) =/braceleftbig
U= (u,v,y,z)⊤∈ H;v,z∈H1
0(0,L),(aux+bvx)x∈L2(0,L),(yx+dzx)x∈L2(0,L)/bracerightbig
and
A(u,v,y,z)⊤= (v,(aux+bvx)x−cz,z,(yx+dzx)x+cv)⊤,∀U= (u,v,y,z)⊤∈D(A).
Now, ifU= (u,ut,y,yt)⊤is the state of system ( 1.1)-(1.4), then it is transformed into the following first order
evolution equation
(2.1) Ut=AU, U(0) =U0,
whereU0= (u0,u1,y0,y1)⊤∈ H.
4Proposition 2.1. If (C1) or (C2) or (C3) holds. Then, the unbounded linear operator Ais m-dissipative in
the Hilbert space H.
Proof.For allU= (u,v,y,z)⊤∈D(A), we have
ℜ/a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH=−/integraldisplayL
0b|vx|2dx−/integraldisplayL
0d|zx|2dx≤0,
which implies that Ais dissipative. Now, similiar to Proposition 2.1 in [ 24] (see also [ 5] and [4]), we can prove
that there exists a unique solution U= (u,v,y,z)⊤∈D(A) of
−AU=F,∀F= (f1,f2,f3,f4)⊤∈ H.
Then 0∈ρ(A) andAis an isomorphism and since ρ(A) is open in C(see Theorem 6.7 (Chapter III) in [ 19]),
we easily get R(λI−A) =Hfor a sufficiently small λ >0. This, together with the dissipativeness of A, imply
thatD(A) is dense in Hand that Ais m-dissipative in H(see Theorems 4.5, 4.6 in [ 22]). /square
According to Lumer-Phillips theorem (see [ 22]), then the operator Agenerates a C0-semigroup of contrac-
tionsetAinHwhich gives the well-posedness of ( 2.1). Then, we have the following result:
Theorem 2.2. For allU0∈ H, system ( 2.1) admits a unique weak solution
U(t) =etAU0∈C0(R+,H).
Moreover, if U0∈D(A), then the system ( 2.1) admits a unique strong solution
U(t) =etAU0∈C0(R+,D(A))∩C1(R+,H).
2.2.Strong Stability. In this subsection, we will prove the strong stability of system ( 1.1)-(1.4). We define
the following conditions:
(SSC1) ( C1) holds and |c0|<min/parenleftbigg√a
c2−c1,1
c2−c1/parenrightbigg
,
(SSC3) ( C3) holds, a= 1 and |c0|<1
c2−c1.
The main result of this section is the following theorem.
Theorem 2.3. Assume that ( SSC1) or(C2) or(SSC3) holds. Then, the C0-semigroupofcontractions/parenleftbig
etA/parenrightbig
t≥0
is strongly stable in H; i.e. for all U0∈ H, the solution of ( 2.1) satisfies
lim
t→+∞/ba∇dbletAU0/ba∇dblH= 0.
According to Theorem A.2, to prove Theorem 2.3, we need to prove that the operator Ahas no pure imaginary
eigenvalues and σ(A)∩iRis countable. Its proof has been divided into the following Lemmas.
Lemma 2.4. Assume that ( SSC1) or (C2) or (SSC3) holds. Then, for all λ∈R,iλI−Ais injective, i.e.
ker(iλI−A) ={0}.
Proof.From Proposition 2.1, we have 0 ∈ρ(A). We still need to show the result for λ∈R∗. For this aim,
suppose that there exists a real number λ/\e}atio\slash= 0 and U= (u,v,y,z)⊤∈D(A) such that
AU=iλU.
Equivalently, we have
v=iλu, (2.2)
(aux+bvx)x−cz=iλv, (2.3)
z=iλy, (2.4)
(yx+dzx)+cv=iλz. (2.5)
Next, a straightforward computation gives
(2.6) 0 = ℜ/a\}b∇acketle{tiλU,U/a\}b∇acket∇i}htH=ℜ/a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH=−/integraldisplayL
0b|vx|2dx−/integraldisplayL
0d|zx|2dx.
5Inserting ( 2.2) and (2.4) in (2.3) and (2.5), we get
λ2u+(aux+iλbux)x−iλcy= 0 in (0 ,L), (2.7)
λ2y+(yx+iλdyx)x+iλcu= 0 in (0 ,L), (2.8)
with the boundary conditions
(2.9) u(0) =u(L) =y(0) =y(L) = 0.
•Case 1: Assume that ( SSC1) holds. From ( 2.2), (2.4) and (2.6), we deduce that
(2.10) ux=vx= 0 in ( b1,b2) andyx=zx= 0 in ( d1,d2).
Using (2.7), (2.8) and (2.10), we obtain
(2.11) λ2u+auxx= 0 in (0 ,c1) and λ2y+yxx= 0 in ( c2,L).
Deriving the above equations with respect to xand using ( 2.10), we get
(2.12)/braceleftiggλ2ux+auxxx= 0 in (0 ,c1),
ux= 0 in ( b1,b2)⊂(0,c1),and/braceleftiggλ2yx+yxxx= 0 in ( c2,L),
yx= 0 in ( d1,d2)⊂(c2,L).
Using the unique continuation theorem, we get
(2.13) ux= 0 in (0 ,c1) and yx= 0 in ( c2,L)
Using (2.13) and the fact that u(0) =y(L) = 0, we get
(2.14) u= 0 in (0 ,c1) and y= 0 in ( c2,L).
Now, ouraim is to provethat u=y= 0 in (c1,c2). For this aim, using ( 2.14) and the fact that u,y∈C1([0,L]),
we obtain the following boundary conditions
(2.15) u(c1) =ux(c1) =y(c2) =yx(c2) = 0.
Multiplying ( 2.7) by−2(x−c2)ux, integrating over ( c1,c2) and taking the real part, we get
(2.16) −/integraldisplayc2
c1λ2(x−c2)(|u|2)xdx−a/integraldisplayc2
c1(x−c2)/parenleftbig
|ux|2/parenrightbig
xdx+2ℜ/parenleftbigg
iλc0/integraldisplayc2
c1(x−c2)yuxdx/parenrightbigg
= 0,
using integration by parts and ( 2.15), we get
(2.17)/integraldisplayc2
c1|λu|2dx+a/integraldisplayc2
c1|ux|2dx+2ℜ/parenleftbigg
iλc0/integraldisplayc2
c1(x−c2)yuxdx/parenrightbigg
= 0.
Multiplying ( 2.8) by−2(x−c1)yx, integrating over ( c1,c2), taking the real part, and using the same argument
as above, we get
(2.18)/integraldisplayc2
c1|λy|2dx+/integraldisplayc2
c1|yx|2dx+2ℜ/parenleftbigg
iλc0/integraldisplayc2
c1(x−c1)uyxdx/parenrightbigg
= 0.
Adding ( 2.17) and (2.18), we get
(2.19)/integraldisplayc2
c1|λu|2dx+a/integraldisplayc2
c1|ux|2dx+/integraldisplayc2
c1|λy|2dx+/integraldisplayc2
c1|yx|2dx≤2|λ||c0|(c2−c1)/integraldisplayc2
c1(|y||ux|+|u||yx|)dx.
Using Young’s inequality in ( 2.19), we get
(2.20)/integraldisplayc2
c1|λu|2dx+a/integraldisplayc2
c1|ux|2dx+/integraldisplayc2
c1|λy|2dx+/integraldisplayc2
c1|yx|2dx≤c2
0(c2−c1)2
a/integraldisplayc2
c1|λy|2dx
+a/integraldisplayc2
c1|ux|2dx+c2
0(c2−c1)2/integraldisplayc2
c1|λu|2dx+/integraldisplayc2
c1|yx|2dx,
consequently, we get
(2.21)/parenleftbigg
1−c2
0(c2−c1)2
a/parenrightbigg/integraldisplayc2
c1|λy|2dx+/parenleftbig
1−c2
0(c2−c1)2/parenrightbig/integraldisplayc2
c1|λu|2dx≤0.
Thus, from the above inequality and ( SSC1), we get
(2.22) u=y= 0 in ( c1,c2).
6Next, we need to prove that u= 0 in (c2,L) andy= 0 in (0 ,c1). For this aim, from ( 2.22) and the fact that
u,y∈C1([0,L]), we obtain
(2.23) u(c2) =ux(c2) = 0 and y(c1) =yx(c1) = 0.
It follows from ( 2.7), (2.8) and (2.23) that
(2.24)/braceleftiggλ2u+auxx= 0 in ( c2,L),
u(c2) =ux(c2) =u(L) = 0,and/braceleftiggλ2y+yxx= 0 in (0 ,c1),
y(0) =y(c1) =yx(c1) = 0.
Holmgren uniqueness theorem yields
(2.25) u= 0 in ( c2,L) andy= 0 in (0 ,c1).
Therefore, from ( 2.2), (2.4), (2.14), (2.22) and (2.25), we deduce that
U= 0.
•Case 2: Assume that ( C2) holds. From ( 2.2), (2.4) and (2.6), we deduce that
(2.26) ux=vx= 0 in ( b1,b2) andyx=zx= 0 in ( d1,d2).
Using (2.7), (2.8) and (2.26), we obtain
(2.27) λ2u+auxx= 0 in (0 ,c1) and λ2y+yxx= 0 in (0 ,c1).
Deriving the above equations with respect to xand using ( 2.26), we get
(2.28)/braceleftiggλ2ux+auxxx= 0 in (0 ,c1),
ux= 0 in ( b1,b2)⊂(0,c1),and/braceleftiggλ2yx+yxxx= 0 in (0 ,c1),
yx= 0 in ( d1,d2)⊂(0,c1).
Using the unique continuation theorem, we get
(2.29) ux= 0 in (0 ,c1) and yx= 0 in (0 ,c1).
From (2.29) and the fact that u(0) =y(0) = 0, we get
(2.30) u= 0 in (0 ,c1) and y= 0 in (0 ,c1).
Using the fact that u,y∈C1([0,L]) and (2.30), we get
(2.31) u(c1) =ux(c1) =y(c1) =yx(c1) = 0.
Now, using the definition of c(x) in (2.7)-(2.8), (2.26) and (2.31) and Holmgren theorem, we get
u=y= 0 in ( c1,c2).
Again, using the fact that u,y∈C1([0,L]), we get
(2.32) u(c2) =ux(c2) =y(c2) =yx(c2) = 0.
Now, using the same argument as in Case 1, we obtain
u=y= 0 in (c2,L),
consequently, we deduce that
U= 0.
•Case 3: Assume that ( SSC3) holds. Using the same argument as in Cases 1 and 2, we obtain
(2.33) u= 0 in (0 ,c1) and u(c1) =ux(c1) = 0.
Step 1. The aim of this step is to prove that
(2.34)/integraldisplayc2
c1|u|2dx=/integraldisplayc2
c1|y|2dx.
7For this aim, multiplying ( 2.7) byyand (2.8) byuand using integration by parts, we get
/integraldisplayL
0λ2uydx−/integraldisplayL
0uxyxdx−iλc0/integraldisplayc2
c1|y|2dx= 0, (2.35)
/integraldisplayL
0λ2yudx−/integraldisplayL
0yxuxdx+iλc0/integraldisplayc2
c1|u|2dx= 0. (2.36)
Adding ( 2.35) and (2.36), taking the imaginary part, we get ( 2.34).
Step 2. Multiplying ( 2.7) by−2(x−c2)ux, integrating over ( c1,c2) and taking the real part, we get
(2.37)−ℜ/parenleftbigg/integraldisplayc2
c1λ2(x−c2)(|u|2)xdx/parenrightbigg
−ℜ/parenleftbigg/integraldisplayc2
c1(x−c2)/parenleftbig
|ux|2/parenrightbig
xdx/parenrightbigg
+2ℜ/parenleftbigg
iλc0/integraldisplayc2
c1(x−c2)yuxdx/parenrightbigg
= 0,
using integration by parts in ( 2.37) and (2.33), we get
(2.38)/integraldisplayc2
c1|λu|2dx+a/integraldisplayc2
c1|ux|2dx+2ℜ/parenleftbigg
iλc0/integraldisplayc2
c1(x−c2)yuxdx/parenrightbigg
= 0.
Using Young’s inequality in ( 2.38), we obtain
(2.39)/integraldisplayc2
c1|λu|2dx+/integraldisplayc2
c1|ux|2dx≤ |c0|(c2−c1)/integraldisplayc2
c1|λy|2dx+|c0|(c2−c1)/integraldisplayc2
c1|ux|2dx.
Inserting ( 2.34) in (2.39), we get
(2.40) (1 −|c0|(c2−c1))/integraldisplayc2
c1/parenleftbig
|λu|2+|ux|2/parenrightbig
dx≤0.
According to ( SSC3) and (2.34), we get
(2.41) u=y= 0 in ( c1,c2).
Step 3. Using the fact that u∈H2(c1,c2)⊂C1([c1,c2]), we get
(2.42) u(c1) =ux(c1) =y(c1) =yx(c1) =y(c2) =yx(c2) = 0.
Now, from ( 2.7), (2.8) and the definition of c, we get
/braceleftbiggλ2u+uxx= 0 in ( c2,L),
u(c2) =ux(c2) = 0,and/braceleftbiggλ2y+yxx= 0 in (0 ,c1)∪(c2,L),
y(c1) =yx(c1) =y(c2) =yx(c2) = 0.
From the above systems and Holmgren uniqueness Theorem, we get
(2.43) u= 0 in ( c2,L) and y= 0 in (0 ,c1)∪(c2,L).
Consequently, using ( 2.33), (2.41) and (2.43), we get U= 0. The proof is thus completed. /square
Lemma 2.5. Assume that ( SSC1) or (C2) or (SSC3) holds. Then, for all λ∈R, we have
R(iλI−A) =H.
Proof.See Lemma 2.5 in [ 24] (see also [ 4]). /square
Proof of Theorems 2.3. From Lemma 2.4, we obtain that the operator Ahas no pure imaginary eigenvalues
(i.e.σp(A)∩iR=∅). Moreover, from Lemma 2.5and with the help of the closed graph theorem of Banach,
we deduce that σ(A)∩iR=∅. Therefore, according to Theorem A.2, we get that the C 0-semigroup ( etA)t≥0
is strongly stable. The proof is thus complete. /square
82.3.Polynomial Stability. In this subsection, we study the polynomial stability of system ( 1.1)-(1.4). Our
main result in this section are the following theorems.
Theorem 2.6. Assume that ( SSC1) holds. Then, for all U0∈D(A), there exists aconstant C >0 independent
ofU0such that
(2.44) E(t)≤C
t4/ba∇dblU0/ba∇dbl2
D(A), t >0.
Theorem 2.7. Assume that ( SSC3) holds . Then, for all U0∈D(A) there exists a constant C >0 independent
ofU0such that
(2.45) E(t)≤C
t/ba∇dblU0/ba∇dbl2
D(A), t >0.
According to Theorem A.3, the polynomial energy decays ( 2.44) and (2.45) hold if the following conditions
(H1) iR⊂ρ(A)
and
(H2) limsup
λ∈R,|λ|→∞1
|λ|ℓ/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble
L(H)<∞withℓ=/braceleftigg1
2for Theorem 2.6,
2 for Theorem 2.7,
aresatisfied. Sincecondition( H1)isalreadyprovedin Subsection 2.2. We stillneedtoprove( H2), let usproveit
byacontradictionargument. Tothisaim,supposethat( H2)isfalse,thenthereexists/braceleftbig/parenleftbig
λn,Un:= (un,vn,yn,zn)⊤/parenrightbig/bracerightbig
n≥1⊂
R∗
+×D(A) with
(2.46) λn→ ∞asn→ ∞and/ba∇dblUn/ba∇dblH= 1,∀n≥1,
such that
(2.47) ( λn)ℓ(iλnI−A)Un=Fn:= (f1,n,f2,n,f3,n,f4,n)⊤→0 inH,asn→ ∞.
For simplicity, we drop the index n. Equivalently, from ( 2.47), we have
iλu−v=f1
λℓ, f1→0 inH1
0(0,L), (2.48)
iλv−(aux+bvx)x+cz=f2
λℓ, f2→0 inL2(0,L), (2.49)
iλy−z=f3
λℓ, f3→0 inH1
0(0,L), (2.50)
iλz−(yx+dzx)x−cv=f4
λℓ, f4→0 inL2(0,L). (2.51)
2.3.1.Proof of Theorem 2.6.In this subsection, we will prove Theorem 2.6by checking the condition ( H2),
by finding a contradiction with ( 2.46) by showing /ba∇dblU/ba∇dblH=o(1). For clarity, we divide the proof into several
Lemmas. By taking the inner product of ( 2.47) withUinH, we remark that
/integraldisplayL
0b|vx|2dx+/integraldisplayL
0d|zx|2dx=−ℜ(/a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH) =λ−1
2ℜ(/a\}b∇acketle{tF,U/a\}b∇acket∇i}htH) =o/parenleftig
λ−1
2/parenrightig
.
Thus, from the definitions of bandd, we get
(2.52)/integraldisplayb2
b1|vx|2dx=o/parenleftig
λ−1
2/parenrightig
and/integraldisplayd2
d1|zx|2dx=o/parenleftig
λ−1
2/parenrightig
.
Using (2.48), (2.50), (2.52), and the fact that f1,f3→0 inH1
0(0,L), we get
(2.53)/integraldisplayb2
b1|ux|2dx=o(1)
λ5
2and/integraldisplayd2
d1|yx|2dx=o(1)
λ5
2.
Lemma 2.8. The solution U∈D(A) of system ( 2.48)-(2.51) satisfies the following estimations
(2.54)/integraldisplayb2
b1|v|2dx=o(1)
λ3
2and/integraldisplayd2
d1|z|2dx=o(1)
λ3
2.
9Proof.We give the proof of the first estimation in ( 2.54), the second one can be done in a similar way. For
this aim, we fix g∈C1([b1,b2]) such that
g(b2) =−g(b1) = 1,max
x∈[b1,b2]|g(x)|=mgand max
x∈[b1,b2]|g′(x)|=mg′.
The proof is divided into several steps:
Step 1. The goal of this step is to prove that
(2.55) |v(b1)|2+|v(b2)|2≤/parenleftigg
λ1
2
2+2mg′/parenrightigg/integraldisplayb2
b1|v|2dx+o(1)
λ.
From (2.48), we deduce that
(2.56) vx=iλux−λ−1
2(f1)x.
Multiplying ( 2.56) by 2gvand integrating over ( b1,b2), then taking the real part, we get
/integraldisplayb2
b1g/parenleftbig
|v|2/parenrightbig
xdx=ℜ/parenleftigg
2iλ/integraldisplayb2
b1guxvdx/parenrightigg
−ℜ/parenleftigg
2λ−1
2/integraldisplayb2
b1g(f1)xvdx/parenrightigg
.
Using integration by parts in the left hand side of the above equation , we get
(2.57) |v(b1)|2+|v(b2)|2=/integraldisplayb2
b1g′|v|2dx+ℜ/parenleftigg
2iλ/integraldisplayb2
b1guxvdx/parenrightigg
−ℜ/parenleftigg
2λ−1
2/integraldisplayb2
b1g(f1)xvdx/parenrightigg
.
Using Young’s inequality, we obtain
2λmg|ux||v| ≤λ1
2|v|2
2+2λ3
2m2
g|ux|2and 2λ−1
2mg|(f1)x||v| ≤mg′|v|2+m2
gm−1
g′λ−1|(f1)x|2.
From the above inequalities, ( 2.57) becomes
(2.58) |v(b1)|2+|v(b2)|2≤/parenleftigg
λ1
2
2+2mg′/parenrightigg/integraldisplayb2
b1|v|2dx+2λ3
2m2
g/integraldisplayb2
b1|ux|2dx+m2
g
mg′λ−1/integraldisplayb2
b1|(f1)x|2dx.
Inserting ( 2.53) in (2.58) and the fact that f1→0 inH1
0(0,L), we get ( 2.55).
Step 2. The aim of this step is to prove that
(2.59) |(aux+bvx)(b1)|2+|(aux+bvx)(b2)|2≤λ3
2
2/integraldisplayb2
b1|v|2dx+o(1).
Multiplying ( 2.49) by−2g/parenleftbig
aux+bvx/parenrightbig
, using integration by parts over ( b1,b2) and taking the real part, we get
|(aux+bvx)(b1)|2+|(aux+bvx)(b2)|2=/integraldisplayb2
b1g′|aux+bvx|2dx+
ℜ/parenleftigg
2iλ/integraldisplayb2
b1gv(aux+bvx)dx/parenrightigg
−ℜ/parenleftigg
2λ−1
2/integraldisplayb2
b1gf2(aux+bvx)dx/parenrightigg
,
consequently, we get
(2.60)|(aux+bvx)(b1)|2+|(aux+bvx)(b2)|2≤mg′/integraldisplayb2
b1|aux+bvx|2dx
+2λmg/integraldisplayb2
b1|v||aux+bvx|dx+2mgλ−1
2/integraldisplayb2
b1|f2||aux+bvx|dx.
By Young’s inequality, ( 2.52), and (2.53), we have
(2.61) 2 λmg/integraldisplayb2
b1|v||aux+bvx|dx≤λ3
2
2/integraldisplayb2
b1|v|2dx+2m2
gλ1
2/integraldisplayb2
b1|aux+bvx|2dx≤λ3
2
2/integraldisplayb2
b1|v|2dx+o(1).
Inserting ( 2.61) in (2.60), then using ( 2.52), (2.53) and the fact that f2→0 inL2(0,L), we get ( 2.59).
10Step 3. The aim of this step is to prove the first estimation in ( 2.54). For this aim, multiplying ( 2.49) by
−iλ−1v, integrating over ( b1,b2) and taking the real part , we get
(2.62)/integraldisplayb2
b1|v|2dx=ℜ/parenleftigg
iλ−1/integraldisplayb2
b1(aux+bvx)vxdx−/bracketleftbig
iλ−1(aux+bvx)v/bracketrightbigb2
b1+iλ−3
2/integraldisplayb2
b1f2vdx/parenrightigg
.
Using (2.52), (2.53), the fact that vis uniformly bounded in L2(0,L) andf2→0 inL2(0,1), and Young’s
inequalities, we get
(2.63)/integraldisplayb2
b1|v|2dx≤λ−1
2
2[|v(b1)|2+|v(b2)|2]+λ−3
2
2[|(aux+bvx)(b1)|2+|(aux+bvx)(b2)|2]+o(1)
λ3
2.
Inserting ( 2.55) and (2.59) in (2.63), we get
/integraldisplayb2
b1|v|2dx≤/parenleftbigg1
2+mg′λ−1
2/parenrightbigg/integraldisplayb2
b1|v|2dx+o(1)
λ3
2,
which implies that
(2.64)/parenleftbigg1
2−mg′λ−1
2/parenrightbigg/integraldisplayb2
b1|v|2dx≤o(1)
λ3
2.
Using the fact that λ→ ∞, we can take λ >4m2
g′. Then, we obtain the first estimation in ( 2.54). Similarly,
we can obtain the second estimation in ( 2.54). The proof has been completed. /square
Lemma 2.9. The solution U∈D(A) of system ( 2.48)-(2.51) satisfies the following estimations
(2.65)/integraldisplayc1
0/parenleftbig
|v|2+a|ux|2/parenrightbig
dx=o(1) and/integraldisplayL
c2/parenleftbig
|z|2+|yx|2/parenrightbig
dx=o(1).
Proof. First, let h∈C1([0,c1]) such that h(0) =h(c1) = 0. Multiplying ( 2.49) by 2a−1h(aux+bvx),
integrating over (0 ,c1), using integration by parts and taking the real part, then using ( 2.52) and the fact that
uxis uniformly bounded in L2(0,L) andf2→0 inL2(0,L), we get
(2.66) ℜ/parenleftbigg
2iλa−1/integraldisplayc1
0vh(aux+bvx)dx/parenrightbigg
+a−1/integraldisplayc1
0h′|aux+bvx|2dx=o(1)
λ1
2.
From (2.48), we have
(2.67) iλux=−vx−λ−1
2(f1)x.
Inserting ( 2.67) in (2.66), using integration by parts, then using ( 2.52), (2.54), and the fact that f1→0 in
H1
0(0,L) andvis uniformly bounded in L2(0,L), we get
(2.68)/integraldisplayc1
0h′|v|2dx+a−1/integraldisplayc1
0h′|aux+bvx|2dx= 2ℜ/parenleftbigg
λ−1
2/integraldisplayc1
0vh(f1)xdx/parenrightbigg
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
=o(λ−1
2)
+ℜ/parenleftigg
2iλa−1b0/integraldisplayb2
b1hvvxdx/parenrightigg
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
=o(1)+o(1)
λ1
2.
Now, we fix the following cut-off functions
p1(x) :=
1 in (0 ,b1),
0 in ( b2,c1),
0≤p1≤1 in (b1,b2),andp2(x) :=
1 in ( b2,c1),
0 in (0 ,b1),
0≤p2≤1 in (b1,b2).
Finally, take h(x) =xp1(x)+(x−c1)p2(x) in (2.68) and using ( 2.52), (2.53), (2.54), we get the first estimation
in (2.65). By using the same argument, we can obtain the second estimation in (2.65). The proof is thus
completed. /square
Lemma 2.10. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimations
(2.69) |λu(c1)|=o(1),|ux(c1)|=o(1),|λy(c2)|=o(1)and|yx(c2)|=o(1).
11Proof.First, from ( 2.48) and (2.49), we deduce that
(2.70) λ2u+auxx=−f2
λ1
2−iλ1
2f1in (b2,c1).
Multiplying ( 2.70) by 2(x−b2)¯ux, integrating over ( b2,c1) and taking the real part, then using the fact that
uxis uniformly bounded in L2(0,L) andf2→0 inL2(0,L), we get
(2.71)/integraldisplayc1
b2λ2(x−b2)/parenleftbig
|u|2/parenrightbig
xdx+a/integraldisplayc1
b2(x−b2)/parenleftbig
|ux|2/parenrightbig
xdx=−ℜ/parenleftbigg
2iλ1
2/integraldisplayc1
b2(x−b2)f1uxdx/parenrightbigg
+o(1)
λ1
2.
Using integration by parts in ( 2.71), then using ( 2.65), and the fact that f1→0 inH1
0(0,L) andλuis uniformly
bounded in L2(0,L), we get
(2.72) 0 ≤(c1−b2)/parenleftbig
|λu(c1)|2+a|ux(c1)|2/parenrightbig
=ℜ/parenleftig
2iλ1
2(c1−b2)f1(c1)u(c1)/parenrightig
+o(1),
consequently, by using Young’s inequality, we get
|λu(c1)|2+|ux(c1)|2≤2λ1
2|f1(c1)||u(c1)|+o(1)
≤1
2|λu(c1)|2+2
λ|f1(c1)|2+o(1).
Then, we get
(2.73)1
2|λu(c1)|2+|ux(c1)|2≤2
λ|f1(c1)|2+o(1).
Finally, from the above estimation and the fact that f1→0 inH1
0(0,L), we get the first two estimations in
(2.69). By using the same argument, we can obtain the last two estimation s in (2.69). The proof has been
completed. /square
Lemma 2.11. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimation
(2.74)/integraldisplayc2
c1|λu|2+a|ux|2+|λy|2+|yx|2dx=o(1).
Proof.Inserting ( 2.48) and (2.50) in (2.49) and (2.51), we get
−λ2u−auxx+iλc0y=f2
λ1
2+iλ1
2f1+c0f3
λ1
2in (c1,c2), (2.75)
−λ2y−yxx−iλc0u=f4
λ1
2+iλ1
2f3−c0f1
λ1
2in (c1,c2). (2.76)
Multiplying ( 2.75) by 2(x−c2)uxand (2.76) by 2(x−c1)yx, integrating over ( c1,c2) and taking the real part,
then using the fact that /ba∇dblF/ba∇dblH=o(1) and/ba∇dblU/ba∇dblH= 1, we obtain
(2.77)−λ2/integraldisplayc2
c1(x−c2)/parenleftbig
|u|2/parenrightbig
xdx−a/integraldisplayc2
c1(x−c2)/parenleftbig
|ux|2/parenrightbig
xdx+ℜ/parenleftbigg
2iλc0/integraldisplayc2
c1(x−c2)yuxdx/parenrightbigg
=
ℜ/parenleftbigg
2iλ1
2/integraldisplayc2
c1(x−c2)f1uxdx/parenrightbigg
+o(1)
λ1
2
and
(2.78)−λ2/integraldisplayc2
c1(x−c1)/parenleftbig
|y|2/parenrightbig
xdx−/integraldisplayc2
c1(x−c1)/parenleftbig
|yx|2/parenrightbig
xdx−ℜ/parenleftbigg
2iλc0/integraldisplayc2
c1(x−c1)uyxdx/parenrightbigg
=
ℜ/parenleftbigg
2iλ1
2/integraldisplayc2
c1(x−c1)f3yxdx/parenrightbigg
+o(1)
λ1
2.
Using integration by parts, ( 2.69), and the fact that f1,f3→0 inH1
0(0,L),/ba∇dblu/ba∇dblL2(0,L)=O(λ−1),/ba∇dbly/ba∇dblL2(0,L)=
O(λ−1), we deduce that
(2.79) ℜ/parenleftbigg
iλ1
2/integraldisplayc2
c1(x−c2)f1uxdx/parenrightbigg
=o(1)
λ1
2andℜ/parenleftbigg
iλ1
2/integraldisplayc2
c1(x−c1)f3yxdx/parenrightbigg
=o(1)
λ1
2.
12Inserting ( 2.79) in (2.77) and (2.78), then using integration by parts and ( 2.69), we get
/integraldisplayc2
c1/parenleftbig
|λu|2+a|ux|2/parenrightbig
dx+ℜ/parenleftbigg
iλc0/integraldisplayc2
c1(x−c2)yuxdx/parenrightbigg
=o(1), (2.80)
/integraldisplayc2
c1/parenleftbig
|λy|2+|yx|2/parenrightbig
dx−ℜ/parenleftbigg
iλc0/integraldisplayc2
c1(x−c1)uyxdx/parenrightbigg
=o(1). (2.81)
Adding ( 2.80) and (2.81), we get
/integraldisplayc2
c1/parenleftbig
|λu|2+a|ux|2+|λy|2+|yx|2/parenrightbig
dx=ℜ/parenleftbigg
2iλc0/integraldisplayc2
c1(x−c1)uyxdx/parenrightbigg
−ℜ/parenleftbigg
2iλc0/integraldisplayc2
c1(x−c2)yuxdx/parenrightbigg
+o(1)
≤2λ|c0|(c2−c1)/integraldisplayc2
c1|u||yx|dx+2λ|c0|
a1
4(c2−c1)a1
4/integraldisplayc2
c1|y||ux|dx+o(1).
Applying Young’s inequalities, we get
(2.82) (1 −|c0|(c2−c1))/integraldisplayc2
c1(|λu|2+|yx|2)dx+/parenleftbigg
1−1√a|c0|(c2−c1)/parenrightbigg/integraldisplayc2
c1(a|ux|2+|λy|2)dx≤o(1).
Finally, using ( SSC1), we get the desired result. The proof has been completed. /square
Lemma 2.12. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimations
(2.83)/integraldisplayc1
0/parenleftbig
|z|2+|yx|2/parenrightbig
dx=o(1)and/integraldisplayL
c2/parenleftbig
|v|2+a|ux|2/parenrightbig
dx=o(1).
Proof.Using the same argument of Lemma 2.9, we obtain ( 2.83). /square
Proof of Theorem 2.6.Using (2.53), Lemmas 2.8,2.9,2.11,2.12, we get /ba∇dblU/ba∇dblH=o(1), which contradicts
(2.46). Consequently, condition (H2) holds. This implies the energy decay estimation ( 2.44).
2.3.2.Proof of Theorem 2.7.In this subsection, we will prove Theorem 2.7by checking the condition ( H2),
that is by finding a contradiction with ( 2.46) by showing /ba∇dblU/ba∇dblH=o(1). For clarity, we divide the proof into
several Lemmas. By taking the inner product of ( 2.47) withUinH, we remark that
/integraldisplayL
0b|vx|2dx=−ℜ(/a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH) =λ−2ℜ(/a\}b∇acketle{tF,U/a\}b∇acket∇i}htH) =o(λ−2).
Then,
(2.84)/integraldisplayb2
b1|vx|2dx=o(λ−2).
Using (2.48) and (2.84), and the fact that f1→0 inH1
0(0,L), we get
(2.85)/integraldisplayb2
b1|ux|2dx=o(λ−4).
Lemma 2.13. Let0< ε <b2−b1
2, the solution U∈D(A)of the system (2.48)-(2.51)satisfies the following
estimation
(2.86)/integraldisplayb2−ε
b1+ε|v|2dx=o(λ−2).
Proof.First, we fix a cut-off function θ1∈C1([0,c1]) such that
(2.87) θ1(x) =
1 if x∈(b1+ε,b2−ε),
0 if x∈(0,b1)∪(b2,L),
0≤θ1≤1 elsewhere .
Multiplying ( 2.49) byλ−1θ1v, integrating over (0 ,c1), using integration by parts, and the fact that f2→0 in
L2(0,L) andvis uniformly bounded in L2(0,L), we get
(2.88) i/integraldisplayc1
0θ1|v|2dx+1
λ/integraldisplayc1
0(ux+bvx)(θ′
1v+θvx)dx=o(λ−3).
13Using (2.84) and the fact that /ba∇dblU/ba∇dblH= 1, we get
1
λ/integraldisplayc1
0(ux+bvx)(θ′
1v+θvx)dx=o(λ−2).
Inserting the above estimation in ( 2.88), we get the desired result ( 2.86). The proof has been completed. /square
Lemma 2.14. The solution U∈D(A)of the system (2.48)-(2.51)satisfies the following estimation
(2.89)/integraldisplayc1
0(|v|2+|ux|2)dx=o(1).
Proof.Leth∈C1([0,c1]) such that h(0) =h(c1) = 0. Multiplying ( 2.49) by 2h(ux+bvx), integrating over
(0,c1) and taking the real part, then using integration by parts and the fact that f2→0 inL2(0,L), we get
(2.90) ℜ/parenleftbigg
2/integraldisplayc1
0iλvh(ux+bvx)dx/parenrightbigg
+/integraldisplayc1
0h′|ux+bvx|2dx=o(λ−2).
Using (2.84) and the fact that vis uniformly bounded in L2(0,L), we get
(2.91) ℜ/parenleftbigg
2/integraldisplayc1
0iλvh(ux+bvx)dx/parenrightbigg
= 2/integraldisplayc1
0iλvhuxdx+o(1).
From (2.48), we have
(2.92) iλux=−vx−/parenleftbig
f1/parenrightbig
x
λ2.
Inserting ( 2.92) in (2.91), using integration by parts and the fact that f1→0 inH1
0(0,L), we get
(2.93) ℜ/parenleftbigg
2/integraldisplayc1
0iλvh(ux+bvx)dx/parenrightbigg
=/integraldisplayc1
0h′|v|2dx+o(1).
Inserting ( 2.93) in (2.90), we obtain
(2.94)/integraldisplayc1
0h′/parenleftbig
|v|2+|ux+bvx|2/parenrightbig
dx=o(1).
Now, we fix the following cut-off functions
θ2(x) :=
1 in (0 ,b1+ε),
0 in ( b2−ε,c1),
0≤θ2≤1 in (b1+ε,b2−ε),andθ3(x) :=
1 in ( b2−ε,c1),
0 in (0 ,b1+ε),
0≤θ3≤1 in (b1+ε,b2−ε).
Takingh(x) =xθ2(x)+(x−c1)θ3(x) in (2.94), then using ( 2.84) and (2.85), we get
(2.95)/integraldisplay
(0,b1+ε)∪(b2−ε,c1)|v|2dx+/integraldisplay
(0,b1)∪(b2,c1)|ux|2dx=o(1).
Finally, from ( 2.85), (2.86) and (2.95), we get the desired result ( 2.89). The proof has been completed. /square
Lemma 2.15. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimations
(2.96) |λu(c1)|=o(1)and|ux(c1)|=o(1),
(2.97)/integraldisplayc2
c1|λu|2dx=/integraldisplayc2
c1|λy|2dx+o(1).
Proof.First, using the same argument of Lemma 2.10, we claim ( 2.96). Inserting ( 2.48), (2.50) in (2.49) and
(2.51), we get
λ2u+(ux+bvx)x−iλcy=−f2
λ2−if1
λ−cf3
λ2, (2.98)
λ2y+yxx+iλcu=−f4
λ2−if3
λ+cf1
λ2. (2.99)
14Multiplying ( 2.98) and (2.99) byλyandλurespectively, integrating over(0 ,L), then using integration by parts,
(2.84), and the fact that /ba∇dblU/ba∇dblH= 1 and /ba∇dblF/ba∇dblH=o(1), we get
λ3/integraldisplayL
0u¯ydx−λ/integraldisplayL
0ux¯yxdx−ic0/integraldisplayc2
c1|λy|2dx=o(1), (2.100)
λ3/integraldisplayL
0y¯udx−λ/integraldisplayL
0yx¯uxdx+ic0/integraldisplayc2
c1|λu|2dx=o(1)
λ. (2.101)
Adding ( 2.100) and (2.101) and taking the imaginary parts, we get the desired result ( 2.97). The proof is thus
completed. /square
Lemma 2.16. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following asymptotic behavior
(2.102)/integraldisplayc2
c1|λu|2dx=o(1),/integraldisplayc2
c1|λy|2dx=o(1)and/integraldisplayc2
c1|ux|2dx=o(1).
Proof.First, Multiplying ( 2.98) by 2(x−c2)¯ux, integrating over ( c1,c2) and taking the real part, using the
fact that /ba∇dblU/ba∇dblH= 1 and /ba∇dblF/ba∇dblH=o(1), we get
(2.103) λ2/integraldisplayc2
c1(x−c2)/parenleftbig
|u|2/parenrightbig
xdx+/integraldisplayc2
c1(x−c2)/parenleftbig
|ux|2/parenrightbig
xdx=ℜ/parenleftbigg
2iλc0/integraldisplayc2
c1(x−c2)y¯uxdx/parenrightbigg
+o(1).
Using integration by parts in ( 2.103) with the help of ( 2.96), we get
(2.104)/integraldisplayc2
c1|λu|2dx+/integraldisplayc2
c1|ux|2dx≤2λ|c0|(c2−c1)/integraldisplayc2
c1|y||ux|+o(1).
Applying Young’s inequality in ( 2.104), we get
(2.105)/integraldisplayc2
c1|λu|2dx+/integraldisplayc2
c1|ux|2dx≤ |c0|(c2−c1)/integraldisplayc2
c1|ux|2dx+|c0|(c2−c1)/integraldisplayc2
c1|λy|2dx+o(1).
Using (2.97) in (2.105), we get
(2.106) (1 −|c0|(c2−c1))/integraldisplayc2
c1/parenleftbig
|λu|2+|ux|2/parenrightbig
dx≤o(1).
Finally, from the above estimation, ( SSC3) and (2.97), we get the desired result ( 2.102). The proof has been
completed. /square
Lemma 2.17. Let0< δ <c2−c1
2. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following
estimations
(2.107)/integraldisplayc2−δ
c1+δ|yx|2dx=o(1).
Proof.First, we fix a cut-off function θ4∈C1([0,L]) such that
(2.108) θ4(x) :=
1 if x∈(c1+δ,c2−δ),
0 if x∈(0,c1)∪(c2,L),
0≤θ4≤1 elsewhere .
Multiplying ( 2.99) byθ4¯y, integrating over (0 ,L) and using integration by parts, we get
(2.109)/integraldisplayc2
c1θ4|λy|2dx−/integraldisplayL
0θ4|yx|2dx−/integraldisplayL
0θ′
4yx¯ydx+iλc0/integraldisplayc2
c1θ4u¯ydx=o(1)
λ2.
Using (2.102) and the definition of θ4, we get
(2.110)/integraldisplayc2
c1θ4|λy|2dx=o(1),/integraldisplayL
0θ′
4yx¯ydx=o(λ−1), iλc 0/integraldisplayc2
c1θ4u¯ydx=o(λ−1).
Finally, Inserting ( 2.110) in (2.109), we get the desired result ( 2.111). The proof has been completed. /square
Lemma 2.18. The solution U∈D(A)of system (2.48)-(2.51)satisfies the following estimations
(2.111)/integraldisplayc1+ε
0|λy|2dx,/integraldisplayc1+ε
0|yx|2dx,/integraldisplayL
c2−ε|λy|2dx,/integraldisplayL
c2−ε|yx|2dx,/integraldisplayL
c2|λu|2dx,/integraldisplayL
c2|ux|2dx=o(1).
15Proof.Letq∈C1([0,L]) such that q(0) =q(L) = 0. Multiplying ( 2.98) by 2q¯yxintegrating over (0 ,L), using
(2.102), and the fact that yxis uniformly bounded in L2(0,L) and/ba∇dblF/ba∇dblH=o(1), we get
(2.112)/integraldisplayL
0q′/parenleftbig
|λy|2+|yx|2/parenrightbig
dx=o(1).
Now, take q(x) =xθ5(x)+(x−L)θ6(x) in (2.112), such that
θ5(x) :=
1 in (0 ,c1+ε),
0 in ( c2−ε,L),
0≤θ1≤1 in (c1+ε,c2−ε),andθ2(x)
1 in ( c2−ε,L),
0 in (0 ,c1+ε),
0≤θ2≤1 in (c1+ε,c2−ε).
Then, we obtain the first four estimations in ( 2.111). Now, multiplying ( 2.98) by 2q/parenleftbig
ux+bvx/parenrightbig
integrating over
(0,L) and using the fact that uxis uniformly bounded in L2(0,L), we get
(2.113)/integraldisplayL
0q′/parenleftbig
|λu|2+|ux|2/parenrightbig
dx=o(1).
By taking q(x) = (x−L)θ7(x), such that
θ7(x) =
1 in ( c2,L),
0 in (0 ,c1),
0≤θ7≤1 in (c1,c2),
we get the the last two estimations in ( 2.111). The proof has been completed. /square
Proof of Theorem 2.7.Using (2.85), Lemmas 2.14,2.16,2.17and2.18, we get /ba∇dblU/ba∇dblH=o(1), which
contradicts ( 2.46). Consequently, condition (H2) holds. This implies the energy decay estimation ( 2.45)
3.Indirect Stability in the multi-dimensional case
In this section, we study the well-posedness and the strong stabilit y of system ( 1.5)-(1.8).
3.1.Well-posedness. In this subsection, we will establish the well-posednessof ( 1.5)-(1.8) by usinf semigroup
approach. The energy of system ( 1.5)-(1.8) is given by
(3.1) E(t) =1
2/integraldisplayL
0/parenleftbig
|ut|2+|∇u|2+|yt|2+|∇y|2/parenrightbig
dx.
Let (u,ut,y,yt) be a regular solution of ( 1.5)-(1.8). Multiplying ( 1.5) and (1.7) byutandytrespectively, then
using the boundary conditions ( 1.9), we get
(3.2) E′(t) =−/integraldisplay
Ωb|∇ut|2dx,
using the definition of b, we get E′(t)≤0. Thus, system ( 1.5)-(1.8) is dissipative in the sense that its energy
is non-increasing with respect to time t. Let us define the energy space Hby
H=/parenleftbig
H1
0(Ω)×L2(Ω)/parenrightbig2.
The energy space His equipped with the inner product defined by
/a\}b∇acketle{tU,U1/a\}b∇acket∇i}htH=/integraldisplay
Ωvv1dx+/integraldisplay
Ω∇u∇u1dx+/integraldisplay
Ωzz1dx+/integraldisplay
Ω∇y·∇y1dx,
for allU= (u,v,y,z)⊤andU1= (u1,v1,y1,z1)⊤inH. We define the unbounded linear operator Ad:D(Ad)⊂
H −→ H by
D(Ad) =/braceleftbig
U= (u,v,y,z)⊤∈ H;v,z∈H1
0(Ω),div(ux+bvx)∈L2(Ω),∆y∈L2(Ω)/bracerightbig
and
AdU=
v
div(∇u+b∇v)−cz
z
∆y+cv
,∀U= (u,v,y,z)⊤∈D(Ad).
16/tildewideΩ
Ωωc ωb •x0
Γ0Γ1 •xν
Figure 4. Geometric description of the sets ωbandωc
IfU= (u,ut,y,yt) is a regular solution of system ( 1.5)-(1.8), then we rewrite this system as the following first
order evolution equation
(3.3) Ut=AdU, U(0) =U0,
whereU0= (u0,u1,y0,y1)⊤∈ H. For all U= (u,v,y,z)⊤∈D(Ad), we have
ℜ/a\}b∇acketle{tAdU,U/a\}b∇acket∇i}htH=−/integraldisplay
Ωb|∇v|2dx≤0,
which implies that Adis dissipative. Now, similar to Proposition 2.1 in [ 7], we can prove that there exists a
unique solution U= (u,v,y,z)⊤∈D(Ad) of
−AdU=F,∀F= (f1,f2,f3,f4)⊤∈ H.
Then 0∈ρ(Ad) andAdis an isomorphism and since ρ(Ad) is open in C(see Theorem 6.7 (Chapter III) in
[19]), we easily get R(λI−Ad) =Hfor a sufficiently small λ >0. This, together with the dissipativeness of Ad,
imply that D(Ad) is dense in Hand that Adis m-dissipative in H(see Theorems 4.5, 4.6 in [ 22]). According
to Lumer-Phillips theorem (see [ 22]), then the operator Adgenerates a C0-semigroup of contractions etAdin
Hwhich gives the well-posedness of ( 3.3). Then, we have the following result:
Theorem 3.1. For allU0∈ H, system ( 2.1) admits a unique weak solution
U(t) =etAdU0∈C0(R+,H).
Moreover, if U0∈D(A), then the system ( 2.1) admits a unique strong solution
U(t) =etAdU0∈C0(R+,D(Ad))∩C1(R+,H).
3.2.Strong Stability. In this subsection, we will prove the strong stability of system ( 1.5)-(1.8). First, we
fix the following notations
/tildewideΩ = Ω−ωc,Γ1=∂ωc−∂Ω and Γ 0=∂ωc−Γ1.
Letx0∈Rdandm(x) =x−x0and suppose that (see Figure 4)
(GC) m·ν≤0 on Γ 0= (∂ωc)−Γ1.
17The main result of this section is the following theorem
Theorem 3.2. Assume that (GC)holds and
(SSC) /ba∇dblc/ba∇dbl∞≤min/braceleftigg
1
/ba∇dblm/ba∇dbl∞+d−1
2,1
/ba∇dblm/ba∇dbl∞+(d−1)Cp,ωc
2/bracerightigg
,
whereCp,ωcis the Poincarr´ e constant on ωc. Then, the C0−semigroup of contractions/parenleftbig
etAd/parenrightbig
is strongly stable
inH; i.e. for all U0∈ H, the solution of (3.3)satisfies
lim
t→+∞/ba∇dbletAdU0/ba∇dblH= 0.
Proof.First, let us prove that
(3.4) ker( iλI−Ad) ={0},∀λ∈R.
Since 0∈ρ(Ad), then we still need to show the result for λ∈R∗. Suppose that there exists a real number
λ/\e}atio\slash= 0 and U= (u,v,y,z)⊤∈D(Ad), such that
AdU=iλU.
Equivalently, we have
v=iλu, (3.5)
div(∇u+b∇v)−cz=iλv, (3.6)
z=iλy, (3.7)
∆y+cv=iλz. (3.8)
Next, a straightforward computation gives
0 =ℜ/a\}b∇acketle{tiλU,U/a\}b∇acket∇i}htH=ℜ/a\}b∇acketle{tAdU,U/a\}b∇acket∇i}htH=−/integraldisplay
Ωb|∇v|2dx,
consequently, we deduce that
(3.9) b∇v= 0 in Ω and ∇v=∇u= 0 in ωb.
Inserting ( 3.5) in (3.6), then using the definition of c, we get
(3.10) ∆u=−λ2uinωb.
From (3.9) we get ∆ u= 0 inωband from ( 3.10) and the fact that λ/\e}atio\slash= 0, we get
(3.11) u= 0 in ωb.
Now, inserting ( 3.5) in (3.6), then using ( 3.9), (3.11) and the definition of c, we get
(3.12)λ2u+∆u= 0 in/tildewideΩ,
u= 0 in ωb⊂/tildewideΩ.
Using Holmgren uniqueness theorem, we get
(3.13) u= 0 in/tildewideΩ.
It follows that
(3.14) u=∂u
∂ν= 0 on Γ 1.
Now, our aim is to show that u=y= 0 inωc. For this aim, inserting ( 3.5) and (3.7) in (3.6) and (3.8), then
using (3.9), we get the following system
λ2u+∆u−iλcy= 0 in Ω , (3.15)
λ2y+∆y+iλcu= 0 in Ω , (3.16)
u= 0 on ∂ωc, (3.17)
y= 0 on Γ 0, (3.18)
∂u
∂ν= 0 on Γ 1. (3.19)
18Let us prove ( 3.4) by the following three steps:
Step 1. The aim of this step is to show that
(3.20)/integraldisplay
Ωc|u|2dx=/integraldisplay
Ωc|y|2dx.
For this aim, multiplying ( 3.15) and (3.16) by ¯yand ¯urespectively, integrating over Ω and using Green’s
formula, we get
λ2/integraldisplay
Ωu¯ydx−/integraldisplay
Ω∇u·∇¯ydx−iλ/integraldisplay
Ωc|y|2dx= 0, (3.21)
λ2/integraldisplay
Ωy¯udx−/integraldisplay
Ω∇y·∇¯udx+iλ/integraldisplay
Ωc|u|2dx= 0. (3.22)
Adding ( 3.21) and (3.22), then taking the imaginary part, we get ( 3.20).
Step 2. The aim of this step is to prove the following identity
(3.23) −d/integraldisplay
ωc|λu|2dx+(d−2)/integraldisplay
ωc|∇u|2dx+/integraldisplay
Γ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u
∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
dΓ−2ℜ/parenleftbigg
iλ/integraldisplay
ωccy(m·∇¯u)dx/parenrightbigg
= 0.
For this aim, multiplying ( 3.15) by 2(m·∇¯u), integrating over ωcand taking the real part, we get
(3.24) 2 ℜ/parenleftbigg
λ2/integraldisplay
ωcu(m·∇¯u)dx/parenrightbigg
+2ℜ/parenleftbigg/integraldisplay
ωc∆u(m·∇¯u)dx/parenrightbigg
−2ℜ/parenleftbigg
iλ/integraldisplay
ωccy(m·∇¯u)dx/parenrightbigg
= 0.
Now, using the fact that u= 0 in∂ωc, we get
(3.25) ℜ/parenleftbigg
2λ2/integraldisplay
ωcu(m·∇¯u)dx/parenrightbigg
=−d/integraldisplay
ωc|λu|2dx.
Using Green’s formula, we obtain
(3.26)2ℜ/parenleftbigg/integraldisplay
ωc∆u(m·∇¯u)dx/parenrightbigg
=−2ℜ/parenleftbigg/integraldisplay
ωc∇u·∇(m·∇¯u)dx/parenrightbigg
+2ℜ/parenleftbigg/integraldisplay
Γ0∂u
∂ν(m·∇¯u)dΓ/parenrightbigg
= (d−2)/integraldisplay
ωc|∇u|2dx−/integraldisplay
∂ωc(m·ν)|∇u|2dx+2ℜ/parenleftbigg/integraldisplay
Γ0∂u
∂ν(m·∇¯u)dΓ/parenrightbigg
.
Using (3.17) and (3.19), we get
(3.27)/integraldisplay
∂ωc(m·ν)|∇u|2dx=/integraldisplay
Γ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u
∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
dΓ andℜ/parenleftbigg/integraldisplay
Γ0∂u
∂ν(m·∇¯u)dΓ/parenrightbigg
=/integraldisplay
Γ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u
∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
dΓ.
Inserting ( 3.27) in (3.26), we get
(3.28) 2 ℜ/parenleftbigg/integraldisplay
ωc∆u(m·∇¯u)dx/parenrightbigg
= (d−2)/integraldisplay
ωc|∇u|2dx+/integraldisplay
Γ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u
∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
dΓ.
Inserting ( 3.25) and (3.28) in (3.24), we get ( 3.23).
Step 3. In this step, we prove ( 3.4). Multiplying ( 3.15) by (d−1)u, integrating over ωcand using ( 3.17), we
get
(3.29) ( d−1)/integraldisplay
ωc|λu|2dx+(1−d)/integraldisplay
ωc|∇u|2dx−ℜ/parenleftbigg
iλ(d−1)/integraldisplay
ωccy¯udx/parenrightbigg
= 0.
Adding ( 3.23) and (3.29), we get
/integraldisplay
ωc|λu|2dx+/integraldisplay
ωc|∇u|2dx=/integraldisplay
Γ0(m·ν)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u
∂ν/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
dΓ−2ℜ/parenleftbigg
iλ/integraldisplay
ωccy(m·∇¯u)dx/parenrightbigg
−ℜ/parenleftbigg
iλ(d−1)/integraldisplay
ωccy¯udx/parenrightbigg
= 0.
Using (GC), we get
(3.30)/integraldisplay
ωc|λu|2dx+/integraldisplay
ωc|∇u|2dx≤2|λ|/integraldisplay
ωc|c||y||m·∇u|dx+|λ|(d−1)/integraldisplay
ωc|c||y||u|dx.
19Using Young’s inequality and ( 3.20), we get
(3.31) 2 |λ|/integraldisplay
ωc|c||y||m·∇u|dx≤ /ba∇dblm/ba∇dbl∞/ba∇dblc/ba∇dbl∞/integraldisplay
ωc/parenleftbig
|λu|2+|∇u|2/parenrightbig
dx
and
(3.32) |λ|(d−1)/integraldisplay
ωc|c(x)||y||u|dx≤(d−1)/ba∇dblc/ba∇dbl∞
2/integraldisplay
ωc|λu|2dx+(d−1)/ba∇dblc/ba∇dbl∞Cp,ωc
2/integraldisplay
ωc|∇u|2dx.
Inserting ( 3.32) in (3.30), we get
/parenleftbigg
1−/ba∇dblc/ba∇dbl∞/parenleftbigg
/ba∇dblm/ba∇dbl∞+d−1
2/parenrightbigg/parenrightbigg/integraldisplay
ωc|λu|2dx+/parenleftbigg
1−/ba∇dblc/ba∇dbl∞/parenleftbigg
/ba∇dblm/ba∇dbl∞+(d−1)Cp,ωc
2/parenrightbigg/parenrightbigg/integraldisplay
ωc|∇u|2dx≤0.
Using (SSC) and (3.20) in the above estimation, we get
(3.33) u= 0 and y= 0 in ωc.
In order to complete this proof, we need to show that y= 0 in/tildewideΩ. For this aim, using the definition of the
function cin/tildewideΩ and using the fact that y= 0 inωc, we get
(3.34)λ2y+∆y= 0 in/tildewideΩ,
y= 0 on ∂/tildewideΩ,
∂y
∂ν= 0 on Γ 1.
Now, using Holmgren uniqueness theorem, we obtain y= 0 in/tildewideΩ and consequently ( 3.4) holds true. Moreover,
similar to Lemma 2.5 in [ 7], we can prove R(iλI− Ad) =H,∀λ∈R. Finally, by using the closed graph
theorem of Banach and Theorem A.2, we conclude the proof of this Theorem. /square
Let us notice that, under the sole assumptions ( GC) and (SSC), the polynomial stability of system ( 1.5)-(1.8)
is an open problem.
Appendix A.Some notions and stability theorems
In order to make this paper more self-contained, we recall in this sh ort appendix some notions and stability
results used in this work.
Definition A.1. Assume that Ais the generator of C0−semigroup of contractions/parenleftbig
etA/parenrightbig
t≥0on a Hilbert space
H. TheC0−semigroup/parenleftbig
etA/parenrightbig
t≥0is said to be
(1) Strongly stable if
lim
t→+∞/ba∇dbletAx0/ba∇dblH= 0,∀x0∈H.
(2) Exponentially (or uniformly) stable if there exists two positive co nstantsMandεsuch that
/ba∇dbletAx0/ba∇dblH≤Me−εt/ba∇dblx0/ba∇dblH,∀t >0,∀x0∈H.
(3) Polynomially stable if there exists two positive constants Candαsuch that
/ba∇dbletAx0/ba∇dblH≤Ct−α/ba∇dblAx0/ba∇dblH,∀t >0,∀x0∈D(A).
/square
To show the strong stability of the C0-semigroup/parenleftbig
etA/parenrightbig
t≥0we rely on the following result due to Arendt-Batty
[9].
Theorem A.2. Assume that Ais the generatorof a C 0−semigroup of contractions/parenleftbig
etA/parenrightbig
t≥0on a Hilbert space
H. IfAhas no pure imaginary eigenvalues and σ(A)∩iRis countable, where σ(A) denotes the spectrum of
A, then the C0-semigroup/parenleftbig
etA/parenrightbig
t≥0is strongly stable. /square
Concerning the characterization of polynomial stability stability of a C0−semigroup of contraction/parenleftbig
etA/parenrightbig
t≥0we
rely on the following result due to Borichev and Tomilov [ 12] (see also [ 11] and [21])
20Theorem A.3. Assume that Ais the generator of a strongly continuous semigroup of contractio ns/parenleftbig
etA/parenrightbig
t≥0
onH. IfiR⊂ρ(A), then for a fixed ℓ >0 the following conditions are equivalent
(A.1) limsup
λ∈R,|λ|→∞1
|λ|ℓ/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble
L(H)<∞,
(A.2) /ba∇dbletAU0/ba∇dbl2
H≤C
t2
ℓ/ba∇dblU0/ba∇dbl2
D(A),∀t >0, U0∈D(A),for some C >0.
/square
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21 |
2009.05372v1.Blow_up_results_for_semilinear_damped_wave_equations_in_Einstein_de_Sitter_spacetime.pdf | arXiv:2009.05372v1 [math.AP] 10 Sep 2020Blow – up results for semilinear damped wave equations in
Einstein – de Sitter spacetime
Alessandro Palmieri
Abstract
We prove by using an iteration argument some blow-up results for a semilinear damped wave equation
in generalized Einstein-de Sitter spacetime with a time-de pendent coefficient for the damping term and
power nonlinearity. Then, we conjecture an expression for t he critical exponent due to the main blow-
up results, which is consistent with many special cases of th e considered model and provides a natural
generalization of Strauss exponent. In the critical case, w e consider a non-autonomous and parameter
dependent Cauchy problem for a linear ODE of second order, wh ose explicit solutions are determined by
means of special functions’ theory.
Keywords Semilinear damped wave equation, Einstein – de Sitter space time, power nonlinearity,
generalized Strauss exponent, lifespan estimates, modifie d Bessel functions
AMS Classification (2020) Primary: 35B44, 35L05, 35L71; Secondary: 35B33, 33C10
1 Introduction
In recent years, the wave equation in Einstein – de Sitter spa cetime has been considered in [9, 10] in the
linear case and in [11, 12, 27] in the semilinear case. Let us c onsider the semilinear wave equation with power
nonlinearity in a generalized Einstein – de Sitter spacetime , that is, the equation with singular coefficients
ϕtt−t−2k∆ϕ+2t−1ϕt=|ϕ|p, (1)
wherek∈[0,1)andp >1. This model is the semilinear wave equation in Einstein – de S itter spacetime
with power nonlinearity for k= 2/3andn= 3. It has been proved in [12, 27] that for
1<p/lessorequalslantmax/braceleftbig
p0/parenleftbig
k,n+2
1−k/parenrightbig
,p1(k,n)/bracerightbig
a local in time solution to the corresponding Cauchy problem (with initial data prescribed at the initial time
t= 1) blows up in finite time, provided that the initial data fulfil l certain integral sign conditions. More
specifically, in [12] the subcritical case for (1) is investi gated, while in [27] the critical case and the upper
bound estimates for the lifespan are studied. Here and throu ghout the paper p0(k,n)is the positive root of
the quadratic equation
/parenleftBig
n−1
2−k
2(1−k)/parenrightBig
p2−/parenleftBig
n+1
2+3k
2(1−k)/parenrightBig
p−1 = 0, (2)
when the coefficient for p2is not positive, we set formally p0(k,n).=∞, while
p1(k,n).= 1+2
(1−k)n. (3)
Note thatp1(k,n)is related to the Fujita exponent pFuj(n).= 1+2
n. Indeed, according to this notation, it
holdsp1(k,n) =pFuj/parenleftbig
(1−k)n/parenrightbig
andp1(0,n) =pFuj(n). On the other hand, p0(k,n)is a generalization of
the Strauss exponent for the classical semilinear wave equa tion, sincep0(0,n) =pStr(n), wherepStr(n)is the
positive root of the quadratic equation (n−1)p2−(n+1)p−2 = 0.
In this paper, we generalize the model (1) with a general mult iplicative constant µfor the damping term.
More specifically, we investigate the blow – up dynamic for th e Cauchy problem
utt−t−2k∆u+µt−1ut=|u|px∈Rn, t∈(1,T),
u(1,x) =εu0(x) x∈Rn,
ut(1,x) =εu1(x) x∈Rn,(4)
1wherek∈[0,1),p>1,µis the nonnegative multiplicative constant in the time – dep endent coefficient for
the damping term and ε >0describes the size of the initial data. Let us point out that t he not damped
caseµ= 0can be treated as well via our approach.
More precisely, we will focus on proving blow-up results whe never the exponent pbelongs to the range
1<p/lessorequalslantmax/braceleftbig
p0/parenleftbig
k,n+µ
1−k/parenrightbig
,p1(k,n)/bracerightbig
,
clearly, under suitable sign assumptions for u0,u1. According to (2), the shift p0/parenleftbig
k,n+µ
1−k/parenrightbig
ofp0(k,n)is
nothing but the positive root to the quadratic equation
/parenleftBig
n−1
2+µ−k
2(1−k)/parenrightBig
p2−/parenleftBig
n+1
2+µ+3k
2(1−k)/parenrightBig
p−1 = 0. (5)
Therefore, the critical exponent p0/parenleftbig
k,n+µ
1−k/parenrightbig
for (4) is obtained by the corresponding exponent in the not
damped case via a formal shift in the dimension of magnitudeµ
1−k.
Let us provide an overview on the methods that we are going to u se to prove the main results in this paper.
In the subcritical case 1< p <max/braceleftbig
p0/parenleftbig
k,n+µ
1−k/parenrightbig
,p1(k,n)/bracerightbig
, we employ a standard iteration argument
based on a multiplier argument (see also [17, 18, 19, 21] for f urther details on the multiplier argument). This
approach is based on the employment of two time – dependent fu nctionals related to a local solution uto
(4) and generalizes the method from [34] for the semilinear w ave equation with scale – invariant damping.
The first functional is the space average of uand its dynamic will be considered for the iterative argumen t.
On the other hand, we will work with a positive solution of the adjoint linear equation in order to prove
the positivity of the second auxiliary functional. Hence, t his second functional will also provide a first lower
bound estimate for the first functional, allowing us to begin with the iteration procedure. In the critical
case we should sharpen our iteration frame by considering a d ifferent time – dependent functional, so that
a slicing procedure may be applied. In comparison to what hap pens in the subcritical case, a more precise
analysis of the adjoint linear equation is necessary in the c ritical case p=p0/parenleftbig
k,n+µ
1−k/parenrightbig
. This approach
follows the one developed in [27] which is in turn a generaliz ation of the ideas introduced by Wakasa and
Yordanov in [36, 37] an developed in different frameworks in [ 29, 30, 21, 3, 4]. Whereas in the other critical
casep=p1(k,n), we can still work with the space average of a local in time sol ution as functional, although
a slicing procedure has to be applied in order to deal with log arithmic factors in the lower bound estimates.
1.1 Notations
Throughout this paper we use the following notations: φk(t).=t1−k
1−kdenotes a distance function produced
by the speed of propagation ak(t) =t−k, while the amplitude of the light cone is given by the functio n
Ak(t).=/integraldisplayt
1τ−kdτ=φk(t)−φk(1); (6)
the ball in Rnwith radius Raround the origin is denoted BR;f/lessorsimilargmeans that there exists a positive
constantCsuch thatf/lessorequalslantCgand, similarly, for f/greaterorsimilarg;IνandKνdenote the modified Bessel function of
first and second kind of order ν, respectively; finally, as in the introduction, p0(k,n)is the positive solution
to (2) andp1(k,n)is defined by (3).
1.2 Main results
Before stating the main theorems, let us introduce a suitabl e notion of energy solution to the semilinear
Cauchy problem (4).
Definition 1.1. Letu0∈H1(Rn)andu1∈L2(Rn). We say that
u∈C/parenleftbig
[1,T),H1(Rn)/parenrightbig
∩C1/parenleftbig
[1,T),L2(Rn)/parenrightbig
∩Lp
loc/parenleftbig
[1,T)×Rn/parenrightbig
is an energy solution to (4) on [1,T)ifufulfillsu(1,·) =εu0inH1(Rn)and the integral relation
/integraldisplay
Rn∂tu(t,x)ψ(t,x)dx−ε/integraldisplay
Rnu1(x)ψ(1,x)dx−/integraldisplayt
1/integraldisplay
Rn∂tu(s,x)ψs(s,x)dxds
+/integraldisplayt
1/integraldisplay
Rns−2k∇u(s,x)·∇ψ(s,x)dxds+/integraldisplayt
1/integraldisplay
Rnµs−1∂tu(s,x)ψ(s,x)dxds
=/integraldisplayt
1/integraldisplay
Rn|u(s,x)|pψ(s,x)dxds (7)
for any test function ψ∈C∞
0([1,T)×Rn)and anyt∈(1,T).
2We point out that performing a further step of integration by parts in (7), we find the integral relation
/integraldisplay
Rn∂tu(t,x)ψ(t,x)dx−/integraldisplay
Rnu(t,x)ψs(t,x)dx+/integraldisplay
Rnµt−1u(t,x)ψ(t,x)dx
−ε/integraldisplay
Rnu1(x)ψ(1,x)dx+ε/integraldisplay
Rnu0(x)ψs(1,x)dx−ε/integraldisplay
Rnµu0(x)ψ(1,x)dx
+/integraldisplayt
1/integraldisplay
Rnu(s,x)/parenleftbig
ψss(s,x)−s−2k∆ψ(s,x)−µs−1ψs(s,x)+µs−2ψ(s,x)/parenrightbig
dxds
=/integraldisplayt
1/integraldisplay
Rn|u(s,x)|pψ(s,x)dxds (8)
for anyψ∈C∞
0([1,T)×Rn)and anyt∈(1,T).
Remark 1.Let us point out that if the Cauchy data have compact support, saysuppuj⊂BRforj= 0,1
and for some R>0, then, for any t∈(1,T)a local solution uto (4) the support condition
suppu(t,·)⊂BR+Ak(t)
is satisfied, where Akis defined by (6). Consequently, in Definition 1.1 it is possib le to consider test functions
which are not compactly supported, i.e., ψ∈C∞([1,T)×Rn).
Theorem 1.2 (Subcritical case) .Letµ/greaterorequalslant0and let the exponent of the nonlinear term psatisfy
1<p<max/braceleftBig
p0/parenleftbig
k,n+µ
1−k/parenrightbig
,p1(k,n)/bracerightBig
.
Let us assume that u0∈H1(Rn)andu1∈L2(Rn)are nonnegative, nontrivial and compactly supported
functions with supports contained in BRfor someR>0. Let
u∈C/parenleftbig
[1,T),H1(Rn)/parenrightbig
∩C1/parenleftbig
[1,T),L2(Rn)/parenrightbig
∩Lp
loc/parenleftbig
[1,T)×Rn/parenrightbig
be an energy solution to (4)according to Definition 1.1 with lifespan T=T(ε)and satisfying the support
condition suppu(t,·)⊂BAk(t)+Rfor anyt∈(1,T).
Then, there exists a positive constant ε0=ε0(u0,u1,n,p,k,µ,R )such that for any ε∈(0,ε0]the energy
solutionublows up in finite time. Moreover, the upper bound estimate for the lifespan
T(ε)/lessorequalslant/braceleftBigg
Cε−p(p−1)
θ(n,k,µ,p ) ifp<p0/parenleftbig
k,n+µ
1−k/parenrightbig
,
Cε−(2
p−1−(1−k)n)−1
ifp<p1(k,n),(9)
holds, where the positive constant Cis independent of εand
θ(n,k,µ,p).= 1−k+/parenleftBig
(1−k)n+1
2+µ+3k
2/parenrightBig
p−/parenleftBig
(1−k)n−1
2+µ−k
2/parenrightBig
p2.
In order to properly state the results in the critical case, l et us explicit provide the threshold for µwhich
yields the transition from a dominant p0/parenleftbig
k,n+µ
1−k/parenrightbig
to the case in which p1(k,n)is the highest exponent.
Due to the fact that p0/parenleftbig
k,n+µ
1−k/parenrightbig
is the biggest solution of (5), we have that p1(k,n)>p0/parenleftbig
k,n+µ
1−k/parenrightbig
if
and only if
/parenleftBig
n−1
2+µ−k
2(1−k)/parenrightBig
p1(k,n)2−/parenleftBig
n+1
2+µ+3k
2(1−k)/parenrightBig
p1(k,n)−1>0.
By straightforward computations, it follows that p1(k,n)>p0/parenleftbig
k,n+µ
1−k/parenrightbig
forµ>µ0(k,n), where
µ0(k,n).=(1−k)2n2+(1−k)(1+2k)n+2
n(1−k)+2. (10)
Note that for k= 0the splitting value µ0(k,n)does coincide with the one for the semilinear wave equation
with scale – invariant damping in the flat case from the work [1 6].
Theorem 1.3 (Critical case: part I) .Let0/lessorequalslantµ/lessorequalslantµ0(k,n)such thatµ/lessorequalslantkorµ/greaterorequalslant2−k. We consider
p=p0/parenleftbig
k,n+µ
1−k/parenrightbig
. Let us assume that u0∈H1(Rn)andu1∈L2(Rn)are nonnegative, nontrivial and
compactly supported functions with supports contained in BRfor someR>0. Let
u∈C/parenleftbig
[1,T),H1(Rn)/parenrightbig
∩C1/parenleftbig
[1,T),L2(Rn)/parenrightbig
∩Lp
loc/parenleftbig
[1,T)×Rn/parenrightbig
3be an energy solution to (4)according to Definition 1.1 with lifespan T=T(ε)and satisfying the support
condition suppu(t,·)⊂BAk(t)+Rfor anyt∈(1,T).
Then, there exists a positive constant ε0=ε0(u0,u1,n,p,k,µ,R )such that for any ε∈(0,ε0]the energy
solutionublows up in finite time. Moreover, the upper bound estimate for the lifespan
T(ε)/lessorequalslantexp/parenleftBig
Cε−p(p−1)/parenrightBig
holds, where the positive constant Cis independent of ε.
Remark 2.It seems that the assumption in Theorem 1.3 for the multiplic ative constant µ/lessorequalslantkorµ/greaterorequalslant2−k
is technical, since it is due to the method we are going to appl y for the proof.
Theorem 1.4 (Critical case: part II) .Letµ/greaterorequalslantµ0(k,n)andp=p1(k,n). Let us assume that u0∈H1(Rn)
andu1∈L2(Rn)are nonnegative, nontrivial and compactly supported functio ns with supports contained in
BRfor someR>0. Let
u∈C/parenleftbig
[1,T),H1(Rn)/parenrightbig
∩C1/parenleftbig
[1,T),L2(Rn)/parenrightbig
∩Lp
loc/parenleftbig
[1,T)×Rn/parenrightbig
be an energy solution to (4)according to Definition 1.1 with lifespan T=T(ε)and satisfying the support
condition suppu(t,·)⊂BAk(t)+Rfor anyt∈(1,T).
Then, there exists a positive constant ε0=ε0(u0,u1,n,p,k,µ,R )such that for any ε∈(0,ε0]the energy
solutionublows up in finite time. Moreover, the upper bound estimate for the lifespan
T(ε)/lessorequalslantexp/parenleftBig
Cε−(p−1)/parenrightBig
holds, where the positive constant Cis independent of ε.
The remaining part of the paper is organized as follows: the p roof of the result in the subcritical case (cf.
Theorem 1.2) is carried out in Section 2; in Section 3 we prove Theorem 1.3 by generalizing the approach
introduced in [36]; finally, we show the proof of Theorem 1.4 i n Section 4 via a standard slicing procedure.
2 Subcritical case
In this section we are going to prove Theorem 1.2. Let ube a local in time solution to (4) and let us assume
that the assumptions from the statement of Theorem 1.2 on pand on the data are fulfilled. We will follow
the multiplier approach introduced by [20] and then improve d by [34], to derive a suitable iteration frame
for the time – dependent functional
U0(t).=/integraldisplay
Rnu(t,x)dx. (11)
In order to obtain a first lower bound estimate for U0we will introduce a second time – dependent functional,
following the main ideas of the pioneering paper [38] and ada pting them to the case with time – depend
coefficients as in [13, 12, 34, 31].
The section is organized as follows: in Section 2.1 we determ ine a suitable positive solution to the
adjoint homogeneous linear equation with separate variabl es, then, we use this function to derive a lower
bound estimate for U0in Section 2.3; in Sections 2.2 and 2.4 the derivation of the i teration frame and its
application in an iterative argument are dealt with, respec tively.
2.1 Solution of the adjoint homogeneous linear equation
In this section, we shall determine a particular positive so lution to the adjoint homogeneous linear equation
Ψss−s−2k∆Ψ−µs−1Ψs+µs−2Ψ = 0. (12)
First of all, we recall the remarkable function
ϕ(x).=/braceleftBigg/integraltext
Sn−1ex·ωdσωifn/greaterorequalslant2,
coshx ifn= 1,(13)
introduced in [38] for the study of the critical semilinear w ave equation. The main properties of this function
that will used throughout this paper are the following: ϕis a positive and smooth function that satisfies
∆ϕ=ϕand asymptotically behaves like cn|x|−n−1
2e|x|as|x| → ∞ .
4If we look for a solution to (12) with separate variables, tha t is, we consider the ansatz Ψ(s,x) =̺(s)ϕ(x),
then, it suffices to find a positive solution to the ODE
̺′′−s−2k̺−µs−1̺′+µs−2̺= 0. (14)
We perform the change of variable τ=φk(s). By using
̺′=t−kd̺
dτ, ̺′′=t−2kd2̺
dτ2−kt−1−kd̺
dτ,
it follows with straightforward computations that ̺solves (14) if and only if
d2̺
dτ2−k+µ
1−k1
τd̺
dτ+/parenleftbiggµ
(1−k)21
τ2−1/parenrightbigg
̺= 0. (15)
To further simplify the previous equation, we carry out the t ransformation ̺(τ) =τσζ(τ), whereσ.=1+µ
2(1−k).
Hence, using
d̺
dτ(τ) =στσ−1ζ(τ)+τσdζ
dτ(τ),d2̺
dτ2=σ(σ−1)τσ−2ζ(τ)+2στσ−1dζ
dτ(τ)+τσd2ζ
dτ2(τ),
we get that ̺is a solution to (15) if and only if ζsolves
τ2d2ζ
dτ2−/parenleftbigg
2σ−k+µ
1−k/parenrightbigg
τdζ
dτ+/bracketleftbigg
σ/parenleftbigg
σ−1−k+µ
1−k/parenrightbigg
+µ
(1−k)2−τ2/bracketrightbigg
ζ= 0. (16)
Due to the choice of the parameter σ, equation (16) is nothing but a modified Bessel equation of or der
γ.=µ−1
2(1−k), that is, (16) can be rewritten as
τ2d2ζ
dτ2−τdζ
dτ−(γ2+τ2)ζ= 0.
If we pick the modified Bessel function of the second kind Kγas solution to the previous equation, then, up
to a negligible multiplicative constant, we found
ρ(s).=s1+µ
2Kγ/parenleftbig
φk(s)/parenrightbig
(17)
as a positive solution to (14) and, in turn,
Ψ(s,x).=ρ(s)ϕ(x) =s1+µ
2Kγ/parenleftbig
φk(s)/parenrightbig
ϕ(x) (18)
as a positive solution of the adjoint equation (12).
In the next sections, we will need to employ the asymptotic be havior of the function ̺=̺(t)fort→ ∞.
SinceKγ(z) =/radicalbig
π/(2z)/parenleftbig
e−z+O(z−1)/parenrightbig
asz→ ∞ (cf. [23]), then, the following asymptotic estimate holds
̺(t) =/radicalbiggπ
2tk+µ
2e−φk(t)/parenleftbig
1+O(t−1+k)/parenrightbig
fort→ ∞. (19)
The solution Ψof the adjoint equation (12) that we determined in this secti on will be employed in
Section 2.3 to introduce a second time – dependent functiona l with the purpose to establish a first lower
bound estimate for U0.
2.2 Derivation of the iteration frame
In this section we are going to determine the iteration frame for the functional U0=U0(t)defined in (11).
Let us choose as test function ψ=ψ(s,x)in the integral relation (7) such that ψ= 1on the forward cone
{(s,x)∈[1,t]×Rn:|x|/lessorequalslantR+Ak(s)}. Then,
/integraldisplay
Rn∂tu(t,x)dx−ε/integraldisplay
Rnu1(x)dx+/integraldisplayt
1/integraldisplay
Rnµs−1∂tu(s,x)dxds=/integraldisplayt
1/integraldisplay
Rn|u(s,x)|pdxds
which can be rewritten as
U′
0(t)−U′
0(1)+/integraldisplayt
1µs−1U′
0(s)ds=/integraldisplayt
1/integraldisplay
Rn|u(s,x)|pdxds.
5Differentiating the last identity with respect to t, we get
U′′
0(t)+µt−1U′
0(t) =/integraldisplay
Rn|u(t,x)|pdx.
Multiplying the previous equation by tµ, it follows
tµU′′
0(t)+µtµ−1U′
0(t) =d
dt/parenleftbig
tµU′
0(t)/parenrightbig
=tµ/integraldisplay
Rn|u(t,x)|pdx.
Integrating twice this relation over [1,t], we find
U0(t) =U0(1)+U′
0(1)/integraldisplayt
1τ−µdτ+/integraldisplayt
1τ−µ/integraldisplayτ
1sµ/integraldisplay
Rn|u(s,x)|pdxdsdτ. (20)
On the one hand , from (20) we derive the lower bound estimate
U0(t)/greaterorsimilarε, (21)
where the unexpressed positive multiplicative constant de pends onu0,u1due to the nonnegativeness of u0,u1
andU(j)(1) =ε/integraltext
Rnuj(x)dxforj∈ {0,1}. On the other hand, we obtain the estimate
U0(t)/greaterorequalslant/integraldisplayt
1τ−µ/integraldisplayτ
1sµ/integraldisplay
Rn|u(s,x)|pdxdsdτ (22)
/greaterorsimilar/integraldisplayt
1τ−µ/integraldisplayτ
1sµ(R+Ak(s))−n(p−1)(U0(s))pdsdτ,
where in the second step we applied Jensen’s inequality and t he support property for u(s,·). Therefore, we
proved the following iteration frame for U0
U0(t)/greaterorequalslantC/integraldisplayt
1τ−µ/integraldisplayτ
1sµ−(1−k)n(p−1)(U0(s))pdsdτ (23)
for a suitable positive constant C=C(n,p,k)and fort/greaterorequalslant1. In Section 2.2 we will employ (23) to
derive iteratively a sequence of lower bound estimates for U0. However, we shall first derive in Section 2.3
another lower bound estimate for U0that will provide, together with (21), the starting point fo r the iteration
procedure.
2.3 First lower bound estimate for the functional
LetΨ = Ψ(t,x)be the function defined by (18). Since this function is smooth and positive, by applying the
integral relation (8) to Ψand using the fact that Ψsolves the adjoint equation (12), we get
0/lessorequalslant/integraldisplayt
1/integraldisplay
Rn|u(s,x)|pΨ(s,x)dxds
=/integraldisplay
Rn∂tu(t,x)Ψ(t,x)dx−/integraldisplay
Rnu(t,x)Ψs(t,x)dx+/integraldisplay
Rnµt−1u(t,x)Ψ(t,x)dx
−ε/integraldisplay
Rn(̺(1)u1(x)+(µ̺(1)−̺′(1))u0(x))ϕ(x)dx.
If we introduce the auxiliary functional
U1(t).=/integraldisplay
Rnu(t,x)Ψ(t,x)dx, (24)
then, from the last estimate we have
U′
1(t)−2̺′(t)
̺(t)U1(t)+µt−1U1(t)/greaterorequalslantε/integraldisplay
Rn/parenleftbig
̺(1)u1(x)+(µ̺(1)−̺′(1))u0(x)/parenrightbig
ϕ(x)dx, (25)
where we applied the relation
U′
1(t) =/integraldisplay
Rn∂tu(t,x)Ψ(t,x)dx+/integraldisplay
Rnu(t,x)ψs(t,x)dx=/integraldisplay
Rn∂tu(t,x)Ψ(t,x)dx+̺′(t)
̺(t)U1(t).
6Let compute more explicitly the term on the right – hand side o f (25) and show its positiveness. By using
the recursive identity
K′
γ(z) =−Kγ+1(z)+γ
zKγ(z)
for the derivative of the modified Bessel function of the seco nd kind and γ=µ−1
2(1−k), it follows
̺′(t) =1+µ
2tµ−1
2Kγ/parenleftbig
φk(t)/parenrightbig
+t1+µ
2−kK′
γ/parenleftbig
φk(t)/parenrightbig
=1+µ
2tµ−1
2Kγ/parenleftbig
φk(t)/parenrightbig
+t1+µ
2−k/parenleftBig
−Kγ+1/parenleftbig
φk(t)/parenrightbig
+µ−1
2t−1+kKγ/parenleftbig
φk(t)/parenrightbig/parenrightBig
=µtµ−1
2Kγ/parenleftbig
φk(t)/parenrightbig
−t1+µ
2−kKγ+1/parenleftbig
φk(t)/parenrightbig
.
In particular, the following relations hold
µ̺(1)−̺′(1) = K γ+1/parenleftbig
φk(1)/parenrightbig
>0, ̺(1) = K γ/parenleftbig
φk(1)/parenrightbig
>0,
so that we may rewrite (25) as
U′
1(t)−2̺′(t)
̺(t)U1(t)+µt−1U1(t)/greaterorequalslantε/integraldisplay
Rn/parenleftbig
Kγ/parenleftbig
φk(1)/parenrightbig
u1(x)+Kγ+1/parenleftbig
φk(1)/parenrightbig
u0(x)/parenrightbig
ϕ(x)dx
/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright.=Ik,µ[u0,u1]. (26)
Multiplying (26) by tµ/̺2(t), we have
d
dt/parenleftbiggtµ
̺2(t)U1(t)/parenrightbigg
=tµ
̺2(t)U′
1(t)−2̺′(t)
̺3(t)tµU1(t)+µtµ−11
̺2(t)U1(t)/greaterorequalslantεIk,µ[u0,u1]tµ
̺2(t).
Integrating the previous inequality over [1,t]and using the sign assumption on u0, we get
U1(t)/greaterorequalslant̺2(t)t−µ
̺2(1)U1(1)+εIk,µ[u0,u1]̺2(t)
tµ/integraldisplayt
1sµ
̺2(s)ds
/greaterorequalslantεIk,µ[u0,u1]̺2(t)
tµ/integraldisplayt
1sµ
̺2(s)ds.
Thanks to (19), there exists T0=T0(k,µ)>1such that
U1(t)/greaterorsimilarεIk,µ[u0,u1]tke−2φk(t)/integraldisplayt
T0s−ke2φk(s)ds
fort/greaterorequalslantT0. Consequently, for for t/greaterorequalslant2T0, shrinking the domain of integration in the last inequality , we have
U1(t)/greaterorsimilarεIk,µ[u0,u1]tke−2φk(t)/integraldisplayt
t/2s−ke2φk(s)ds= 2−1εIk,µ[u0,u1]tke−2φk(t)/parenleftBig
e2φk(t)−e2φk(t
2)/parenrightBig
= 2−1εIk,µ[u0,u1]tk/parenleftBig
1−e2φk(t
2)−2φk(t)/parenrightBig
= 2−1εIk,µ[u0,u1]tk/parenleftBig
1−e−2
1−k(1−2k−1)t1−k/parenrightBig
/greaterorequalslant2−1εIk,µ[u0,u1]tk/parenleftBig
1−e−2
1−k(21−k−1)T1−k
0/parenrightBig
/greaterorsimilarεtk. (27)
By repeating exactly the same computations as in [28, Sectio n 3] (which are completely independent of
the amplitude function Ak), we obtain
/integraldisplay
BR+Ak(t)(Ψ(t,x))p′dx= (̺(t))p′/integraldisplay
BR+Ak(t)(ϕ(x))p′dx/lessorsimilar(̺(t))p′ep′(R+Ak(t))(R+Ak(t))n−1−n−1
2p′.
Therefore, by using (19), for t/greaterorequalslantT0we get
/integraldisplay
BR+Ak(t)(Ψ(t,x))p′dx/lessorsimilarep′(R−φk(1))tk+µ
2p′(R+Ak(t))n−1−n−1
2p′
/lessorsimilart(1−k)(n−1)+[k+µ
2−(1−k)n−1
2]p′. (28)
Then, combining Hölder’s inequality, (27) and (28), it foll ows
/integraldisplay
Rn|u(t,x)|pdx/greaterorequalslant(U1(t))p/parenleftBigg/integraldisplay
BR+Ak(t)(Ψ(t,x))p′dx/parenrightBigg−(p−1)
/greaterorsimilarεptkp−(1−k)(n−1)(p−1)+[(1−k)n−1
2−k+µ
2]p
/greaterorsimilarεpt(1−k)(n−1)+k
2p−((1−k)n−1
2+µ
2)p(29)
7fort/greaterorequalslantT1.= 2T0. Finally, plugging (29) in (20), for t/greaterorequalslantT1it holds
U0(t)/greaterorequalslant/integraldisplayt
T1τ−µ/integraldisplayτ
T1sµ/integraldisplay
Rn|u(s,x)|pdxdsdτ/greaterorsimilarεp/integraldisplayt
T1τ−µ/integraldisplayτ
T1sµ+(1−k)(n−1)+k
2p−((1−k)n−1
2+µ
2)pdsdτ
/greaterorsimilarεpt−((1−k)n−1
2+µ
2)p−µ/integraldisplayt
T1/integraldisplayτ
T1(s−T1)µ+(1−k)(n−1)+k
2pdsdτ
/greaterorsimilarεpt−((1−k)n−1
2+µ
2)p−µ(t−T1)µ+(1−k)(n−1)+k
2p+2.
Summarizing we proved the lower bound estimate for the funct ionalU0
U0(t)/greaterorequalslantKεpt−a0(t−T1)b0(30)
fort/greaterorequalslantT1, whereK=K(n,k,µ,p,R,u 0,u1)is a suitable positive constant and
a0.=/parenleftbig
(1−k)n−1
2+µ
2/parenrightbig
p+µ, b 0.=µ+(1−k)(n−1)+k
2p+2. (31)
2.4 Iteration argument
In this section we will use the iteration frame (23) to prove t hatU0blows up in finite time under the
assumptions of Theorem 1.2. More precisely, we are going to p rove the sequence of lower bound estimates
U0(t)/greaterorequalslantDjt−aj(t−T1)bj(32)
fort/greaterorequalslantT1, where{Dj}j∈N,{aj}j∈Nand{bj}j∈Nare sequences of nonnegative real numbers that will be
determined iteratively during the proof.
Clearly, for j= 0the estimate in (32) is nothing but (30) with D0=Kεpanda0,b0defined by (31). In
order to prove (32) via an inductive argument, it remains jus t to prove the inductive step. Let us assume
the validity of (32) for j. We prove now its validity for j+1too.
Plugging (32) into (23), for t>T1we get
U0(t)/greaterorequalslantC/integraldisplayt
T1τ−µ/integraldisplayτ
T1sµ−(1−k)n(p−1)(U0(s))pdsdτ
/greaterorequalslantCDp
j/integraldisplayt
T1τ−µ/integraldisplayτ
T1sµ−(1−k)n(p−1)−ajp(s−T1)bjpdsdτ
/greaterorequalslantCDp
jt−(1−k)n(p−1)−µ−ajp/integraldisplayt
T1/integraldisplayτ
T1(s−T1)µ+bjpdsdτ
=CDp
j
(1+µ+bjp)(2+µ+bjp)t−(1−k)n(p−1)−µ−ajp(t−T1)2+µ+bjp,
which is exactly (32) for j+1provided that
Dj+1.=CDp
j
(1+µ+bjp)(2+µ+bjp), (33)
aj+1.= (1−k)n(p−1)+µ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright.=α+paj, bj+1.= 2+µ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright.=β+pbj. (34)
Employing recursively (34), we may express explicitly ajandbjas follows
aj=α+paj−1=···=αj−1/summationdisplay
k=0pk+a0pj=/parenleftBig
α
p−1+a0/parenrightBig
pj−α
p−1, (35)
bj=β+pbj−1=···=βj−1/summationdisplay
k=0pk+b0pj=/parenleftBig
β
p−1+b0/parenrightBig
pj−β
p−1. (36)
Combining (34) and (36), we find
bj= 2+µ+pbj−1</parenleftBig
β
p−1+b0/parenrightBig
pj,
8that implies, in turn,
Dj/greaterorequalslantCDp
j−1
(2+µ+pbj−1)2=CDp
j−1
b2
j/greaterorequalslantC
/parenleftBig
β
p−1+b0/parenrightBig2
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright.=/tildewideCDp
j−1p−2j=/tildewideCDp
j−1p−2j.
Applying the logarithmic function to both sides of the last i nequality and using the resulting inequality
iteratively, we get
logDj/greaterorequalslantplogDj−1−2jlogp+log/tildewideC
/greaterorequalslantp2logDj−2−2(j+(j−1)p)logp+(1+p)log/tildewideC
/greaterorequalslant···/greaterorequalslantpjlogD0−2logpj−1/summationdisplay
k=0(j−k)pk+log/tildewideCj−1/summationdisplay
k=0pk.
Using the formulas
j−1/summationdisplay
k=0(j−k)pk=1
p−1/parenleftbiggpj+1−p
p−1−j/parenrightbigg
andj−1/summationdisplay
k=0pk=pj−1
p−1, (37)
that can be shown via an inductive argument, we obtain
logDj/greaterorequalslantpjlogD0−2logp
p−1/parenleftbiggpj+1−p
p−1−j/parenrightbigg
+(pj−1)log/tildewideC
p−1
=pj/parenleftBigg
logD0−2plogp
(p−1)2+log/tildewideC
p−1/parenrightBigg
+2jlogp
p−1+2plogp
(p−1)2−log/tildewideC
p−1.
Let us denote by j0=j0(n,p,k,µ)∈Nthe smallest integer greater thanlog/tildewideC
2logp−p
p−1. Then, for any j/greaterorequalslantj0
we have
logDj/greaterorequalslantpj/parenleftBigg
logD0−2plogp
(p−1)2+log/tildewideC
p−1/parenrightBigg
=pjlog/parenleftBig
Kp−(2p)/(p−1)2/tildewideC1/(p−1)εp/parenrightBig
=pjlog(E0εp),(38)
whereE0.=Kp−(2p)/(p−1)2/tildewideC1/(p−1). Combining (32), (35), (36) and (38), for j/greaterorequalslantj0andt/greaterorequalslantT1it holds
U0(t)/greaterorequalslantexp/parenleftbig
pjlog(E0εp)/parenrightbig
t−aj(t−T1)bj
= exp/parenleftBig
pj/parenleftBig
log(E0εp)−/parenleftBig
α
p−1+a0/parenrightBig
logt+/parenleftBig
β
p−1+b0/parenrightBig
log(t−T1)/parenrightBig/parenrightBig
tα/(p−1)(t−T1)−β/(p−1).
Fort/greaterorequalslant2T1, we have log(t−T1)/greaterorequalslantlog(t/2), so forj/greaterorequalslantj0
U0(t)/greaterorequalslantexp/parenleftBig
pj/parenleftBig
log(E0εp)+/parenleftBig
β−α
p−1+b0−a0/parenrightBig
logt−/parenleftBig
β
p−1+b0/parenrightBig
log2/parenrightBig/parenrightBig
tα/(p−1)(t−T1)−β/(p−1)
= exp/parenleftBig
pj/parenleftBig
log/parenleftBig
2−b0−β/(p−1)E0εptθ(n,k,µ,p )
p−1/parenrightBig/parenrightBig/parenrightBig
tα/(p−1)(t−T1)−β/(p−1), (39)
where for the exponent of tin the last equality we used
β−α
p−1+b0−a0=2
p−1−(1−k)n+(1−k)(n−1)+k
2p+2−/parenleftbig
(1−k)n−1
2+µ
2/parenrightbig
p
=2p
p−1−(1−k)−/parenleftBig
(1−k)n−1
2+µ−k
2/parenrightBig
p
=1
p−1/braceleftBig
1−k+/parenleftBig
(1−k)n+1
2+µ+3k
2/parenrightBig
p−/parenleftBig
(1−k)n−1
2+µ−k
2/parenrightBig
p2/bracerightBig
=θ(n,k,µ,p)
p−1.(40)
Note thatθ(n,k,µ,p)is a positive quantity for p < p0/parenleftbig
k,n+µ
1−k/parenrightbig
. Let us fix ε0>0sufficiently small so
that
ε−p(p−1)
θ(n,k,µ,p )
0 /greaterorequalslant21−(b0(p−1)+β)/θ(n,k,µ,p)T1.
Then, for any ε∈(0,ε0]and fort/greaterorequalslant2(b0(p−1)+β)/θ(n,k,µ,p)ε−p(p−1)
θ(n,k,µ,p )it results
t/greaterorequalslant2T1and2−b0−β/(p−1)E0εptθ(n,k,µ,p )
p−1>1,
9also, letting j→ ∞ in (39) it turns out that U0(t)blows up. Consequently, we proved the blowing – up of
U0in finite time for any ε∈(0,ε0]wheneverp<p0/parenleftbig
k,n+µ
1−k/parenrightbig
and, moreover, as byproduct we found the
upper bound estimate for the lifespan T(ε)/lessorsimilarε−p(p−1)
θ(n,k,µ,p )as well.
So far we applied only the lower bound estimate in (30) for U0. Nevertheless, we also proved another
lower bound estimate for U0, namely, (21). Using (21) instead of (30), the initial value s for the parameters
in (32) are a0=b0= 0andD0≈ε. Repeating the computations analogously as in the previous case and
using
logDj/greaterorequalslantpjlog(E1ε)
forj/greaterorequalslantj1, wherej1is a suitable nonnegative integer and E1is a suitable positive constant, in place of (38)
and
β−α
p−1+b0−a0=2
p−1−(1−k)n
instead of (40), we obtain immediately the blow – up of U0in finite time for p < p1(k,n)and the corre-
sponding upper bound estimate for the lifespan in (9).
3 Critical case: part I
In order to study the critical case p=p0/parenleftbig
k,n+µ
1−k/parenrightbig
, we will follow an approach which is based on the
technique introduced in [36] and subsequently applied to di fferent frameworks in [37, 29, 30, 21, 3, 4, 27].
From (39) it is clear that we can no longer employ U0as functional to study the blow – up dynamic.
Therefore, we need to sharpen the choice of the functional. M ore precisely, we are going to consider a weighted
space average of a local in time solution to (4). Hence, the bl ow – up result will be proved by applying the
so – called slicing procedure in an iteration argument to show a sequence of lower bound est imates for the
above mentioned functional. Throughout this section we wor k under the assumptions of Theorem 1.3.
The section is organized as follows: in Section 3.1 we determ ine a pair of auxiliary functions which have
a fundamental role in the definition of the time – dependent fu nctional and in the determination of the
iteration frame, while in Section 3.2 we establish some fund amental properties for these functions; finally,
in Section 3.3 we determine the iteration frame for the weigh ted space average whose dynamic provides the
blow – up result.
3.1 Auxiliary functions
In this section, we introduce two auxiliary functions (see ξqandηqbelow). These auxiliary functions represent
a generalization of the solution to the classical free wave e quation given in [39] and are defined by using the
remarkable function ϕintroduced in [38], that we have already used in the section f or the subcritical case
(the definition of this function is given in (13)).
According to our purpose of introducing the auxiliary funct ions, we begin by determining the solutions
yj=yj(t,s;λ,k,µ),j∈ {0,1}of the non – autonomous, parameter – dependent, ordinary Cau chy problems
∂2
tyj(t,s;λ,k,µ)−λ2t−2kyj(t,s;λ,k,µ)+µt−1yj(t,s;λ,k,µ) = 0, t>s,
yj(s,s;λ,k,µ) =δ0j,
∂tyj(s,s;λ,k,µ) =δ1j,(41)
whereδijdenotes the Kronecker delta, s/greaterorequalslant1is the initial time and λ >0is a real parameter. To find a
system of independent solutions to
d2y
dt2−λ2t−2ky+µt−1dy
dt= 0 (42)
we start by performing the change of variable τ=τ(t;λ,k).=λφk(t). By the straightforward relations
dy
dt=λt−kdy
dτ,d2y
dt2=λ2t−2kd2y
dτ2−λkt−k−1dy
dτ,
it follows that ysolves (42) if and only if
τd2y
dτ2+µ−k
1−kdy
dτ−τy= 0. (43)
10Carrying out the transformation y(τ) =τνw(τ)withν=ν(k,µ).=1−µ
2(1−k), it turns out that ysolves (43) if
and only if wsolves the modified Bessel equation of order ν
τ2d2w
dτ2+τdw
dτ−/parenleftbig
ν2+τ2/parenrightbig
w= 0. (44)
Employing the modified Bessel function of first and second kin d of orderν, denoted, respectively, by Iν(τ)
andKν(τ), as independent solutions to (44), then, we obtain
V0(t;λ,k,µ).=τνIν(τ) = (λφk(t))νIν(λφk(t)),
V1(t;λ,k,µ).=τνKν(τ) = (λφk(t))νKν(λφk(t))
as basis for the space of solutions to (42).
Proposition 3.1. The functions
y0(t,s;λ,k,µ).=λφk(s)sµ−1
2t1−µ
2/bracketleftbig
Iν−1(λφk(s))Kν(λφk(t))+K ν−1(λφk(s))Iν(λφk(t))/bracketrightbig
, (45)
y1(t,s;λ,k,µ).= (1−k)−1s1+µ
2t1−µ
2/bracketleftbig
Kν(λφk(s))Iν(λφk(t))−Iν(λφk(s))Kν(λφk(t))/bracketrightbig
, (46)
solve the Cauchy problems (41)forj= 0andj= 1, respectively, where ν=1−µ
2(1−k)andIν,Kνdenote the
modified Bessel function of order νof the first and second kind, respectively.
Proof. Since we proved that V0,V1form a system of independent solutions to (42), we may expres s the
solutions to (41) as linear combinations of V0,V1in the following way
yj(t,s;λ,k,µ) =aj(s;λ,k,µ)V0(t;λ,k,µ)+bj(s;λ,k,µ)V1(t;λ,k,µ) (47)
for suitable coefficients aj(s;λ,k,µ),bj(s;λ,k,µ), withj∈ {0,1}.
We can describe the initial conditions ∂i
tyj(s,s;λ,k) =δijthrough the system
/parenleftbiggV0(s;λ,k,µ)V1(s;λ,k,µ)
∂tV0(s;λ,k,µ)∂tV1(s;λ,k,µ)/parenrightbigg/parenleftbigga0(s;λ,k,µ)a1(s;λ,k,µ)
b0(s;λ,k,µ)b1(s;λ,k,µ)/parenrightbigg
=I,
whereIdenotes the identity matrix. Also, to determine the coefficie nts in (47), we calculate the inverse
matrix
/parenleftbiggV0(s;λ,k,µ)V1(s;λ,k,µ)
∂tV0(s;λ,k,µ)∂tV1(s;λ,k,µ)/parenrightbigg−1
= (W(V0,V1)(s;λ,k,µ))−1/parenleftbigg∂tV1(s;λ,k,µ)−V1(s;λ,k,µ)
−∂tV0(s;λ,k,µ)V0(s;λ,k,µ)/parenrightbigg
,
(48)
whereW(V0,V1)denotes the Wronskian of V0,V1. Next, we compute explicitly the function W(V0,V1).
Thanks to
∂tV0(t;λ,k,µ) =ν(λφk(t))ν−1λφ′
k(t)Iν(λφk(t))+(λφk(t))νI′
ν(λφk(t))λφ′
k(t),
∂tV1(t;λ,k,µ) =ν(λφk(t))ν−1λφ′
k(t)Kν(λφk(t))+(λφk(t))νK′
ν(λφk(t))λφ′
k(t),
recallingφ′
k(t) =t−kand2ν−1 =k−µ
1−k, we can express W(V0,V1)as follows:
W(V0,V1)(t;λ,k,µ) = (λφk(t))2ν(λφ′
k(t))/braceleftbig
K′
ν(λφk(t))Iν(λφk(t))−I′
ν(λφk(t))Kν(λφk(t))/bracerightbig
= (λφk(t))2ν(λφ′
k(t))W(Iν,Kν)(λφk(t)) =−(λφk(t))2ν−1(λφ′
k(t))
=−λ2ν(φk(t))2ν−1φ′
k(t) =−c−1
k,µλ2νt−µ,
whereck,µ.= (1−k)k−µ
1−kand in the third equality we used the value of the Wronskian of Iν,Kν
W(Iν,Kν)(z) = Iν(z)∂Kν
∂z(z)−Kν(z)∂Iν
∂z(z) =−1
z.
Plugging the previously determined representation of W(V0,V1)in (48), we have
/parenleftbigg
a0(s;λ,k,µ)a1(s;λ,k,µ)
b0(s;λ,k,µ)b1(s;λ,k,µ)/parenrightbigg
=ck,µλ−2νsµ/parenleftbigg
−∂tV1(s;λ,k,µ)V1(s;λ,k,µ)
∂tV0(s;λ,k,µ)−V0(s;λ,k,µ)/parenrightbigg
.
11Let us begin by showing (45). Using the above representation ofa0(s;λ,kµ),b0(s;λ,k,µ)in (47), we find
y0(t,s;λ,k,µ) =ck,µλ−2νsµ/braceleftbig
∂tV0(s;λ,k,µ)V1(t;λ,k,µ)−∂tV1(s;λ,k,µ)V0(t;λ,k,µ)/bracerightbig
=ck,µνsµφ′
k(s)(φk(s))ν−1(φk(t))ν/braceleftbig
Iν(λφk(s))Kν(λφk(t))−Kν(λφk(s))Iν(λφk(t))/bracerightbig
+ck,µλsµφ′
k(s)(φk(s))ν(φk(t))ν/braceleftbig
I′
ν(λφk(s))Kν(λφk(t))−K′
ν(λφk(s))Iν(λφk(t))/bracerightbig
.
Using the following recursive relations for the derivative s of the modified Bessel functions
∂Iν
∂z(z) =−ν
zIν(z)+Iν−1(z),
∂Kν
∂z(z) =−ν
zKν(z)−Kν−1(z),
there is a cancellation in the last relation, so, we arrive at
y0(t,s;λ,k,µ) =ck,µλsµφ′
k(s)(φk(s)φk(t))ν/braceleftbig
Iν−1(λφk(s))Kν(λφk(t))+K ν−1(λφk(s))Iν(λφk(t))/bracerightbig
.(49)
Thanks to
ck,µsµφ′
k(s)(φk(s)φk(t))ν= (1−k)−1sµ−k(st)1−µ
2=φk(s)sµ−1
2t1−µ
2,
from (49) it follows immediately (45). Let us show now the rep resentation for y1. Plugging the above
determined expressions for a1(s;λ,k,µ),b1(s;λ,k,µ)in (47), we get
y1(t,s;λ,k,µ) =ck,µλ−2νsµ/braceleftbig
V1(s;λ,k,µ)V0(t;λ,k,µ)−V0(s;λ,k,µ)V1(t;λ,k,µ)/bracerightbig
=ck,µλ−2νsµ(λφk(s))ν(λφk(t))ν/braceleftbig
Kν(λφk(s))Iν(λφk(t))−Iν(λφk(s))Kν(λφk(t))/bracerightbig
=ck,µsµ(φk(s)φk(t))ν/braceleftbig
Kν(λφk(s))Iν(λφk(t))−Iν(λφk(s))Kν(λφk(t))/bracerightbig
. (50)
Hence, due to ck,µsµ(φk(s)φk(t))ν= (1−k)−1s1+µ
2t1−µ
2, from (50) it results (46). The proof is complete.
Lemma 3.2. Lety0,y1be the functions defined in (45)and(46), respectively. Then, the following identities
are satisfied for any t/greaterorequalslants/greaterorequalslant1
∂y1
∂s(t,s;λ,k,µ) =−y0(t,s;λ,k,µ)+µs−1y1(t,s;λ,k,µ), (51)
∂2y1
∂s2(t,s;λ,k,µ)−λ2s−2ky1(t,s;λ,k,µ)−µs−1∂y1
∂s(t,s;λ,k,µ)+µs−2y1(t,s;λ,k,µ) = 0.(52)
Remark 3.As the operator ∂2
s−λ2s−2k−µs−1∂s+µs−2is the formal adjoint of ∂2
t−λ2t−2k+µt−1∂t, in
particular, (51) and (52) tell us that y1solves also the adjoint problem to (42) with final conditions (0,−1).
Proof. Let us introduce the pair of independent solutions to (42)
z0(t;λ,k,µ).=y0(t,1;λ,k,µ),
z1(t;λ,k,µ).=y1(t,1;λ,k,µ).
Since the Wronskian W(z0,z1)(t;λ,k,µ)solves the differential equation W′(z0,z1) =−µt−1W(z0,z1)with
initial condition W(z0,z1)(1;λ,k,µ) = 1 , then,W(z0,z1)(t;λ,k,µ) =t−µ. Therefore, repeating similar
computations as in the proof of Proposition 3.1, we may show t he representations
y0(t,s;λ,k,µ) =sµ{z′
1(s;λ,k,µ)z0(t;λ,k,µ)−z′
0(s;λ,k,µ)z1(t;λ,k,µ)},
y1(t,s;λ,k,µ) =sµ{z0(s;λ,k,µ)z1(t;λ,k,µ)−z1(s;λ,k,µ)z0(t;λ,k,µ)}.
Let us prove (51). Differentiating the second one of the previ ous representations with respect to s, we find
∂y1
∂s(t,s;λ,k) =µsµ−1{z0(s;λ,k,µ)z1(t;λ,k,µ)−z1(s;λ,k,µ)z0(t;λ,k,µ)}
+sµ{z′
0(s;λ,k,µ)z1(t;λ,k,µ)−z′
1(s;λ,k,µ)z0(t;λ,k,µ)}
=µs−1y1(t,s;λ,k,µ)−y0(t,s;λ,k,µ).
12On the other hand, due to the fact that z0,z1satisfy (42), then,
∂2y1
∂s2(t,s;λ,k) =sµ{z′′
0(s;λ,k,µ)z1(t;λ,k,µ)−z′′
1(s;λ,k,µ)z0(t;λ,k,µ)}
+2µsµ−1{z′
0(s;λ,k,µ)z1(t;λ,k,µ)−z′
1(s;λ,k,µ)z0(t;λ,k,µ)}
+µ(µ−1)sµ−2{z0(s;λ,k,µ)z1(t;λ,k,µ)−z1(s;λ,k,µ)z0(t;λ,k,µ)}
=sµ/braceleftbig/bracketleftbig
λ2s−2kz0(s;λ,k,µ)−µs−1z′
0(s;λ,k,µ)/bracketrightbig
z1(t;λ,k,µ)
−/bracketleftbig
λ2s−2kz1(s;λ,k,µ)−µs−1z′
1(s;λ,k,µ)/bracketrightbig
z0(t;λ,k,µ)/bracerightbig
+2µsµ−1{z′
0(s;λ,k,µ)z1(t;λ,k,µ)−z′
1(s;λ,k,µ)z0(t;λ,k,µ)}
+µ(µ−1)sµ−2{z0(s;λ,k,µ)z1(t;λ,k,µ)−z1(s;λ,k,µ)z0(t;λ,k,µ)}
=λ2s−2ksµ/braceleftbig
z0(s;λ,k,µ)z1(t;λ,k,µ)−z1(s;λ,k,µ)z0(t;λ,k,µ)/bracerightbig
+µsµ−1{z′
0(s;λ,k,µ)z1(t;λ,k,µ)−z′
1(s;λ,k,µ)z0(t;λ,k,µ)}
+µ(µ−1)sµ−2{z0(s;λ,k,µ)z1(t;λ,k,µ)−z1(s;λ,k,µ)z0(t;λ,k,µ)}
=λ2s−2ky1(t,s;λ,k,µ)−µs−1y0(t,s;λ,k,µ)+µ(µ−1)s−2y1(t,s;λ,k,µ).
Applying (51), from the last chain of equalities we get
∂2y1
∂s2(t,s;λ,k) =λ2s−2ky1(t,s;λ,k,µ)+µs−1/parenleftbigg∂y1
∂s(t,s;λ,k)−µs−1y1(t,s;λ,k,µ)/parenrightbigg
+µ(µ−1)s−2y1(t,s;λ,k,µ)
=λ2s−2ky1(t,s;λ,k,µ)+µs−1∂y1
∂s(t,s;λ,k)−µs−2y1(t,s;λ,k,µ).
Thus, we proved (52) too. This completes the proof.
Proposition 3.3. Letu0∈H1(Rn)andu1∈L2(Rn)be functions such that suppuj⊂BRforj= 0,1
and for some R>0and letλ>0be a parameter. Let ube a local in time energy solution to (4)on[1,T)
according to Definition 1.1. Then, the following integral id entity is satisfied for any t∈[1,T)
/integraldisplay
Rnu(t,x)ϕλ(x)dx=εy0(t,1;λ,k)/integraldisplay
Rnu0(x)ϕλ(x)dx+εy1(t,1;λ,k)/integraldisplay
Rnu1(x)ϕλ(x)dx
+/integraldisplayt
1y1(t,s;λ,k)/integraldisplay
Rn|u(s,x)|pϕλ(x)dxds, (53)
whereϕλ(x).=ϕ(λx)andϕis defined by (13).
Proof. Assumingu0,u1compactly supported, we can consider a test function ψ∈C∞([1,T)×Rn)in
Definition 1.1 according to Remark 1. Hence, we take ψ(s,x) =y1(t,s;λ,k,µ)ϕλ(x)(heret,λcan be treated
as fixed parameters). Consequently, ψsatisfies
ψ(t,x) =y1(t,t;λ,k,µ)ϕλ(x) = 0, ψ(1,x) =y1(t,1;λ,k,µ)ϕλ(x),
ψs(t,x) =∂sy1(t,t;λ,k,µ)ϕλ(x) =/parenleftbig
µt−1y1(t,t;λ,k,µ)−y0(t,t;λ,k,µ)/parenrightbig
ϕλ(x) =−ϕλ(x),
ψs(1,x) =∂sy1(t,1;λ,k,µ)ϕλ(x) = (µy1(t,1;λ,k,µ)−y0(t,1;λ,k,µ))ϕλ(x),
and
ψss(s,x)−s−2k∆ψ(s,x)−µ∂s(s−1ψ(s,x)) =/parenleftbig
∂2
s−λ2s−2k−µs−1∂s+µs−2/parenrightbig
y1(t,s;λ,k,µ)ϕλ(x) = 0,
where we used (51), (52) and the property ∆ϕλ=λ2ϕλ. Then, employing the above defined ψin (8), we
find immediately (53). This completes the proof.
Proposition 3.4. Lety0,y1be the functions defined in (45)and(46), respectively. Then, the following
estimates are satisfied for any t/greaterorequalslants/greaterorequalslant1
y0(t,s;λ,k,µ)/greaterorequalslantsµ−k
2tk−µ
2cosh/parenleftbig
λ(φk(t)−φk(s))/parenrightbig
ifµ∈[2−k,∞), (54)
y1(t,s;λ,k,µ)/greaterorequalslantsµ+k
2tk−µ
2sinh/parenleftbig
λ(φk(t)−φk(s))/parenrightbig
λifµ∈[0,k]∪[2−k,∞). (55)
13Proof. The proof of the inequalities (54) and (55) is based on the fol lowing minimum type principle:
letw=w(t,s;λ,k,µ)be a solution of the Cauchy problem
/braceleftBigg
∂2
tw−λ2t−2kw+µt−1∂tw=h,fort>s/greaterorequalslant1,
w(s) =w0, ∂tw(s) =w1,(56)
whereh=h(t,s;λ,k,µ)is a continuous function; if h/greaterorequalslant0andw0=w1= 0(i.e.wis asupersolution of
the homogeneous problem with trivial initial conditions), t hen,w(t,s;λ,k,µ)/greaterorequalslant0for anyt>s.
In order to prove this minimum principle, we apply the contin uous dependence on initial conditions (note
that fort/greaterorequalslant1the functions t−2kandµt−1are smooth). Indeed, if we denote by wǫthe solution to (56) with
w0=ǫ>0andw1= 0, then,wǫsolves the integral equation
wǫ(t,s;λ,k,µ) =ǫ+/integraldisplayt
sτ−µ/integraldisplayτ
sσµ/parenleftbig
λ2σ−2kwǫ(σ,s;λ,k,µ)+h(σ,s;λ,k,µ)/parenrightbig
dσdτ.
By contradiction, one can prove easily that wǫ(t,s;λ,k,µ)>0for anyt > s. Hence, by the continuous
dependence on initial data, letting ǫ→0, we find that w(t,s;λ,k,µ)/greaterorequalslant0for anyt>s.
Let us prove the validity of (55). Denoting by w1=w1(t,s;λ,k,µ)the function on the right – hand side
of (55), we find immediately w1(s,s;λ,k,µ) = 0 and∂tw1(s,s;λ,k,µ) = 1. Moreover,
∂2
tw1(t,s;λ,k,µ) =λ−1sk+µ
2tk−µ
2/bracketleftBig
k−µ
2/parenleftBig
k−µ
2−1/parenrightBig
t−2sinh/parenleftbig
λ(φk(t)−φk(s))/parenrightbig
+(k−µ)t−1cosh/parenleftbig
λ(φk(t)−φk(s))/parenrightbig
λφ′
k(t)
+sinh/parenleftbig
λ(φk(t)−φk(s))/parenrightbig
(λφ′
k(t))2+cosh/parenleftbig
λ(φk(t)−φk(s))/parenrightbig
λφ′′
k(t)/bracketrightBig
=/bracketleftBig
k−µ
2/parenleftBig
k−µ
2−1/parenrightBig
t−2+λ2t−2k/bracketrightBig
w1(t,s;λ,k,µ)−µsk+µ
2t−1−k+µ
2cosh/parenleftbig
λ(φk(t)−φk(s))/parenrightbig
and
∂tw1(t,s;λ,k,µ) =λ−1sk+µ
2tk−µ
2/bracketleftBig
k−µ
2t−1sinh/parenleftbig
λ(φk(t)−φk(s))/parenrightbig
+λt−kcosh/parenleftbig
λ(φk(t)−φk(s))/parenrightbig/bracketrightBig
=k−µ
2t−1w1(t,s;λ,k,µ)+sk+µ
2t−k+µ
2cosh/parenleftbig
λ(φk(t)−φk(s))/parenrightbig
imply that
∂2
tw1(t,s;λ,k,µ)−λ2t−2kw1(t,s;λ,k,µ)+µt−1∂tw1(t,s;λ,k,µ) =k−µ
2/parenleftBig
k+µ
2−1/parenrightBig
w1(t,s;λ,k,µ)/lessorequalslant0,
where in the last step we employ the assumption µ /∈(k,2−k)to guarantee that the multiplicative constant
is negative. Therefore, y1−w1is a supersolution of (56) with h= 0andw0=w1= 0. Thus, applying the
minimum principle we have that (y1−w1)(t,s;λ,k)/greaterorequalslant0for anyt>s, that is, we showed (55).
In a completely analogous way, one can prove (54), repeating the previous argument based on the mini-
mum principle with w0(t,s;λ,k,µ).=sµ−k
2tk−µ
2cosh/parenleftbig
λ(φk(t)−φk(s))/parenrightbig
in place of w1(t,s;λ,k,µ)andy0in
place ofy1, respectively. However, in order to guarantee that w0(s,s;λ,k,µ) = 1 and∂tw0(s,s;λ,k,µ)/lessorequalslant0,
we are forced to require µ/greaterorequalslantk, which provides, together with the condition µ /∈(k,2−k)that is necessary
to ensure that w0is actually a subsolution of the homogeneous equation, the r ange forµin (54).
Remark 4.Although (54) might be restrictive from the viewpoint of the range forµin the statement of
Theorem 1.3, we can actually overcome this difficulty by showi ng a transformation which allows to link the
caseµ∈[0,k]to the case µ∈[2−k,2], when a lower bound estimate for y0is available. Indeed, if we
perform the transformation v=v(t,x).=tµ−1u(t,x), then,uis a solution to (4) if and only if vsolves
vtt−t−2k∆v+(2−µ)t−1vt=t(1−µ)(p−1)|v|px∈Rn, t∈(1,T),
v(1,x) =εu0(x) x∈Rn,
ut(1,x) =εu1(x)+ε(1−µ)u0(x) x∈Rn.(57)
Let us point out that in (57) a time – dependent factor which de cays with polynomial order appears in the
nonlinear term on the right – hand side. Therefore, we will re duce the case µ∈[0,k]to the case µ/greaterorequalslant2−k,
up to the time – dependent factor t(1−µ)(p−1)in the nonlinearity.
We can introduce now for t/greaterorequalslants/greaterorequalslant1andx∈Rnthe definition of the following auxiliary function
ξq(t,s,x;k,µ).=/integraldisplayλ0
0e−λ(Ak(t)+R)y0(t,s;λ,k,µ)ϕλ(x)λqdλ, (58)
ηq(t,s,x;k,µ).=/integraldisplayλ0
0e−λ(Ak(t)+R)y1(t,s;λ,k,µ)
φk(t)−φk(s)ϕλ(x)λqdλ, (59)
14whereq>−1,λ0>0is a fixed parameter and Akis defined by (6).
Combining Proposition 3.3 and (58) and (59), we establish a f undamental equality, whose role will be
crucial in the next sections in order to prove the blow – up res ult.
Corollary 3.5. Letu0∈H1(Rn)andu1∈L2(Rn)such that suppuj⊂BRforj= 0,1and for some R>0.
Letube a local in time energy solution to (4)on[1,T)according to Definition 1.1. Let q >−1and let
ξq(t,s,x;k),ηq(t,s,x;k)be the functions defined by (58)and(59), respectively. Then,
/integraldisplay
Rnu(t,x)ξq(t,t,x;k,µ)dx=ε/integraldisplay
Rnu0(x)ξq(t,1,x;k,µ)dx+ε(φk(t)−φk(1))/integraldisplay
Rnu1(x)ηq(t,s,x;k,µ)dx
+/integraldisplayt
1(φk(t)−φk(s))/integraldisplay
Rn|u(s,x)|pηq(t,s,x;k,µ)dxds (60)
for anyt∈[1,T).
Proof. Multiplying both sides of (53) by e−λ(Ak(t)+R)λq, integrating with respect to λover[0,λ0]and
applying Fubini’s theorem, we get easily (60).
3.2 Properties of the auxiliary functions
In this section, we establish lower and upper bound estimate s for the auxiliary functions ξq,ηqunder suitable
assumptions on q. In the lower bound estimates, we may restrict our considera tions to the case µ/greaterorequalslant2−k
thanks to Remark 4, even though the estimate for ηqthat will be proved thanks to (55) clearly would be
true also for µ∈[0,k].
Lemma 3.6. Letn/greaterorequalslant1,k∈[0,1),µ/greaterorequalslant2−kandλ0>0. If we assume q >−1, then, fort/greaterorequalslants/greaterorequalslant1and
|x|/lessorequalslantAk(s)+Rthe following lower bound estimates are satisfied:
ξq(t,s,x;k,µ)/greaterorequalslantB0sµ−k
2tk−µ
2/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−q−1; (61)
ηq(t,s,x;k,µ)/greaterorequalslantB1sµ+k
2tk−µ
2/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−q. (62)
HereB0,B1are positive constants depending only on λ0,q,R,k and we employ the notation /an}b∇acketle{ty/an}b∇acket∇i}ht.= 3+|s|.
Proof. We adapt the main ideas in the proof of Lemma 3.1 in [36] to our m odel. Since
/an}b∇acketle{t|x|/an}b∇acket∇i}ht−n−1
2e|x|/lessorsimilarϕ(x)/lessorsimilar/an}b∇acketle{t|x|/an}b∇acket∇i}ht−n−1
2e|x|(63)
holds for any x∈Rn, there exists a constant B=B(λ0,R,k)>0independent of λandssuch that
B/lessorequalslant inf
λ∈/bracketleftBigλ0
/angbracketleftAk(s)/angbracketright,2λ0
/angbracketleftAk(s)/angbracketright/bracketrightBiginf
|x|/lessorequalslantAk(s)+Re−λ(Ak(s)+R)ϕλ(x).
Let us begin by proving (61). Using the lower bound estimate i n (54), shrinking the domain of integration
in (58) to/bracketleftBig
λ0
/angbracketleftAk(s)/angbracketright,2λ0
/angbracketleftAk(s)/angbracketright/bracketrightBig
and applying the previous inequality, we arrive at
ξq(t,s,x;k,µ)/greaterorequalslantsµ−k
2tk−µ
2/integraldisplay2λ0//angbracketleftAk(s)/angbracketright
λ0//angbracketleftAk(s)/angbracketrighte−λ(Ak(t)−Ak(s))cosh/parenleftbig
λ(φk(t)−φk(s))/parenrightbig
e−λ(Ak(s)+R)ϕλ(x)λqdλ
/greaterorequalslantBsµ−k
2tk−µ
2/integraldisplay2λ0//angbracketleftAk(s)/angbracketright
λ0//angbracketleftAk(s)/angbracketrighte−λ(Ak(t)−Ak(s))cosh/parenleftbig
λ(φk(t)−φk(s))/parenrightbig
λqdλ
=B
2sµ−k
2tk−µ
2/integraldisplay2λ0//angbracketleftAk(s)/angbracketright
λ0//angbracketleftAk(s)/angbracketright/parenleftBig
1+e−2λ(φk(t)−φk(s))/parenrightBig
λqdλ
/greaterorequalslantB
2sµ−k
2tk−µ
2/integraldisplay2λ0//angbracketleftAk(s)/angbracketright
λ0//angbracketleftAk(s)/angbracketrightλqdλ=B(2q+1−1)λq+1
0
2(q+1)/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−q−1.
15Repeating similar steps as before, thanks to (55) we obtain
ηq(t,s,x;k,µ)/greaterorequalslantsµ+k
2tk−µ
2/integraldisplay2λ0//angbracketleftAk(s)/angbracketright
λ0//angbracketleftAk(s)/angbracketrighte−λ(Ak(t)−Ak(s))sinh/parenleftbig
λ(φk(t)−φk(s))/parenrightbig
λ(φk(t)−φk(s))e−λ(Ak(s)+R)ϕλ(x)λqdλ
/greaterorequalslantB
2sµ+k
2tk−µ
2/integraldisplay2λ0//angbracketleftAk(s)/angbracketright
λ0//angbracketleftAk(s)/angbracketright1−e−2λ(φk(t)−φk(s))
φk(t)−φk(s)λq−1dλ
/greaterorequalslantB
2sµ+k
2tk−µ
21−e−2λ0φk(t)−φk(s)
/angbracketleftAk(s)/angbracketright
φk(t)−φk(s)/integraldisplay2λ0//angbracketleftAk(s)/angbracketright
λ0//angbracketleftAk(s)/angbracketrightλq−1dλ
=B(2q−1)λq
0
2qsµ+k
2tk−µ
2/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−q1−e−2λ0φk(t)−φk(s)
/angbracketleftAk(s)/angbracketright
φk(t)−φk(s),
with obvious modifications in the case q= 0. The previous inequality implies (62), provided that we sho w
the validity of the inequality
1−e−2λ0φk(t)−φk(s)
/angbracketleftAk(s)/angbracketright
φk(t)−φk(s)/greaterorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1.
Hence, we need to prove this inequality. For φk(t)−φk(s)/greaterorequalslant1
2λ0/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht, it holds
1−e−2λ0φk(t)−φk(s)
/angbracketleftAk(s)/angbracketright/greaterorequalslant1−e−1
and, consequently,
1−e−2λ0φk(t)−φk(s)
/angbracketleftAk(s)/angbracketright
φk(t)−φk(s)/greaterorsimilar/parenleftbig
φk(t)−φk(s)/parenrightbig−1/greaterorequalslantAk(t)−1/greaterorequalslant/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1.
On the other hand, when φk(t)−φk(s)/lessorequalslant1
2λ0/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht, using the estimate 1−e−σ/greaterorequalslantσ/2forσ∈[0,1], we
get easily
1−e−2λ0φk(t)−φk(s)
/angbracketleftAk(s)/angbracketright
φk(t)−φk(s)/greaterorequalslantλ0
/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/greaterorequalslantλ0
/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht.
Therefore, the proof of (62) is completed.
Next we prove an upper bound estimate in the special case s=t.
Lemma 3.7. Letn/greaterorequalslant1,k∈[0,1),µ/greaterorequalslant0andλ0>0. If we assume q >(n−3)/2, then, for t/greaterorequalslant1and
|x|/lessorequalslantAk(t)+Rthe following upper bound estimate holds:
ξq(t,t,x;k,µ)/lessorequalslantB2/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−n−1
2/an}b∇acketle{tAk(t)−|x|/an}b∇acket∇i}htn−3
2−q. (64)
HereB2is a positive constant depending only on λ0,q,R,k and/an}b∇acketle{ty/an}b∇acket∇i}htdenotes the same function as in the
statement of Lemma 3.6.
Proof. Due to the representation
ξq(t,t,x;k,µ) =/integraldisplayλ0
0e−λ(Ak(t)+R)ϕλ(x)λqdλ,
the proof is exactly the same as in [27, Lemma 2.7].
3.3 Derivation of the iteration frame
In this section, we define the time – dependent functional who se dynamic is studied in order to prove the
blow – up result. Then, we derive the iteration frame for this functional and a first lower bound estimate of
logarithmic type.
Fort/greaterorequalslant1we introduce the functional
U(t).=tµ−k
2/integraldisplay
Rnu(t,x)ξq(t,t,x;k,µ)dx (65)
for someq>(n−3)/2.
16From (60), (61) and (62), it follows
U(t)/greaterorsimilarB0ε/integraldisplay
Rnu0(x)dx+B1εAk(t)
/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht/integraldisplay
Rnu1(x)dx.
As we assume both u0,u1nonnegative and nontrivial, then, we find that
U(t)/greaterorsimilarε (66)
for anyt∈[1,T), where the unexpressed multiplicative constant depends on u0,u1. In the next proposition,
we derive the iteration frame for the functional Ufor a given value of q.
Proposition 3.8. Letn/greaterorequalslant1,k∈[0,1)andµ∈[0,k]∪[2−k,∞). Let us consider u0∈H1(Rn)and
u1∈L2(Rn)such that suppuj⊂BRforj= 0,1and for some R >0and letube a local in time energy
solution to (4)on[1,T)according to Definition 1.1. If Uis defined by (65)withq= (n−1)/2−1/p, then,
there exists a positive constant C=C(n,p,R,k,µ )such that
U(t)/greaterorequalslantC/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt
1φk(t)−φk(s)
s/parenleftbig
log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)(U(s))pds (67)
for anyt∈(1,T).
Proof. By (65), applying Hölder’s inequality we find
sk−µ
2U(s)≤/parenleftbigg/integraldisplay
Rn|u(s,x)|pηq(t,s,x;k,µ)dx/parenrightbigg1/p
/integraldisplay
BR+Ak(s)/parenleftbig
ξq(s,s,x;k,µ)/parenrightbigp′
/parenleftbig
ηq(t,s,x;k,µ)/parenrightbigp′/pdx
1/p′
.
Hence,
/integraldisplay
Rn|u(s,x)|pηq(t,s,x;k,µ)dx/greaterorequalslant/parenleftbig
sk−µ
2U(s)/parenrightbigp/parenleftBigg/integraldisplay
BR+Ak(s)/parenleftbig
ξq(s,s,x;k,µ)/parenrightbigp/(p−1)
/parenleftbig
ηq(t,s,x;k,µ)/parenrightbig1/(p−1)dx/parenrightBigg−(p−1)
. (68)
Let us determine an upper bound for the integral on the right – hand side of (68). By using (64) and (62),
we obtain
/integraldisplay
BR+Ak(s)/parenleftbig
ξq(s,s,x;k,µ)/parenrightbigp/(p−1)
/parenleftbig
ηq(t,s,x;k,µ)/parenrightbig1/(p−1)dx
/lessorequalslantB−1
p−1
1Bp
p−1
2s−µ+k
2(p−1)t−k−µ
2(p−1)/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−n−1
2p
p−1+q
p−1/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht1
p−1/integraldisplay
BR+Ak(s)/an}b∇acketle{tAk(s)−|x|/an}b∇acket∇i}ht(n−3
2−q)p
p−1dx
/lessorequalslantB−1
p−1
1Bp
p−1
2s−µ+k
2(p−1)tµ−k
2(p−1)/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht1
p−1/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht1
p−1(−n−1
2p+n−1
2−1
p)/integraldisplay
BR+Ak(s)/an}b∇acketle{tAk(s)−|x|/an}b∇acket∇i}ht−1dx
/lessorequalslantB−1
p−1
1Bp
p−1
2s−µ+k
2(p−1)tµ−k
2(p−1)/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht1
p−1/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht1
p−1(−n−1
2p+n−1
2−1
p)+n−1log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht,
where in the second inequality we used value of qto get exactly −1as power for the function in the integral.
Consequently, from (68) we have
/integraldisplay
Rn|u(s,x)|pηq(t,s,x;k,µ)dx/greaterorsimilar/parenleftbig
sk−µ
2U(s)/parenrightbigpsµ+k
2tk−µ
2/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−n−1
2(p−1)+1
p/parenleftbig
log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)
/greaterorsimilartk−µ
2/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1sk
2(p+1)+µ
2(1−p)/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−n−1
2(p−1)+1
p/parenleftbig
log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)/parenleftbig
U(s)/parenrightbigp.
Combining the previous lower bound estimate and (60), we arr ive at
U(t)/greaterorequalslanttµ−k
2/integraldisplayt
1(φk(t)−φk(s))/integraldisplay
Rn|u(s,x)|pηq(t,s,x;k,µ)dxds
/greaterorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt
1(φk(t)−φk(s))sk
2(p+1)+µ
2(1−p)/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−n−1
2(p−1)+1
p/parenleftbig
log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)/parenleftbig
U(s)/parenrightbigpds
/greaterorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt
1(φk(t)−φk(s))/an}b∇acketle{tAk(s)/an}b∇acket∇i}htk(p+1)
2(1−k)−µ(p−1)
2(1−k)−n−1
2(p−1)+1
p/parenleftbig
log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)/parenleftbig
U(s)/parenrightbigpds
/greaterorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt
1(φk(t)−φk(s))/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−(n−1
2+µ−k
2(1−k))p+(n−1
2+µ+k
2(1−k))+1
p/parenleftbig
log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)/parenleftbig
U(s)/parenrightbigpds,
17where in third step we used s= (1−k)1
1−k(Ak(s)+φk(1))1
1−k≈ /an}b∇acketle{tAk(s)/an}b∇acket∇i}ht1
1−kfors/greaterorequalslant1. Sincep=p0/parenleftbig
k,n+µ
1−k/parenrightbig
from (5) it follows
−/parenleftBig
n−1
2+µ−k
2(1−k)/parenrightBig
p+/parenleftBig
n−1
2+µ+k
2(1−k)/parenrightBig
+1
p=−1−k
1−k=−1
1−k, (69)
then, plugging (69) in the above lower bound estimate for U(t)it yields
U(t)/greaterorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt
1(φk(t)−φk(s))/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−1
1−k/parenleftbig
log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)/parenleftbig
U(s)/parenrightbigpds
/greaterorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt
1φk(t)−φk(s)
s/parenleftbig
log/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht/parenrightbig−(p−1)/parenleftbig
U(s)/parenrightbigpds,
which is exactly (67). Therefore, the proof is completed.
Lemma 3.9. Letn/greaterorequalslant1,k∈[0,1)andµ∈[0,k]∪[2−k,∞). Let us consider u0∈H1(Rn)andu1∈L2(Rn)
such that suppuj⊂BRforj= 0,1and for some R>0and letube a local in time energy solution to (4)
on[1,T)according to Definition 1.1. Then, there exists a positive co nstantK=K(u0,u1,n,p,R,k,µ )such
that the lower bound estimate
/integraldisplay
Rn|u(t,x)|pdx/greaterorequalslantKεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht(n−1)(1−p
2)+(k−µ)p
2(1−k) (70)
holds for any t∈(1,T).
Proof. We modify of the proof of Lemma 5.1 in [36] accordingly to our m odel. Let us fix q>(n−3)/2+1/p′.
Combining (65), (66) and Hölder’s inequality, it results
εtk−µ
2/lessorsimilartk−µ
2U(t) =/integraldisplay
Rnu(t,x)ξq(t,t,x;k,µ)dx
/lessorequalslant/parenleftbigg/integraldisplay
Rn|u(t,x)|pdx/parenrightbigg1/p/parenleftBigg/integraldisplay
BR+Ak(t)/parenleftbig
ξq(t,t,x;k,µ/parenrightbigp′
dx/parenrightBigg1/p′
.
Hence,
/integraldisplay
Rn|u(t,x)|pdx/greaterorsimilarεptk−µ
2p/parenleftBigg/integraldisplay
BR+Ak(t)/parenleftbig
ξq(t,t,x;k,µ/parenrightbigp′
dx/parenrightBigg−(p−1)
. (71)
Let us determine an upper bound estimates for the integral of ξq(t,t,x;k,µ)p′. By using (64), we have
/integraldisplay
BR+Ak(t)/parenleftbig
ξq(t,t,x;k,µ/parenrightbigp′
dx/lessorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−n−1
2p′/integraldisplay
BR+Ak(t)/an}b∇acketle{tAk(t)−|x|/an}b∇acket∇i}ht(n−3)p′/2−p′qdx
/lessorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−n−1
2p′/integraldisplayR+Ak(t)
0rn−1/an}b∇acketle{tAk(t)−r/an}b∇acket∇i}ht(n−3)p′/2−p′qdr
/lessorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−n−1
2p′+n−1/integraldisplayR+Ak(t)
0/an}b∇acketle{tAk(t)−r/an}b∇acket∇i}ht(n−3)p′/2−p′qdr.
Performing the change of variable Ak(t)−r=̺, one gets
/integraldisplay
BR+Ak(t)/parenleftbig
ξq(t,t,x;k,µ/parenrightbigp′
dx/lessorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−n−1
2p′+n−1/integraldisplayAk(t)
−R(3+|̺|)(n−3)p′/2−p′qd̺
/lessorsimilar/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−n−1
2p′+n−1
because of (n−3)p′/2−p′q<−1. If we combine this upper bound estimates for the integral of ξq(t,t,x;k,µ)p′,
the inequality (71) and we employ t≈ /an}b∇acketle{tAk(t)/an}b∇acket∇i}ht1
1−kfort/greaterorequalslant1, then, we arrive at (70). This completes the
proof.
In Proposition 3.8, we derive the iteration frame for U. In the next result, we shall prove a first lower
bound estimate of logarithmic type for U, as base case for the iteration argument.
18Proposition 3.10. Letn/greaterorequalslant1,k∈[0,1)andµ∈[0,k]∪[2−k,∞). Let us consider u0∈H1(Rn)and
u1∈L2(Rn)such that suppuj⊂BRforj= 0,1and for some R >0and letube a local in time energy
solution to (4)on[1,T)according to Definition 1.1. Let Ube defined by (65)withq= (n−1)/2−1/p.
Then, fort/greaterorequalslant3/2the functional U(t)fulfills
U(t)/greaterorequalslantMεplog/parenleftbig2t
3/parenrightbig
, (72)
where the positive constant Mdepends on u0,u1,n,p,R,k,µ .
Proof. From (60) it results
U(t)/greaterorequalslanttµ−k
2/integraldisplayt
1(φk(t)−φk(s))/integraldisplay
Rn|u(s,x)|pηq(t,s,x;k,µ)dxds.
Consequently, applying (62) first and then (70), we find
U(t)/greaterorequalslantB1/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt
1(φk(t)−φk(s))sµ+k
2/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−q/integraldisplay
Rn|u(s,x)|pdxds
/greaterorequalslantB1Kεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt
1(φk(t)−φk(s))sµ+k
2/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−q+(n−1)(1−p
2)+(k−µ)p
2(1−k)ds
/greaterorsimilarεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt
1(φk(t)−φk(s))/an}b∇acketle{tAk(s)/an}b∇acket∇i}htµ+k
2(1−k)−n−1
2+1
p+(n−1)(1−p
2)+(k−µ)p
2(1−k)ds
/greaterorsimilarεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt
1(φk(t)−φk(s))/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−(n−1
2+µ−k
2(1−k))p+(n−1
2+µ+k
2(1−k))+1
pds
/greaterorsimilarεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt
1(φk(t)−φk(s))/an}b∇acketle{tAk(s)/an}b∇acket∇i}ht−1
1−kds/greaterorsimilarεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1/integraldisplayt
1φk(t)−φk(s)
sds.
Integrating by parts, we obtain
/integraldisplayt
1φk(t)−φk(s)
sds=/parenleftbig
φk(t)−φk(s)/parenrightbig
logs/vextendsingle/vextendsingle/vextendsingles=t
s=1+/integraldisplayt
1φ′
k(s)logsds
=/integraldisplayt
1s−klogsds/greaterorequalslantt−k/integraldisplayt
1logsds.
Consequently, for t/greaterorequalslant3/2
U(t)/greaterorsimilarεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1t−k/integraldisplayt
1logsds/greaterorequalslantεp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1t−k/integraldisplayt
2t/3logsds/greaterorequalslant(1/3)εp/an}b∇acketle{tAk(t)/an}b∇acket∇i}ht−1t1−klog(2t/3)
/greaterorsimilarεplog(2t/3),
where in the last line we applied t≈ /an}b∇acketle{tAk(t)/an}b∇acket∇i}ht1
1−kfort/greaterorequalslant1. Thus, the proof is over.
In order to conclude the proof of Theorem 1.3 it remains to use an iteration argument together with a
slicing procedure for the domain of integration. This proce dure consists in determining a sequence of lower
bound estimates for U(t)(indexes by j∈N) and, then, proving that U(t)may not be finite for tover a
certainε– dependent threshold by taking the limit as j→ ∞. Since the iteration frame (67) and the first
lower bound estimate (72) are formally identical to those in [27, Section 2.3] (of course, for different values
of the critical exponent p), the iteration argument can be rewritten verbatim as in [27 , Section 2.4].
Finally, we show how the previous steps can be adapted to the t reatment of the case µ∈[0,k]. According
to Remark 4 , through the transformation v(t,x) =tµ−1u(t,x), we may consider the transformed semilinear
Cauchy problem (57) for v. Note that v0.=u0andv1.=u1+(1−µ)u0satisfies the same assumptions for u0
andu1in the statement of Theorem 1.3 in this case (nonnegativenes s and nontriviality, compactly supported
and belongingness to the energy space H1(Rn)×L2(Rn)). Of course, we may introduce the auxiliary function
ξq(t,s,x;k,2−µ),ξq(t,s,x;k,2−µ)as in (58), (59) replacing µby2−µ. In Corollary 3.5, nevertheless, we
have to replace the fundamental identity (53) by
/integraldisplay
Rnv(t,x)ξq(t,t,x;k,2−µ)dx=ε/integraldisplay
Rnv0(x)ξq(t,1,x;k,2−µ)dx+εAk(t)/integraldisplay
Rnv1(x)ηq(t,s,x;k,2−µ)dx
+/integraldisplayt
1(φk(t)−φk(s))s(1−µ)(p−1)/integraldisplay
Rn|v(s,x)|pηq(t,s,x;k,2−µ)dxds.
19As we have already pointed out in Remark 4, the estimates in (5 4) and (55) holds true in this case with
2−µinstead ofµ(we recall that this was the actual reason to consider the tra nsformed problem in place of
the original one). Moreover, also the lower bound estimate i n (70) is valid for v, provided that we replace µ
by2−µ. Accordingly to what we have just remarked, the suitable tim e – dependent functional to study for
the transformed problem is
V(t).=t1−µ+k
2/integraldisplay
Rnv(t,x)ξq(t,t,x;k,2−µ)dx.
In fact, Vsatisfies V(t)/greaterorsimilarεfort∈[1,T)and, furthermore, it is possible to derive for Vcompletely
analogous iteration frame and first logarithmic lower bound , respectively, as the ones for Uin (67) and
(72), respectively. We point out that both for the iteration frame and for the first logarithmic lower bound
estimate the time – dependent factor t(1−µ)(p−1)in the nonlinearity compensates the modifications due to
the replacement of µby2−µin the proofs of Propositions 3.8 and 3.10.
4 Critical case: part II
In Section 2, we derived the upper bound for the lifespan in th e subcritical case, whereas in Section 3 we
studied the critical case p=p0/parenleftbig
k,n+µ
1−k/parenrightbig
. It remains to consider the critical case p=p1(k,n), that is,
whenµ/greaterorequalslantµ0(k,n). In this section, we are going to prove Theorem 1.4. In this cr itical case, our approach
will be based on a basic iteration argument combined with the slicing procedure introduced for the first time
in the paper [1]. The parameters characterizing the slicing procedure are given by the sequence {ℓj}j∈N,
whereℓj.= 2−2−(j+1).
As time – depending functional we consider the same one studi ed in Section 2, namely, U0defined in
(11). Hence, since p=p1(k,n)is equivalent to the condition
(1−k)n(p−1) = 2, (73)
we can rewrite (23) as
U0(t)/greaterorequalslantC/integraldisplayt
1τ−µ/integraldisplayτ
1sµ−2(U0(s))pdsdτ (74)
for anyt∈(1,T)and for a suitable positive constant C >0. Let us underline that (74) will be the iteration
frame in the iteration procedure for the critical case p=p1(k,n).
We know that U0(t)/greaterorequalslantKεfor anyt∈(1,T)and for a suitable positive constant K, provided that u0,u1
are nonnegative, nontrivial and compactly supported (cf. t he estimate in (21)). Thus,
U0(t)/greaterorequalslantCKpεp/integraldisplayt
1τ−µ/integraldisplayτ
1sµ−2dsdτ/greaterorequalslantCKpεp/integraldisplayt
1τ−µ−2/integraldisplayτ
1(s−1)µdsdτ
=CKpεp
µ+1/integraldisplayt
1τ−µ−2(τ−1)µ+1dτ/greaterorequalslantCKpεp
µ+1/integraldisplayt
ℓ0τ−µ−2(τ−1)µ+1dτ
/greaterorequalslantCKpεp
3µ+1(µ+1)/integraldisplayt
ℓ0τ−1dτ/greaterorequalslantCKpεp
3µ+1(µ+1)log/parenleftbiggt
ℓ0/parenrightbigg
(75)
fort/greaterorequalslantℓ0= 3/2, where we used τ/lessorequalslant3(τ−1)forτ/greaterorequalslantℓ0in the second last step.
Therefore, by using recursively (74), we prove now the seque nce of lower bound estimates
U0(t)/greaterorequalslantKj/parenleftbigg
log/parenleftbiggt
ℓj/parenrightbigg/parenrightbiggσj
fort/greaterorequalslantℓj (76)
for anyj∈N, where{Kj}j∈N,{σ}j∈Nare sequences of positive reals that we determine afterward s in the
inductive step.
Clearly (76) for j= 0holds true thanks to (75), provided that K0= (CKpεp)/(3µ+1(µ+1)) andσ0= 1.
Next we show the validity of (76) by using an inductive argume nt. Assuming that (76) is satisfied for some
j/greaterorequalslant0, we prove (76) for j+1. According to this purpose, we plug (76) in (74), so, after sh rinking the domain
of integration, we get
U0(t)/greaterorequalslantCKp
j/integraldisplayt
ℓjτ−µ/integraldisplayτ
ℓjsµ−2/parenleftBig
log/parenleftBig
s
ℓj/parenrightBig/parenrightBigσjp
dsdτ
20fort/greaterorequalslantℓj+1. If we shrink the domain of integration to [(ℓj/ℓj+1)τ,τ]in thes– integral (this operation is
possible for τ/greaterorequalslantℓj+1), we find
U0(t)/greaterorequalslantCKp
j/integraldisplayt
ℓj+1τ−µ−2/integraldisplayτ
ℓjτ
ℓj+1sµ/parenleftBig
log/parenleftBig
s
ℓj/parenrightBig/parenrightBigσjp
dsdτ
/greaterorequalslantCKp
j/integraldisplayt
ℓj+1τ−µ−2/parenleftBig
log/parenleftBig
τ
ℓj+1/parenrightBig/parenrightBigσjp/integraldisplayτ
ℓjτ
ℓj+1/parenleftBig
s−ℓj
ℓj+1τ/parenrightBigµ
dsdτ
=CKp
j(µ+1)−1/parenleftBig
1−ℓj
ℓj+1/parenrightBigµ+1/integraldisplayt
ℓj+1τ−1/parenleftBig
log/parenleftBig
τ
ℓj+1/parenrightBig/parenrightBigσjp
dτ
/greaterorequalslant2−(j+3)(µ+1)CKp
j(µ+1)−1(1+pσj)−1/parenleftBig
log/parenleftBig
t
ℓj+1/parenrightBig/parenrightBigσjp+1
fort/greaterorequalslantℓj+1, where in the last step we applied the inequality 1−ℓj/ℓj+1>2−(j+3). Hence, we proved (76)
forj+1provided that
Kj+1.= 2−(j+3)(µ+1)C(µ+1)−1(1+pσj)−1Kp
jandσj+1.= 1+σjp.
Let us establish a suitable lower bound for Kj. Using iteratively the relation σj= 1 +pσj−1and the
initial exponent σ0= 1, we have
σj=σ0pj+j−1/summationdisplay
k=0pk=pj+1−1
p−1. (77)
In particular, the inequality σj−1p+1 =σj/lessorequalslantpj+1/(p−1)yields
Kj/greaterorequalslantL/parenleftbig
2µ+1p/parenrightbig−jKp
j−1 (78)
for anyj/greaterorequalslant1, whereL.= 2−2(µ+1)C(µ+1)−1(p−1)/p. Applying the logarithmic function to both sides of
(78) and using the resulting inequality iteratively, we obt ain
logKj/greaterorequalslantplogKj−1−jlog/parenleftbig
2µ+1p/parenrightbig
+logL
/greaterorequalslant.../greaterorequalslantpjlogK0−/parenleftBiggj−1/summationdisplay
k=0(j−k)pk/parenrightBigg
log/parenleftbig
2µ+1p/parenrightbig
+/parenleftBiggj−1/summationdisplay
k=0pk/parenrightBigg
logL
=pj/parenleftBigg
log/parenleftbiggCKpεp
3µ+1(µ+1)/parenrightbigg
−plog/parenleftbig
2µ+1p/parenrightbig
(p−1)2+logL
p−1/parenrightBigg
+/parenleftbiggj
p−1+p
(p−1)2/parenrightbigg
log/parenleftbig
2µ+1p/parenrightbig
−logL
p−1,
where we applied again the identities in (37). Let us define j2=j2(n,p,k,µ)as the smallest nonnegative
integer such that
j2/greaterorequalslantlogL
log/parenleftbig
2µ+1p/parenrightbig−p
p−1.
Consequently, for any j/greaterorequalslantj2the following estimate holds
logKj/greaterorequalslantpj/parenleftBigg
log/parenleftbiggCKpεp
3µ+1(µ+1)/parenrightbigg
−plog/parenleftbig
2µ+1p/parenrightbig
(p−1)2+logL
p−1/parenrightBigg
=pjlog(Nεp), (79)
whereN.= 3−(µ+1)CKp(µ+1)−1/parenleftbig
2µ+1p/parenrightbig−p/(p−1)2
L1/(p−1).
Combining (76), (77) and (79), we arrive at
U0(t)/greaterorequalslantexp/parenleftbig
pjlog(Nεp)/parenrightbig/parenleftBig
log/parenleftBig
t
ℓj/parenrightBig/parenrightBigσj
/greaterorequalslantexp/parenleftbig
pjlog(Nεp)/parenrightbig/parenleftbig1
2logt/parenrightbig(pj+1−1)/(p−1)
= exp/parenleftBig
pjlog/parenleftBig
2−p/(p−1)Nεp(logt)p/(p−1)/parenrightBig/parenrightBig/parenleftbig1
2logt/parenrightbig−1/(p−1)
fort/greaterorequalslant4and for any j/greaterorequalslantj2, where we employed the inequality log(t/ℓj)/greaterorequalslantlog(t/2)/greaterorequalslant(1/2)logtfort/greaterorequalslant4.
Introducing the notation H(t,ε).= 2−p/(p−1)Nεp(logt)p/(p−1), the previous estimate may be rewritten as
U0(t)/greaterorequalslantexp/parenleftbig
pjlogH(t,ε)/parenrightbig/parenleftbig1
2logt/parenrightbig−1/(p−1)(80)
21fort/greaterorequalslant4and anyj/greaterorequalslantj2.
If we fixε0=ε0(n,p,k,µ,R,u 0,u1)such that
exp/parenleftBig
2N−(1−p)/pε−(p−1)
0/parenrightBig
/greaterorequalslant4,
then, for any ε∈(0,ε0]and fort>exp/parenleftbig
2N−(1−p)/pε−(p−1)/parenrightbig
we havet/greaterorequalslant4andH(t,ε)>1. Therefore, for
anyε∈(0,ε0]and fort>exp/parenleftbig
2N−(1−p)/pε−(p−1)/parenrightbig
lettingj→ ∞ in (80) we see that the lower bound for
U0(t)blows up and, consequently, U0(t)may not be finite as well. Summarizing, we proved that U0blows
up in finite time and, moreover, we showed the upper bound esti mate for the lifespan
T(ε)/lessorequalslantexp/parenleftBig
2N−(1−p)/pε−(p−1)/parenrightBig
.
Hence the proof of Theorem 1.4 in the critical case p=p1(k,n)is complete.
5 Final remarks
According to the results we obtained in Theorems 1.2, 1.3 and 1.4 it is quite natural to conjecture that
max/braceleftbig
p0/parenleftbig
k,n+µ
1−k/parenrightbig
,p1(k,n)/bracerightbig
is the critical exponent for the semilinear Cauchy problem ( 4), although the global existence of small data
solutions is completely open in the supercritical case. Fur thermore, this exponent is consistent with other
models studied in the literature.
In the flat case k= 0, this exponent coincide with max{pStr(n+µ),pFuj(n)}which in many subcases has
been showed to be optimal in the case of semilinear wave equat ion with time – dependent scale – invariant
damping, see [5, 8, 7, 22, 16, 34, 28, 31, 24, 25, 6] and referen ces therein for further details.
On the other hand, in the undamped case µ= 0(that is, for the semilinear wave equation with speed
of propagation t−k) the exponent max{p0(k,n),p1(k,n)}is consistent with the result for the generalized
semilinear Tricomi equation (i.e., the semilinear wave equ ation with speed of propagation tℓ, whereℓ >0)
obtained in the recent works [13, 14, 15, 21].
Clearly, in the very special case µ= 0andk= 0, our result is nothing but a blow-up result for the
classical semilinear wave equation for exponents below pStr(n), which is well – known to be optimal (for a
detailed historical overview on Strauss’ conjecture and a c omplete list of references we address the reader to
the introduction of the paper [33]).
As we have already explained in the introduction, for µ= 2andk= 2/3the equation in (4) is the semi-
linear wave equation in the Einstein – de Sitter spacetime. I n particular, our result is a natural generalization
of the results in [12, 27].
Furthermore, we underline explicitly the fact that the expo nentp0/parenleftbig
k,n+µ
1−k/parenrightbig
for (4) is obtained by the
corresponding exponent in the not damped case µ= 0via a formal shift in the dimension of magnitudeµ
1−k.
This phenomenon is due to the threshold nature of the time – de pendent coefficient of the damping term
and it has been widely observed in the special case k= 0not only for the semilinear Cauchy problem with
power nonlinearity but also with nonlinarity of derivative type|ut|p(see [32]) or weakly coupled system (see
[2, 26, 32]).
Finally, we have to point out that after the completion of the final version of this work, we found out the
existence of the paper [35], where the same model is consider ed. We stress that the approach we used in the
critical case is completely different, and that we slightly i mproved their result, by removing the assumption
on the size of the support of the Cauchy data (cf. [35, Theorem 2.3]), even though we might not cover the
full rangeµ∈[0,µ0(k,n)]in the critical case due to the assumption µ/ne}ationslash∈(k,2−k).
Acknowledgments
A. Palmieri is supported by the GNAMPA project ‘Problemi sta zionari e di evoluzione nelle equazioni di
campo nonlineari dispersive’. The author would like to ackn owledge Karen Yagdjian (UTRGV), who first
introduced him to the model considered in this work.
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24 |
1609.01063v2.Remarks_on_an_elliptic_problem_arising_in_weighted_energy_estimates_for_wave_equations_with_space_dependent_damping_term_in_an_exterior_domain.pdf | arXiv:1609.01063v2 [math.AP] 23 Nov 2016REMARKS ON AN ELLIPTIC PROBLEM ARISING IN
WEIGHTED ENERGY ESTIMATES FOR WAVE EQUATIONS
WITH SPACE-DEPENDENT DAMPING TERM IN AN
EXTERIOR DOMAIN
MOTOHIRO SOBAJIMA AND YUTA WAKASUGI
Abstract. This paper is concerned with weighted energy estimates and d if-
fusion phenomena for the initial-boundary problem of the wa ve equation with
space-dependent damping term in an exterior domain. In this analysis, an el-
liptic problem was introduced by Todorova and Yordanov. Thi s attempt was
quite useful when the coefficient of the damping term is radial ly symmetric. In
this paper, by modifying their elliptic problem, we establi sh weighted energy
estimates and diffusion phenomena even when the coefficient of the damping
term is not radially symmetric.
1.Introduction
LetN≥2. We consider the wave equation with space-dependent damping te rm
in an exterior domain Ω ⊂RNwith a smooth boundary:
utt−∆u+a(x)ut= 0, x∈Ω, t >0,
u(x,t) = 0, x ∈∂Ω, t >0,
(u,ut)(x,0) = (u0,u1)(x), x∈Ω,(1.1)
where we denote by ∆ the usual Laplacian in RNand byutanduttthe first and
second derivative of uwith respect to the variable t, andu=u(x,t) is a real-valued
unknown function. The coefficient of the damping term a(x) satisfies a∈C2(Ω),
a(x)>0 onΩ and
lim
|x|→∞/parenleftBig
/a\}b∇acketle{tx/a\}b∇acket∇i}htαa(x)/parenrightBig
=a0 (1.2)
with some constants α∈[0,1) anda0∈(0,∞), where /a\}b∇acketle{ty/a\}b∇acket∇i}ht= (1+|y|2)1
2fory∈RN.
In this moment, the initial data ( u0,u1) are assumed to have compact supports in
Ω and to satisfy the compatibility condition of order k≥1:
(uℓ−1,uℓ)∈(H2∩H1
0(Ω))×H1
0(Ω),for allℓ= 1,...,k (1.3)
whereuℓis successively defined by uℓ= ∆uℓ−2−a(x)uℓ−1(ℓ= 2,...,k). We note
that existence and uniqueness of solution to the problem (1.1) have been discussed
(see e.g., Ikawa [2, Theorem 2]).
It is proved in Matsumura [4] that if Ω = RNanda(x)≡1, then the solution u
of (1.1) satisfies the energy decay estimate/integraldisplay
RN(|∇u(x,t)|2+|ut(x,t)|2)dx≤C(1+t)−N
2−1/ba∇dbl(u0,u1)/ba∇dbl2
H1×L2,
Key words and phrases. Damped wave equation; elliptic problem; exterior domain; w eighted
energy estimates; diffusion phenomena.
12 MOTOHIRO SOBAJIMA AND YUTA WAKASUGI
where the constant Cdepends on the size of the supprot of initial data. Moreover,
it is shown in Nishihara [7] that uhas the same asymptotic behavior as the one of
the problem/braceleftbigg
vt−∆v= 0, x ∈RN, t >0,
v(x,0) =u0(x)+u1(x), x∈RN.(1.4)
In particular, we have
/ba∇dblu(·,t)−v(·,t)/ba∇dblL2=o(t−N
4)
ast→ ∞. Energy decay properties of solutions to (1.1) for general cases with
a(x)≥ /a\}b∇acketle{tx/a\}b∇acket∇i}ht−α(0≤α≤1) have been dealt with by Matsumura [5]. On the other
hand, Mochizuki [6] proved that if 0 ≤a(x)≤C/a\}b∇acketle{tx/a\}b∇acket∇i}ht−αfor some α >1, then the
energy of the solution to (1.1) does not vanish as t→ ∞for suitable initial data.
(The solution has an asymptotic behavior similar to the solution of the usual wave
equation without damping). Therefore one can expect that diffusio n phenomena
occur only when a(x)≥C/a\}b∇acketle{tx/a\}b∇acket∇i}ht−αforα≤1.
In this paper, we discuss precise decay rates of the weighted ener gy/integraldisplay
RN(|∇u(x,t)|2+|ut(x,t)|2)Φ(x,t)dx
with a special weight function
Φ(x,t) = exp/parenleftbigg
βA(x)
1+t/parenrightbigg
(for some A∈C2(RN) andβ >0) which is introduced by Todorova and Yordanov
[12] based on the ideas in [11] and in [3]. They proved weighted energy e stimates/integraldisplay
RNa(x)|u(x,t)|2Φ(x,t)dx≤C(1+t)−N−α
2−α+ε,
/integraldisplay
RN(|∇u(x,t)|2+|ut(x,t)|2)Φ(x,t)dx≤C(1+t)−N−α
2−α−1+ε
whena(x) is radially symmetric and satisfies (1.2). After that, Radu, Todoro va
and Yordanov [8] extended it to higher-order derivatives. In [13], t he second author
proved diffusion phenomena for (1.1) with Ω = RNanda(x) =/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α(α∈[0,1))
by comparing the solution of the following problem
a(x)vt−∆v= 0, x ∈RN, t >0,
v(x,0) =u0(x)+1
a(x)u1(x), x∈RN.(1.5)
In [10], diffusion phenomena for (1.1) with an exterior domain and for g eneral
radially symmetric damping term are obtained. However, the weighte d energy esti-
matesand diffusion phenomenafor (1.1) with non-radially symmetric damping
are still remaining open. The difficulty seems to come from the choice o f auxiliary
function Ain the weighted energy, which strongly depends on the existence of
positive solution to the Poisson equation ∆ A(x) =a(x). In fact, an example of
non-existence of positive solution to ∆ A=afor non-radial a(x) is shown in [10].
Radu, Todorova and Yordanov [9] considered the case Ω = RNand used a solu-
tionA∗(x) of ∆A∗=a1(1 +|x|)−αwitha1>0 satisfying a1(1 +|x|)−α≥a(x)
forx∈RN, that is, A∗(x) is a subsolution of the equation ∆ A=a. In general
one cannot obtain the optimal decay estimate via this choice becaus e of the luck
of the precise behavior of a(x) at the spatial infinity which can be expected toWAVE EQUATION WITH SPACE-DEPENDENT DAMPING 3
determine the precise decay late of weighted energy estimates. Ou r main idea to
overcome this difficulty is to weaken the equality ∆ A=aand consider the in-
equality (1 −ε)a≤∆A≤(1+ε)a, and to construct a solution having appropriate
behavior, we employ a cut-off argument.
The aim of this paper is to give a proof of Ikehata–Todorova–Yorda nov type
weighted energy estimates for (1.1) with non-radially symmetric dam ping and to
obtain diffusion phenomena for (1.1) under the compatibility condition of order 1
and the condition (1.2) (without any restriction).
This paper is originated as follows. In Section 2, we discuss related ellip tic
and parabolic problems. The weighted energy estimates for (1.1) ar e established
in Section 3 (Proposition 3.5). Section 4 is devoted to show diffusion ph enomena
(Proposition 4.1).
2.Related elliptic and parabolic problems
2.1.An elliptic problem for weighted energy estimates. As we mentioned
above, in general, existence of positive solutions to the Poisson equ ation ∆A(x) =
a(x) is false for non-radial a(x). Thus, we weaken this equation and consider the
following inequality
(1−ε)a(x)≤∆A(x)≤(1+ε)a(x), x∈Ω, (2.1)
whereε∈(0,1) is a parameter. Here we construct a positive solution Aof (2.1)
satisfying
A1ε/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α≤A(x)≤A2ε/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α, (2.2)
|∇A(x)|2
a(x)A(x)≤2−α
N−α+ε (2.3)
for some constants A1ε,A2ε>0.
Lemma 2.1. For every ε∈(0,1), there exists Aε∈C2(Ω)such that Aεsatisfies
(2.1)–(2.3).
Proof.Firstly, we extend a(x) as a positive function in C2(RN); note that this
is possible by virtue of the smoothness of ∂Ω. To simplify the notation, we use
the same symbol a(x) as a function defined on RN. We construct a solution of
approximated equation
∆Aε(x) =aε(x), x∈RN
for some aε∈C2(RN) satisfying
(1−ε)a(x)≤aε(x)≤(1+ε)a(x), x∈RN. (2.4)
Noting (1.2), we divide a(x) asa(x) =b1(x)+b2(x) with
b1(x) = ∆/parenleftbigga0
(N−α)(2−α)/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α/parenrightbigg
=a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α+a0α
N−α/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α−2,
b2(x) =a(x)−a0/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α−a0α
N−α/a\}b∇acketle{tx/a\}b∇acket∇i}ht−α−2.
Then we have
lim
|x|→∞/parenleftbiggb2(x)
a(x)/parenrightbigg
= lim
|x|→∞/bracketleftbigg1
/a\}b∇acketle{tx/a\}b∇acket∇i}htαa(x)/parenleftbigg
/a\}b∇acketle{tx/a\}b∇acket∇i}htαa(x)−a0−a0α
N−α/a\}b∇acketle{tx/a\}b∇acket∇i}ht−2/parenrightbigg/bracketrightbigg
= 0.(2.5)4 MOTOHIRO SOBAJIMA AND YUTA WAKASUGI
Letε∈(0,1) be fixed. Then by (2.5) there exists a constant Rε>0 such
that|b2(x)| ≤εa(x) forx∈RN\B(0,Rε). Here we introduce a cut-off function
ηε∈C∞
c(RN,[0,1]) such that ηε≡1 onB(0,Rε). Define
aε(x) :=b1(x)+ηε(x)b2(x) =a(x)−(1−ηε(x))b2(x), x∈RN.
Thenaε(x) =a(x) onB(0,Rε) and for x∈RN\B(0,Rε),
/vextendsingle/vextendsingle/vextendsingle/vextendsingleaε(x)
a(x)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle= (1−ηε(x))|b2(x)|
a(x)≤ε
and therefore (2.4) is verified.
Next we define
B1ε(x) :=a0
(N−α)(2−α)/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α, x∈RN,
B2ε(x) :=−/integraldisplay
RNN(x−y)ηε(y)b2(y)dy, x∈RN,
whereNis the Newton potential given by
N(x) =
1
2πlog1
|x|ifN= 2,
Γ(N
2+1)
N(N−2)πN
2|x|2−NifN≥3.
Then we easily see that ∆ B1ε(x) =b1(x) and ∆ B2ε=ηε(x)b2(x). Moreover,
noting that supp( ηεb2) is compact, we see from a direct calculation that there exist
a constant Mε>0 such that
|B2ε(x)| ≤/braceleftBigg
Mε(1+log/a\}b∇acketle{tx/a\}b∇acket∇i}ht) ifN= 2,
Mε/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−NifN≥3,|∇B2ε(x)| ≤Mε/a\}b∇acketle{tx/a\}b∇acket∇i}ht1−N, x∈RN.
This yields that Bε:=B1ε+B2εis bounded from below and positive for x∈RN
with sufficiently large |x|. Moreover, we have
lim
|x|→∞/parenleftBig
/a\}b∇acketle{tx/a\}b∇acket∇i}htα−2Bε(x)/parenrightBig
=a0
(N−α)(2−α)
and
lim
|x|→∞/parenleftbigg|∇Bε(x)|2
a(x)Bε(x)/parenrightbigg
= lim
|x|→∞/parenleftBigg
1
/a\}b∇acketle{tx/a\}b∇acket∇i}htαa(x)·1
/a\}b∇acketle{tx/a\}b∇acket∇i}htα−2Bε(x)/vextendsingle/vextendsingle/vextendsingle/vextendsinglea0
N−α/a\}b∇acketle{tx/a\}b∇acket∇i}ht−1x+/a\}b∇acketle{tx/a\}b∇acket∇i}htα−1∇B2ε(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg
=2−α
N−α.
Using the same argument as in the proof of [10, Lemma 3.1], we can see that there
exists a constant λε≥0 such that Aε(x) :=λε+Bε(x) satisfies (2.1)-(2.3). /squareWAVE EQUATION WITH SPACE-DEPENDENT DAMPING 5
2.2.A parabolic problem for diffusion phenomena. Here we consider Lp-Lq
type estimates for solutions to the initial-boundary value problem of the following
parabolic equation
a(x)wt−∆w= 0, x∈Ω, t >0,
w(x,t) = 0, x ∈∂Ω, t >0,
w(x,0) =f(x), x∈Ω.(2.6)
Here we introduce a weighted Lp-spaces
Lp
dµ:=/braceleftBigg
f∈Lp
loc(Ω) ;/ba∇dblf/ba∇dblLp
dµ:=/parenleftbigg/integraldisplay
Ω|f(x)|pa(x)dx/parenrightbigg1
p
<∞/bracerightBigg
,1≤p <∞
which is quite reasonable because the corresponding elliptic operato ra(x)−1∆ can
be regarded as a symmetric operator in L2
dµ.
TheLp-Lqtype estimates for the semigroup associated with the Friedrichs’
extension −L∗(inL2
dµ) of−a(x)−1∆ are stated in [10]. The proof is based on
Beurling–Deny’s criterion and Gagliardo–Nirenberg inequality.
Proposition 2.2 ([10, Proposition 2.6]) .LetetL∗be a semigroup generated by L∗.
For every f∈L1
dµ∩L2
dµ, we have
/ba∇dbletL∗f/ba∇dblL2
dµ≤Ct−N−α
2(2−α)/ba∇dblf/ba∇dblL1
dµ(2.7)
and
/ba∇dblL∗etL∗f/ba∇dblL2
dµ≤Ct−N−α
2(2−α)−1/ba∇dblf/ba∇dblL1
dµ. (2.8)
3.Weighted energy estimates
In this section we establish weighted energy estimates for solutions of (1.1) by
introducing Ikehata–Todorova–Yordanov type weight function w ith an auxiliary
function Aεconstructed in Subsection 2.1.
To begin with, let us recall the finite speed propagation property of the wave
equation (see [2]).
Lemma 3.1 (Finite speed of propagation) .Letube the solution of (1.1)with the
initial data (u0,u1)satisfying supp(u0,u1)⊂B(0,R0) ={x∈Ω;|x| ≤R0}. Then,
one has
suppu(·,t)⊂ {x∈Ω ;|x| ≤R0+t}
and therefore |x|/(R0+1+t)≤1fort≥0andx∈suppu(·,t).
Before introducing a weight function, we also recall two identities fo r partial
energy functionals proved in [10].
Lemma 3.2 ([10, Lemma 3.7]) .LetΦ∈C2(Ω×[0,∞))satisfyΦ>0and∂tΦ<0
and letube a solution of (1.1). Then
d
dt/bracketleftbigg/integraldisplay
Ω/parenleftBig
|∇u|2+|ut|2/parenrightBig
Φdx/bracketrightbigg
=/integraldisplay
Ω(∂tΦ)−1/vextendsingle/vextendsingle∂tΦ∇u−ut∇Φ/vextendsingle/vextendsingle2dx
+/integraldisplay
Ω/parenleftBig
−2a(x)Φ+∂tΦ−(∂tΦ)−1|∇Φ|2/parenrightBig
|ut|2dx.6 MOTOHIRO SOBAJIMA AND YUTA WAKASUGI
Lemma 3.3 ([10, Lemma 3.9]) .LetΦ∈C2(Ω×[0,∞))satisfyΦ>0and∂tΦ<0
and letube a solution to (1.1). Then, we have
d
dt/bracketleftbigg/integraldisplay
Ω/parenleftBig
2uut+a(x)|u|2/parenrightBig
Φdx/bracketrightbigg
= 2/integraldisplay
Ωuut(∂tΦ)dx+2/integraldisplay
Ω|ut|2Φdx−2/integraldisplay
Ω|∇u|2Φdx
+/integraldisplay
Ω/parenleftbig
a(x)∂tΦ+∆Φ/parenrightbig
|u|2dx.
Here we introduce a weight function for weighted energy estimates , which is a
modification of the one in Todorova-Yordanov [12].
Definition 3.4. Defineh:=2−α
N−αand forε∈(0,1),
Φε(x,t) = exp/parenleftbigg1
h+2εAε(x)
1+t/parenrightbigg
, (3.1)
whereAεis given in Lemma 2.1. And define for t≥0,
E∂x(t;u) :=/integraldisplay
Ω|∇u|2Φεdx, E ∂t(t;u) :=/integraldisplay
Ω|ut|2Φεdx, (3.2)
Ea(t;u) :=/integraldisplay
Ωa(x)|u|2Φεdx, E ∗(t;u) := 2/integraldisplay
ΩuutΦεdx, (3.3)
and also define E1(t;u) :=E∂x(t;u)+E∂t(t;u)andE2(t;u) :=E∗(t;u)+Ea(t;u).
Now we are in a position to state our main result for weighted energy e stimates
for solutions of (1.1).
Proposition 3.5. Assume that (u0,u1)satisfies supp(u0,u1)⊂B(0,R0)and the
compatibility condition of order k0≥1. Letube a solution of the problem (1.1).
For every δ >0and0≤k≤k0−1, there exist ε >0andMδ,k,R0>0such that
for every t≥0,
(1+t)N−α
2−α+2k+1−δ/parenleftBig
E∂x(t;∂k
tu)+E∂t(t;∂k
tu)/parenrightBig
+(1+t)N−α
2−α+2k−δEa(t;∂k
tu)
≤Mδ,k,R0/ba∇dbl(u0,u1)/ba∇dbl2
Hk+1×Hk(Ω).
To prove, this, we prepare the following two lemmas.
Lemma 3.6. Fort≥0, we have
1−ε
h+2ε1
1+tEa(t;u)≤E∂x(t;u). (3.4)
Proof.As in the proof of [10, Lemma 3.6], by integration by parts we have
/integraldisplay
Ω∆(logΦ ε)|u|2Φεdx=/integraldisplay
Ω/parenleftbigg
∆Φε−|∇Φε|2
Φε/parenrightbigg
|u|2dx≤/integraldisplay
Ω|∇u|2Φεdx.
Noting that
∆(logΦ ε(x)) =1
h+2ε∆Aε(x)
1+t≥1−ε
h+2εa(x)
1+t,
we have (3.4). /square
In order to clarify the effect of the finite propagation property, w e now put
a1:= inf
x∈Ω/parenleftBig
/a\}b∇acketle{tx/a\}b∇acket∇i}htαa(x)/parenrightBig
.
ThenWAVE EQUATION WITH SPACE-DEPENDENT DAMPING 7
Lemma 3.7. Fort≥0, we have
E∂t(t;u)≤1
a1(R0+1+t)αEa(t;∂tu), (3.5)
/integraldisplay
ΩAε(x)
a(x)|ut|2Φεdx≤A2ε
a1(R0+1+t)2E∂t(t;u), (3.6)
|E∗(t;u)| ≤2√a1(R0+1+t)α
2/radicalbig
Ea(t;u)E∂t(t;u).(3.7)
Proof.Bya(x)−1≤a−1
1/a\}b∇acketle{tx/a\}b∇acket∇i}htα≤a−1
1(1+|x|)αand the finite propagation property
we have/integraldisplay
Ω|ut|2Φεdx=/integraldisplay
Ωa(x)
a(x)|ut|2Φεdx≤1
a1(R0+1+t)αEa(t;∂tu).
Using the Cauchy-Schwarz inequality and the above inequality yields ( 3.6):
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
ΩuutΦεdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
≤/parenleftbigg/integraldisplay
Ω|u|2Φεdx/parenrightbigg/parenleftbigg/integraldisplay
Ω|ut|2Φεdx/parenrightbigg
≤(R0+1+t)α
a1/parenleftbigg/integraldisplay
Ωa(x)|u|2Φεdx/parenrightbigg
E∂t(t;u)
≤(R0+1+t)α
a1Ea(t;u)E∂t(t;u).
We can prove (3.7) in a similar way. /square
Lemma 3.8. (i) For every t≥0, we have
d
dtE1(t;u)≤ −Ea(t;∂tu). (3.8)
(ii)For every ε∈(0,1
3)andt≥0,
d
dtE2(t;u)≤ −1−3ε
1−εE∂x(t;u)+/parenleftbigg2
a1+A2ε(R0+1)2
εa2
1/parenrightbigg
(R0+1+t)αEa(t;∂tu).
(3.9)
Proof.Noting (2.3), we have
−2a(x)Φε+∂tΦε−(∂tΦε)−1|∇Φε|2
=/parenleftbigg
−2a(x)−Aε(x)
(h+2ε)(1+t)2+1
h+2ε|∇Aε(x)|2
Aε(x)/parenrightbigg
Φε
≤/parenleftbigg
−2a(x)+h+ε
h+2εa(x)/parenrightbigg
Φε
≤ −a(x)Φε.
This implies (3.8). On the other hand, from (2.3) and (2.1) we see
a(x)∂tΦε+∆Φε=1
h+2ε/parenleftbigg
−a(x)Aε(x)
(1+t)2+|∇Aε(x)|2
(h+2ε)(1+t)2+∆Aε(x)
1+t/parenrightbigg
Φε
≤1
h+2ε/parenleftbigg
−a(x)Aε(x)
(1+t)2+(h+ε)a(x)Aε(x)
(h+2ε)(1+t)2+(1+ε)a(x)
1+t/parenrightbigg
Φε
≤/parenleftbigg
−ε
(h+2ε)2a(x)Aε(x)
(1+t)2+1+ε
h+2εa(x)
1+t/parenrightbigg
Φε.8 MOTOHIRO SOBAJIMA AND YUTA WAKASUGI
Therefore combining it with Lemma 3.6, we have
/integraldisplay
Ω/parenleftbig
a(x)∂tΦε+∆Φε/parenrightbig
|u|2dx
≤1+ε
1−ε/integraldisplay
Ω|∇u|2Φεdx−ε
(h+2ε)21
(1+t)2/integraldisplay
Ωa(x)Aε(x)|u|2Φεdx.
Using (3.6), we have
2/integraldisplay
Ωuut(∂tΦε)dx
=−2
h+2ε1
(1+t)2/integraldisplay
ΩuutAε(x)Φεdx
≤2
h+2ε1
(1+t)2/parenleftbigg/integraldisplay
Ωa(x)Aε(x)|u|2Φεdx/parenrightbigg1
2/parenleftbigg/integraldisplay
ΩAε(x)
a(x)|ut|2Φεdx/parenrightbigg1
2
≤2(R0+1)
h+2ε1
1+t/parenleftbigg/integraldisplay
Ωa(x)Aε(x)|u|2Φεdx/parenrightbigg1
2/parenleftbiggA2ε
a1E∂t(t;u)/parenrightbigg1
2
≤ε
(h+2ε)21
(1+t)2/integraldisplay
Ωa(x)Aε(x)|u|2Φεdx+A2ε(R0+1)2
εa1E∂t(t;u).
Applying (3.5), we obtain (3.9). /square
Lemma 3.9. The following assertions hold:
(i)Sett∗(R0,α,m) := max/braceleftbigg/parenleftBig
2m
a1/parenrightBig1
1−α,R0+1/bracerightbigg
. Then for every t,m≥0and
t1≥t∗(R0,α,m),
d
dt/parenleftBig
(t1+t)mE1(t;u)/parenrightBig
≤m(t1+t)m−1E∂x(t;u)−1
2(t1+t)mEa(t;∂tu).(3.10)
(ii)for every t,λ≥0andt2≥R0+1,
d
dt/parenleftBig
(t2+t)λE2(t;u)/parenrightBig
≤λ(1+ε)(t2+t)λ−1Ea(t;u)−1−3ε
1−ε(t2+t)λE∂x(t;u)
+/parenleftbigg2
a1+A2ε(R0+1)2
εa2
1+λ
2εa2
1t1−α
2/parenrightbigg
(t2+t)λ+αEa(t;∂tu).(3.11)
(iii)In particular, setting
ν:=4
a1+2A2ε(R0+1)2
εa2
1+1
4εa1,
t∗∗(ε,R0,α,λ) := max/braceleftBigg/parenleftbigg(1−ε)(λ+α)ν
ε/parenrightbigg1
1−α
,/parenleftbigg2(λ+α)
a1/parenrightbigg1
1−α
,R0+1/bracerightBigg
,
one has that for t,λ≥0andt3≥t∗∗(ε,R0,α,λ),
d
dt/parenleftBig
ν(t3+t)λ+αE1(t;u)+(t3+t)λE2(t;u)/parenrightBig
≤ −1−4ε
1−ε(t3+t)λE∂x(t;u)+λ(1+ε)(t3+t)λ−1Ea(t;u).(3.12)WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 9
Proof.(i)Letm≥0 be fixed and let t1≥t∗(R0,α,m). Using (3.8) and (3.5), we
have
(t1+t)−md
dt/parenleftBig
(t1+t)mE1(t;u)/parenrightBig
≤m
t1+tE∂x(t;u)+m
t1+tE∂t(t;u)+d
dtE1(t;u)
≤m
t1+tE∂x(t;u)+m
t1+tE∂t(t;u)−Ea(t;∂tu)
≤m
t1+tE∂x(t;u)+/parenleftbiggm(R0+1+t)α
a1(t1+t)−1/parenrightbigg
Ea(t;∂tu).
Therefore we obtain (3.10).
(ii)Fort≥0, andt≥R0+1,
(t2+t)−λd
dt/parenleftBig
(t2+t)λE2(t;u)/parenrightBig
≤λ
t2+tE∗(t;u)+λ
t2+tEa(t;u)+d
dtE2(t;u)
≤λ
t2+tE∗(t;u)+λ
t2+tEa(t;u)−1−3ε
1−εE∂x(t;u)
+/parenleftbigg2
a1+A2ε(R0+1)2
εa2
1/parenrightbigg
(R0+1+t)αEa(t;∂tu).
Noting that by (3.7) and (3.5),
λ
t2+tE∗(t;u)≤2λ(R0+1+t)α
a1(t2+t)/radicalbig
Ea(t;u)Ea(t;∂tu)
≤λε
t2+tEa(t;u)+λ
εa2
1(R0+1+t)2α
t2+tEa(t;∂tu)
≤λε
t2+tEa(t;u)+λ
εa2
1t1−α
2(t2+t)αEa(t;∂tu),
we deduce (3.11).
(iii)Combining (3.10) with m=λ+αand (3.11), we have for t3≥t∗∗(ε,R0,α,λ)
andt≥0,
d
dt/parenleftBig
ν(t3+t)λ+αE1(t;u)+(t3+t)λE2(t;u)/parenrightBig
≤/parenleftbigg
ν(λ+α)(t3+t)α−1−1−3ε
1−ε/parenrightbigg
(t3+t)λE∂x(t;u)+λ(1+ε)(t3+t)λ−1Ea(t;u)
+/parenleftbigg2
a1+A2ε(R0+1)2
εa2
1+λ
2εa2
1t1−α
3−ν
2/parenrightbigg
(t3+t)λ+αEa(t;∂tu)
≤ −1−4ε
1−ε(t3+t)λE∂x(t;u)+λ(1+ε)(t3+t)λ−1Ea(t;u).
This proves the assertion. /square10 MOTOHIRO SOBAJIMA AND YUTA WAKASUGI
Proof of Proposition 3.5. Firstly, by (3.7) we observe that
ν(t3+t)αE1(t;u)+E2(t;u)≥4
a1(t3+t)αE1(t;u)−|E∗(t;u)|+Ea(t;u)
≥4
a1(t3+t)αE∂t(t;u)
−2√a1(t3+t)α
2/radicalbig
Ea(t;u)E∂t(t;u)+Ea(t;u)
≥3
4Ea(t;u).
By using the above estimate, we prove the assertion via mathematic al induction.
Step 1 ( k= 0).By (3.12) using Lemma 3.6 implies that
d
dt/parenleftBig
ν(t3+t)λ+αE1(t;u)+(t3+t)λE2(t;u)/parenrightBig
≤/parenleftbigg
−1−4ε
1−ε+λ(1+ε)(h+2ε)
1−ε/parenrightbigg
(t3+t)λE∂x(t;u).
Therefore taking λ0=(1−ε)(1−4ε)
(1+ε)(h+2ε), (λ0↑h−1asε↓0) we have
d
dt/parenleftBig
ν(t3+t)λ0+αE1(t;u)+(t3+t)λ0E2(t;u)/parenrightBig
≤ −ε(1−4ε)
1−ε(t3+t)λ0E∂x(t;u).
Integrating over (0 ,t) with respect to t, we see
3
4(t3+t)λ0Ea(t;u)+ε(1−4ε)
1−ε/integraldisplayt
0(t3+s)λ0E∂x(s;u)ds
≤ν(t3+t)λ0+αE1(t;u)+(t3+t)λ0E2(t;u)+ε(1−4ε)
1−ε/integraldisplayt
0(t3+s)λ0E∂x(s;u)ds
≤νtλ0+α
3E1(0;u)+tλ0
3E2(0;u).
Using (3.10) with m=λ0+1 and integrating over (0 ,t), we obtain
(t3+t)λ0+1E1(t;u)+1
2/integraldisplayt
0(t3+s)λ0+1Ea(s;∂tu)ds
≤tλ0+1
3E1(0;u)+(λ0+1)/integraldisplayt
0(t3+s)λ0E∂x(s;u)ds
≤tλ0+1
3E1(0;u)+(λ0+1)(1−ε)
ε(1−4ε)/parenleftBig
νtλ0+α
3E1(0;u)+tλ0
3E2(0;u)/parenrightBig
.
This proves the desired assertion with k= 0 and also the integrability of ( t3+
s)λ0+1Ea(s;∂tu).
Step 2 ( 1< k≤k0−1).Suppose that for every t≥0,
(1+t)λ0+2k−1E1(t;∂k−1
tu)+(1+t)λ0+2k−2Ea(t;∂k−1
tu)≤Mε,k−1/ba∇dbl(u0,u1)/ba∇dbl2
Hk×Hk−1(Ω)
and additionally,
/integraldisplayt
0(1+s)λ0+2k−1Ea(s;∂k
tu)ds≤M′
ε,k−1/ba∇dbl(u0,u1)/ba∇dbl2
Hk×Hk−1(Ω).
Since the initial value ( u0,u1) satisfies the compatibility condition of order k,∂k
tu
is also a solution of (1.1) with replaced ( u0,u1) with (uk−1,uk). Applying (3.12)WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 11
withλ=λ0+ 2k, putting t3k=t∗∗(ε,R0,α,λ0+ 2k) (see Lemma 3.9 (iii)) and
integrating over (0 ,t), we have
3
4(t3k+t)λ0+2kEa(t;∂k
tu)+1−4ε
1−ε/integraldisplayt
0(t3k+s)λ0+2kE∂x(s;∂k
tu)ds
≤ν(t3k+t)λ0+2k+αE1(t;∂k
tu)+(t3k+t)λ0+2kE2(t;∂k
tu)
+1−4ε
1−ε/integraldisplayt
0(t3k+s)λ0+2kE∂x(s;∂k
tu)ds
≤νtλ0+2k+α
3kE1(0;∂k
tu)+tλ0+2k
3kE2(0;∂k
tu)
+(λ0+2k)(1+ε)/integraldisplayt
0(t3k+s)λ0+2k−1Ea(s;∂k
tu)ds
≤νtλ0+2k+α
3kE1(0;∂k
tu)+tλ0+2k−1
3kE2(0;∂k
tu)
+(λ0+2k)(1+ε)M′
ε,k−1/ba∇dbl(u0,u1)/ba∇dbl2
Hk×Hk−1(Ω).
Moreover, from (3.10) with m=λ0+2k+1 we have
(t3k+t)λ0+2k+1E1(t;∂k
tu)+1
2/integraldisplayt
0(t3k+s)λ0+2k+1Ea(s;∂k+1
tu)ds
≤tλ0+2k+1
3kE1(0;∂k
tu)+(λ0+2k+1)/integraldisplayt
0(t3k+s)λ0+2kE∂x(s;∂k
tu)ds
≤M′′
ε,k/parenleftBig
E1(0;∂k
tu)+E2(0;∂k
tu)+/ba∇dbl(u0,u1)/ba∇dbl2
Hk×Hk−1(Ω)/parenrightBig
with some constant M′′
ε,k>0. By induction we obtain the desired inequalities for
allk≤k0−1. /square
4.Diffusion phenomena as an application of weighted energy
estimates
Proposition 4.1. Assume that (u0,u1)∈(H2∩H1
0(Ω))×H1
0(Ω)and suppose that
supp(u0,u1)⊂B(0,R0). Letube the solution of (1.1). Then for every ε >0,
there exists a constant Cε,R0>0such that
/vextenddouble/vextenddouble/vextenddoubleu(·,t)−etL∗[u0+a(·)−1u1]/vextenddouble/vextenddouble/vextenddouble
L2
dµ≤Cε,R0(1+t)−N−α
2(2−α)−1−α
2−α+ε/ba∇dbl(u0,u1)/ba∇dblH2×H1.
To prove Proposition 4.1 we use the following lemma stated in [10, Sectio n 4].
Lemma 4.2. Assume that (u0,u1)∈(H2∩H1
0(Ω))×H1
0(Ω)and suppose that
supp(u0,u1)⊂ {x∈Ω;|x| ≤R0}. Then for every t≥0,
u(x,t)−etL∗[u0+a(·)−1u1] =−/integraldisplayt
t/2e(t−s)L∗[a(·)−1utt(·,s)]ds
−et
2L∗[a(·)−1ut(·,t/2)]
−/integraldisplayt/2
0L∗e(t−s)L∗[a(·)−1ut(·,s)]ds, (4.1)
whereL∗is the (negative) Friedrichs extension of −L=−a(x)−1∆inL2
dµ.12 MOTOHIRO SOBAJIMA AND YUTA WAKASUGI
Proof of Proposition 4.1. First we show the assertion for ( u0,u1) satisfying the
compatibility condition of order 2. Taking L2
dµ-norm of both side, we have
/vextenddouble/vextenddouble/vextenddoubleu(x,·)−etL∗[u0+a(·)−1u1]/vextenddouble/vextenddouble/vextenddouble
L2
dµ≤ J1(t)+J2(t)+J3(t),
where
J1(t) :=/integraldisplayt
t/2/vextenddouble/vextenddoublee(t−s)L∗[a(·)−1utt(·,s)]/vextenddouble/vextenddouble
L2
dµds,
J2(t) :=/vextenddouble/vextenddoubleet
2L∗[a(·)−1ut(·,t/2)]/vextenddouble/vextenddouble
L2
dµ,
J3(t) :=/integraldisplayt/2
0/vextenddouble/vextenddoubleL∗e(t−s)L∗[a(·)−1ut(·,s)]/vextenddouble/vextenddouble
L2
dµds.
Noting that for x∈Ω,
a(x)−1Φε(x,t)−1≤1
a1/a\}b∇acketle{tx/a\}b∇acket∇i}htαexp/parenleftbigg
−A1ε
h+2ε/a\}b∇acketle{tx/a\}b∇acket∇i}ht2−α
1+t/parenrightbigg
≤1
a1/parenleftbiggα(h+2ε)
(2−α)eA1ε/parenrightbiggα
2−α
(1+t)α
2−α,
we see that for k= 0,1,
/vextenddouble/vextenddoublea(·)−1∂k+1
tu(·,s)/vextenddouble/vextenddouble2
L2
dµ=/integraldisplay
Ωa(x)−1|∂k+1
tu(·,s)|2dx
≤ /ba∇dbla(·)−1Φε(·,t)−1/ba∇dblL∞(Ω)/integraldisplay
Ω|∂k+1
tu(·,s)|2Φεdx
≤/tildewideC(1+t)α
2−αE∂t(t,∂k
tu)
≤/tildewideCMε,k(1+t)−λ0−2−2α
2−α−2k/ba∇dbl(u0,u1)/ba∇dbl2
Hk+1×Hk.
Therefore from Proposition 3.5 with k= 1 and k= 0 we have
J1(t)≤/integraldisplayt
t/2/vextenddouble/vextenddoublea(·)−1utt(·,s)/vextenddouble/vextenddouble
L2
dµds
≤/radicalBig
/tildewideCM1/ba∇dbl(u0,u1)/ba∇dblH2×H1/integraldisplayt
t/2(1+s)−λ0
2−1−α
2−α−1ds
≤2(2−α)
λ0(2−α)+1−α/radicalBig
/tildewideCMε,1(1+t)−λ0
2−1−α
2−α/ba∇dbl(u0,u1)/ba∇dblH2×H1
and
J2(t)≤/vextenddouble/vextenddoublea(·)−1ut(·,t/2)/vextenddouble/vextenddouble
L2
dµ≤/radicalBig
/tildewideCMε,0(1+t)−λ0
2−1−α
2−α/ba∇dbl(u0,u1)/ba∇dblH1×L2.
Moreover, by Lemma 2.2, we see by Cauchy–Schwarz inequality that fort≥1,
J3(t)≤C/integraldisplayt/2
0(t−s)−N−α
2(2−α)−1/vextenddouble/vextenddoublea(·)−1ut(·,s)/vextenddouble/vextenddouble
L1
dµds
≤C/parenleftbiggt
2/parenrightbigg−N−α
2(2−α)−1/integraldisplayt/2
0/radicalBig
/ba∇dblΦ−1ε(·,s)/ba∇dblL1(Ω)E∂t(s;u)ds.WAVE EQUATION WITH SPACE-DEPENDENT DAMPING 13
Since
/ba∇dblΦ−1(·,t)/ba∇dblL1(Ω)≤/integraldisplay
RNexp/parenleftbigg
−A1ε
h+2ε|x|2−α
1+t/parenrightbigg
dx
= (1+t)N
2−α/integraldisplay
RNexp/parenleftbigg
−A1ε
h+2ε|y|2−α/parenrightbigg
dy,
we deduce
J3(t)≤C′(1+t)−N−α
2(2−α)−1/ba∇dbl(u0,u1)/ba∇dblH1×L2/integraldisplayt/2
0(1+s)N−α
2(2−α)−λ0
2−1−α
2−αds
≤C′/parenleftbiggN−α
2(2−α)−λ0
2+1
2−α/parenrightbigg
(1+t)−N−α
2(2−α)−1(1+t/2)N−α
2(2−α)−λ0
2−1−α
2−α+1
×/ba∇dbl(u0,u1)/ba∇dblH1×L2
≤C′′(1+t)−λ0
2−1−α
2−α/ba∇dbl(u0,u1)/ba∇dblH1×L2.
Consequently, we obtain
/vextenddouble/vextenddouble/vextenddoubleu(·,t)−etL∗[u0+a(·)−1u1]/vextenddouble/vextenddouble/vextenddouble
L2
dµ≤C′′′(1+t)−λ0
2−1−α
2−α/ba∇dbl(u0,u1)/ba∇dblH2×H1.
Next we show the assertion for ( u0,u1) satisfying ( u0,u1)∈(H2×H1
0(Ω))×
H1
0(Ω) (the compatibility condition of order 1) via an approximation argu ment.
Fixφ∈C∞
c(RN,[0,1]) such that φ≡1 onB(0,R0) andφ≡0 onRN\B(0,R0+1)
and define for n∈N,
/parenleftbigg
u0n
u1n/parenrightbigg
=/parenleftbigg
φ˜u0n
φ˜u1n/parenrightbigg
,/parenleftbigg
˜u0n
˜u1n/parenrightbigg
=/parenleftbigg
1+1
nA/parenrightbigg−1/parenleftbigg
u0
u1/parenrightbigg
,
whereAis anm-accretive operator in H=H1
0(Ω)×L2(Ω) associated with (1.1),
that is,
A=/parenleftbigg
0−1
−∆a(x)/parenrightbigg
endowed with domain D(A) = (H2∩H1
0(Ω))×H1
0(Ω). Then ( u0n,u1n) satisfies
supp(u0n,u1n)⊂B(0,R0+1) and the compatibility condition of order 2. Let vn
be a solution of (1.1) with ( u0n,u1n). Observe that
/ba∇dbl(u0n,u1n)/ba∇dbl2
H2×H1≤C2/ba∇dblφ/ba∇dbl2
W2,∞/ba∇dbl(˜u0,˜u1)/ba∇dbl2
H2×H1
≤C′2/ba∇dblφ/ba∇dbl2
W2,∞(/ba∇dbl(˜u0,˜u1)/ba∇dbl2
H+/ba∇dblA(˜u0,˜u1)/ba∇dbl2
H)
≤C′2/ba∇dblφ/ba∇dbl2
W2,∞(/ba∇dbl(u0,u1)/ba∇dbl2
H+/ba∇dblA(u0,u1)/ba∇dbl2
H)
≤C′′2/ba∇dblφ/ba∇dbl2
W2,∞/ba∇dbl(u0,u1)/ba∇dbl2
H2×H1
with suitable constants C,C′,C′′>0, and
/parenleftbiggu0n
u1n/parenrightbigg
→/parenleftbiggφu0
φu1/parenrightbigg
=/parenleftbiggu0
u1/parenrightbigg
inH
asn→ ∞and also u0n+a−1u1n→u0+a−1u1inL2
dµasn→ ∞. Using the result
of the previous step, we deduce
/vextenddouble/vextenddouble/vextenddoublevn(·,t)−etL∗[u0n+a(·)−1u1n]/vextenddouble/vextenddouble/vextenddouble
L2
dµ≤˜C(1+t)−λ0
2−1−α
2−α/ba∇dbl(u0,u1)/ba∇dblH2×H114 MOTOHIRO SOBAJIMA AND YUTA WAKASUGI
with some constant ˜C >0. Letting n→ ∞, by continuity of the C0-semigroup
e−tAinHwe also obtain diffusion phenomena for initial data in ( H2∩H1
0(Ω))∩
H1
0(Ω). /square
Acknowledgments
This work is supported by Grant-in-Aid for JSPS Fellows 15J01600 of Japan
Society for the Promotion of Science and also partially supported by Grant-in-Aid
for Young Scientists Research (B), No. 16K17619. The authors w ould like to thank
the referee for giving them valuable comments and suggestions.
References
[1] M. Ikawa, Mixed problems for hyperbolic equations of sec ond order, J. Math. Soc. Japan
20(1968), 580–608.
[2] M. Ikawa, Hyperbolic partial differential equations and wave phenomena, American Math-
ematical Society (2000).
[3] R. Ikehata, Some remarks on the wave equation with potent ial type damping coefficients,
Int. J. Pure Appl. Math. 21(2005), 19–24.
[4] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ.
Res. Inst. Math. Sci. 12(1976), 169–189.
[5] A. Matsumura, Energy decay of solutions of dissipative w ave equations, Proc. Japan Acad.,
Ser. A53(1977), 232–236.
[6] K. Mochizuki, Scattering theory for wave equations with dissipative terms, Publ. Res. Inst.
Math. Sci. 12(1976), 383–390.
[7] K. Nishihara, Lp-Lqestimates of solutions to the damped wave equation in 3-dime nsional
space and their application, Math. Z. 244(2003), 631–649.
[8] P. Radu, G. Todorova, B. Yordanov, Higher order energy de cay rates for damped wave
equations with variable coefficients, Discrete Contin. Dyn. Syst. Ser. S 2(2009), 609–629.
[9] P. Radu, G. Todorova, B. Yordanov, Decay estimates for wa ve equations with variable
coefficients, Trans. Amer. Math. Soc. 362(2010), 2279–2299.
[10] M. Sobajima and Y. Wakasugi, Diffusion phenomena for the wave equation with space-
dependent damping in an exterior domain, J. Differential Equations 261(2016), 5690–5718.
[11] G. Todorova, B. Yordanov, Critical exponent for a nonli near wave equation with damping,
J. Differential Equations 174(2001), 464–489.
[12] G. Todorova, B. Yordanov, Weighted L2-estimates for dissipative wave equations with vari-
able coefficients, J. Differential Equations 246(2009), 4497–4518.
[13] Y. Wakasugi, On diffusion phenomena for the linear wave e quation with space-dependent
damping, J. Hyp. Diff. Eq. 11(2014), 795–819.
(M.Sobajima) Departmentof Mathematics, Faculty of Science andTechnolo gy, Tokyo
University of Science, 2641 Yamazaki, Noda-shi, Chiba-ken 27 8-8510, Japan
E-mail address :msobajima1984@gmail.com
(Y.Wakasugi) Graduate School of Mathematics, NagoyaUniversity, Furocho, Chikusaku,
Nagoya 464-8602 Japan
E-mail address :yuta.wakasugi@math.nagoya-u.ac.jp |
2004.04840v3.Magnetic_Damping_in_Epitaxial_Fe_Alloyed_with_Vanadium_and_Aluminum.pdf | 1
Magnetic Damping in Epitaxial Fe Alloyed with Vanadium and Aluminum
David A. Smith1, Anish Rai2,3, Youngmin Lim1, Timothy Hartnett4, Arjun Sapkota2,3, Abhishek
Srivastava2,3, Claudia Mewes2,3, Zijian Jiang1, Michael Clavel5, Mantu K. Hudait5, Dwight D.
Viehland6, Jean J. Heremans1, Prasanna V. Balachandran4,7, Tim Mewes2,3, Satoru Emori1
1Department of Physics, Virginia Tech, Blacksburg, VA 24061, U.S.A.
2Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, U.S.A.
3Center for Materials for Information Technology (MINT), University of Alabama, Tuscaloosa,
AL 35487, U.S.A .
4Department of Material Science and Engineering, University of Virginia,
Charlottesville, VA 22904, U.S.A.
5Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 24061,
U.S.A.
6Department of Materials Science and Engineering, Virginia Tech, Blacksburg, VA 24061,
U.S.A.
7Department of Mechanical and Aerospace Engineering, University of Virginia,
Charlottesville, VA 22904, U.S.A.
2
To develop low -moment, low -damping metallic ferromagnets for power -efficient spintronic
devices, it is crucial to understand how magnetic relaxation is impacted by the addition of
nonmagnetic elements. Here, we compare magnetic relaxation in epitaxial Fe films alloyed
with light nonmagnetic elements of V and Al. FeV alloys exhibit lower intrinsic damping
compared to pure Fe, reduced by nearly a factor of 2, whereas damping in FeAl alloys
increases with Al content . Our experimental and computat ional results indicate that
reducing the density of states at the Fermi level , rather than the average atomic number,
has a more significant impact in lowering damping in Fe alloyed with light elements .
Moreover, FeV is confirmed to exhibit an intrinsic Gi lbert damping parameter of ≃0.001,
among the lowest ever reported for ferromagnetic metals.
I. INTRODUCTION
The relaxation of magnetization dynamics (e.g., via Gilbert damping) plays important
roles in many spintronic applications, including those based on magnetic switching1,2, domain
wall motion3,4, spin wave propagation5,6, and su perfluid -like spin transport7,8. For devices driven
by spin -torque precessional dynamics1,9,10, the critical current density for switching is predicted
to scale with the produ ct of the Gilbert damping parameter and the saturation magnetization 2,11.
Thus, it is desirable to engineer magnetic materials that possess both low damping and low
moment for energy -efficient operation . While some electrically insulating magnetic oxides have
been considered for certain applications5,12,13, it is essential to engineer low -damping, low -
moment metallic ferromagnets for robust electrical readout via giant magnetoresistance and
tunnel magnetoresistance. Fe is the elemental ferromagnet with the lowest intrinsic Gilbert
damping parameter ( ≃0.002)14,15, albeit with the highest saturation magnetization ( ≃2.0 T). 3
Recent experiments have reported that Gilbert damping can be further reduced by alloy ing Fe
with Co (also a ferromagnetic element), with Fe 75Co25 yielding an ultralow intrinsic Gilbert
damping parameter of ≃0.00116,17. However, Fe 75Co25 is close to the top of the Slater -Pauling
curve , such that its saturation magnetization is greater than that of Fe by approximately 20 %18.
There is thus an unmet need to engineer ferromagnetic alloys tha t simultaneously exhibit lower
damping and lower moment than Fe.
A promising approach towards low -damping, low -moment ferromagnetic metals is to
introduce nonmagnetic elements into Fe . In addition to diluting the magnetic moment,
nonmagnetic elements int roduced into Fe could influence the spin -orbit coupling strength ξ,
which underlies spin relaxation via orbital and electronic degrees of freedom19–21. Simple atomic
physics suggests that ξ is related to the average atomic number <Z> of the alloy so that,
conceivably, damping might be lowered by alloying Fe with lighter (lower -Z) elements. Indeed,
motivated by the premise of lowering damping through a reduced <Z> and presumably ξ, prior
experiments have explored Fe thin films alloyed with V20,22,23, Si24, and Al25. However, the
experimentally reported damping parameters for these alloys are often a factor of >2 higher22,23 ,25
than the theoretically predicted intrinsic Gilbert damping parameter of ≃0.002 in Fe26 and do not
exhibit a significant dependence on the alloy composition20,23,24. A possible issue is that the
reported damping parameters – obtained from the frequency dependence of ferromagnetic
resonance (FMR) linewidth with the film magnetized in -plane – may include contributions from
non-Gilbert relaxation induced by inhomogeneity and defects (e.g., two -magnon scattering)27–36,
which can be affected by the alloying. Therefore, how Gilbert damping in Fe is impacted by
alloying with low -Z elements remains an open question. 4
Here, we investigate the compositiona l dependence of magnetic relaxation at room
temperature in epitaxial thin films of ferromagnetic FeV and FeAl alloys. Both alloys are
crystalline bcc solid solutions and hence constitute excellent model systems. We employ two
configurations of FMR measurem ents to gain complementary insights: (1) FMR with samples
magnetized in the film plane (similar to the prior experiments) to derive the “effective” Gilbert
damping parameter, 𝛼𝑒𝑓𝑓𝐼𝑃, which is found to include extrinsic magnetic relaxation due to two -
magnon scattering, and (2) FMR with samples magnetized perpendicular to the film plane to
quantify the intrinsic Gilbert damping parameter, 𝛼𝑖𝑛𝑡, which is free of the two -magnon
scattering contribution.
Since Al ( Z = 13) is a much lighter element than V ( Z = 23), we might expect lower
magnetic relaxation in FeAl than FeV, if the smaller < Z> lowers intrinsic Gilbert damping via
reduced ξ. Instead, we find a significant decrease in magnetic relaxation by alloying Fe w ith V –
i.e., yielding an intrinsic Gilbert damping parameter of ≃0.001, on par with the lowest values
reported for ferromagnetic metals – whereas damping in FeAl alloys increases with Al content .
These experimental results , combined with density functi onal theory calculations, point to the
density of states at the Fermi level D(EF) as a plausible dominant factor for the lower (higher)
Gilbert damping in FeV (FeAl). We thus find that incorporating a low -Z element does not
generally lower damping and that, rather, reducing D(EF) is an effective route for lower damping
in Fe alloyed wi th a nonmagnetic element. Our findings confirm that FeV is an intrinsically
ultralow -damping alloy, as theoretically predicted by Mankovsky et al.26, which also possesses a
lower saturation magnetization than Fe and FeCo. The combination of low damping and low
moment makes FeV a highly promising material for practical metal -based spintronic
applications. 5
II. FILM DEPOSITION AND STRUCTURAL PROPERTIES
Epitaxial Fe 100-xVx and Fe 100-xAlx thin films were grown using dc magnetron sputtering
on (001) -oriented MgO substrates. Prior to deposition, the substrates were annealed at 600 oC for
2 hours37. The base pressure prior to deposition was < 5×10-8 Torr, and all film s were grown with
an Ar pressure of 3 mTorr. Fe and V (Al) 2” targets were dc co -sputtered to deposit Fe 100-xVx
(Fe 100-xAlx) films at a substrate temperature of 200 oC. By adjusting the deposition power, we
tuned the deposition rate of each material (calibrated by X -ray reflectivity) to achieve the desired
atomic percentage x of V (Al). All FeV and FeAl films had a thickness of 25 nm, which is well
above the thickness regime where interfacial effects dominate31,38. The FeV (FeAl) films were
capped with 3 -nm-thick V (Al) deposited at room temperature to protect against oxidation,
yielding a film structure of MgO/Fe 100-xVx(25nm)/V(3nm) or MgO/Fe 100-xAlx(25nm)/Al(3nm).
We confirmed the epitaxial bcc structure of our thi n films using high resolution X -ray
diffraction. 2θ -ω scans show only the (002) peak of the film and the (002) and (004) peaks of the
substrate, as shown in Fig ure 1. Rocking curve scans of the film peaks show similar full -width -
at-half-maximum values of ≃ 1.3o irrespective of composition . The epitaxial relation between
bcc Fe and MgO is well known16,39: the bcc film crystal is rotated 45o with respect to the
substrate crystal , such that the [100] axis of the film lies parallel to the [110] axis of the
substrate. The absence of the (001) film peak indicates that our epitaxial FeV and FeAl films are
solid sol utions rather than B2 -ordered compounds40.
6
III. MAGNETIC RELAXATION
3.1. In -Plane Ferromagnetic Resonance
Many spintronic devices driven by precessional magnetization dynamics are based on in -
plane magnetized thin films. The equilibrium magnetization also lies in -plane for soft
ferromagnetic thin films dominated by shape anisotropy (i.e., negligible perpendicular magnetic
anisotropy), as is the case for our epitaxial FeV and FeAl films. We therefore first discuss FMR
results w ith films magnetized in -plane. The in -plane FMR results further provide a basis for
comparison with previous studies20,22,23,25.
Samples were placed with the film side facing a coplanar waveguid e (maximum
frequency 50 GHz) and magnetized by an e xternal field H (from a conventional electromagnet,
maximum field 1.1 T) along the in -plane [100] and [110] axes of the films. Here, unless
otherwise stated, we show results for H || [110] of the film. FMR spectra were acquired via field
modulation by sweeping H and fixing the microwave excitation frequency.
Exemplary spectra for Fe, Fe 80V20, and Fe 80Al20 are shown in Fig ure 2, where we
compare the peak -to-peak linewidths at a microwave excitation frequency of 20 GHz. We see
that the linewidth for Fe 80V20 shows a ≃ 25 % reduction compared to Fe. We further note that
the linewidth for the Fe 80V20 sample here is a factor of ≃ 2 narrower than that in previously
reported FeV20; a possible origin of the narrow linewidth is discussed later . In contrast, Fe 80Al20
shows an enhancement in linewidth over Fe, which is contrar y to the expectation of lower
magnetic relaxation with a lower average atomic number.
The FMR linewidth is generally governed not only by magnetic relaxation, but also by
broadening contributions from magnetic inhomogeneities28,41,42. To disentangle the magnetic 7
relaxation and inhomogeneous broadening contributions to the linewidth, the typical prescription
is to fit the frequency f dependence of linewidth ∆𝐻𝑝𝑝𝐼𝑃 with the linear relation41
∆𝐻𝑝𝑝𝐼𝑃=∆𝐻0𝐼𝑃+ℎ
𝑔𝜇𝐵𝜇02
√3𝛼𝑚𝑒𝑎𝑠𝐼𝑃𝑓, (1)
where h is the Planck constant, 𝜇𝐵 is the Bohr magneton, 𝜇0 is the permeability of free space,
and 𝑔 is the g-factor obtained from the frequency dependence of the resonance field (see Section
IV and Supplementa l Material). In Eq. (1), the slope is attributed to viscous magnetic damping,
captured by the measured damping parameter 𝛼𝑚𝑒𝑎𝑠𝐼𝑃, while t he zero -frequency linewidth ∆𝐻0𝐼𝑃 is
attributed to inhomogeneo us broadening. The fitting with Eq. (1) was carried out for f 10 GHz,
where H was sufficiently large to saturate the films. As is evident from the results in Fig ure 3,
Fe80V20 has lower linewidths across all frequencies and a slightly lower slope, i.e., 𝛼𝑚𝑒𝑎𝑠𝐼𝑃. On the
other hand, Fe 80Al20 shows higher linewidths and a higher slope.
The measured viscous damping includes a small contribution from eddy currents,
parameter ized by 𝛼𝑒𝑑𝑑𝑦 (Supplemental Material) , and a contribution due to radiative damping43,
given by 𝛼𝑟𝑎𝑑 (Supplemental Material). Together these contributions make up ≃20 % of the total
𝛼𝑚𝑒𝑎𝑠𝐼𝑃 for pure Fe and decrease in magnitude with increasing V or Al content . We subtract these
to obtain the effective in -plane Gilbert damping parameter,
𝛼𝑒𝑓𝑓𝐼𝑃=𝛼𝑚𝑒𝑎𝑠𝐼𝑃−𝛼𝑒𝑑𝑑𝑦 − 𝛼𝑟𝑎𝑑. (2)
As shown in Fig ure 4a, 𝛼𝑒𝑓𝑓𝐼𝑃 remains either invariant or slightly decreases in Fe 100-xVx up to x =
25, whereas we observe a monotonic enhancement of 𝛼𝑒𝑓𝑓𝐼𝑃 with Al content in Figure 4b . These
results point to lower (higher) damping in FeV (FeAl) and suggest a factor other than the average
atomic number governing magnetic relaxation in these alloys. However, such a conclusion
assumes that 𝛼𝑒𝑓𝑓𝐼𝑃 is a reliable measure of intrinsic Gilbert damping . In reality, 𝛼𝑒𝑓𝑓𝐼𝑃 may include 8
a contribution from defect -induced two -magnon scattering27–31,35,36, a well -known non -Gilbert
relaxation mechanism in in -plane magnetized epitaxial films27,32 –34,44. We show in the next
subsection that substantial two -magnon scattering is indeed present in our FeV and FeAl alloy
thin films.
Although Eq. (1) is not necessarily the correct framework for quantifying Gilbert
damping in in -plane magnetized thin films, we can gain insight into the quality (homogeneity) of
the films from ∆𝐻0𝐼𝑃. For our samples, μ0∆𝐻0𝐼𝑃 is below ≈ 1 mT (see Fig ure 4c,d), which implies
higher film quality for our FeV samples than previously reported20. For example, Fe 73V27 in
Scheck et al. exhibits μ0∆𝐻0𝐼𝑃 ≃ 2.8 mT20, whereas Fe 75V25 in our study exhibits μ0∆𝐻0𝐼𝑃 ≃ 0.8
mT. Although 𝛼𝑒𝑓𝑓𝐼𝑃 is comparable between Scheck et al. and our study, the small ∆𝐻0𝐼𝑃 leads to
overall much narrower linewidths in our FeV films (e.g., as shown in Figs. 2 and 3) . We
speculate that the annealing of the MgO substrate prior to film deposition37 – a common practice
for molecular beam epitaxy – facilitates high -quality epitaxial film growth and hence small ∆𝐻0𝐼𝑃
even by sputtering.
3.2. Out -of-Plane Ferromagnetic Resonance
To quantify intrinsic Gilbert damping, we performed broadband FMR with the film
magnetized out -of-plane, which is the configuration that suppresses two -magnon scattering28–31.
Samples were placed in side a W-band shorted -waveguide spectrometer (frequency range 70 -110
GHz) in a superconducting electromagnet that enabled measurements at fields > 4 T. This high
field range is well above the shape anisotropy field of ≤2 T for our films and hence sufficient to
completely saturate the film out -of-plane. 9
The absence of two -magnon scattering in broadband out -of-plane FMR allows us to
reliably obtain the measured viscous damping parameter 𝛼𝑚𝑒𝑎𝑠𝑂𝑃 by fitting the linear frequency
dependence of the linewidth ∆𝐻𝑝𝑝𝑂𝑃, as shown in Figure 5, with
∆𝐻𝑝𝑝𝑂𝑃=∆𝐻0𝑂𝑃+ℎ
𝑔𝜇𝐵𝜇02
√3𝛼𝑚𝑒𝑎𝑠𝑂𝑃𝑓. (3)
We note that the zero -frequency linewidth for the out -of-plane configuration ∆𝐻0𝑂𝑃 (Figure 6c,d)
is systematically g reater than that for the in -plane configuration ∆𝐻0𝐼𝑃 (Figure 4c,d). Such a trend
of ∆𝐻0𝑂𝑃>∆𝐻0𝐼𝑃, often seen in epitaxial films15,33,45, may be explained by the stronger
contribution of inhomogeneity to the FMR field when the magnetic precessional orbit is circular,
as is the case for out -of-plane FMR, compared to the case of the highly elliptical precession in
in-plane FMR41; however, the detailed mechanisms contributing to the zero -frequency linewidth
remain the subject of future work . The larger ∆𝐻0𝑂𝑃 at high V and Al concentrations may be due
to broader distributions o f anisotropy fields and saturation magnetization, or the presence of a
secondary crystal phase that is below the resolution of our X -ray diffraction results.
The absence of two -magnon scattering in out -of-plane FMR allows us to quantify the
intrinsic Gilbert damping parameter,
𝛼𝑖𝑛𝑡=𝛼𝑚𝑒𝑎𝑠𝑂𝑃−𝛼𝑒𝑑𝑑𝑦, (4)
by again subtracting the eddy current contribution 𝛼𝑒𝑑𝑑𝑦. Since we utilize a shorted waveguide,
the contribution due to radiative damping does not apply.
From the compositional dependence of 𝛼𝑖𝑛𝑡 as summarized in Figure 6a1, a reduction in
intrinsic Gilbert damping is evidenced with V alloying. Our observation is in contrast to the
previous experiments on FeV alloys20,22,23 where the reported damping parameters remain >0.002
1 We were unable to carry out out -of-plane FMR measurements for FeV with x = 20 (Fig. 2(c,d )) as the sample had
been severely damaged during transit. 10
and depend weakly on the V concentration. In particular, the observed minimum of 𝛼𝑖𝑛𝑡≃0.001
at x ≃ 25-30 is approximately half of the lowest Gilbert damping parameter previously reported
for FeV20 and that of pure Fe15. The low 𝛼𝑖𝑛𝑡 here is also comparable to the lowest damping
parameters reported for ferromagnetic metals, such as Fe75Co2516,17 and Heusler compounds46–48.
Moreover, t he reduced intrinsic damping by alloying Fe w ith V is qualitatively consistent with
the computational prediction by Mankov sky et al.26, as shown by the curve in Figure 6a. Our
experimental finding therefore confirms that FeV is indeed an intrinsically ultralow -damping
ferromagnet that possesses a smaller saturation magnetization than Fe.
In contrast to the reduction of 𝛼𝑖𝑛𝑡 observed in FeV alloys, FeAl shows an increase in
intrinsic damping with increasing Al concentration, as seen in Figure 6b. Recalling that Al has an
atomic number of Z = 13 that is lower than Z = 23 for V, this trend clashes with the expectation
that lower < Z> red uces the intrinsic Gilbert damping through a reduction of the atomic spin -orbit
coupling. Thus, we are required to consider an alternative mechanism to explain the higher
(lower) damping in FeAl (FeV), which we discuss further in Section V.
3.3. Magnetic Relaxation: Practical Consideration s
For both FeV and FeAl alloys, 𝛼𝑖𝑛𝑡 derived from out -of-plane FMR (Figure 6a,b) is
consistently lower than 𝛼𝑒𝑓𝑓𝐼𝑃 derived from in -plane FMR (Fig ure 4a,b). Th is discrepancy
between 𝛼𝑖𝑛𝑡 and 𝛼𝑒𝑓𝑓𝐼𝑃 implies a two-magnon scattering contribution to magnetic relaxation in
the in-plane configuration (Figure 4a,b). For many applications including spin -torque oscillators
and magnonic devices , it is crucial to minimize magnetic relaxation in in-plane magnetized thin
films. While the in -plane magnetic relaxation ( 𝛼𝑒𝑓𝑓𝐼𝑃≃0.002) is already quite low for the FeV
alloys shown here, the low intrinsic Gilbert damping ( 𝛼𝑖𝑛𝑡≃0.001) points to the possibility of 11
even lower relaxation and narrow er FMR linewidths by minimizing two -magnon scattering and
inhomogeneous linewidth broadening. Such ultralow magnetic relaxation in FeV alloy thin films
may be achieved by optimizing structural properties through growth conditions16 or seed layer
engineering49.
While ultralow intrinsic Gilbert damping values have been confirmed in high -quality
epitaxial FeV, it would be desirable for device integration to understand how magnetic relaxation
in FeV would be impacted by the presence of grain boundaries, i.e. in polycrystalline thin films.
Reports on polycrystalline FeCo49 suggest intrinsic damping values comparable to those seen in
epitaxial FeCo16,17. While beyond the scope of this study, our future work will explore the
possibility of low damping in polycrystalline FeV thin films.
IV. SPECTROSCOPIC PARAMETERS
The results presented so far reveal that magnetic relaxation is reduced by alloying Fe with
V, whereas it is increased by alloying Fe with Al. On the other hand, FeV and FeAl alloys
exhibit similar compositional dependence of the spectroscopic parameters: effective
magnetization Meff (here, equivalent to saturation magnetization Ms), magnetocrystalline
anisotropy field Hk, and the g-factor 𝑔 – all of which are quantified by fitting the frequency
dependence of resonance field (Supplemental Material) . As shown in Fig ure 7a, there is a
systematic reduction in Meff with increasing concentration of V and Al. We also note in Fig ure 7b
a gradual reduction in magnitude of the in -plane cubic anisotropy. Both of these trends are
expected as magnetic Fe atoms are r eplaced with nonmagnetic atoms of V and Al. The reduction
of Meff by ≃20% in the ultralow -damping Fe 100-xVx alloys with x = 25-30, compared to pure Fe,
is of particular practical interest. The saturation magnetization of these FeV alloys is on par with 12
commonly used soft ferromagnetic alloys (e.g., Ni 80Fe2050, CoFeB51), but the damping parameter
of FeV is several ti mes lower. Further, w hile FeV and FeCo in the optimal composition window
show similarly low intrinsic damping parameters, FeV provides the advantage of lower moment .
With the product 𝛼𝑖𝑛𝑡𝑀𝑒𝑓𝑓 approximately proportional to the critical current densi ty to excite
precessional dynamics by spin torque2,11, FeV is expected to be a superior material platform for
low-power spin tronic devices .
The g-factor 𝑔=2(1+𝜇𝐿/𝜇𝑆) is related to the orbital moment 𝜇𝐿 and spin moment 𝜇𝑆;
the deviation from the spin -only value of 𝑔= 2.00 provides insight into the strength of spin -orbit
coupling ξ52. As seen in Figure 7c, 𝑔 increases by 1-2% with both V and Al alloying, which
suggests that ξ increases slightly with the addition of these low -Z elements. This finding verifies
that < Z> is not necessarily a good predictor of ξ in a solid. Moreover, the higher 𝑔 for FeV is
inconsistent with the scenario for lower damping linked to a reduced spin -orbit coupling. Thus,
spin-orbit coupling alone cannot explain the observed behavior of Gilbert damping in Fe alloyed
with low -Z elements.
V. DISCUSSION
In contrast to what has been suggested by prior experimental studies20,22 –25, we have
shown that the reduction of average atomic number by alloying with a light element (e.g., Al in
this case) does not generally lower the intrinsic Gilbert dampin g of Fe. A possible source for the
qualitatively distinct dependencies of damping on V and Al contents is the density of states at the
Fermi level, D(EF): it has been predicted theoretically that the intrinsic Gilbert damping
parameter is reduced with decr easing D(EF), since D(EF) governs the availability of states for
spin-polarized electrons to scatter into21,26,53 –55. Such a correlation between lower damping and 13
smaller D(EF) has been reported by recent experiments on FeCo alloys17,50, FeRh alloys40, CoNi
alloys56, and Heusler compounds46,48,57. The similarity in the predicted composition dependence
of the Gilbert damping parameter for FeCo and FeV26 suggests that the low damping of FeV may
be correlated with reduced D(EF). However, no prior experiment has corroborated this
correlation for FeV or other alloys of Fe and light elements.
We therefore e xamine whether the lower (higher) damping in FeV (FeAl) compared to Fe
can be qualitatively explained by D(EF). Utilizing the Quantum ESPRESSO58 package to
perform density functional theory calculations (details in Supplemental Material) , we calculated
the density of states for Fe, Fe 81.25V18.75, and Fe 81.25Al18.75. It should be recalled that although
FeV and FeAl films measured experimentally her e are single -crystalline, they are solid solutions
in which V or Al atoms replace Fe atoms at arbitrary bcc lattice sites. Therefore, f or each of the
binary alloys, we computed 6 distinct atomic configurations in a 2×2×2 supercell , as shown in
Figure 8 . The spin -split density of states for each unique atomic configuration is indicated by a
curve in Figure 9. Here, D(EF) is the sum of the states for the spin -up and spin -down bands,
averaged over results from the 6 distinct atomic configurations.
As summari zed in Fig ure 9 and Table 1, FeV has a smaller D(EF) than Fe, whereas FeAl
has a larger D(EF). These calculation results confirm a smaller (larger ) availability of states for
spin-polarized electrons to scatter into in FeV (FeAl), qualitatively consistent with the lower
(higher) intrinsic Gilbert damping in FeV (FeAl).
We remark that this correlation between damping and D(EF) is known to hold parti cularly
well in the limit of low electronic scattering rates 𝜏−1, where intra band scattering dominates21,54.
Gilmore et al. have pointed out that at sufficiently high electronic scattering rates, i.e., when
ℏ𝜏−1 is large enough that inter band scattering is substantial, the simple correlation between the 14
strength of Gilbert damping and D(EF) breaks down. It is unclear whether our FeV and FeAl
alloy films at room temperature are in the intraband - or interband -dominated regime. Schoen et
al. have argued that polycrystalline FeCo alloy films – with higher degree of structural disorder
and likely higher electronic scattering rates than our epitaxial films – at room temperature are
still well within the intraband -dominated regime17. On the other hand, a recent temperature -
dependent study on epitaxial Fe suggests coexistence of the intraband and interband
contributions at room temperature15. A consistent explanation for the observed room -temperature
intrinsic damping in our alloy films is that the interband contribution depends weakly on alloy
composition; it appears re asonable to conclude that D(EF), primarily through the intraband
contribution, governs the difference in intrinsic Gilbert damping among Fe, FeV, and FeAl .
VI. SUMMARY
We have experimentally in vestigated magnetic relaxation in epitaxial thin films of Fe
alloyed with low -atomic -number nonmagnetic elements V and Al . We observe a reduction in the
intrinsic Gilbert damping parameter to 𝛼𝑖𝑛𝑡≃0.001 in FeV films , comparable to the lowest -
damping ferromagnetic metals reported to date. In contrast, an increase in damping is observed
with the addition of Al, demonstrating that a smaller average atomic number does not necessarily
lower intrinsic damping in an alloy . Furthermore, our results on FeV and FeAl cannot be
explained by the change in spin -orbit coupling through alloying . Instead, we conclude that the
density of states at the Fermi level plays a larger role in determining the magnitude of damping
in Fe alloyed w ith lighter elements. Our work also confirms FeV alloys as promising ultra low-
damping , low-moment metallic materials for practical power -efficient spin -torque devices.
15
Acknowledgements:
This research was funded in part by 4 -VA, a collaborative partnership for advancing the
Commonwealth of Virginia, as well as by the ICTAS Junior Faculty Program. D.A.S.
acknowledges support of the Virginia Tech Graduate School Doctoral Assistantship. A. Sapkota
and C. M . would like to acknowledge support by NSF -CAREER Award No. 1452670, A.R. and
T.M. would like to acknowledge support by DARPA TEE Award No. D18AP00011, and A.
Srivastava would like to acknowledge support by NASA Award No. CAN80NSSC18M0023.
We thank M.D. Stiles for helpful input regarding intrinsic damping mechanisms in alloys.
The data that support the findings of this study are available from the corresponding author upon
reasonable request.
Number of Spin -Up States (eV-1)
at EF Number of Spin -Down States
(eV-1) at EF
Fe 10.90 3.44
Fe81.25V18.75 6.28 ± 1.80 4.61 ± 0.43
Fe81.25Al18.75 6.81 ± 1.58 10.20 ± 3.03
Table 1: Number of spin -up and spin -down states at EF. For Fe81.25V18.75 and
Fe81.25Al18.75, the average and standard deviation of values for the 6 distinct atomic
configurations (cf. Figure 8) are shown. 16
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21
15 30 45 60 75 90
MgO (004) MgO (004) MgO (004)MgO (002) MgO (002)
MgO (002)BCC
Fe
(002)
Log(Intensity) (arb. units)BCC
Fe80V20
(002)
2q (deg)BCC
Fe80Al20
(002)
Figure 1: (a) 2θ-ω X-ray diffraction scans showing (00 2) and (004) substrate and (002) film
peaks for bcc Fe, Fe 80V20, and Fe 80Al20.
22
-15 -10 -5 0 5 10 15
Fe 2.70 mT
FMR Signal (arb. units)Fe80V20 2.04 mT
m0(H - HFMR) (mT)Fe80Al203.20 mT
Figure 2: FMR spectra at f = 20 GHz with the magnetic field H applied in the film plane, fitted
using a Lorentzian derivative (solid curve ) for Fe, Fe 80V20 and Fe 80Al20. 23
0 10 20 30 40 5002468 Fe
Fe80V20
Fe80Al20
Scheck et al.m0DHIP
PP (mT)
Frequency (GHz)
Figure 3: FMR linewidths versus microwave frequency for the magnetic field applied within the
plane of the film for three distinct alloys. The solid lines are linear fit s, described by Eq. (1),
from which the effective damping parameter and zero frequency linewidth are determined. The
dashed line represents the result for Fe 73V27 from Scheck et al.20
24
0 10 20 30 40246
0 10 20 302468
0 10 20 30 40024
0 5 10 15 20 25 3001 Fe100-xVx
Scheck et al.aIP
eff x 103
aIP
eff x 103 Fe100-xAlxm0DHIP
0 (mT)
Alloy Composition, x (%)(a) (b)
(c) (d)m0DHIP
0 (mT)
Alloy Composition, x (%)
Figure 4: The effective damping parameter 𝛼𝑒𝑓𝑓𝐼𝑃 for (a) Fe 100-xVx and (b) Fe 100-xAlx and zero
frequency linewidth 𝜇0Δ𝐻0𝐼𝑃 for (c) Fe 100-xVx and (d) Fe 100-xAlx, obtained from in -plane FMR.
The solid symbols in (a) and (c) represent results reported by Scheck et al.20
25
0 20 40 60 80 100 12001020304050
Fe
Fe70V30
Fe70Al30m0DHOP
PP (mT)
Frequency (GHz)
Figure 5: FMR linewidths versus applied microwave frequency for the magnetic field applied
perpendicular to the plane of the film for three distinct alloys. The line is a linear fit, described
by Eq. (3), from which the intrinsic Gilbert damping parameter and zero frequency linewidth are
determined.
26
0 10 20 30 400123
0 10 20 300246
0 10 20 30 4001020
0 10 20 3001020 Fe100-xVx
Mankovsky et al.aint x 103 Fe100-xAlxaint x 103(a) (b)
(c) (d)m0DHOP
0 (mT)
Alloy Composition, x (%)
m0DHOP
0 (mT)
Alloy Composition, x (%)
Figure 6: The intrinsic Gilbert damping parameter 𝛼𝑖𝑛𝑡 for (a) Fe 100-xVx and (b) Fe 100-xAlx and
zero frequency linewidth 𝜇0Δ𝐻0𝑂𝑃 for (c) Fe 100-xVx and (d) Fe 100-xAlx, obtained from out -of-
plane FMR. In (a), the dashed curve show s the predicted intrinsic damping parameter computed
by Mankovsky et al.26
27
0.81.21.62.02.4
204060
0 10 20 30 402.082.102.122.14 Fe
Fe100-xVx
Fe100-xAlx
m0Meff (T)(a)
|m0Hk| (mT)(b)
g-factor
Alloy Composition, x (%)(c)
Figure 7: (a) Effective magnetization, (b) in -plane cubic anisotropy field, and (c) g-factor versus
V and Al concentration. The solid (open) markers represent data from in -plane (out -of-plane)
measurements .
28
Figure 8: The six unique atomic configurations from the supercell program for mimicking the
Fe81.25V18.75 or Fe81.25Al18.75 solid solution.
29
-10010-10010
-1.0 -0.5 0.0 0.5 1.0-10010
(a)
Fe81.25V18.75
Density of States (eV-1)
(b)Fe
E - EF (eV)(c)Fe81.25Al18.75
Figure 9: Calculated spin-up (positive) and spin -down (negative) densit ies of states for (a) Fe,
(b) Fe 81.25V18.75 and (c) Fe 81.25Al18.75. Results from the 6 distinct atomic configurations are shown
in (b,c); the average densities of states at EF for Fe81.25V18.75 and Fe81.25Al18.75 are shown in
Table 1.
|
2210.08429v1.Magnetic_damping_anisotropy_in_the_two_dimensional_van_der_Waals_material_Fe__3_GeTe__2__from_first_principles.pdf | Magnetic damping anisotropy in the two-dimensional van der Waals material
Fe3GeTe 2from rst principles
Pengtao Yang, Ruixi Liu, Zhe Yuan, and Yi Liu
The Center for Advanced Quantum Studies and Department of Physics,
Beijing Normal University, 100875 Beijing, China
(Dated: October 18, 2022)
Magnetization relaxation in the two-dimensional itinerant ferromagnetic van der Waals ma-
terial, Fe 3GeTe 2, below the Curie temperature is fundamentally important for applications to
low-dimensional spintronics devices. We use rst-principles scattering theory to calculate the
temperature-dependent Gilbert damping for bulk and single-layer Fe 3GeTe 2. The calculated damp-
ing frequency of bulk Fe 3GeTe 2increases monotonically with temperature because of the dominance
of resistivitylike behavior. By contrast, a very weak temperature dependence is found for the damp-
ing frequency of a single layer, which is attributed to strong surface scattering in this highly con ned
geometry. A systematic study of the damping anisotropy reveals that orientational anisotropy is
present in both bulk and single-layer Fe 3GeTe 2. Rotational anisotropy is signi cant at low tem-
peratures for both the bulk and a single layer and is gradually diminished by temperature-induced
disorder. The rotational anisotropy can be signi cantly enhanced by up to 430% in gated single-layer
Fe3GeTe 2.
I. INTRODUCTION
Newly emerged intrinsic two-dimensional (2D) ferro-
magnetic (FM) van der Waals (vdW) materials1{6have
become the subject of intense research. Weak vdW
bonding facilitates the extraction of thin layers down to
atomic thicknesses, whereas strong magnetocrystalline
anisotropy protects long-range magnetic order. These
materials provide an exciting arena to perform funda-
mental investigations on 2D magnetism and promis-
ing applications of low-dimensional spintronics devices.
Among these materials, Fe 3GeTe 2(FGT) is especially
attractive for its itinerant ferromagnetism and metal-
licity, such that both spin and charge degrees of free-
dom can be exploited for designing functional devices.
Bulk FGT has a relatively high Curie temperature ( TC)
of approximately 220-230 K.7{11Atomically thin lay-
ers of FGT have lower TCs, which, however, have been
raised to room temperature (by ionic gating4) and be-
yond (by patterning12). As a FM metal at reasonably
high temperature, FGT opens up vast opportunities for
applications.13{23
The dynamical properties of FGT critically a ect the
applicability and performance of these proposed low-
dimensional spintronics devices. The most salient of
these properties is the dynamical dissipation of mag-
netization. It is usually described using a phenomeno-
logical parameter called Gilbert damping, which char-
acterizes the eciency of the instantaneous magneti-
zation to align eventually with the e ective magnetic
eld during its precessional motion. Although this pa-
rameter has been extensively studied in conventional
FM materials, such as 3 dtransition metals and alloys,
two key issues with the Gilbert parameter of FGT re-
main to be addressed: the temperature dependence and
anisotropy (one naturally expects anisotropic damping
in FGT because of its layered structure and the strong
magnetocrystalline anisotropy). Temperature-dependentGilbert damping was rst observed in Fe24and later more
systematically in Fe, Co and Ni.25{27A nonmonotonic
temperature dependence has been found, for which a so-
called \conductivitylike" component decreases with in-
creasing temperature, usually at low temperatures, and a
\resistivitylike" component increases with temperature,
usually at high temperatures. This nonmonotonic be-
havior has been successfully described by the torque-
correlation model28and reproduced by rst-principles
computations.29{32Anisotropic damping was rst theo-
retically predicted in FM metals33and in noncollinear
magnetic textures.34With di erent orientation of the
equilibrium magnetization with respect to the crystal-
lographic axes, the damping parameter can be quanti-
tatively di erent in general. This is referred to as the
orientational anisotropy. Even for the same equilibrium
magnetization orientation in a single crystalline lattice,
the magnetization may precess instantaneously along
di erent directions resulting in the so-called rotational
anisotropy.33The orientational anisotropy of damping
has been observed in recent experiments on single-crystal
FM alloys,35{37but the underlying physical mechanism
remains unclear.
The dimensionless Gilbert damping parameter can
be expressed in terms of a frequency via=
M ,38
whereM=jMjis the magnetization magnitude and
is
the gyromagnetic ratio. Despite of the di erent dimen-
sions, these two parameters are equivalent39and both
present in literature for experimental24{27,35{37and the-
oretical studies.28,29,31,33,34,40{42
In this study, we systematically investigate
temperature-dependent Gilbert damping in single-
layer (SL) and bulk FGT using rst-principles scattering
theory. Considering that the magnetization perpen-
dicular to the 2D atomic planes is favored by the
strong magnetocrystalline anisotropy, we calculate the
damping as a function of temperature below TCand
nd nearly temperature-independent damping in thearXiv:2210.08429v1 [cond-mat.mes-hall] 16 Oct 20222
(a)
FeⅠ
FeⅡ
Ge
Te(b)
FIG. 1. (a) Side and (b) top view of the lattice structure
for bulk Fe 3GeTe 2. The black dashed frame delineates the
in-plane unit cell.
SL and damping dominated by resistivitylike behavior
in the bulk. Varying the equilibrium direction of the
FGT magnetization produces a twofold symmetry in
damping. When the magnetization is aligned inside
the 2D planes, a remarkable rotational anisotropy in
the Gilbert damping is present for in- and out-of-plane
rotating magnetization.
This paper is organized as follows. The crystalline
structure of SL and bulk FGT is brie
y introduced in Sec.
II, followed by a description of our theoretical methods
and computational details. The calculated temperature-
dependent damping in SL and bulk FGT is presented in
Sec. III. The two types of damping anisotropy, i.e., orien-
tational and rotational anisotropy, are analyzed in Sect.
IV. Conclusions are drawn in Sec. V.
II. GEOMETRIC STRUCTURE OF FGT AND
COMPUTATIONAL METHODS
The lattice structure of FGT is shown in Fig. 1. Two
di erent types of Fe atoms occupy inequivalent Wycko
sites and are denoted as FeI and FeII. Five atomic layers
stack along the caxis to form an SL of FGT: Ge and
FeII constitute the central atomic layer perpendicular to
thecaxis, and two FeI layers and two Te layers are lo-
cated symmetrically above and beneath the central layer,
respectively. Single layers with ABAB :::stacking form
the bulk FGT, where Layer A is translated in plane with
respect to Layer B, such that the Ge atoms in Layer A
lie on top of the Te and FeII atoms in Layer B.
The electronic structure of bulk and SL FGT has
been determined using the linear augmented plane wave
method43within the local density approximation (LDA).
Di erent types of exchange-correlation functionals have
been investigated in the literature, among which LDA
was found to yield satisfactory structural and magnetic
properties for FGT.44We employ experimentally ob-
tained lattice constants7for bulk FGT calculations and
obtain magnetic moments of 1.78 Band 1.13Bfor
the two types of Fe, respectively. The initial structure
of a single layer is taken from the bulk lattice and fully
relaxed, resulting in an in-plane constant a= 3:92A.
A vacuum spacing of 11.76 A is chosen to exclude theinterlayer interaction under periodic boundary condi-
tions. The magnetic moments for the Fe atoms in SL
FGT are obtained as 1.72 Band 1.01B. All the
calculated magnetic moments are in good agreement
with experimental7,8,45,46and calculated values44,47,48re-
ported in the literature.
The Gilbert damping calculation is performed using
the scattering theory of magnetization dissipation pro-
posed by Brataas et al.49Within this theory, a single do-
main FM metal is sandwiched between two nonmagnetic
(NM) metal leads. The Gilbert damping that charac-
terizes the energy dissipation during magnetization dy-
namics can be expressed in terms of a scattering ma-
trix and its derivative with respect to the magnetiza-
tion direction. We thus construct a two-terminal trans-
port structure as Au jFGTjAu, where the Au lattice is
slightly deformed to match that of FGT: we use 3 1 and
41 unit cells (UCs) of Au (001) to match the UCs of
SL and bulk FGT, respectively. To investigate the ef-
fect of temperature on Gilbert damping, we use a frozen
thermal lattice and spin disorder31,40,50to mimic lattice
vibration and spin
uctuation at nite temperatures in
FGT. The measured Debye temperature D= 232 K
and temperature-dependent magnetization for the bulk8
and SL3are employed to model the lattice and spin disor-
der. In the scattering calculations, lateral supercells are
employed to satisfy periodic boundary conditions perpen-
dicular to the transport direction. The electronic poten-
tials required for the transport calculation are calculated
self-consistently using a minimal basis of tight-binding
linear mun-tin orbitals (TB-LMTOs), and the result-
ing band structures for SL and bulk FGT e ectively re-
produce those obtained using the linear augmented plane
wave method. Then, the scattering matrices consisting
of re
ection and transmission probability amplitudes for
the Bloch wave functions incident from the NM leads are
determined by the so-called \wave function matching"
method, which is also implemented using TB-LMTOs.40
Other computational details can be found in our previous
publications.31,40{42In this work, we focus on the damp-
ing with collective magnetization dynamics in the long-
wave limit corresponding to the reported values in ex-
periment via ferromagnetic resonance and time-resolved
magneto-optical Kerr e ect. The damping with a -
nite wavelength can be determined in our framework of
scattering calculation42or using the torque-correlation
model,51but the wavelength dependence of damping is
beyond the scope of the current study.
III. TEMPERATURE-DEPENDENT DAMPING
The strong magnetocrystalline anisotropy of FGT re-
sults in the equilibrium magnetization being naturally
perpendicular to the atomic layers. Slightly excited mag-
netization deviates from the plane normal (denoted as ^ z)
and relaxes back by dissipating energy and angular mo-
mentum, as schematized in the inset of Fig. 2(a). The3
𝛂∥𝛂∥𝑴(𝑡)𝑥𝑦𝑧
5101520F|| (10-3)
0 0.2 0.4 0.6 0.8 1T/TC468Q|| (108 Hz)Lattice disorder only(b)(a)
BulkSingle layer
BulkSingle layer
FIG. 2. The calculated dimensionless Gilbert damping pa-
rameter k(a) and corresponding damping frequency k(b)
for single-layer and bulk Fe 3GeTe 2as a function of tempera-
ture. The relaxation of the instantaneous magnetization M(t)
results in a change in the in-plane magnetization component,
which is parallel to the atomic planes, as schematized in the
inset of (a). The empty symbols in (b) denote the damping
frequencies that are calculated considering only thermal lat-
tice disorder. The green line indicates the linear temperature
dependence.
Gilbert damping parameter kdescribes the eciency
of such a dissipative process. The calculated kof SL
and bulk FGT is plotted in Fig. 2(a) as a function of
temperature. The damping for both increases monoton-
ically with the temperature. This behavior resembles
the so-called \resistivitylike" damping observed in many
single-crystal FM metals.24{26However, the damping k
for the bulk tends to diverge as the temperature ap-
proachesTC. This divergence originates from vanishing
magnetization, as has been found in three-dimensional
FM alloys.42Therefore, as temperatures approaching TC,
it is more appropriate to use the damping frequency pa-
rameter=
M .
The calculated damping frequencies are shown in
Fig. 2(b). The damping of a SL FGT, S
k, is larger
than the damping of the bulk, B
k, especially at low
temperatures. This di erence can be attributed to the
strong surface e ect of highly con ned SL FGT. The
lowered symmetry at the surface signi cantly enhances
spin-orbit coupling (SOC),52which enables the dissipa-
tion of angular momentum from electronic spins to the
orbital degree of freedom and then into the lattice reser-voir. In addition, as the thickness of a single layer is
considerably smaller than the electronic mean free path,
conduction electrons in FGT are strongly scattered by
the surface. Therefore, the two necessary ingredients for
Gilbert damping, namely, SOC and electronic scattering,
are both enhanced in the SL compared with the bulk, re-
sulting in a larger damping for the SL.
The calculated damping frequency S
kremains nearly
constant with increasing temperature, except for a mi-
nor increase at T > 0:6TC. To gain further insight into
the temperature e ect, we perform the damping calcu-
lation considering only lattice disorder, where the calcu-
latedS
latare plotted as red empty circles in Fig. 2(b).
Lattice-disorder-induced damping in the SL FGT, S
lat,
exhibits a very weak temperature dependence, indicating
that increasing lattice vibration does not in
uence the
damping frequency. The di erence between S
latandS
k
increases slightly only near TC, which can be attributed
to the strong spin
uctuation. The overall weak tem-
perature dependence in the damping for a single layer
indicates that a non-thermal disorder scattering mecha-
nism is dominant: the strong surface scattering in such
a thin layer (only a few A) combined with the enhanced
SOC at the surfaces is the main channel for the magnetic
damping in the SL FGT instead of spin
uctuation and
lattice vibration. Gilbert damping with a similarly weak
temperature dependence has also been found in a permal-
loy,40,53where chemical disorder scattering overwhelms
thermally induced disorder.
The temperature dependence of the bulk damping fre-
quency is signi cantly di erent from that of the SL. The
calculated bulk damping, B
k, (shown by the black solid
diamonds in Fig. 2(b)) increases linearly with the temper-
ature. This typical resistivitylike behavior suggests that
the interband transition in bulk FGT is the dominant
damping mechanism.54We also calculate the damping
frequencyB
latconsidering only lattice disorder, as shown
as the black empty diamonds in Fig. 2(b). Comparing the
results corresponding to the solid and empty diamonds
leads us to conclude that both lattice and spin disorder
substantially contribute to damping in bulk FGT. As the
temperature approaches TC, the bulk damping is compa-
rable with that in the single layer.
IV. ANISOTROPIC DAMPING
The damping torque exerted on the magnetization
in the Landau-Lifshitz-Gilbert equation has the general
form of M(t)[~ _M(t)], where the Gilbert damping
parameter ~ or the corresponding frequency is a tensor.
This tensor and its elements depend on both the instan-
taneous M(t) and its time derivative _M(t), where the
anisotropy has been extensively analyzed using theoret-
ical models55and rst-principles calculations.33,34Fol-
lowing the de nition given by Gilmore et al. ,33we call
the anisotropic damping that depends on the equilibrium
orientation of Meqthe orientational anisotropy and that4
𝑴𝐞𝐪𝑥𝑦𝑧𝜃
-U/2 -U/40U/4U/2V81216F|| (10-3)Single layerBulk
FIG. 3. The calculated Gilbert damping parameter kfor SL
(red circles) and bulk FGT (black diamonds) as a function
of the angle between the equilibrium magnetization Meqand
the atomic layer normal (^ z) of Fe 3GeTe 2. The lines are tted
usingC0+C2cos 2.
depending on _M(t) the rotational anisotropy. Consider-
ing the layered structure of vdW materials, the lowered
symmetry should result in remarkable anisotropy for the
magnetization relaxation. Both the orientational and ro-
tational anisotropy in bulk and SL FGT have been sys-
tematically analyzed in this section. Notably, the damp-
ing tensor is reduced to a scalar for the con guration
shown in Fig. 2.
Under a large in-plane magnetic eld, the perpendicu-
lar magnetization of FGT can be tilted toward the exter-
nal eld direction, which is de ned as the y-axis without
loss of generality. Thus, the angle between the equilib-
rium magnetization Meqand the plane normal ^ zis re-
ferred to as , as shown in the inset of Fig. 3. At = 0,
as studied in Sec. III, xx= yy= k. For6= 0,
xx= kstill holds, whereas the other diagonal element
yydepends on speci c values of . Here, we focus on k
to study the orientational anisotropy of damping. The
calculated in-plane damping kis plotted as a function
ofin Fig. 3 for a SL at 77 K and bulk FGT at 100
K. The temperature is chosen in this way to obtain the
same relative magnetization for the two systems, namely,
M=M s= 88%, according to the experimentally measured
magnetization as a function of temperature.3,8The same
twofold symmetry is found for the damping parameters
of both SL and bulk FGT, which can be e ectively tted
using a cos 2 term. As the magnetization rotates away
from the easy axis, kincreases and reaches a maximum
when the magnetization aligns inside the FGT layer. The
changes, [ (==2) (= 0)]= (= 0), are 62% for
the SL and 39% for the bulk. A similar dependence of
the damping on the magnetization orientation has been
recently observed in single-crystal CoFe alloys.35,36The
predicted anisotropic damping of FGT shown in Fig. 3
should analogously be experimentally observable.
The rotational anisotropy of damping33in FGT is most
signi cant when the equilibrium magnetization lies in-
side the atomic plane of FGT (along the hard axis), i.e.,
0 0.20.40.6 0.81T/TC120150180QC/Q|| (%)
0 0.2 0.4 0.6 0.8 1T/TC10152025Q (108 Hz)Bulk
Single layerBulkQCQ||Single layer
BulkBulk
𝛂∥𝑴(𝐭)
𝑥𝑦𝑧(a)𝜶"(b)FIG. 4. (a) Schematic of damping with the equilibrium mag-
netization Meqlying inside the atomic plane. Then, the in-
stantaneous magnetization M(t) dissipates both the in- and
out-of-plane spin angular momentum. The two types of dis-
sipation are denoted as k(k) and ?(?). (b) The calcu-
lated Gilbert damping frequency k(?)as a function of tem-
perature for single-layer and bulk Fe 3GeTe 2. The inset shows
the ratio of the two frequencies ?=k.
==2. As schematized in Fig. 4(a), the magne-
tization M(t) loses its in- or out-of-plane components
depending on the instantaneous precessional direction
_M(t). In this case, one has xx= kand zz= ?,
whereas the o -diagonal elements of the damping ten-
sor are guaranteed to remain zero by symmetry.40The
calculatedkand?for SL and bulk FGT are shown
as a function of temperature in Fig. 4(b). For SL FGT,
k(as shown by the circles with horizontal hatching) is
nearly independent of temperature, which is the same as
forMeqalong the easy axis. This result suggests that de-
spite the sizable orientational anisotropy in the damping
of SL FGT, the temperature has very little in
uence on
the speci c values of the damping frequency. The calcu-
lated?for the SL (shown by the red circles with vertical
hatching) is considerably larger than kat low tempera-
tures but decreases with increasing temperature. ?be-
comes comparable with knear the Curie temperature,
indicating that the rotational anisotropy is signi cantly
diminished by temperature.
The calculated kfor bulk FGT with Meqalong the
hard axis (shown by the black diamonds with horizontal
hatching) is temperature-independent, in sharp contrast
to the linear temperature dependence of kwith Meq
along the easy axis shown in Fig. 2(b). This result sug-
gests that the damping is already saturated in this case5
-0.4 -0.2 0 0.2 0.4
E-EF (eV)100200300400500λ⊥/λ|| (%)50 K
77 K
100 KSingle layer
FIG. 5. The calculated rotational damping anisotropy for
single-layer Fe 3GeTe 2as a function of the Fermi energy at
di erent temperatures.
at a suciently large scattering rate, where saturated
damping has also been found in FM Ni.25The calculated
?of bulk FGT is also larger than kat low tempera-
tures and slightly decreases with increasing temperature.
We summarize the results for the rotationally anisotropic
damping frequency by plotting the ratio between ?and
kin the inset of Fig. 4(b). The ratio for both SL and
bulk FGT decreases with increasing temperature and
approaches unity near TC. This behavior is consistent
with the results calculated using the torque-correlation
model,33where rotationally anisotropic damping disap-
pears gradually as the scattering rate increases. In highly
disordered systems, the damping is more isotropic, as in-
tuitively expected.
We emphasize that the calculated ?values are dis-
tinct from those reported in previous studies in the
literature,55that is,?was found to vanish in single-
crystal monoatomic FM layers based on the breathing
Fermi surface model.56{58Interband scattering is ne-
glected in the breathing Fermi surface model. However,
the resistivitylike behavior of our calculated kfor bulk
FGT shows that interband scattering plays an important
role in this vdW FM material.
One of the unique advantages of 2D vdW materials
is the tunability of the electronic structure via electri-
cal gating.4,59To simulate such a scenario, we slightly
adjust the Fermi level EFof SL FGT without changing
the band structure for simplicity. The calculated rota-
tional anisotropy in the damping ?=kof SL FGT is
shown as a function of the Fermi energy in Fig. 5. At all
the temperatures considered, the anisotropy ratio ?=k
increases dramatically as EFis lowered by 0.3 eV, es-
pecially at low temperatures, and only exhibits minor
changes when EFis increased. At 50 K, the ratio ?=k
becomes as high as 430%, which is almost three times
larger than that obtained without gating. This result
suggests that a small quantity of holes doped into SLFGT at low temperatures remarkably enhances the rota-
tional damping anisotropy.
V. CONCLUSIONS
We have systematically studied Gilbert damping in a
2D vdW FM material Fe 3GeTe 2by using rst-principles
scattering calculations where the temperature-induced
lattice vibration and spin
uctuation are modeled by
frozen thermal lattice and spin disorder. When the mag-
netization is perpendicular to the 2D atomic plane, the
damping frequency of bulk FGT increases linearly with
the temperature, whereas that of SL FGT exhibits a
weak temperature dependence. The di erence can be
attributed to surface scattering (which is absent in the
bulk) dominating scattering due to temperature-induced
disorder in SLs, which have a thickness smaller than the
electronic mean free path. The anisotropy of Gilbert
damping in this 2D vdW material has also been thor-
oughly investigated. The orientational anisotropy, which
depends on the direction of the equilibrium magnetiza-
tion with respect to the atomic planes, exhibits twofold
rotational symmetry in both the bulk and SL. When
the equilibrium magnetization is parallel to the atomic
plane, the damping is signi cantly enhanced compared to
that with the magnetization perpendicular to the atomic
plane. The rotational anisotropic damping depending on
the direction of motion of the instantaneous magnetiza-
tion is remarkable with the equilibrium magnetization ly-
ing inside the atomic plane. With an out-of-plane compo-
nent in the timederivative of the precessional magnetiza-
tion, the damping frequency ( ?) is much larger than the
one where only in-plane magnetization is varying ( k).
The ratio?=kis larger than unity for both the bulk
and a single layer and decreases with increasing temper-
ature. In SL FGT, ?=kcan be enhanced up to 430%
by slight holedoping at 50 K.
Antiferromagnetic order has recently been discovered
in 2D vdW materials (as reviewed in Ref. 60 and the ref-
erences therein) and some intriguing properties are found
in their damping behaviors.61,62Owing to the more com-
plex magnetic order, more than a single parameter is
necessary in describing the damping in antiferromagnetic
dynamics.63,64It would be very interesting to study the
magnetization relaxation in these 2D materials with more
complex magnetic order.
ACKNOWLEDGMENTS
The authors are grateful to Professor Xiangang Wan
at Nanjing University for his support and helpful dis-
cussions. Financial support for this study was provided
by the National Natural Science Foundation of China
(Grants No. 11734004 and No. 12174028).6
yiliu@bnu.edu.cn
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1802.02415v1.Breaking_the_current_density_threshold_in_spin_orbit_torque_magnetic_random_access_memory.pdf | arXiv:1802.02415v1 [cond-mat.mes-hall] 7 Feb 2018Breaking the current density threshold in spin-orbit-torq ue magnetic random access
memory
Yin Zhang,1,2H. Y. Yuan,3,∗X. S. Wang,4,1and X. R. Wang1,2,†
1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
2HKUST Shenzhen Research Institute, Shenzhen 518057, China
3Department of Physics, Southern University of Science and T echnology of China, Shenzhen 518055, China
4School of Microelectronics and Solid-State Electronics,
University of Electronic Science and Technology of China, C hengdu, Sichuan 610054, China
(Dated: March 25, 2022)
Spin-orbit-torque magnetic random access memory (SOT-MRA M) is a promising technology
for the next generation of data storage devices. The main bot tleneck of this technology is the
high reversal current density threshold. This outstanding problem of SOT-MRAM is now solved
by using a current density of constant magnitude and varying flow direction that reduces the
reversal current density threshold by a factor of more than t he Gilbert damping coefficient. The
Euler-Lagrange equation for the fastest magnetization rev ersal path and the optimal current
pulse are derived for an arbitrary magnetic cell. The theore tical limit of minimal reversal current
density and current density for a GHz switching rate of the ne w reversal strategy for CoFeB/Ta
SOT-MRAMs are respectively of the order of 105A/cm2and 106A/cm2far below 107A/cm2and
108A/cm2in the conventional strategy. Furthermore, no external mag netic field is needed for a
deterministic reversal in the new strategy.
Subject Areas: Magnetism, Nanophysics, Spintronics
I. INTRODUCTION
Fast and efficient magnetization reversal is of not only
fundamentally interesting, but also technologically im-
portant for high density data storage and massive in-
formation processing. Magnetization reversal can be in-
duced by magnetic field [1–3], electric current through
direct [4–9] and/or indirect [10–22] spin angular mo-
mentum transfer from polarized itinerant electrons to
magnetization, microwaves [23], laser light [24], and
even electric fields [25]. While the magnetic field in-
duced magnetization reversal is a matured technology,
it suffers from scalability and field localization problems
[8, 26] for nanoscale devices. Spin transfer torque mag-
netic random-access memory is an attractive technol-
ogy in spintronics [26] although Joule heating, device
durability and reliability are challenging issues [11, 26].
In an spin-orbit-torque magnetic random access mem-
ory (SOT-MRAM) whose central component is a heavy-
metal/ferromagnet bilayer, an electric current in the
heavy-metal layer generates a pure spin current through
the spin-Hall effect [10, 11] that flows perpendicularly
into the magnetic layer. The spin current, in turn, pro-
duces spin-orbit torques (SOT) through spin angular
momentum transfer [4, 5] and/or Rashba effect [16–22].
SOT-MRAM is a promising technology because writing
charge current does not pass through the memory cells
so that the cells do not suffer from the Joule heating
and associated device damaging. In principle, such de-
vices are infinitely durable due to negligible heating from
∗[Corresponding author:]yuanhy@sustc.edu.cn
†[Corresponding author:]phxwan@ust.hkspin current [11]. However, the reversal current density
threshold (above 107A/cm2[14, 15] for realistic materi-
als) in the present SOT-MRAM architecture is too high.
To have a reasonable switching rate (order of GHz), the
current density should be much larger than 108A/cm2
[14, 15] that is too high for devices. In order to lower the
minimalreversalcurrentdensityaswellastoswitchmag-
netization states at GHz rate at a tolerable current den-
sity in SOT-MRAM, it is interesting to find new reversal
schemes (strategies) that can achieveabove goals. In this
paper, we show that a proper current density pulse of
time-dependent flow direction and constant magnitude,
much lower than the conventional threshold, can switch
a SOT-MRAM at GHz rate. Such a time-dependent cur-
rent pulse can be realized by using two perpendicular
currentspassingthrough the heavy-metallayer. The the-
oretical limit of minimal reversal current density of the
new reversal strategy for realistic materials can be of the
order of 105A/cm2, far below 107A/cm2in the con-
ventional strategy that uses a direct current (DC), both
based on macrospin approximation. The validity of the
macrospin model is also verified by micromagnetic simu-
lations.
II. MACROSPIN MODEL AND RESULTS
A. Model
Our new reversal strategy for an SOT-MRAM, whose
central component is a ferromagnetic/heavy-metal bi-
layer lying in the xy-plane with initial spin along the
+z-direction as shown in Fig. 1, uses a current den-
sityJ=JcosΦˆx+JsinΦˆygenerated from two time-2
m
FM
HM JxJx
JyJy
FIG. 1. Schematic illustration of new reversal scheme for
SOT-MRAMs. Two perpendicular currents flowin the heavy-
metal layer of a ferromagnet/heavy-metal bilayer to genera te
a current whose direction can vary in the xy-plane.
dependent electric currents flowing along the x- and the
y-directions, where Φ is a time-dependent angle between
Jand the x-axis and Jis a constant total current den-
sity. The magnetic energy density is ε=−Kcos2θwith
Kbeing the anisotropy coefficient and θbeing the polar
angle of the magnetization. In the absence of an electric
current, the system has two stable states m= +ˆzand
m=−ˆzwheremis the unit direction of magnetization
M=Mmof magnitude M. The electric current gen-
erates a transverse spin current perpendicularly flowing
into the ferromagnetic layer via the spin-Hall effect [10],
andthenproducesaneffectiveSOTonthemagnetization[4, 5, 16], i.e.
/vector τ=−am×(m׈s)+βam׈s, (1)
where the first term on the right-hand-side is the
Slonczewski-liketorquewhile thesecondtermis thefield-
like torque. The spin-polarization direction is ˆ s=ˆJ׈z
(for other type of spin-Hall effect, see Note [27]) with ˆJ
being the unit vector of current density. a=¯h
2edθSHJ
measures SOT where ¯ h,e, anddare respectively the
Plank constant, the electron charge, and the sample
thickness. θSHis the spin Hall angle which measures
the conversion efficiency between the spin current and
charge current. βmeasures the field-like torque and can
be an arbitrary real number since this torque may also
be directly generated from the Rashba effect [16].
Themagnetizationdynamicsunderanin-planecurrent
densityJis governed by the generalized dimensionless
Landau-Lifshitz-Gilbert (LLG) equation,
∂m
∂t=−m×heff+αm×∂m
∂t+/vector τ, (2)
whereαis the Gilbert damping constant that is typically
much smaller than unity. The effective field is heff=
−∇mεfrom energy density ε. Time, magnetic field and
energy density are respectively in units of ( γM)−1,M
andµ0M2, where γandµ0are respectively the gyro-
magnetic ratio and vacuum magnetic permeability. In
this unit system, a=¯h
2edµ0M2θSHJbecomes dimension-
less.
The magnetization mcan be conveniently described
by a polar angle θand an azimuthal angle φin thexyz-
coordinate. In terms of θandφ, the generalized LLG
equation becomes
(1+α2)˙θ=−αKsin2θ+a(1−αβ)cosθsin(Φ−φ)+a(α+β)cos(Φ−φ)≡F1, (3a)
(1+α2)˙φsinθ=Ksin2θ−a(1−αβ)cos(Φ−φ)+a(α+β)cosθsin(Φ−φ)≡F2. (3b)
B. Derivation of the Euler-Lagrange equation
The goal is to reverse the initial state θ= 0 to the
target state θ=πby SOT. There are an infinite number
of paths that connect the initial state θ= 0 with the
target state θ=π, and each of these paths can be used
as a magnetization reversal route. For a given reversal
route, there are an infinite number of current pulses that
can reverse the magnetization. The theoretical limit of
minimal current density Jcis defined as the smallest val-
ues of minimal reversal current densities of all possible
reversal routes. Then it comes two interesting and im-
portant questions: 1) What is Jcabove which there is at
least one reversal route that the current density can re-
verse the magnetization along it? 2) For a given J > Jc,what are the optimal reversal route and the optimal cur-
rent pulse Φ( t) that can reverse the magnetization at the
highest speed?
Dividing Eq. (3b) by Eq. (3a), one can obtain the
following constraint,
G≡∂φ
∂θsinθF1−F2= 0. (4)
The magnetization reversal time Tis
T=/integraldisplayπ
0dθ
˙θ=/integraldisplayπ
01+α2
F1dθ. (5)
The optimization problem here is to find the optimal
reversal route φ(θ) and the optimal current pulse Φ( t)3
such that Tis minimum under constraint (4). Using the
Lagrange multiplier method, the optimal reversal route
and the optimal current pulse satisfy the Euler-Lagrange
equations [28, 29],
∂F
∂φ=d
dθ(∂F
∂(∂φ/∂θ)),∂F
∂Φ=d
dθ(∂F
∂(∂Φ/∂θ)),(6)
whereF= (1 +α2)/F1+λGandλis the Lagrange
multipliers which can be determined self-consistently by
Eq. (6) and constrain (4). Given a current density of
constant magnitude J, Eq. (6) may or may not have a
solution of φ(θ) that continuously passing through θ=
0 andθ=π. If such a solution exists, then φ(θ) is
the optimal path for the fastest magnetization reversal
and the corresponding solution of Φ( t) is the optimal
current pulse. The theoretical limit of minimal reversal
current density is then the smallest current density Jc
below which the optimal reversal path does not exist.
C. The optimal current pulse and theoretical limit
of minimal reversal current density
From Eqs. (3a), (3b) and (4) as well as F= (1 +
α2)/F1+λG, theEuler-Lagrangeequationof (6)becomes
λd
dθ(F1sinθ) = 0, (7a)
1+α2
F2
1∂F1
∂φ−λ∂G
∂φ=−1+α2
F2
1∂F1
∂Φ+λ∂G
∂Φ= 0.(7b)
From Eq. (7a), one has λ/ne}ationslash= 0 orλ= 0. Ifλ/ne}ationslash= 0,F1must
beF1=C/sinθ(C/ne}ationslash= 0) so that (1+ α2)˙θ=C/sinθ→
∞asθ→0 orπ. This solution is not physical, and
shouldbe discarded. Therefore, the only allowedsolution
must be λ= 0, and one has ∂F1/∂Φ = 0 according to
Eq. (7b). Interestingly, this is exactly the condition of
maximal ˙θ=F1/(1+α2) as Φ varies. Φ satisfies tan(Φ −
φ) =1−αβ
α+βcosθ, or
Φ = tan−1(1−αβ
α+βcosθ)+φ+π(β <−α) (8a)
Φ = tan−1(1−αβ
α+βcosθ)+φ (β >−α).(8b)
Substituting Eq. (8) into the LLG equation (3), θ(t)
andφ(t) are determined by the following equations,
˙θ=1
1+α2[aP(θ)−αKsin2θ], (9a)
˙φ=1
1+α2[2Kcosθ−a(α+β)(1−αβ)sinθ
P(θ)],(9b)
whereP(θ) =/radicalbig
(α+β)2+(1−αβ)2cos2θ. To reverse
magnetization from θ= 0 toθ=π,amust satisfy a >
αKsin(2θ)/P(θ) according to Eq. (9a) so that ˙θis no
negative for all θ. Obviously, ˙θ= 0 atθ=π/2 whenFIG. 2. The log α-dependence of Jcfor various βare plotted
as the solid curves for model parameters of M= 3.7×105
A/m,K= 5.0×103J/m3,θSH= 0.084 and d= 0.6 nm. As
a comparison, Jdc
cis also plotted as the dashed lines.
β=−α. The magnetization reversal is not possible in
this case, and β=−αisasingularpoint. The theoretical
limit of minimal reversal current density Jcforβ/ne}ationslash=−α
is
Jc=2αeKd
θSH¯hQ, (10)
whereQ≡max{sin2θ/P(θ)}forθ∈[0,π].
In comparison with the current density threshold [13,
14, 18] (Jdc
c) in the conventional strategy for β= 0,
Jdc
c=2eKd
θSH¯h(1−H√
2K), (11)
the minimal reversal current density is reduced by more
than a factor of α. HereH(≃22 Oe in experiments)
is a small external magnetic needed for a deterministic
reversal in conventional strategy. Using CoFeB/Ta pa-
rameters of M= 3.7×105A/m,K= 5.0×103J/m3,
θSH= 0.084 and d= 0.6 nm [11, 14, 15], Fig. 2 shows
logα-dependenceof Jc(solidlines)and Jdc
c(dashedlines)
forβ= 0 (black), 0 .3 (red) and −0.3 (blue), respectively.
BothJdc
candJcdepend on β. The lower the damping of
a magnetic material is, the smaller our minimum switch-
ing current density will be. For a magnetic material of
α= 10−5, the theoretical limit of minimal reversal cur-
rent density can be five order of magnitude smaller than
the value in the conventional strategy.
For a given J > Jc, the shortest reversal time is given
by Eqs. (5) and (9a):
T=/integraldisplayπ
01+α2
aP(θ)−αKsin2θdθ. (12)
The optimal reversal path is given by φ(θ) =/integraltextθ
0˙φ
˙θdθ′
where˙θand˙φare given by Eqs. (9a) and (9b). Eq.
(9a) gives t(θ) =/integraltextθ
0(1+α2)/(aP(θ)−αKsin2θ)dθ′and
thenθ(t) is just θ(t) =t−1(θ). Thus, Φ( θ,φ),φ(θ)
andθ(t) giveφ(t) =φ(θ(t)) and Φ( t) = Φ(θ(t),φ(t)).4
(d) (e) mz
mxmymz
my
mx(a) (b) (c)
(f)
mymz
mx
FIG. 3. Model parameters of M= 3.7×105A/m,K= 5.0×103J/m3,θSH= 0.084,α= 0.008 and d= 0.6 nm are used to
mimic CoFeB/Ta bilayer, and β= 0.3 for (a), (c), (d) and (f) while β= 0.1 for (b) and (e). The theoretical limit of minimum
reversal current density is Jc= 1.56×105A/cm2forβ= 0.1 andJc= 1.28×105A/cm2forβ= 0.3. Optimal current pulses
((a)-(c)) and fastest reversal routes ((d)-(f)) are for J= 1.92×106A/cm2((a) and (d)), and for J= 9.0×106A/cm2((b),
(c)), (e) and (f).
Using the same parameters as those for Fig. 2 with
α= 0.008 and various β, Fig. 3 shows the optimal
current pulses ((a)-(c)) and the corresponding fastest
magnetization reversal routes ((d)-(f)) for β= 0.3 and
J= 1.92×106A/cm2≈15Jc((a) and (d)), for β= 0.1
andJ= 9.0×106A/cm2≈58Jc((b) and (e)), and for
β= 0.3 andJ= 9.0×106A/cm2≈70Jc((c) and (f)). It
isknownthatTahaslesseffecton α[11]. Theminimalre-
versal current density Jcunder the optimal current pulse
is 1.56×105A/cm2forβ= 0.1 and 1.28×105A/cm2
forβ= 0.3 which is far below Jdc
c= 9.6×106A/cm2for
the same material parameters [15]. The multiple oscilla-
tions ofmxandmyreveal that the reversal is a spinning
process and optimal reversal path winds around the two
stable states many times. Correspondingly, the driving
current makes also many turns as shown by the multiple
oscillations of JxandJy. The number of spinning turns
dependsonhowfar JisfromJc. Thecloser JtoJcis, the
number of turns is larger. The number of turns is about
5 in Figs. 3(a) and 3(d) for J≈15Jcand one turn for
J >50Jcas shown in Figs. 3(b), 3(c), 3(e) and 3(f), so
that the reversal is almost ballistic. The reversaltime for
β= 0.3 andJ= 1.92×106A/cm2is about 10 nanosec-
onds, for β= 0.1 andJ= 9.0×106A/cm2is about 3.3
nanoseconds, and for β= 0.3 andJ= 9.0×106A/cm2is
about 2.1 nanoseconds. Figure 4 is the reversaltime Tas
a function of current density Junder the optimal current
pulse for the same parameters as those for Fig. 2. TheFIG. 4. Magnetization reversal time Tunder the optimal
current pulses as a function of Jfor various αandβ.
reversaltime quickly decreases to nanoseconds as current
density increases. In a real experiment, there are many
uncertainties so that the current pulse may be different
from the optimal one. To check whether our strategy is
robust again small fluctuations, we let the current pulse
in Fig. 3(c) deviate from its exact value. Numerical
simulations show that the magnetization reversal is not
significantly influenced at least when the deviation be-
tween the real current and optimal current is less than
five percents.5
III. VERIFICATION OF MACROSPIN MODEL
BY MICROMAGNETIC SIMULATION
In our analysis, the memory cell is treated as a
macrospin. A nature question is how good the macrospin
model is for a realistic memory device. To answer this
question, we carried out micromagnetic simulations by
using Newton-Raphson algorithm [30] for two memory
cells of 150 nm ×150nm×0.6 nm (Figs. 5(a), (b), (d) and
(e)) and 250 nm ×250 nm×0.6 nm (Figs. 5(c) and (f)).
To model the possible edge pinning effect due to mag-
netic dipole-dipole interaction, we consider square-shape
devices instead of cylinder shape device whose edge pin-
ning is negligible. To make a quantitative comparison,
the material parameters are the same as those used in
Fig. 3. In our simulations, the unit cell size is 2 nm ×2
nm×0.6 nm. For a fair comparison, the optimal current
pulses shown in Figs. 3(a) and (c) of respective current
densityJ= 1.92×106A/cm2andJ= 9.0×106A/cm2
were applied to the memory cell of 150 nm ×150 nm×0.6
nm. The symbols in Figs. 5(a) and (b) are the time evo-
lution of averaged magnetization mx,myandmzwhile
thesolidlinesarethetheoreticalpredictionsofmacrospin
model shown in Figs. 3(d) and (f). The perfect agree-
ments prove the validity of the macrospin approximation
for our device of such a size. To further verify that the
memory device can be treated as a macrospin, Figs. 5(d)
and (e) are the spin configurations in the middle of the
reversal at t= 5.5 ns for Fig. 5(a) and at t= 1.2 ns for
Fig. 5(b). Thefactthatallspinsalignalmostinthesame
direction verifies the validity of the macro spin model. In
real experiments, non-uniformity of current density is in-
evitable. To demonstrate the macrospin model is still
valid, we let current density linearly varies from 9 .5×106
A/cm2on the leftmost column of cells to 8 .5×106A/cm2
onthe rightmostcolumn ofcells. As expected, thereis no
noticeable difference with the data shown in Figs. 5(b)
and (e).
For the large memory device of 250 nm ×250 nm×0.6
nm, the optimal current pulse shown in Fig. 3(c) of cur-
rent density J= 9.0×106A/cm2was considered. The
time evolution of averaged magnetization mx,myand
mzare plotted in Fig. 5(c), with the symbols for simula-
tions andsolid linesfor the macrospinmodel. They agree
very well although there is a small deviation for device
of such a large size. Figure 5(f) is the spin configurations
in the middle of the reversal at t= 1.2 ns for Fig. 5(c).
The marcospin model is not too bad although all spins
are not perfectly aligned in this case.
In summary, for a normal SOT-MRAM device of size
less than 300 nm [11, 15], macrospin model describes
magnetization reversal well. However, for a larger sam-
ple size and lower current density ( J <106A/cm2for
the same material parameters as those used in Fig. 3),
only the spins in sample center can be reversed while the
spins near sample edges are pinned.(a) (b) (c)
(d) (e) (f) t = 5.5 ns t = 1.2 ns t = 1.2 ns
FIG. 5. (a)-(c) Time evolution of the average magnetization :
cycles for micromagnetic simulations and solid lines are th e-
oretical predictions from macrospin model. (a) and (b) are
for the memory cell of 150 nm ×150 nm×0.6 nm and optimal
current pulse of current density of J= 1.92×106A/cm2and
J= 9.0×106A/cm2, respectively. (c) is for the memory
cell of 250 nm ×250 nm×0.6 nm and optimal current pulse of
current density of J= 9.0×106A/cm2. (d)-(f) Spin configu-
rations respectively corresponding to (a)-(c) in the middl e of
magnetization reversal at t= 5.5 ns and 1.2 ns. The cell size
in micromagnetic simulation is 2 nm ×2 nm×0.6 nm.
IV. DISCUSSION
Obviously, the strategy present here can easily be gen-
eralized to the existing spin-transfer torque MRAM. The
mathematics involved are very similar, and one expects
a substantial current density reduction is possible there
if a proper optimal current pulse is used. Of course, how
to generate such a current pulse should be much more
challenge than that for SOT-MRAM where two perpen-
dicular currents can be used. In the conventional strat-
egy that uses a DC-current, a static magnetic field along
current flow is required for a deterministic magnetiza-
tion reversal [13, 14, 18]. Although several field-free de-
signs have been proposed [19, 20], an antiferromagnet is
needed to create an exchange bias which plays the role
of an applied magnetic field. As we have shown, such a
requirement or complication is not needed in our strat-
egy. Ourstrategydoesnothaveanotherproblemexisting
in the conventional strategy in which the magnetization
can only be placed near θ=π/2 [13, 14, 18] so that
the system falls into the target state by itself through
the damping. Therefore, one would like to use materials
with largerdamping in the conventionalstrategyin order
to speed up this falling process. In contrast, our strategy
preferslowdampingmaterials,andreversalisalmostbal-
listic when current density is large enough ( >50Jcin the
current case). To reverse the magnetization from θ=π
toθ= 0, one only needs to reverse the current direction
of the optimal current pulse. One should notice that
the Euler-Lagrange equation allows us to easily obtain
the optimal reversal current pulse and theoretical limit6
of the minimal reversal current density for an arbitrary
magnetic cell such as in-plane magnetized layer [11] and
biaxial anisotropy.
V. CONCLUSION
In conclusion, weinvestigatedthe magnetizationrever-
sal of SOT-MRAMs, and propose a new reversal strat-
egy whose minimal reversal current density is far be-
low the existing current density threshold. For popular
CoFeB/Ta system, it is possible to use a current densityless than 106A/cm2to reversethe magnetizationat GHz
rate, in comparison with order of J≈108A/cm2in the
conventional strategy.
ACKNOWLEDGMENTS
This work was supported by the National Natural Sci-
ence Foundation of China (Grant No. 11774296 and
No. 61704071) as well as Hong Kong RGC Grants No.
16300117 and No. 16301816. X.R.W. acknowledges the
hospitalities of Beijing Normal University and Beijing
ComputationalScience ResearchCenterduringhisvisits.
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1010.0268v2.Ferromagnetic_resonance_study_of_Co_Pd_Co_Ni_multilayers_with_perpendicular_anisotropy_irradiated_with_Helium_ions.pdf | Ferromagnetic resonance study of Co/Pd/Co/Ni multilayers with perpendicular
anisotropy irradiated with Helium ions
J-M. L. Beaujour, A. D. Kent,1D. Ravelosona,2and I. Tudosa and E. E. Fullerton3
1Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA
2Institut d'Electronique Fondamentale, UMR CNRS 8622,
Universite Paris Sud, 91405 Orsay Cedex, France
3University of California, San Diego, Center for Magnetic Recording Research, La Jolla, CA 92093-0401, USA
(Dated: April 24, 2022)
We present a ferromagnetic resonance (FMR) study of the e ect of Helium ion irradiation on the
magnetic anisotropy, the linewidth and the Gilbert damping of a Co/Ni multilayer coupled to Co/Pd
bilayers. The perpendicular magnetic anisotropy decreases linearly with He ion
uence, leading to
a transition to in-plane magnetization at a critical
uence of 5 1014ions/cm2. We nd that the
damping is nearly independent of
uence but the FMR linewidth at xed frequency has a maximum
near the critical
uence, indicating that the inhomogeneous broadening of the FMR line is a non-
monotonic function of the He ion
uence. Based on an analysis of the angular dependence of the
FMR linewidth, the inhomogeneous broadening is associated with spatial variations in the magnitude
of the perpendicular magnetic anisotropy. These results demonstrate that ion irradiation may be
used to systematically modify the magnetic anisotropy and distribution of magnetic anisotropy
parameters of Co/Pd/Co/Ni multilayers for applications and basic physics studies.
PACS numbers:
Co/Ni multilayers are of great interest in informa-
tion technology and in spin-transfer devices because they
combine high spin polarization with large perpendicu-
lar magnetic anisotropy (PMA) [1{3]. Perpendicular
anisotropy was predicted in Co/Ni multilayers and has
been shown experimentally to be a function of layer com-
position and thin lm growth conditions [4]. Recently
it has been shown that the coercivity and perpendicu-
lar anisotropy of a multilayer can be tailored by Helium
ion irradiation [5], making it possible to modify lms af-
ter growth to tune their magnetic properties. This is
of great interest for applications and basic physics stud-
ies, as in many cases the perpendicular anisotropy of a
structure sets importance device metrics, and ion irradi-
ation o ers the possibility of changing these properties,
both locally and globally, after device fabrication. For
instance, in spin-transfer magnetic random access mem-
ories (STT-MRAM) the current threshold for switching
is proportional to the PMA [2, 6].
Light ion irradiation has been used to vary the mag-
netic properties of multilayer lms in many earlier studies
[7{11]. For instance, the coercivity of Co/Pt multilayers
was found to decrease with ion dose [8]. This behavior
was attributed to interface mixing and strain relaxation
reducing the PMA. Very recently, it was reported that
the coercive eld of Co/Ni multilayers decreases linearly
with increasing He+irradiation
uence up to F= 1015
ions/cm2, suggesting changes in the magnetic anisotropy
of the lm [10]. The e ect of ion irradiation on the FMR
linewidth has also been studied in Au/Fe multilayer lms
with PMA [11]. The PMA is reduced by He+irradi-
ation and the authors explained this by a reduction of
the inhomogeneous contribution to the FMR linewidth.
In a recent paper, we presented a FMR study of the
anisotropy and the linewidth of a Co/Ni multilayer lmexposed to a relatively high He+irradiation
uence ( F
= 1015ions/cm2) [12]. In addition to a strong decrease
of the PMA, the contribution to the linewidth from spa-
tial variation of the anisotropy, was reduced compared
to that of a non irradiated Co/Ni multilayer. Further-
more, a correlation between the anisotropy distribution
and the linewidth broadening from two-magnon scatter-
ing (TMS) mechanism was observed. However, a system-
atic study of the e ect of the He+irradiation on the FMR
spectra as a function of
uence has yet to be reported.
In this paper, we present a FMR study of a Co/Ni
multilayer coupled to Co/Pd bilayers exposed to Helium
ion irradiation of
uence up to 1015ions/cm2. The PMA
and the contributions to the FMR linewidth, including
those from Gilbert damping ( ), are studied as a function
of
uence.
The samples had the following layer structure: jj3 Taj1
Pdj0.3 Coj1 Pdj0.14 Coj[0.8 Nij0.14 Co]3j1 Pdj0.3 Coj1
Pdj3 Tajj(layer thickness in nm) and was fabricated by dc
magnetron sputtering. The Co/Ni multilayer is embed-
ded between Co/Pd bilayers to enhance the overall PMA
of the lm and to have resonance frequencies in which the
full angular dependence of the FMR response could be
investigated in a 1 T electromagnet. The substrate was
cleaved into several pieces that were then exposed to dif-
ferent doses of Helium ion irradiation of energy 20 keV
with
uence in the range 1014F1015ions/cm2.
FMR measurements were conducted at room tempera-
ture using a coplanar waveguide (CPW). Details of the
experimental setup can be found in [13]. The eld swept
CPW transmission signal ( S21) was recorded as a func-
tion of frequency for dc magnetic elds normal to the lm
plane and as a function of the out-of-plane eld angle at
20 GHz. The magnetization density of the lm at room
temperature was measured with a SQUID magnetome-arXiv:1010.0268v2 [cond-mat.mes-hall] 5 Oct 20102
03 06 09 00.40.60.8K
K0
5 1 001020H
/s615490Hres (T) 0
1
2.5
F
ield angle /s61542H (deg)20 GHz 5
7.5
10HH
-0.40.00.40.80
102030f (GHz)(a)
(b) /s615490Hres (T)
/s61549 0Meff (
c)
F (1014 ions/cm2)K (105 J/m3)21
FIG. 1: a) Angular dependence of the resonance eld for dif-
ferent irradiation
uence ( 1014ions/cm2). The solid lines
are guides to the eye. b) Frequency dependence of the reso-
nance eld when the applied eld is normal to the lm plane
( H= 90o) at selected
uences. The solid lines are the ts
to Eq. 1. The zero frequency intercept gives the e ective de-
magnetization eld, 0Me . c) The second and fourth order
perpendicular anisotropy constants, K1andK2, versus
u-
ence.
ter:Ms'4:75105A/m. Within the measurement
uncertainty, Msremains unchanged after irradiation.
Fig. 1a shows the out of plane angular dependence
of the resonance eld at 20 GHz for di erent
uences.
For the non-irradiated lm, the resonance eld when H
is normal to the lm plane ( H= 90o) is smaller than
that when the eld is in the lm plane ( H= 0o). This
shows that the magnetic easy axis is normal to the lm
plane. As the
uence increases, Hresat H= 90oin-
creases whereas that at H= 0odecreases. In the high
uence range, F>51014ions/cm2,Hresis the larger
when the eld is normal to the lm plane, i.e. the mag-
netic easy axis is in the lm plane. Fig. 1b shows the
frequency dependence of the resonance eld for di erent
uences when the dc eld is normal to the lm plane.
This data is tted to the resonance condition [14]:
f=1
2
0(Hres Me ); (1)
where
is the gyromagnetic ratio. 0Me , the e ec-
tive easy plane anisotropy, is given by: Me =Ms
2K1=(0Ms), whereK1is the second order anisotropy
constant. We nd that 0Me is negative at low
u-
ence which implies that the PMA is sucient to over-
come the demagnetizing energy and hence the easy axis
is normal to the lm plane. As the
uence is further
increased,0Me becomes positive. These results con-
rm that there is a re-orientation of the easy axis, as was
inferred indirectly through magnetic hysteresis loop mea-
surements in Ref. [10]. 0Me changes sign for
uence
between 5 and 7.5 1014ions/cm2. Therefore, by expos-
ing the lm to a speci c
uence, it is posible to engineer
01000
1000
3 06 09 001000
102030020406080
F=0Δ
Hα ΔHinh ΔHtot
F=5/s615490ΔH (mT)f
/s61472 ( GHz ) F=10F
ield angle /s61542H ( deg )HH /s615490ΔH (mT)
F=7.5H
FIG. 2: On the left, the linewidth as a function of frequency
for the lm irradiated at 7.5 1014ions/cm2. The solid line
is a linear t to the experimental data. On the right, the
angular dependence of the linewidth at 20 GHz for a selec-
tion of
uences. The solid lines represent the ts to the total
linewidth Htot= H +Hinh, , where the intrinsic damp-
ing and the inhomogeneous contribution are represented by
the dashed line and the dotted line respectively.
the anisotropy so that the PMA eld just compensates
the demagnetization eld.
The second order perpendicular anisotropy constant
K1decreases linearly with
uence (Fig. 1c). The lm
irradiated at 1015ions/cm2has an anisotropy constant
40% smaller than that of the non-irradiated lm. The
4thorder anisotropy constant K2is determined from
the angular dependence of Hresfor magnetization angles
45 90o[13].K2is smaller than K1by a factor 10,
and is nearly independent of
uence.
The FMR linewidth 0Hwhen the dc eld is ap-
plied normal the lm plane was measured as a function
of frequency. Fig. 2 shows 0Hversusffor the lm
irradiated at F= 7:51014ions/cm2. The linewidth in-
creases linearly with frequency, characteristic of Gilbert
damping, an intrinsic contribution to the linewidth H
[15]:
H =4
0
f: (2)
From a linear t to the experimental data, the magnetic
damping constant is estimated from the slope of the line:
= 0:0370:004. The lms irradiated at F= 0;1 and
101014ions/cm2shows a similar frequency dependence
of the linewidth and have about the same damping con-
stant, 0:04. At intermediate
uence F= 2:5, 51014
ions/cm2, the linewidth is enhanced and is frequency in-
dependent, i.e. the linewidth is dominated by an inhomo-
geneous contribution, Hinh. The angular dependence
of the linewidth measured at 20 GHz is shown in Fig.
2 for lms irradiated at selected
uences. For the non-
irradiated lm and the lm irradiated at 1015ions/cm2,
the linewidth is practically independent of the eld an-3
05 1 01 53060901200
5100.030.04µ0 Δ/s61512/s61472
⊥
(mT)F
(1014 ions/cm2)ΔK1 ( 105 J/m3 )H
0.00.10.20.30.4
α-dampingF
(1014 ions/cm2)
FIG. 3: The
uence dependence of the linewidth at 20 GHz
when the dc eld is normal to the lm plane (squares). The
solid circles represent the
uence dependence of the distribu-
tion in the PMA constant K1determined from tting Hvs.
H. The inset shows the Gilbert damping constant as a
function of
uence.
gle from about 30oup to 90o. For the lm irradiated at
51014ions/cm2, His clearly angular dependent and
shows a minimum at an intermediate eld angle.
The angular dependence of the linewidth was t to
a sum of the intrinsic linewidth H and an inhomo-
geneous contribution Hinhfor magnetization angles
45o 90o, an angular range in which TMS does
not contribute to the linewidth [13]. The inhomogeneous
linewidth is given by:
Hinh:( H) =j@Hres=@K 1jK1+j@Hres=@ j ;(3)
where K1is the width of the distribution of anisotropies
and is the distribution of the angles of the magnetic
easy axis relative to the lm normal. The computed
linewidth contributions are shown for the lm irradiated
atF= 51014ions/cm2in Fig. 2. Note that the intrin-
sic contribution H is practically independent of eld
angles, as expected when the angle between the magne-
tization and the applied eld is small. For this sample,
the maximum angle is about 5oand it is due to the fact
that the resonance eld ( Hres'0:6 T) is much larger
than the e ective demagnetization eld ( Me '0). Theinhomogeneous contribution from the distribution in the
anisotropy eld directions does not signi cantly a ect the
t. For the lm irradiated at the lower and upper
uence
range, the angular dependence of the intrinsic linewidth
is computed xing the value of to that obtained from
the t of the frequency dependence of the linewidth. For
the other lms ( F=2.5 and 51014ions/cm2), was
a tting parameter.
The
uence dependence of K1and the linewidth at
20 GHz are shown in Fig. 3. The inset shows the Gilbert
damping constant as a function of
uence. The linewidth
at 20 GHz when the eld is normal to the lm plane is
a non monotonic function of
uence. Hincreases as
the
uence increases, reaching a maximum value at F
51014ions/cm2. Then, as the
uence is further in-
creased, Hdecreases and falls slightly below the range
of values at the lower
uence range. Interestingly, the
larger linewidth is observed just at the
uence for which
0Me = 0. The magnetic damping is practically not
a ected by irradiation within the error bars: 0:04.
The distribution of PMA constants, K1, shows a similar
uence dependence as the total linewidth, with a maxi-
mum at F51014ions/cm2, clearly indicating that
this is at the origin of the
uence dependence of the mea-
sured linewidth. The distribution in PMA anisotropy is
almost zero when the
uence is above 7 1014ions/cm2.
The largest value of K1corresponds to variation of K1
ofabout 8%, which is much larger than that of non irradi-
ated lm and the highly irradiated lm, K1=K12%
and 0.3% respectively.
In summary, irradiation of Co/Pd/Co/Ni lms with
Helium ions leads to clear changes in its magnetic char-
acteristics, a signi cant decrease in magnetic anisotropy
and a change in the distribution of magnetic anisotropies.
Importantly, this is achieved without a ecting the lm
magnetization density and magnetic damping, which re-
main virtually unchanged. It would be of interest to have
a better understanding of the origin of the maximum in
the distribution of magnetic anisotropy at the critical
uence, the
uence needed to produce a reorientation of
the magnetic easy axis. Nonetheless, these results clearly
demonstrate that ion irradiation may be used to system-
atically tailor the magnetic properties of Co/Pd/Co/Ni
multilayers for applications and basic physics studies.
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1709.03775v1.Green_s_function_formalism_for_spin_transport_in_metal_insulator_metal_heterostructures.pdf | Green’s function formalism for spin transport in metal-insulator-metal
heterostructures
Jiansen Zheng,1Scott Bender,1Jogundas Armaitis,2Roberto E. Troncoso,3,4and Rembert A. Duine1,5
1Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,
Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands
2Institute of Theoretical Physics and Astronomy,
Vilnius University, Saul˙ etekio Ave. 3, LT-10222 Vilnius, Lithuania
3Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
4Departamento de Física, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile
5Department of Applied Physics, Eindhoven University of Technology,
PO Box 513, 5600 MB Eindhoven, The Netherlands
(Dated: September 13, 2017)
We develop a Green’s function formalism for spin transport through heterostructures that contain
metallic leads and insulating ferromagnets. While this formalism in principle allows for the inclusion
of various magnonic interactions, we focus on Gilbert damping. As an application, we consider
ballistic spin transport by exchange magnons in a metal-insulator-metal heterostructure with and
without disorder. For the former case, we show that the interplay between disorder and Gilbert
damping leads to spin current fluctuations. For the case without disorder, we obtain the dependence
of the transmitted spin current on the thickness of the ferromagnet. Moreover, we show that the
results of the Green’s function formalism agree in the clean and continuum limit with those obtained
from the linearized stochastic Landau-Lifshitz-Gilbert equation. The developed Green’s function
formalism is a natural starting point for numerical studies of magnon transport in heterostructures
that contain normal metals and magnetic insulators.
PACS numbers: 05.30.Jp, 03.75.-b, 67.10.Jn, 64.60.Ht
I. INTRODUCTION
Magnons are the bosonic quanta of spin waves, oscil-
lations in the magnetization orientation in magnets1,2.
Interest in magnons has recently revived as enhanced ex-
perimental control has made them attractive as potential
data carriers of spin information over long distances and
withoutOhmicdissipation3. Ingeneral, magnonsexistin
two regimes. One is the dipolar magnon with long wave-
lengths that is dominated by long-range dipolar interac-
tions and which can be generated e.g. by ferromagnetic
resonance4,5. The other type is the exchange magnon6,
dominated by exchange interactions and which generally
has higher frequency and therefore perhaps more poten-
tial for applications in magnon based devices3. In this
paper, we focus on transport of exchange magnons.
Thermally driven magnon transport has been widely
investigated, and is closely related to spin pumping of
spin currents across the interface between insulating fer-
romagnets (FMs) and normal metals (NM)7–9and de-
tection of spin current by the inverse spin Hall Effect10.
The most-often studied thermal effect in this context is
the spin Seebeck effect, which is the generation of a spin
current by a temperature gradient applied to a magnetic
insulator that is detected in an adjacent normal metal
via the inverse spin Hall effect11,12. Here, thermal fluc-
tuations in the NM contacts drive spin transport into the
FM, while the dissipation of spin back into the NM by
magnetic dynamics is facilitated by the above mentioned
spin-pumping mechanism.
The injection of spin into a FM can also be accom-plished electrically, via the interaction of spin polarized
electrons in the NM and the localized magnetic mo-
ments of the FM. Reciprocal to spin-pumping is the spin-
transfer torque, which, in the presence of a spin accu-
mulation (typically generated by the spin Hall effect)
in the NM, drives magnetic dynamics in the FM13,14.
Spin pumping likewise underlies the flow of spin back
into the NM contacts, which serve as magnon reservoirs.
In two-terminal set-ups based on YIG and Pt, the char-
acteristic length scales and device-specific parameter de-
pendence of magnon transport has attracted enormous
attention, both in experiments and theory. Cornelis-
senet al.15studied the excitation and detection of high-
frequency magnons in YIG and measured the propagat-
ing length of magnons, which reaches up to 10m in
a thin YIG film at room temperature. Other experi-
ments have shown that the polarity reversal of detected
spins of thermal magnons in non-local devices of YIG
are strongly dependent on temperature, YIG film thick-
ness, and injector-detector separation distance16. That
the interfaces are crucial can e.g. be seen by changing the
interface electron-magnon coupling, which was found to
significantly alter the longitudinal spin Seebeck effect17.
A linear-response transport theory was developed for
diffusive spin and heat transport by magnons in mag-
netic insulators with metallic contacts. Among other
quantities, this theory is parameterized by relaxation
lengths for the magnon chemical potential and magnon-
phonon energy relaxation18,19. In a different but closely-
related development, Onsager relations for the magnon
spin and heat currents driven by magnetic field andarXiv:1709.03775v1 [cond-mat.mes-hall] 12 Sep 20172
temperature differences were established for insulating
ferromagnet junctions, and a magnon analogue of the
Wiedemann-Franz law was is also predicted20,21. Wang
et al.22consider ballistic transport of magnons through
magnetic insulators with magnonic reservoirs — rather
thanthemoreexperimentallyrelevantsituationofmetal-
lic reservoirs considered here — and use a nonequilib-
rium Green’s function formalism (NEGF) to arrive at
Landuaer-Bütikker-type expressions for the magnon cur-
rent. Theabove-mentionedworksareeitherinthelinear-
response regime or do not consider Gilbert damping
and/or metallic reservoirs. So far, a complete quantum
mechanical framework to study exchange magnon trans-
port through heterostructures containing metallic reser-
voirs that can access different regimes, ranging from bal-
listic to diffusive, large or small Gilbert damping, and/or
small or large interfacial magnon-electron coupling, and
that can incorporate Gilbert damping, is lacking.
Figure 1: Illustration of the system where magnon transport
in a ferromagnet (orange region) is driven by a spin accu-
mulation difference L Rand temperature difference
TL TRbetween two normal-metal leads (blue regions). Spin-
flip scattering at the interface converts electronic to magnonic
spin current. Here, Sis the local spin density in equilibrium.
In this paper we develop the non-equilibrium Green’s
functionformalism23forspintransportthroughNM-FM-
NM heterostructures (see Fig. 1). In principle, this for-
malism straightforwardly allows for adding arbitrary in-
teractions, such as scattering of magnons with impuri-
ties and phonons, Gilbert damping, and magnon-magnon
interactions, and provides a suitable platform to study
magnon spin transport numerically, in particular beyond
linear response. Here, we apply the formalism to ballistic
magnon transport through a low-dimensional channel in
the presence of Gilbert damping. For that case, we com-
pute the magnon spin current as a function of channel
length both numerically and analytically. For the clean
case in the continuum limit we show how to recover our
results from the linearized stochastic Landau-Lifshitz-
Gilbert (LLG) equation24used previously to study ther-
mal magnon transport in the ballistic regime25that ap-
plies to to clean systems at low temperatures such that
Gilbert damping is the only relaxation mechanism. Us-
ing this formalism we also consider the interplay between
Gilbert damping and disorder and show that it leads to
spin-current fluctuations.
This paper is organized as follows. In Sec. II, we
discuss the non-equilibrium Green’s function approachto magnon transport and derive an expression for the
magnon spin current. Additionally a Landauer-Büttiker
formula for the magnon spin current is derived. In
Sec. III, we illustrate the formalism by numerically con-
sidering ballistic magnon transport and determine the
dependence of the spin current on thickness of the ferro-
magnet. To further understand these numerical results,
we consider the formalism analytically in the continuum
limit in Sec. IV, and also show that in that limit we ob-
tain the same results using the stochastic LLG equation.
We give a further discussion and outlook in section V.
II. NON-EQUILIBRIUM GREEN’S FUNCTION
FORMALISM
In this section we describe our model and, using
Keldysh theory, arrive at an expression for the density
matrix of the magnons from which all observables can be
calculated. The reader interested in applying the final re-
sult of our formalism may skip ahead to Sec. IIE where
we give a summary on how to implement it.
A. Model
j j/primej j/primej j/prime∆µL
TL∆µR
TR
TFMNNM FM NM
JLJRρj,j/prime G(±)
j,j/prime(t,t/prime)
Gk,k/prime(t,t/prime) Gk,k/prime(t,t/prime)
ΣFM,(±)ΣL,(±)ΣR,(±)
Self-energy
Figure 2: Schematic for the NM-FM-NM heterostructure and
notationfortheGreen’sfunctionsandself-energies. Thearray
of circles denotes the localized magnetic moments, while the
two regions outside the parabolic lines denote the leads, i.e.,
reservoirs of polarized electrons. Moreover, JL=R
j;kk0denotes the
interface coupling, and TL=RandL=Rdenote the temper-
ature and spin accumulation for the leads. The properties of
the magnons are encoded in G(+)
j;j0(t;t0), the retarded magnon
Green’s function, and the magnon density matrix j;j0. The
number of sites in the spin-current direction is N. The self-
energies FM; (),L;(),R;()are due to Gilbert damping,
and the left and right lead, respectively.
We consider a magnetic insulator connected to two
nonmagnetic metallic leads, as shown in Fig. 2. For3
our formalism it is most convenient to consider both the
magnons and the electrons as hopping on the lattice for
the ferromagnet. Here, we consider the simplest versions
of such cubic lattice models; extensions, e.g. to multi-
ple magnon and/or electron bands, and multiple leads
are straightforward. The leads have a temperature TL=R
and a spin accumulation L=Rthat injects spin cur-
rent from the non-magnetic metal into the magnetic in-
sulator. This nonzero spin accumulation could, e.g., be
established by the spin Hall effect.
The total Hamiltonian is a sum of the uncoupled
magnon and lead Hamiltonians together with a coupling
term:
^Htot=^HFM+^HNM+^HC: (1)
Here, ^HFMdenotesthefreeHamiltonianforthemagnons,
^HFM= X
<j;j0>Jj;j0by
j0bj+X
jjby
jbjX
<j;j0>hj0;jby
j0bj:
(2)
wherebj(by
j)is a magnon annihilation (creation) opera-
tor. This hamiltonian can be derived from a spin hamil-
tonian using the Holstein-Primakoff transformation26,27
and expanding up to second order in the bosonic fields.
Eq. (2) describes hopping of the magnons with amplitude
Jj;j0between sites labeled by jandj0on the lattice, with
an on-site potential energy jthat, if taken to be homo-
geneous, would correspond to the magnon gap induced
by a magnetic field and anisotropy. We have taken the
external field in the zdirection, so that one magnon,
created at site jby the operator ^by
j, corresponds to spin
+~.
The Hamiltonian for the electrons in the leads is
^HNM= X
r2fL;RgX
<k;k0>X
2";#tr^ y
kr^ k0r+h:c:(3)
where the electron creation ( y
kr) and annihilation
( kr) operators are labelled by the lattice position k,
spin, and an index rdistinguishing (L)eft and (R)ight
leads. The hopping amplitude for the electrons is de-
notedbytrandcouldinprinciplebedifferentfordifferent
leads. Moreover, terms to describe hopping beyond near-
estneighborcanbestraightforwardlyincluded. Belowwe
will show that microscopic details will be incorporated in
a single parameter per lead that describes the coupling
between electrons and magnons.
Finally, the Hamiltonian that describes the coupling
between metal and insulator, ^HC, is given by28
^HC=X
r;j;kk0
Jr
j;kk0^by
j^ y
k#r^ k0"r+ h:c:
;(4)
with the matrix elements Jr
j;kk0that depend on the mi-
croscopic details of the interface. An electron spin that
flips from up to down at the interface creates one magnon
withspin +~inthemagneticinsulator. Thisformofcou-
pling between electrons and magnons derives from inter-
face exchange coupling between spins in the insulators
with electronic spins in the metal28.
Gk/prime,k/prime/prime;↑
Gk/prime/prime/prime,k;↓t/prime, j/primet, jFigure 3: Feynman diagram for the spin-flip processes emit-
ting and absorbing magnons that are represented by the wavy
lines. The two vertices indicate the exchange coupling at one
of the interfaces of the magnetic insulator (sites j;j0) and nor-
mal metal (sites k;k0;k00;k000).Gk0k00;"andGk000k;#denotes
the electron Keldysh Green’s function of one of the leads.
B. Magnon density matrix and current
Our objective is to calculate the steady-state magnon
Green’s function iG<
j;j0(t;t0) =h^by
j0(t0)^bj(t)i, from which
all observables are calculated (note that time-dependent
operators refer to the Heisenberg picture). This “lesser”
Green’s function follows from the Keldysh Green’s func-
tion
iGj;j0(t;t0)Trh
^(t0)TC1
^bj(t)^by
j0(t0)i
;(5)
with ^(t0)the initial (at time t0) density matrix, and
C1the Keldysh contour, and Tr[:::]stands for perform-
ing a trace average. The time-ordering operator on this
contour is defined by
TC1
^O(t)^O0(t0)
(t;t0)^O(t)^O0(t0)(t0;t)^O0(t0)^O(t);
(6)
with(t;t0)the corresponding Heaviside step function
and the +( )sign applies when the operators have
bosonic (fermionic) commutation relations. In Fig. 2
we schematically indicate the relevant quantities enter-
ing our theory.
Att= 0, the spin accumulation in the two leads is
We compute the magnon self energy due the coupling
between magnons and electrons to second order in the
coupling matrix elements Jj;kk0. This implies that the
magnons acquire a Keldysh self-energy due to lead r
given by
~r
j;j0(t;t0) = i
~X
kk0k00k000Jr
j;kk0(Jr)
j0;k00k000
Gk0k00;r"(t;t0)Gk000k;r#(t0;t);(7)
whereGk0k00;r(t;t0)denotes the electron Keldysh
Green’s function of lead r, that reads
Gkk0;r(t;t0) = ihTC1^ kr(t)^ y
k0r(t0)i:(8)4
The Feynman diagram for this self-energy is shown in
Fig. 3. While this self-energy is computed to second
order inJr
j;kk0, the magnon Green’s function and the
magnon spin current, both of which we evaluate below,
contain all orders in Jr
j;kk0, which therefore does not need
to be small. In this respect, our approach is different
from the work of Ohnuma et al.[29], who evaluate the
interfacial spin current to second order in the electron-
magnon coupling. Irreducible diagrams other than that
in Fig. 3 involve one or more magnon propagators as in-
ternal lines and therefore correspond to magnon-magnon
interactions at the interface induced by electrons in the
normal metal. For the small magnon densities of interest
to use here these can be safely neglected and the self-
energy in Eq. (7) thus takes into account the dominant
process of spin transfer between metal and insulator.
The lesser and greater component of the electronic
Green’s functions can be expressed in terms of the spec-
tral functions Akk0;r()via
iG<
kk0;r=Akk0;r()NF r
kBTr
;
iG>
kk0;r=Akk0;r()
1 NF r
kBTr
;(9)
withNF(x) = [ex+ 1] 1the Fermi distribution function,
Trthe temperature of lead r(kBbeing Boltzmann’s con-
stant) and;rthe chemical potential of spin projection
in leadr. As we will see later on, the lead chemical
potential are taken spin-dependent to be able to imple-
ment nonzero spin accumulation. The spectral function
is related to the retarded Green’s function via
Akk0;r() = 2Imh
G(+)
kk0;r()i
; (10)
which does not depend on spin as the leads are taken to
be normal metals. While the retarded Green’s function
of the leads can be determined explicitly for the model
that we consider here, we will show below that such a
level of detail is not needed but that, instead, we can pa-
rameterize the electron-magnon coupling by an effective
interface parameter.
As mentioned before, all steady-state properties of the
magnon system are determined by the magnon lesser
Green’s function leading to the magnon density matrix.
It is ultimately given by the kinetic equation23,30
j;j0h^by
j0(t)^bj(t)i=Zd
(2)h
G(+)()i~<()G( )()i
j;j0;
(11)
where ~j;j0(t;t0)is the total magnon self-energy dis-
cussed in detail below, of which the "lesser" component
enters in the above equation. In the above and what
follows, quantities with suppressed site indexes are in-
terpreted as matrices, and matrix multiplication applies
for products of these quantities. The retarded (+)and
advanced ( )magnon Green’s functions satisfy
h
h ~()()i
G()() = 1;(12)where=i0. The magnon self-energies have con-
tributions from the leads, as well as a contribution from
the bulk denoted by ~FM:
~() =~FM() +X
r2fL;Rg~r():(13)
From Eq. (7) we find that for the retarded and advanced
component, the contribution due to the leads is given by
~r;()
j;j0() =X
kk0k00k000Jr
j;kk0(Jr)
j0;k00k000Zd0
(2)Zd00
(2)
Ak0k00;r(0)Ak000k;r(00)NF
0 r"
kBTr
NF
00 r#
kBTr
+0 00
;
(14)
whereas the "lesser" self-energy can be shown to be of
the form:
~r;<
j;j0() = 2iNB r
kBTr
Imh
~r;(+)
j;j0()i
;(15)
withNB(x) = [ex 1] 1the Bose-Einstein distribution
function and r=r" r#the spin accumulation in
leadr.
Having established the contributions due to the leads,
we consider the bulk self-energy ~FM, which in princi-
ple could include various contributions, such as magnon
conserving and nonconsering magnon-phonon interac-
tions, or magnon-magnon interactions. Here, we consider
magnon non-conserving magnon-phonon coupling as the
source of the bulk self-energy and use the Gilbert damp-
ing phenomenology to parameterize it by the constant
which for the magnetic insulator YIG is of the order
of10 4. Gilbert damping corresponds to a decay of the
magnons into phonons with a rate proportional to their
energy. This thus leads to the contributions
~FM;<
j;j0() = 2NB
kBTFM
~FM;(+)
j;j0() ;
~FM;(+)
j;j0() = i j;j0; (16)
whereTFMis the temperature of phonon bath. We note
that in principle the temperature could be taken position
dependent to implement a temperature gradient, but we
do not consider this situation here.
With the results above, the density-matrix elements
j;j0can be explicitly computed from the magnon re-
tarded and advanced Green’s function and the “lesser”
component of the total magnon self-energy using
Eq. (11). The magnon self-energy is evaluated using the
explicitexpressionfortheretardedandadvancedmagnon
self-energies due to leads and Gilbert damping ~FM, see
Eq. (16).
We are interested in the computation of the magnon
spin currenthjm;jj0iin the bulk of the FM from site j5
to sitej0, which in terms of the magnon density matrix
reads,
hjm;jj0i= i(hj;j0j0;j c:c:); (17)
and follows from evaluating the change in time of the lo-
cal spin density, ~dh^by
j^bji=dt, using the Heisenberg equa-
tions of motion. The magnon spin current in the bulk
thus follows straightforwardly from the magnon density
matrix.
While the formalism presented so far provides a com-
plete description of the magnon spin transport driven by
metallic reservoirs, we discuss two simplifying develop-
ments below. First, we derive a Landauer-Bütikker-like
formula for the spin current from metallic reservoirs to
the magnon system. Second, we discuss how to replace
the matrix elements Jr
j;k;k0by a single phenomenologi-
cal parameter that characterizes the interface between
metallic reservoirs and the magnetic insulator.
C. Landauer-Büttiker formula
In this section we derive a Landauer-Büttiker formula
for the magnon transport. Using the Heisenberg equa-
tions of motion for the local spin density, we find that
the spin current from the left reservoir into the magnon
system is given by
jL
s ~
2*
d
dtX
k
^ y
k"L k"L y
k#L k#L+
= 2
~X
j;kk0Re[
JL
j;kk0g<
j;kk0(t;t0)];(18)
in terms of the Green’s function
g<
j;kk0(t;t0)ih^ y
k0"L(t0)^ k#L(t0)^bj(t)i:(19)
This “lesser” coupling Green’s function g<
j;kk0(t;t0)is cal-
culated using Wick’s theorem and standard Keldysh
methods as described below.
We introduce the spin-flip operator for lead r
^dy
kk0;r(t) =^ y
k0"r(t)^ k#r(t); (20)
so that the coupling Green’s function becomes
g<
j;kk0(t;t0)ih^dy
kk0;L(t0)^bj(t)i: (21)
The Keldysh Green’s function for the spin-flip operator
is given by
r
kk0k00k000(t;t0) = ihTC1^dkk0;r(t)^dy
k00k000;r(t0)i(22)
and using Wick’s theorem we find that
r;>
kk0k00k000(t;t0) = iG>
kk000;r#(t;t0)G<
k0k00;r"(t0;t) ;
r;<
kk0k00k000(t;t0) = iG>
k0k00;r"(t0;t)G<
kk000;r#(t;t0) ;
r;(+)
kk0k00k000(t;t0)
= i(t t0)h
G>
kk000;r#(t;t0)G<
k0k00;r"(t0;t)
G>
k0k00;r"(t0;t)G<
kk000;r#(t;t0)i
;(23)where we used the definition for the electron Green’s
function in Eq. (8).
Applying the Langreth theorem30and Fourier trans-
forming, we write down the lesser coupling Green’s
function in terms of the spin-flip Green’s function and
magnon Green’s function
g<
j;kk0() =X
j0;k00k000JL
j0;k00k000
G(+)
j;j0()L;<
kk0k00k000()
+G<
j0;j()L;( )
kk0k00k000()
; (24)
where the retarded and “lesser" magnon Green’s function
are given by Eq. (11) and Eq. (12). Using these results,
we ultimately find that
jL
s=Zd
2
NB L
kBTL
NB R
kBTR
T()
+Zd
2
NB L
kBTL
NB
kBTFM
Trh
~ L()G(+)()~ FM()G( )()i
; (25)
with the transmission function
T()Trh
~ L()G(+)()~ R()G( )()i
:(26)
In the above, the rates ~ L=R()are defined by
~ r() 2Imh
~r;(+)()i
; (27)
and
~ FM() 2Imh
~FM;(+)()i
;(28)
and correspond to the decay rates of magnons with en-
ergydue to interactions with electrons in the normal
metal at the interfaces, and phonons in the bulk, respec-
tively. This result is similar to the Laudauer-Büttiker
formalism23for electronic transport using single-particle
scattering theory. In the present context, a Landauer-
Büttiker-like for spin transport was first derived by Ben-
deret al.[28] for a single NM-FM interface. In the
absence of Gilbert damping, the spin current would cor-
respond to the expected result from Landauer-Bütikker
theory, i.e., the spin current from left to the right lead is
then given by the first line of Eq. (25). The presence of
damping gives leakage of spin current due to the coupling
with the phononic reservoir, as the second term shows.
Finally,wenotethatthespincurrentfromtherightreser-
voir into the system is obtained by interchanging labels L
and R in the first term, and the label L replaced by R in
the second one. Due to the presence of Gilbert damping,
however, we have in general that jL
s6= jR
s.
D. Determining the interface coupling
We now proceed to express the magnon spin current
(Eq.(25))intermsofamacroscopic,measurablequantity6
ratherthantheinterfacialexchangeconstants Jr
j;k;k0. For
r F(withFthe Fermi energy of the metallic
leads), which is in practice always obeyed, we have for
low energies and temperatures that
~r;()
j;j0()'i1
4X
kk0k00k000Jr
j;kk0(Jr)
j0;k00k000
Ak0;k00;r(F)Ak000;k;r(F)( r):(29)
Here, we also neglected the real part of this self-energy
which provides a small renormalization of the magnon
energies but is otherwise unimportant. The expansion
for small energies in Eq. (29) is valid as long as F,
which applies since is a magnon energy, and therefore
at most on the order of the thermal energy. Typically,
the above self-energy is strongly peaked for j;j0at the
interface because the magnon-electron interactions occur
at the interface. For j;j0at the interface we have that
the self-energy depends weakly on varying j;j0along the
interface provided that the properties of the interface do
not vary substantially from position to position. We can
thus make the identification:
~r;()
j;j0()'ir( r)j;j0j;jr;(30)
withjrthe positions of the sites at the r-th interface,
andrparametrizing the coupling between electrons and
magnons at the interface. Note that rcan be read off
from Eq. (29). Rather than evaluating this parameter
in terms of the matrix elements Jr
j;kk0and the electronic
spectral functions of the leads Ak;k0;r(), we determine it
in terms of the real part of the spin-mixing conductance
g"#;r, a phenomenological parameter that characterizes
thespin-transferefficiencyattheinterface31. Thiscanbe
donebynotingthatintheclassicallimittheself-energyin
Eq. (30) leads to an interfacial contribution, determined
by the damping constant r=N, to the Gilbert damping
ofthehomogeneousmode, where Nisthenumberofsites
of the system perpendicular to the leads, as indicated in
Fig. 2. In terms of the spin-mixing conductance, we have
that this contribution is given by32g"#;r=4srN, withsr
the saturation spin density per area of the ferromagnet
at the interface with the r-th lead. Hence, we find that
r=g"#;r
4sr; (31)
which is used to express the reservoir contributions to the
magnon self-energies in terms of measurable quantities.
The spin-mixing conductance can be up to 5~nm 2for
YIG-Pt interfaces33, leading to the conclusion that can
be of the order 1 10for that case.
E. Summary on implementation
We end this section with some summarizing remarks
on implementation that may facilitate the reader who is
interested in applying the formalism presented here.Table I: Parameters chosen for numerical calculations based
on the NEGF formalism (unless otherwise noted).
Quantity Value
J 0:05eV
L=J 2:010 5
R=J 0:0
8
=J 2:010 3
kBTFM=J0:60
First, one determines the retarded and advanced
magnon Green’s functions. This can be done given
a magnon hamiltonian characterized by matrix ele-
mentshj;j0in Eq. (2), mixing conductances for the
metal-insulator interfaces g"#;r, and a value for the
Gilbert damping constant , from which one computes
the retarded self-energies at the interfaces in Eq. (30)
with Eq. (31), and Eq. (16). The retarded and ad-
vanced magnon Green’s functions are then computed via
Eq. (12), which amounts to a matrix inversion. The next
stepistocalculatethedensitymatrixforthemagnonsus-
ing Eq. (11), with as input the expressions for the “lesser”
self-energies in Eqs. (15) and (16). Finally, the spin cur-
rent is evaluated using Eq. (17) in the bulk of the FM or
Eq. (25) at the NM-FM interface. In the next sections,
we discuss some applications of our formalism.
III. NUMERICAL RESULTS
In this section, we present results of numerical calcula-
tions using the formalism presented in the previous sec-
tion.
A. Clean system
For simplicity, we consider now the situation where
the leads and magnetic insulators are one dimensional.
The values of various parameters are displayed in Ta-
ble I, where we take the hopping amplitudes Jj;j0=
J(j;j0+1+j;j0 1), i.e.,Jj;j0is equal toJbetween near-
est neighbours, and zero otherwise. We focus on trans-
port driven by spin accumulation in the leads and set
all temperatures equal, i.e., TL=TR=TFMT.
We also assume both interfaces to have equal proper-
ties, i.e., for the magnon-electron coupling parameters to
obeyL=R. First we consider the case without
disorder and take j= .
We are interested in how the Gilbert damping affects
the magnon spin current. In particular, we calculate the
spincurrentinjectedintherightreservoirasafunctionof
system size. The results of this calculation are shown in
Fig. 4 for various temperatures, which indicates that for7
a certain fixed spin accumulation, the injected spin cur-
rent decays with the thickness of the system for N > 25,
for the parameters we have chosen. We come back to the
various regimes of thickness dependence when we present
analyticalresultsforcleansystemsinthecontinuumlimit
in Sec. IV. From these results we define a magnon relax-
0 20 40 60 80 100 120 140 160
system/uni00A0size/uni00A0(d/a)10/uni00AD410/uni00AD310/uni00AD210/uni00AD1100magnon/uni00A0spin/uni00A0current/uni00A0(J)kBT/J=0.012
kBT/J=0.024
kBT/J=0.048
kBT/J=0.108
kBT/J=0.192
Figure 4: System-size dependence of spin current ejected in
the right reservoir for = 6:910 2;= 8:0and various
temperatures.
ation length drelaxusing the definition
jm(d)/exp( d=drelax); (32)
applied to the region N > 25and where d= Nawith
athe lattice constant. The magnon relaxation length
depends on system temperature and is shown in Fig. 5.
We attempt to fit the temperature dependence with
drelax(T) =a(
0+
1p
T+
2
T);(33)
with
0;
1;
2constantsand Tdefinedasthedimension-
less temperature TkBT=J. The term proportional to
1is expected for quadratically dispersing magnons with
Gilbert damping as the only relaxation mechanism15,25.
The terms proportional to
0and
2are added to charac-
terize the deviation from this expected form. Our results
show that the relaxation length has not only 1=p
T
behaviour. This is due to the finite system size, the con-
tact resistance that the spin current experiences at the
interface between metal and magnetic insulator, and the
deviation of the magnon dispersion from a quadratic one
due to the presence of the lattice.
B. Disordered system
We now consider the effects of disorder on the spin
current as a function of the thickness of the FM. We con-
sider a one-dimensional system with a disorder potential
drelax/a=γ0+γ1
T*+γ2
T*
Numerical result
Fitted curve
0.0 0.2 0.4 0.6 0.8115120125130135140145150
T*=kBT/Jdrelax/aFigure 5: Magnon relaxation length as a function of dimen-
sionlesstemperature Tfor = 6:910 2;= 8:0. Thefitted
parameters are obtained as
0= 114:33;
1= 0:96;
2= 0:32.
implemented by taking j= (1+j), wherejis a ran-
dom number evenly distributed between and(with
1andpositive)thatisuncorrelatedbetweendifferent
sites. In one dimension, all magnon states are Anderson
localized34. Since this is an interference phenomenon, it
is expected that Gilbert damping diminishes such local-
ization effects. The effect of disorder on spin waves was
investigated using a classical model in Ref. [35], whereas
Ref. [36] presents a general discussion of the effect of
dissipation on Anderson localization. Very recently, the
effect of Dzyaloshinskii-Moriya interactions on magnon
localization was studied37. Here we consider how the in-
terplay between Gilbert damping and the disorder affects
the magnon transport.
For a system without Gilbert damping the spin current
carried by magnons is conserved and therefore indepen-
dent of position regardless of the presence or absence of
disorder. DuetothepresenceofGilbertdampingthespin
current decays as a function of position. Adding disorder
on top of the dissipation due to Gilbert damping causes
the spin current to fluctuate from position to position.
For large Gilbert damping, however, the effects of dis-
order are suppressed as the Gilbert damping suppresses
the localization of magnon states. In Fig. 6 we show nu-
merical results of the position dependence of the magnon
current for different combinations of disorder and Gilbert
damping constants. The plots clearly show that the spin
current fluctuates in position due to the combined ef-
fect of disorder and Gilbert damping, whereas it is con-
stant without Gilbert damping, and decays in the case
with damping but without disorder. Note that for the
two cases without Gilbert damping the magnitude of the
spin current is different because the disorder alters the
conductance of the system and each curve in Fig. 6 cor-
responds to a different realization of disorder.
To characterize the fluctuations in the spin current, we8
0 20 40 60 80
site j1.01.52.02.53.03.54.04.5magnon spin current (J)1e7
=0.0,=0.0
=0.0,=0.0015
=0.0069,=0.0
=0.0069,=0.0015
Figure 6: Spatial dependence of local magnon current for the
case without Gilbert damping and disorder ( = 0;= 0),
without disorder ( = 6:910 3;= 0), without Gilbert
damping ( = 0;= 1:510 3), and both disorder and
Gilbert damping ( = 6:910 3;= 1:510 3). The inter-
face coupling parameter is taken equal to = 0:8.
define the correlation function
Cj=vuut
jm;j;j+1 jm;j;j+12
jm;j;j+12; (34)
where the bar stands for performing averaging over the
realizations of disorder. Fig. (7) shows this correlation
function for j=N 1as a function of Gilbert damp-
ing for various strengths of the disorder. As we expect,
basedonthepreviousdiscussion, thefluctuationsbecome
small as the Gilbert damping becomes very large or zero,
leaving an intermediate range where there are sizeable
fluctuations in the spin current.
IV. ANALYTICAL RESULTS
In this section we analytically compute the magnon
transmission function in the continuum limit a!0
for a clean system. We consider again the situation
of a magnon hopping amplitude Jj;j0that is equal to
Jand nonzero only for nearest neighbors, and a con-
stant magnon gap j= . We compute the magnon
density matrix, denoted by (x;x0), and retarded and
advanced Green’s functions, denoted by G()(x;x00;).
Here, the spatial coordinates in the continuum are de-
noted byx;x0;x00;. We take the system to be trans-
lationally invariant in the y z-plane and the current
direction as shown in Fig. 1 to be x.
In the continuum limit, the imaginary part of the vari-
0.000 0.005 0.010 0.015 0.020
α012345CN−1
δ=0.0005
δ=0.001
δ=0.0015Figure 7: Correlation function Cjthat characterizes the fluc-
tuations in the spin curent for j=N 1as a function of the
Gilbert damping constant, for three strengths of the disorder
potential. The curves are obtained by performing averaging
over 100 realizations of the disorder. The interface coupling
parameter is taken equal to = 0:8.
ous self-energies acquired by the magnons have the form:
Imh
~r;(+)(x;x0;)i
=
~r( r)(x xr)(x x0) ;
Imh
~FM;(+)(x;x0;)i
= (x x0);(35)
wherexris the position of the r-th lead, and where ~ris
the parameter that characterizes the interfacial coupling
between magnons and electrons. We use a different nota-
tion for this parameter as in the continuum situation its
dimension is different with respect to the discrete case.
To express ~rin terms of the spin-mixing conductance
we have that ~r=g"#=4~srwhere ~sris now the three-
dimensional saturated spin density of the ferromagnet.
We proceed by evaluating the magnon transmission
function from Eq. (26). We compute the rates in Eq. (27)
from the self-energies Eqs. (35) and find for the transmis-
sion function in the first instance that
T() = 4~L~R( L)( R)
Zdq
(2)2g(+)(xL;xR;q;)g( )(xR;xL;q;);(36)
where qis the two-dimensional momentum that results
fromFouriertransforminginthe y z-plane. TheGreen’s
functionsg()(x;x0;q;)obey [compare Eq. (12)]
(1i )+Ad2
dx2 Aq2
iX
r2fL;Rg~r( r)(x xr)3
5g()(x;x0;q;)
=(x x0); (37)9
whereA=Ja2. This Green’s function is evaluated using
standard techniques for inhomogeneous boundary value
problems (see Appendix A) to ultimately yield
T() = 4~2( L)( R)Zdq
(2)2jt(q;)j2;(38)
with
t(q;) =A
A22 ~2( L)( R)
sinh(d)
iA~(2 L R) cosh(d)] 1; (39)
with=p
(Aq2+ i )=Aand whered=xR
xL. Note that we have at this point taken both interfaces
equal for simplicity, so that ~L= ~R~. In terms of an
interfacial Gilbert damping parameter 0we have that
~=d 0.
Let us identify the magnon decay length
l
;
where=p
A=kBTis proportional to the thermal de
Broglie wavelength. Equipped with a closed, analytic
expression, we may now, in an analogous way as Hoffman
et al.[25], investigate the behavior of Eq. (38) in the thin
FM (d l) and thick FM ( d l) regimes. In order
to do so, we take L= 0so that the second term in
Eq. (25) vanishes and the spin current is fully determined
by the transmission coefficient T(). Before analyzing
the result for the spin current more closely, we remark
that the result for the transmission function may also be
obtained from the linearized stochastic Landau-Lifshitz-
Gilbert equation, as shown in Appendix B.
A. Thin film regime ( d l)
In the thin film regime, the transmission coefficient
T()exhibits scattering resonances near =nqfor given
q, where
nq
A=q2+1
2+n22
d2
andnisanintegerandwhere =p
A=isthecoherence
length of the ferromagnet. When the ferromagnet is suf-
ficiently thin ( d = 1=2=p l), one finds that these
peaks are well separated, and the transmission coefficient
is approximated as a sum of Lorentzians: T =P1
n=0Tn,
where:
Tn()Anq L
n R
n
Ln+ Rn+ FMn(40)
with
Anq() = n
( nq)2+ ( n=2)2(41)as the spin wave spectral density. The broadening rates
are given by FM
n= 2 , L
0= 2 0, R
0= 2 0( R),
L
n6=0= 4 0, R
n6=0= 4 0( R)and n= FM
n+ L
n+
R
n. Intheextremesmalldissipationlimit(i.e. neglecting
spectral broadening by the Gilbert damping), one has:
Anq()!2( nq); (42)
and the current has the simple form, jL
s=P1
n=0jn,
where
jn=a2Zd2q
(2)2 L
n R
n
n
NBnq
kBT
NBnq R
kBT
(43)
where L
n, R
nand FM
nare all evaluated at =nq.
Eq. (43) allows one to estimate the thickness dependence
of the signal. Supposing R.nq, whend g"#=s ,
then 0 , and L
n R
n= FM
njL
s;cl1=d; whend
g"#=s , then 0 , andjL
s;cl1=d2. The enhancement
of the spin current for small dis in rough agreement with
our numerical results in the previous section as shown in
Fig. 4.
B. Thick film regime ( d l)
In the thick film regime, the transmission function be-
comes
T()(4Ad)2 L
x R
xp
( 0q)2+ ( FMx=2)2e 2rd
j(4A)2 (d)2 Lx Rx i4dA RxS(r)j2
where L=R= FM
x = L=R= FM
n6=0,r= Re[], andS(r)isthe
sign ofr. For 1, we have=ikx
1 +i =2Ak2
x
,
wherekx=p
q2+ 2 =A. For energies > A (q2+
2),kxis imaginary, and the contribution to the spin
current decays rapidly with d. When, however, <
A(q2+ 2),kxis real, and r=
q2+ 2
=2kx
=(for thermal magnons), so that the signal decays
over a length scale l/1=p
T, in agreement with our
numerical results as shown in Fig. 5.
C. Comparison with numerical results
In order to compare the numerical with the analyti-
cal results we plot in Fig. 8 the transmission function
as a function of energy. Here, the numerical result is
evaluated for a clean system using Eq. (26) while the an-
alytical result is that of Eq. (38). While they agree in
the appropriate limit ( N!1;a!0), for finite Nthere
are substantial deviations that are due to the increased
importance of interfacing coupling relative to the Gilbert
damping for small systems and the deviations of the dis-
persion from a quadratic one.10
Δ/J=0.2N=20
Analytic
Numerical
0 1 2 3 4 50.000.020.040.060.08
ϵ/JT(ϵ)
Figure 8: Magnon transmission function as a function of en-
ergy. The parameters are chosen to be =J= 0:2; =
0:069;= 8:0.
V. DISCUSSION AND OUTLOOK
We have developed a NEGF formalism for exchange
magnon transport in a NM-FM-NM heterostructure. We
have illustrated the formalism with numerical and ana-
lytical calculations and determined the thickness depen-
dence of the magnon spin current. We have also con-
sidered magnon disorder scattering and shown that the
interplay between disorder and Gilbert damping leads to
spin-current fluctuations.
Wehavealsodemonstratedthatforacleansystem,i.e.,
without disorder, in the continuum limit the results ob-
tained from the NEGF formalism agree with those fromthe stochastic LLG formalism. The latter is suitable for
a clean system in the continuum limit where the vari-
ous boundary conditions on the solutions of the stochas-
tic equations are easily imposed. The NEGF formal-
ism is geared towards real-space implementation, such
that, e.g., disorder scattering due to impurities are more
straightforwardly included as illustrated by our example
application. The NEGF formalism is also more flexi-
ble for systematically including self-energies due to ad-
ditional physical processes, such as magnon-conserving
magnon-phonon scattering and magnon-magnon scatter-
ing, or, for example, for treating strong-coupling regimes
intowhichthestochasticLandau-Lifshitz-Gilbertformal-
ism has no natural extension.
Using our formalism, a variety of mesoscopic transport
features of magnon transport can be investigated includ-
ing, e.g., magnon shot noise38. The generalization of our
formalism to elliptical magnons and magnons in antifer-
romagnets is an attractive direction for future research.
Acknowledgments
This work was supported by the Stichting voor Funda-
menteel Onderzoek der Materie (FOM), the Netherlands
Organization for Scientific Research (NWO), and by the
European Research Council (ERC) under the Seventh
Framework Program (FP7). J. Z. would like to thank
the China Scholarship Council. J. A. has received fund-
ing from the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-
Curie grant agreement No 706839 (SPINSOCS).
Appendix A: Evaluation of magnon Green’s function in the continuum limit
In this appendix we evaluate the magnon Green’s function in the continuum limit that is determined by Eq. (37).
For simplicity we take the momentum qequal to zero and suppress it in the notation, as it can be trivially restored
afterwards. The Green’s function is then determined by
2
4i iX
r2fL;Rg~r( r)(x xr) +Ad2
dx2 H3
5g()(x;x0;) =(x x0): (A1)
To determine this Green’s function we first solve for the states (x)that obey:
2
4i iX
r2fL;Rg~r( r)(x xr) +Ad2
dx2 H3
5 (x) = 0: (A2)
Integrating this equation across x=xLandx=xRleads to the boundary conditions:
x=xL:i~L( L) (xL) +Ad (x)
dxjx=xL= 0; (A3)
x=xR:i~R( R) (xR) Ad (x)
dxjx=xR= 0: (A4)11
ForxL<x<xR, the general solution is:
(x) =Beikx+Ce ikx; (A5)
withk=p
(i H)=A. We write the solution obeying the boundary condition at x=xLas
L(x) =eikx+Ce ikx; (A6)
With the boundary condition at x=xL( Eq.A4), we find that
C=Ak~L( L)
Ak~L( L)
e2ikxL:
For the solution obeying the boundary condition at x=xR, we write
R(x) =Beikx+e ikx: (A7)
With the boundary condition at x=xR( Eq.A4), we have:
A(iBkeikxR ike ikxR) =i~R( R)(BeikxR+e ikxR);
so that
B=Ak~R( R)
Ak~R( R)
e 2ikxR:
The Green’s function is now given by39
g()(x;x0;) =8
<
: ()
L(x0) ()
R(x)
AW()(x0)forx>x0;
()
L(x) ()
R(x0)
AW()(x0)forx<x0:(A8)
with the Wronskian
W(x0) =
L(x0)d
R(x0)
dx0
R(x0)d
L(x0)
dx0:
Inserting the result for the Green’s function in Eq. (36) and using that k+=iandk = (k+), we obtain Eqs. (38)
and (39) after restoring the q-dependence and taking ~R= ~L= ~.
Appendix B: Stochastic Formalism for Spin Transport in a Ferromagnet
Here, we show how to recover our analytical results from the stochastic Landau-Lifshitz-Gilbert equation, general-
izing the results of Ref. [25] to the case of nonzero spin accumulation in the metallic reservoirs. The dynamics of the
spin density unit vector nis governed by:
(1 + n)~_n+n(H+h) Anr2n= 0; (B1)
where H= ^zis the effective applied magnetic field (in units of energy) and his the bulk stochastic field24. We
assume a spin accumulation 0= Rzin the right normal metal, while the spin accumulation in the left lead is
taken zero. The boundary condition at x= 0reads
js(x= 0) = A~sn@xnjx=0
=g"#
4(n(n0) +n~_n) +nh0
L
x=0(B2)
and atx=d:
js(x=d) = A~sn@xnjx=d
= g"#
4(n~_n) +nh0
R
x=d: (B3)12
Defining (x;t) =n(x;t)p
~s=2, wherennx iny, we linearize the dynamics around the equilibrium orientation
n= z. Fourier transforming:
(x;q;) =Zdt
2~Zd2r?
2eit=~e ir?q (x;r?;t);
the bulk equation of motion reads:
A
@2
x 2
=hp
~s : (B4)
The bulk transformed stochastic field h=hx ihyobeys the fluctuation dissipation theorem:
hh(x;q;)h(x0;q0;0)i= 2 (2)3 (~2=~s)
(x x0)(q q0)( 0)
tanh [=2kBT]: (B5)
The boundary conditions, Eqs. (B2) and (B3), become respectively:
A@x ig"#
4~s( R) =hRp
2~s(B6)
atx= 0and
A@x +ig"#
4~s =hLp
2~s(B7)
atx=d, where we have taken the coupling at both interfaces equal. Similarly, the interfacial stochastic fields obey:
D
h0
R(q;)h0
R(q0;0)E
=2 (2)3 0d~2~s( R)(q q0)( 0)
tanh [( R)=2kBT](B8)
and
D
h0
L(q;0)h0
L(q0;)E
=2 (2)3 0d~2~s(q q0)( 0)
tanh [=2kBT]: (B9)
Using Eqs. (B4)-(B9), one finds the current on the left side of the structure: jL
szjs(x= 0)to be of the form:
jL
s=Zd
2
NB
kBT
NB R
kBT
T() (B10)
where T()is the transmission coefficient in Eq. (38). Hence, for a clean system and in the continuum limit the results
of the stochastic Landau-Lifshitz-Gilbert equation coincide with those of the NEGF formalism given by Eq. (25).
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1101.5522v1.Entanglement_between_two_atoms_in_a_damping_Jaynes_Cummings_model.pdf | |
1705.07489v2.Dynamical_depinning_of_chiral_domain_walls.pdf | Dynamical depinning of chiral domain walls
Simone Moretti,Michele Voto, and Eduardo Martinez
Department of Applied Physics, University of Salamanca, Plaza de los Caidos, Salamanca 37008, Spain.
The domain wall depinning eld represents the minimum magnetic eld needed to move a domain
wall, typically pinned by samples' disorder or patterned constrictions. Conventionally, such eld
is considered independent on the Gilbert damping since it is assumed to be the eld at which the
Zeeman energy equals the pinning energy barrier (both damping independent). Here, we analyse
numerically the domain wall depinning eld as function of the Gilbert damping in a system with per-
pendicular magnetic anisotropy and Dzyaloshinskii-Moriya interaction. Contrary to expectations,
we nd that the depinning eld depends on the Gilbert damping and that it strongly decreases for
small damping parameters. We explain this dependence with a simple one-dimensional model and
we show that the reduction of the depinning eld is related to the nite size of the pinning barriers
and to the domain wall internal dynamics, connected to the Dzyaloshinskii-Moriya interaction and
the shape anisotropy.
I. INTRODUCTION
Magnetic domain wall (DW) motion along ferromag-
netic (FM) nanostructures has been the subject of in-
tense research over the last decade owing to its po-
tential for new promising technological applications1,2
and for the very rich physics involved. A consider-
able e ort is now focused on DW dynamics in systems
with perpendicular magnetic anisotropy (PMA) which
present narrower DWs and a better scalability. Typ-
ical PMA systems consist of ultrathin multi-layers of
heavy metal/FM/metal oxide (or heavy metal), such as
Pt=Co=Pt3,4or Pt=Co=AlOx5{7, where the FM layer has
a thickness of typically 0 :6 1 nm. In these systems,
PMA arises mainly from interfacial interactions between
the FM layer and the neighbouring layers (see Ref.8and
references therein). Another important interfacial ef-
fect is the Dzyaloshinskii-Moriya interaction (DMI)9,10,
present in systems with broken inversion symmetry such
as Pt/Co/AlOx. This e ect gives rise to an internal in-
plane eld that xes the DW chirality (the magnetization
rotates always in the same direction when passing from
up to down and from down to up domains) and it can
lead to a considerably faster domain wall motion10and to
new magnetic patterns such as skyrmions11or helices12.
Normally, DWs are pinned by samples' intrinsic disorder
and a minimum propagation eld is needed in order to
overcome such pinning energy barrier and move the DW.
Such eld is the DW depinning eld ( Hdep) and it repre-
sents an important parameter from a technological point
of view since a low depinning eld implies less energy
required to move the DW and, therefore, a energetically
cheaper device.
From a theoretical point of view, DW motion can be
described by the Landau-Lifshitz-Gilbert (LLG) equa-
tion13which predicts, for a perfect sample without dis-
order, the velocity vs eld curve depicted in Fig. 1 and
labelled as Perfect . In a disordered system, experi-
ments have shown that a DW moves as a general one-
dimensional (1D) elastic interface in a two-dimensional
disordered medium3,4and that it follows a theoreticalvelocityvsdriving force curve, predicted for such inter-
faces14,15(also shown in Fig. 1 for T= 0 andT= 300K).
Moreover, this behaviour can be reproduced by including
disorder in the LLG equation16{18. At zero temperature
(T= 0) the DW does not move as long as the applied
eld is lower than Hdep, while, at T6= 0, thermal ac-
tivation leads to DW motion even if H < H dep(the so
called creep regime). For high elds ( H >> H dep) the
DW moves as predicted by the LLG equation in a per-
fect system. Within the creep theory, the DW is con-
sidered as a simple elastic interface and all its internal
dynamics are neglected. Conventionally, Hdepis consid-
ered independent of the Gilbert damping because it is as-
sumed to be the eld at which the Zeeman energy equals
the pinning energy barrier19,20(both damping indepen-
dent). Such assumption, consistently with the creep the-
ory, neglects any e ects related to the internal DW dy-
namics such as DW spins precession or vertical Bloch
lines (VBL) formation21. The damping parameter, for
its part, represents another important parameter, which
controls the energy dissipation and a ects the DW veloc-
ity and Walker Breakdown22. It can be modi ed by dop-
ing the sample23or by a proper interface choice as a con-
sequence of spin-pumping mechanism24. Modi cations of
the DW depinning eld related to changes in the damping
parameter were already observed in in-plane systems23,25
and attributed to a non-rigid DW motion23,25. Oscilla-
tions of the DW depinning eld due to the internal DW
dynamics were also experimentally observed in in-plane
similar systems26. Additional dynamical e ects in soft
samples, such as DW boosts in current induced motion,
were numerically predicted and explained in terms of DW
internal dynamics and DW transformations27,28.
Here, we numerically analyse the DW depinning eld
in a system with PMA and DMI as function of the Gilbert
damping. We observe a reduction of Hdepfor low damp-
ing and we explain this behaviour by adopting a simple
1D model. We show that the e ect is due to the nite
size of pinning barriers and to the DW internal dynam-
ics, related to the DMI and shape anisotropy elds. This
article is structured as follows: in Section II we present
the simulations method, the disorder implementation andarXiv:1705.07489v2 [cond-mat.mes-hall] 25 Aug 20172
theHdepcalculations. The main results are outlined and
discussed in Section III, where we also present the 1D
model. Finally, the main conclusions of our work are
summarized in Section IV.
●●●●
●
●
●
●●●●●●●●●●�=��
�=����
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FIG. 1. DW velocity vsapplied eld as predicted by the LLG
equation in a perfect system and by the creep law atT= 0
andT= 300K.
II. MICROMAGNETIC SIMULATIONS
We consider a sample of dimensions
(102410240:6) nm3with periodic bound-
ary conditions along the ydirection, in order to simulate
an extended thin lm. Magnetization dynamics is
analysed by means of the LLG equation13:
dm
dt=
0
1 + 2(mHe )
0
1 + 2[m(mHe )];
(1)
where m(r;t) =M(r;t)=Msis the normalized magneti-
zation vector, with Msbeing the saturation magnetiza-
tion.
0is the gyromagnetic ratio and is the Gilbert
damping. He =Hexch+HDMI+Han+Hdmg+Hz^uz
is the e ective eld, including the exchange, DMI, uni-
axial anisotropy, demagnetizing and external eld con-
tributions13respectively. Typical PMA samples param-
eters are considered: A= 1710 12J=m,Ms= 1:03
106A=m,Ku= 1:3106J=m3andD= 0:9 mJ=m2,
whereAis the exchange constant, Dis the DMI constant
andKuis the uniaxial anisotropy constant. Disorder is
taken into account by dividing the sample into grains
by Voronoi tessellation29,30, as shown in Fig. 2(a). In
each grain the micromagnetic parameters fMs;Dc;Kug
change in a correlated way in order to mimic a normally
distributed thickness31:
tG=N(t0;)!8
<
:MG= (MstG)=t0
KG= (Kut0)=tG
DG= (Dct0)=tG; (2)
where the subscript Gstands for grain, t0is the aver-
age thickness ( t0= 0:6nm) andis the standard devi-
ation of the thickness normal distribution. The sample
is discretized in cells of dimensions (2 20:6)nm3,smaller than the exchange length lex5nm. Grain size
is GS=15 nm, reasonable for these materials, while the
thickness
uctuation is = 7%. Eq. (1) is solved by the
nite di erence solver MuMax 3.9.329.
A DW is placed and relaxed at the center of the sample
as depicted in Fig. 2(b). Hdepis calculated by applying
a sequence of elds and running the simulation, for each
eld, until the DW is expelled from the sample, or until
the system has reached an equilibrium state (i.e. the DW
remains pinned): max<( ). maxindicates the maxi-
mum torque, which rapidly decreases when the system is
at equilibrium. It only depends on the system parame-
ters and damping. For each value of , we choose a spe-
ci c threshold, ( ), in order to be sure that we reached
an equilibrium state (see Supplementary Material32for
more details). The simulations are repeated for 20 dif-
ferent disorder realizations. Within this approach, Hdep
corresponds to the minimum eld needed to let the DW
propagate freely through the whole sample. In order to
avoid boundaries e ects, the threshold for complete de-
pinning is set tohmzi>0:8, wherehmziis averaged over
all the realizations, i.e. hmzi=PN
i=1hmzii=N, where
N= 20 is the number of realizations. We checked that,
in our case, this de nition of Hdepcoincides with tak-
ingHdep= MaxfHi
depg, withHi
depbeing the depinning
eld of the single realization. In other words, Hdepcor-
responds to the minimum eld needed to depin the DW
from any possible pinning site considered in the 20 real-
izations33.
Following this strategy, the DW depinning eld is nu-
merically computed with two di erent approaches:
(1) by Static simulations, which neglect any precessional
dynamics by solving
dm
dt=
0
1 + 2[m(mHe )]: (3)
This is commonly done when one looks for a minimum
of the system energy and it corresponds to the picture
in whichHdepsimply depends on the balance between
Zeeman and pinning energies.34
(2) by Dynamic simulations, which include precessional
dynamics by solving the full Eq. (1). This latter method
corresponds to the most realistic case. Another way to
estimate the depinning eld is to calculate the DW veloc-
ityvs eld curve at T= 0 and look for minimum eld at
which the DW velocity is di erent from zero. For these
simulations we use a moving computational region and
we run the simulations for t= 80ns (checking that longer
simulations do not change the DW velocity, meaning that
we reached a stationary state). This second setup re-
quires more time and the calculations are repeated for
only 3 disorder realizations.
Using these methods, the depinning eld Hdepis cal-
culated for di erent damping parameters .3
(a) (b)
xy
(c)
FIG. 2. (a) Grains structure obtained by Voronoi tassellation.
(b) Initial DW state. (c) Sketch of the internal DW angle .
III. RESULTS AND DISCUSSION
A. Granular system
Our rst result is shown in Fig. 3(a)-(b), which depicts
the nal average magnetization hmzias function of the
applied eld for di erent damping parameters. In the
Static simulations (Fig. 3(a)) Hdepdoes not depend on
damping, so that a static depinning eld can be de ned.
Conversely, in the Dynamic simulations (Fig. 3(b)), Hdep
decreases for low damping parameters. The depinning
eld is indicated by a star in each plot and the static
depinning eld is labelled as Hs. The same result is ob-
tained by calculating Hdepfrom the DW velocity vsap-
plied eld plot, shown in Fig. 3(c). The stars in Fig. 3(c)
correspond to the depinning elds calculated in the pre-
vious simulations and they are in good agreement with
the values predicted by the velocity vs eld curve. The
dynamical depinning eld 0Hd, normalized to the static
depinning eld 0Hs= (871)mT, with 0being the
vacuum permeability, is shown in Fig. 3(d) as function of
the damping parameter .Hdsaturates for high damp-
ing (in this case 0:5) while it decreases for low damp-
ing untilHd=Hs0:4 at = 0:02. This reduction must
be related to the precessional term, neglected in the static
simulations. The same behaviour is observed with di er-
ent grain sizes (GS=5 and 30 nm) and with a di erent
disorder model, consisting of a simple variation of the Ku
module in di erent grains. This means that the e ect is
not related to the grains size or to the particular disorder
model we used.
Additionally, Fig. 4 represents the DW energy35as
function of DW position and damping parameter for
0Hz= 70 mT. At high damping, the average DW en-
ergy density converges to 110 mJ=m2, in good agree-
ment with the analytical value 0= 4pAK0 D=
10:4 mJ=m2, whereK0is the e ective anisotropy K0=
Ku 0M2
s=2. On the contrary, for low damping, the
DW energy increases up to (0:02)14 mJ=m2. This
increase, related to DW precessional dynamics, reduces
the e ective energy barrier and helps the DW to over-
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��������/��(�)FIG. 3. Average hmzias function of applied eld for dif-
ferent damping parameters for the (a) Static simulations and
(b)Dynamic simulations. (c) DW velocity vs applied eld for
di erent damping. (d) Dynamical depinning eld, normalized
toHs, as function of damping.
come the pinning barriers. Fig. 4(c) shows the total en-
ergy of the system (including Zeeman). As expected36,
the energy decreases as the DW moves.
Finally, Fig. 5 shows the DW motion as function of
time for = 0:02 and = 0:5, along the same grain
pattern (and therefore along the same pinning barriers).
The applied eld is 0Hz= 70mT, which satis es
Hd(0:02)<Hz<Hd(0:5). The initial DW con guration
is the same but, for = 0:02, VBL start to nucleate and
the DW motion is much more turbulent (see Supplemen-
tary Material32for a movie of this process). At t= 4 ns
the DW has reached an equilibrium position for = 0:5,
while it has passed through the (same) pinning barriers
for = 0:02. Thus, one might think that the reduction
of the depinning eld could be related to the presence4
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FIG. 4. (a) DW energy density as function of DW posi-
tion for di erent damping. The nal drop corresponds to
the expulsion of the DW. (b) Average DW density as funci-
ton of damping. Dashed line represents the analytical value
110 mJ=m2. (c) Total energy density of the system as
function of DW position for di erent damping parameters.
of VBL and their complex dynamics21. Further insights
about this mechanism are given by analysing the DW
depinning at a single energy barrier as described in the
next subsection.
B. Single barrier
In order to understand how the DW precessional dy-
namics reduces Hdep, we micromagnetically analysed the
DW depinning from a single barrier as sketched in Fig. 6.
We considered a strip of dimensions (1024 2560:6)nm3
and we divided the strip into two regions, R1andR2,
which are assumed to have a thickness of t1= 0:58 and
t2= 0:62 nm respectively. Their parameters vary ac-
cordingly (see Sec. II), generating the DW energy bar-
rier () shown in Fig. 6(b). A DW is placed and re-
laxed just before the barrier. The nite size of the DW
(DW15 nm, with DWbeing the DW width pa-
rameter) smooths the abrupt energy step and, in fact,
the energy pro le can be successfully tted by using theBloch pro le22
DW=0+
+
2
1 + cos
2 arctan
expx0 x
DW
;
(4)
wherex0= 20 nm is the step position, while 0and
1are the DW energies at the left and right side of the
barrier as represented in Fig. 6(b). This means that
the pinning energy barrier has a spatial extension which
is comparable to the DW width. By performing the
same static and dynamic simulations, we obtain a static
depinning eld of 0Hs= 120 mT and, when decreasing
the damping parameter, we observe the same reduction
of the depinning eld as in the granular system (see
Fig. 6(c)). In this case the DW behaves like a rigid
object whose spins precess coherently and no VBL
nucleation is observed. Hence, Hdepreduction does not
depend directly on the presence of VBL but on the more
general mechanism of spins' precession already present
in this simpli ed case.
Nevertheless, an important characteristic of these single
barrier simulations is that the barrier is localized and it
has a nite size which is of the order of the DW width.
Note that the same holds for the granular system:
despite a more complex barrier structure, the dimension
of the single barrier between two grains has the size of
the DW width.
Thus, in order to understand the interplay between the
DW precessional dynamics and the nite size of the bar-
rier, we considered a 1D collective-coordinate model with
a localized barrier. The 1D model equations, describing
the dynamics of the DW position qand the internal angle
(sketched in Fig. 2(c)), are given by16
(1 + 2)_ =
0[(Hz+Hp(q))
HKsin 2
2
2HDMIsin
|{z }
Hint( )];(5)
(1 + 2)_q
DW=
0[ (Hz+Hp(q))
+
HKsin 2
2
2HDMIsin
;(6)
whereHK=MsNxis the shape anisotropy eld, favour-
ing Bloch walls, with Nx=t0log 2=(DW)37being the
DW demagnetizing factor along the xaxis.HDMI =
D=(0MsDW) is the DMI eld. Hint( ) represents
the internal DW eld, which includes DMI and shape
anisotropy. Hintfavours Bloch ( ==2) or N eel wall
( = 0 or =) depending on the relative strength
ofHKandHDMI. In our system, the DMI dominates
over shape anisotropy since 0HDMI170 mT while
0HK30 mT. Hence, the DW equilibrium angle is5
Out[64]=
Out[60]=
… (a) 𝜶=𝟎.𝟎𝟐time 0 0.1 ns 0.2 ns 0.3 ns 4 ns
time 0 0.1 ns 0.2 ns 0.3 ns 4 ns(b) 𝜶=𝟎.𝟓
…
Out[395]=mx
Out[395]=mx
Out[62]=
Out[65]=
FIG. 5. (a) Snapshots of the magnetization dynamics at subsequent instants under 0Hz= 70mT, for two di erent damping:
(a) = 0:02 and (b) = 0:5. The grains pattern, and therefore the energy barrier, is the same for both cases. In order to let
the DW move across more pinning sites, these simulations were performed on a larger sample with Lx= 2048 nm.
=( = 0 or =additionally depends on the sign
of the DMI). Hp(q) is the DW pinning eld, obtained
from the DW energy pro le (Eq. (4)) as follows: the max-
imum pinning eld is taken from the static simulations
while the shape of the barrier is taken as the normalized
DW energy gradient (see Supplementary Material32for
more details),
Hp(q) =Hs@DW(x)
@x
N=
= 2Hsexp
x0 q
DW
sinh
2 arctan
exp
x0 q
DWi
1 + exp
2(x0 q)
DW :(7)
The corresponding pinning eld is plotted in Fig. 7(a).38
The results for the dynamical Hdep, obtained with this
modi ed 1D model, are plotted in Fig. 6(c) and they
show a remarkable agreement with the single barrier mi-
cromagnetic simulations. This indicates that the main
factors responsible for the reduction of Hdepare already
included in this simple 1D model. Therefore, additional
insights might come from analysing the DW dynamics
within this 1D model. Fig. 7(b) and (c) represents the
DW internal angle and the DW position qas function
of time for di erent damping. The plots are calculated
with0Hz= 55 mT which satis es Hdep(0:02)< Hz<
Hdep(0:1)< H dep(0:5). As shown in Fig. 7(b) and (c),
below the depinning eld ( = 0:1, = 0:5), both the
internal angle and the DW position oscillate before reach-
ing the same nal equilibrium state. However, the am-plitude of these oscillations (the maximum displacement)
depends on the damping parameter. Fig. 7(d) shows the
nal equilibrium position as function of the applied eld
for di erent damping. The equilibrium position is the
same for all damping and it coincides with the position
at whichHz=Hp(q). Conversely, the maximum dis-
placement, shown in Fig. 7(e), strongly increases for low
damping parameters. For applied eld slightly smaller
than the depinning eld, the DW reaches the boundary
of the pinning barrier, meaning that a further increase
of the eld is enough to have a maximum displacement
higher than the barrier size and depin the DW. In other
words, the decrease of the depinning eld, observed in
the single barrier simulations, is due to DW oscillations
that depend on and that can be larger than the bar-
rier size, leading to DW depinning for lower eld. The
DW dynamics and the depinning mechanism are further
clari ed in Fig. 7(f) and Fig. 7(g). Fig. 7(f) represents
the DW coordinates fq; gfor0Hz= 55 mT and dif-
ferent damping. Before reaching the common equilib-
rium state, the DW moves in orbits (in the fq; gspace)
whose radius depends on the damping parameter. For
= 0:5 (black line) the DW rapidly collapse into the -
nal equilibrium state. Conversely, for = 0:1 (red open
circles), the DW orbits around the equilibrium state be-
fore reaching it. If the radius of the orbit is larger than
the barrier size the DW gets depinned, as in the case
of = 0:02 (blue full circles). This mechanism is also
represented in Fig. 7(g), where the DW orbits are placed
in the energy landscape. The energy is calculated as6
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(c)
R1R2yx(a)
FIG. 6. (a) Sketch of the two regions implemented for the
single barrier (SB) micromagnetic simulations. (b) DW en-
ergy as function of DW position along the strip. Blue solid
line represents the analytical value, red points the DW con-
voluted energy (due to the nite size of the DW) while black
dashed line a t using Eq. 4. (c) Dynamical depinning eld,
normalized to the static depinning eld, for the single bar-
rier simulations as function of damping, obtained from full
micromagnetic simulations and the 1D model.
(q; ) =DW(q; ) 20MsHzq, whereDWis given by
Eq. (4). Fig. 7(g) shows that the equilibrium state cor-
responds to the new minimum of the energy landscape.
Furthermore, it con rms that the applied eld is below
the static depinning eld, at which the pinning barrier
would have been completely lifted. Nevertheless, while
reaching the equilibrium state, the DW moves inside the
energy potential and, if the radius of the orbit is larger
than the barrier size, the DW can overcome the pinning
barrier, as shown for = 0:02 in Fig. 7(g).
At this point we need to understand why the amplitude
of the DW oscillations depends on damping. By solving
Eq. (5) and Eq.(6) for the equilibrium state ( _ q= 0, _ =
0) we obtain
_q= 0)jHp(q)j=Hz+Hint( )
Hz
2HDMI
sin ; (8)
_ = 0)jHp(q)j=Hz Hint( )
Hz+
2HDMIsin ; (9)
since0HDMI 0HKand, therefore, Hint
(=2)HDMIsin . These equations have a single com-
mon solution which corresponds to jHp(q)j=Hzand
= 0=(at whichHint() = 0). However, at t= 0,the DW starts precessing under the e ect of the applied
eld and, if 6=whenjHp(q)j=Hz, the DW does not
stop at the nal equilibrium position but it continues its
motion, as imposed by Eq. (8) and (9). In other words,
the DW oscillations in Fig. 7(b) are given by oscillations
of the DW internal angle , around its equilibrium value
0=. These oscillations lead to a modi cation of the
DW equilibrium position due to the DW internal eld
(Hint( )), which exerts an additional torque on the DW
in order to restore the equilibrium angle. As previously
commented, if the amplitude of these oscillations is large
enough, the DW gets depinned. From Eq. (8) we see
that the new equilibrium position (and therefore the am-
plitude of the oscillations) depends on the DMI eld, the
value of the DW angle and the damping parameter.
In particular, damping has a twofold in
uence on this
dynamics: one the one hand, it appears directly in
Eq. (8), dividing the internal eld, meaning that for the
same deviation of from equilibrium, we have a stronger
internal eld for smaller damping. On the other hand,
the second in
uence of damping is on the DW internal
angle: once the DW angle has deviated from equilibrium,
the restoring torque due to DMI is proportional to the
damping parameter (see Eq. (9)). Hence, a lower damp-
ing leads to lower restoring torque and a larger deviation
of from equilibrium. The maximum deviation of from
equilibrium ( = max 0) is plotted in Fig. 8(b) as
function of damping for 0Hz= 40 mT. As expected, a
lower damping leads to a larger deviation .
In this latter section, the DW was set at rest close to
the barrier and, therefore, the initial DW velocity is zero.
Nevertheless, one might wonder what happens when the
DW reaches the barrier with a nite velocity. We simu-
lated this case by placing the DW at an initial distance
d1= 200 nm from the barrier. The depinning is further
reduced in this case (see Supplementary Material32for
more details). However, in the static simulations, the de-
pinning eld remains constant, independently from the
velocity at which the DW reaches the barrier, meaning
that the reduction of Hdepis again related to the DW
precession. When the DW starts from d1it reaches the
barrier precessing, thus with a higher deviation from its
equilibrium angle, leading to a higher e ect of the inter-
nal eld.
C. Di erent DMI and pinning barriers
Finally, by using the 1D model it is possible to ex-
plore the dependence of Hdepon the pinning potential
amplitudeHs(related to the disorder strength) and on
the DMI constant D. The depinning eld as function of
damping for di erent values of Hsis plotted in Fig. 9(a).
The reduction of Hdepis enhanced for larger values of
Hs(strong disorder). This is consistent with our expla-
nation, since for strong disorder we need to apply larger
elds that lead to larger oscillations of .
Fig. 9(b) represents the dynamical Hdepas function of7
●●●●●●●●●●●●●●●●●●●●●●●●
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Max Displacement
Eq. Position(a)(b)(c)(d)
(e)(f)
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▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼●α=����○α=���α=������������������������������������
�(��)ϕ(°)
FIG. 7. (a) Pinning eld obtained from Eq. (7) as function of DW position. DW position internal angle as function of
time for di erent damping parameter and 0Hz= 55 mT. (c) DW position qas function of time for di erent damping and
0Hz= 55 mT. (d) Equilibrium position as function of applied eld for di erent damping. (e) Maximum DW displacement as
function of the applied eld for di erent damping. (f) DW coordinates fq; gfor0Hz= 55 mT and di erent damping. (g)
DW coordinatesfq; ginside the energy landscape: =DW(q; ) 20MsHzq.
������������������������������
�������αδϕ(°)
FIG. 8. Maximum deviation of from its equilibrium posi-
tion as function of damping.
damping for 0Hs= 120 mT and di erent DMI con-
stants (expressed in term of the critical DMI constant
Dc= 4pAK0== 3:9 mJ=m2)39. In this case, the reduc-
tion ofHdepis enhanced for low DMI, until D= 0:05Dc,
but a negligible reduction is observed for D= 0. This
non-monotonic behaviour can be explained by looking at
the dependence of andHinton the DMI constant.
Fig. 10(a) shows the maximum
uctuation as func-
tion of DMI for 0Hz= 30 mT. increases for low
DMI and it has a maximum at HDMI =HK, which
in our case corresponds to D= 0:014Dc. The increase
of for small values of Dis due to the smaller restor-
ing torque in Eq. (9). This holds until HDMI =HK,
where shape anisotropy and DMI are comparable and
they both a ect the DW equilibrium con guration. As a
consequence, the reduction of Hdepis enhanced by de-creasingDuntilD0:014Dc, while it is reduced if
0< D < 0:014Dc. Another contribution is given by
the amplitude of the internal eld, Hint. Fig. 10(b) de-
picts0Hintas function of andD. The maximum
, obtained at 0Hz= 30 mT, is additionally marked
in the plot. The internal eld decreases with the DMI
but this reduction is compensated by an increase in ,
which leads to an overall increase of 0Hint, as discussed
in the previous part. However, at very low DMI, the in-
ternal eld is dominated by shape anisotropy and, inde-
pendently on the DW angle displacement, it is too small
to have an e ect on the depinning mechanism. Note,
however, that the amplitude of Hintshould be compared
with the amplitude of the pinning barrier Hs. Fig. 9(b)
is calculated with 0Hs= 120 mT and the internal eld,
given by shape anisotropy ( HK=215 mT), has indeed
a negligible e ect. However, larger e ects are observed,
in the caseD= 0, for smaller Hs, with reduction of Hdep
up toHd=Hs0:6, as shown in Fig. 9(c), which is calcu-
lated with0Hs= 30 mT. In other words, the reduction
of the depinning eld depends on the ratio between the
pinning barrier and the internal DW eld.
Finally, it is interesting to see what happens for
weaker disorder and di erent DMI in the system with
grains. Fig. 11 shows the dynamical Hdep, for di erent
pinning potential and di erent DMI, obtained in the
granular system. The results are in good agreement with
what predicted by the 1D model for di erent disorder
strengths. However, we observe a smaller dependence
on the DMI parameter. This is due to two reasons:8
●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ●
■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
◆◆◆◆◆◆◆◆◆◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆
▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲
●μ���=�� ��
■μ���=�� ��
◆μ���=�� ��
▲μ���=��� ��
������������������������������������
��������/��(�)
●●●●●●●●●●●●●●●●●●●●
■■■■■■■■■■■■■■■■■■■■
◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆
▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼
○○○○○○○○○○○○○○○○○○○○
□□□□□□□□□□□□□□□□□□□□
●�=���� � �■�=��� � �
◆�=��� � �▲�=��� � �
▼�=��� � �○�=��� � � □�=���
���������������������������������������
��������/��(�) ��=��� ��
●●●●●●●●●●●●●●●●●●●●
■■■■■■■■■■■■■■■■■■■■
◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆
▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼
○○○○○○○○○○○○○○○○○○○○
□□□□□□□□□□□□□□
□□□□□□
●�=���� � �■�=���� � �
◆�=���� � �▲�=���� � �
▼�=���� � �○�=���� � � □�=���
���������������������������������������
��������/��(�) ��=�� ��
FIG. 9. (a) Dynamical Hdepas function of damping for di er-
entHs(disorder strength). (b) Dynamical Hdepas function
of damping for di erent DMI constant and 0Hs= 120 mT.
(c) Dynamical Hdepas function of damping for di erent DMI
constant and 0Hs= 30 mT.
(1) in the system with grains the static pinning barrier
is0Hs= 87 mT and the dependence of the depinning
eld with DMI is smaller for smaller barriers, as shown
in Fig. 9(c). (2) The DW motion in the granular
system presents the formation of VBL which might also
contribute to the reduction of the depinning eld. The
mechanism is the same: a VBL is a non-equilibrium
con guration for the DW (as a deviation of from
equilibrium) that generates additional torques on the
DW, which contribute to the DW depinning.
����� ���� ��� �������������
�/��δϕ(°)(�)
��=��
����(��)
� ��� ��� ��� ���
������������������������������
�/��δϕ(°)(�)FIG. 10. (a) Max DW angle
uctuation = max eq
as function of DMI for 0Hz= 30 mT. (b) Internal DW
eld0Hintas function of DMI and . The green points
correspond the max
uctuation plotted in (a). Note that the
scale is logarithmic in (a).
●●●●● ● ●
□□□□□ □
●μ���=�� ��
□μ���=�� ����������������/��(�)
●●●●● ● ●
◇◇◇◇◇ ◇
○○○○○ ○
●�=��� ��/��~���� �
◇�=��� ��/��~���� �
○�=�
������������������������������
��������/��(�)
FIG. 11. (a) Dynamical Hdepas function of damping for dif-
ferentHs(disorder strength). (b) Dynamical Hdepas function
of damping for di erent DMI constants.9
IV. CONCLUSIONS
To summarize, we have analysed the DW depinning
eld in a PMA sample with DMI and we found that Hdep
decreases with the damping parameter with reductions
up to 50%. This decrease is related to the DW inter-
nal dynamics and the nite size of the barrier: due to
DW precession, the DW internal angle ( ) deviates from
equilibrium and triggers the internal DW eld (DMI and
shape anisotropy) which tries to restore its original value.
At the same time, the internal eld pushes the DW above
its equilibrium position within the energy barrier. This
mechanism leads to DW oscillations and, if the ampli-
tude of the oscillations is higher than the barrier size,
the DW gets depinned for a lower eld. Deviations of
from equilibrium and DW oscillations are both damping
dependent and they are enhanced at low damping.
In the system with grains the mechanism is the same
but deviations from the internal DW equilibrium include
the formation of VBL with more complex dynamics.
The e ect is enhanced for low DMI (providing thatHDMI> H K) and for stronger disorder since we need
to apply larger external elds, which lead to larger DW
oscillations. These results are relevant both from a tech-
nological and theoretical point of view, since they rstly
suggest that a low damping parameter can lead to a
lowerHdep. Furthermore, they show that micromagnetic
calculations of the depinning eld, neglecting the DW
precessional dynamics can provide only an upper limit
forHdep, which could actually be lower due to the DW
precessional dynamics.
V. ACKNOWLEDGEMENT
S.M. would like to thank K. Shahbazi, C.H. Mar-
rows and J. Leliaert for helpful discussions. This work
was supported by Project WALL, FP7- PEOPLE-2013-
ITN 608031 from the European Commission, Project No.
MAT2014-52477-C5-4-P from the Spanish government,
and Project No. SA282U14 and SA090U16 from the
Junta de Castilla y Leon.
Corresponding author: simone.moretti@usal.es
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32See Appendices.
33This de nition is preferred over the average of Hi
depsince
it is more independent on the sample size. In fact, by in-creasing the sample dimension along the xdirection, we
increase the probability of nding the highest possible hj
in the single realization and the average of Hi
depwill tend
to the maximum.
34This is solved by the Relax solver of MuMax with the as-
sumption =(1 + 2) = 1.
35The DW energy is calculated as the energy of the system
with the DW minus the energy of the system without the
DW (uniform state). The pro le is obtained by moving the
DW with an external applied eld and then subtracting the
Zeeman energy.
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38The same results are obtained with a Gaussian barrier,
meaning that the key point is the nite size of the barrier
rather than its shape.
39ForD >D c, DW have negative energies and the systems
spontaneously breaks into non-uniform spin textures.
Appendix A: Maximum torque and equilibrium state
In this section we show in more detail how the maximum torque represents an indicator of the equilibrium state.
Maximu torque is de ned as
max
0= Maxf 1
1 + 2miHe ;i
1 + 2mi(miHe ;i)g=1
0Maxdmi
dt
; (A1)
over all cells with label i=f1;:::;N =NxNyg. MuMax3.9.329can provide this output automatically if selected.
We perform the same simulations as indicated in the main text, without any stopping condition, but simply running
fort= 20 ns. Fig. 12(a) shows the average mzcomponent for = 0:2 andBz= 10 mT, while Fig. 12(b) depicts the
corresponding maximum torque. We can see that, once the system has reached equilibrium, the maximum torque has
dropped to a minimum value. The same results is obtained for di erent damping but the nal maximum torque is
di erent. Numerically this value is never zero since it is limited by the code numerical precision and by the system
parameters, in particular by damping.
Fig. 12(c) represents the maximum torque as function of applied eld for di erent damping. The maximum torque
is clearly independent on the applied eld but depends on the damping value. Finally, Fig. 12(d) shows the max
torque as function of damping. The maximum torque decreases with damping and it saturates for 0:5 since we
have reached the minimum numerical precision of the code29. For higher damping the maximum torque oscillates
around this minimum sensibility value, as shown in the inset of Fig. 12(d). The value obtained with these preliminary
simulations is used to set a threshold ( ) for the depinning eld simulations in order to identify when the system has
reached an equilibrium. Furthermore, additional tests were performed, without putting any max torque condition,
but simply running the simulations for a longer time ( t= 80;160 ns) and calculating the depinning eld in order
to ensure that the results obtained with these two method were consistent, i.e., that we have actually reached an
equilibrium state with the maximum torque condition.11
������������������-���-���-���
����(��)<��>⨯��-�α=���(�)
���������������������������
����(��)��� ������/γ �(��)α=���(�)
○ ○ ○ ○ ○ ○ ○ ○
□ □ □ □ □ □ □ □
◇◇◇◇◇◇◇◇
○α=����□α=���◇α=���
� �� �� �� ��������������������
μ���(��)��� ������/γ �(��)(�)
●
●●
��������������������-�������������������
�������α��� ������/γ �(��)(�)
���������������������
����(��)τ���/γ�(��)α=���
FIG. 12. (a) average mzas function of time. (b) Max torque/
0( max) as function of time. maxrapidly decreases when the
system is at equilibrium. (c) Max torque as function of applied eld for di erent damping. (d) Max torque at equilibrium as
function of damping. The inset shows the max torque as function of time for = 0:5.
Appendix B: 1D energy barrier
As commented in the main text, the pinning eld implemented in the 1D model simulations is obtained by using the
shape of the DW energy pro le derivative @(x)=@x(beingxthe DW position) and the amplitude of the depinning
eld obtained in the full micromagnetic simulations Hsfor the single barrier case. Namely
Hdep=Hs@(x)
@x
N; (B1)
where we recall that Nstands for the normalized value. This choice might sound unusual and needs to be justi ed.
In fact, having the DW energy pro le, the depinning eld could be simply calculated as20
Hdep=1
20Ms@(x)
@x: (B2)
This expression is derived by imposing that the derivative of the total DW energy E(x) = 20MsHzx+(x) (Zeeman
+ internal energy) must be always negative. However, in our case also Ms(x) depends on the DW position and the
results obtained with Eq. B2 is di erent from the depinning eld measured in the static single barrier simulations.
For this reason we use Eq. B1 which keep the correct barrier shape and has the measured static value.
Finally, we recall that equivalent results are obtained by using a simple Gaussian shape for the pinning eld, meaning
that the key point is the localized shape of the barrier, rather than its exact form.
Appendix C: Dynamical depinning for a moving Domain Wall
In this section we show the results for the dynamical depinning eld when the DW is placed at an initial distance of
d1= 200 nm from the barrier. In this way the DW hits the pinning with an initial velocity. The d0case corresponds to
the DW at rest relaxed just before the barrier and extensively analysed in the main text. Also for this con guration
we performed static and dynamic simulations, neglecting or including the DW precessional dynamics respectively.
The depinning eld for the d1case is further reduces at small damping, reaching Hd=Hs0:08 (Hd= 9 mT and
Hs= 120 mT) at = 0:02. Nevertheless, the depinning eld remains constant in the static simulations independently
on the velocity at which the DW hits the barrier. This suggests that, rather than related to the DW velocity, the
reduction is again related to the DW precession. When the DW starts from d1it reaches the barrier precessing, thus
with a higher displacement from its equilibrium angle, leading to a higher e ect of the internal eld.12
●●●●●●
○○○○○○ ◆◆◆◆◆ ◆ □□□□□ □
●��(�������)
○��(�������)◆��(������)
□��(������)
��� ��� ��� ��� ������������������������
��������/��
FIG. 13. Dynamical depinning eld as function of damping for static and dynamic simulations for the d0andd1cases. |
1409.2340v1.Self_similar_solutions_of_the_one_dimensional_Landau_Lifshitz_Gilbert_equation.pdf | arXiv:1409.2340v1 [math.AP] 8 Sep 2014Self-similar solutions of the one-dimensional
Landau–Lifshitz–Gilbert equation
Susana Gutiérrez1and André de Laire2
Abstract
We consider the one-dimensional Landau–Lifshitz–Gilbert (LLG) equation, a model des-
cribing the dynamics for the spin in ferromagnetic material s. Our main aim is the analytical
study of the bi-parametric family of self-similar solution s of this model. In the presence of
damping, our construction provides a family of global solut ions of the LLG equation which
are associated to a discontinuous initial data of infinite (t otal) energy, and which are smooth
and have finite energy for all positive times. Special emphas is will be given to the behaviour
of this family of solutions with respect to the Gilbert dampi ng parameter.
We would like to emphasize that our analysis also includes th e study of self-similar so-
lutions of the Schrödinger map and the heat flow for harmonic m aps into the 2-sphere as
special cases. In particular, the results presented here re cover some of the previously known
results in the setting of the 1d-Schrödinger map equation.
Keywords and phrases: Landau–Lifshitz–Gilbert equation, Landau–Lifshitz equa tion, ferro-
magnetic spin chain, Schrödinger maps, heat-flow for harmon ic maps, self-similar solutions,
asymptotics.
Contents
1 Introduction and statement of results 2
2 Self-similar solutions of the LLG equation 10
3 Integration of the Serret–Frenet system 12
3.1 Reduction to the study of a second order ODE . . . . . . . . . . . . . . . . . . . 12
3.2 The second-order equation. Asymptotics . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 The second-order equation. Dependence on the parameter s . . . . . . . . . . . . 28
3.3.1 Dependence on α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.2 Dependence on c0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Proof of the main results 34
5 Some numerical results 37
6 Appendix 41
1School of Mathematics, University of Birmingham, Edgbasto n, Birmingham, B15 2TT, United Kingdom.
E-mail:s.gutierrez@bham.ac.uk
2Laboratoire Paul Painlevé, Université Lille 1, 59655 Ville neuve d’Ascq Cedex, France. E-mail:
andre.de-laire@math.univ-lille1.fr
11 Introduction and statement of results
In this work we consider the one-dimensional Landau–Lifshi tz–Gilbert equation (LLG)
∂t/vectorm =β/vectorm×/vector mss−α/vectorm×(/vectorm×/vectormss), s∈R, t>0, (LLG)
where/vectorm = (m 1,m2,m3) :R×(0,∞)−→S2is the spin vector, β≥0,α≥0,×denotes the
usual cross-product in R3, andS2is the unit sphere in R3.
Here we have not included the effects of anisotropy or an exter nal magnetic field. The first term
on the right in (LLG) represents the exchange interaction, w hile the second one corresponds to the
Gilbert damping term and may be considered as a dissipative t erm in the equation of motion. The
parameters β≥0andα≥0are the so-called exchange constant and Gilbert damping coe fficient,
and take into account the exchange of energy in the system and the effect of damping on the
spin chain respectively. Note that, by considering the time -scaling/vectorm(s,t)→/vectorm(s,(α2+β2)1/2t),
in what follows we will assume w.l.o.g. that
α, β∈[0,1] andα2+β2= 1. (1.1)
The Landau–Lifshitz–Gilbert equation was first derived on p henomenological grounds by L. Lan-
dau and E. Lifshitz to describe the dynamics for the magnetiz ation or spin /vectorm(s,t)in ferromag-
netic materials [24, 11]. The nonlinear evolution equation (LLG) is related to several physical
and mathematical problems and it has been seen to be a physica lly relevant model for several
magnetic materials [19, 20]. In the setting of the LLG equati on, of particular importance is to
consider the effect of dissipation on the spin [27, 7, 6].
The Landau–Lifshitz family of equations includes as specia l cases the well-known heat-flow
for harmonic maps and the Schrödinger map equation onto the 2-sphere. Precisely, when β= 0
(and therefore α= 1) the LLG equation reduces to the one-dimensional heat-flow equation for
harmonic maps
∂t/vectorm =−/vectorm×(/vectorm×/vectormss) =/vectormss+|/vectorms|2/vectorm (HFHM)
(notice that |/vectorm|2= 1, and in particular /vectorm·/vectormss=−|/vectorms|2). The opposite limiting case of the
LLG equation (that is α= 0, i.e. no dissipation/damping and therefore β= 1) corresponds to
theSchrödinger map equation onto the sphere
∂t/vectorm =/vectorm×/vectormss. (SM)
Both special cases have been objects of intense research and w e refer the interested reader to
[21, 14, 25, 13] for surveys.
Of special relevance is the connection of the LLG equation wi th certain non-linear Schrödinger
equations. This connection is established as follows: Let u s suppose that /vectormis the tangent vector
of a curve in R3, that is/vectorm =/vectorXs, for some curve /vectorX(s,t)∈R3parametrized by the arc-length. It
can be shown [7] that if /vectormevolves under (LLG) and we define the so-called filament funct ionu
associated to /vectorX(s,t)by
u(s,t) = c(s,t)ei/integraltexts
0τ(σ,t)dσ, (1.2)
in terms of the curvature cand torsion τassociated to the curve, then usolves the following
non-local non-linear Schrödinger equation with damping
iut+(β−iα)uss+u
2/parenleftbigg
β|u|2+2α/integraldisplays
0Im(¯uus)−A(t)/parenrightbigg
= 0, (1.3)
whereA(t)∈Ris a time-dependent function defined in terms of the curvatur e and torsion
and their derivatives at the point s= 0. The transformation (1.2) was first considered in the
2undamped case by Hasimoto in [18]. Notice that if α= 0, equation (1.3) can be transformed
into the well-known completely integrable cubic Schröding er equation.
The main purpose of this paper is the analytical study of self -similar solutions of the LLG
equation of the form
/vectorm(s,t) =/vector m/parenleftbiggs√
t/parenrightbigg
, (1.4)
for some profile /vector m:R→S2, with emphasis on the behaviour of these solutions with resp ect to
the Gilbert damping parameter α∈[0,1].
Forα= 0, self-similar solutions have generated considerable inte rest [22, 21, 4, 15, 9]. We are
not aware of any other study of such solutions for α >0in the one dimensional case (see [10]
for a study of self-similar solutions of the harmonic map flow in higher dimensions). However,
Lipniacki [26] has studied self-similar solutions for a rel ated model with nonconstant arc-length.
On the other hand, little is known analytically about the effe ct of damping on the evolution
of a one-dimensional spin chain. In particular, Lakshmanan and Daniel obtained an explicit
solitary wave solution in [7, 6] and demonstrated the dampin g of the solution in the presence
of dissipation in the system. In this setting, we would like t o understand how the dynamics of
self-similar solutions to this model is affected by the intro duction of damping in the equations
governing the motion of these curves.
As will be shown in Section 2 self-similar solutions of (LLG) of the type (1.4) constitute a
bi-parametric family of solutions {/vector mc0,α}c0,αgiven by
/vectormc0,α(s,t) =/vector mc0,α/parenleftbiggs√
t/parenrightbigg
, c 0>0, α∈[0,1], (1.5)
where/vector mc0,αis the solution of the Serret–Frenet equations
/vector m′=c/vector n,
/vector n′=−c/vector m+τ/vectorb,
/vectorb′=−τ/vector n,(1.6)
with curvature and torsion given respectively by
cc0,α(s) =c0e−αs2
4, τc0,α(s) =βs
2, (1.7)
and initial conditions
/vector mc0,α(0) = (1,0,0), /vector nc0,α(0) = (0,1,0),/vectorbc0,α(0) = (0,0,1). (1.8)
The first result of this paper is the following:
Theorem 1.1. Letα∈[0,1],c0>01and/vector mc0,αbe the solution of the Serret–Frenet system
(1.6)with curvature and torsion given by (1.7)and initial conditions (1.8). Define/vectormc0,α(s,t)by
/vectormc0,α(s,t) =/vector mc0,α/parenleftbiggs√
t/parenrightbigg
, t> 0.
Then,
1The case c0= 0corresponds to the constant solution for (LLG), that is
/vectormc0,α(s,t) =/vector m/parenleftbiggs√
t/parenrightbigg
= (1,0,0),∀α∈[0,1].
3(i) The function /vectormc0,α(·,t)is a regular C∞(R;S2)-solution of (LLG) fort>0.
(ii) There exist unitary vectors /vectorA±
c0,α= (A±
j,c0,α)3
j=1∈S2such that the following pointwise
convergence holds when tgoes to zero:
lim
t→0+/vectormc0,α(s,t) =
/vectorA+
c0,α,ifs>0,
/vectorA−
c0,α,ifs<0,(1.9)
where/vectorA−
c0,α= (A+
1,c0,α,−A+
2,c0,α,−A+
3,c0,α).
(iii) Moreover, there exists a constant C(c0,α,p)such that for all t>0
/bardbl/vectormc0,α(·,t)−/vectorA+
c0,αχ(0,∞)(·)−/vectorA−
c0,αχ(−∞,0)(·)/bardblLp(R)≤C(c0,α,p)t1
2p, (1.10)
for allp∈(1,∞). In addition, if α>0,(1.10) also holds for p= 1. Here,χEdenotes the
characteristic function of a set E.
The graphics in Figure 1 depict the profile /vector mc0,α(s)for fixedc0= 0.8and the values of
α= 0.01,α= 0.2, andα= 0.4. In particular it can be observed how the convergence of /vector mc0,α
to/vectorA±
c0,αis accelerated by the diffusion α.
m1m2m3
(a)α= 0.01
m1m2m3
(b)α= 0.2
m1m2m3
(c)α= 0.4
Figure 1: The profile /vector mc0,αforc0= 0.8and different values of α.
Notice that the initial condition
/vectormc0,α(s,0) =/vectorA+
c0,αχ(0,∞)(s)+/vectorA−
c0,αχ(−∞,0)(s), (1.11)
has a jump singularity at the point s= 0whenever the vectors /vectorA+
c0,αand/vectorA−
c0,αsatisfy
/vectorA+
c0,α/ne}ationslash=/vectorA−
c0,α.
In this situation (and we will be able to prove analytically t his is the case at least for certain ranges
of the parameters αandc0, see Proposition 1.5 below), Theorem 1.1 provides a bi-para metric
family of global smooth solutions of (LLG) associated to a di scontinuous singular initial data
(jump-singularity).
4As has been already mentioned, in the absence of damping ( α= 0), singular self-similar
solutions of the Schrödinger map equation were previously o btained in [15], [22] and [4]. In this
framework, Theorem 1.1 establishes the persistence of a jum p singularity for self-similar solutions
in the presence of dissipation.
Some further remarks on the results stated in Theorem 1.1 are in order. Firstly, from the
self-similar nature of the solutions /vectormc0,α(s,t)and the Serret–Frenet equations (1.6), it follows
that the curvature and torsion associated to these solution s are of the self-similar form and given
by
cc0,α(s,t) =c0√
te−αs2
4t andτc0,α(s,t) =βs
2√
t. (1.12)
As a consequence, the total energy E(t)of the spin /vectormc0,α(s,t)found in Theorem 1.1 is expressed
as
E(t) =1
2/integraldisplay∞
−∞|/vectorms(s,t)|2ds=1
2/integraldisplay∞
−∞c2
c0,α(s,t)ds
=1
2/integraldisplay∞
−∞/parenleftbiggc0√
te−αs2
4t/parenrightbigg2
ds=c2
0/radicalbiggπ
αt, α> 0, t>0. (1.13)
It is evident from (1.13) that the total energy of the spin cha in at the initial time t= 0is infinite,
while the total energy of the spin becomes finite for all posit ive times, showing the dissipation
of energy in the system in the presence of damping.
Secondly, it is also important to remark that in the setting o f Schrödinger equations, for fixed
α∈[0,1]andc0>0, the solution /vectormc0,α(s,t)of (LLG) established in Theorem 1.1 is associated
through the Hasimoto transformation (1.2) to the filament fu nction
uc0,α(s,t) =c0√
te(−α+iβ)s2
4t, (1.14)
which solves
iut+(β−iα)uss+u
2/parenleftbigg
β|u|2+2α/integraldisplays
0Im(¯uus)−A(t)/parenrightbigg
= 0,withA(t) =βc2
0
t(1.15)
and is such that at initial time t= 0
uc0,α(s,0) = 2c0/radicalbig
π(α+iβ)δ0.
Hereδ0denotes the delta distribution at the point s= 0and√zdenotes the square root of a
complex number zsuch that Im(√z)>0.
Notice that the solution uc0,α(s,t)is very rough at initial time, and in particular uc0,α(s,0)
does not belong to the Sobolev class Hsfor anys≥0. Therefore, the standard arguments (that
is a Picard iteration scheme based on Strichartz estimates a nd Sobolev-Bourgain spaces) cannot
be applied at least not in a straightforward way to study the l ocal well-posedness of the initial
value problem for the Schrödinger equations (1.15). The exi stence of solutions of the Scrödinger
equations (1.15) associated to an initial data proportiona l to a Dirac delta opens the question
of developing a well-posedness theory for Schrödinger equa tions of the type considered here to
include initial data of infinite energy. This question was ad dressed by A. Vargas and L. Vega
in [29] and A. Grünrock in [12] in the case α= 0and whenA(t) = 0 (see also [2] for a related
problem), but we are not aware of any results in this setting w henα >0(see [14] for related
well-posedness results in the case α >0for initial data in Sobolev spaces of positive index).
Notice that when α>0, the solution (1.14) has infinite energy at the initial time, however the
5energy becomes finite for any t>0. Moreover, as a consequence of the exponential decay in the
space variable when α>0,uc0,α(t)∈Hm(R), for allt>0andm∈N. Hence these solutions do
not fit into the usual functional framework for solutions of t he Schrödinger equations (1.15).
As already mentioned, one of the main goals of this paper is to study both the qualitative and
quantitative effect of the damping parameter αand the parameter c0on the dynamical behaviour
of the family {/vectormc0,α}c0,αof self-similar solutions of (LLG) found in Theorem 1.1. Pre cisely, in an
attempt to fully understand the regularization of the solut ion at positive times close to the initial
timet= 0, and to understand how the presence of damping affects the dyn amical behaviour of
these self-similar solutions, we aim to give answers to the f ollowing questions:
Q1: Can we obtain a more precise behaviour of the solutions /vector mc0,α(s,t)at positive times tclose
to zero?
Q2: Can we understand the limiting vectors /vectorA±
c0,αin terms of the parameters c0andα?
In order to address our first question, we observe that, due to the self-similar nature of these
solutions (see (1.5)), the behaviour of the family of soluti ons/vectormc0,α(s,t)at positive times close to
the initial time t= 0is directly related to the study of the asymptotics of the ass ociated profile
/vector mc0,α(s)for large values of s. In addition, the symmetries of /vector mc0,α(s)(see Theorem 1.2 below)
allow to reduce ourselves to obtain the behaviour of the profi le/vector mc0,α(s)for large positive values
of the space variable. The precise asymptotics of the profile is given in the following theorem.
Theorem 1.2 (Asymptotics) .Letα∈[0,1],c0>0and{/vector mc0,α,/vector nc0,α,/vectorbc0,α}be the solution of
the Serret–Frenet system (1.6)with curvature and torsion given by (1.7)and initial conditions
(1.8). Then,
(i) (Symmetries). The components of /vector mc0,α(s),/vector nc0,α(s)and/vectorbc0,α(s)satisfy respectively that
•m1,c0,α(s)is an even function, and mj,c0,α(s)is an odd function for j∈ {2,3}.
•n1,c0,α(s)andb1,c0,α(s)are odd functions, while nj,c0,α(s)andbj,c0,α(s)are even func-
tions forj∈ {2,3}.
(ii) (Asymptotics). There exist an unit vector /vectorA+
c0,α∈S2and/vectorB+
c0,α∈R3such that the following
asymptotics hold for all s≥s0= 4/radicalbig
8+c2
0:
/vector mc0,α(s) =/vectorA+
c0,α−2c0
s/vectorB+
c0,αe−αs2/4(αsin(/vectorφ(s))+βcos(/vectorφ(s)))
−2c2
0
s2/vectorA+
c0,αe−αs2/2+O/parenleftBigg
e−αs2/4
s3/parenrightBigg
, (1.16)
/vector nc0,α(s) =/vectorB+
c0,αsin(/vectorφ(s))+2c0
s/vectorA+
c0,ααe−αs2/4+O/parenleftBigg
e−αs2/4
s2/parenrightBigg
, (1.17)
/vectorbc0,α(s) =/vectorB+
c0,αcos(/vectorφ(s))+2c0
s/vectorA+
c0,αβe−αs2/4+O/parenleftBigg
e−αs2/4
s2/parenrightBigg
. (1.18)
Here,sin(/vectorφ)andcos(/vectorφ)are understood acting on each of the components of /vectorφ= (φ1,φ2,φ3),
with
φj(s) =aj+β/integraldisplays2/4
s2
0/4/radicalbigg
1+c2
0e−2ασ
σdσ, j∈ {1,2,3}, (1.19)
6for some constants a1,a2,a3∈[0,2π), and the vector /vectorB+
c0,αis given in terms of /vectorA+
c0,α=
(A+
j,c0,α)3
j=1by
/vectorB+
c0,α= ((1−(A+
1,c0,α)2)1/2,(1−(A+
2,c0,α)2)1/2,(1−(A+
3,c0,α)2)1/2).
As we will see in Section 2, the convergence and rate of conver gence of the solutions /vectormc0,α(s,t)
of the LLG equation established in parts (ii)and(iii)of Theorem 1.1 will be obtained as a con-
sequence of the more refined asymptotic analysis of the assoc iated profile given in Theorem 1.2.
With regard to the asymptotics of the profile established in p art(ii)of Theorem 1.2, it is
important to mention the following:
(a) The errors in the asymptotics in Theorem 1.2- (ii)depend only on c0. In other words,
the bounds for the errors terms are independent of α∈[0,1]. More precisely, we use the
notationO(f(s))to denote a function for which exists a constant C(c0)>0depending on
c0, but independent on α, such that
|O(f(s))| ≤C(c0)|f(s)|,for alls≥s0. (1.20)
(b) The terms /vectorA+
c0,α,/vectorB+
c0,α,B+
jsin(aj)andB+
jcos(aj),j∈ {1,2,3}, and the error terms in
Theorem 1.2- (ii)depend continuously on α∈[0,1](see Subsection 3.3 and Corollary 3.14).
Therefore, the asymptotics (1.16)–(1.18) show how the profi le/vector mc0,αconverges to /vector mc0,0as
α→0+and to/vector mc0,1asα→1−. In particular, we recover the asymptotics for /vector mc0,0given
in [15].
(c) We also remark that using the Serret–Frenet formulae and the asymptotics in Theorem 1.2-
(ii), it is straightforward to obtain the asymptotics for the der ivatives of/vectormc0,α(s,t).
(d) Whenα= 0and for fixed j∈ {1,2,3}, we can write φjin (1.19) as
φj(s) =aj+s2
4+c2
0ln(s)+C(c0)+O/parenleftbigg1
s2/parenrightbigg
,
and we recover the logarithmic contribution in the oscillat ion previously found in [15].
Moreover, in this case the asymptotics in part (ii)represents an improvement of the one
established in Theorem 1 in [15].
Whenα>0,φjbehaves like
φj(s) =aj+βs2
4+C(α,c0)+O/parenleftBigg
e−αs2/2
αs2/parenrightBigg
, (1.21)
and there is no logarithmic correction in the oscillations i n the presence of damping.
Consequently, the phase function /vectorφdefined in (1.19) captures the different nature of the
oscillatory character of the solutions in both the absence a nd the presence of damping in
the system of equations.
(e) Whenα= 1, there exists an explicit formula for /vector mc0,1,/vector nc0,1and/vectorbc0,1, and in particular
we have explicit expressions for the vectors /vectorA±
c0,1in terms of the parameter c0>0in the
asymptotics given in part (ii). See Appendix.
7(f) At first glance, one might think that the term −2c2
0/vectorA+
c0,αe−αs2/2/s2in (1.16) could be
included in the error term O(e−αs2/4/s3). However, we cannot do this because
e−αs2/2
s2>e−αs2/4
s3, for all2≤s≤/parenleftbigg2
3α/parenrightbigg1/2
, α∈(0,1/8], (1.22)
and in our notation the big- Omust be independent of α. (The exact interval where the
inequality in (1.22) holds can be determined using the so-ca lled Lambert Wfunction.)
(g) Let/vectorB+
c0,α,sin= (Bjsin(aj))3
j=1,/vectorB+
c0,α,cos= (Bjcos(aj))3
j=1. Then the orthogonality of
/vector mc0,α,/vector nc0,αand/vectorbc0,αtogether with the asymptotics (1.16)–(1.18) yield
/vectorA+
c0,α·/vectorB+
c0,α,sin=/vectorA+
c0,α·/vectorB+
c0,α,cos=/vectorB+
c0,α,sin·/vectorB+
c0,α,cos= 0,
which gives relations between the phases.
(h) Finally, the amplitude of the leading order term control ling the wave-like behaviour of the
solution/vector mc0,α(s)around/vectorA±
c0,αfor values of ssufficiently large is of the order c0e−αs2/4/s,
from which one observes how the convergence of the solution t o its limiting values /vectorA±
c0,αis
accelerated in the presence of damping in the system. See Fig ure 1.
We conclude the introduction by stating the results answeri ng the second of our questions. Pre-
cisely, Theorems 1.3 and 1.4 below establish the dependence of the vectors /vectorA±
c0,αin Theorem 1.1
with respect to the parameters αandc0. Theorem 1.3 provides the behaviour of the limiting
vector/vectorA+
c0,αfor a fixed value of α∈(0,1)and “small” values of c0>0, while Theorem 1.4 states
the behaviour of /vectorA+
c0,αfor fixedc0>0andαclose to the limiting values α= 0andα= 1. Recall
that/vectorA−
c0,αis expressed in terms of the coordinates of /vectorA+
c0,αas
/vectorA−
c0,α= (A+
1,c0,α,−A+
2,c0,α,−A+
3,c0,α) (1.23)
(see part (ii)of Theorem 1.1).
Theorem 1.3. Letα∈[0,1],c0>0, and/vectorA+
c0,α= (A+
j,c0,α)3
j=1be the unit vector given in
Theorem 1.2. Then /vectorA+
c0,αis a continuous function of c0>0. Moreover, if α∈(0,1]the
following inequalities hold true:
|A+
1,c0,α−1| ≤c2
0π
α/parenleftbigg
1+c2
0π
8α/parenrightbigg
, (1.24)
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA+
2,c0,α−c0/radicalbig
π(1+α)√
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤c2
0π
4+c2
0π
α√
2/parenleftBigg
1+c2
0π
8+c0/radicalbig
π(1+α)
2√
2/parenrightBigg
+/parenleftbiggc2
0π
2√
2α/parenrightbigg2
,(1.25)
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA+
3,c0,α−c0/radicalbig
π(1−α)√
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤c2
0π
4+c2
0π
α√
2/parenleftBigg
1+c2
0π
8+c0/radicalbig
π(1−α)
2√
2/parenrightBigg
+/parenleftbiggc2
0π
2√
2α/parenrightbigg2
.(1.26)
The following result provides an approximation of the behav iour of/vectorA+
c0,αfor fixedc0>0and
values of the Gilbert parameter close to 0and1.
Theorem 1.4. Letc0>0,α∈[0,1]and/vectorA+
c0,αbe the unit vector given in Theorem 1.2. Then
/vectorA+
c0,αis a continuous function of αin[0,1], and the following inequalities hold true:
|/vectorA+
c0,α−/vectorA+
c0,0| ≤C(c0)√α|ln(α)|,for allα∈(0,1/2], (1.27)
|/vectorA+
c0,α−/vectorA+
c0,1| ≤C(c0)√
1−α,for allα∈[1/2,1]. (1.28)
Here,C(c0)is a positive constant depending on c0but otherwise independent of α.
8As a by-product of Theorems 1.3 and 1.4, we obtain the followi ng proposition which asserts
that the solutions /vectormc0,α(s,t)of the LLG equation found in Theorem 1.1 are indeed associate d
to a discontinuous initial data at least for certain ranges o fαandc0.
Proposition 1.5. With the same notation as in Theorems 1.1 and 1.2, the followi ng statements
hold:
(i) For fixed α∈(0,1)there exists c∗
0>0depending on αsuch that
/vectorA+
c0,α/ne}ationslash=/vectorA−
c0,α for allc0∈(0,c∗
0).
(ii) For fixed c0>0, there exists α∗
0>0small enough such that
/vectorA+
c0,α/ne}ationslash=/vectorA−
c0,α for allα∈(0,α∗
0).
(iii) For fixed 0<c0/ne}ationslash=k√πwithk∈N, there exists α∗
1>0with1−α∗
1>0small enough such
that
/vectorA+
c0,α/ne}ationslash=/vectorA−
c0,α for allα∈(α∗
1,1).
Remark 1.6. Based on the numerical results in Section 5, we conjecture th at/vectorA+
c0,α/ne}ationslash=/vectorA−
c0,αfor
allα∈[0,1)andc0>0.
We would like to point out that some of our results and their pr oofs combine and extend
several ideas previously introduced in [15] and [16]. The ap proach we use in the proof of the
main results in this paper is based on the integration of the S erret–Frenet system of equations
via a Riccati equation, which in turn can be reduced to the stu dy of a second order ordinary
differential equation given by
f′′(s)+s
2(α+iβ)f′(s)+c2
0
4e−αs2
2f(s) = 0 (1.29)
when the curvature and torsion are given by (1.7).
Unlike in the undamped case, in the presence of damping no exp licit solutions are known
for equation (1.29) and the term containing the exponential in the equation (1.29) makes it
difficult to use Fourier analysis methods to study analytical ly the behaviour of the solutions to
this equation. The fundamental step in the analysis of the be haviour of the solutions of (1.29)
consists in introducing new auxiliary variables z,handydefined by
z=|f|2, y= Re(¯ff′)andh= Im(¯ff′)
in terms of solutions fof (1.29), and studying the system of equations satisfied by t hese key
quantities. As we will see later on, these variables are the “ natural” ones in our problem, in the
sense that the components of the tangent, normal and binorma l vectors can be written in terms
of these quantities. It is important to emphasize that, in or der to obtain error bounds in the
asymptotic analysis independent of the damping parameter α(and hence recover the asymptotics
whenα= 0andα= 1as particular cases), it will be fundamental to exploit the c ancellations
due to the oscillatory character of z,yandh.
The outline of this paper is the following. Section 2 is devot ed to the construction of the family
of self-similar solutions {/vectormc0,α}c0,αof the LLG equation. In Section 3 we reduce the study of the
properties of this family of self-similar solutions to that of the properties of the solutions of the
complex second order complex ODE (1.29). This analysis is of independent interest. Section 4
contains the proofs of the main results of this paper as a cons equence of those established in
9Section 3. In Section 5 we give provide some numerical result s for/vectorA+
c0,α, as a function of α∈[0,1]
andc0>0, which give some inside for the scattering problem and justi fy Remark 1.6. Finally,
we have included the study of the self-similar solutions of t he LLG equation in the case α= 1
in Appendix.
Acknowledgements. S. Gutiérrez and A. de Laire were supported by the British proj ect
“Singular vortex dynamics and nonlinear Schrödinger equat ions” (EP/J01155X/1) funded by
EPSRC. S. Gutiérrez was also supported by the Spanish projec ts MTM2011-24054 and IT641-
13.
Both authors would like to thank L. Vega for many enlightening conversations and for his
continuous support.
2 Self-similar solutions of the LLG equation
First we derive what we will refer to as the geometric represe ntation of the LLG equation. To
this end, let us assume that /vectorm(s,t) =/vectorXs(s,t)for some curve /vectorX(s,t)inR3parametrized with
respect to the arc-length with curvature c(s,t)and torsion τ(s,t). Then, using the Serret–Frenet
system of equations (1.6), we have
/vectormss= cs/vectorn+c(−c/vectorn+τ/vectorb),
and thus we can rewrite (LLG) as
∂t/vectorm =β(cs/vectorb−cτ/vectorn)+α(cτ/vectorb+cs/vectorn), (2.1)
in terms of intrinsic quantities c,τand the Serret–Frenet trihedron {/vectorm,/vectorn,/vectorb}.
We are interested in self-similar solutions of (LLG) of the f orm
/vectorm(s,t) =/vector m/parenleftbiggs√
t/parenrightbigg
(2.2)
for some profile /vector m:R−→S2. First, notice that due to the self-similar nature of /vectorm(s,t)in (2.2),
from the Serret–Frenet equations (1.6) it follows that the u nitary normal and binormal vectors
and the associated curvature and torsion are self-similar a nd given by
/vectorn(s,t) =/vector n/parenleftbiggs√
t/parenrightbigg
,/vectorb(s,t) =/vectorb/parenleftbiggs√
t/parenrightbigg
, (2.3)
c(s,t) =1√
tc/parenleftbiggs√
t/parenrightbigg
andτ(s,t) =1√
tτ/parenleftbiggs√
t/parenrightbigg
. (2.4)
Assume that /vectorm(s,t)is a solution of the LLG equation, or equivalently of its geom etric version
(2.1). Then, from (2.2)–(2.4) it follows that the Serret–Fr enet trihedron {/vector m(·),/vector n(·),/vectorb(·)}solves
−s
2c/vector n=β(c′/vectorb−cτ/vector n)+α(cτ/vectorb+c′/vector n), (2.5)
As a consequence,
−s
2c=αc′−βcτ andβc′+αcτ= 0.
Thus, we obtain
c(s) =c0e−αs2
4andτ(s) =βs
2, (2.6)
10for some positive constant c0(recall that we are assuming w.l.o.g. that α2+β2= 1). Therefore,
in view of (2.4), the curvature and torsion associated to a se lf-similar solution of (LLG) of the
form (2.2) are given respectively by
c(s,t) =c0√
te−αs2
4tandτ(s,t) =βs
2t, c 0>0. (2.7)
Notice that given (c,τ)as above, for fixed time t>0one can solve the Serret–Frenet system of
equations to obtain the solution up to a rigid motion in the sp ace which in general may depend
ont. As a consequence, and in order to determine the dynamics of t he spin chain, we need
to find the time evolution of the trihedron {/vectorm(s,t),/vectorn(s,t),/vectorb(s,t)}at some fixed point s∗∈R.
To this end, from the above expressions of the curvature and t orsion associated to /vectorm(s,t)and
evaluating the equation (2.1) at the point s∗= 0, we obtain that /vectormt(0,t) =/vector0. On the other
hand, differentiating the geometric equation (2.1) with res pect tos, and using the Serret–Frenet
equations (1.6) together with the compatibility condition /vectormst=/vectormts, we get the following relation
for the time evolution of the normal vector
c/vectornt=β(css/vectorb+c2τ/vectorm−cτ2/vectorb)+α((cτ)s/vectorb−ccs/vectorm+csτ/vectorb).
The evaluation of the above identity at s∗= 0together with the expressions for the curvature
and torsion in (2.7) yield /vectornt(0,t) =/vector0. The above argument shows that
/vectormt(0,t) =/vector0, /vectornt(0,t) =/vector0and/vectorbt(0,t) = (/vectorm×/vectorn)t(0,t) =/vector0.
Therefore we can assume w.l.o.g. that
/vectorm(0,t) = (1,0,0), /vectorn(0,t) = (0,1,0)and/vectorb(0,t) = (0,0,1),
and in particular
/vector m(0) =/vectorm(0,1) = (1,0,0), /vector n(0) =/vectorn(0,1) = (0,1,0),and/vectorb(0) =/vectorb(0,1) = (0,0,1).(2.8)
Givenα∈[0,1]andc0>0, from the theory of ODE’s, it follows that there exists a uniq ue
{/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)} ∈/parenleftbig
C∞(R;S2)/parenrightbig3solution of the Serret–Frenet equations (1.6) with
curvature and torsion (2.6) and initial conditions (2.8) su ch that
/vector mc0,α⊥/vector nc0,α, /vector mc0,α⊥/vectorbc0,α, /vector nc0,α⊥/vectorbc0,α
and
|/vector mc0,α|2=|/vector nc0,α|2=|/vectorbc0,α|2= 1.
Define/vectormc0,α(s,t)as
/vectormc0,α(s,t) =/vector mc0,α/parenleftbiggs√
t/parenrightbigg
. (2.9)
Then,/vector mc0,α(·,t)∈ C∞/parenleftbig
R;S2/parenrightbig
for allt>0, and bearing in mind both the relations in (2.3)–(2.4)
and the fact that the vectors {/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)}satisfy the identity (2.5), a straightfor-
ward calculation shows that /vector mc0,α(·,t)is a regular C∞(R;S2)-solution of the LLG equation for
allt>0. Notice that the case c0= 0yields the constant solution /vector m0,α(s,t) = (1,0,0). Therefore
in what follows we will assume that c0>0.
The rest of the paper is devoted to establish analytical prop erties of the solutions {/vectormc0,α(s,t)}c0,α
defined by (2.9) for fixed α∈[0,1]andc0>0. As already mentioned, due to the self-similar
nature of these solutions, it suffices to study the properties of the associated profile /vector mc0,α(·)or,
equivalently, of the solution {/vector mc0,α,/vector nc0,α,/vectorbc0,α}of the Serret–Frenet system (1.6) with curvature
and torsion given by (2.6) and initial conditions (2.8). As w e will continue to see, the analysis
of the profile solution {/vector mc0,α,/vector nc0,α,/vectorbc0,α}can be reduced to the study of the properties of the
solutions of a certain second order complex differential equ ation.
113 Integration of the Serret–Frenet system
3.1 Reduction to the study of a second order ODE
Classical changes of variables from the differential geomet ry of curves allow us to reduce the nine
equations in the Serret–Frenet system into three complex-v alued second order equations (see
[8, 28, 23]). Theses changes of variables are related to ster eographic projection and this approach
was also used in [15]. However, their choice of stereographi c projection has a singularity at the
origin, which leads to an indetermination of the initial con ditions of some of the new variables.
For this reason, we consider in the following lemma a stereog raphic projection that is compatible
with the initial conditions (2.8). Although the proof of the lemma below is a slight modification
of that in [23, Subsections 2.12 and 7.3], we have included it s proof here both for the sake of
completeness and to clarify to the unfamiliar reader how the integration of the Frenet equations
can be reduced to the study of a second order differential equa tion.
Lemma 3.1. Let/vector m= (mj(s))3
j=1,/vector n= (nj(s))3
j=1and/vectorb= (bj(s))3
j=1be a solution of the Serret–
Frenet equations (1.6)with positive curvature cand torsion τ. Then, for each j∈ {1,2,3}the
function
fj(s) =e1
2/integraltexts
0c(σ)ηj(σ)dσ,withηj(s) =(nj(s)+ibj(s))
1+mj(s),
solves the equation
f′′
j(s)+/parenleftbigg
iτ(s)−c′(s)
c(s)/parenrightbigg
f′
j(s)+c2(s)
4fj(s) = 0, (3.1)
with initial conditions
fj(0) = 1, f′
j(0) =c(0)(nj(0)+ibj(0))
2(1+mj(0)).
Moreover, the coordinates of /vector m,/vector nand/vectorbare given in terms of fjandf′
jby
mj(s) = 2/parenleftBigg
1+4
c(s)2/vextendsingle/vextendsingle/vextendsingle/vextendsinglef′
j(s)
fj(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg−1
−1, nj(s)+ibj(s) =4f′
j(s)
c(s)fj(s)/parenleftBigg
1+4
c(s)2/vextendsingle/vextendsingle/vextendsingle/vextendsinglef′
j(s)
fj(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg−1
.
(3.2)
The above relations are valid at least as long as mj>−1and|fj|>0.
Proof. For simplicity, we omit the index j. The proof relies on several transformations that are
rather standard in the study of curves. First we define the com plex function
N= (n+ib)ei/integraltexts
0τ(σ)dσ. (3.3)
ThenN′=iτN+ (n′+ib′)ei/integraltexts
0τ(σ)dσ. On the other hand, the Serret–Frenet equations imply
that
n′+ib′=−cm−iτNe−i/integraltexts
0τ(σ)dσ.
Therefore, setting
ψ=cei/integraltexts
0τ(σ)dσ,
we get
N′=−ψm. (3.4)
Using again the Serret–Frenet equations, we also obtain
m′=1
2(ψN+ψN). (3.5)
12Let us consider now the auxiliary function
ϕ=N
1+m. (3.6)
Differentiating and using (3.4), (3.5) and (3.6)
ϕ′=N′
1+m−Nm′
(1+m)2
=N′
1+m−ϕm′
1+m
=−ϕ2ψ
2−ψ
2(1+m)(2m+ϕN).
Noticing that we can recast the relation m2+n2+b2= 1asNN= (1−m)(1+m)and recalling
the definition of ϕin (3.6), we have ϕN= 1−m, so that
ϕ′+ϕ2ψ
2+ψ
2= 0. (3.7)
Finally, define the stereographic projection of (m,n,b)by
η=n+ib
1+m. (3.8)
Observe that from the definitions of Nandϕ, respectively in (3.3) and (3.6), we can rewrite η
as
η=ϕe−i/integraltexts
0τ(σ)dσ,
and from (3.7) it follows that ηsolves the Riccati equation
η′+iτη+c
2(η2+1) = 0, (3.9)
(recall that ψ=cei/integraltexts
0τ(σ)dσ). Finally, setting
f(s) =e1
2/integraltexts
0c(σ)η(σ)dσ, (3.10)
we get
η=2f′
cf(3.11)
and equation (3.1) follows from (3.9). The initial conditio ns are an immediate consequence of
the definition of ηandfin (3.8) and (3.10).
A straightforward calculation shows that the inverse trans formation of the stereographic pro-
jection is
m=1−|η|2
1+|η|2, n=2Reη
1+|η|2, b=2Imη
1+|η|2,
so that we obtain (3.2) using (3.11) and the above identities .
Going back to our problem, Lemma 3.1 reduces the analysis of t he solution {/vector m,/vector n,/vectorb}of the
Serret–Frenet system (1.6) with curvature and torsion give n by (2.6) and initial conditions (2.8)
to the study of the second order differential equation
f′′(s)+s
2(α+iβ)f′(s)+c2
0
4e−αs2/2f(s) = 0, (3.12)
13with three initial conditions: For (m1,n1,b1) = (1,0,0)the associated initial condition for f1is
f1(0) = 1, f′
1(0) = 0, (3.13)
for(m2,n2,b2) = (0,1,0)is
f2(0) = 1, f′
2(0) =c0
2, (3.14)
and for(m3,n3,b3) = (0,0,1)is
f3(0) = 1, f′
3(0) =ic0
2. (3.15)
It is important to notice that, by multiplying (3.12) by ¯f′and taking the real part, it is easy to
see that
d
ds/bracketleftbigg1
2/parenleftbigg
eαs2
2|f′|2+c2
0
4|f|2/parenrightbigg/bracketrightbigg
= 0.
Thus,
E(s) :=1
2/parenleftbigg
eαs2
2|f′|2+c2
0
4|f|2/parenrightbigg
=E0,∀s∈R, (3.16)
withE0a constant defined by the value of E(s)at some point s0∈R. The conservation of the
energyE(s)allows us to simplify the expressions of mj,njandbjforj∈ {1,2,3}in the formulae
(3.2) in terms of the solution fjto (3.12) associated to the initial conditions (3.13)–(3.1 5).
Indeed, on the one hand notice that the energies associated t o the initial conditions (3.13)–
(3.15) are respectively
E0,1=c2
0
8, E 0,2=c2
0
4andE0,3=c2
0
4. (3.17)
On the other hand, from (3.16), it follows that
/parenleftBigg
1+4
c2
0e−αs2
2|f′
j|2(s)
|fj|2(s)/parenrightBigg−1
=c2
0
8E0,j|fj|2(s), j∈ {1,2,3}.
Therefore, from (3.17), the above identity and formulae (3. 2) in Lemma 3.1, we conclude that
m1(s) = 2|f1(s)|2−1, n 1(s)+ib1(s) =4
c0eαs2/4¯f1(s)f′
1(s), (3.18)
mj(s) =|fj(s)|2−1, n j(s)+ibj(s) =2
c0eαs2/4¯fj(s)f′
j(s), j∈ {2,3}. (3.19)
The above identities give the expressions of the tangent, no rmal and binormal vectors in terms
of the solutions {fj}3
j=1of the second order differential equation (3.12) associated to the initial
conditions (3.13)–(3.15).
By Lemma 3.1, the formulae (3.18) and (3.19) are valid as long a smj>−1, which is equivalent
to the condition |fj| /ne}ationslash= 0. As shown in Appendix, for α= 1there is˜s>0such thatmj(˜s) =−1
and then (3.18) and (3.19) are (a priori) valid just in a bound ed interval. However, the trihedron
{/vector m,/vector n,/vectorb}is defined globally and fjcan also be extended globally as the solution of the linear
equation (3.12). Then, it is simple to verify that the functi ons given by the l.h.s. of formulae
(3.18) and (3.19) satisfy the Serret–Frenet system and henc e, by the uniqueness of the solution,
the formulae (3.18) and (3.19) are valid for all s∈R.
143.2 The second-order equation. Asymptotics
In this section we study the properties of the complex-value d equation
f′′(s)+s
2(α+iβ)f′(s)+c2
0
4f(s)e−αs2/2= 0, (3.20)
for fixedc0>0,α∈[0,1),β >0such thatα2+β2= 1. We begin noticing that in the
caseα= 0, the solution can be written explicitly in terms of paraboli c cylinder functions or
confluent hypergeometric functions (see [1]). Another anal ytical approach using Fourier analysis
techniques has been taken in [15], leading to the asymptotic s
f(s) =C1ei(c2
0/2)ln(s)+C2e−is2/4
se−i(c2
0/2)ln(s)+O(1/s2), (3.21)
ass→ ∞, where the constants C1,C2andO(1/s2)depend on the initial conditions and c0.
Forα= 1, equation (3.20) can be also solved explicitly and the solut ion is given by
f(s) =2f′(0)
c0sin/parenleftbiggc0
2/integraldisplays
0e−σ2/4dσ/parenrightbigg
+f(0)cos/parenleftbiggc0
2/integraldisplays
0e−σ2/4dσ/parenrightbigg
.
In the case α∈(0,1), one cannot compute the solutions of (3.20) in terms of known functions
and we will follow a more analytical analysis. In contrast wi th the situation when α= 0, it is
far from evident to use Fourier analysis to study (3.20) when α>0.
For the rest of this section we will assume that α∈[0,1). In addition, we will also assume that
s>0and we will develop the asymptotic analysis necessary to est ablish part (ii)of Theorem 1.2.
At this point, it is important to recall the expressions give n in (3.18)–(3.19) for the coordinates
of the tangent, normal and binormal vectors associated to ou r family of solutions of the LLG
equation in terms f. Bearing this in mind, we observe that the study of the asympto tic behaviour
of these vectors are dictated by the asymptotic behaviour of the variables
z=|f|2, y= Re(¯ff′),andh= Im(¯ff′) (3.22)
associated to the solution fof (3.20).
As explained in the remark (a) after Theorem 1.2, we need to wo rk with remainder terms that
are independent of α. To this aim, we proceed in two steps: first we found uniform es timates
forα∈[0,1/2]in Propositions 3.2 and 3.3, then we treat the case α∈[1/2,1)in Lemma 3.6. In
Subsection 3.3 we provide some continuity results that allo ws us to take α→1−and give the
full statement in Corollary 3.14. Finally, notice that thes e asymptotics lead to the asymptotics
for the original equation (3.20) (see Remark 3.9).
We begin our analysis by establishing the following:
Proposition 3.2. Letc0>0,α∈[0,1),β >0such thatα2+β2= 1, andfbe a solution of
(3.20). Define z,yandhasz=|f|2andy+ih=¯ff′. Then
(i) There exists E0≥0such that the identity
1
2/parenleftbigg
eαs2
2|f′|2+c2
0
4|f|2/parenrightbigg
=E0
holds true for all s∈R. In particular, f,f′,z,yandhare bounded functions. Moreover,
for alls∈R
|f(s)| ≤√8E0
c0,|f′(s)| ≤/radicalbig
2E0e−αs2/4, (3.23)
|z(s)| ≤8E0
c2
0and|h(s)|+|y(s)| ≤8E0
c0e−αs2/4. (3.24)
15(ii) The limit
z∞:= lim
s→∞z(s)
exists.
(iii) Letγ:= 2E0−c2
0z∞/2ands0= 4/radicalbig
8+c2
0. For alls≥s0, we have
z(s)−z∞=−4
s(αy+βh)−4γ
s2e−αs2/2+R0(s), (3.25)
where
|R0(s)| ≤C(E0,c0)e−αs2/4
s3. (3.26)
Proof. Part(i)is just the conservation of energy proved in (3.16). Next, us ing the conservation
law in part (i), we obtain that the variables {z,y,h}solve the first-order real system
z′= 2y, (3.27)
y′=βs
2h−αs
2y+e−αs2/2/parenleftbigg
2E0−c2
0
2z/parenrightbigg
, (3.28)
h′=−βs
2y−αs
2h. (3.29)
To show (ii), plugging (3.27) into (3.29) and integrating from 0to somes>0we obtain
z(s)−1
s/integraldisplays
0z(σ)dσ=−4
βs/parenleftbigg
h(s)−h(0)+α
2/integraldisplays
0σh(σ)dσ/parenrightbigg
. (3.30)
Also, using the above identity,
d
ds/parenleftbigg1
s/integraldisplays
0z(σ)dσ/parenrightbigg
=−4
βs2/parenleftbigg
h(s)−h(0)+α
2/integraldisplays
0σh(σ)dσ/parenrightbigg
. (3.31)
Now, since from part (i)|h(s)| ≤8E0
c0e−αs2/4, bothhandα/integraltexts
0σh(σ)dσare bounded functions,
thus from (3.31) it follows that the limit of1
s/integraltexts
0zexists, ass→ ∞. Hence (3.30) and previous
observations conclude that the limit z∞:= lims→∞z(s)exists and furthermore
z∞:= lim
s→∞z(s) = lim
s→∞1
s/integraldisplays
0z(σ). (3.32)
We continue to prove (iii). Integrating (3.31) between s>0and+∞and using integration
by parts, we obtain
z∞−1
s/integraldisplays
0z(σ)dσ=−4
β/integraldisplay∞
sh(σ)
σ2dσ+4
βh(0)
s−2α
β/bracketleftbigg1
s/integraldisplays
0σh(σ)dσ+/integraldisplay∞
sh(σ)dσ/bracketrightbigg
.(3.33)
From (3.30) and (3.33), we get
z(s)−z∞=−4
βh(s)
s+2α
β/integraldisplay∞
sh(σ)dσ+4
β/integraldisplay∞
sh(σ)
σ2. (3.34)
In order to compute the integrals in (3.34), using (3.27) and (3.28), we write
h=2
β/parenleftbiggy′
s+α
4z′−2E0
se−αs2/2+c2
0
2sze−αs2/2/parenrightbigg
.
16Then, integrating by parts and using the bound for yin (3.24),
/integraldisplay∞
sh(σ) =2
β/parenleftBigg
−y
s+/integraldisplay∞
sy
σ2+α
4(z∞−z)−2E0/integraldisplay∞
se−ασ2/2
σ+c2
0
2/integraldisplay∞
sz
σe−ασ2/2/parenrightBigg
.(3.35)
Also, from (3.27) and (3.34), we obtain
/integraldisplay∞
sh(σ)
σ2=2
β/parenleftBigg/integraldisplay∞
sy′
σ3+α
2/integraldisplay∞
sy
σ2−2E0/integraldisplay∞
se−ασ2/2
σ3+c2
0
2/integraldisplay∞
sz
σ3e−ασ2/2/parenrightBigg
.(3.36)
Multiplying (3.34) by β2, using (3.35), (3.36) and the identity
α/integraldisplay∞
se−ασ2/2
σn=e−αs2/2
sn+1−(n+1)/integraldisplay∞
se−ασ2/2
σn+2,for allα≥0, n≥1,
we conclude that
(α2+β2)(z−z∞) =−4
s(αy+βh)−8E0
s2e−αs2/2
+8α/integraldisplay∞
sy
σ2+8/integraldisplay∞
sy′
σ3+2c2
0/integraldisplay∞
se−ασ2/2z/parenleftbiggα
σ+2
σ3/parenrightbigg
. (3.37)
Finally, using (3.27) and the boundedness of zandy, an integration by parts argument shows
that
8α/integraldisplay∞
sy
σ2+8/integraldisplay∞
sy′
σ3=−4αz
s2−8y
s3−12z
s4+8/integraldisplay∞
sz/parenleftbiggα
σ3−6
σ5/parenrightbigg
. (3.38)
Bearing in mind that α2+β2= 1, from (3.37) and (3.38), we obtain the following identity
z−z∞=−4
s(αy+βh)−8E0
s2e−αs2/2−4αz
s2−8y
s3−12z
s4+8/integraldisplay∞
sz/parenleftbiggα
σ3+6
σ5/parenrightbigg
dσ
+2c2
0/integraldisplay∞
se−ασ2/2z/parenleftbiggα
σ+2
σ3/parenrightbigg
dσ,(3.39)
for alls>0. In order to prove (iii), we first write z=z−z∞+z∞and observe that
8α/integraldisplay∞
sz
σ3= 8α/integraldisplay∞
sz−z∞
σ3+4αz∞
s2,
/integraldisplay∞
sz
σ5=/integraldisplay∞
sz−z∞
σ5+z∞
4s4and
/integraldisplay∞
se−ασ2/2z/parenleftbiggα
σ+2
σ3/parenrightbigg
=/integraldisplay∞
se−ασ2/2(z−z∞)/parenleftbiggα
σ+2
σ3/parenrightbigg
+z∞
s2e−αs2/2.
Therefore, we can recast (3.39) as (3.25) with
R0(s) =−4α(z−z∞)
s2−8y
s3−12(z−z∞)
s4+8/integraldisplay∞
s(z−z∞)/parenleftbiggα
σ3+6
σ5/parenrightbigg
dσ
+2c2
0/integraldisplay∞
se−ασ2/2(z−z∞)/parenleftbiggα
σ+2
σ3/parenrightbigg
dσ.(3.40)
Let us take s0≥1to be fixed in what follows. For t≥s0, we denote /bardbl · /bardbltthe norm of
L∞([t,∞)). From the definition of R0in (3.40) and the elementary inequalities
α/integraldisplay∞
se−ασ2/2
σn≤e−αs2/2
sn+1,for allα≥0, n≥1, (3.41)
17and/integraldisplay∞
se−ασ2/2
σn≤e−αs2/2
(n−1)sn−1,for allα≥0, n>1, (3.42)
we obtain
/bardblR0/bardblt≤8/bardbly/bardblt
t3+4
t2/parenleftBig
8+c2
0e−αt2/2/parenrightBig
/bardblz−z∞/bardblt.
Hence, choosing s0= 4/radicalbig
8+c2
0, so that4
t2/parenleftBig
8+c2
0e−αt2/2/parenrightBig
≤1/2, from (3.24) and (3.25) we
conclude that there exists a constant C(E0,c0)>0such that
/bardblz−z∞/bardblt≤C(E0,c0)
te−αt2/4,for allα∈[0,1)andt≥s0,
which implies that
|z(s)−z∞| ≤C(E0,c0)
se−αs2/4,for allα∈[0,1), s≥s0. (3.43)
Finally, plugging (3.24) and (3.43) into (3.40) and bearing in mind the inequalities (3.41) and
(3.42), we deduce that
|R0(s)| ≤C(E0,c0)e−αs2/4
s3,∀s≥s0= 4/radicalBig
8+c2
0, (3.44)
and the proof of (iii)is completed.
Formula (3.25) in Proposition 3.2 gives zin terms of yandh. Therefore, we can reduce our
analysis to that of the variables yandhor, in other words, to that of the system (3.27)–(3.29).
In fact, a first attempt could be to define w=y+ih, so that from (3.28) and (3.29), we have
thatwsolves/parenleftBig
we(α+iβ)s2/4/parenrightBig′
=e(−α+iβ)s2/4/parenleftbigg
γ−c2
0
2(z−z∞)/parenrightbigg
. (3.45)
From (3.43) in Proposition 3.2 and (3.45), we see that the lim itw∗= lims→∞w(s)e(α+iβ)s2/4
exists (at least when α/ne}ationslash= 0), and integrating (3.45) from some s>0to∞we find that
w(s) =e−(α+iβ)s2/4/parenleftbigg
w∗−/integraldisplay∞
se(−α+iβ)σ2/4/parenleftbigg
γ−c2
0
2(z−z∞)/parenrightbigg
dσ/parenrightbigg
.
In order to obtain an asymptotic expansion, we need to estima te/integraltext∞
se(−α+iβ)σ2/4(z−z∞), fors
large. This can be achieved using (3.43),
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞
se(−α+iβ)σ2/4(z−z∞)dσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)/integraldisplay∞
se−ασ2/2
σdσ (3.46)
and the asymptotic expansion
/integraldisplay∞
se−ασ2/2
σdσ=e−αs2/2/parenleftbigg1
αs2−2
α2s4+8
α3s6+···/parenrightbigg
.
However this estimate diverges as α→0. The problem is that the bound used in obtaining
(3.46) does not take into account the cancellations due to th e oscillations. Therefore, and in
order to obtain the asymptotic behaviour of z,yandhvalid for all α∈[0,1), we need a more
refined analysis. In the next proposition we study the system (3.27)–(3.29), where we consider
the cancellations due the oscillations (see Lemma 3.5 below ). The following result provides
estimates that are valid for s≥s1, for somes1independent of α, ifαis small.
18Proposition 3.3. With the same notation and terminology as in Proposition 3.2 , let
s1= max/braceleftBigg
4/radicalBig
8+c2
0,2c0/parenleftbigg1
β−1/parenrightbigg1/2/bracerightBigg
.
Then for all s≥s1,
y(s) =be−αs2/4sin(φ(s1;s))−2αγ
se−αs2/2+O/parenleftBigg
e−αs2/2
β2s2/parenrightBigg
, (3.47)
h(s) =be−αs2/4cos(φ(s1;s))−2βγ
se−αs2/2+O/parenleftBigg
e−αs2/2
β2s2/parenrightBigg
, (3.48)
where
φ(s1;s) =a+β/integraldisplays2/4
s2
1/4/radicalbigg
1+c2
0e−2αt
tdt,
a∈[0,2π)is a real constant, and bis a positive constant given by
b2=/parenleftbigg
2E0−c2
0
4z∞/parenrightbigg
z∞. (3.49)
Proof. First, notice that plugging the expression for z(s)−z∞in (3.25) into (3.28), the system
(3.28)–(3.29) for the variables yandhrewrites equivalently as
y′=s
2(βh−αy)+2c2
0
se−αs2/2(βh+αy)+γe−αs2/2+R1(s), (3.50)
h′=−s
2(βy+αh), (3.51)
where
R1(s) =−c2
0
2e−αs2/2R0(s)+2c2
0γe−αs2
s2, (3.52)
andR0is given by (3.40).
Introducing the new variables,
u(t) =eαty(2√
t), v(t) =eαth(2√
t), (3.53)
we recast (3.50)–(3.51) as
/parenleftbiggu
v/parenrightbigg′
=/parenleftbiggαK β(1+K)
−β0/parenrightbigg/parenleftbiggu
v/parenrightbigg
+/parenleftbiggF
0/parenrightbigg
, (3.54)
with
K=c2
0e−2αt
t, F=γe−αt
√
t+e−αt
√
tR1(2√
t),
whereR1is the function defined in (3.52). In this way, we can regard (3 .54) as a non-autonomous
system. It is straightforward to check that the matrix
A=/parenleftbiggαK β(1+K)
−β0/parenrightbigg
is diagonalizable, i.e. A=PDP−1, with
D=/parenleftbiggλ+0
0λ−/parenrightbigg
, P=/parenleftbigg−αK
2β−i∆1/2−αK
2β+i∆1/2
1 1/parenrightbigg
,
19λ±=αK
2±iβ∆1/2,and∆ = 1+K−α2K2
4β2. (3.55)
At this point we remark that the condition t≥t1, witht1:=s2
1/4ands1≥2c0(1
β−1)1/2, implies
that
0<K/parenleftbigg1
β−1/parenrightbigg
≤1,∀t≥t1, (3.56)
so that
∆ = 1+K−(1−β2)
4β2K2=/parenleftbigg
1+K
2+K
2β/parenrightbigg/parenleftbigg
1+K
2/parenleftbigg
1−1
β/parenrightbigg/parenrightbigg
≥1
2,∀t≥t1.(3.57)
Thus, defining
w= (w1,w2) =P−1(u,v), (3.58)
we get/parenleftBig
e−/integraltextt
t1Dw/parenrightBig′
=e−/integraltextt
t1D/parenleftBig
(P−1)′Pw+P−1˜F/parenrightBig
, (3.59)
with˜F= (F,0). From the definition of wand taking into account that uandvare real functions,
we have that w1= ¯w2and therefore the study of (3.59) reduces to the analysis of t he equation:
/parenleftBig
e−/integraltextt
t1λ+w1/parenrightBig′
=e−/integraltextt
t1λ+G(t), (3.60)
with
G(t) =iαK′
4β∆1/2(w1+ ¯w1)−∆′
4∆(w1−¯w1)+iF
2∆1/2.
From (3.60) we have
w1(t) =e/integraltextt
t1λ+/parenleftbigg
w1(t1)+w∞−/integraldisplay∞
te−/integraltextτ
t1λ+G(τ)dτ/parenrightbigg
, (3.61)
with
w∞=/integraldisplay∞
t1e−/integraltextτ
t1λ+G(τ).
Since
w1=iu
2∆1/2+v
2+iαKv
4β∆1/2, (3.62)
we recastGasG=i(G1+G2+G3)with
G1=αK′v
4β∆1/2−∆′
4∆3/2/parenleftbigg
u+αKv
2β/parenrightbigg
, G2=γe−αt
2t1/2∆1/2andG3=e−αt
2t1/2∆1/2R1(2t1/2).
Now, from the definition of Kand∆, we have
K′=−K/parenleftbigg
2α+1
t/parenrightbigg
, K′′=K/parenleftBigg/parenleftbigg
2α+1
t/parenrightbigg2
+1
t2/parenrightBigg
,
∆′=K′/parenleftbigg
1−α2K
2β2/parenrightbigg
and∆′′=K/parenleftBigg/parenleftbigg
2α+1
t/parenrightbigg2
+1
t2/parenrightBigg/parenleftbigg
1−α2K
2β2/parenrightbigg
−α2
2β2K2/parenleftbigg
2α+1
t/parenrightbigg2
.
Also, since s1= max{4/radicalbig
8+c2
0,2c0(1/β−1)1/2}, for allt≥t1=s2
1/4, we have in particular
thatt≥8+c2
0andt≥c2
0(1/β−1), hence
c2
0
tβ=c2
0
t/parenleftbigg1
β−1/parenrightbigg
+c2
0
t≤2 (3.63)
20and /vextendsingle/vextendsingle/vextendsingle/vextendsingle1−α2K
4β2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1+1
4β/parenleftbiggc2
0
tβ/parenrightbigg
≤2
β. (3.64)
Therefore
|K′| ≤c2
0e−2αt/parenleftbigg2α
t+1
t2/parenrightbigg
,|∆′| ≤2c2
0e−2αt
β/parenleftbigg2α
t+1
t2/parenrightbigg
(3.65)
and
|∆′′| ≤24c2
0
βe−2αt/parenleftbiggα
t+1
t2/parenrightbigg
. (3.66)
From Proposition 3.2, uandvare bounded in terms of the energy. Thus, from the definition o f
G1and the estimates (3.56), (3.57) and (3.65), we obtain
|G1(t)| ≤C(E0,c0)e−2αt
β2/parenleftbiggα
t+1
t2/parenrightbigg
.
Since /vextendsingle/vextendsingle/vextendsinglee±/integraltextτ
t1λ+/vextendsingle/vextendsingle/vextendsingle≤2, (3.67)
we conclude that
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞
te−/integraltextτ
t1λ+G1(τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)
β2/integraldisplay∞
te−2ατ/parenleftbiggα
τ+1
τ2/parenrightbigg
≤C(E0,c0)e−2αt
β2t. (3.68)
Here we have used the inequality
α/integraldisplay∞
te−2ασ
σndσ≤e−2αt
2tn, n≥1, (3.69)
which follows by integrating by parts.
In order to handle the terms involving G2andG3, we need to take advantage of the oscillatory
character of the involved integrals, which is exploited in L emma 3.5. From (3.57), (3.65) and
(3.66), straightforward calculations show that the functi on defined by f=γ/(2t1/2∆1/2)satisfies
the hypothesis in part (ii)of Lemma 3.5 with a= 1/2andL=C(E0,c0)/β. Thus invoking this
lemma with f=γ/(2t1/2∆1/2)and noticing that
1
∆1/2= 1+/parenleftbigg1
∆1/2−1/parenrightbigg
and that
/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
∆1/2−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−∆
∆1/2(∆1/2+1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤|K|/vextendsingle/vextendsingle/vextendsingle1−α2K
4β2/vextendsingle/vextendsingle/vextendsingle
|∆1/2(∆1/2+1)|≤2√
2c2
0
βt,
where we have used (3.57) and (3.64), we conclude that
/integraldisplay∞
te−/integraltextτ
t1λ+G2(τ)dτ=γ
2(α+iβ)t1/2e−/integraltextt
t1λ+e−αt+R2(t), (3.70)
with
|R2(t)| ≤C(E0,c0)e−αt
β2t3/2.
ForG3, we first write explicitly (recall the definition of R1in (3.52))
G3(t) =−c2
0R0(2√
t)e−3αt
4t1/2∆1/2+c2
0γe−5αt
4t3/2∆1/2:=G3,1(t)+G3,2(t). (3.71)
21Using (3.44) and (3.57), we see that |G3,1(t)| ≤C(E0,c0)e−4αt/t2, so that we can treat this term
as we did for G1to obtain
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞
te−/integraltextτ
t1λ+G3,1(τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)e−4αt
t. (3.72)
For the second term, using (3.57), (3.65) and (3.63), it is ea sy to see that the function fdefined
byf= (c2
0γ)/(4t3/2∆1/2)satisfies
|f(t)| ≤C(E0,c0)
t3/2and|f′(t)| ≤C(E0,c0)/parenleftbiggα
t3/2+1
t5/2/parenrightbigg
,
as a consequence, invoking part (i)of Lemma 3.5, we obtain
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞
te−/integraltextτ
t1λ+G3,2(τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)e−5αt
βt3/2. (3.73)
From (3.61), (3.67), (3.68), (3.72) and (3.73), we deduce th at
w1(t) =e/integraltextt
t1λ+(w1(t1)+w∞)−γ(β+iα)
2t1/2e−αt+R3(t)with|R3(t)| ≤C(E0,c0)e−αt
β2t.(3.74)
Now we claim that
e/integraltextt
t1λ+=Cα,c0eiβI(t)+H(t),withI(t) =/integraldisplayt
t1/radicalbig
1+K(σ)dσ,|H(t)| ≤3c2
0e−2αt
t,(3.75)
and
Cα,c0= exp/parenleftbiggα
2/integraldisplay∞
t1Kdσ/parenrightbigg
exp/parenleftbigg
−iα2
4β/integraldisplay∞
t1K2
∆1/2+(1+K)1/2dσ/parenrightbigg
.
Indeed, recall that λ+=αK
2+iβ∆1/2so that
e/integraltextt
t1λ+=eα/integraltextt
t1K
2eiβ/integraltextt
t1∆1/2
. (3.76)
First, we notice that
α/integraldisplayt
t1K
2=c2
0α/integraldisplay∞
t1e−2ασ
2σ−c2
0α/integraldisplay∞
te−2ασ
2σ,
where both integrals are finite in view of (3.69). Moreover, b y combining with the fact that
|1−e−x| ≤x, forx≥0, we can write
exp/parenleftbigg
−c2
0α/integraldisplay∞
te−2ασ
2σ/parenrightbigg
= 1+H1(t),
with
|H1(t)| ≤c2
0e−2αt
4t,for allt≥c2
0/4. (3.77)
The above argument shows that
eα/integraltextt
t1K
2=eα/integraltext∞
t1K
2(1+H1(t)), (3.78)
withH1(t)satisfying (3.77).
22For the second term of the eigenvalue, using the definition of ∆in (3.55), we write
iβ/integraldisplayt
t1∆1/2=iβ/integraldisplayt
t1/parenleftBig
∆1/2−√
1+K/parenrightBig
+iβ/integraldisplayt
t1√
1+K
=−iα2
4β/integraldisplayt
t1K2
∆1/2+(1+K)1/2+iβ/integraldisplayt
t1√
1+K.
Proceeding as before and using that |1−eix| ≤ |x|, forx∈R, and that
α/integraldisplay∞
tK2
∆1/2+√
1+K≤α/integraldisplay∞
tK2(σ)dσ=αc4
0/integraldisplay∞
te−4ατ
τ2≤c4
0e−4αt
4t2,
we conclude that
eiβ/integraltextt
t1∆1/2
=eiβI(t)e−iα2
4β/integraltext∞
t1K2
∆1/2+(1+K)1/2(1+H2), (3.79)
with
|H2(t)| ≤c4
0e−4αt
16βt2≤c2
0e−4αt
8t,
bearing in mind (3.63). Therefore, from (3.76), (3.78) and ( 3.79),
e/integraltextt
t1λ+=Cα,c0eiβI(t)(1+H1(t))(1+H2(t)).
The claim follows from the above identity, the bounds for H1andH2, and the fact that Cα,c0
satisfies that |Cα,c0|=|e/integraltext∞
t1λ+| ≤2(see (3.67)). From (3.74), the claim and writing
Cα,c0(w1(t1)+w∞) = (beia)/2 (3.80)
for some real constants aandbsuch thatb≥0anda∈[0,2π), it follows that
w1(t) =b
2ei(βI(t)+a)−γ(β+iα)
2t1/2+Rw1(t)with|Rw1(t)| ≤C(E0,c0)e−αt
β2t. (3.81)
The above bound for Rw1(t)easily follows from the bounds for R3(t)andH(t)in (3.74) and
(3.75) respectively, and the fact that
|w1(t)| ≤C(E0,c0),∀t≥t1. (3.82)
This last inequality is a consequence of (3.53), (3.57), (3. 62), (3.63) and the bounds for yandh
established in (3.24) in Proposition 3.2.
Going back to the definition of win (3.58), we have (u,v) =P(w1,w2), that is
u=−αK
2β(w1+ ¯w1)−i∆1/2(w1−¯w1) = 2Im(w1)+R4(t),
v= (w1+ ¯w1) = 2Re(w1),(3.83)
with
|R4(t)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle−αK
βRe(w1)+2(∆1/2−1)Im(w1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
≤K
β|Re(w1)|+2|∆−1|
∆1/2+1|Im(w1)|
≤2c2
0e−2αt
βt(|Re(w1)|+|Im(w1)|)≤C(E0,c0)e−2αt
βt,
23where we have used (3.57), (3.64), and (3.82). From (3.81) an d (3.83), we obtain
u(t) =bsin(βI(t)+a)−αγ
t1/2e−αt+R5(t),
v(t) =bcos(βI(t)+a)−βγ
t1/2e−αt+R6(t),
with
|R5(t)|+|R6(t)| ≤C(E0,c0)e−αt/(β2t).
The asymptotics for yandhgiven in (3.47) and (3.48) are a direct consequence of (3.53) and
the above identities and bounds.
Finally, we compute the value of b. In fact, from (3.47) and (3.48)
lim
s→∞(y2(s)+h2(s))eαs2/2=b2.
On the other hand, since y+ih=¯ff′and using the conservation of energy (3.16)
/parenleftbig
y2(s)+h2(s)/parenrightbig
eαs2/2=|y+ih|2(s)eαs2/2=|f′|2|f|2eαs2/2= (2E0−c2
0
4|f|2)|f|2,
so that, taking the limit as s→ ∞ and recalling that z=|f|2, (3.49) follows.
Remark 3.4. From the definitions of bin(3.49), andbeiain(3.80) (in terms of Cα,c0,w1(t1)
andw∞in(3.80)), it is simple to verify that bandbeiadepend continuously on α∈[0,1),
provided that z∞is a continuous function of α. In Subsection 3.3 we will prove that z∞depends
continuously on α, forα∈[0,1], and establish the continuous dependence of the constants band
beiawith respect to the parameter αin Lemma 3.13 above.
In the proof of Proposition 3.3, we have used the following ke y lemma that establishes the
control of certain integrals by exploiting their oscillato ry character.
Lemma 3.5. With the same notation as in the proof of Proposition 3.2.
(i) Letf∈C1((t1,∞))such that
|f(t)| ≤L/taand|f′(t)| ≤L/parenleftbiggα
ta+1
ta+1/parenrightbigg
,
for some constants L,a>0. Then, for all t≥t1andl≥1
/integraldisplay∞
te−/integraltextτ
t1λ+e−lατf(τ)dτ=1
(α+iβ)e−/integraltextt
t1λ+e−lαtf(t)+F(t),
with
|F(t)| ≤C(l,a,c0)Le−lαt
βta. (3.84)
(ii) If in addition f∈C2((t1,∞)),
|f′(t)| ≤L/ta+1and|f′′(t)| ≤L/parenleftbiggα
ta+1+1
ta+2/parenrightbigg
, (3.85)
then
|F(t)| ≤C(l,a,c0)Le−lαt
βta+1. (3.86)
24HereC(l,a,c0)is a positive constant depending only on l,aandc0.
Proof. Defineλ=λ+. Recall (see proof of Proposition 3.2) that
λ+=αK
2+iβ∆1/2and∆ = 1+K−α2K2
4β2,withK=c2
0e−2αt
t.
SettingRλ= 1/λ−1/(iβ)and integrating by parts, we obtain
/parenleftbigg
1+lα
iβ/parenrightbigg/integraldisplay∞
te−/integraltextτ
t1λe−lατf(τ)dτ=e−/integraltextt
t1λe−lαtf(t)/parenleftbigg1
iβ+Rλ/parenrightbigg
+/integraldisplay∞
te−/integraltextτ
t1λe−lατ/parenleftbigg
−lαfRλ+f′
λ−fλ′
λ2/parenrightbigg
dτ,
or, equivalently,
/integraldisplay∞
te−/integraltextτ
t1λe−ατf(τ)dτ=1
lα+iβe−/integraltextt
t1λe−αtf(t)+F(t),
with
F(t) =iβ
lα+iβ/parenleftbigg
e−/integraltextt
t1λe−lαtRλf+/integraldisplay∞
te−/integraltextτ
t1λe−lατ/parenleftbigg
−lαfRλ+f′
λ−fλ′
λ2/parenrightbigg
dτ/parenrightbigg
.
Using (3.57), (3.63) and (3.65), it is easy to check that for a llt≥t1
|λ| ≥β√
2and|λ′| ≤3c2
0/parenleftbigg2α
t+1
t2/parenrightbigg
. (3.87)
On the other hand,
|Rλ|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleiβ−λ
iβλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤√
2
β2/parenleftbigg
β|1−∆1/2|+αK
2/parenrightbigg
,
with, using the definition of ∆in (3.57) and (3.63),
αK
2≤c2
0
2tand|1−∆1/2|=|1−∆|
1+∆1/2≤ |1−∆| ≤c2
0
t+c2
0
4βt/parenleftbiggc2
0
βt/parenrightbigg
≤2c2
0
βt.
Previous lines show that
|Rλ| ≤10c2
0
β2t. (3.88)
The estimate (3.84) easily follows from the bounds (3.67), ( 3.69), (3.87), (3.88) and the hypothe-
ses onf. To obtain part (ii)we only need to improve the estimate for the term
/integraldisplay∞
te−/integraltextτ
t1λe−lατf′
λdτ
in the above argument. In particular, it suffices to prove that
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞
te−/integraltextτ
t1λe−lατf′
λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(l,c0,a)Le−lαt
β2ta+1.
Now, consider the function g=f′/λ. Notice that from (3.63), (3.87) and the hypotheses on f
in (3.85), we have
|g(t)| ≤√
2L
βta+1
25and
|g′(t)| ≤√
2
βL/parenleftbiggα
ta+1+1
ta+2/parenrightbigg
+6L
β/parenleftbiggc2
0
βt/parenrightbigg/parenleftbigg2α
ta+1+1
ta+2/parenrightbigg
≤14L
β/parenleftbigg2α
ta+1+1
ta+2/parenrightbigg
.
Therefore, from part (i), we obtain
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞
te−/integraltextτ
t1λe−lατf′
λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(l,c0,a)Le−lαt/parenleftbigg1
βta+1+1
β2ta+1/parenrightbigg
≤C(l,c0,a)Le−lαt
β2ta+1,
as desired.
We remark that if α∈[0,1/2], the asymptotics in Proposition 3.3 are uniform in α. Indeed,
max
α∈[0,1/2]/braceleftBigg
4/radicalBig
8+c2
0,2c0/parenleftbigg1
β−1/parenrightbigg1/2/bracerightBigg
= 4/radicalBig
8+c2
0=s0.
Therefore in this situation we can omit the dependence on s1in the function φ(s1;s), because
the asymptotics are valid with
φ(s) :=φ(s0;s) =a+β/integraldisplays2/4
s2
0/4/radicalbigg
1+c2
0e−2αt
tdt. (3.89)
We continue to show that the factor 1/β2in the big-Oin formulae (3.47) and (3.48) are due
to the method used and this factor can be avoided if αis far from zero. More precisely, we have
the following:
Lemma 3.6. Letα∈[1/2,1). With the same notation as in Propositions 3.2 and 3.3, we hav e
the following asymptotics: for all s≥s0,
y(s) =be−αs2/4sin(φ(s))−2αγ
se−αs2/2+O/parenleftBigg
e−αs2/2
s2/parenrightBigg
, (3.90)
h(s) =be−αs2/4cos(φ(s))−2βγ
se−αs2/2+O/parenleftBigg
e−αs2/2
s2/parenrightBigg
. (3.91)
Here, the function φis defined by (3.89) and the bounds controlling the error terms depend on
c0, and the energy E0, and are independent of α∈[1/2,1)
Proof. Letα∈[1/2,1)and define w=y+ih. From Proposition 3.3 and (1.21), we have that
for allα∈[1/2,1)
lim
s→∞we(α+iβ)s2/4=bie−i˜a, (3.92)
where˜a:=a+C(α,c0),aandbare the constants defined in Proposition 3.3 and C(α,c0)is the
constant in (1.21). Then, since wsatisfies
/parenleftBig
we(α+iβ)s2/4/parenrightBig′
=e(−α+iβ)s2/4/parenleftbigg
γ−c2
0
2(z−z∞)/parenrightbigg
, (3.93)
integrating the above identity between sand infinity,
we(α+iβ)s2/4=ibe−i˜a−/integraldisplay∞
se(−α+iβ)σ2/4/parenleftbigg
γ−c2
0
2(z−z∞)/parenrightbigg
dσ.
26Now, integrating by parts and using (3.41) (recall that 1≤2α), we see that
/integraldisplay∞
se(−α+iβ)σ2/4dσ= 2(α+iβ)e(−α+iβ)s2/4
s+O/parenleftBigg
e−αs2/4
s3/parenrightBigg
,∀s≥s0.
Next, notice that from (3.43) in Proposition 3.2, we also obt ain
/integraldisplay∞
se(−α+iβ)σ2/4(z−z∞)dσ=O/parenleftBigg
e−αs2/2
s2/parenrightBigg
,∀s≥s0.
The above argument shows that for all s≥s0
w(s) =ibe−αs2/4e−i(˜a+βs2/4)−2(α+iβ)γ
se−αs2/2+O/parenleftBigg
e−αs2/2
s2/parenrightBigg
. (3.94)
The asymptotics for yandhin the statement of the lemma easily follow from (3.94) beari ng in
mind thatw=y+ihand recalling that the function φbehaves like (1.21) when α>0.
In the following corollary we summarize the asymptotics for z,yandhobtained in this section.
Precisely, as a consequence of Proposition 3.2- (iii), Proposition 3.3 and Lemma 3.6, we have the
following:
Corollary 3.7. Letα∈[0,1). With the same notation as before, for all s≥s0= 4/radicalbig
8+c2
0,
y(s) =be−αs2/4sin(φ(s))−2αγ
se−αs2/2+O/parenleftBigg
e−αs2/2
s2/parenrightBigg
, (3.95)
h(s) =be−αs2/4cos(φ(s))−2βγ
se−αs2/2+O/parenleftBigg
e−αs2/2
s2/parenrightBigg
, (3.96)
z(s) =z∞−4b
se−αs2/4(αsin(φ(s))+βcos(φ(s)))+4γe−αs2/2
s2+O/parenleftBigg
e−αs2/4
s3/parenrightBigg
, (3.97)
where
φ(s) =a+β/integraldisplays2/4
s2
0/4/radicalbigg
1+c2
0e−2αt
tdt,
for some constant a∈[0,2π),
b=z1/2
∞/parenleftbigg
2E0−c2
0
4z∞/parenrightbigg1/2
, γ= 2E0−c2
0
2z∞ andz∞= lim
s→∞z(s).
Here, the bounds controlling the error terms depend on c0and the energy E0, and are independent
ofα∈[0,1).
Remark 3.8. In the case when s<0, the same arguments to the ones leading to the asymptotics
in the above corollary will lead to an analogous asymptotic b ehaviour for the variables z,hand
yfors<0. As mentioned at the beginning of Subsection 3.2, here we hav e reduced ourselves to
the case of s >0when establishing the asymptotic behaviour of the latter qu antities due to the
parity of the solution we will be applying these results to.
27Remark 3.9. The asymptotics in Corollary 3.7 lead to the asymptotics for the solutions fof the
equation (3.20), at least if |f|∞:=z1/2
∞is strictly positive. Indeed, this implies that there exist s
s∗≥s0such thatf(s)/ne}ationslash= 0for alls≥s∗. Then writing fin its polar form f=ρexp(iθ), we
haveρ2θ′= Im(¯ff′). Hence, using (3.22), we obtain ρ=z1/2andθ′=h/z. Therefore, for all
s≥s∗,
θ(s)−θ(s∗) =/integraldisplays
s∗h(σ)
z(σ)dσ. (3.98)
Hence, using the asymptotics for zandhin Corollary 3.7, we can obtain the asymptotics for f.
In the case that α∈(0,1], we can also show that the phase converges. Indeed, the asymp totics
in Corollary 3.7 yield that the integral in (3.98) converges as s→ ∞forα>0, and we conclude
that there exists a constant θ∞∈Rsuch that
f(s) =z(s)1/2exp/parenleftbigg
iθ∞−i/integraldisplay∞
sh(σ)
z(σ)dσ/parenrightbigg
,for alls≥s∗.
The asymptotics for fis obtained by plugging the asymptotics in Corollary 3.7 int o the above
expression.
3.3 The second-order equation. Dependence on the parameter s
The aim of this subsection is to study the dependence of the f,z,yandhon the parameters
c0>0andα∈[0,1]. This will allow us to pass to the limit α→1−in the asymptotics in
Corollary 3.7 and will give us the elements for the proofs of T heorems 1.3 and 1.4.
3.3.1 Dependence on α
We will denote by f(s,α)the solution of (3.20) with some initial conditions f(0,α),f′(0,α)that
are independent of α. Indeed, we are interested in initial conditions that depen d only onc0(see
(3.13)–(3.15)). Moreover, in view of (3.17), we assume that the energyE0in (3.16) is a function
ofc0. In order to simplify the notation, we denote with a subindex αthe derivative with respect
toαand by′the derivative with respect to s. Analogously to Subsection 3.2, we define
z(s,α) =|f(s,α)|2, y(s,α) = Re(¯f(s,α)f′(s,α)), h(s,α) = Im(¯f(s,α)f′(s,α)) (3.99)
and
z∞(α) = lim
s→∞|f(s,α)|2.
Observe that in Proposition 3.2- (ii), we proved the existence of z∞(α), forα∈[0,1). For
α∈(0,1], the estimates in (3.24) hold true and hence z(s,α)is a bounded function whose
derivative decays exponentially. Therefore, it admits a li mit at infinity for all α∈[0,1]and
z∞(1)is well-defined.
The next lemma provides estimates for zα,hαandyα.
Lemma 3.10. Letα∈(0,1). There exists a constant C(c0), depending on c0but not onα, such
that for all s≥0,
|zα(s,α)| ≤C(c0)min/braceleftBigg
s2
√1−α+s3,s2
/radicalbig
α(1−α),1
α2√1−α/bracerightBigg
, (3.100)
|yα(s,α)|+|hα(s,α)| ≤C(c0)e−αs2/4min/braceleftBigg
s2
√1−α+s3,s2
/radicalbig
α(1−α)/bracerightBigg
. (3.101)
28Proof. Differentiating (3.12) with respect to α,
f′′
α+s
2(α+iβ)f′
α+c2
0
4fαe−αs2/2=g, (3.102)
where
g(s,α) =−/parenleftbigg
1−iα
β/parenrightbiggs
2f′+c2
0s2
8fe−αs2/2.
Also, since the initial conditions do not depend on α,
fα(0,α) =f′
α(0,α) = 0. (3.103)
Using the estimates in (3.23) and that α2+β2= 1, we obtain
|g| ≤C(c0)/parenleftbiggs
βe−αs2/4+s2e−αs2/2/parenrightbigg
,for alls≥0. (3.104)
Multiplying (3.102) by ¯f′
αand taking real part, we have
1
2/parenleftbig
|f′
α|2/parenrightbig′+αs
2|f′
α|2+c2
0
8/parenleftbig
|fα|2/parenrightbig′e−αs2/2= Re(g¯f′
α). (3.105)
Multiplying (3.105) by 2eαs2/2and integrating, taking into account (3.103),
|f′
α|2eαs2/2+c2
0
4|fα|2= 2/integraldisplays
0eασ2/2Re(g¯f′
α)dσ. (3.106)
Let us define the real-valued function η=|f′
α|eαs2/4. Then (3.106) yields
η2(s)≤2/integraldisplays
0eασ2/4|g|ηdσ, for alls≥0.
Thus, by the Gronwall inequality (see e.g. [3, Lemma A.5]),
η(s)≤/integraldisplays
0eασ2/4|g|,dσ, for alls≥0. (3.107)
From (3.104), (3.106) and (3.107), we conclude that
(|f′
α|eαs2/4+c0
2|fα|)2≤2(|fα|2eαs2/2+c2
0
4|fα|2)
≤4/integraldisplays
0eασ2/4|g|ηdσ≤4/parenleftBigg
sup
σ∈[0,s]η(σ)/parenrightBigg/parenleftbigg/integraldisplays
0eασ2/4|g|dσ/parenrightbigg
≤/parenleftbigg/integraldisplays
0eασ2/4|g|dσ/parenrightbigg2
.
Thus, using (3.104), from the above inequality it follows
|f′
α|eαs2/4+c0
2|fα| ≤C(c0)/integraldisplays
0/parenleftbiggσ
β+σ2e−ασ2/4/parenrightbigg
dσ, for alls≥0. (3.108)
In particular, for all s≥0,
|fα(s)| ≤C(c0)min/braceleftBigg
s2
√1−α+s3,s2
/radicalbig
α(1−α)/bracerightBigg
,
|f′
α(s)| ≤C(c0)e−αs2/4min/braceleftBigg
s2
√1−α+s3,s2
/radicalbig
α(1−α)/bracerightBigg
,(3.109)
29where we have used that
/integraldisplays
0σ2e−ασ2/4dσ≤s2/integraldisplays
0e−ασ2/4dσ≤s2/radicalbig
π/α.
Notice that from (3.103) and (3.109),
|fα(s)| ≤/integraldisplays
0|f′
α|dσ≤C(c0)/radicalbig
α(1−α)/integraldisplays
0σ2e−ασ2/4dσ,
and /integraldisplay∞
0σ2e−ασ2/4dσ=2√π
α3/2, (3.110)
so that
|fα(s)| ≤C(c0)
α2√1−α. (3.111)
On the other hand, differentiating the relations in (3.99) wi th respect to α,
|zα| ≤2|fα||f|,|yα+ihα| ≤ |fα||f′|+|f||f′
α|. (3.112)
By putting together (3.23), (3.109), (3.111) and (3.112), we obtain (3.100) and (3.101).
Lemma 3.11. The function z∞is continuous in (0,1]. More precisely, there exists a constant
C(c0)depending on c0but not onα, such that
|z∞(α2)−z∞(α1)| ≤C(c0)
L(α2,α1)|α2−α1|,for allα1,α2∈(0,1], (3.113)
where
L(α2,α1) :=α2
1α3/2
2/parenleftBig
α3/2
1√
1−α2+α3/2
2√
1−α1/parenrightBig
.
In particular,
|z∞(1)−z∞(α)| ≤C(c0)√
1−α,for allα∈[1/2,1]. (3.114)
Proof. Letα1,α2∈(0,1],α1< α2. By classical results from the ODE theory, the functions
y(s,α),h(s,α)andz(s,α)are smooth in R×[0,1)and continuous in R×[0,1](see e.g. [5, 17]).
Hence, integrating (3.27) with respect to s, we deduce that
z∞(α2)−z∞(α1) = 2/integraldisplay∞
0(y(s,α2)−y(s,α1))ds= 2/integraldisplay∞
0/integraldisplayα2
α1dy
dµ(s,µ)dµds. (3.115)
To estimate the last integral, we use (3.101)
/integraldisplayα2
α1|dy
dµ(s,µ)|dµ≤C(c0)s2
√α1/integraldisplayα2
α1e−µs2/4
√1−µdµ. (3.116)
Now, integrating by parts,
/integraldisplayα2
α1e−µs2/4
√1−µdµ= 2/parenleftBig√
1−α1e−α1s2/4−√
1−α2e−α2s2/4/parenrightBig
−s2
2/integraldisplayα2
α1/radicalbig
1−µe−µs2/4dµ.
Therefore, by combining with (3.115) and (3.116),
|z∞(α2)−z∞(α1)| ≤C(c0)√α1/parenleftbigg√
1−α1/integraldisplay∞
0s2e−α1s2/4ds−√
1−α2/integraldisplay∞
0s2e−α2s2/4ds/parenrightbigg
,
30and bearing in mind (3.110), we conclude that
|z∞(α2)−z∞(α1)| ≤C(c0)√α1/parenleftBigg√1−α1
α3/2
1−√1−α2
α3/2
2/parenrightBigg
,
which, after some algebraic manipulations and using that α1,α2∈(0,1], leads to (3.113).
The estimate for z∞near zero is more involved and it is based in an improvement of the
estimate for the derivative of z∞.
Lemma 3.12. The function z∞is continuous in [0,1]. Moreover, there exists a constant
C(c0)>0, depending on c0but not onαsuch that for all α∈(0,1/2],
|z∞(α)−z∞(0)| ≤C(c0)√α|ln(α)|. (3.117)
Proof. As in the proof of Lemma 3.11, we recall that the functions y(s,α),h(s,α)andz(s,α)
are smooth in any compact subset of R×[0,1). From now on we will use the identity (3.39)
fixings= 1. We can verify that the two integral terms in (3.39) are conti nuous functions at
α= 0, which proves that z∞is continuous in 0. In view of Lemma 3.11, we conclude that z∞is
continuous in [0,1].
Now we claim that
/vextendsingle/vextendsingle/vextendsingle/vextendsingledz∞
dα(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)|ln(α)|√α,for allα∈(0,1/2]. (3.118)
In fact, once (3.118) is proved, we can compute
|z∞(α)−z∞(0)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayα
0dz∞
dµ(µ)dµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)/integraldisplayα
0|ln(µ)|√µdµ= 2C(c0)√α(|ln(α)|+2),
which implies (3.117).
It remains to prove the claim. Differentiating (3.39) (recal l thats= 1) with respect to α,
and using that y(1,·),h(1,·)andz(1,·)are continuous differentiable in [0,1/2], we deduce that
there exists a constant C(c0)>0such that
/vextendsingle/vextendsingle/vextendsingle/vextendsingledz∞
dα(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)+8|I1(α)|+2c2
0|I2(α)|, (3.119)
with
I1(α) =/integraldisplay∞
1z
σ3+α/integraldisplay∞
1zα
σ3+6/integraldisplay∞
1zα
σ5(3.120)
and
I2(α) =−α
2/integraldisplay∞
1e−ασ2/2zσ+α/integraldisplay∞
1e−ασ2/2zα
σ+2/integraldisplay∞
1e−ασ2/2zα
σ3. (3.121)
By (3.24) and (3.100), zis uniformly bounded and zαgrows at most as a cubic polynomial,
so that the first and the last integral in the r.h.s. of (3.120) are bounded independently of
α∈[0,1/2]. In addition, (3.100) also implies that
|zα|=|zα|1/2|zα|1/2≤C(c0)(s3)1/2/parenleftbigg1
α2/parenrightbigg1/2
=C(c0)s3/2
α, (3.122)
which shows that the remaining integral in (3.120) is bounde d.
31Thus, the above argument shows that
|I1(α)| ≤C(c0)for allα∈[0,1/2]. (3.123)
The same arguments also yield that the first two integrals in t he r.h.s. of (3.121) are bounded
byC(c0)α−1/2.Using once more that |zα| ≤C(c0)s2α−1/2, we obtain the following bounds for
the remaining two integrals in (3.121)
/vextendsingle/vextendsingle/vextendsingle/vextendsingleα/integraldisplays
1e−ασ2/2zα
σ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)√α/integraldisplay∞
1ασe−ασ2/2dσ=C(c0)√αe−α/2≤C(c0)√α
and /vextendsingle/vextendsingle/vextendsingle/vextendsingle2/integraldisplay∞
1e−ασ/2zα
σ3dσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)√α/integraldisplay∞
1e−ασ2/2
σdσ≤C(c0)|ln(α)|√α.
In conclusion, we have proved that
|I2(α)| ≤C(c0)|ln(α)|√α,
which combined with (3.119) and (3.123), completes the proo f of claim.
We end this section showing that the previous continuity res ults allow us to “pass to the limit”
α→1−in Corollary 3.7. Using the notation b(α) =banda(α) =afor the constants defined for
α∈[0,1)in Proposition 3.3 in Subsection 3.2, we have
Lemma 3.13. The valueb(α)is a continuous function of α∈[0,1]and the value b(α)eia(α)is
continuous function of α∈[0,1)that can be continuously extended to [0,1]. The function a(α)
has a (possible discontinuous) extension for α∈[0,1]such thata(α)∈[0,2π).
Proof. By Lemma 3.12, we have the continuity of z∞in [0,1]. Therefore, in view of Remark 3.4,
the function beiais a continuous function of α∈[0,1)and by (3.49) bis actually well-defined
and continuous in α∈[0,1].
It only remains to prove that the limit
L:= lim
α→1−b(α)eia(α)(3.124)
exists. Ifb(1) = 0 , it is immediate that L= 0and we can give any arbitrary value in [0,2π)to
a(1). Let us suppose that b(1)>0. Integrating (3.93), we get
w(s)e(α+iβ)s2/4=w(s0)e(α+iβ)s2
0/4+/integraldisplays
s0e(−α+iβ)σ2/4/parenleftbigg
γ−c2
0
2(z−z∞)/parenrightbigg
dσ,
and this relation is valid for any α∈(0,1]. Letα∈(0,1). In view of (3.92), letting s→ ∞, we
have
ibei(a+C(α,c0))=w(s0)e(α+iβ)s2
0/4+/integraldisplay∞
s0e(−α+iβ)σ2/4/parenleftbigg
γ−c2
0
2(z−z∞)/parenrightbigg
dσ, (3.125)
whereC(α,c0)is the constant in (1.21). Notice that the r.h.s. of (3.125) i s well-defined for any
α∈(0,1]and by the arguments given in the proof of Lemma 3.11 and the do minated convergence
theorem, the r.h.s. is also continuous for any α∈(0,1]. Therefore, the limit Lin (3.124) exists
and is given by the r.h.s. of (3.125) evaluated in α= 1and divided by ieiC(1,c0). Moreover,
lim
α→1−eia(α)=L
b(1),
so that by the compactness of the the unit circle in C, there exists θ∈[0,2π)such thateiθ=
L/b(1)and we can extend aby defining a(1) =θ.
32The following result summarizes an improvement of Corollar y 3.7 to include the case α= 1
and the continuous dependence of the constants appearing in the asymptotics on α. Precisely,
we have the following:
Corollary 3.14. Letα∈[0,1],β≥0withα2+β2= 1andc0>0. Then,
(i) The asymptotics in Corollary 3.7 holds true for all α∈[0,1].
(ii) Moreover, the values bandbeiaare continuous functions of α∈[0,1]and each term in the
asymptotics for z,yandhin Corollary 3.7 depends continuously on α∈[0,1].
(iii) In addition, the bounds controlling the error terms de pend onc0and are independent of
α∈[0,1].
Proof. Lets≥s0fixed. As noticed in the proof of Lemma 3.11, the functions y(s,α),h(s,α),
z(s,α)are continuous in α= 1. In addition, by Lemma 3.13 beiais continuous in α= 1, using
the definition of φ, it is immediate that bsin(φ(s))andbcos(φ(s))are continuous in α= 1.
Therefore the big- Oterms in (3.95), (3.96) and (3.97) are also are continuous in α= 1. The
proof of the corollary follows by letting α→1−in (3.95), (3.96) and (3.97).
3.3.2 Dependence on c0
In this subsection, we study the dependence of z∞as a function of c0, for a fixed value of α.
To this aim, we need to take into account the initial conditio ns given in (3.13)–(3.15). More
generally, let us assume that fis a solution of (3.20) with initial conditions f(0)andf′(0)that
depend smoothly on c0, for anyc0>0, and that E0>0is the associated energy defined in
(3.16). To keep our notation simple, we omit the parameter c0in the functions fandz∞. Under
these assumptions, we have
Proposition 3.15. Letα∈[0,1]andc0>0. Thenz∞is a continuous function of c0∈(0,∞).
Moreover if α∈(0,1], the following estimate hold
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglez∞−/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(0)+f′(0)√π√α+iβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤√2E0c0π
α/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(0)+f′(0)√π√α+iβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/parenleftbigg√2E0c0π
2α/parenrightbigg2
. (3.126)
Proof. Since we are assuming that the initial conditions f(0)andf′(0)depend smoothly on c0,
by classical results from the ODE theory, the functions f,y,handzare smooth with respect to s
andc0. From (3.39) with s= 1, we have that z∞can be written in terms of continuous functions
ofc0(the continuity of the integral terms follows from the domin ated convergence theorem), so
thatz∞depends continuously on c0.
To prove (3.126), we multiply (3.20) by e(α+iβ)s2/4, so that
(f′e(α+iβ)s2/4)′=−c2
0
4f(s)e(−α+iβ)s2/4.
Hence, integrating twice, we have
f(s) =f(0)+G(s)+F(s), (3.127)
with
G(s) =f′(0)/integraldisplays
0e−(α+iβ)σ2/4dσandF(s) =−c2
0
4/integraldisplays
0e−(α+iβ)σ2/4/integraldisplayσ
0e(−α+iβ)τ2/4f(τ)dτdσ.
33Since by Proposition 3.2 |f(s)| ≤2√2E0
c0, we obtain
|F(s)| ≤√2E0c0
2/integraldisplays
0e−ασ2/4/integraldisplayσ
0e−ατ2/4dτdσ≤√2E0c0
2·π
α. (3.128)
Using (3.127) and the identity,
|z1+z2|2=|z1|2+2Re(¯z1z2)+|z2|2, z1,z2∈C,
we conclude that z(s) =|f(s)|2satisfies
z(s) =|f(0)+G(s)|2+2Re(¯F(s)(f(0)+G(s)))+|F(s)|2.
Therefore, for all s≥0,
|z(s)−|f(0)+G(s)|2| ≤2|F(s)||f(0)+G(s)|+|F(s)|2.
Hence we can use the bound (3.128) and then let s→ ∞. Noticing that
lim
s→∞G(s) =f′(0)/integraldisplay∞
0e−(α+iβ)σ2/4dσ=f′(0)√π√α+iβ,
the estimate (3.126) follows.
4 Proof of the main results
In Section 3 we have performed a careful analysis of the equat ion (3.12), taking also into con-
sideration the initial conditions (3.13)–(3.15). Therefo re, the proofs of our main theorem consist
mainly in coming back to the original variables using the ide ntities (3.18) and (3.19). For the
sake of completeness, we provide the details in the followin g proofs.
Proof of Theorem 1.2 .Letα∈[0,1],c0>0and{/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)}be the unique
C∞(R;S2)-solution of the Serret–Frenet equations (1.6) with curvat ure and torsion (2.6) and
initial conditions (2.8). In order to simplify the notation , in the rest of the proof we drop the
subindexes c0andαand simply write {/vector m(·),/vector n(·),/vectorb(·)}for{/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)}.
First observe that if we define {/vectorM,/vectorN,/vectorB}in terms of {/vector m,/vector n,/vectorb}by
/vectorM(s) = (m(−s),−m(−s),−m(−s)),
/vectorN(s) = (−n(−s),n(−s),n(−s)),
/vectorB(s) = (−b(−s),b(−s),b(−s)), s∈R,
then{/vectorM,/vectorN,/vectorB}is also a solution of the Serret system (1.6) with curvature a nd torsion (2.6).
Notice also that
{/vectorM(0),/vectorN(0),/vectorB(0)}={/vector m(0),/vector n(0),/vectorb(0)}.
Therefore, from the uniqueness of the solution we conclude t hat
/vectorM(s) =/vector m(s),/vectorN(s) =/vector n(s)and/vectorB(s) =/vectorb(s),∀s∈R.
This proves part (i)of Theorem 1.2.
34Second, in Section 3 we have seen that one can write the compon ents of the Frenet trihedron
{/vector m,/vector n,/vectorb}as
m1(s) = 2|f1(s)|2−1, n 1(s)+ib1(s) =4
c0eαs2/4¯f1(s)f′
1(s), (4.1)
mj(s) =|fj(s)|2−1, n j(s)+ibj(s) =2
c0eαs2/4¯fj(s)f′
j(s), j∈ {2,3}, (4.2)
withfjsolution of the second order ODE (3.12) with initial conditi ons (3.13)-(3.15) respectively,
and associated initial energies (see (3.17))
E0,1=c2
0
8andEj,1=c2
0
8,forj∈ {2,3}. (4.3)
Notice that the identities (4.1)–(4.2) rewrite equivalent ly as
m1,c0,α= 2z1−1, n1,c0,α=4
c0eαs2/4y1, b1,c0,α=4
c0eαs2/4h1,
mj,c0,α=zj−1, nj,c0,α=2
c0eαs2/4yj, bj,c0,α=2
c0eαs2/4hj, j∈ {2,3},(4.4)
in terms of the quantities {zj,yj,hj}defined by
zj=|fj|2, yj= Re(¯fjf′
j)andhj= Im(¯fjf′
j).
Denote by zj,∞,aj,bj,γjandφjthe constants and function appearing in the asymptotics of
{yj,hj,zj}proved in Section 3 in Corollary 3.14.
Taking the limit as s→+∞in (4.1)–(4.2), and since |/vector m(s)|= 1, we obtain that there exists
/vectorA+= (A+
j)3
j=1∈S2with
A+
1= 2z1,∞−1, A+
j=zj,∞−1,forj∈ {2,3}. (4.5)
The asymptotics stated in part (ii)of Theorem 1.2 easily follows from formulae (4.1)–(4.2) and the
asymptotics for {zj,yj,hj}established in Corollary 3.14. Indeed, it suffices to observe that from
the formulae for bjandγjin terms of the initial energies E0,jandzj,∞given in Corollary 3.14,
(4.3) and (4.5) we obtain
b2
1=c2
0
16(1−(A+
1)2), b2
2=c2
0
4(1−(A+
2)2), b2
3=c2
0
4(1−(A+
3)2), (4.6)
γ1=−c2
0
4A+
1, γ2=−c2
0
2A+
2, γ3=−c2
0
2A+
3. (4.7)
Substituting these constants in (3.95), (3.96) and (3.97) i n Corollary 3.14, we obtain (1.16),
(1.17) and (1.18). This completes the proof of Theorem 1.2- (ii).
Proof of Theorem 1.1 .Letα∈[0,1], andc0>0. As before, dropping the subindexes, we
will denote by {/vector m,/vector n,/vectorb}the unique solution of the Serret–Frenet equations (1.6) wi th curvature
and torsion (2.6) and initial conditions (2.8). Define
/vectorm(s,t) =/vector m/parenleftbiggs√
t/parenrightbigg
. (4.8)
35As has been already mentioned (see Section 2), part (i)of Theorem 1.1 follows from the fact
that the triplet {/vector m,/vector n,/vectorb}is a regular- (C∞(R;S2))3solution of (1.6)-(2.6)-(2.8) and satisfies the
equation
−s
2c/vector n=β(c′/vectorb−cτ/vector n)+α(cτ/vectorb+c′/vector n).
Next, from the parity of the components of the profile /vector m(·)and the asymptotics established in
parts(i)and(ii)in Theorem 1.2, it is immediate to prove the pointwise conver gence (1.9). In
addition,/vectorA−= (A+
1,−A+
2,−A+
3)in terms of the components of the vector /vectorA+= (A+
j)3
j=1.
Now, using the symmetries of /vector m(·), the change of variables η=s/√
tgives us
/bardbl/vectorm(·,t)−/vectorA+χ(0,∞)(·)−/vectorA−χ(−∞,0)(·)/bardblLp(R)=3/summationdisplay
j=1/parenleftbigg
2t1/2/integraldisplay∞
0|mj(η)−A+
j|pdη/parenrightbigg1/p
.(4.9)
Therefore, it only remains to prove that the last integral is finite. To this end, let s0= 4/radicalbig
8+c2
0.
On the one hand, notice that since /vector mand/vectorA+are unitary vectors,
/integraldisplays0
0|mj(s)−Aj|pds≤2ps0. (4.10)
On the other hand, from the asymptotics for /vector m(·)in (1.16), (1.20), and the fact that the vectors
/vectorA+and/vectorB+satisfy|/vectorA+|2= 1and|/vectorB+|2= 2, we obtain
/parenleftbigg/integraldisplay∞
s0|mj(s)−A+
j|pds/parenrightbigg1/p
≤2√
2c0(α+β)/parenleftBigg/integraldisplay∞
s0e−αs2p/4
sp/parenrightBigg1/p
+2c2
0/parenleftBigg/integraldisplay∞
s0e−αs2p/2
s2p/parenrightBigg1/p
+C(c0)/parenleftBigg/integraldisplay∞
s0e−αs2p/4
s3p/parenrightBigg1/p
. (4.11)
Since the r.h.s. of (4.11) is finite for all p∈(1,∞)ifα∈[0,1], and for all p∈[1,∞)if
α∈(0,1], inequality (1.10) follows from (4.9), (4.10) and (4.11). T his completes the proof of
Theorem 1.1.
Proof of Theorem 1.3 .The proof is a consequence of Proposition 3.15. In fact, reca ll the
relations (4.5) and (3.17), that is
A+
1= 2z1,∞−1,andA+
j=zj,∞−1,forj∈ {2,3},
and
E0,1=c2
0
8, E 0,j=c2
0
4,forj∈ {2,3},
Thus the continuity of /vectorA+
c0,αwith respect to c0, follows from the continuity of z∞in Proposi-
tion 3.15.
Using the initial conditions (3.13)–(3.15), the values for the energies E0,jforj∈ {1,2,3}, and
the identity√π√α+iβ=√π√
2/parenleftbig√
1+α−i√
1−α/parenrightbig
,
we now compute
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglefj(0)+f′
j(0)√π√α+iβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
=
1, ifj= 1,
1+c2
0π
4+c0√π√
2√1+α,ifj= 2,
1+c2
0π
4+c0√π√
2√1−α,ifj= 3.(4.12)
36Then, substituting the values (4.12) in (3.126) and using th e above relations together with the
inequality√1+x≤1+x/2forx≥0, we obtain the estimates (1.24)–(1.26).
Proof of Theorem 1.4 .Recall that the components of /vectorA+
c0,αare given explicitly in (4.5) in
terms of the functions zj,∞, forj∈ {1,2,3}. The continuity on [0,1]ofA+
j,c0,αas a function
ofαforj∈ {1,2,3}follows from that of zj,∞established in Lemma 3.12. Notice also that the
estimates (1.27) and (1.28) are an immediate consequence of (3.117) in Lemma 3.12 and (3.114)
in Lemma 3.11, respectively.
Before giving the proof of Proposition 1.5, we recall that whe nα= 0orα= 1, the vector
/vectorA+
c0,α= (Aj,c0,α)3
j=1is determined explicitly in terms of the parameter c0(see [15] for the case
α= 0and Appendix for the case α= 1). Precisely,
A1,c0,0=e−πc2
0
2, (4.13)
A2,c0,0= 1−e−πc2
0
4
8πsinh(πc2
0/2)|c0Γ(ic2
0/4)+2eiπ/4Γ(1/2+ic2
0/4)|2, (4.14)
A3,c0,0= 1−e−πc2
0
4
8πsinh(πc2
0/2)|c0Γ(ic2
0/4)−2e−iπ/4Γ(1/2+ic2
0/4)|2(4.15)
and
/vectorA+
c0,1= (cos(c0√π),sin(c0√π),0). (4.16)
Proof of Proposition 1.5 .Recall that (see Theorem 1.1)
/vectorA−
c0,α= (A+
1,c0,α,−A+
2,c0,α,−A+
3,c0,α), (4.17)
withA+
j,c0,αthe components of /vectorA+
c0,α. Therefore /vectorA+
c0,α/ne}ationslash=/vectorA−
c0,αiffA+
1,c0,α/ne}ationslash= 1or−1.
Parts (ii)and(iii)follow from the continuity of A+
1,c0,αin[0,1]established in Theorem 1.4
bearing in mind that, from the expressions for A+
1,c0,0in (4.13) and A+
1,c0,1in (4.16), we have that
A+
1,c0,0/ne}ationslash=±1for allc0>0andA+
1,c0,1/ne}ationslash=±1ifc0/ne}ationslash=k√πwithk∈N.
In order to proof part (i), we will argue by contradiction. Assume that for some α∈(0,1),
there exists a sequence {c0,n}n∈Nsuch thatc0,n>0,c0,n−→0asn→ ∞ and/vectorA+
c0,n,α=/vectorA−
c0,nα.
Hence from (4.17) the second and third component of /vectorA+
c0,n,αare zero. Thus the estimate (1.25)
in Theorem 1.3 yields
c0,n/radicalbig
π(1+α)√
2≤c2
0,nπ
4+c2
0,nπ
α√
2/parenleftBigg
1+c2
0,nπ
8+c0,n/radicalbig
π(1+α)
2√
2/parenrightBigg
+/parenleftBigg
c2
0,nπ
2√
2α/parenrightBigg2
.
Dividing by c0,n>0and letting c0,n→0asn→ ∞, the contradiction follows.
5 Some numerical results
As has been already pointed out, only in the cases α= 0andα= 1we have an explicit formula
for/vectorA+
c0,α(see (4.13)–(4.16)). Theorems 1.3 and 1.4 give information about the behaviour of /vectorA+
c0,α
for small values of c0for a fixed valued of α, and for values of αnear to 0 or 1 for a fixed valued of
c0. The aim of this section is to give some numerical results tha t allow us to understand the map
37(α,c0)∈[0,1]×(0,∞)/ma√sto→/vectorA±
c0,α∈S2. For a fixed value of α, we will discuss first the injectivity
and surjectivity (in some appropriate sense) of the map c0/ma√sto→/vectorA±
c0,αand second the behaviour of
/vectorA+
c0,αasc0→ ∞.
For fixedα, defineθc0,αto be the angle between the unit vectors /vectorA+
c0,αand−/vectorA−
c0,αassociated
to the family of solutions /vectormc0,α(s,t)established in Theorem 1.1, that is θc0,αsuch that
cos(θc0,α) = 1−2(A+
1,c0,α)2. (5.1)
It is pertinent to ask whether θc0,αmay attain any value in the interval [0,π]by varying the
parameterc0>0.
In Figure 2 we plot the function θc0,αassociated to the family of solutions /vectormc0,α(s,t)estab-
lished in Theorem 1.1 for α= 0,α= 0.4andα= 1, as a function of c0>0. The curves θc0,0
andθc0,1are exact since we have explicit formulae for A+
1,c0,αwhenα= 0andα= 1(see (4.13)
and (4.16)). We deduce that in the case α= 0, there is a bijective relation between c0>0and
the angles in (0,π). In the case α= 1, there are infinite values of c0>0that allow to reach
any angle in [0,π]. Ifα∈(0,1), numerical simulations show that there exists θ∗
α∈(0,π)such
that the angles in (θ∗
α,π)are reached by a unique value of c0, but for angles in [0,θ∗
α]there are
at least two values of c0>0that produce them (See θc0,0.4in Figure 2).
θc0,0
π
c0
θc0,0.4
π
c0
θc0,1
π
c0
Figure 2: The angles θc0,αas a function of c0forα= 0,α= 0.4andα= 1.
These numerical results suggest that, due to the invariance of (LLG) under rotations2, for a
fixedα∈[0,1)one can solve the following inverse problem: Given any disti nct vectors /vectorA+,/vectorA−∈
S2there exists c0>0such that the associated solution /vectormc0,α(s,t)given by Theorem 1.1 (possibly
multiplied by a rotation matrix) provides a solution of (LLG ) with initial condition
/vectorm(·,0) =/vectorA+χ(0,∞)(·)+/vectorA−χ(−∞,0)(·). (5.2)
Note that in the case α= 1the restriction /vectorA+/ne}ationslash=/vectorA−can be dropped.
In addition, Figure 2 suggests that /vectorA+
c0,α/ne}ationslash=/vectorA−
c0,αfor fixedα∈[0,1)andc0>0. Indeed,
notice that /vectorA+
c0,α/ne}ationslash=/vectorA−
c0,αif and only if A1/ne}ationslash=±1or equivalently cosθc0,α/ne}ationslash=−1, that isθc0,α/ne}ationslash=π,
which is true if α∈[0,1)for anyc0>0(See Figure 2). Notice also that when α= 1, then the
valueπis attained by different values of c0.
The next natural question is the injectivity of the applicat ionc0−→θc0,α, for fixed α.
Precisely, can we generate the same angle using different val ues ofc0? In the case α= 0, the
2In fact, using that
(M/vector a)×(M/vectorb) = (det M)M−T(/vector a×/vectorb),for allM∈ M3,3(R), /vector a,/vectorb∈R3,
it is easy to verify that if /vectorm(s,t)is a solution of (LLG) with initial condition /vectorm0, then/vectormR:=R/vectormis a solution
of (LLG) with initial condition /vectorm0
R:=R/vectorm0, for any R∈SO(3).
38plot ofθc0,0in Figure 2 shows that the value of c0is unique, in fact one has following formula
sin(θc0,0/2) =A1,c0,0=e−c2
0
2π(see [15]). In the case α= 1, we have sin(θc0,1/2) =A1,c0,1=
cos(c0/radicalbig
π), moreover
/vectorA+
c0,1=/vectorA+
c0+2k√π,1,for anyk∈Z. (5.3)
As before, if α∈(0,1)we do not have an analytic answer and we have to rely on numeric al
simulations. However, it is difficult to test the uniqueness o fc0numerically. Using the command
FindRoot in Mathematica, we have found such values. For instance, for α= 0.4, we obtain that
c0≈2.1749andc0≈6.6263give the same value of /vectorA+
c0,0.4. The respective profiles /vector mc0,0.4(·)are
shown in Figure 3. This multiplicity of solutions suggests t hat the Cauchy problem for (LLG)
with initial condition (5.2) is ill-posed, at least for cert ain values of c0. This interesting problem
will be studied in a forthcoming paper.
m1m2m3
(a)/vector mc0,0.4(·), withc0≈2.1749
m1m2m3
(b)/vector mc0,0.4(·), withc0≈6.6263
Figure 3: Two profiles /vector mc0,0.4(·), with the same limit vector /vectorA+
c0,0.4.
The rest of this section is devoted to give some numerical res ults on the behaviour of the
limiting vector /vectorA+
c0,α. In particular, the results below aim to complement those es tablished in
Theorem 1.3 on the behaviour of /vectorA+
c0,αfor small values of c0, whenαis fixed.
We start recalling what it is known in the extremes cases α= 0andα= 1. Precisely, if
α= 0, the explicit formulae (4.13)–(4.15) for /vectorA+
c0,0allow us to prove that
lim
c0→0+A+
3,c0,0= 0 andlim
c0→∞A+
3,c0,1= 1, (5.4)
and also that {A+
3,c0,0:c0∈(0,∞)}= (0,1). Whenα= 1the picture is completely different. In
factA+
3,c0,1= 0for allc0>0, and the limit vectors remain in the equator plane S1×{0}. The
natural question is what happens with /vectorA+
c0,αwhenα∈(0,1)as a function of c0.
Although we do not provide a rigorous answer to this question , in Figure 4 we show some
numerical results. Precisely, Figure 4 depicts the curves /vectorA+
c0,0.01,/vectorA+
c0,0.4and/vectorA+
c0,0.8as functions
ofc0, forc0∈[0,1000]. We see that the behaviour of /vectorA+
c0,αchanges when αincreases in the sense
that the first and second coordinates start oscillating more and more as αgoes to 1. In all the
cases the third component remains monotonically increasin g withc0, but the value of A+
3,1000,α
seems to be decreasing with α. At this point it is not clear what the limit value of A+
3,c0,αas
39c0→ ∞ is. For this reason, we perform a more detailed analysis of A+
3,c0,αand we show the
curvesA+
3,1,α,A+
3,10,α,A+
3,1000,α(for fixedα∈[0,1]) in Figure 5. From these results we conjecture
that{A+
3,c0,·}c0>0is a pointwise nondecreasing sequence of functions that con verges to 1for any
α<1asc0→ ∞. This would imply that, for α∈(0,1)fixed,A1,c0,α→0asc0→ ∞, and since
A1,c0,α→1asc0→0(see (1.24)), we could conclude by continuity (see Theorem 1 .3) that for
any angleθ∈(0,π)there exists c0>0such thatθis the angle between /vectorA+
c0,αand−/vectorA+
c0,α(see
(5.1)). This provides an alternative way to justify the surj ectivity of the map c0/ma√sto→/vectorA+
c0,α(in the
sense explained above).
A+
1A+
2A+
3
(a)/vectorA+
c0,0.01
A+
1A+
2A+
3
(b)/vectorA+
c0,0.4
A+
1A+
2A+
3
(c)/vectorA+
c0,0.8
Figure 4: The curves /vectorA+
c0,0.01,/vectorA+
c0,0.4and/vectorA+
c0,0.8as functions of c0, forc0∈[0,1000].
01
1αA+
3,1,αA+
3,10,αA+
3,1000,α
Figure 5: The curves A+
3,1,α,A+
3,10,α,A+
3,1000,αas functions of α, forα∈[0,1].
The curves in Figure 5 also allow us to discuss further the res ults in Theorem 1.4. In fact,
whenαis close to 1 the slope of the functions become unbounded and, roughly speaking, the
behaviour of A+
3,c0,αis in agreement with the result in Theorem 1.4, that is
A+
3,c0,α∼C(c0)√
1−α,asα→1−.
Numerically, the analysis is more difficult when α∼0, because the number of computations
needed to have an accurate profile of A+
3,c0,αincreases drastically as α→0+. In any case,
Figure 5 suggests that A+
3,c0,αconverges to A+
3,c0,0faster than√α|ln(α)|. We think that this rate
of convergence can be improved to α|ln(α)|. In fact, in the proof of Lemma 3.10 we only used
energy estimates. Probably, taking into account the oscill ations in equation (3.102) (as did in
Proposition 3.3), it would be possible to establish the nece ssary estimates to prove the following
conjecture:
|/vectorA+
c0,α−/vectorA+
c0,0| ≤C(c0)α|ln(α)|,forα∈(0,1/2].
406 Appendix
In this appendix we show how to compute explicitly the soluti on/vectormc0,α(s,t)of the LLG equation
in the case α= 1. As a consequence, we will obtain an explicit formula for the limiting vector
/vectorA+
c0,1and the other constants appearing in the asymptotics of the a ssociated profile established
in Theorem 1.2 in terms of the parameter c0in the case when α= 1.
We start by recalling that if α= 1thenβ= 0. We need to find the solution {/vector m,/vector n,/vectorb}of the
Serret–Frenet system (1.6) with c(s) =c0e−s2/4,τ≡0and the initial conditions (1.8). Hence,
it is immediate that
m3=n3≡0, b1=b2≡0andb3≡1.
To compute the other components, we use the Riccati equation (3.9) satisfied by the stereographic
projection of {mj,nj,bj}
ηj=nj+ibj
1+mj,forj∈ {1,2}, (6.1)
found in the proof of Lemma 3.1. For the values of curvature an d torsionc(s) =c0e−s2/4and
τ(s) = 0 the Riccati equation (3.9) reads
η′
j+iβs
2ηj+c0
2e−αs2/4(η2
j+1) = 0. (6.2)
We see that when α= 1, and thusβ= 0, (6.2) is a separable equation that we write as:
dηj
η2
j+1=−c0
2e−αs2/4,
so integrating, we get
ηj(s) = tan/parenleftBig
arctan(ηj(0))−c0
2Erf(s)/parenrightBig
, (6.3)
whereErf(s)is the non-normalized error function
Erf(s) =/integraldisplays
0e−σ2/4dσ.
Also, using (1.8) and (6.1) we get the initial conditions η1(0) = 0 andη2(0) = 1 . In particular,
ifc0is small (6.3) is the global solution of the Riccati equation , but it blows-up in finite time if
c0is large. As long as ηjis well-defined, by Lemma 3.1,
fj(s) =ec0
2/integraltexts
0e−ασ2/4ηj(σ)dσ.
The change of variables
µ= arctan(ηj(0))−c0
2Erf(s)
yields/integraldisplays
0e−ασ2/4ηj(σ)dσ=2
c0ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos/parenleftbig
arctan(ηj(0))−c0
2Erf(s)/parenrightbig
cos(arctan( ηj(0)))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,
and after some simplifications, we obtain
f1(s) =/vextendsingle/vextendsingle/vextendsinglecos/parenleftBigc0
2Erf(s)/parenrightBig/vextendsingle/vextendsingle/vextendsingleandf2(s) =/vextendsingle/vextendsingle/vextendsinglecos/parenleftBigc0
2Erf(s)/parenrightBig
+sin/parenleftBigc0
2Erf(s)/parenrightBig/vextendsingle/vextendsingle/vextendsingle.
In view of (3.18) and (3.19), we conclude that
m1(s) = 2|f1(s)|2−1 = cos(c0Erf(s))andm2(s) =|f2(s)|2−1 = sin(c0Erf(s)).(6.4)
41A priori, the formulae in (6.4) are valid only as long as ηis well-defined, but a simple verification
show that these are the global solutions of (1.6), with
n1(s) =−sin(c0Erf(s))andn2(s) = cos(c0Erf(s)).
In conclusion, we have proved the following:
Proposition 6.1. Letα= 1, and thusβ= 0. Then, the trihedron {/vector mc0,1,/vector nc0,1,/vectorbc0,1}solution
of(1.6)–(1.8)is given by
/vector mc0,1(s) = (cos(c0Erf(s)),sin(c0Erf(s)),0),
/vector nc0,1(s) =−(sin(c0Erf(s)),cos(c0Erf(s)),0),
/vectorbc0,1(s) = (0,0,1),
for alls∈R. In particular, the limiting vectors /vectorA+
c0,1and/vectorA−
c0,1in Theorem 1.2 are given in
terms ofc0as follows:
/vectorA±
c0,1= (cos(c0√π),±sin(c0√π),0).
Proposition 6.1 allows us to give an alternative explicit pr oof of Theorem 1.2 when α= 1.
Corollary 6.2. [Explicit asymptotics when α= 1] With the same notation as in Proposition 6.1,
the following asymptotics for {/vector mc0,1,/vector nc0,1,/vectorbc0,1}holds true:
/vector mc0,1(s) =/vectorA+
c0,1−2c0
s/vectorB+
c0,1e−s2/4sin(/vector a)−2c2
0
s2/vectorA+
c0,1e−s2/2+O/parenleftBigg
e−s2/4
s3/parenrightBigg
,
/vector nc0,1(s) =/vectorB+
c0,1sin(/vector a)+2c0
s/vectorA+
c0,1e−s2/4−2c2
0
s2/vectorB+
c0,1e−s2/2sin(/vector a)+O/parenleftBigg
e−s2/4
s3/parenrightBigg
,
/vectorbc0,1(s) =/vectorB+
c0,1cos(/vector a),
where the vectors /vectorA+
c0,1,/vectorB+
c0,1and/vector a= (aj)3
j=1are given explicitly in terms of c0by
/vectorA+
c0,1= (cos(c0√π),sin(c0√π),0),/vectorB+
c0,1= (|sin(c0√π)|,|cos(c0√π)|,1),
a1=/braceleftBigg
3π
2,ifsin(c0√π)≥0,
π
2,ifsin(c0√π)<0,a2=/braceleftBigg
π
2,ifcos(c0√π)≥0,
3π
2,ifcos(c0√π)<0,anda3= 0.
Here, the bounds controlling the error terms depend on c0.
Proof. By Proposition 6.1,
/vector mc0,1(s) = (cos(c0√π−c0Erfc(s)),sin(c0√π−c0Erfc(s)),0),
/vector nc0,1(s) =−(sin(c0√π−c0Erfc(s)),cos(c0√π−c0Erfc(s)),0),
/vectorbc0,1(s) = (0,0,1),(6.5)
where the complementary error function is given by
Erfc(s) =/integraldisplay∞
se−σ2/4dσ=√π−Erf(s).
It is simple to check that
sin(c0Erfc(s)) =e−s2/4/parenleftbigg2c0
s−4c0
s3+24c0
s5+O/parenleftBigc0
s7/parenrightBig/parenrightbigg
,
cos(c0Erfc(s)) = 1+e−s2/2/parenleftbigg
−2c2
0
s2+8c2
0
s4−56c2
0
s6+O/parenleftbiggc2
0
s8/parenrightbigg/parenrightbigg
,
42so that, using (6.5), we obtain that
m1(s) =n2(s) = cos(c0√π)+2c0
se−s2/4sin(c0√π)−2c2
0
s2e−s2/2cos(c0√π)+O/parenleftBigg
e−s2/4
s3/parenrightBigg
,
m2(s) =−n1(s) = sin(c0√π)−2c0
se−s2/4cos(c0√π)−2c2
0
s2e−s2/2sin(c0√π)+O/parenleftBigg
e−s2/4
s3/parenrightBigg
.
The conclusion follows from the definitions of /vectorA+
c0,1,/vectorB+
c0,1and/vector a.
Remark 6.3. Notice that /vector ais not a continuous function of c0, but the vectors (B+
jsin(aj))3
j=1
and(B+
jcos(aj))3
j=1are.
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45 |
1104.1625v1.Magnetization_Dissipation_in_Ferromagnets_from_Scattering_Theory.pdf | arXiv:1104.1625v1 [cond-mat.mes-hall] 8 Apr 2011Magnetization Dissipation in Ferromagnets from Scatterin g Theory
Arne Brataas∗
Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
Yaroslav Tserkovnyak
Department of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA
Gerrit E. W. Bauer
Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan and
Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands
The magnetization dynamicsofferromagnets are often formu lated intermsof theLandau-Lifshitz-
Gilbert (LLG) equation. The reactive part of this equation d escribes the response of the magnetiza-
tion in terms of effective fields, whereas the dissipative par t is parameterized by the Gilbert damping
tensor. We formulate a scattering theory for the magnetizat ion dynamics and map this description
on the linearized LLG equation by attaching electric contac ts to the ferromagnet. The reactive part
can then be expressed in terms of the static scattering matri x. The dissipative contribution to the
low-frequency magnetization dynamics can be described as a n adiabatic energy pumping process
to the electronic subsystem by the time-dependent magnetiz ation. The Gilbert damping tensor
depends on the time derivative of the scattering matrix as a f unction of the magnetization direction.
By the fluctuation-dissipation theorem, the fluctuations of the effective fields can also be formulated
in terms of the quasistatic scattering matrix. The theory is formulated for general magnetization
textures and worked out for monodomain precessions and doma in wall motions. We prove that the
Gilbert damping from scattering theory is identical to the r esult obtained by the Kubo formalism.
PACS numbers: 75.40.Gb,76.60.Es,72.25.Mk
I. INTRODUCTION
Ferromagnets develop a spontaneous magnetization
below the Curie temperature. The long-wavelengthmod-
ulations of the magnetization direction consist of spin
waves, the low-lying elementary excitations (Goldstone
modes) of the ordered state. When the thermal energy is
much smaller than the microscopic exchange energy, the
magnetization dynamics can be phenomenologically ex-
pressed in a generalized Landau-Lifshitz-Gilbert (LLG)
form:
˙ m(r,t) =−γm(r,t)×[Heff(r,t)+h(r,t)]+
m(r,t)×/integraldisplay
dr′[˜α[m](r,r′)˙ m(r′,t)],(1)
where the magnetization texture is described by m(r,t),
the unit vector along the magnetization direction at po-
sitionrand timet,˙ m(r,t) =∂m(r,t)/∂t,γ=gµB//planckover2pi1is
the gyromagnetic ratio in terms of the g-factor (≈2 for
free electrons) and the Bohr magneton µB. The Gilbert
damping ˜αis a nonlocal symmetric 3 ×3 tensor that is
a functional of m. The Gilbert damping tensor is com-
monly approximated to be diagonal and isotropic (i), lo-
cal (l), and independent of the magnetization m, with
diagonal elements
αil(r,r′) =αδ(r−r′). (2)
The linearized version of the LLG equation for small-
amplitude excitations has been derived microscopically.1It has been used very successfully to describe the mea-
sured response of ferromagnetic bulk materials and thin
films in terms of a small number of adjustable, material-
specific parameters. The experiment of choice is fer-
romagnetic resonance (FMR), which probes the small-
amplitude coherent precession of the magnet.2The
Gilbertdampingmodelinthelocalandtime-independent
approximationhasimportantramifications, suchasalin-
ear dependence of the FMR line width on resonance fre-
quency, that have been frequently found to be correct.
The damping constant is technologically important since
it governs the switching rate of ferromagnets driven by
external magnetic fields or electric currents.3In spatially
dependent magnetization textures, the nonlocal charac-
ter of the damping can be significant as well.4–6Moti-
vated by the belief that the Gilbert damping constant is
animportantmaterialproperty, weset outheretounder-
stand its physical origins from first principles. We focus
on the well studied and technologically important itiner-
ant ferromagnets, although the formalism can be used in
principle for any magnetic system.
The reactive dynamics within the LLG Eq. (1) is de-
scribed by the thermodynamic potential Ω[ M] as a func-
tional of the magnetization. The effective magnetic field
Heff[M](r)≡ −δΩ/δM(r) is the functional derivative
with respect to the local magnetization M(r) =Msm(r),
including the external magnetic field Hext, the magnetic
dipolar field Hd, the texture-dependent exchange energy,
and crystal field anisotropies. Msis the saturation mag-
netization density. Thermal fluctuations can be included
by a stochastic magnetic field h(r,t) with zero time av-2
left
reservoirF N Nright
reservoir
FIG. 1: Schematic picture of a ferromagnet (F) in contact
with a thermal bath (reservoirs) via metallic normal metal
leads (N).
erage,/an}b∇acketle{th/an}b∇acket∇i}ht= 0, and white-noise correlation:7
/an}b∇acketle{thi(r,t)hj(r′,t′)/an}b∇acket∇i}ht=2kBT
γMs˜αij[m](r,r′)δ(t−t′),(3)
whereMsis the magnetization, iandjare the Cartesian
indices, and Tis the temperature. This relation is a con-
sequence ofthe fluctuation-dissipation theorem (FDT) in
the classical (Maxwell-Boltzmann) limit.
The scattering ( S-) matrix is defined in the space of
the transport channels that connect a scattering region
(the sample) to real or fictitious thermodynamic (left
and right) reservoirs by electric contacts with leads that
are modeled as ideal wave guides. Scattering matri-
ces are known to describe transport properties, such as
the giant magnetoresistance, spin pumping, and current-
inducedmagnetizationdynamicsinlayerednormal-metal
(N)|ferromagnet (F).8–10When the ferromagnet is part
of an open system as in Fig. 1, also Ω can be expressed
in terms of the scattering matrix, which has been used
to express the non-local exchange coupling between fer-
romagnetic layers through conducting spacers.11We will
show here that the scattering matrix description of the
effective magnetic fields is valid even when the system is
closed, provided the dominant contribution comes from
the electronic band structure, scattering potential disor-
der, and spin-orbit interaction.
Scattering theory can also be used to compute the
Gilbert damping tensor ˜ αfor magnetization dynamics.15
The energy loss rate of the scattering region can be ex-
pressedin termsofthe time-dependent S-matrix. To this
end, the theory of adiabatic quantum pumping has to be
generalizedtodescribedissipationinametallicferromag-
net. The Gilbert damping tensor is found by evaluating
the energy pumping out of the ferromagnet and relat-
ing it to the energy loss that is dictated by the LLG
equation. In this way, it is proven that the Gilbert phe-
nomenology is valid beyond the linear response regime
of small magnetization amplitudes. The key approxima-
tion that is necessary to derive Eq. (1) including ˜ αis the
(adiabatic) assumption that the ferromagnetic resonance
frequencyωFMRthat characterizesthe magnetizationdy-
namics is small compared to internal energy scale set by
the exchange splitting ∆ and spin-flip relaxation rates
τs. The LLG phenomenology works well for ferromag-
nets for which ωFMR≪∆//planckover2pi1, which is certainly the case
for transition metal ferromagnets such as Fe and Co.
Gilbert damping in transition-metal ferromagnets is
generally believed to stem from the transfer of energy
fromthemagneticorderparametertotheitinerantquasi-particle continuum. This requires either magnetic disor-
der or spin-orbit interactions in combination with impu-
rity/phonon scattering.2Since the heat capacitance of
the ferromagnet is dominated by the lattice, the energy
transferred to the quasiparticles will be dissipated to the
lattice as heat. Here we focus on the limit in which elas-
tic scattering dominates, such that the details of the heat
transfer to the lattice does not affect our results. Our ap-
proachformallybreaks down in sufficiently clean samples
at high temperatures in which inelastic electron-phonon
scattering dominates. Nevertheless, quantitative insight
can be gained by our method even in that limit by mod-
elling phonons by frozen deformations.12
In the present formulation, the heat generated by the
magnetization dynamics can escape only via the contacts
to the electronic reservoirs. By computing this heat cur-
rent through the contacts we access the total dissipa-
tion rate. Part of the heat and spin current that es-
capes the sample is due to spin pumping that causes
energy and momentum loss even for otherwise dissipa-
tion less magnetization dynamics. This process is now
wellunderstood.10For sufficiently largesamples, the spin
pumping contribution is overwhelmed by the dissipation
in the bulk of the ferromagnet. Both contributions can
be separated by studying the heat generation as a func-
tion of the length of a wire. In principle, a voltage can be
added to study dissipation in the presence of electric cur-
rents as in 13,14, but we concentrate here on a common
and constant chemical potential in both reservoirs.
Although it is not a necessity, results can be simpli-
fied by expanding the S-matrix to lowest order in the
amplitude of the magnetization dynamics. In this limit
scattering theory and the Kubo linear response formal-
ism for the dissipation can be directly compared. We
will demonstrate explicitly that both approaches lead to
identical results, which increases our confidence in our
method. The coupling to the reservoirs of large samples
is identified to play the same role as the infinitesimals in
the Kubo approach that guarantee causality.
Our formalism was introduced first in Ref. 15 lim-
ited to the macrospin model and zero temperature. An
extension to the friction associatedwith domain wall mo-
tion was given in Ref. 13. Here we show how to handle
general magnetization textures and finite temperatures.
Furthermore, we offer an alternative route to derive the
Gilbert damping in terms of the scattering matrix from
the thermal fluctuations of the effective field. We also
explain in more detail the relation of the present theory
to spin and charge pumping by magnetization textures.
Our paper is organized in the following way. In Sec-
tion II, we introduce our microscopic model for the fer-
romagnet. In Section III, dissipation in the Landau-
Lifshitz-Gilbert equation is exposed. The scattering the-
ory of magnetization dynamics is developed in Sec. IV.
We discuss the Kubo formalism for the time-dependent
magnetizationsin Sec. V, before concluding our article in
Sec. VI. The Appendices provide technical derivations of
spin, charge, and energy pumping in terms of the scat-3
tering matrix of the system.
II. MODEL
Our approach rests on density-functional theory
(DFT), which is widely and successfully used to describe
the electronic structure and magnetism in many fer-
romagnets, including transition-metal ferromagnets and
ferromagnetic semiconductors.16In the Kohn-Sham im-
plementation of DFT, noninteracting hypothetical par-
ticles experience an effective exchange-correlationpoten-
tial that leads to the same ground-statedensity as the in-
teractingmany-electronsystem.17Asimpleyetsuccessful
scheme is the local-densityapproximationto the effective
potential. DFT theory can also handle time-dependent
phenomena. We adopt here the adiabatic local-density
approximation (ALDA), i.e. an exchange-correlationpo-
tential that is time-dependent, but local in time and
space.18,19As the name expresses, the ALDA is valid
when the parametric time-dependence of the problem is
adiabatic with respect to the electron time constants.
Here we consider a magnetization direction that varies
slowly in both space and time. The ALDA should be
suited to treat magnetization dynamics, since the typical
time scale ( tFMR∼1/(10 GHz) ∼10−10s) is long com-
paredtothethat associatedwith theFermi andexchange
energies, 1 −10 eV leading to /planckover2pi1/∆∼10−13s in transition
metal ferromagnets.
In the ALDA, the system is described by the time-
dependent effective Schr¨ odinger equation
ˆHALDAΨ(r,t) =i/planckover2pi1∂
∂tΨ(r,t), (4)
where Ψ( r,t) is the quasiparticle wave function at posi-
tionrand timet. We consider a generic mean-field elec-
tronic Hamiltonian that depends on the magnetization
direction ˆHALDA[m] and includes the periodic Hartree,
exchange and correlation potentials and relativistic cor-
rectionssuchasthe spin-orbitinteraction. Impurityscat-
tering including magnetic disorder is also represented by
ˆHALDA.The magnetization mis allowed to vary in time
and space. The total Hamiltonian depends additionally
on the Zeeman energy of the magnetization in external
Hextand dipolar Hdmagnetic fields:
ˆH=ˆHALDA[m]−Ms/integraldisplay
drm·(Hext+Hd).(5)
For this general Hamiltonian (5), our task is to de-
duce an expression for the Gilbert damping tensor ˜ α. To
this end, from the form of the Landau-Lifshitz-Gilbert
equation (3), it is clear that we should seek an expansionin terms of the slow variations of the magnetizations in
time. Such an expansion is valid provided the adiabatic
magnetization precession frequency is much less than the
exchange splitting ∆ or the spin-orbit energy which gov-
erns spin relaxation of electrons. We discuss first dissi-
pation in the LLG equation and subsequently compare
it with the expressions from scattering theory of electron
transport. This leads to a recipe to describe dissipation
by first principles. Finally, we discuss the connection to
the Kubo linear response formalism and prove that the
two formulations are identical in linear response.
III. DISSIPATION AND
LANDAU-LIFSHITZ-GILBERT EQUATION
The energy dissipation can be obtained from the solu-
tion of the LLG Eq. (1) as
˙E=−Ms/integraldisplay
dr[˙ m(r,t)·Heff(r,t)] (6)
=−Ms
γ/integraldisplay
dr/integraldisplay
dr′˙ m(r)·˜α[m](r,r′)·˙ m(r′).(7)
Thescatteringtheoryofmagnetizationdissipationcanbe
formulated for arbitrary spatiotemporal magnetization
textures. Much insight can be gained for certain special
cases. In small particles or high magnetic fields the col-
lective magnetization motion is approximately constant
in space and the “macrospin” model is valid in which
all spatial dependences are disregarded. We will also
consider special magnetization textures with a dynamics
characterized by a number of dynamic (soft) collective
coordinates ξa(t) counted by a:20,21
m(r,t) =mst(r;{ξa(t)}), (8)
wheremstis the profile at t→ −∞.This representation
has proven to be very effective in handling magnetiza-
tion dynamics of domain walls in ferromagnetic wires.
The description is approximate, but (for few variables)
it becomes exact in special limits, such as a transverse
domain wall in wires below the Walker breakdown (see
below); it becomes arbitrarily accurate by increasing the
number of collective variables. The energy dissipation to
lowest (quadratic) order in the rate of change ˙ξaof the
collective coordinates is
˙E=−/summationdisplay
ab˜Γab˙ξa˙ξb, (9)
The (symmetric) dissipation tensor ˜Γabreads4
˜Γab=Ms
γ/integraldisplay
dr/integraldisplay
dr′∂mst(r)
∂ξaα[m](r,r′)·∂mst(r′)
∂ξb. (10)
The equation of motion of the collective coordinates un-
der a force
F=−∂Ω
∂ξ(11)
are20,21
˜η˙ξ+[F+f(t)]−˜Γ˙ξ= 0, (12)
introducing the antisymmetric and time-independent gy-
rotropic tensor:
˜ηab=Ms
γ/integraldisplay
drmst(r)·/bracketleftbigg∂mst(r)
∂ξa×∂mst(r)
∂ξb/bracketrightbigg
.(13)
We show below that Fand˜Γ can be expressed in terms
of the scattering matrix. For our subsequent discussions
it is necessary to include a fluctuating force f(t) (with
/an}b∇acketle{tf(t)/an}b∇acket∇i}ht= 0),which has not been considered in Refs. 20,21.
From Eq. (3) if follows the time correlation of fis white
and obeys the fluctuation-dissipation theorem:
/an}b∇acketle{tfa(t)fb(t′)/an}b∇acket∇i}ht= 2kBT˜Γabδ(t−t′). (14)
In the following we illustrate the collective coordinate
description of magnetization textures for the macrospin
model and the Walker model for a transverse domain
wall. The treatment is easily extended to other rigid
textures such as magnetic vortices.
A. Macrospin excitations
When high magnetic fields are applied or when the
system dimensions are small the exchange stiffness dom-
inates. In both limits the magnetization direction and
its low energy excitations lie on the unit sphere and its
magnetization dynamics is described by the polar angles
θ(t) andϕ(t):
m= (sinθcosϕ,sinθsinϕ,cosθ).(15)
The diagonal components of the gyrotropic tensor vanish
by (anti)symmetry ˜ ηθθ= 0, ˜ηϕϕ= 0.Its off-diagonal
components are
ηθϕ=MsV
γsinθ=−ηϕθ. (16)
Vis the particle volume and MsVthe total magnetic
moment. We now have two coupled equations of motion
MsV
γ˙ϕsinθ−∂Ω
∂θ−/parenleftBig
˜Γθθ˙θ+˜Γθϕ˙ϕ/parenrightBig
= 0,(17)
−MsV
γ˙θsinθ−∂Ω
∂ϕ−/parenleftBig
˜Γϕθ˙θ+˜Γϕϕ˙ϕ/parenrightBig
= 0.The thermodynamic potential Ω determines the ballistic
trajectories of the magnetization. The Gilbert damping
tensor˜Γabwill be computed below, but when isotropic
and local,
˜Γ =˜1δ(r−r′)Msα/γ, (18)
where˜1 is a unit matrix in the Cartesian basis and α
is the dimensionless Gilbert constant, Γ θθ=MsVα/γ,
Γθϕ= 0 = Γ ϕθ, and Γ ϕϕ= sin2θMsVα/γ.
B. Domain Wall Motion
We focus on a one-dimensional model, in which the
magnetization gradient, magnetic easy axis, and external
magnetic field point along the wire ( z) axis. The mag-
netic energy of such a wire with transverse cross section
Scan be written as22
Ω =MsS/integraldisplay
dzφ(z), (19)
in terms of the one-dimensional energy density
φ=A
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m
∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
−Hamz+K1
2/parenleftbig
1−m2
z/parenrightbig
+K2
2m2
x,(20)
whereHais the applied field and Ais the exchange stiff-
ness. Here the easy-axis anisotropy is parametrized by
an anisotropy constant K1. In the case of a thin film
wire, there is also a smaller anisotropy energy associated
with the magnetization transverse to the wire governed
byK2. In a cylindrical wire from a material without
crystal anisotropy (such as permalloy) K2= 0.
When the shape of such a domain wall is pre-
served in the dynamics, three collective coordinates
characterize the magnetization texture: the domain
wall position ξ1(t) =rw(t), the polar angle ξ2(t) =
ϕw(t), and the domain wall width λw(t). We con-
sider a head-to-head transverse domain wall (a tail-
to-tail wall can be treated analogously). m(z) =
(sinθwcosϕw,sinθwsinϕw,cosθw), where
cosθw= tanhrw−z
λw(21)
and
cscθw= coshrw−z
λw(22)
minimizes the energy (20) under the constraint that the
magnetization to the far left and right points towardsthe5
domain wall. The off-diagonal elements are then ˜ ηrl=
0 = ˜ηlrand ˜ηrϕ=−2Ms/γ=−˜ηϕr.The energy (20)
reduces to
Ω =MsS/bracketleftbig
A/λw−2Har+K1λw+K2λwcos2ϕw/bracketrightbig
.
(23)
Disregarding fluctuations, the equation of motion Eq.
(12) can be expanded as:
2˙rw+αϕϕ˙ϕ+αϕr˙rw+αϕλ˙λw=γK2λwsin2ϕw,
(24)
−2 ˙ϕ+αrr˙rw+αrϕ˙ϕ+αrλ˙λw= 2γHa, (25)
A/λ2
w+αλr˙rw+αλϕ˙ϕ+αλλ˙λw=K1+K2cos2ϕw,
(26)
whereαab=γΓab/MsS.
When the Gilbert dampingtensorisisotropicandlocal
in the basis of the Cartesian coordinates, ˜Γ =˜1δ(r−
r′)Msα/γ
αrr=2α
λw;αϕϕ= 2αλw;αλλ=π2α
6λw.(27)
whereas all off-diagonal elements vanish.
Most experiments are carried out on thin film ferro-
magnetic wires for which K2is finite. Dissipation is es-
pecially simple below the Walker threshold, the regime
in which the wall moves with a constant drift velocity,
˙ϕw= 0 and23
˙rw=−2γHa/αrr. (28)
The Gilbert damping coefficient αrrcan be obtained di-
rectly from the scattering matrix by the parametric de-
pendence of the scattering matrix on the center coordi-
nate position rw. When the Gilbert damping tensor is
isotropic and local, we find ˙ rw=λwγHa/α. The domain
wall width λw=/radicalbig
A/(K1+K2cos2ϕw) and the out-
of-plane angle ϕw=1
2arcsin2γHa/αK2. At the Walker-
breakdownfield ( Ha)WB=αK2/(2γ) the sliding domain
wall becomes unstable.
In a cylindrical wire without anisotropy, K2= 0,ϕwis
time-dependent and satisfies
˙ϕw=−(2+αϕr)
αϕϕ˙rw (29)
while
˙rw=2γHa
2/parenleftBig
2+αϕr
αϕϕ/parenrightBig
+αrr. (30)
For isotropic and local Gilbert damping coefficients,22
˙rw
λw=αγHa
1+α2. (31)
Inthe nextsection, weformulatehowthe Gilbert scatter-
ing tensor can be computed from time-dependent scat-
tering theory.IV. SCATTERING THEORY OF MESOSCOPIC
MAGNETIZATION DYNAMICS
Scattering theory of transport phenomena24has
proven its worth in the context of magnetoelectronics.
It has been used advantageously to evaluate the non-
local exchange interactions multilayers or spin valves,11
the giantmagnetoresistance,25spin-transfertorque,9and
spin pumping.10We first review the scattering theory
of equilibrium magnetic properties and anisotropy fields
and then will turn to non-equilibrium transport.
A. Conservative forces
Considering only the electronic degrees of freedom in
our model, the thermodynamic (grand) potential is de-
fined as
Ω =−kBTlnTre−(ˆHALDA−µˆN), (32)
whileµis the chemical potential, and ˆNis the number
operator. The conservative force
F=−∂Ω
∂ξ. (33)
can be computed for an open systems by defining a scat-
teringregionthat isconnectedby idealleadstoreservoirs
at common equilibrium. For a two-terminal device, the
flow of charge, spin, and energy between the reservoirs
can then be described in terms of the S-matrix:
S=/parenleftbigg
r t′
t r′/parenrightbigg
, (34)
whereris the matrix of probability amplitudes of states
impinging from and reflected into the left reservoir, while
tdenotes the probability amplitudes of states incoming
from the left and transmitted to the right. Similarly,
r′andt′describes the probability amplitudes for states
that originate from the right reservoir. r,r′,t, andt′are
matricesin the space spanned by eigenstates in the leads.
We areinterested in the free magnetic energymodulation
by the magnetic configuration that allows evaluation of
the forces Eq. (33). The free energy change reads
∆Ω =−kBT/integraldisplay
dǫ∆n(ǫ)ln/bracketleftBig
1+e(ǫ−µ)/kBT/bracketrightBig
,(35)
where ∆n(ǫ)dǫis the change in the number of states at
energyǫand interval dǫ, which can be expressed in terms
of the scattering matrix45
∆n(ǫ) =−1
2πi∂
∂ǫTrlnS(ǫ). (36)
Carrying out the derivative, we arrive at the force
F=−1
2πi/integraldisplay
dǫf(ǫ)Tr/parenleftbigg
S†∂S
∂ξ/parenrightbigg
,(37)6
wheref(ǫ) is the Fermi-Dirac distribution function with
chemical potential µ. This established result will be re-
producedandgeneralizedtothedescriptionofdissipation
and fluctuations below.
B. Gilbert damping as energy pumping
Here we interpretGilbert damping asan energypump-
ing process by equating the results for energy dissipa-
tion from the microscopic adiabatic pumping formalism
with the LLG phenomenology in terms of collective co-
ordinates, Eq. (9). The adiabatic energy loss rate of a
scattering region in terms of scattering matrix at zero
temperature has been derived in Refs. 26,27. In the ap-
pendices, we generalize this result to finite temperatures:
˙E=/planckover2pi1
4π/integraldisplay
dǫ/parenleftbigg
−∂f
∂ǫ/parenrightbigg
Tr/bracketleftbigg∂S(ǫ,t)
∂t∂S†(ǫ,t)
∂t/bracketrightbigg
.(38)
Since we employ the adiabatic approximation, S(ǫ,t) is
the energy-dependent scattering matrix for an instanta-
neous (“frozen”)scattering potential at time t. In a mag-
netic system, the time dependence arises from its magne-
tization dynamics, S(ǫ,t) =S[m(t)](ǫ). In terms of the
collective coordinates ξ(t),S(ǫ,t) =S(ǫ,{ξ(t)})
∂S[m(t)]
∂t≈/summationdisplay
a∂S
∂ξa˙ξa, (39)
where the approximate sign has been discussed in the
previous section. We can now identify the dissipation
tensor (10) in terms of the scattering matrix
Γab=/planckover2pi1
4π/integraldisplay
dǫ/parenleftbigg
−∂f
∂ǫ/parenrightbigg
Tr/bracketleftbigg∂S(ǫ)
∂ξa∂S†(ǫ)
∂ξb/bracketrightbigg
.(40)In the macrospin model the Gilbert damping tensor can
then be expressed as
˜αij=γ/planckover2pi1
4πMs/integraldisplay
dǫ/parenleftbigg
−∂f
∂ǫ/parenrightbigg
Tr/bracketleftbigg∂S(ǫ)
∂mi∂S†(ǫ)
∂mj/bracketrightbigg
,(41)
wheremiis a Cartesian component of the magnetization
direction..
C. Gilbert damping and fluctuation-dissipation
theorem
At finite temperatures the forces acting on the mag-
netization contain thermal fluctuations that are related
to the Gilbert dissipation by the fluctuation-dissipation
theorem, Eq. (14). The dissipation tensor is therefore ac-
cessible via the stochastic forces in thermal equilibrium.
The time dependence of the force operators
ˆF(t) =−∂ˆHALDA(m)
∂ξ(42)
is caused by the thermal fluctuations of the magneti-
zation. It is convenient to rearrange the Hamiltonian
ˆHALDAinto an unperturbed part that does not de-
pend on the magnetization and a scattering potential
ˆHALDA(m) =ˆH0+ˆV(m). In the basis of scattering
wave functions of the leads, the force operator reads
ˆF=−/integraldisplay
dǫ/integraldisplay
dǫ′/an}b∇acketle{tǫα|∂ˆV
∂ξ|ǫ′β/an}b∇acket∇i}htˆa†
α(ǫ)ˆaβ(ǫ′)ei(ǫ−ǫ′)t//planckover2pi1, (43)
where ˆaβannihilates an electron incident on the scatter-
ing region, βlabels the lead (left or right) and quantum
numbers of the wave guide mode, and |ǫ′β/an}b∇acket∇i}htis an associ-
ated scatteringeigenstateat energy ǫ′. We takeagainthe
left and rightreservoirsto be in thermal equilibrium with
the same chemical potentials, such that the expectation
values
/angbracketleftbig
ˆa†
α(ǫ)ˆaβ(ǫ′)/angbracketrightbig
=δαβδ(ǫ−ǫ′)f(ǫ).(44)
Therelationbetweenthematrixelementofthescattering
potential and the S-matrix
/bracketleftbigg
S†(ǫ)∂S(ǫ)
∂ξ/bracketrightbigg
αβ=−2πi/an}b∇acketle{tǫα|∂ˆV
∂ξ|ǫβ/an}b∇acket∇i}ht(45)follows from the relation derived in Eq. (61) below as
well as unitarity of the S-matrix,S†S= 1. Taking these
relationsintoaccount,the expectationvalueof ˆFisfound
to be Eq. (37). We now consider the fluctuations in the
forceˆf(t) =ˆF(t)− /an}b∇acketle{tˆF(t)/an}b∇acket∇i}ht, which involves expectation
values
/angbracketleftbig
ˆa†
α1(ǫ1)ˆaβ1(ǫ′
1)ˆa†
α2(ǫ2)ˆaβ2(ǫ′
2)/angbracketrightbig
−/angbracketleftbig
ˆa†
α1(ǫ1)ˆaβ1(ǫ′
1)/angbracketrightbig/angbracketleftbig
ˆa†
α2(ǫ2)ˆaβ2(ǫ′
2)/angbracketrightbig
=δα1β2δ(ǫ1−ǫ′
2)δβ1α2δ(ǫ′
1−ǫ2)f(ǫ1)[1−f(ǫ2)],
(46)
where we invoked Wick’s theorem. Putting everything7
together, we finally find
/an}b∇acketle{tfa(t)fb(t′)/an}b∇acket∇i}ht= 2kBTδ(t−t′)Γab, (47)
where Γ abhas been defined in Eq. (40). Comparing with
Eq. (14), we conclude that the dissipation tensor Γ ab
governingthe fluctuationsisidentical tothe oneobtained
from the energy pumping, Eq. (40), thereby confirming
the fluctuation-dissipation theorem.
V. KUBO FORMULA
The quality factor of the magnetization dynamics of
most ferromagnets is high ( α/lessorsimilar0.01). Damping can
therefore often be treated as a small perturbation. In
the presentSectionwedemonstratethat the dampingob-
tained from linear response (Kubo) theory agrees28with
that ofthe scattering theory ofmagnetization dissipation
in this limit. At sufficiently low temperatures or strong
elastic disorder scattering the coupling to phonons may
be disregarded and is not discussed here.
The energy dissipation can be written as
˙E=/angbracketleftBigg
dˆH
dt/angbracketrightBigg
, (48)
where/an}b∇acketle{t/an}b∇acket∇i}htdenotes the expectation value for the non-
equilibrium state. We are interested in the adiabatic
response of the system to a time-dependent perturba-
tion. In the adiabatic (slow) regime, we can at any time
expand the Hamiltonian around a static configuration at
the reference time t= 0,
ˆH=ˆHst+/summationdisplay
aδξa(t)/parenleftBigg
∂ˆH
∂ξa/parenrightBigg
m(r)→mst(r).(49)
The static part, ˆHst, is the Hamiltonian for a magneti-
zation for a fixed and arbitrary initial texture mst, as,
without loss of generality, described by the collective
coordinates ξa. Since we assume that the variation of
the magnetization in time is small, a linear expansion in
terms of the small deviations of the collective coordinate
δξi(t) is valid for sufficiently short time intervals. We can
then employ the Kubo formalism and express the energy
dissipation as
˙E=/summationdisplay
aδ˙ξa(t)/parenleftBigg
∂ˆH
∂ξa/parenrightBigg
m(r)→mst(r),(50)
where the expectation value of the out-of-equilibrium
conservative force
/parenleftBigg
∂ˆH
∂ξa/parenrightBigg
m(r)→mst(r)≡∂aˆH (51)consists of an equilibrium contribution and a term linear
in the perturbed magnetization direction:
/angbracketleftBig
∂aˆH/angbracketrightBig
(t) =/angbracketleftBig
∂aˆH/angbracketrightBig
st+/summationdisplay
b/integraldisplay∞
−∞dt′χab(t−t′)δξb(t′).
(52)
Here, we introduced the retarded susceptibility
χab(t−t′) =−i
/planckover2pi1θ(t−t′)/angbracketleftBig/bracketleftBig
∂aˆH(t),∂bˆH(t′)/bracketrightBig/angbracketrightBig
st,(53)
where/an}b∇acketle{t/an}b∇acket∇i}htstis the expectation value for the wave functions
of the static configuration. Focussing on slow modula-
tions we can further simplify the expression by expand-
ing
δξa(t′)≈δξa(t)+(t′−t)δ˙ξa(t), (54)
so that
/angbracketleftBig
∂aˆH/angbracketrightBig
=/angbracketleftBig
∂aˆH/angbracketrightBig
st+/integraldisplay∞
−∞dt′χab(t−t′)δξb(t)+
/integraldisplay∞
−∞dt′χab(t−t′)(t′−t)δ˙ξb(t). (55)
The first two terms in this expression, /an}b∇acketle{t∂aˆH/an}b∇acket∇i}htst+/integraltext∞
−∞dt′χab(t−t′)δξb(t),correspond to the energy vari-
ation with respect to a change in the static magnetiza-
tion. These terms do not contribute to the dissipation
since the magnetic excitations are transverse, ˙ m·m= 0.
Only the last term in Eq. (55) gives rise to dissipation.
Hence, the energy loss reduces to29
˙E=i/summationdisplay
ijδ˙ξaδ˙ξb∂χS
ab
∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle
ω=0, (56)
whereχS
ab(ω) =/integraltext∞
−∞dt[χab(t)+χba(t)]eiωt/2. The
symmetrized susceptibility can be expanded as
χS
ab=/summationdisplay
nm(fn−fm)
2/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}ht+(a↔b)
/planckover2pi1ω+iη−(ǫn−ǫm),
(57)
where|n/an}b∇acket∇i}htis an eigenstate of the Hamiltonian ˆHstwith
eigenvalueǫn,fn≡f(ǫn),f(ǫ) is the Fermi-Dirac distri-
bution function at energy ǫ, andηis a positive infinites-
imal constant. Therefore,8
i/parenleftbigg∂χS
ab
∂ω/parenrightbigg
ω=0=π/summationdisplay
nm/parenleftbigg
−∂fn
∂ǫ/parenrightbigg
/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}htδ(ǫn−ǫm), (58)
and the dissipation tensor
Γab=π/summationdisplay
nm/parenleftbigg
−∂fn
∂ǫ/parenrightbigg
/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}htδ(ǫn−ǫm). (59)
We nowdemonstratethatthe dissipationtensorobtained
from the Kubo linear response formula, Eq. (59), is
identical to the expression from scattering theory, Eq.
(40), following the Fisher and Lee proof of the equiv-
alence of linear response and scattering theory for the
conductance.36
The static Hamiltonian ˆHst(ξ) =ˆH0+ˆV(ξ) can be
decomposed into a free-electron part ˆH0=−/planckover2pi12∇2/2m
and a scattering potential ˆV(ξ). The eigenstates of ˆH0
are denoted |ϕs,q(ǫ)/an}b∇acket∇i}ht,with eigenenergies ǫ, wheres=±
denotes the longitudinal propagation direction along the
system (say, to the left or to the right), and qa trans-
verse quantum number determined by the lateral con-
finement. The potential ˆV(ξ) scatters the particles be-tween the propagating states forward or backward. The
outgoing (+) and incoming ( −) scattering eigenstates
of the static Hamiltonian ˆHstare written as/vextendsingle/vextendsingle/vextendsingleψ(±)
s,q(ǫ)/angbracketrightBig
,
whichform anothercomplete basiswith orthogonalityre-
lations/angbracketleftBig
ψ(±)
s,q(ǫ)/vextendsingle/vextendsingle/vextendsingleψ(±)
s′,q′(ǫ′)/angbracketrightBig
=δs,s′δq,q′δ(ǫ−ǫ′).33These
wave functions can be expressed as/vextendsingle/vextendsingle/vextendsingleψ(±)
s,q(ǫ)/angbracketrightBig
= [1 +
ˆG(±)
stˆV]|ϕs,q/an}b∇acket∇i}ht, where the retarded (+) and advanced ( −)
Green’s functions read ˆG(±)
st(ǫ) = (ǫ±iη−ˆHst)−1. By
expanding Γ abin the basis of outgoing wave functions,
|ψ(+)
s,q/an}b∇acket∇i}ht, the energy dissipation (59) becomes
Γab=π/summationdisplay
sq,s′q′/integraldisplay
dǫ/parenleftbigg
−∂fs,q
∂ǫ/parenrightbigg/angbracketleftBig
ψ(+)
s,q/vextendsingle/vextendsingle/vextendsingle∂aˆH/vextendsingle/vextendsingle/vextendsingleψ(+)
s′,q′/angbracketrightBig/angbracketleftBig
ψ(+)
s′,q′/vextendsingle/vextendsingle/vextendsingle∂bˆH/vextendsingle/vextendsingle/vextendsingleψ(+)
s,q/angbracketrightBig
, (60)
where wave functions should be evaluated at the energy ǫ.
Let us now compare this result, Eq. (60), to the direct scattering matrix expression for the energy dissipation,
Eq. (40). The S-matrix operator can be written in terms of the T-matrix as ˆS(ǫ;ξ) = 1−2πiˆT(ǫ;ξ), where the
T-matrix is defined recursively by ˆT=ˆV[1+ˆG(+)
stˆT]. We then find
∂ˆT
∂ξa=/bracketleftBig
1+ˆVˆG(+)
st/bracketrightBig
∂aˆH/bracketleftBig
1+ˆG(+)
stˆV/bracketrightBig
.
The change in the scattering matrix appearing in Eq. (40) is then
∂Ss′q′,sq
∂ξa=−2πi/an}b∇acketle{tϕs,q|/bracketleftBig
1+ˆVˆG(+)
st/bracketrightBig
∂aˆH/bracketleftBig
1+ˆG(+)
stˆV/bracketrightBig
|ϕs′,q′/an}b∇acket∇i}ht=−2πi/angbracketleftBig
ψ(−)
s′,q′/vextendsingle/vextendsingle/vextendsingle∂aˆH/vextendsingle/vextendsingle/vextendsingleψ(+)
s′,q′/angbracketrightBig
. (61)
Since
/angbracketleftBig
ψ(−)
s,q(ǫ)/vextendsingle/vextendsingle/vextendsingle=/summationdisplay
s′q′Ssq,s′q′/angbracketleftBig
ψ(+)
s′q′(ǫ)/vextendsingle/vextendsingle/vextendsingle(62)
andSS†= 1, we can write the linear response result,
Eq. (60), as energy pumping (40). This completes our
proof of the equivalence between adiabatic energy pump-
ingintermsofthe S-matrixandtheKubolinearresponse
theory.VI. CONCLUSIONS
We have shown that most aspects of magnetization
dynamics in ferromagnets can be understood in terms of
the boundary conditions to normal metal contacts, i.e.
a scattering matrix. By using the established numerical
methods to compute electron transport based on scatter-
ing theory, this opens the way to compute dissipation in
ferromagnets from first-principles. In particular, our for-9
malism should work well for systems with strong elastic
scattering due to a high density of large impurity poten-
tials or in disordered alloys, including Ni 1−xFex(x= 0.2
represents the technologically important “permalloy”).
The dimensionless Gilbert damping tensors (41) for
macrospin excitations, which can be measured directly
in terms of the broadening of the ferromagnetic reso-
nance, havebeen evaluated for Ni 1−xFexalloysby ab ini-
tiomethods.42Permalloy is substitutionally disordered
and damping is dominated by the spin-orbit interaction
in combination with disorder scattering. Without ad-
justable parameters good agreement has been obtained
with the available low temperature experimental data,
which is a strong indication of the practical value of our
approach.
In clean samples and at high temperatures, the
electron-phonon scattering importantly affects damping.
Phonons are not explicitly included here, but the scat-
tering theory of Gilbert damping can still be used for
a frozen configuration of thermally displaced atoms, ne-
glecting the inelastic aspect of scattering.12
While the energy pumping by scattering theory has
been applied to described magnetization damping,15it
can be used to compute other dissipation phenomena.
This has recently been demonstrated for the case of
current-induced mechanical forces and damping,43with
a formalism analogous to that for current-induced mag-
netization torques.13,14
Acknowledgments
We would like to thank Kjetil Hals, Paul J. Kelly, Yi
Liu, Hans Joakim Skadsem, Anton Starikov, and Zhe
Yuan for stimulating discussions. This work was sup-
ported by the EC Contract ICT-257159 “MACALO,”
theNSFunderGrantNo.DMR-0840965,DARPA,FOM,
DFG, and by the Project of Knowledge Innovation Pro-
gram(PKIP) of Chinese Academy of Sciences, Grant No.
KJCX2.YW.W10
Appendix A: Adiabatic Pumping
Adiabatic pumping is the current response to a time-
dependent scattering potential to first order in the time-
variation or “pumping” frequency when all reservoirsare
at the same electro-chemical potential.38A compact for-
mulation of the pumping charge current in terms of the
instantaneous scattering matrix was derived in Ref. 39.
In the same spirit, the energy current pumped out of the
scattering region has been formulated (at zero tempera-
ture) in Ref. 27. Some time ago, we extended the charge
pumping concept to include the spin degree of free-
domandascertainedits importancein magnetoelectronic
circuits.10More recently, we demonstrated that the en-
ergyemitted byaferromagnetwith time-dependentmag-
netizations into adjacent conductors is not only causedby interface spin pumping, but also reflects the energy
loss by spin-flip processes inside the ferromagnet15and
therefore Gilbert damping. Here we derive the energy
pumping expressions at finite temperatures, thereby gen-
eralizing the zero temperature results derived in Ref. 27
and used in Ref. 15. Our results differ from an earlier ex-
tension to finite temperature derived in Ref. 40 and we
point out the origin of the discrepancies. The magneti-
zation dynamics must satisfy the fluctuation-dissipation
theorem, which is indeed the case in our formulation.
We proceed by deriving the charge, spin, and energy
currentsintermsofthetimedependenceofthescattering
matrix of a two-terminal device. The transport direction
isxand the transverse coordinates are ̺= (y,z). An
arbitrary single-particle Hamiltonian can be decomposed
as
H(r) =−/planckover2pi12
2m∂2
∂x2+H⊥(x,̺), (A1)
where the transverse part is
H⊥(x,̺) =−/planckover2pi12
2m∂2
∂̺2+V(x,̺).(A2)
V(̺) is an elastic scattering potential in 2 ×2 Pauli
spin space that includes the lattice, impurity, and
self-consistent exchange-correlation potentials, including
spin-orbit interaction and magnetic disorder. The scat-
teringregionisattachedtoperfect non-magneticelectron
wave guides (left α=Land rightα=R) with constant
potential and without spin-orbit interaction. In lead α,
the transverse part of the 2 ×2 spinor wave function
ϕ(n)
α(x,̺) and its corresponding transverse energy ǫ(n)
α
obey the Schr¨ odinger equation
H⊥(̺)ϕ(n)
α(̺) =ǫ(n)
αϕ(n)
α(̺), (A3)
wherenis the spin and orbit quantum number. These
transverse wave guide modes form the basis for the ex-
pansion of the time-dependent scattering states in lead
α=L,R:
ˆΨα=/integraldisplay∞
0dk√
2π/summationdisplay
nσϕ(n)
α(̺)eiσkxe−iǫ(nk)
αt//planckover2pi1ˆc(nkσ)
α,(A4)
where ˆc(nkσ)
αannihilates an electron in mode nincident
(σ= +) or outgoing ( σ=−) in leadα. The field opera-
tors satisfy the anticommutation relation
/braceleftBig
ˆc(nkσ)
α,ˆc†(n′k′σ′)
β/bracerightBig
=δαβδnn′δσσ′δ(k−k′).
The total energy is ǫ(nk)
α=/planckover2pi12k2/2m+ǫ(n)
α. In the leads
the particle, spins, and energy currents in the transport10
direction are
ˆI(p)=/planckover2pi1
2mi/integraldisplay
d̺Trs/parenleftBigg
ˆΨ†∂ˆΨ
∂x−∂ˆΨ†
∂xˆΨ/parenrightBigg
,(A5a)
ˆI(s)=/planckover2pi1
2mi/integraldisplay
d̺Trs/parenleftBigg
ˆΨ†σ∂ˆΨ
∂x−∂ˆΨ†
∂xσˆΨ/parenrightBigg
,(A5b)
ˆI(e)=/planckover2pi1
4mi/integraldisplay
d̺Trs/parenleftBigg
ˆΨ†H∂ˆΨ
∂x−∂ˆΨ†
∂xHˆΨ/parenrightBigg
+H.c.,
(A5c)
where we suppressed the time tand lead index α,σ=
(σx,σy,σz) is a vector of Pauli matrices, and Tr sdenotes
the trace in spin space. Note that the spin current Is
flows in the x-direction with polarization vector Is/Is.
To avoid dependence on an arbitrary global potential
shift, it is convenient to work with heat ˆI(q)rather than
energy currents ˆI(ǫ):
ˆI(q)(t) =ˆI(ǫ)(t)−µˆI(p)(t), (A6)
whereµis the chemical potential. Inserting the waveg-uide representation (A4) into (A5), the particle current
reads41
ˆI(p)
α=/planckover2pi1
4πm/integraldisplay∞
0dkdk′/summationdisplay
nσσ′(σk+σ′k′)×
ei(σk−σ′k′)xe−i/bracketleftBig
ǫ(nk)
α−ǫ(nk′)
α/bracketrightBig
t//planckover2pi1ˆc†(nk′σ′)
αˆc(nkσ)
α.(A7)
Weareinterestedinthelow-frequencylimitoftheFourier
transforms I(x)
α(ω) =/integraltext∞
−∞dteiωtI(x)
α(t). Following Ref.
41 we assume long wavelengths such that only the inter-
vals withk≈k′andσ=σ′contribute. In the adiabatic
limitω→0 this approach is correct to leading order in
/planckover2pi1ω/ǫF,whereǫFis the Fermi energy. By introducing the
(current-normalized) operator
ˆc(nσ)
α(ǫ(nk)
α) =1/radicalBig
dǫ(nkσ)
α
dkˆc(nkσ)
α, (A8)
which obey the anticommutation relations
/braceleftBig
ˆc(nσ)
α(ǫα),ˆc†(n′σ′)
β(ǫβ)/bracerightBig
=δαβδnn′δσσ′δ(ǫα−ǫβ). (A9)
The charge current can be written as
ˆI(c)
α(t) =1
2π/planckover2pi1/integraldisplay∞
ǫ(n)
αdǫdǫ′/summationdisplay
nσσe−i(ǫ−ǫ′)t//planckover2pi1ˆc†(nσ)
α(ǫ′)ˆc(nσ)
α(ǫ). (A10)
Weoperateinthe linearresponseregimeinwhichapplied
voltages and temperature differences as well as the exter-
nally induced dynamics disturb the system only weakly.
Transport is then governed by states close to the Fermi
energy. We may therefore extend the limits of the en-
ergy integration in Eq. (A10) from ( ǫ(n)
α,∞) to (−∞
to∞). We relabel the annihilation operators so that
ˆa(nk)
α= ˆc(nk)
α+denotes particles incident on the scattering
region from lead αandˆb(nk)
α= ˆc(nk)
α−denotes particles
leavingthe scatteringregionbylead α. Using the Fourier
transforms
ˆc(nσ)
α(ǫ) =/integraldisplay∞
−∞dtˆc(nσ)
α(t)eiǫt//planckover2pi1, (A11)
ˆc(nσ)
α(t) =1
2π/planckover2pi1/integraldisplay∞
−∞dǫˆc(nσ)
α(ǫ)e−iǫt//planckover2pi1,(A12)
we obtain in the low-frequency limit41
ˆI(p)
α(t) = 2π/planckover2pi1/bracketleftBig
ˆa†
α(t)ˆaα(t)−ˆb†
α(t)ˆbα(t)/bracketrightBig
,(A13)
whereˆbαis a column vector of the creation operators forall wave-guidemodes {ˆb(n)
α}. Analogouscalculations lead
to the spin current
ˆI(s)
α= 2π/planckover2pi1/parenleftBig
ˆa†
ασˆaα−ˆb†
ασˆbα/parenrightBig
(A14)
and the energy current
ˆI(e)
α=iπ/planckover2pi12/parenleftBigg
ˆa†
α∂ˆaα
∂t−ˆb†
α∂ˆbα
∂t/parenrightBigg
+H.c..(A15)
Next, we express the outgoing operators ˆb(t) in terms
of the incoming operators ˆ a(t) via the time-dependent
scattering matrix (in the space spanned by all waveguide
modes, including spin and orbit quantum number):
ˆbα(t) =/summationdisplay
β/integraldisplay
dt′Sαβ(t,t′)ˆaβ(t′).(A16)
When the scattering region is stationary, Sαβ(t,t′) only
depends on the relative time difference t−t′, and its
Fourier transform with respect to the relative time is
energy independent, i.e.transport is elastic and can11
be computed for each energy separately. For time-
dependent problems, Sαβ(t,t′) also depends on the total
timet+t′and there is an inelastic contribution to trans-
port as well. An electron can originate from a lead with
energyǫ, pick up energy in the scattering region and end
up in the same or the other lead with different energy ǫ′.
The reservoirs are in equilibrium with controlled lo-
cal chemical potentials and temperatures. We insert the
S-matrix (A16) into the expressions for the currents,Eqs. (A13), (A14), (A15), and use the expectation value
at thermal equilibrium
/angbracketleftBig
ˆa†(n)
α(t2)ˆa(m)
β(t1)/angbracketrightBig
eq=δnmδαβfα(t1−t2)/2πℏ,(A17)
wherefβ(t1−t2) = (2π/planckover2pi1)−1/integraltext
dǫ−iǫ(t1−t2)//planckover2pi1fα(ǫ) and
fα(ǫ) is the Fermi-Dirac distribution of electrons with
energyǫin theα-th reservoir. We then find
2π/planckover2pi1/angbracketleftBig
ˆb†
α(t)ˆbα(t)/angbracketrightBig
eq=/summationdisplay
β/integraldisplay
dt1dt2S∗
αβ(t,t2)Sαβ(t,t1)fβ(t1−t2), (A18)
2π/planckover2pi1/angbracketleftBig
ˆb†
α(t)σˆbα(t)/angbracketrightBig
eq=/summationdisplay
β/integraldisplay
dt1dt2S∗
αβ(t,t2)σSαβ(t,t1)fβ(t1−t2), (A19)
2π/planckover2pi1/angbracketleftBig
/planckover2pi1∂tˆb†
α(t)ˆbα(t)/angbracketrightBig
eq=/summationdisplay
β/integraldisplay
dt1dt2/bracketleftbig
/planckover2pi1∂tS∗
αβ(t,t2)/bracketrightbig
Sαβ(t,t1)fβ(t1−t2). (A20)
Next, we use the Wigner representation (B1):
S(t,t′) =1
2π/planckover2pi1/integraldisplay∞
−∞dǫS/parenleftbiggt+t′
2,ǫ/parenrightbigg
e−iǫ(t−t′)//planckover2pi1, (A21)
and by Taylor expanding the Wigner represented S-matrix S((t+t′)/2,ǫ) aroundS(t,ǫ), S((t+t′)/2,ǫ) =/summationtext∞
n=0∂n
tS(t,ǫ)(t′−t)n/(2nn!), we find
S(t,t′) =1
2π/planckover2pi1/integraldisplay∞
−∞dǫe−iǫ(t−t′)//planckover2pi1ei/planckover2pi1∂ǫ∂t/2S(t,ǫ) (A22)
and
/planckover2pi1∂tS(t,t′) =1
2π/planckover2pi1/integraldisplay∞
−∞dǫe−iǫ(t−t′)//planckover2pi1ei/planckover2pi1∂ǫ∂t/2/parenleftbigg1
2/planckover2pi1∂t−iǫ/parenrightbigg
S(t,ǫ). (A23)
The factor 1 /2 scaling the term /planckover2pi1∂tS(t,ǫ) arises from commuting ǫwithei/planckover2pi1∂ǫ∂t/2. The currents can now be evaluated
as
I(c)
α(t) =−1
2π/planckover2pi1/summationdisplay
β/integraldisplay∞
−∞dǫ/bracketleftBig/parenleftBig
e−i∂ǫ∂t/planckover2pi1/2S†
βα(ǫ,t)/parenrightBig/parenleftBig
ei∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig
fβ(ǫ)−fα(ǫ)/bracketrightBig
(A24a)
I(s)
α(t) =−1
2π/planckover2pi1/summationdisplay
β/integraldisplay∞
−∞dǫ/bracketleftBig/parenleftBig
e−i∂ǫ∂t/planckover2pi1/2S†
βα(ǫ,t)/parenrightBig
σ/parenleftBig
ei∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig
fβ(ǫ)/bracketrightBig
(A24b)
I(ǫ)
α(t) =−1
4π/planckover2pi1/summationdisplay
β/integraldisplay∞
−∞dǫ/bracketleftBig/parenleftBig
e−i∂ǫ∂t/2/planckover2pi1(−i/planckover2pi1∂t/2+ǫ)S†
βα(ǫ,t)/parenrightBig/parenleftBig
e+i∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig
fβ(ǫ)−ǫfα(ǫ)/bracketrightBig
−1
4π/planckover2pi1/integraldisplay∞
−∞dǫ/bracketleftBig/parenleftBig
e−i∂ǫ∂t/2/planckover2pi1S†
βα(ǫ,t)/parenrightBig/parenleftBig
ei∂ǫ∂t/2/planckover2pi1(i/planckover2pi1∂t/2+ǫ)Sαβ(ǫ,t)/parenrightBig
fβ(ǫ)−ǫfα(ǫ)/bracketrightBig
,(A24c)
where the adjoint of the S-matrix has elements S†(n′,n)
βα=S∗(n,n′)
αβ.
We are interested in the average (DC) currents, where simplified ex pressions can be found by partial integration
over energy and time intervals. We will consider the total DC curren tsout ofthe scattering region, I(out)=−/summationtext
αIα,
when the electrochemical potentials in the reservoirs are equal, fα(ǫ) =f(ǫ) for allα. The averaged pumped spin and12
energy currents out of the system in a time interval τcan be written compactly as
I(c)
out=1
2π/planckover2pi1τ/integraldisplayτ
0dt/integraldisplay
dǫTr/braceleftbigg/bracketleftbigg
f/parenleftbigg
ǫ−i/planckover2pi1
2∂
∂t/parenrightbigg
S/bracketrightbigg
S†−f(ǫ)/bracerightbigg
, (A25a)
I(s)
out=1
2π/planckover2pi1τ/integraldisplayτ
0dt/integraldisplay
dǫTr/braceleftbigg
σ/bracketleftbigg
f/parenleftbigg
ǫ−i/planckover2pi1
2∂
∂t/parenrightbigg
S/bracketrightbigg
S†/bracerightbigg
, (A25b)
I(ǫ)
out=1
2π/planckover2pi1τ/integraldisplayτ
0dt/integraldisplay
dǫTr/braceleftbigg/bracketleftbigg/parenleftbigg
ǫ−i/planckover2pi1
2∂
∂t/parenrightbigg
f/parenleftbigg
ǫ−i/planckover2pi1
2∂
∂t/parenrightbigg
S/bracketrightbigg
S†−ǫf(ǫ)/bracerightbigg
+1
2π/planckover2pi1τ/integraldisplayτ
0dt/integraldisplay
dǫTr/braceleftbigg/bracketleftbigg
f/parenleftbigg
ǫ−i/planckover2pi1
2∂
∂t/parenrightbigg
S/bracketrightbigg/parenleftbigg
−i/planckover2pi1∂S†
∂t/parenrightbigg/bracerightbigg
, (A25c)
where Tr is the trace over all waveguide modes (spin
and orbital quantum numbers). As shown in Ap-
pendix C the charge pumped into the reservoirs vanishes
for a scattering matrix with a periodic time dependence
when,integrated over one cycle:
I(p)
out= 0. (A26)
This reflects particle conservation; the number of elec-
trons cannot build up in the scattering region for peri-
odic variations ofthe system. We can showthat a similar
contribution to the energy current, i.e.the first line in
Eq. (A25c), vanishes, leading to to the simple expression
I(e)
out=−i
2π/integraldisplayτ
0dt
τ/integraldisplay
dǫTr/braceleftbigg/bracketleftbigg
f/parenleftbigg
ǫ−i/planckover2pi1
2∂
∂t/parenrightbigg
S/bracketrightbigg∂S†
∂t/bracerightbigg
.
(A27)
Expanded to lowest order in the pumping frequency the
pumped spin current (A25b) becomes
I(s)
out=1
2π/planckover2pi1/integraldisplayτ
0dt
τ/integraldisplay
dǫTr/braceleftbigg/parenleftbigg
SS†f−i/planckover2pi1
2∂S
∂tS†∂ǫf/parenrightbigg
σ/bracerightbigg
(A28)
This formula is not the most convenient form to com-
pute the current to specified order. SS†also contains
contributions that are linear and quadratic in the pre-
cession frequency since S(t,ǫ) is theS-matrix for a time-
dependent problem. Instead, wewouldliketoexpressthe
current in terms of the frozenscattering matrix Sfr(t,ǫ).
The latter is computed for an instantaneous, static elec-
tronic potential. In our case this is determined by a mag-
netization configuration that depends parametrically on
time:Sfr(t,ǫ) =S[m(t),ǫ]. Using unitarity of the time-dependentS-matrix (as elaborated in Appendix C), ex-
pand it to lowest order in the pumping frequency, and
insert it into (A28) leads to39
I(s)
out=i
2π/summationdisplay
β/integraldisplayτ
0dt
τ/integraldisplay
dǫ/parenleftbigg
−∂f
∂ǫ/parenrightbigg
Tr/braceleftbigg∂Sfr
∂tS†
frσ/bracerightbigg
.
(A29)
We evaluate the energy pumping by expanding (A27)
to second order in the pumping frequency:
I(e)
out=/planckover2pi1
4π/integraldisplayτ
0dt
τ/integraldisplay
dǫTr/braceleftbigg
−ifS∂S†
∂t−(∂ǫf)1
2∂S
∂t∂S†
∂t/bracerightbigg
.
(A30)
As a consequence of unitarity of the S-matrix (see Ap-
pendix C), the first term vanishes to second order in the
precession frequency:
I(e)
out=/planckover2pi1
4π/integraldisplayτ
0dt
τ/integraldisplay
dǫ/parenleftbigg
−∂f
∂ǫ/parenrightbigg
Tr/braceleftBigg
∂Sfr
∂t∂S†
fr
∂t/bracerightBigg
,(A31)
where,at this point , we may insert the frozen scattering
matrix since the current expression is already propor-
tional to the square of the pumping frequency. Further-
more, since there is no net pumped charge current in
one cycle (and we are assuming reservoirs in a common
equilibrium), the pumped heat current is identical to the
pumped energy current, I(q)
out=I(e)
out.
Our expression for the pumped energy current (A31)
agrees with that derived in Ref. 27 at zero temperature.
Our result (A31) differs from Ref. 40 at finite tempera-
tures. The discrepancy can be explained as follows. In-
tegration by parts over time tin Eq. (A27), using
/bracketleftbigg
f/parenleftbigg
ǫ−i/planckover2pi1
2∂
∂t/parenrightbigg
i/planckover2pi1∂S
∂t/bracketrightbigg
S†= 2/bracketleftbigg
ǫf/parenleftbigg
ǫ−i/planckover2pi1
2∂
∂t/parenrightbigg
S/bracketrightbigg
S†−2/bracketleftbigg/parenleftbigg
ǫ−i/planckover2pi1
2∂
∂t/parenrightbigg
f/parenleftbigg
ǫ−i/planckover2pi1
2∂
∂t/parenrightbigg
S/bracketrightbigg
S†,(A32)
and the unitarity condition from Appendix C,
/integraldisplayτ
0dt
τ/integraldisplay
dǫ/bracketleftbigg/parenleftbigg
ǫ−i/planckover2pi1
2∂
∂t/parenrightbigg
f/parenleftbigg
ǫ−i/planckover2pi1
2∂
∂t/parenrightbigg
S/bracketrightbigg
S†=/integraldisplayτ
0dt
τ/integraldisplay
dǫǫf(ǫ), (A33)13
the DC pumped energy current can be rewritten as
I(ǫ)
out=1
π/planckover2pi1/integraldisplayτ
0dt
τ/integraldisplay
dǫTr/braceleftbigg/bracketleftbigg
ǫf/parenleftbigg
ǫ−i/planckover2pi1
2∂
∂t/parenrightbigg
S/bracketrightbigg
S†−ǫf(ǫ)/bracerightbigg
. (A34)
Next, we expand this to the second order in the pumping frequency and find
I(ǫ)
out=1
π/planckover2pi1/integraldisplayτ
0dt
τ/integraldisplay
dǫTr/braceleftbigg
ǫf(ǫ)/parenleftbig
SS†−1/parenrightbig
−ǫ(∂ǫf)i/planckover2pi1
2∂S
∂tS†−ǫ(∂2
ǫf)/planckover2pi12
8∂2S
∂t2S†/bracerightbigg
. (A35)
This form of the pumped energy current, Eq. (A35),
agrees with Eq. (10) in Ref. 40 if one ( incorrectly ) as-
sumesSS†= 1. Although for the frozen scattering ma-
trix,SfrS†
fr= 1, unitarity does not hold for the Wigner
representation of the scattering matrix to the second or-
der in the pumping frequency. ( SS†−1) therefore does
not vanish but contributes to leading order in the fre-
quency to the pumped current, which may not be ne-
glected at finite temperatures. Only when this term is
included our new result Eq. (A31) is recovered.
Appendix B: Fourier transform and Wigner
representation
There is a long tradition in quantum theory to trans-
form the two-time dependence of two-operator correla-
tion functions such as scattering matrices by a mixed
(Wigner)representationconsistingofaFouriertransform
over the time difference and an average time, which has
distinct advantages when the scattering potential varies
slowlyintime.44Inordertoestablishconventionsandno-
tations, we present here a short exposure how this works
in our case.
The Fourier transform of the time dependent annihi-
lation operators are defined in Eqs. (A11) and (A12).Consider a function Athat depends on two times t1
andt2,A=A(t1,t2). The Wigner representation with
t= (t1+t2)/2 andt′=t1−t2is defined as:
A(t1,t2) =1
2π/planckover2pi1/integraldisplay∞
−∞dǫA(t,ǫ)e−iǫ(t1−t2)//planckover2pi1,(B1)
A(t,ǫ) =/integraldisplay∞
−∞dt′A/parenleftbigg
t+t′
2,t−t′
2/parenrightbigg
eiǫt′//planckover2pi1.(B2)
We also need the Wigner representation of convolutions,
(A⊗B)(t1,t2) =/integraldisplay∞
−∞dt′A(t1,t′)B(t′,t2).(B3)
By a series expansion, this can be expressed as44
(A⊗B)(t,ǫ) =e−i(∂A
t∂B
ǫ−∂B
t∂A
ǫ)/2A(t,ǫ)B(t,ǫ) (B4)
which we use in the following section.
Appendix C: Properties of S-matrix
Here we discuss some general properties of the two-
point time-dependent scattering matrix. Current conser-
vation is reflected by the unitarity of the S-matrix which
can be expressed as
/summationdisplay
βn′s′/integraldisplay
dt′S(α1β)
n1s1,n′s′(t1,t′)S(α2β)∗
n2s2,n′s′(t′,t2) =δn1n2δs1s2δα1α2δ(t1−t2). (C1)
Physically, this means that a particle entering the scattering region from a lead αat some time tis bound to exit the
scattering region in some lead βat another (later) time t′. Using Wigner representation (B1) and integrating over
the local time variable, this implies (using Eq. (B4))
1 =/parenleftbig
S⊗S†/parenrightbig
(t,ǫ) =e−i/parenleftBig
∂S
t∂S†
ǫ−∂S†
t∂S
ǫ/parenrightBig
/2S(t,ǫ)S†(t,ǫ), (C2)
where 1 is a unit matrix in the space spanned by the wave guide modes ( labelled by spin sand orbital quantum
numbern). Similary, we find
1 =/parenleftbig
S†⊗S/parenrightbig
(t,ǫ) =e+i/parenleftBig
∂S
t∂S†
ǫ−∂S†
t∂S
ǫ/parenrightBig
/2S†(t,ǫ)S(t,ǫ). (C3)
To second order in the precession frequency, by respectively sub tracting and summing Eqs. (C2) and (C3) give
Tr/braceleftbigg∂S
∂t∂S†
∂ǫ−∂S
∂ǫ∂S†
∂t/bracerightbigg
= 0 (C4)14
and
Tr/braceleftbig
SS†−1/bracerightbig
= Tr/braceleftbigg∂2S
∂t2∂2S†
∂ǫ2−2∂2S
∂t∂ǫ∂2S†
∂t∂ǫ+∂2S
∂ǫ2∂2S†
∂t2/bracerightbigg
. (C5)
Furthermore, foranyenergydependent function Z(ǫ)andarbitrarymatrixin thespacespannedbyspinandtransverse
waveguide modes Y, Eq. (C2) implies
1
τ/integraldisplayτ
0dt/integraldisplay
dǫZ(ǫ)Tr/braceleftbigg/bracketleftbigg
e−i/parenleftBig
∂S
t∂S†
ǫ−∂S†
t∂S
ǫ/parenrightBig
/2S(t,ǫ)S†(t,ǫ)−1/bracketrightbigg
Y/bracerightbigg
= 0. (C6)
Integration by parts with respect to tandǫgives
1
τ/integraldisplayτ
0dt/integraldisplay
dǫTr/braceleftbigg/bracketleftbigg
e−i/parenleftBig
∂S
t∂S†
ǫ−∂S
t∂ZS†
ǫ/parenrightBig
/2S(t,ǫ)Z(ǫ)S†(t,ǫ)−Z(ǫ)/bracketrightbigg
Y/bracerightbigg
= 0, (C7)
which can be simplified to
1
τ/integraldisplayτ
0dt/integraldisplay
dǫTr/braceleftbigg/parenleftbigg/bracketleftbigg
Z/parenleftbigg
ǫ+i
2∂
∂t/parenrightbigg
S(t,ǫ)/bracketrightbigg
S†(t,ǫ)−Z(ǫ)/parenrightbigg
Y/bracerightbigg
= 0. (C8)
Similarly from (C3), we find
1
τ/integraldisplayτ
0dt/integraldisplay
dǫTr/braceleftbigg/parenleftbigg
S†(t,ǫ)/bracketleftbigg
Z/parenleftbigg
ǫ−i
2∂
∂t/parenrightbigg
S(t,ǫ)/bracketrightbigg
−1/parenrightbigg
Y/bracerightbigg
= 0. (C9)
Using this result for Y= 1 andZ(ǫ) =f(ǫ) in the
expression for the DC particle current (A25a), we see
that unitarity indeed implies particle current conserva-
tion,/summationtext
αI(c)
α= 0 for a time-periodic potential. However,
such a relation does not hold for spins. Choosing Y=σ,
we cannot rewrite Eq. (C9) in the form (A25b), unless
theS-matrix and the Pauli matrices commute. Unless
theS-matrix is time or spin independent, a net spin cur-
rent can be pumped out of the system, simultaneously
exerting a torque on the scattering region.Furthermore, choosing Z(ǫ) =/integraltextǫ
0dǫ′f(ǫ′),Y= 1 and
expanding the difference between (C9) and (C8) to sec-
ond order in frequency gives
1
τ/integraldisplayτ
0dt/integraldisplay
dǫTr/braceleftbigg
f(ǫ)∂S(t,ǫ)
∂tS†(t,ǫ)/bracerightbigg
= 0,
which we use to eliminate the first term in the expression
for the energy pumping, Eq. (A30).
∗Electronic address: Arne.Brataas@ntnu.no
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1409.6900v2.Dissipationless_Multiferroic_Magnonics.pdf | arXiv:1409.6900v2 [cond-mat.mes-hall] 17 Apr 2015Dissipationless Multiferroic Magnonics
Wei Chen1and Manfred Sigrist2
1Max-Planck-Institut f ¨ur Festk¨orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, G ermany
2Theoretische Physik, ETH-Z¨ urich, CH-8093 Z¨ urich, Switz erland
(Dated: October 15, 2018)
We propose that the magnetoelectric effect in multiferroic i nsulators with coplanar antiferromag-
netic spiral order, such as BiFeO 3, enables electrically controlled magnonics without the ne ed of
a magnetic field. Applying an oscillating electric field in th ese materials with frequency as low
as household frequency can activate Goldstone modes that ma nifests fast planar rotations of spins,
whose motion is essentially unaffected by crystalline aniso tropy. Combining with spin ejection mech-
anisms, such a fast planar rotation can deliver electricity at room temperature over a distance of
the magnetic domain, which is free from energy loss due to Gil bert damping in an impurity-free
sample.
PACS numbers: 85.75.-d, 72.25.Pn, 75.85.+t
Introduction.- A primary goal of spintronic research
is to seek for mechanisms that enable electric ( E) field
controlled spin dynamics, since, in practice, Efields are
much easier to manipulate than magnetic ( B) fields. As
spinsdonotdirectlycoupleto Efield, incorporatingspin-
orbit coupling seems unavoidable for this purpose. Along
this line came the landmark proposals such as spin field
effect transistor [1] and spin-orbit torque [2–5], the real-
izations of which suggest the possibility of spin dynamics
with low power consumption. On the other hand, in an-
other major category of spintronics, namely magnonics,
which aims at the generation, propagation, and detection
of magnons, a mechanism that enables electrically con-
trolled magnonics without the aid of a magnetic field has
yet been proposed.
Raman scattering experiments [6, 7] on the room tem-
perature multiferroic BiFeO 3(BFO) shed light on this
issue. The magnetic order of BFO is a canted antiferro-
magnetic(AF) spiralontheplanespannedbythe electric
polarization Palong[111]andoneofthethreesymmetry-
equivalent wave vectors on a rhombohedral lattice [8, 9].
The spins have only a very small out-of-plane component
[10, 11]. Applying a static Efield∼100kV/cm signif-
icantly changes the cyclon (in-plane) and extra-cyclon
(out-of-plane) magnons because of the magnetoelectric
effect [7]. Indeed, spin-orbit coupling induced magneto-
electric effects are a natural way to connect Efield to
the spin dynamics of insulators [12, 13]. Motivated by
the Raman scattering experiments on BFO, in this Let-
ter we propose that applying an oscillating Efield to a
coplanar multiferroic insulator (CMI) that has AF spiral
order can achieve electrically controlled dissipationless
magnonics, which can deliver electricity with frequency
as low as household frequency up to the range of mag-
netic domains. Compared to the magnonics that uses B
field, microwave, or spin torques to generate spin dynam-
ics in prototype Y 3Fe5O12(YIG) [14–16], the advantage
of using CMI is that a single domain sample up to mm
size is available [17], and Raman scattering data indicatewell-defined magnons in the absence of Bfield [7], so an
external Bfield is not required in the proposed mecha-
nism.
Spin dynamics in CMI.- We start from the AF spiral
on a square lattice shown in Fig. 1 (a), described by
H=/summationdisplay
i,αJSi·Si+α−Dα·(Si×Si+α) (1)
whereα={a,c}are the unit vectors defined on
thexz-plane,J >0, andDα=Dαˆ y>0 is the
Dzyaloshinskii-Moriya (DM) interaction. The staggered
moment ( −1)iSiin the ground state shown in Fig. 1
(a) is characterized by the angle θα=Q·α=
−sin−1/parenleftig
Dα/˜Jα/parenrightig
betweenneighboringspins,where ˜Jα=
/radicalbig
J2+D2α. The DM interaction
Dα=D0
α+wE×α (2)
can be controlled by an Efield [18], where D0
αrepresents
the intrinsic value due to the lack of in version symmetry
of theα-bond. In the rotated reference frame S′defined
by
S′z
i=Sz
icosQ·ri+Sx
isinQ·ri,
S′x
i=−Sz
isinQ·ri+Sx
icosQ·ri,(3)
andS′y
i=Sy
i, the Hamiltonian is
H=/summationdisplay
i,α˜Jα/parenleftbig
S′x
iS′x
i+α+S′z
iS′z
i+α/parenrightbig
+JS′y
iS′y
i+α.(4)
Since˜Jα>J, the spins have collinear AF order and all
S′z
i= (−1)iSlie inxz-plane.
Thespin dynamicsin the absenceof Bfieldisgoverned
by the Landau-Lifshitz-Gilbert (LLG) equation
dS′
i
dt=∂H
∂S′
i×S′
i+αGS′
i×/parenleftbigg∂H
∂S′
i×S′
i/parenrightbigg
(5)2
expressed in the S′frame, where αGis the phenomeno-
logical damping parameter. Eq. (5) can be solved by the
spin wave ansatz for the even ( e) and odd ( o) sites [19]
/parenleftbiggS′x
e,o
S′y
e,o/parenrightbigg
=/parenleftbiggux
e,o
vy
e,o/parenrightbigg
ei(k·re,o−ωt). (6)
Ignoringthe dampingterm inEq.(5) yieldseigenenergies
ω±
k
2S=
/parenleftigg/summationdisplay
α˜Jα±γα−(k)/parenrightigg2
−/parenleftigg/summationdisplay
αγα+(k)/parenrightigg2
1/2
,(7)
whereγα±(k) =/parenleftig
˜Jα/2±J/2/parenrightig
cosk·α. Their eigenval-
ues and eigenvectors near k= (0,0) andk= (π,π) are
summarized below
/braceleftig
ω+
k→(0,0),ω−
k→(π,π)/bracerightig
= 2S/radicalbig
2(D2a+D2c),
ue
ve
uo
vo
∝
0
1
0
∓1
+O/parenleftbiggD
J/parenrightbigg
.
/braceleftig
ω−
k→(0,0),ω+
k→(π,π)/bracerightig
= 0,
ue
ve
uo
vo
∝
1
0
∓1
0
.(8)
The in-plane magnon dS′
i/dt= (dS′x
i/dt,0,0) is gapless,
while the out-of-plane magnon dS′
i/dt= (0,dS′y
i/dt,0)
develops a gap, as displayed in Fig. 1 (c). Even includ-
ing the damping term in Eq. (5), the in-plane magnons
very near the Goldstone modes ω−
k→(0,0)andω+
k→(π,π)re-
mainunchangedanddamping-free. Awayfromthe Gold-
stone limit, the eigenenergies become complex, hence the
magnons are subject to the damping and decay within a
time scale set by α−1
G.
Spin dynamics induced by oscillating Efield.-We an-
alyze now the spin dynamics in the damping-free in-
plane magnonchannel induced bymagnetoelectriceffects
(Eq. (8)). Unlike the spin injection by using the spin Hall
effect (SHE) to overcome the damping torque [16], our
design does not require an external Bfield, and is fea-
sible over a broad range of frequencies. Consider the
device shown in Fig. 2, where an oscillating electric field
E=E0cosωtis applied parallel to the ferroelectric mo-
ment over a region of length L=Na, such that the DM
interaction in Eq. (2) oscillates in this region. Thus, the
wave length of the spiral changes with time yielding an
oscillation of the number of spirals inside this region,
nQ=L
2π/|Q|≈N
2πJ/bracketleftbig
D0
a+wE0acosωt/bracketrightbig
,(9)
assumingDa=D0
a+wE0a≪J,Dc= 0, and E⊥a.
Suppose the spin S0at one boundary is fixed by, for
FIG. 1: (color online) Schematics of 2D AF spiral in the (a)
originalS-frame and the (b) rotated S′-frame. Red and blue
arrows indicate the spins on the two sublattices. (c) Spin
wave dispersion ω+
k(dashed line) and ω−
k(solid line) solved
in theS′frame, with Da/J= 0.14,Dc= 0. Inserts show
their eigen modes in the S′frame near k= (0,0) and (π,π),
where the spin dynamics dS′
i/dtis indicated by black arrows
or symbols.
instance, surface anisotropy because of specific coating.
ThenSNat the other boundary rotates by
∂θN
∂t=−N
JwE0aωsinωt, (10)
because whenever the number of waves nQchanges by 1,
SNrotates 2πin orderto to wind or unwind the spin tex-
ture in the Efield region. The significance of this mecha-
nism is that although the Efield is driven by a very small
frequencyω, the spin dynamics ∂tθNat the boundary is
manyordersofmagnitude enhanced because of the wind-
ing process. The rotation of SNserves as a driving force
for the spin dynamics in the field-free region from SNto
SN+M. As long as the spin dynamics is slower than the
energy scale of the DM interaction ∂tθi<|D0|//planckover2pi1∼THz,
one can safely consider the Efield region as adiabatically
changing its wave length but remaining in the ground
state. The spins in the field-free region rotate coherently
∂tθN=∂tθN+1=...=∂tθN+M, synonymous to exciting
theω−
k→(0,0)mode in Eq.(8), hence the spin dynamics
in the field-free region remains damping-free in an ideal
situation.
In real materials, crystalline anisotropy and impuri-
ties are the two major sources to spoil the spin rota-
tional symmetry implicitly assumed here. In the supple-
mentary material[20], their effects are discussed by draw-
ing analogy with similar situations in the atom absorp-
tion on periodic substrates and the impurity pinning of
charge density wave states. It is found that crystalline3
quantity symbol magnitude
lattice constant a nm
s−dexchange Γ 0.1eV
s−dexchange time τex 10−14s
spin relaxation time τsf 10−12s
spin diffusion length λN 10nm
spin density n01027/m3
spin Hall angle θH 0.1
intrinsic DM D0
α10−3eV
superexchange J 0.1eV
Eq. (2) w 10−19C
electric flux quantum ˜Φ0
E 1V
TABLE I: List of material parameters and their order of mag-
nitude values.
anisotropy remains idle because of the long spiral wave
length and the smallness of crystalline anisotropy com-
pared to exchange coupling. The impurities that tend to
pin the spins alongcertaincrystalline directionopen up a
gap in the Goldstone mode and cause energy dissipation,
which nevertheless do not obstruct the coherent rotation
of spins generated by Eq. (10).
FIG. 2: (color online) Experimental proposal of using oscil -
latingEfield to induce spin dynamics in CMI. The AF spiral
order is shown in the S′frame. The Efield is applied between
S′
0andS′
N, causing dynamics in the whole spin texture. Two
ways for spin ejection out of S′
N+Mare proposed: (a) Using
SHE to converted it into a charge current. (b) Using time-
varying spin accumulation and inductance.
Spin ejection and delivery of electricity.- We now ad-
dress the spin ejection from the CMI to an attached
normal metal (NM). A spin current is induced in the
NM when a localized spin Siat the NM/CMI inter-
face rotates [16, 21]. Defining the conduction electron
spinm(r,t) =−∝an}b∇acketle{tσ∝an}b∇acket∇i}ht/2, thes-dcoupling at the interface
Hsd= Γσ·Sidefines a time scale τex=/planckover2pi1/2S|Γ|, withΓ<0 [21]. The Bloch equation in the NM reads
∂m
∂t+∇·Js=1
τexm׈Si−δm
τsf(11)
whereJs=JNM
s/varotimesσ/planckover2pi1/2 is the spin current tensor, and
τsfis the spin relaxation time in the NM. In equilib-
rium, we assume mhybridizes with each Sion the spi-
ral texture locally. If the dynamics of Siis slow com-
pared to 1/τex, which is true for the proposed mechanism
and also for other usual means such as ferromagnetic
resonance[16], mfollows−ˆSiat any time with a very
small deviation m=m0+δm=−n0ˆSi+δm, wheren0
is the local equilibrium spin density. The spin current
tensorJs=−D0∇δmis obtained from the diffusion of
δm, whereD0is the spin diffusion constant. Under such
an adiabatic process, the small deviation is[21]
δm=τex
1+ξ2/braceleftigg
−ξn0∂ˆSi
∂t−n0ˆSi×∂ˆSi
∂t/bracerightigg
,(12)
whereξ=τex/τsf<1 so one can drop the first term on
the right hand side, and replace ˆSi×∂tˆSi→δ(r)ˆSi×∂tˆSi
sinceˆSiis located at the NM/CMI interface r= 0 (ras
coordinate perpendicular to the interface). The resulting
equation solves the time dependence of δm. Away from
r= 0, Eq. (11) yields D0∇2δm=δm/τsf, which solves
the spatial dependence of δm. The spin current caused
by a particular Sithen follows
JNM
sδˆm=δmD0
λN=−τexn0D0
(1+ξ2)λNˆSi×∂ˆSi
∂te−r/λN,(13)
whereλN=/radicalbig
D0τsf, similar to results obtained previ-
ously[16]. Ifonlythein-planeGoldstonemodeisexcited,
as shown in Fig. 2, it is equivalent to a global rotation of
spinsˆSi= (−1)i(sin(θ(t) +Q·ri),0,cos(θ(t) +Q·ri))
in the field-free region. Thus the time dependence in
Eq. (13), ˆSi×∂tˆSi=ˆy∂θ/∂t, is that described by
Eq. (10), and is the same for every Siat the NM/CMI
interface, even though each Sipoint at a different polar
angle. In other words, the spin current ejected from each
Siof the AF spiral, described by Eq. (13), is the same,
so a uniform spin current flows into the NM.
We propose two setups to convert the ejected spin cur-
rent into an electric signal. The first device uses inverse
SHE[16]inaNMdepositedatthesideofthespiralplane,
yieldingδˆmperpendicular to JNM
sand consequently a
voltage in the transverse direction, as shown in Fig. 2
(a). The second design ejects spin into a NM film de-
posited on top of the spiral plane, as shown in Fig. 2
(b), causing δˆmparallel to JNM
s. A spin accumulation
in the NM develops and oscillates with time, producing
an oscillatingmagnetic flux Φ Bthrougha coil that wraps
around the NM, hence a voltage E=−∂ΦB/∂t.
Experimental realizations.- TheRamanscatteringdata
on BFO [7] show that applying |E| ∼100kV/cm can4
change the spin wave velocity by δv0/v0∼1%. We
can make use of this information to estimate the field-
dependence win Eq. (2). The ω−
kmode in Eq. (7) near
k= (0,0) is
ω−
k→0= 2√
2SJka/bracketleftbigg
1+5
16/parenleftbiggD2
ak2
a+D2
ck2
c
J2k2/parenrightbigg/bracketrightbigg
= (v0+δv0)k, (14)
wherev0= 2√
2SJais the spin wave velocity in the
absence of DM interaction. Assuming Da∝ne}ationslash= 0,Dc=
0, andE⊥a, the Raman scattering data gives w∼
10−19C∼ |e|. We remark that a coplanar magnetic order
can be mapped into a spin superfluid [36, 37] ψiby
∝an}b∇acketle{tSi∝an}b∇acket∇i}ht=S(sinθi,0,cosθi) =√v(Imψi,0,Reψi),(15)
wherevis the volumeofthe 3Dunit cell. Within this for-
malism, the Efield can induce quantum interference of
the spin superfluid via magnetoelectric effect, in which
the electric flux vector ΦE=/contintegraltext
E×dlis quantized
[24, 25]. The flux quantum is ˜Φ0
E= 2πJ/w, which is
˜Φ0
E∼1V for BFO, close to that ( ∼10V) obtained from
current-voltage characteristics of a spin field-effect tran-
sistor [24], indicating that strong spin-orbit interaction
reduces the flux quantum to an experimentally accessi-
ble regime. For instance, BFO has a spiral wave length
2π/Q∼100nm, so in a BFO ring of µm size, the num-
ber of spirals at zero field is nQ∼10, and applying
|E| ∼1kV/cm can change nQby 1. Besides changing
the winding number, we remarkthat the magnetoelectric
effect can also be used to affect the topological proper-
ties of a magnet in a different respect[26]. Table I lists
the parameters and their order of magnitude values by
assuming CMI has similar material properties as other
magnetic oxide insulators such as YIG, and we adopt
lattice constant a∼1nm for both CMI and the NM for
simplicity.
For the device in Fig. 2, consider the field |E0| ∼
100kV/cm oscillating with a household frequency ω∼
100Hz is applied to a range L∼1mm. This region covers
N=L/a∼106sites with a number of spirals nQ∼104
at zero field. The Efield changes the number of spi-
rals tonQ∼105within time period 1 /ω∼0.01s, so
the spins at the boundary SNwind with angular speed
∂tθN∼107sinωtwhich is enhanced by 5 orders of mag-
nitude from the driving frequency ω. To estimate the
ejected spin current in Eq. (13), we use the typical spin
relaxation time τsf∼10−12s and length λN∼10nm
for heavy metals [16]. The s-dcoupling can range be-
tween [16] 0 .01eV to 1eV. We choose Γ ∼0.1eV, which
givesτex∼10−14s. The spin Hall angle θH∼0.1 has
been achieved [27, 28]. To estimate n0, we use the fact
that thes-dhybridization Γ σ·Siis equivalent to ap-
plying a magnetic field H= 2ΓSi/µ0gµBlocally at the
interface atomic layer of the NM. Given the typical mo-
lar susceptibility χm∼10−4cm3/mol and molar volumeVm∼10cm3/mol, the interface magnetization of the NM
isn0µB=χmH/Vm∼104C/sm, thus n0∼1027/m3.
Theoscillating Efieldgives ˆSi×∂tˆSi=∂tθNˆy∼ˆy107Hz,
so the ejected spin current is JNM
s∼1024/planckover2pi1/m2s. Using
the design in in Fig. 2(a) to convert JNM
sinto a charge
current via inverse SHE yields JNM
c∼104A/m2, hence
a voltage ∼µV oscillating with ωin a mm-wide sample.
To use the setup in Fig. 2(b), a NM film of area ∼1 mm2
and thickness ∼10nm yields E ∼mV oscillating with ω.
In summary, we propose that for multiferroics that
have coplanar AF spiral order, such as BFO, applying
an oscillating Efield with frequency as low as house-
hold frequency generates a coherent planar rotation of
the spin texture whose frequency is many orders of mag-
nitude enhanced. This coherent rotation activates the
Goldstone mode of multiferroic insulators that remains
unaffected by the crystalline anisotropy. The Goldstone
mode can be used to deliver electricity at room tempera-
ture up to the extensions of magnetic domains, in a way
that is free from the energy loss due to Gilbert damping
if the sample is free from impurities. The needlessness
ofBfield greatly reduces the energy consumption and
increases the scalability of the proposed device, pointing
to its applications in a wide range of length scales.
We thank exclusively P. Horsch, J. Sinova, H. Naka-
mura, Y. Tserkovnyak, D. Manske, M. Mori, C. Ulrich,
J. Seidel, and M. Kl¨ aui for stimulating discussions.
Supplementary material
I. Crystalline anisotropy in multiferroics
First we demonstrate that because the wave length of
the spiral order in multiferroics is typically 1 ∼2 orders
longer than the lattice constant, and the exchange cou-
pling is typically few orders larger than the crystalline
anisotropy energy, the spiral order remains truly incom-
mensurate and very weakly affected by the crystalline
anisotropy. For simplicity, we consider a spiral state with
wave vector Q∝ba∇dbl(1,0) and translationally invariant per-
pendicular to Qsuch that the geometry can be reduced
to a 1D problem. The classical elastic energy for a 1D
antiferromagnetic (AF) spiral is
E0=/summationdisplay
n−˜JaS2cos(θn+1−θn−θa)
≈ −N˜JaS2+/summationdisplay
n1
2˜JaS2(θn+1−θn−θa)2,(16)
whereθn=Q·rnis the angle relative to the staggered
spin (−1)iSi, andθa=Q·ais the natural pitch an-
gle between neighboring spins ( a= (a,0)). The square
lattice symmetry of our model yields a 4-fold degener-
ate crystalline spin anisotropy[38], leading to the total5
energy
E=/summationdisplay
n1
2˜JaS2(θn+1−θn−θa)2
+/summationdisplay
nVani(1−cos4θn), (17)
whereVaniis the anisotropy energy per site. This is the
well-known Frenkel-Kontorowa(FK) model[30, 31] that
has been discussed extensively owing to its rich physics.
FIG. 3: (color online) Schematics of mapping the AF spi-
ral order in the presence of crystalline anisotropy into FK
model. The angles θiof staggered spins ( −1)iSi(blue ar-
rows) are mapped into displacements xiof particles (orange
dots). The width of the 4-fold degenerate pinning potential
V(1−cos2πxi/b) isb=π/2, and the spacing of particles in
the absence of the pinning potential is a0=Q·a.
We consider the limit of weak anisotropy V=
Vani/˜JaS2a2≪1 and the case of long wavelength of
the spiral, θa≪π/2 whereπ/2 is the angle between
two minima of the anisotropy potential. In the spirit of
Ref.[32, 33] we assume now that there are prime num-
bers,MandLwithM˜θa=Lπ/2 andM≫Lwhich is
the average pitch in the ground state of Eq.(17). Then
we introduce the parametrization
θn=n˜θa+ϕn
4(18)
and the misfit parameter δ= 4(θa−˜θa). Turning to
the continuous limit one can derive the effective en-
ergy functional based on expanding the first harmonic
approximation[32–34],
˜E[ϕ] =/integraldisplay
dx/bracketleftigg
1
2/parenleftbiggdϕ
dx−δ/parenrightbigg2
+VMcos(Mϕ)/bracketrightigg
(19)
withVM∼VMwhich can become extremely small for
M≫1. The commensurate-incommensurate transition
happens ifδis large enough to stabilize the formation of
solitonsδ > δc(M)∼4√VM/π. Deep inside the incom-
mensurate phase, ϕ(x)≈δxsuch thatθn≈θanfollows
esentially the natural spiral pitch.
In our system, BFO, the spiral wave length ℓ≈
60nm∼100awhich yields M∼100/4 = 25, i.e. every
25thspin could be pinned along one of the 4 anisotropy
minima (assuming L= 1). Typical anisotropy ener-
gies for ferrites[35] lead to Vani∼10−3eV while theexchange energy is Ja∼0.1eV, from which we obtain
V∼Vani/Ja∼10−2and consequently VM∼10−50is
a negligible number. The misfit parameter may be as
large asδ= 4(θa−˜θa)∼π/M2such thatδ≫δc(M) is
well satisfied, even if by an electrical field Mshrinks by
one order of magnitude. Thus, the electric field-driven
oscillations of the spin spiral remains most likely unaf-
fected by the spin anisotropy. The small VMrenders
the energy gap due to the anisotropy energy irrelevant,
hence the in-plane magnon mode remains essentially un-
damped. Another important consequence of this analy-
sis is that although the concept of spin superfluidity, i.e.,
treating the spin texture as a quantum condensate, has
been proposed long ago, its realization in collinear mag-
nets is problematic because of the crystalline anisotropy
and subsequently the formation of domain walls. We
demonstrate explicitly that multiferroics are not sub-
ject to these problems because of the noncollinear or-
der, hence a room temperature macroscopic condensate
of mm size can be realized.
II. Phase-pinning impurities in multiferroics
We proceed to show that dilute, randomly distributed
impurities, exist either in the bulk of the multiferroic or
at the metal/multiferroic interface, do not obstruct the
proposed electrically controlled multiferroic magnonics.
Drawing analogyfrom the FK model, impurities that pin
the spins along certain crystalline directions, denoted by
phase-pinningimpurities, arethe impuritiesto be consid-
ered because they tend to impede the coherent motion of
spins[39]. Since we propose to use an oscillating Efield
to drive the spin rotation from the boundary, each cross
section channel is equivalent, which reduces the problem
from 2D to 1D. This leads us to consider the following
1D classical model similar to Eq. (17) for the field-free
region (S′
N+1toS′
N+Min the Fig. 2 of the main text).
E=/summationdisplay
i1
2˜JaS2(θi+a−θi−θa)2−/summationdisplay
i∈impVimpcos4θi,(20)
whereVimp>0 is the pinning potential, and/summationtext
i∈imp
sums over impurity sites. The total length of the chain is
L′=MawithManinteger. Inthepresenceofoscillating
Efield that causes the winding of boundary spins ( S′
N
in the Fig. 2 of the main text), the angle of spins in the
disordered field-free region has three contributions
θi=θ0
i+∆θi+ηi, (21)
whereθ0
irepresents the spiral texture in the unstretched
cleanlimitsatisfying θ0
i+a−θ0
i−θa= 0,∆θiisthestretch-
ing of the spin texture caused by winding of boundary
spins, and ηiis the distortion due to impurities. Only
the later two contribute to the elastic energy, so Eq. (20)6
becomes
E=/summationdisplay
i1
2˜JaS2(∆θi+a−∆θi+ηi+a−ηi)2
−/summationdisplay
i∈impVimpcos4θi. (22)
In this analysis we consider weak impurities Vimp≪
˜JaS2, andassumethatthe windingofthe boundaryspins
is slow such that the winding spreads through the whole
field-free region evenly, causing every pair of neighboring
spins to stretch by the same amount ∆ θi+a−∆θi= ∆θ.
For the electrically driven magnonics proposed in the
main text, which can achieve winding of boundary spins
byθN∼nQ∼105within half-period, a field-free region
of lengthL′∼mm has ∆θ∼0.1, so our numerics is done
with ∆θlimited within this value.
In the weak impurity limit, the length scale L0over
whichηichanges by O(1) can be calculated in the fol-
lowing way. The elastic energy part in Eq. (22) within
L0is, in the continuous limit,
K(L0) =1
a/integraldisplayL0
0dx1
2˜JaS2a2/parenleftbigg∆θ
a+∂xη/parenrightbigg2
=L0
2a˜JaS2∆θ2+˜JaS2∆θ
α1+˜JaS2a
2α0L0,(23)
whereα0andα1are numerical constants of O(1), and
are set to be unity without loss of generality. Denoting
impurity density as nimp=Nimp/L′whereNimpis the
total number of impurities in the sample, the impurity
potential energy within L0is calculated by
V(L0) =−VimpRe
/summationdisplay
i∈impe4i(θ0
i+∆θ+η)
=−Vimp/radicalbig
nimpL0. (24)
Note that the contribution comes not from the zeroth
order impurity averaging, but its fluctuation that mimics
a random walk in the complex plane[40]. The phase ηis
assumed to be constant within L0and chosen to give
Eq. (24) and hence the total energy E(L0) =K(L0) +
V(L0) withinL0. Minimizing the total energy per site
E(L0)/L0gives the most probable pinning length L0. In
the unstretched case ∆ θ= 0,
L0=/parenleftigg˜JaS2a
α0Vimpn1/2
imp/parenrightigg2/3
(25)
is similar to the Fukuyama-Lee-Rice (FLR) length that
characterizes the impurity pinning of a charge density
wave ground state[40, 41]. Putting Eq. (25) back to
Eqs. (24) and (23), the corresponding E(L0)a/L0<0
can be viewed as the pinning energy per site that im-
pedes the coherent rotation of spins, and equivalently
represents the gap opened at the Goldstone mode.00.020.040.060.08 0.10246810
/CapDΕltaΘLog10/LParen1L0/Slash1a/RParen1/LParen1a/RParen1 Aimp/EΘual10/Minus5
10/Minus4
10/Minus3
10/Minus2
00.020.040.060.08 0.1/Minus1/Minus0.500.51
/CapDΕltaΘΕ/Multiply103/LParen1b/RParen1
Aimp/EΘual10/Minus4
10/Minus3
5/Multiply10/Minus3
10/Minus2
FIG. 4: (color online) (a) The logrithmic of the dimensionle ss
FLR length log10(L0/a) versus winding angle per site ∆ θ, in
several values of the dirtiness parameter Aimp. Dashed line
indicates the threshold when L0∼mm. (b) The dimensionless
pinning energy ǫversus winding per site.
In the presence of the stretching ∆ θ, the expression
ofL0is rather lengthy. It is convenient to define two
dimensionless parameters
Aimp=Vimp
˜JaS2/radicalbigg
Nimpa
L′,
ǫ=E(L0)
˜JaS2/parenleftbigga
L0/parenrightbigg
, (26)
whereAimp(the ”dirtiness parameter”) is the impurity
potential measured in unit of the elastic constant times
the square root of the impurity density, and ǫis the to-
tal energy per site measured in unit of the elastic con-
stant. Figure 4 shows the logrithmic of the dimensionless
pinning length L0/aand the dimensionless total energy
ǫ, plotted as functions of the stretching ∆ θ. There are
two evidences showing that the spin texture, originally
pinned by impurities with the pinning length in Eq. (25),
is depinned by the stretching ∆ θ: Firstly, the pinning
lengthL0increases as increasing ∆ θ. For a particular
sample size, for instance L′∼mm, the spin texture is
depinned when the pinning length exceeds the sample
sizeL0> L′, or equivalently when ∆ θis greater than
a certain threshold (intercept of the dashed line and the
coloredlinesinFig.4(a)). Secondly, thepinningenergy ǫ
becomes positive at large ∆ θ, indicating that the elastic
energy from stretching overcomes the impurity pinning
energy, so the spin texture is depinned. From Fig. 4, it
is also evident that the cleaner is the sample, the easier
it is to depin the spins by stretching, as smaller Aimpre-
quires smaller threshold value of ∆ θ. We conclude that
thephasepinning impuritiesdonot hampertheproposed
electrically driven multiferroic magnonics as long as the
dirtiness of the sample is limited, the winding speed of
the boundary spin is sufficient, and the sample size is
short enough.
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1706.01185v1.Consistent_microscopic_analysis_of_spin_pumping_effects.pdf | Consistent microscopic analysis of spin pumping e ects
Gen Tatara
RIKEN Center for Emergent Matter Science (CEMS),
2-1 Hirosawa, Wako, Saitama, 351-0198 Japan
Shigemi Mizukami
WPI - Advanced Insitute for Materials Research,
Tohoku University Katahira 2-1-1, Sendai, Japan
(Dated: October 20, 2018)
Abstract
We present a consistent microscopic study of spin pumping e ects for both metallic and insulating
ferromagnets. As for metallic case, we present a simple quantum mechanical picture of the e ect as
due to the electron spin
ip as a result of a nonadiabatic (o -diagonal) spin gauge eld. The e ect of
interface spin-orbit interaction is brie
y discussed. We also carry out eld-theoretic calculation to
discuss on the equal footing the spin current generation and torque e ects such as enhanced Gilbert
damping constant and shift of precession frequency both in metallic and insulating cases. For thick
ferromagnetic metal, our study reproduces results of previous theories such as the correspondence
between the dc component of the spin current and enhancement of the damping. For thin metal
and insulator, the relation turns out to be modi ed. For the insulating case, driven locally by
interfacesdexchange interaction due to magnetic proximity e ect, physical mechanism is distinct
from the metallic case. Further study of proximity e ect and interface spin-orbit interaction would
be crucial to interpret experimental results in particular for insulators.
1arXiv:1706.01185v1 [cond-mat.mes-hall] 5 Jun 2017I. INTRODUCTION
Spin current generation is of a fundamental importance in spintronics. A dynamic method
using magnetization precession induced by an applied magnetic eld, called the spin pumping
e ect, turns out to be particularly useful1and is widely used in a junction of a ferromagnet
(F) and a normal metal (N)(Fig. 1). The generated spin current density (in unit of A/m2)
has two independent components, proportional to _nandn_n, wherenis a unit vector
describing the direction of localized spin, and thus is represented phenomenologically as
js=e
4(Arn_n+Ai_n); (1)
whereeis the elementally electric charge and ArandAiare phenomenological constants
having unit of 1 =m2. Spin pumping e ect was theoretically formulated by Tserkovnyak et al.2
by use of scattering matrix approach3. This approach, widely applied in mesoscopic physics,
describes transport phenomena in terms of transmission and re
ection amplitudes (scattering
matrix), and provides quantum mechanical pictures of the phenomena without calculating
explicitly the amplitudes. Tserkovnyak et al. applied the scattering matrix formulation
of general adiabatic pumping4,5to the spin-polarized case. The spin pumping e ect was
described in Ref.2in terms of spin-dependent transmission and re
ection coecients at the
FN interface, and it was demonstrated that the two parameters, ArandAi, are the real
and the imaginary part of a complex parameter called the spin mixing conductance. The
spin mixing conductance, which is represented by transmission and re
ection coecients,
turned out to be a convenient parameter for discussing spin current generation and other
e ects like the inverse spin-Hall e ect. Nevertheless, scattering approach hides microscopic
physical pictures of what is going on, as the scattering coecients are not fundamental
material parameters but are composite quantities of Fermi wave vector, electron e ective
mass and the interface properties.
E ects of slowly-varying potential is described in a physically straightforward and clear
manner by use of a unitary transformation that represents the time-dependence. (See Sec.
II A for details.) The laboratory frame wave function under time-dependent potential, j (t)i,
is written in terms of a static ground state ('rotated frame' wave function) j iand a unitary
matrixU(t) asj (t)i=U(t)j i. The time-derivative @tis then replaced by a covariant
derivative,@t+ (U 1@tU), and the e ects of time-dependence are represented by (the time-
component of) an e ective gauge eld, A i(U 1@tU) (See Eq. (12)). In the same
2FIG. 1. Spin pumping e ect in a junction of ferromagnet (F) and normal metal (N). Dynamic
magnetization n(t) generates a spin current jsthrough the interface.
manner as the electromagnetic gauge eld, the e ective gauge eld generates a current if
spatial homogeneity is present (like in junctions) and this is a physical origin of adiabatic
pumping e ect in metals.
In the perturbative regime or in insulators, a simple picture instead of e ective gauge eld
can be presented. Let us focus on the case driven by an sdexchange interaction, Jsdn(t),
whereJsdis a coupling constant and is the electron spin. Considering the second-order
e ect of the sdexchange interaction, the electron wave function has a contribution of a
time-dependent amplitude
U(t1;t2) = (Jsd)2(n(t1))(n(t2)) = (Jsd)2[(n(t1)n(t2)) +i[n(t1)n(t2)]];(2)
wheret1andt2are the time of the interactions. The rst term on the right-hand side,
representing the amplitude for charge degrees of freedom, is neglected. The spin contribution
vanishes for static spin con guration, as is natural, while for slowly varying case, it reads
U(t1;t2)' i(t1 t2)(Jsd)2(n_n)(t1): (3)
As a result of this amplitude, spin accumulation and spin current is induced proportional to
n_n. The fact indicates that n_nplays a role of an e ective scalar potential or voltage in
electromagnetism, as we shall demonstrate in Sec. VII B for insulators. (The factor of time
di erence is written in terms of derivative with respect to energy or angular frequency in a
rigorous derivation. See for example, Eqs. (129)(132).) The essence of spin pumping e ect
is therefore the non-commutativity of spin operators. The above picture in the perturbative
regime naturally leads to an e ective gauge eld in the strong coupling limit6.
3The same scenario applies for cases of spatial variation of spin, and an equilibrium spin
current proportional to nrinemerges, where idenotes the direction of spatial variation7.
The spin pumping e ect is therefore the time analog of the equilibrium spin current induced
by vector spin chirality. Moreover, charge current emerges from the third-order process from
the identity6
tr[(n1)(n2)(n3)] = 2in1(n2n3); (4)
and this factor, a scalar spin chirality, is the analog of the spin Berry phase in the pertur-
bative regime. The spin pumping e ect and spin Berry's phase and spin motive force have
the same physical root, namely the non-commutative spin algebra.
From the scattering matrix theory view point the cases of metallic and insulating fer-
romagnet make no di erence as what conduction electrons in the normal metal see is the
interface. From physical viewpoints, such treatment appears too crude. Unlike the metallic
case discussed above, in the case of insulator ferromagnet, the coupling between the mag-
netization and the conduction electron in normal metal occurs due to a magnetic proximity
e ect at the interface. Thus the spin pumping by an insulator ferromagnet seems to be a
locally-induced perturbative e ect rather than a transport induced by a driving force due to
a generalized gauge eld. We therefore need to apply di erent approaches for the two cases
as brie
y argued above. In the insulating case, one may think that magnon spin current is
generated inside the ferromagnet because magnon itself couples to an e ective gauge eld8
similarly to the electrons in metallic case. This is not, however, true, because the gauge eld
for magnon is abelian (U(1)). Although scattering matrix approach apparently seems to
apply to both metallic and insulating cases, it would be instructive to present in this paper
a consistent microscopic description of the e ects to see di erent physics governing the two
cases.
A. Brief overview of theories and scope of the paper
Before carrying out calculation, let us overview history of theoretical studies of spin
pumping e ect. Spin current generation in a metallic junction was originally discussed by
Silsbee9before Tserkovnyak et al. It was shown there that dynamic magnetization induces
spin accumulation at the interface, resulting in a di usive
ow of spin in the normal metal.
4Although of experimental curiosity at that time was the interface spin accumulation, which
enhances the signal of conduction electron spin resonance, it would be fair to say that Silsbee
pointed out the `spin pumping e ect'.
In Ref.2, spin pumping e ect was originally argued in the context of enhancement of
Gilbert damping in FN junction, which had been a hot issue after the study by Berger10,
who studied the case of FNF junction based on a quantum mechanical argument. Berger
discussed that when a normal metal is attached to a ferromagnet, the damping of ferromagnet
is enhanced as a result of spin polarization formed in the normal metal, and the e ect was
experimentally con rmed by Mizukami11. Tserkovnyak et al. pointed out that the e ect has
a di erent interpretation of the counter action of spin current generation, because the spin
current injected into the normal metal indicates a change of spin angular momentum or a
torque on ferromagnet. In fact, the equation of motion for the magnetization of ferromagnet
reads
_n=
Bn n_n a3
eSdjs; (5)
where
is the gyromagnetic ratio, is the Gilbert damping coecient, dis the thickness
of the ferromagnet, Sis the magnetude of localized spin, and ais the lattice constant. Spin
current of Eq. (1) thus indicates that the gyromagnetic ratio and the the Gilbert damping
coecient are modi ed by the spin pumping e ect to be2
~ = +a3
4SdAr
~
=
1 +a3
4SdAi 1
: (6)
The spin pumping e ect is therefore detected by measuring the e ective damping constant
and gyromagnetic ratio. The formula (6) is, however, based on a naive picture neglecting
the position-dependence of the damping torque and the relation between the pumped spin
current amplitude and damping or
would not be so simple in reality. (See Sec. V.)
The issue of damping in FN junction was formulated based on linear-response theory by
Simanek and Heinirch12,13. They showed that the damping coecient is given by the rst-
order derivative with respect to the angular frequency !of the imaginary part of the spin
correlation function and argued that the damping e ect is consistent with the Tserkovnyak's
spin pumping e ect. Recently, a microscopic formulation of spin pumping e ect in metallic
junction was provided by Chen and Zhang14and one of the author15by use of the Green's
5functions, and a transparent microscopic picture of pumping e ect was provided. Scattering
representation and Green's function one are related14because the asymptotic behaviors of
the Green's functions at long distance are governed by transmission coecient16. In the
study of Ref.15, the uniform ferromagnet was treated as a dot having only two degrees
of freedom of spin. Such simpli cation neglects the dependence on electron wave vectors
in ferromagnets and thus cannot discuss the the case of inhomogeneous magnetization or
position-dependence of spin damping.
The aim of this paper is to provide a microscopic and consistent theoretical formula-
tion of spin pumping e ect for metallic and insulating ferromagnets. We do not rely on
the scattering approach. Instead we provide elementary quantum mechanical argument to
demonstrated that spin current generation is a natural consequence of magnetization dy-
namics (Sec. II). Based on the formulation, the e ect of interface spin-orbit interaction is
discussed in Sec. III. We also provide a rigorous formulation based on eld-theoretic ap-
proach emploied in Ref.15in Sec. IV. We also reproduce within the same framework Berger's
result10that the spin pumping e ect is equivalent to the enhancement of the spin damping
(Sec. V). E ect of inhomogeneous magnetization is brie
y discussed in Sec. VI.
Case of insulating ferromagnet is studied in Sec. VII assuming that the pumping is
induced by an interface exchange interaction between the magnetization and conduction
electron in normal metal, namely, by magnetic proximity e ect. The interaction is treated
perturbatively similarly to Refs.17,18. The dominant contribution to the spin current, the
one linear in the interface exchange interaction, turns out to be proportional to _n, while the
one proportional to n_nis weaker if the proximity e ect is weak.
The contribution from the magnon, magnetization
uctuation, is also studied. As has
been argued8, a gauge eld for magnon emerges from magnetization dynamics. It is, however,
an adiabatic one diagonal in spin, which acts as chemical potential for magnon giving rise
only to adiabatic spin polarization proportional to n. This is in sharp contrast to the
metallic case, where electrons are directly driven by spin-
ip component of spin gauge eld,
resulting in perpendicular spin accumulation, i.e., along _nandn_n. The excitation
in ferromagnet when magnetization is time-dependent is therefore di erent for metallic and
insulating cases. We show that magnon excitation nevertheless generates perpendicular spin
current,n_n, in the normal metal as a result of annihilation and creation at the interface,
which in turn
ips electron spin. The result of magnon-driven contribution agrees with the
6one in previous study19carried out in the context of thermally-driven spin pumping ('spin
Seebeck' e ect). It is demonstrated that the magnon-induced spin current depends linearly
on the temperature at high temperature compared to magnon energy. The amplitude of
magnon-driven spin current provides the magnitude of magnetic proximity e ect.
In our analysis, we calculate consistently the pumped spin current and change of the
Gilbert damping and resonant frequency and obtain the relations among them. It is shown
that the spin mixing conductance scenario saying that the magnitude of spin current pro-
portional ton_nis given by the enhancement factor of the Gilbert damping constant2,
applies only the case of thick ferromagnetic metal. For thin metallic case and insulator case,
di erent relations hold (See Sec. VIII.).
II. QUANTUM MECHANICAL DESCRIPTION OF METALLIC CASE
In this section, we derive the spin current generated by the magnetization dynamics
of metallic ferromagnet by a quantum mechanical argument. It is sometimes useful for
intuitive understanding, although the description may lack clearness as it cannot handle
many-particle nature like particle distributions. In Sec. IV we formulate the problem in the
eld-theoretic language.
A. Electrons in ferromagnet with dynamic magnetization
The model we consider is a junction of metallic ferromagnet (F) and a normal metal (N).
The magnetization (or localized spins) in the ferromagnet is treated as spatially uniform but
changing with time slowly. As a result of strong sdexchange interaction, the conduction
electron's spin follows instantaneous directions of localized spins, i.e., the system is in the
adiabatic limit. The quantum mechanical Hamiltonian for the ferromagnet is
HF= r2
2m F Mn(t); (7)
wheremis the electron's mass, is a vector of Pauli matrices, Mrepresents the energy split-
ting due to the sdexchange interaction and n(t) is a time-dependent unit vector denoting
the localized spin direction. The energy is measured from the Fermi energy F.
7As a result of the sdexchange interaction, the electron's spin wave function is given by20
jnicos
2j"i+ sin
2ei j#i (8)
wherej"iandj#irepresent the spin up and down states, respectively, and ( ; ) are polar
coordinates for n. To treat slowly varying localized spin, we switch to a rotating frame where
the spin direction is de ned with respect to instantaneous direction n7. This corresponds
to diagonalizing the Hamiltonian at each time by introducing a unitary matrix U(t) as
jn(t)iU(t)j"i; (9)
where
U(r) =0
@cos
2sin
2e i
sin
2ei cos
21
A; (10)
where states are in vector representation, i.e., j"i=0
@1
01
Aandj#i=0
@0
11
A. The rotated
Hamiltonian is diagonalized as (in the momentum representation)
eHFU 1HFU=k Mz; (11)
wherekk2
2m Fis the kinetic energy in the momentum representation (Fig. 2). In general,
FIG. 2. Unitary transformation Ufor conduction electron in ferromagnet converts the original
Hamiltonian HFinto a diagonalized uniformly spin-polarized Hamiltonian eHFand an interaction
with spin gauge eld, As;t.
when a statej ifor a time-dependent Hamiltonian H(t), satisfying the Schr odinger equation
i@
@tj i=H(t)j i, is written in terms of a state j iconnected by a unitary transformation
j iU 1j i, the new state satis es a modi ed Schr odinger equation
i@
@t+iU 1@
@tU
j i=~Hj i; (12)
8where ~HU 1HU. Namely, there arises a gauge eld iU 1@
@tUin the new frame j i. In
the present case of dynamic localized spin, the gauge eld has three components (sux t
denotes the time-component);
As;t iU 1@
@tUAs;t; (13)
explicitly given as7
As;t=1
20
BBB@ @tsin sincos @t
@tcos sinsin @t
(1 cos)@t 1
CCCA: (14)
Including the gauge eld in the Hamiltonian, the e ective Hamiltonian in the rotated frame
reads
eHe
FeHF+As;t=0
@k M Az
s;tA
s;t
A+
s;tk+M+Az
s;t1
A (15)
whereA
s;tAx
s;tiAy
s;t. We see that the adiabatic ( z) component of the gauge eld, Az
s;t,
acts as a spin-dependent chemical potential (spin chemical potential) generated by dynamic
magnetization, while non-adiabatic ( xandy) components causes spin mixing. In the case
of uniform magnetization we consider, the mixing is between the electrons with di erent
spin"and#but having the same wave vector k, because the gauge eld A
s;tcarries no
momentum. This leads to a mixing of states having an excitation energy of Mas shown in
Fig. 3. In low energy transport e ects, what concern are the electrons at the Fermi energy;
The wave vector kshould be chosen as kF+andkF , the Fermi wave vectors for "and#
electrons, respectively. (E ects of nite momentum transfer is discussed in Sec. VI. )
The Hamiltonian Eq. (15) is diagonalized to obtain energy eigenvalues of ~ k=k
q
(M+Az
s;t)2+jA?
s;tj2, wherejA?
s;tj2A+
s;tA
s;tand=represents spin ("and#cor-
respond to + and , respectively). We are interested in the adiabatic limit, and so the
contribution lowest-order, namely, the rst order, in the perpendicular component, A?
s;t, is
sucient. In the present rotating-frame approach, the gauge eld is treated as a static po-
tential, since it already include time-derivative to the linear order (Eq. (14)). Moreover, the
adiabatic component of the gauge eld, Az
s;t, is neglected, as it modi es the spin pumping
only at the second-order of time-derivative. The energy eigenvalues, k'k M, are
9thus una ected by the gauge eld, while the eigenstates to the linear order read
jk"iFjk"i A+
s;t
Mjk#i
jk#iFjk#i+A
s;t
Mjk"i; (16)
corresponding to energy of k+andk , respectively. For low energy transport, states we
need to consider are the following two having spin-dependent Fermi wave vectors, kFfor
=";#, namely
jkF""iF=jkF""i A+
s;t
MjkF"#i
jkF##iF=jkF##i+A
s;t
MjkF#"i: (17)
FIG. 3. For uniform magnetization, the non-adiabatic components of the gauge eld, A
s;t, induces
a spin
ip conserving the momentum.
B. Spin current induced in the normal metal
Spin pumping e ect is now studied by taking account of the interface hopping e ects on
states in Eq. (17). The interface hopping amplitude of electron in F to N with spin is
denoted by ~tand the amplitude from N to F is ~t
. We assume that the spin-dependence
of electron state in F is governed by the relative angle to the magnetization vector, and
hence the spin is the one in the rotated frame. Assuming moreover that there is no spin
ip scattering at the interface, the amplitude ~tis diagonal in spin. (Interface spin-orbit
interaction is considered in Sec. III.) The spin wave function formed in the N region at the
10interface as a result of the state in F (Eq. (17)) is then
jkF"iN~tjkF"i=~t"jkF"i ~t#A+
s;t
MjkF#i
jkF#iN~tjkF#i=~t#jkF#i+~t"A
s;t
MjkF"i; (18)
wherekFis the Fermi wave vector of N electron. The spin density induced in N region at
the interface is therefore
es(N)=1
2(NhkF"jjkF"iN"+NhkF#jjkF#iN#) (19)
whereis the spin-dependent density of states of F electron at the Fermi energy. It reads
es(N)=1
2X
T^z " #
M
Re[T"#]A?
s;t+ Im[T"#](^zA?
s;t)
(20)
whereA?
s;t= (Ax
s;t;Ay
s;t;0) =As;t ^zAz
s;tis the transverse (non-adiabatic) components of
spin gauge eld and
T0~t
~t0: (21)
Spin density of Eq. (20) is in the rotated frame. The spin polarization in the laboratory
frame is obtained by a rotation matrix Rij, de ned by
U 1iURijj; (22)
as
s(N)
i=Rijes(N)
j: (23)
Explicitly,Rij= 2mimj ij, wherem
sin
2cos ;sin
2sin ;cos
27. Using
Rij(A?
s;t)j= 1
2(n_n)i
Rij(^zA?
s;t)j= 1
2_ni; (24)
andRiz=ni, the induced interface spin density is nally obtained as
s(N)=s
0n+ Re[s](n_n) + Im[s]_n (25)
11where
s
01
2X
T
s" #
2MT"#: (26)
Since the N electrons contributing to induced spin density is those at the Fermi energy,
the spin current is simply proportional to the induced spin density as jsN=kF
ms(N), resulting
in
j(N)
s=kF
ms
0n+kF
mRe[s](n_n) +kF
mIm[s]_n: (27)
This is the result of spin current at the interface. The pumping eciency is determined
by the product of hopping amplitudes t"andt
#. The spin mixing conductance de ned in
Ref.2corresponds to iT"#. If spin mixing e ects due to spin-orbit interaction is neglected at
the interface, the hopping amplitudes tare chosen as real, and Im[ s] = 0. If spin current
proportional to _nis measured, it would be useful tool to estimate the strength of interface
spin-orbit interaction, as discussed in Sec. III.
It should be noted that the spin pumping e ect at the linear order in time-derivative is
mapped to a static problem of spin polarization formed by a static spin-mixing potential in
the rotated frame as was mentioned in Ref.15. The rotate frame approach employed here
provides clear physical picture, as it grasps the low energy dynamics in a mathematically
proper manner. In this approach, as we have seen, it is clearly seen that pumping of spin
current arises as a result of o -diagonal components of the spin gauge eld that cause
electron spin
ip. Important role of nonadiabaticity is also indicated in a recent analysis
based on the full counting statistics21. In the strict sense, spin pumping e ect is a result of a
non-adiabatic process including state change. The same goes for general adiabatic pumping;
Some sort of state change is necessary for current generation, although the nonadiabaticity
is obscured in the conventional \adiabatic\ argument focusing on the wave function in
the laboratory frame. In the case of slowly-varying external potential with frequency
acting on electrons, the state change is represented by the Fermi distribution di erence,
f(!+
) f(!)'
f0(!), where!is the electron frequency3,4. The existence of a factor of
f0clearly indicates that a state change or nonadiabaticity is necessary for current pumping.
12III. EFFECTS OF INTERFACE SPIN-ORBIT INTERACTION
In this section, we discuss the e ect of spin-orbit interaction at the interface, which
modi es hopping amplitude ~t. We particularly focus on that linear in the wave vector,
namely the interaction represented in the continuum representation by a Hamiltonian
Hso=a2(x)X
ij
ijkij; (28)
where
ijis a coecient having the unit of energy representing the spin-orbit interaction,
ais the lattice constant, and the interface is chosen as at x= 0. Assuming that spin-
orbit interaction is weaker than the sdexchange interaction in F, we carry out a unitary
transformation to which diagonalize the sdinteraction to obtain
Hso=a2(x)X
ije
ijkij; (29)
wheree
ijP
l
ilRlj, withRijbeing a rotation matrix de ned by Eq. (22). This spin-
orbit interaction modi es diagonal hopping amplitude ~tiin the direction iat the interface
to become a complex as
eti=~t0
i iX
je
ijj: (30)
(In this section, we denote the total hopping amplitude including the interface spin-orbit
interaction by etand the one without by et0.) We consider the hopping amplitude perpendic-
ular to the interface, i.e., along the xdirection, and suppress the sux irepresenting the
direction. In the matrix representation for spin the hopping amplitude is
et(etx) =0
@et"et"#
et#"et#1
A; (31)
where
et"=~t0
" ie
xzet#=~t0
#+ie
xz
et"#=i(e
xx+ie
xy) et#"=i(e
xx ie
xy): (32)
Let us discuss how the spin pumping e ect discussed in Sec. II B is modi ed when the
hopping amplitude is a matrix of Eq. (31). The spin pumping eciency is written as in Eqs.
13(21)(26). In the absence of spin-orbit interaction hopping amplitude ~tis chosen as real, and
thus the contribution proportional to n_nin Eq. (27) is dominant. Spin-orbit interaction
enhances the other contribution proportional to _nbecause it gives rise to an imaginary part.
Moreover, it leads to spin mixing at the interface, modifying the spin accumulation formed
in the N region at the interface.
The electron states in the N region at the interface are now given instead of Eq. (18) by
the following two states (choosing basis as0
@jkF"i
jkF#i1
A)
jkF"iNetjkF""iF=0
@et" et"#A+
s;t
M
et#" et#A+
s;t
M1
A
jkF#iNetjkF##iF=0
@et"#+et"A
s;t
M
et#+et#"A
s;t
M1
A: (33)
The pumped (i.e., linear in the gauge eld) spin density for these two states are
NhkF"jjkF"iN= 2
M(A?
s;tRe[Ttot
"#] + (^zA?
s;t)Im[Ttot
"#]
+Re[(et"#)et#"]0
BBB@Ax
s;t
Ay
s;t
01
CCCA+ Im[(et"#)et#"]0
BBB@Ay
s;t
Ax
s;t
01
CCCA1
CCCA
^z(Ax
s;tRe[(et")et"# et#(et#")] Ay
s;tIm[(et")et"# et#(et#")]) (34)
NhkF#jjkF#iN=2
M(A?
s;tRe[Ttot
"#] + (^zA?
s;t)Im[Ttot
"#]
+Re[(et"#)et#"]0
BBB@Ax
s;t
Ay
s;t
01
CCCA+ Im[(et"#)et#"]0
BBB@Ay
s;t
Ax
s;t
01
CCCA1
CCCA
+^z(Ax
s;tRe[(et")et"# et#(et#")] Ay
s;tIm[(et")et"# et#(et#")]) (35)
We here focus on the linear e ect of interface spin-orbit interaction and neglect the
spin polarization along the magnetization direction, n. The expression for the pumped
spin current then agrees with Eq. (27) with the amplitude swritten in terms of hopping
including the interface spin-orbit,
T"#= ((~t0
")+i(e
xz))(~t0
#+ie
xz): (36)
14If bulk spin-orbit interaction is neglected, bare hopping amplitude ~t0
is real and we may
reasonably assume that e
ijis real. The interface spin-orbit then leads to an imaginary part
as (usinge
xz=ni
xi)
Im[s] =" #
2M(~t0
"+~t0
#)
xini: (37)
The amplitude of spin current proportional to _nthus works as a probe for interface spin-orbit
interaction strength,
xi.
Let us discuss some examples. Of recent particular interest is the interface Rashba
interaction, represented by antisymmetric coecient
(R)
ij=ijk R
k; (38)
where Ris a vector representing the Rashba eld. In the case of interface, Ris perpen-
dicular to the interface, i.e., Rk^x. Therefore the interface Rashba interaction leads to
(R)
xj= 0 and does not modify spin pumping e ect at the linear order. (It contributes at
the second order as discussed in Ref.14.) In other words, vector coupling between the wave
vector and spin in the form of kexists only along the x-direction, and does not a ect
the interface hopping (i.e., does not include kx).
In contrast, a scalar coupling (D)(k) ((D)is a coecient), called the Dirac type
spin-orbit interaction, leads to
(D)
ij=(D)ij. The spin current along _nthen reads
j_n
s=(D)kF(" #)
2mM(~t0
"+~t0
#)nx_n: (39)
For the case of in-plane easy axis along the zdirection and magnetization precession given
byn(t) = (sincos!t;sinsin!t;cos), whereis the precession angle and !is the angular
frequency, we expect to have a dc spin current along the ydirection, as nx_n= !
2sin2^y
(nx_ndenotes time average).
IV. FIELD THEORETIC DESCRIPTION OF METALLIC CASE
Here we present a eld-theoretic description of spin pumping e ect of metallic ferromag-
net. The many-body approach has an advantage of taking account of particle distributions
automatically. Moreover, it describes propagation of particle density in terms of the Green's
15functions, and thus is suitable for studying spatial propagation as well as for intuitive under-
standing of transport phenomena. All the transport coecients are determined by material
constants.
The formalism presented here is essentially the same as in Ref.15, but treating the fer-
romagnet of a nite size and taking account of electron states with di erent wave vectors.
Interface spin-orbit interaction is not considered here.
Conduction electron in ferromagnetic and normal metals are denoted by eld operators
d,dyandc,cy, respectively. These operators are vectors with two spin components, i.e.,
d(d";d#). The Hamiltonian describing the F and N electrons is HF+HN, where
HFZ
Fd3rdy
r2
2m F Mn(t)
d
HNZ
Nd3rcy
r2
2m F
c: (40)
We set the Fermi energies for ferromagnet and normal metal equal. The hopping through
the interface is described by the Hamiltonian
HIZ
IFd3rZ
INd3r0
cy(r0)t(r0;r;t)d(r) +dy(r)t(r0;r;t)c(r0)
; (41)
wheret(r0;r;t) represents the hopping amplitude of electron from rin ferromagnetic regime
to a siter0in the normal region and the integrals are over the interface (denoted by I F
and I Nfor F and N regions, respectively). The hopping amplitude is generally a matrix
depending on magnetization direction n(t), and thus depends on time t. Hopping is treated
as energy-conserving. Assuming sharp interface at x= 0, the momentum perpendicular to
the interface is not conserved on hopping.
We are interested in the spin current in the normal region, given by
j
s;i(r;t) = 1
4m(r(r) r(r0))itr[ G<
N(r;t;r0;t)jr0=r; (42)
whereG<
N(r;t;r0;t0)i
c(r;t)cy(r0;t0)
denotes the lesser Green's function for the normal
region. It is calculated from the Dyson's equation for the path-ordered Green's function
de ned for a complex time along a complex contour C
GN(r;t;r0;t0) =gN(r r0;t t0)
+Z
cdt1Z
cdt2Z
d3r1Z
d3r2gN(r r1;t t1)N(r1;t1;r2;t2)GN(r2;t2;r0;t0);
(43)
16whereg<
Ndenotes the Green's function without interface hopping and N(r1;t1;r2;t2) is the
self-energy for N electron, given by the contour-ordered Green's function in the ferromagnet
as
N(r1;t1;r2;t2)Z
IFd3r3Z
IFd3r4t(r1;r3;t1)G(r3;t1;r4;t2)t(r2;r4;t2): (44)
Herer1andr2are coordinates at the interface I Nin N region and r3andr4are those in
IFfor F.Gis the contour-ordered Green's function for F electron in the laboratory frame
including the e ect of spin gauge eld. We denote Green's functions of F electron by G
andgwithout sux and those of N electron with sux N. The lesser component of the
normal metal Green's function is obtained from Eq. (43) as (suppressing the time and space
coordinates)
G<
N= (1 +Gr
Nr
N)g<
N(1 + a
NGa
N) +Gr
N<
NGa
N: (45)
For pumping e ects, the last term on the right-hand side is essential, as it contains the
information of excitation in F region. We thus consider the second term only;
G<
N'Gr
N<
NGa
N; (46)
and neglect spin-dependence of the normal region Green's functions, Gr
NandGa
N. The
contribution is diagramatically shown in Fig. 4.
A. Rotated frame
To solve for the Green's function in the ferromagnet, rotated frame we used in Sec. II A
is convenient. In the eld representation, the unitary transformation is represented as (Fig.
5(c))
d=U~d; c =U~c; (47)
whereUis the same 22 matrix de ned in Eq. (10). We rotate N electrons as well as F
electrons, to simplify the following expressions. The hopping interaction Hamiltonian reads
HI=Z
IFd3rZ
INd3r0
~cy(r0)~t(r0;r)~d(r) +~dy(r)~t(r0;r)~c(r0)
; (48)
17FIG. 4. (a) Schematic diagramatic representations of the lessor Green's function for N electron
connecting the same position r,G<
N(r;r)'Gr
N<
NGa
Nrepresenting propagation of electron density.
It is decomposed into a propagation of N electron from rto the interface at r2, then hopping to r4
in the F side, a propagation inside F, followed by a hopping to N side (to r1) and propagation back
tor. (Position labels are as in Eqs. (43)(44).) (b): The self energy <
Nrepresents all the e ects
of the ferromagnet. (c) Standard Feynman diagram representation of lessor Green's function for
N atr, Eqs. (46) and (44).
FIG. 5. Unitary transformation Uof F electron converts the original system with eld operator
d(shown as (a)) to the rotated one with eld operator ~dU 1d(b). The hopping amplitude
for representation in (b) is modi ed by U. If N electrons are also rotated as ~ cU 1c, hopping
becomes ~tU 1tU, while the N electron spin rotates with time, as shown as (c).
where
~t(r0;r)Uy(t)t(r0;r;t)U(t); (49)
is the hopping amplitude in the rotated frame. The rotated amplitude (neglecting interface
spin-orbit interaction) is diagonal in spin;
~t=0
@~t"0
0~t#1
A: (50)
Including the interaction with spin gauge eld, the Hamiltonian for F and N electrons in
18the momentum representation is
HF+HN=X
k~dy
k0
@k M Az
s;tA
s;t
A+
s;tk+M+Az
s;t1
A~dk+X
k(N)
k~cy
k~ck (51)
As for the hopping, we consider the case the interface is atomically sharp. The hopping
Hamiltonian is then written in the momentum space as
HI=X
kk0
~cy(k)~t(k;k0)~d(k0) +~dy(k0)~t(k;k0)~c(k)
; (52)
wherek= (kx;ky;kz),k0= (k0
x;ky;kz), choosing the interface as the plane of x= 0. Namely,
the wave vectors parallel to the interface are conserved while kxandk0
xare uncorrelated.
B. Spin density induced by magnetization dynamics in F
Pumped spin current in N is calculated by evaluating <
Nand using Eqs. (42)(45)(46).
Before discussing the spin current, let us calculate spin density in ferromagnet induced by
magnetization dynamics neglecting the e ect of interface, HI. (E ects of HIare discussed
in Sec. V.) The spin accumulation in N is discussed by extending the calculation here as
shown in Sec. IV C.
The lessor Green's function in F in the rotated frame including the spin gauge eld to
the linear order is calculated from the Dyson's equation
G<=g<+gr(As;t)g<+g<(As;t)ga; (53)
whereg ( =<;r,a) represents Green's functions without spin gauge eld. The lessor
Green's function satis es for static case g<=F(ga gr), whereF0
@f"0
0f#1
Ais spin-
dependent Fermi distribution function. We thus obtain the Green's function at the linear
order as15
G<=gr[As;t;F]ga+gaF(As;t)ga gr(As;t)Fgr: (54)
The last two terms of the right-hand side are rapidly oscillating as function of position and
are neglected. The commutator is calculated as (sign denotes spin"and#)
[As;t;F] = (f+ f )X
()A
s;t: (55)
19FIG. 6. Feynman diagram for electron spin density of ferromagnet induced by magnetization
dynamics (represented by spin gauge eld As) neglecting the e ect of normal metal. The amplitude
is essentially given by the spin
ip correlation function (Eq. (58)).
In the rotated frame, the spin density in F pumped by the spin gauge eld is therefore
(diagrams shown in Fig. 6)
~s(F)
(k;k0) iZd!
2tr[ G<(k;k0;!)]
= iZd!
2X
k00(fk00+ fk00 )X
()A
s;ttr[ gr(k;k00;!)ga(k00;k0;!)]
=8
<
:iRd!
2P
k00(fk00+ fk00 )A
s;tgr
(k;k00;!)ga
(k00;k0;!) ( =)
0 ( =z): (56)
Let us here neglect the e ects of interface in dicussing spin polarization of F electrons; Then
the Green's functions are translationally invariant, i.e., ga(k;k0) =k;k0ga(k) (a= r;a).
Using the explicit form of the free Green's function, ga
(k;!) =1
! k; i0, and
Zd!
2gr
(k;k00;!)ga
(k00;k0;!) =i
k; k;+i0; (57)
the spin density in the rotated frame then reduces to
~s(F)
(k) = A
s;t ; (58)
where
X
kfk; fk;
k; k;+i0; (59)
is the spin correlation function with spin
ip, + i0 meaning an in nitesimal positive imaginary
part. Since we focus on adiabatic limit and spatially uniform magnetization, the correlation
function is at zero momentum- and frequency-transfer. We thus easily see that
=n+ n
2M; (60)
20wheren=P
kfkis spin-resolved electron density.
The spin polarization of Eq. (58) in the rotated frame is proportional to A?
s;t, and
represents a renormalization of total spin in F. In fact, it corresponds in the laboratory
frame tos(F)/n_n, and exerts a torque proportional to _nonn.
It may appear from Eq. (60) that a damping of spin, i.e., a torque proportional to
n_n, arises when the imaginary part for the Green's function becomes nite, because
1
Mis replaced by1
Mii, whereiis the imaginary part. This is not always the case. For
example, nonmagnetic impurities introduce a nite imaginary part inversely proportional to
the elastic lifetime ( ),i
2 . They should not, however, cause damping of spin. The solution
to this apparent controversy is that Eq. (56) is not enough to discuss damping even including
lifetime. In fact, there is an additional process called vertex correction contributing to the
lesser Green's function, and it gives rise to the same order of small correction as the lifetime
does, and the sum of the two contributions vanishes. Similarly, we expect damping does not
arise from spin-conserving component of spin gauge eld, Az
s;t. This is indeed true as we
explicitly demonstrate in Appendix A. We shall show in Sec. V that damping arises from
the spin-
ip components of the self energy.
C. Spin polarization and current in N
FIG. 7. Feynman diagram for electron spin density of normal metal driven by the spin gauge eld
of ferromagnetic metal, As. The spin current is represented by the same diagram but with spin
current vertex.
The spin polarization of N electron lesser Green's function including the self-energy to
21the linear order is calculated from Eqs. (46) (54)(55) as (diagram shown in Fig. 7)
itr[G<
N(r;t;r0;t)] = iX
kk0k00eikre ik0r0gr
N(k;!)ga
N(k0;!)
X
(fk00 fk00)A
s;t~t(k;k00)~t
(k00;k0)gr
(k00;!)ga
(k00;!): (61)
We assume that dependence of N Green's functions on !is weak and useP
keikrgr
N(k;!) =
iNeikFxe jxj=`gr
N(r), where`is elastic mean free path, NandkFare the density of
states at the Fermi energy and Fermi wave vector, respectively, whose !-dependences are ne-
glected. (For in nitely wide interface, the Green's function becomes one-dimensional.) As a
result of summation over wave vectors, the product of hopping amplitudes ~t(k;k00)~t
(k00;k0)
is replaced by the average over the Fermi surface, ~t~t
T, i.e.,
~t(k;k00)~t
(k00;k0)!T: (62)
The spin polarization of N electron induced by magnetization dynamics (the spin gauge
eld) is therefore obtained in the rotated frame as
~s(N)
(r;t) = jgr
N(r)j2X
A
s;t T; (63)
or using
+=
~s(N)(r;t) = 2jgr
N(r)j2
A?
s;tRe[ +T+ ] + (^zA?
s;t)Im[ +T+ ]
: (64)
In the laboratory frame, we have (using s(N)
i=Rij~s(N)
j)
s(N)(r;t) =jgr
N(r)j2
Re[ +T+ ](n_n) + Im[ +T+ ]_n
: (65)
The spin current induced in N region is similarly given by (neglecting the contribution
proportional to n)
js(r;t) =kF
mjgr
N(r)j2
Re[ +T+ ](n_n) + Im[ +T+ ]_n
=e jxj=`(Re[s](n_n) + Im[s]_n); (66)
where
s2kF2
N
2mM(n+ n )T+ : (67)
22The coecient sis essentially the same as the one in Eq. (27) derived by quantum mechan-
ical argument, as quantum mechanical dimensionless hopping amplitude corresponds to N~t
of eld representation.
For 3d ferromagnet, we may estimate the spin current by approximating roughly M
1=NF1eV andnkF3. The hopping amplitude jT+ jin metallic case would be
order ofF. The spin current density then is of the order of (including electric charge eand
recovering ~),jse~kF
mh~!
F51011A/m2if precession frequency is 10 GHz.
V. SPIN ACCUMULATION IN FERROMAGNET
The spin current pumping is equivalent to the increase of spin damping due to magne-
tization precession, as was discussed in Refs.2,10. In this section, we con rm this fact by
calculating the torque by evaluating the spin polarization of the conduction electron spin in
F region.
There are several ways to evaluate damping of magnetization. One way is to calculate the
spin-
ip probability of the electron as in Ref.10, which leads to damping of localized spin in
the presence of strong sdexchange interaction. The second is to estimate the torque on the
electron by use of equation motion22. The relation between the damping and spin current
generation is clearly seen in this approach. In fact, the total torque acting on conduction
electron is ( ~times) the time-derivative of the electron spin density,
ds
dt=i
[H;dy]d
+
dy[H;d]
: (68)
At the interface, the right-hand side arises from the interface hopping. Using the hopping
Hamiltonian of Eq. (41), we have
ds
dt
interface=i
cytd
dytyc
; (69)
as the interface contribution. As is natural, the the right -hand side agrees with the de nition
of the spin current passing through the interface. Evaluating the right-hand side, we obtain in
general a term proportional to n_n, which gives the Gilbert damping, and term proportional
to_n, which gives a renormalization of magnetization. In contrast, away from the interface,
the commutator [ H;d] arises from the kinetic term H0R
d3rjrdj2
2mdescribing electron
23propagation, resulting in
ds
dt=i
[H0;dy]d
+
dy[H;d]
=rj
s (70)
wherej
s(r) i
2m(rr r r0)
dy(r0) d(r)
jr0=ris the spin current. Away from the inter-
face, the damping therefore occurs if the spin current has a source or a sink at the site of
interest.
Here we use the third approach and estimate the torque on the localized spin by calculat-
ing the spin polarization of electrons as was done in Refs.7,23. The electron spin polarization
at positionrin the ferromagnet at time tiss(F)(r;t)
dyd
, which reads in the rotated
frames(F)
=R ~s(F)
, where
~s(F)
(r;t) = itr[ G<(r;r;t;t)]; (71)
whereG<
0(r;r0;t;t0)iD
~dy
0~dE
is the lesser Green's function in F region, which is a
matrix in spin space ( ;0=). We are interested in the e ect of the N region arising from
the hopping. We must note that the hopping interaction of Eq. (48) is not convenient for
integrating out N electrons, since the ~ celectrons' spins are time-dependent as a result of a
unitary transformation, U(t). We thus use the following form (Fig. 5(b)),
HI=Z
IFd3rZ
INd3r0
cy(r0)U~t(r0;r)~d(r) +~dy(r)~t(r0;r)Uyc(r0)
; (72)
namely, the hopping amplitude between ~dandcelectrons includes unitary matrix U.
Let us argue in the rotated frame why the e ect of damping arising from the interface.
In the totally rotated frame of Fig. 5(c), the spin of F electron is static, while that of N
electron varies with time. When F electron hops to N region and comes back, therefore,
electron spin gets rotated with the amount depending on the time it stayed in N region. This
e ect is in fact represented by a retardation e ect of the matrices UandU 1in Eq. (72). If
o -diagonal nature of UandU 1are neglected, the interface e ects are all spin-conserving
and do not induce damping for F electron (See Sec. A).
We now proceed calculation of induced spin density in the ferromagnetic metal. Diagra-
matic representation of the contribution is in Fig. 8. Writing spatial and temporal positions
explicitly, the self-energy of F electron arising from the hopping to N region reads ( r1and
24FIG. 8. Diagramatic representation of the spin accumulation in ferromagnetic metal induced as
a result of coupling to the normal metal (Eqs. (71)(73)). Conduction electron Green's functions
in ferromagnet and normal metal are denoted by gandgN, respectively. Time-dependent matrix
U(t), de ned by Eq. (10), represents the e ect of dynamic magnetization. Expanding UandU 1
with respect to slow time-dependence of magnetization, we obtain gauge eld representation, Eq.
(75).
r2are in F)
a(r1;r2;t1;t2) =Z
INd3r0
1Z
INd3r0
2~t(r1;r0
1)U 1(t1)ga
N(r0
1;r0
2;t1 t2)U(t2)~ty(r2;r0
2) (73)
wherea= r;a;<. We assume the Green's function in N region is spin-independent; i.e., we
neglect higher order contribution of hopping. Moreover, we treat the hopping to occur only
at the interface, i.e., at x= 0. The self-energy is then represented as
a(r1;r2;t1;t2) =a2(x1)(x2)~tU 1(t1)U(t2)~tyX
kga
N(k;t1 t2); (74)
whereais the interface thickness, which we assume to be the order of the lattice constant.
Diagramatic representation of Eqs. (71)(73) are in Fig. 8. Expanding the matrix using
spin gauge eld as U 1(t1)U(t2) = 1 i(t1 t2)As;t+O((As;t)2), we obtain the gauge eld
contribution of the self-energy as
a(r1;r2;t1;t2) =a2(x1)(x2)Zd!
2de i!(t1 t2)
d!~tAs;t~tyX
kga
N(k;!)
= a2(x1)(x2)Zd!
2e i!(t1 t2)~tAs;t~tyX
kd
d!ga
N(k;!) (75)
25The linear contribution of the lessor component of the o -diagonal self-energy is
G<(r;t;r0;t) =grrga+gr<ga+g<aga
=a2Zd!
2X
k
gr(r;!)dgr
N(k;!)
d!~tAs;t~tyg<( r;!)
+gr(r;!)dg<
N(k;!)
d!~tAs;t~tyga( r;!) +g<(r;!)dga
N(k;!)
d!~tAs;t~tyga( r;!)
(76)
For nite distance from the interface, r, dominant contribution arises from the terms contain-
ing bothgr(r;!) andga( r;!), as they do not contain a rapid oscillation like ei(kF++kF )r
ande2ikFr. Using an approximationP
kgr
N(k;!) iNand partial integration with
respect to!, Eq. (76) reduces to
G<(r;t;r0;t) = 2iNa2Zd!
2f0
N(!)gr(r;!)~tAs;t~tyga( r;!) (77)
For damping, o -diagonal contributions, A
s;t, are obviously essential. The result of the spin
density in F in the rotated frame, Eq. (71), is
~s(F)
(r;t) = 2iNa2Zd!
2f0
N(!)A
s;ttr[ gr(r;!)~t ~tyga( r;!)]
= 2iNa2Zd!
2f0
N(!)A
s;tX
kk0ei(k k0)rtr[ gr(k;!)~t ~tyga(k0;!)] (78)
Evaluating the trace in spin space, we obtain
~s(F)(r;t) = N
A?
s;t
1(r) + ( ^zA?
s;t)
2(r)
(79)
where
1(r)X
~t ~ty
gr
(r)ga
( r)
2(r)X
( i)~t ~ty
gr
(r)ga
( r): (80)
We consider an interface with in nite area and consider spin accumulation averaged over
the plane parallel to the interface. The wave vector contributing is then those with nite kx
but withky=kz= 0 and Green's function become one-dimensional like
X
keikrgr
(k) =im
kFeikFjxje jxj=(2`); (81)
26where`vF (vFkF=m) is electron mean free path for spin . The induced spin
density in the ferromagnet is nally obtained from Eq. (79) as
s(F)(r;t) =m2Na2
2kF+kF X
(n_n)T; e i(kF+ kF )x+_n( i)T; e i(kF+ kF )x
=m2Na2
2kF+kF X
(n_n)
Re[T";#] cos((kF+ kF )x) + Im[T";#] sin((kF+ kF )x)
+_n
Im[T";#] cos((kF+ kF )x) Re[T";#] sin((kF+ kF )x)
(82)
and the torque on localized spin, Mns(F), is
(r;t) = m2Na2M
2kF+kF X
_n
Re[T";#] cos((kF+ kF )x) + Im[T";#] sin((kF+ kF )x)
+ (n_n)
Im[T";#] cos((kF+ kF )x) Re[T";#] sin((kF+ kF )x)
: (83)
A. Enhanced damping and spin renormalization of ferromagnetic metal
The total induced spin accumulation density in ferromagnet is
s(F)1
dZ0
ddxs(F)(x)
=1
M
(n_n)h
Im[](1 cos~d) + Re[] sin~di
+_nh
Re[](1 cos~d) + Im[] sin~di
;
(84)
where ~d(kF+ kF )d,dis the thickness of ferromagnet and
m2Na2M
kF+kF (kF+ kF )dT";#: (85)
As a result of this induced electron spin density, s(F), the equation of motion for the averaged
magnetization is modi ed to be10
_n= n_n
Bn Mns(F); (86)
whereBis the external magnetic eld.
Let us rst discuss thin ferromagnet case, d jkF+ kF j 1, where oscillating part with
respect to ~dis neglected. The spin density then reads s(F)'1
M( Im[](n_n) + Re[]_n)
and the equation of motion becomes
(1 + Im)_n= ~ n_n
Bn; (87)
27where
~ + Re; (88)
is the Gilbert damping including the enhancement due to the spin pumping e ect. The
precession angular frequency !Bis modi ed by the imaginary part of T";#, i.e., by the spin
current proportional to _n, as
!B=
B
1 + Im: (89)
This is equivalent to the modi cation of the gyromagnetic ratio (
) or theg-factor.
For most 3d ferromagnets, we may approximatem2NaMF2
2kF+kF (kF+ kF )'O(1) (askF+
kF /M), resulting in a
dT";#. As discussed in Sec. III, when interface spin-orbit
interaction is taken into account, we have T";#=~t0
"~t0
#+ie
xz(~t0
"+~t0
#) +O((e
)2), where ~t0
ande
xzare assumed to be real. Moreover, ~t0
can be chosen as positive and thus T";#>0.
(~t0
here is eld-representaion, and has unit of energy.) Equations (88) and (89) indicates
that the strength of the hopping amplitude ~t0
and interface spin-orbit interaction e
xzare
experimentally accessible by measuring Gilbert damping and shift of resonance frequency
as has been known2. A signi cant consequence of Eq. (88) is that the enhancement of the
Gilbert damping,
a
d1
F2~t0
"~t0
#; (90)
can exceed in thin ferromagnets the intrinsic damping parameter , as the two contributions
are governed by di erent material parameters. In contrast to the positive enhancement of
damping, the shift of the resonant frequency or g-factor can be positive or negative, as it is
linear in the interface spin-orbit parameter e
xz.
Experimentally, enhancement of the Gilbert damping and frequency shift has been mea-
sured in many systems11. In the case of Py/Pt junction, enhancement of damping is observed
to be proportional to 1 =din the range of 2nm <d< 10nm, and the enhancement was large,
= '4 atd= 2 nm11. These results appears to be consistent with our analysis. Same 1 =d
dependence was observed in the shift of g-factor. The shift was positive and magnitude is
about 2% for Py/Pt and Py/Pd with d= 2nm, while it was negative for Ta/Pt11. The exis-
tence of both signs suggests that the shift is due to the linear e ect of spin-orbit interaction,
and the interface spin-orbit interaction we discuss is one of possible mechanisms.
28For thin ferromagnet, ~d.1, the spin accumulation of Eq. (84) reads
s(F)=1
M((n_n)Re[thin] +_nIm[thin]); (91)
where
thin~d=m2Na2M
2kF+kF T";#: (92)
Equation (91) indicates that the roles of imaginary and real part of T";#are interchanged
for thick and thin ferromagnet, resulting in
~ = + Imthin
!B=
B
1 Rethin; (93)
for thin ferromagnet. Thus, for weak interface spin-orbit interaction, positive shift of reso-
nance frequency is expected (as Re thin>0). Signi cant feature is that the damping can be
smallened or even be negative if strong interface spin-orbit interaction exists with negative
sign of Imthin. Our result indicates that 'spin mixing conductance' description of Ref.2
breaks down in thin metallic ferromagnet (and insulator case as we shall see in Sec. VII D).
In this section, we have discussed spin accumulation and enhanced Gilbert damping in
ferromagnet attached to a normal metal. In the eld-theoretic description, the damping
enhancement arises from the imaginary part of the self-energy due to the interface. Thus
a randomness like the interface scattering changing the electron momentum is essential for
the damping e ect, which sounds physically reasonable. The same is true for the reaction,
namely, spin current pumping e ect into N region, and thus spin current pumping requires
randomness, too. (In the quantum mechanical treatment of Sec. II, change of electron
wave vector at the interface is essential.) The spin current pumping e ect therefore ap-
pears di erent from general pumping e ects, where randomness does not play essential roles
apparently3.
Spin accumulation and enhanced Gilbert damping was discussed by Berger10based on a
quantum mechanical argument. There 1 =ddependence was pointed out and the damping
e ect was calculated by evaluating the decay rate of magnons. Comparison of enhanced
Gilbert damping with experiments was carried out in Ref.2but in a phenomenological man-
ner.
29VI. CASE WITH MAGNETIZATION STRUCTURE
Field theoretic approach has an advantage that generalization of the results is straightfor-
ward. Here we discuss brie
y the case of ferromagnet with spatially-varying magnetization.
The excitations in metallic ferromagnet consist of spin waves (magnons) and Stoner excita-
tion. While spin waves usually have gap as a result of magnetic anisotropy, Stoner excitation
is gapless for nite wave vector, ( kF+ kF )<jqj<(kF++kF ), and it may be expected
to have signi cant contribution for magnetization structures having wavelength larger than
kF+ kF . Let us look into this possibility.
Our result of spin accumulation in ferromagnet, represented in the rotated frame, Eq.
(63), indicates that when the magnetization has a spatial pro le, the accumulation is deter-
mined by the spin gauge eld and spin correlation function depending on the wave vector q
as
X
qA
s;t(q) (q;0); (94)
where
(q;
) X
kfk+q; fk;
k+q; k;+
+i0; (95)
is the correlation function with nite momentum transfer qand nite angular frequency
.
For the case of free electron with quadratic dispersion, the correlation function is24
(q;
) =Aq+i
Bqst(q) +O(
2); (96)
where
Aq=ma3
82
(kF++kF )
1 +(kF+ kF )2
q2
+1
2q3((kF++kF )2 q2)(q2 (kF+ kF )2) ln q+ (kF++kF )
q (kF++kF )
Bq=m2a3
4jqj; (97)
and
st(q)8
<
:1 (kF+ kF )<jqj<(kF++kF )
0 otherwise: (98)
30describes the wave vectors where Stoner excitation exists. As we see from Eq. (96), the
Stoner excitation contribution vanishes to the lowest order in
, and thus the spin pumping
e ect in the adiabatic limit (
!0) is not a ected. Moreover, the real part of the correlation
function,Aq, is a decreasing function of qand thus the spin pumping eciency would
decrease when ferromagnet has a structure. However, for rigorous argument, we need to
include the spatial component of the spin gauge eld arising form the spatial derivative of
the magnetization pro le.
As for the e ect of the Stoner excitation on spin damping (Gilbert damping), it was
demonstrated for the case of a domain wall that the e ect is negligibly small for a wide wall
with thickness (kF+ kF ) 1(Refs.24,25). Simanek and Heinrich presented a result of
the Gilbert damping as the linear term in the frequency of the imaginary part of the spin
correlation function integrated over the wave vector12. The result is, however, obtained for a
model where ferromagnet is atomically thin layer (a sheet), and would not be applicable for
most experimental situations. Discussion of Gilbert damping including nite wave vector
and the impurity scattering was given in Ref.26. Inhomogenuity e ects of damping of a
domain wall was studied recently in detail27.
VII. INSULATOR FERROMAGNET
In this section, we discuss the case of ferromagnetic insulator. It turns out that the
generation mechanisms for spin current in the insulating and metallic cases are distinct.
A. Magnon and adiabatic gauge eld
The Lagrangian for the insulating ferromagnet is
LIF=Z
d3r
S_ (cos 1) J
2(rS)2
HK; (99)
whereJis the exchange interaction between the localized spin, S, andHKdenotes the
magnetic anisotropy energy.
We rst study low energy magnon dynamics induced by slow magnetization dynamics.
For separating the classical variable and
uctuation (magnon), rotating coordinate descrip-
tion used in the metallic case is convenient. For magnons described by the Holstein-Primakov
31boson, the unitary transformation is a 3 3 matrix de ned as follows28.
S=UeS; (100)
where
U=0
BBB@coscos sin sincos
cossin cos sinsin
sin 0 cos1
CCCA=
ee n
: (101)
The diagonalized spin eSis represented in terms of annihilation and creation operators for
the Holstein-Primakov boson, bandby, as29
eS=0
BBB@q
S
2(by+b)
iq
S
2(by b)
S byb1
CCCA: (102)
We neglect the terms that are third- and higher-order in boson operators. Derivatives of the
localized spin then read
@S=U(@+iAU;)eS; (103)
where
AU; iU 1rU; (104)
is the spin gauge eld represented as a 3 3 matrix. The spin Berry's phase of the Lagrangian
(99) is written in terms of magnon as (derivation is in Sec. B)
Lm= 2S
2Z
d3ri[by(@t+iAz
s;t)b by(
@t iAz
s;t))b]; (105)
namely, magnons interacts with the adiabatic component of the same spin gauge eld for
electrons,Az
s;t, de ned in Eq. (14). As magnon is a single-component eld, the gauge eld
is also single-component, i.e., a U(1) gauge eld. This is a signi cant di erence between
insulating and metallic ferromagnet; In the metallic case, conduction electron couples to
an SU(2) gauge eld with spin-
ip components, which turned out to be essential for spin
current generation. In contrast, in the insulating case, the magnon has diagonal gauge eld,
i.e., a spin chemical potential, which simply induces diagonal spin polarization. Pumping of
magnon was discussed in a di erent approach by evaluating magnon source term in Ref.30.
32The exchange interaction at the interface is represented by a Hamiltonian
HI=JIZ
d3rIS(r)cyc; (106)
whereJIis the strength of the interface sdexchange interaction and the integral is over
the interface. We consider a sharp interface at x= 0. Using Eq. (100), the interaction is
represented in terms of magnon operators up to the second order as
HI=JIZ
d3rI"
(S byb)cy(n)c+r
S
2
bycyc+bcyc#
; (107)
where
e+ie =0
BBB@coscos isin
cossin +icos
sin1
CCCA: (108)
Equation (107) indicates that there are two mechanisms for spin current generation; namely,
the one due to the magnetization at the interface (the term proportional to n) and the one
due to the magnon spin scattering at the interface (described by the term linear in magnon
operators).
Let us brie
y demonstrate based on the expression of Eq. (107) that spin-
ip processes
due to magnon creation or annihilation lead to generation of spin current in the normal
metal. At the second order, the interaction induces a factor on the electron wave function
((t))((t0)) for magnon creation and ( (t))((t0)) for annihilation (we
allow an in nitesimal di erence in time tandt0). The factor for the creation has charge
and spin contributions, ( (t))((t0)) =(t)(t0) +i((t)(t0)). For
magnon annihilation, we have ( (t)(t0)), and thus the sum of the magnon creation
and annihilation processes give arise to a factor
X
q[(nq+ 1)((t)(t0)) +nq((t)(t0))] =X
q[(2nq+ 1)Re[ (t)(t0)] +iIm[(t)(t0)]:
(109)
For adiabatic change the amplitude is expanded as
((t)(t0)) = 2i(1 +i(t t0) cos_ )n (t t0)(n_n i_n) +O((@t)2); (110)
33where we see that an retardation e ect from the adiabatic change of magnetization (rep-
resented by the second term on the right-hand side) gives rise to a magnon state change
proportional to n_nand _n. The retardation contribution for the spin part (Eq. (109)) is
(t t0)X
q[ (2nq+ 1)(n_n) +i_n]: (111)
We therefore expect that a spin current proportional to n_nemerges proportional to the
magnon creation and annihilation number,P
q(2nq+ 1). (As we shall see below, the factor
t t0reduces to a derivative with respect to angular frequency of the Green's function.) A
rigorous estimation using Green's function method is presented in Sec. VII C.
In Eq. (111), the last term proportional to _nis an imaginary part arising from the
di erence of magnon creation and annihilation probabilities of vacuum, nq+ 1 andnq.
The term is, however, unphysical one corresponding to a real energy shift due to magnon
interaction, and is removed by rede nition of the Fermi energy.
B. Spin current pumped by the interface exchange interaction
Here we study the spin current pumped by the classical magnetization at the interface,
namely, the one driven by the term proportional to Snin Eq. (107). We treat the ex-
change interaction perturbatively to the second order as the exchange interaction between
conduction electron and insulator ferromagnet is localized at the interface and is expected
to be weak. The weak coupling scheme employed here is in the opposite limit as the strong
coupling (adiabatic) approach used in the metallic ferromagnet (Sec. IV).
In the perturbative regime, the issue of adiabaticity needs to be argued carefully. In
the strong sdcoupling limit, the adiabaticity is trivially satis ed, as the time needed for
the electron spin to follow the localized spin is the fastest timescale. In the weak coupling
limit, this timescale is long. Nevertheless, the adiabatic condition is satis ed if the electron
spin relaxation is strong so that the electron spin relaxes quickly to the local equilibrium
state determined by the localized spin. Thus the adiabatic condition is expected to be
MI sf=~ 1, whereMIand sfare the interface spin splitting energy, and conduction
electron spin relaxation time, respectively. In the following calculation, we consider the case
ofF sf=~ 1, i.e., ~( sf) 1 F, as the spin
ip lifetime is by de nition longer than the
elastic electron lifetime , which satis es F =~ 1 in metal. Our results therefore cover
34both adiabatic and nonadiabatic limits.
The calculation is carried out by evaluating Feynmann diagrams of Fig. 9, similar to the
study of Refs.17,18. A di erence is that while Refs.17,18assumed a smooth magnetization
structure and used a gradient expansion, the exchange interaction we consider is localized.
FIG. 9. T
he Feynmann diagrams for spin current pumped by interface sdexchange interaction.
The lesser Green's function for normal metal including the interface exchange interaction
to the linear order is
G(1)<
N(r;t;r;t) =MIZd!
2Zd
2X
kk0e i
tei(k0 k)r
(f(!+
) f(!))gr
k0;!+
ga
k! f(!)gr
k0;!+
gr
k!+f(!+
)ga
k0;!+
ga
k!
(n
);
(112)
whereMIJISis the local spin polarization at the interface. Expanding the expression
with respect to
and keeping the dominant contribution at long distance, i.e., the terms
containing both gaandgr. UsingP
kga
k!eikr'im
kFeikre jxj
`(ga(r)), the result of spin
current is
j(1)
s(r;t) = MIm
kF_ne jxj=`: (113)
The second-order contribution is similarly calculated to obtain
G(2)<
N(r;t;r;t)'(MI)2Zd!
2Zd
1
2Zd
2
2
X
kk0k00e i(
1+
2)tei(k0 k)rf0(!)gr
k0;!ga
k!(
1ga
k00!+
2gr
k00!)(n
1)(n
2)
= 2i(MI)2jgr(r)j2(n_n): (114)
The corresponding spin current at the interface ( x= 0) is thus
j(2)
s(x= 0;t) =(MI)2m
kF(n_n); (115)
35and the total spin current reads
js(x= 0;t) = MIm
kF_n 2(MI)2m
kF(n_n): (116)
In the perturbation regime, the spin current proportional to _nis dominant (larger by a
factor of (M I) 1) compared to the one proportional to n_n.
Expression of spin current induced by the interface exchange interaction was presented
in Ref.31in the limit of strong spin relaxation, MI sf 1, where sfis the spin relaxation
time of electron. By solving the Landau-Lifshitz-Gilbert equation for the electron spin, they
obtained an result of Eq. (116) with M Ireplaced by MI sf.
C. Calculation of magnon-induced spin current
Here magnon-induced spin current due to the magnon interaction in Eq. (107) is cal-
culated. As magnon is a small
uctuation of magnetization, the contribution here is a
small correction to the contribution of Eq. (116). Nevertheless, the magnon contribution
has a typical linear dependence on the temperature, and is expected to be experimentally
identi ed easily.
Spin current induced in normal metal is evaluated by calculating the self-energy arising
from the interface magnon scattering of Eq. (107). The contribution to the path-ordered
Green's function of N electron from the magnon scattering to the second order is
GN(r;t;r0:t0) =Z
Cdt1Z
Cdt2X
r1r2gN(r;t;r1;t1)I(r1;t1;r2;t2)gN(r2;t2;r0;t0); (117)
where
I(r1;t1;r2;t2)iSJ2
I
2D (r1;t1;r2;t2) gN(r1;t1;r2;t2) ; (118)
represents the self energy. Here
D (r1;t1;r2;t2) ihTCB (r1;t1)B (r2;t2)i; (119)
is the Green's function for magnon dressed by the magnetization structure ( is de ned in
Eq. (108)),
B (r;t) (t)by(r;t) + y
(t)b(r;t): (120)
36FIG. 10. T
he Feynmann diagrams for spin current pumped by magnons at the interface. Green's functions
for magnons and electrons in the normal metal are denoted by DandgN, respectively.
represents the e ects of magnetization dynamics (Eq. (108)).
Diagramatic representation is in Fig. 10. In the present approximation including the inter-
face scattering to the second order, the electron Green's function in Eq. (118) is treated as
spin-independent, resulting in a self energy (de ned on complex time contour)
I(r1;t1;r2;t2) =iSJ2
I
2( +i
)D (r1;t1;r2;t2)gN(r1;t1;r2;t2): (121)
We focus on the spin-polarized contribution,
I;
(r1;t1;r2;t2) SJ2
I
2eD
(r1;t1;r2;t2)gN(r1;t1;r2;t2); (122)
whereeD
D . The spin-dependent contribution of lessor Green's function, Eq. (117),
reads (time and spatial coordinates partially suppressed)
G<
N(r;t;r0;t0) =
Z1
1dt1Z1
1dt2
gr
N(t t1)r
I;
(t1;t2)g<
N(t2 t0) +gr
N<
I;
ga
N+g<
Na
I;
ga
N
G<
N;
(r;t;r0:t0):
(123)
For the self energy type of the Green's functions, depending on two time as g(t1 t2)D(t1 t2)
(Eq. (122)), real-time components are written as (suppressing time and sux of N) (See
Sec. C)
[g(t1 t2)D(t1 t2)]r=grD<+g>Dr=g<Dr+grD>
[g(t1 t2)D(t1 t2)]a=gaD>+g<Da=gaD<+g>Da
[g(t1 t2)D(t1 t2)]<=g<D<: (124)
The Green's function eDis that of a composite eld B de ned in Eq. (120), and is decom-
37posed to elementary magnon Green's function, D, as
eD
(r1;t1;r2;t2) = [y(t1)(t2)]
D(r1;t1;r2;t2) [y(t2)(t1)]
D(r2;t2;r1;t1);
(125)
where
D(r1;t1;r2;t2) i
TCb(r1;t1)by(r2;t2)
: (126)
The spin-dependent factor in Eq. (125) is calculated for adiabatic dynamics as
y(t1)(t2) = 2in(t1) + (t2 t1)[ +i_n] +O((@t)2); (127)
where
2 cos_ n+n_n: (128)
The real-time Green's functions are therefore ( D(1;2)D(r1;t1;r2;t2))
eD<
(r1;t1;r2;t2) = 2in(t1)[D<(r1;t1;r2;t2) D>(r2;t2;r1;t1)]
+ (t2 t1)
[D<(r1;t1;r2;t2) +D>(r2;t2;r1;t1)] +i_n[D<(r1;t1;r2;t2) D>(r2;t2;r1;t1)]
eDr
(1;2) =(t1 t2)(eD<
(1;2) eD>
(1;2))
eDa
(1;2) = (t2 t1)
(D<
(1;2) D>
(1;2)); (129)
andeD<
is obtained by exchanging <and>ineD<
. Elementary Green's functions are
calculated as
D<(r1;t1;r2;t2) = iX
qeiq(r1 r2)nqe i!q(t1 t2)
D>(r1;t1;r2;t2) = iX
qeiq(r1 r2)(nq+ 1)e i!q(t1 t2); (130)
where!qis magnon energy and nq1
e !q 1. In our model, the interface is atomically
at
and has an in nite area, and thus ri(i= 1;2) are atx= 0. Fourier components de ned as
(a= r;a;<;> )
eDa
(x1= 0;t1;x2= 0;t2)X
qZd
2e i
(t1 t2)eDa
(q;
); (131)
38are calculated from Eq. (129) as
eD<
(q;
) = i
2n(D<
D>
+) +d
d
(D<
+D>
+) +i_n(D<
D>
+)
eDr
(q;
) = i
2n(Dr
+Dr
+) +d
d
(Dr
Dr
+) +i_n(Dr
+Dr
+)
eDa
(q;
) = i
2n(Da
+Da
+) +d
d
(Da
Da
+) +i_n(Da
+Da
+)
; (132)
where
Da
1
!q i0; Dr
1
!q+i0
D<
nq(Da
Dr
); D>
+(1 +nq)(Da
+ Dr
+): (133)
The spin part of the Green's function, Eq. (123), is
G<
N;
(r;t;r0;t) = SJ2
I
2Zd!
2Zd
2X
kk0X
k00q
gr
N;k!
eDr
(q;
)g>
N;k00;!
+eD<
(q;
)gr
N;k00;!
g<
N;k0!
+gr
N;k!eDr
(q;
)g>
N;k00;!
ga
N;k0!+g<
N;k!
eDa
(q;
)g>
N;k00;!
+eD<
(q;
)ga
N;k00;!
ga
N;k0!
:
(134)
The contribution survives at long distance is the one containing gr
N;!(r) andga
N;!( r), i.e.,
G<
N;
(r;t;r0;t)'Zd!
2X
kk0gr
N;k!ga
N;k0!eikre ik0r0eI;
; (135)
where
eI;
SJ2
I
2Zd
2X
k00q
fk0eDr
(q;
) fkeDa
(q;
)
(fk00 1)(ga
N;k00;!
gr
N;k00;!
)
+eD<
(q;
)(fk0gr
N;k00;!
fkga
N;k00;!
+fk00(ga
N;k00;!
gr
N;k00;!
))
: (136)
We focus on the pumped contribution, containing derivative with respect to
in Eq. (132).
The result is, using partial integration with respect to
( eIis a vector representation of
eI;
),
eI' iSJ2
I
2Zd
2X
k00q
fk0[ (Dr
Dr
+) +i_n(Dr
+Dr
+)] fk[ (Da
Da
+) +i_n(Da
+Da
+)]
(fk00 1)d
d
(ga
N;k00;!
gr
N;k00;!
)
+ [ (D<
+D>
+) +i_n(D<
D>
+)]d
d
[(fk00 fk)ga
N;k00;!
(fk00 fk0)gr
N;k00;!
]
:
(137)
39Usingd
d
ga
k00;!
= (ga
k00;!)2+O(
) and an approximation, we obtainP
k00(ga
k00;!)2' i
2F,
eI'
FSJ2
I
2Zd
2X
qk00
(fk00 1)[fk0(Dr
Dr
+) fk(Da
Da
+)] +1
2(2fk00 fk fk0)(D<
+D>
+)
+i_n
(fk00 1)[fk0(Dr
+Dr
+) fk(Da
+Da
+)] +1
2(2fk00 fk fk0)(D<
D>
+)
:
(138)
As argued for Eq. (111), only the imaginary part of self energy contributes to the induced
spin current, as the real part, the shift of the chemical potential, is compensated by redis-
tribution of electrons. The result is thus
eI'i
FSJ2
I
2X
qk00(1 + 2nq)(2fk00 fk fk0): (139)
We further note that the component of proportional to n(Eq. (128)) does not contribute
to the current generation, as a result of gauge invariance. (In other words, the contribution
cancels with the one arising from the e ective gauge eld for magnon.)
The nal result of the spin current pumped by the magnon scattering is therefore
jm
s(r;t) =
FSJ2
I
2jgr(r)j2X
q(1 + 2nq)(n_n): (140)
At high temperature compared to magnon energy, !q 1, 1 + 2nq'2kBT
!q, and the
magnon-induced spin current depends linearly on temperature. The result (140) agrees
with previous study carried out in the context of thermally-induced spin current19.
D. Correction to Gilbert damping in the insulating case
In this subsection, we calculate the correction to the Gilbert damping and g-factor of
insulating ferromagnet as a result of spin pumping e ect. We study the torque on the
ferromagnetic magnetization arising from the e ect of conduction electron of normal metal,
given by
I=BIn=MI(nsI); (141)
40where
BI HI
n= MIsI; (142)
is the e ective magnetic eld arising from the interface electron spin polarization, sI(t)
itr[G<
N(0;t)]. The contribution to the electron spin density linear in the interface ex-
change interaction, Eq. (106), is
s(1);
I(t) = iZ
dt1MIn (t1)tr[ gN(t;t1) gN(t1;t)]<; (143)
where the Green's functions connect positions at the interface, i.e., from x= 0 tox= 0,
and are spin unpolarized. (The Feynman diagrams for the spin density are the same as the
one for the spin current, Fig. 9 with the vertex jsreplaced by the Pauli matrix.) Pumped
contribution proportional to the time variation of magnetization is obtained as
s(1)
I(t) = MI_nZd!
2X
kk0f0(!)(ga
N;k0 gr
N;k0)(ga
N;k gr
N;k)
= MI()2_n: (144)
The second order contribution similarly reads
s(2);
I(t) = i
2Z
dt1Z
dt2(MI)2n (t1)n
(t2)tr[ gN(t;t1) gN(t1;t2)
gN(t2;t)]<
' 2(MI)2()3(n_n): (145)
The interface torque is therefore
I= (MI)2(n_n) + 2(MI)3_n: (146)
Including this torque in the LLG equation, _n= n_n
Bn+ , we have
(1 I)_n= I(n_n)
Bn; (147)
where
I= 2d(M I)3
I= +d(M I)2; (148)
whereddmp=dis the ratio of the length of magnetic proximity ( dmp) and thickness of
the ferromagnet, d. The Gilbert damping constant therefore increases as far as the interface
41spin-orbit interaction is neglected. The resonance frequency is !B=
B
1 I, and the shift can
have both signs depending on the sign of interface exchange interaction, MI.
There may be a possibility that magnon excitation induce torque that corresponds to
e ective damping. In fact, such torque arises of hbior
by
are nite, i.e., if magnon Bose
condensation glows, as seen from Eq. (102). Such condensation can in principle develop from
the interface interaction of magnon creation or annihilation induced by electron spin
ip,
Eq. (107). However, conventional spin relaxation processes arising from the second order
of random spin scattering do not contribute to such magnon condensation and additional
damping.
Comparing the result of pumped spin current, Eq. (116), and that of damping coecient,
Eq. (148), we notice that the 'spin mixing conductance' argument2, where the coecients
for the spin current component proportional to n_nand the enhancement of the Gilbert
damping constant are governed by the same quantity (the imaginary part of 'spin mixing
conductance') does not hold for the insulator case. In fact, our result indicates that the
spin current component proportional to n_narises from the second order correction to
the interaction (the second diagram of Fig. 9), while the damping correction arises from the
rst order process (the rst diagram of Fig. 9). Although the magnitudes of the two e ects
happen to be both second order of interface spin splitting, MI, physical origins appear to
be distinct. From our analysis, we see that the 'spin mixing conductance' description is not
general and applies only to the case of thick metallic ferromagnet (see Sec. V A for metallic
case).
VIII. DISCUSSION
Our results are summarized in table I. Let us discuss experimental results in the light of
our results. In the early ferromagnetic resonance (FMR) experiments, consistent studies of
g-factor and the Gilbert damping were carried out on metallic ferromagnets11. The results
appear to be consistent with theories (Refs.2,10and the present paper). Both the damping
constant and the gfactor have 1 =d-dependence on the thickness of ferromagnet in the range
of 2nm<d< 10nm11. The maximum additional damping reaches 0:1 atd=2nm, which
exceeds the original value of 0:01. Theg-factor modulation is about 1% at d= 2nm, and
its sign depends on the material; the g-factor increases for Pd/Py/Pd and Pt/Py/Pt while
42Ferromagnet(F) Ai Ar ! B Assumption Equations
Metal ImT+ ReT+ ReT+
ImT+ ImT+
ReT+ Thick F
Thin F(27)(66) (88)(89)
(93)
Insulator MI (MI)2(MI)2(MI)3Weak spin relaxation(116) (148)
- (MI)2P
q(1 + 2nq) - - Magnon (140)
TABLE I. Summary of essential parameters determining spin current js, corrections to the Gilbert
damping and resonance frequency shift !Bfor metallic and insulating ferromagnets. Coe-
cientsAiandArare for the spin current, de ned by Eq. (1). Label indicates that it is not
discussed in the present paper.: For strong spin relaxation case, the density of states is
replaced by inverse of electron spin-
ip time, sf.31
decreases for Ta/Py/Ta. These results appear consistent with ours, because !Bis governed
by ImT+ , whose sign depends on the sign of interface spin-orbit interaction. In contrast,
damping enhancement proportional to Re T+ is positive for thick metals. However, other
possibilities like the e ect of a large interface orbital moment playing a role in the gfactor,
cannot be ruled out at present.
Recently, inverse spin Hall measurement has become common for detecting the spin cur-
rent. In this method, however, only the dc component proportional to n_nis accessible so
far and there remains an ambiguity for qualitative estimates because another phenomeno-
logical parameter, the conversion eciency from spin to charge, enters. Qualitatively, the
values ofArobtained by the inverse spin Hall measurements32and FMR measurements are
consistent with each other.
The cases of insulating ferromagnets have been studied recently. In the early experiments,
orders of magnitude smaller value of Arcompared to metallic cases were reported31, while
those small values are now understood as due to poor interface quality. In fact, FMR
measurements on epitaxially grown samples like YIG/Au/Fe turned out to show Arof 1
51018m 2(Refs.33,34), which is the same order as in the metallic cases. Inverse spin Hall
measurements on YIG/Pt reports similar values35, and the value is consistent with the rst
principles calculation36. Systematic studies of YIG/NM with NM=Pt, Ta, W, Au, Ag, Cu,
Ti, V, Cr, Mn etc. were carried out with the result of Ar1017 1018m 2(Refs.37{40). If we
use naive phenomenological relation, Eq. (6), Ar= 1018m 2corresponds to = 310 4if
43a= 2A,S= 1 andd= 20A. Assuming interface sdexchange interaction, the value indicates
MI0:01, which appears reasonable at least by the order of magnitude from the result of
x-ray magnetic circular dichroism (XMCD) suggesting spin polarization of interface Pt of
0:05B41.
On the other hand, FMR frequency shift of insulators cannot be explained by our theory.
In fact, the shift for YIG/Pt is !B=!B1:610 2, which is larger than 210 3,
while our perturbation theory assuming weak interface sdinteraction predicts !B=!B< .
We expect that the discrepancy arises from the interface spin-orbit interaction that would
be present at insulator-metal interface, which modi es the magnetic proximity e ect and
damping torque signi cantly. It would be necessary to introduce anomalous sdcoupling at
the interface like the one discussed in Ref.42. Experimentally, in
uence of interface spin-
orbit interaction43and proximity e ect needs to be carefully characterized by using the
microscopic technique, such as MCD, to compare with theories.
IX. SUMMARY
We have presented a microscopic study of spin pumping e ects, generation of spin cur-
rent in ferromagnet-normal metal junction by magnetization dynamics, for both metallic and
insulating ferromagnets. As for the case of metallic ferromagnet, a simple quantum mechan-
ical picture was developed using a unitary transformation to diagonal the time-dependent
sdexchange interaction. The problem of dynamic magnetization is thereby mapped to the
one with static magnetization and o -diagonal spin gauge eld, which mixes the electron
spin. In the slowly-varying limit, spin gauge eld becomes static, and the conventional spin
pumping formula is derived simply by evaluating the spin accumulation formed in the nor-
mal metal as a result of interface hopping. The e ect of interface spin-orbit interaction was
discussed. Rigorous eld-theoretical derivation was also presented, and the enhancement of
spin damping (Gilbert damping) in the ferromagnet as a result of spin pumping e ect was
discussed. The case of insulating ferromagnet was studied based on a model where spin
current is driven locally by the interface exchange interaction as a result of magnetic prox-
imity e ect. The dominant contribution turns out to be the one proportional to _n, while
magnon contribution leads to n_n, whose amplitude depends linearly on the temperature.
Our analysis clearly demonstrate the di erence in the spin current generation mechanism
44for metallic and insulating ferromagnet.
The in
uence of atomic-scale interface structure on the spin pumping e ect are open and
urgent issues, in particular for the case of ferrimagnetic insulators which have two sub-lattice
magnetic moments.
ACKNOWLEDGMENTS
GT thanks H. Kohno, C. Uchiyama, K. Hashimoto and A. Shitade for valuable discus-
sions. This work was supported by a Grant-in-Aid for Exploratory Research (No.16K13853)
and a Grant-in-Aid for Scienti c Research (B) (No. 17H02929) from the Japan Society for
the Promotion of Science and a Grant-in-Aid for Scienti c Research on Innovative Areas
(No.26103006) from The Ministry of Education, Culture, Sports, Science and Technology
(MEXT), Japan.
Appendix A: E ect of spin-conserving spin gauge eld on spin density
Here we calculate contribution of spin-conserving spin gauge eld, Az
s;t, on the interface
e ects of spin density in F. It turns out that spin-conserving spin gauge eld combined with
interface e ects does not induce damping. This result is consistent with a naive expectation
that only the nonadiabatic components of spin current should contribute to damping.
FIG. 11. Diagramatic representation of the contribution to the lessor Green's function for F
electron arising from the interface hopping (represented by tandt) and spin gauge eld ( As;t).
The diagram (c) includes the spin gauge eld implicitly in unitary matrices UandU 1.
The contribution to the lesser Green's function in F from the interface hopping (lowest,
the second-order in the hopping) at the linear order in the spin gauge eld reads (diagra-
45matically shown in Fig. 11)
G<=G<
(a)+G<
(b)+G<
(c)
G<
(a)=gr(As;t)grr
0g<+gr(As;t)gr<
0ga+gr(As;t)g<a
0ga+g<(As;t)gaa
0ga
G<
(b)=grr
0gr(As;t)g<+gr<
0g<(As;t)ga+gra
0ga(As;t)ga+g<a
0ga(As;t)ga
G<
(c)=grrg<+gr<ga+g<aga: (A1)
Here
a~tU 1ga
NU~ty;(a= a;r;<)
a
0~tga
N; (A2)
are self energy due to the interface hopping, where ais the full self energy including the
time-dependent unitary matrix U, which includes spin gauge eld. a
0is the contribution of
awith the spin gauge eld neglected. We here focus on the contribution of the adiabatic ( z)
component,Az
s;t. Usingg<=F(ga gr) for F (Fis a 22 matrix of the spin-polarized Fermi
distribution function) and g<
N=fN(ga
N gr
N) and noting that all the angular frequencies of
the Green's function are equal, we obtain
G<
(a)+G<
(b)'Az
s;tz
2F[(gr)3r
0 (ga)3a
0] (F fN)[(gr)2ga+gr(ga)2](a
0 r
0)
:
(A3)
The contribution G<
(c)is calculated noting that
~tU 1ga
NU~ty=ga
N~t~ty dga
N
d!~t(As;t)~ty+O((As;t)2): (A4)
The linear contribution with respect to the zcomponent of the gauge eld turns out to be
G<
(c)'Az
s;tz
F
(gr)2@
@!r
0 (ga)2@
@!a
0
+ (F fN)grga@
@!(a
0 r
0)
: (A5)
We therefore obtain the e ect of spin-conserving gauge eld as
G<=Az
s;tz@
@!
F
(gr)2r
0 (ga)2a
0
+ (F fN)grga(a
0 r
0)
; (A6)
which vanishes after integration over !. Therefore, contribution from spin-conserving gauge
eld and interface hopping vanishes in the spin density, leaving the damping una ected.
46Appendix B: Magnon representation of spin Berry's phase term
Here we derive the expression for the spin Berry's phase term of the Lagrangian (99)
in terms of magnon operator. The time-integral of the term is written by introducing an
arti cial variable uas44
Z
dtL B=SZ
dt_ (cos 1) =S 2Z
dtZ1
0duS(@tS@uS); (B1)
whereS(t;u) is extended to a function of tandu, but onlyS(t;u= 1) is physical. Noting
that the unitary transformation matrix element of Eq. (101) is written as
Uij= (ej)i; (B2)
wherer1e,e2e ande3n, we obtain
S(@tS@uS) =eS[(@t+iAU;t)eS(@u+iAU;u)eS)]: (B3)
Evaluating to the second order in the magnon operators, we have
@teS@ueS= 2i
^z[(@uby)(@tb) (@tby)(@ub)]: (B4)
Using the explicit form of AU;, the gauge eld contribution is
@ueS[eSiAU;teS)] =S2
[(@uby)( sin_ +i_) + (@ub)( sin_ i_)] 2S
2cos(@t )@u(byb):
(B5)
The terms linear in the boson operators vanish by the equation of motion, and the second-
order contribution is
S(@tS@uS) = 2S
2
i@u[by(@tb) (@tby)b]
@u[cos(@t )byb] +@t[cos(@u )byb] + sin((@t)(@u ) (@u)(@t ))byb
:
(B6)
Integrating over tandu, the term total derivative with respect to tof Eq. (B6) vanishes,
resulting in
Z
dtZ1
0duS(@tS@uS) = 2S
2Z
dt
i[by(@tb) (@tby)b] cos(@t )byb
+ sin((@t)(@u ) (@u)(@t ))byb
: (B7)
47The last term of Eq. (B7) represents the renormalization of spin Berry's phase term, i.e.,
the e ectS!S byb, which we neglect below. The Lagrangian for magnon thus reads
Lm= 2S
2Z
d3ri[by(@t+iAz
s;t)b by(
@t iAz
s;t))b]; (B8)
namely, magnons interacts with the adiabatic component of spin gauge eld, Az
s;t.
Appendix C: Decomposition of contour-ordered self energy
Here we summarize decomposition formula of self energy. Obviously, we have
[gD]<=g<D<: (C1)
Retarded component is de ned as
[gD]r[gD]t [gD]<; (C2)
where the time-ordered one is
[g(t1 t2)D(t1 t2)]t(t1 t2)g>D>+(t2 t1)g<D<
=grDr+grD<+g<Dr+g<D<: (C3)
We thus obtain
[gD]r=grDr+grD<+g<Dr: (C4)
Noting that grDa= 0, we can write it as
[gD]r=grD<+g>Dr=g<Dr+grD>: (C5)
The advanced component is similarly written as
[gD]a= gaDa+gaD<+g<Da
=gaD>+g<Da=gaD<+g>Da: (C6)
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50 |
2302.08910v1.Control_of_magnon_photon_coupling_by_spin_torque.pdf | Control of magnon-photon coupling by spin torque
Anish Rai1,and M. Benjamin Jung
eisch1,y
1Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, United States
(Dated: February 20, 2023)
We demonstrate the in
uence of damping and eld-like torques in the magnon-photon coupling
process by classically integrating the generalized Landau-Lifshitz-Gilbert equation with RLC equa-
tion in which a phase correlation between dynamic magnetization and microwave current through
combined Amp ere and Faraday e ects are considered. We show that the gap between two hybridized
modes can be controlled in samples with damping parameter in the order of 10 3by changing the
direction of the dc current density Jif a certain threshold is reached. Our results suggest that an
experimental realization of the proposed magnon-photon coupling control mechanism is feasible in
yttrium iron garnet/Pt hybrid structures.
I. INTRODUCTION
Coherent magnon-photon coupling in hybrid cavity-
spintronics contributed to the advancement of magnon-
based quantum information and technologies [1{15]. The
collective excitations of an electron spin system in mag-
netically ordered media called magnons can couple to mi-
crowave photons via dipolar interaction, demonstrating
level repulsion and Rabi oscillations [3]. Strongly cou-
pled magnon-photon systems have been explored to bring
many exotic e ects into the limelight, some of which in-
clude the manipulation of spin currents [16], and bidi-
rectional microwave-to-optical transduction [17, 18]. In
addition to the coherent magnon-photon coupling, there
exists an exciting domain of dissipative magnon-photon
coupling where level attraction can be observed, which is
characterized by a coalescence of the hybridized magnon-
photon modes [19{25].
The theoretical framework of magnon-photon coupling
is given by the following dispersion relation [26] of the
hybridized modes:
e!=1
2
(e!m+e!c)q
(e!m e!c)2+ 4g2
;(1)
wheree!m=!m i !mande!c=!c i !care the
complex resonance frequencies of the magnon and pho-
ton (cavity) modes, respectively. gis the coupling be-
tween the two modes. and are the intrinsic damping
rates of the magnon and photon modes, respectively. The
real and imaginary parts of e!represent the dispersion
shape and the linewidth of the coupled modes, respec-
tively. The second term of the square root in Eq. (1) not
only gives the strength of the coupling but also reveals
the nature of the coupling. Harder and co-workers [19]
carefully introduced a coupling term based on the cavity
Lenz e ect to mitigate the Amp ere e ect. However, the
on-demand manipulation of the magnon-photon polari-
ton by spin torques has not been addressed so far.
arai@udel.edu
ymbj@udel.eduIn this work, we examine the in
uence of damping-
and eld-like torques in the magnon-photon coupling pro-
cess. Our results indicate that the magnitude of the
level repulsion (manifested by the frequency gap of the
hybridized modes) and, hence, the magnon-photon cou-
pling strength can eciently be controlled by varying the
magnitude and the direction of dc current density Jfor
realistic parameters of the magnetic properties. By cou-
pling the generalized Landau-Lifshitz Gilbert equation
with the RLC equation of the cavity, we show that an
on-demand manipulation of the magnon-photon coupling
strength can be achieved for current densities of the order
as small as 105A/cm2.
This article is structured in the following fashion. In
section II, we discuss the classical description to model
our system, in which the ferromagnetic resonance of the
magnetic system is strongly coupled to photon resonator
mode of the microwave cavity. In section III, we intro-
duce the parameters used for the analysis followed by
a detailed discussion of our ndings. In section IV, we
summarize our work.
FIG. 1. The schematic of the experimental setup. A pat-
terned YIG/platinum(Pt) bilayer is the sample under consid-
eration. The dc current is passed through the platinum layer.
The microwave current is passed through the cavity and an-
alyzed using a Vector Network Analyzer (VNA). Here, the
external magnetic eld is applied along bzdirection.arXiv:2302.08910v1 [cond-mat.mes-hall] 17 Feb 20232
FIG. 2. Magnon-photon coupling control for an intermediate value of the Gilbert damping parameter and a continuous, low
current density J: The dispersion ( ! !c) in (a-c) and the linewidth ( !) in (d-f) are plotted as a function of the eld detuning
(!m !c) for = 2:2710 3. The hybridization of magnon and photon modes is compared for di erent dc current densities
J: (a), (d) J = 5105A=cm2, (b), (e) J = 0 A =cm2and (c), (f) J =5 105A=cm2. The blue and red line represent the two
hybridized modes. The inset in (b) shows that for larger eld detuning ( !m !c), the uncoupled photon mode approaches !c
making (! !c) approach zero.
II. CLASSICAL DESCRIPTION
The magnetization dynamics in ferromagnetic systems
can be described by the generalized Landau-Lifshitz-
Gilbert equation [27{29] given by:
d~M
dt=
~M~He
Ms
~Md~M
dt!
+
aJ
Ms
~M
~M~ p
bJ
~M~ p
;(2)
where~Mis the magnetization vector, Msis the satu-
ration magnetization, ~He is the e ective magnetic eld
including external eld ~H, anisotropy, microwave, and
demagnetization elds,
is the gyromagnetic ratio, is
the Gilbert damping parameter, ~ pis the spin polarization
unit vector. Furthermore, the terms proportional to aJ
andbJare the damping-like torque and eld-like torque,
respectively. The coecients aJandbJare de ned as
[30]:
aJ=aJ~
2eMsd;bJ=bJ~
2eMsd; (3)whereaandbare the damping-like torque eciency
and eld-like torque eciency, respectively. Jis the
dc current density, whose polarity determines the direc-
tions of eld-like and damping-like torque terms through
Eqs. (2) and (3), ~is the reduced Planck's constant, e
is the electron charge, and dis the thickness of the fer-
romagnetic sample. We de ne the magnetic eld, mag-
netization, and spin polarization unit vectors as ~Ht=
hx(t)bx+hy(t)by+Hbz,~M=mx(t)bx+my(t)by+Msbzand
~ p=bz, whereHandMsare the dc magnetic eld and
saturation magnetization, respectively, and hx;y(t) and
mx;y(t) are the dynamic magnetic eld and magnetiza-
tion.
If we de ne the dynamic components, h=hx+ihy
andm=mx+imy, then Eq. (2) can be reduced to:
(! ~!m+
~cJ)m+!sh= 0; (4)
where ~!mis the complex ferromagnetic resonance fre-
quency de ned by ~ !m=!m i !(where!m'
His
the ferromagnetic resonance frequency), !s=
Ms, and
~cJ=bJ iaJ. The e ective RLC circuit for the cavity3
FIG. 3. Magnon-photon coupling control for an intermediate value of the Gilbert damping parameter and a pulsed, high
current density J: The dispersion ( ! !c) in (a-c) and the linewidth ( !) in (d-f) are plotted as a function of the eld detuning
(!m !c) for = 2:510 3. The hybridization of magnon and photon modes is compared for di erent dc current densities
J: (a), (d) J = 5106A=cm2, (b), (e) J = 0 A =cm2and (c), (f) J =5 106A=cm2. The blue and red line represent the two
hybridized modes.
can be written as [19]:
Rjx;y(t) +1
CZ
jx;y(t)dt+Ldjx;y(t)
dt=V0x;y(t);(5)
where R, L, and C represent the resistance, inductance,
and capacitance, respectively. V0x;yis the voltage that
drives the microwave current. For j=jx+ijyandV0=
V0x+iV0y, we have [19]
!2 !2
c+i2!!c
j=i!
LV0; (6)
where!c= 1=p
LCis the cavity resonance frequency and
= (R=2)p
C=L is the intrinsic damping of the cavity-
photon mode.
The microwave magnetic eld will exert a torque on
the magnetization through Amp ere's law. The relation
can be expressed as:
hx=KAjy;hy= KAjx; (7)
whereKAis the positive coupling term associated with
a phase relation between jx;yandhx;y. In a similar way,
the precessional magnetization will induce a voltage inthe RLC circuit through Faraday induction:
Vx= KFLdmy
dt;Vy=KFLdmx
dt; (8)
whereKFis the positive coupling term associated with
a phase relation between Vx;yandmx;y. Combining
Eqs. (4)-(8) gives us the coupled equations of the form:
!2 !2
c+i2 !c! i!2KF
i!sKA! ~!m+
~cJ
j
m
=
i!!cj0
0
;(9)
wherej0=V0p
C=L. The hybridized eigenmodes are
calculated by solving the determinant of Eq. (9). This
yields the following analytical form [see Supplemental
Material (SM)]:
~!=
!c
1+i +!m
1+i
r
!c
1+i !m
1+i 2
+2!c!sKFKA
(1+i )(1+i )
2;
(10)
where=
~cJ. Here,
is the gyromagnetic ratio and ~ cJ
is a complex term associated with bJandaJde ned by
~cJ=bJ iaJ.4
FIG. 4. Variation of coherent magnon-photon coupling (minimum frequency gap between two hybridized modes) for di erent
values of andJ. For (a), (b), and (c) is varied from 3 10 3to 510 5andJ(continuous) is varied from 5105A=cm2
to 5105A=cm2and for (d), (e), and (f) is varied from 4 10 3to 510 5andJ(pulsed) is varied from 5106A=cm2to
5106A=cm2. Based on our model, we can distinguish between eld-like contribution (a) and (d), damping-like contribution
(b) and (e), and a combination of eld-like and damping-like contribution to the manipulation of the anticrossing gap (c) and
(f). For (a) and (d) a= 0 andb= 0:05 (pure eld-like torque e ect), for (b) and (e) a= 0:2 andb= 0 (pure damping-like
torque e ect), and for (c) and (f) a= 0:2 andb= 0:05 (combination of damping-like and eld-like torque e ects).
III. RESULTS AND DISCUSSION
For our model we choose the following realistic pa-
rameters [18, 31{34]. The frequency of the cavity mode
is selected at !c=2= 10 GHz with a cavity damping
= 110 4(corresponding to quality factor Q5000).
The reduced gyromagnetic ratio (
=2), damping-like
torque eciency ( a), and eld-like torque eciency ( b)
are taken as 2 :8106Hz=Oe, 0:2, and 0:05, respectively.
For a Pt/FM bilayer, the typical range of damping-like
torque eciency ( a) is 0.10 to 0.20 [32, 35, 36] and the
typical value of eld-like torque eciency ( b) is0.05
[33, 37{39]. Due to its low Gilbert damping parame-
ter and high spin density, we choose yttrium iron garnet
(YIG) as magnetic material. In particular, we consider
a YIG lm with a thickness t= 210 5cm (smallest
thickness available commercially) and saturation magne-
tization,Ms= 144 emu =cm3[18]. For the calculation,
the termKFKAis taken as 510 6[19]. For YIG lms,
depending upon the thickness and preparation method,
varies from order 10 3to 10 5[34, 38, 40{47]. Therefore,
we vary in our model from 3 10 3to 510 5. Further-more, we vary Jfrom 5105A=cm2to 5105A=cm2.
The maximum value of chosen current density is at least
one order of magnitude smaller than what is used for
magnetic tunnel junctions (MTJs) [48, 49]. Note that, a
current density of this order of magnitude has previously
been reported for YIG/Pt systems [50] to thermally con-
trol magnon-photon coupling in experiment. Reference
[50] reports that such current density leads to a rise of the
system temperature above 40C. Negative e ects of heat-
ing on the magnetic properties can be drastically reduced
by using a pulsed dc current through the Pt layer [51] in-
stead of using a continuous current. For instance, using a
pulsed current with duty cycle of 50%, heating e ects can
be mitigated while reaching reasonable high levels of cur-
rent density between 5106A=cm2to 5106A=cm2
. Such a high value of current density will create an
Oersted eld and, hence, modify the resonance condi-
tion. The generated Oersted eld can be considered as a
contribution to the e ective magnetic eld presented in
Eq. (2). Hence, this eld will modify the resonance posi-
tion of the magnon modes in the following way: for one
polarity of the current density (J), the resonance eld5
shifts up, while for the other, it shifts down. Experi-
mentally, this a ect can be compensated by tuning the
biasing magnetic eld so the resonance frequency remains
the same. In the following analysis, we consider two sce-
narios: (1) a relatively low continuous current density
and (2) a higher pulsed current density. The e ects of
both conditions on the magnon-photon coupling process
are compared below. The proposed experimental set up
and measurement con guration is shown in Fig. 1.
A. Dispersion and Linewidth
In Fig. 2 (intermediate value of Gilbert damping pa-
rameter and continuous, low value of current density
J) and Fig. 3 (intermediate value of and pulsed, high
value ofJ), the hybridized mode frequency ( ! !c) and
linewidth ( !) are plotted as a function of the eld de-
tuning (!m !c).
We rst focus on the former, shown in Fig. 2, top pan-
els: For = 2:2710 3and forJ= 0 A=cm2, we observe
a level attraction of the real part of the eigenvalues [Fig.
2 (b)] in a small region. For J= 5105A=cm2, a
similar behavior is found [Fig. 2(a)]. However, the be-
havior drastically changes for reversed current polarity:
forJ= 5105(A=cm2), a gap (a prominent level repul-
sion) is seen between the hybridized modes. This clearly
shows that depending upon the strength and direction of
the dc current density Jone can tune the gap between
the hybridized modes, i.e., transitioning the system into
the strong coupling regime. Let us now consider Fig. 3,
top panels: A similar but enhanced behavior can be ob-
served for higher values of J(i.e.,jJj= 5106A/cm2)
and = 2:510 3[Fig. 3]. As is obvious from Figs. 2
and 3, there is a shift in the position where the coher-
ent coupling occurs. For negative and positive values of
J, the resonance shifts towards the negative and positive
sides of the eld detuning ( !m !c), respectively. This
shift can be understood by the fact that di erent mag-
nitudes of eld-like torques directly a ect the resonance
condition as will be discussed in Sec. III B.
Next, we discuss the lower panels of Figs. 2 and 3. The
linewidths of the two hybridized modes distinctly cross
each other for J= 5105A=cm2andJ= 5106A=cm2
as is expected for a broad coupling region [Fig. 2(f) and
Fig. 3(f)]. This e ect is less distinct for the cases J=
5105A=cm2,J= 5106A=cm2andJ= 0 A=cm2.
However, despite the lower number of region in the cross-
ing regime (the crossing is less spread), we emphasize that
level crossings in the linewidths of the hybridized modes
are are also observed here. We note that the coupling re-
gion broadens as the current density increases from neg-
ative values to positive values [Figs. 2(d,e,f) and Figs. 3
(d,e,f)] and nally a distinct crossing of linewidths is ob-
served over a broad range [Fig. 2(f) and Fig. 3 (f)].
For special cases discussed in Sec. II of the SM, a
level attraction [Fig. S1 (a,b)] in the real part and level
repulsion [Fig. S1 (d,e)] in the imaginary part of theeigenvalues are observed along with exceptional points
(EPs) [52{55]. For more details on the observed EP we
refer to the SM.
B. Anticrossing gap between hybridized modes
Figure 4 shows the variation of the anticrossing gap
between the hybridized modes for di erent values of
andJ. It is clear that the variation is nonlinear in nature.
As is evident from the gure, the gap between the two
hybridized modes becomes smaller for a larger value of
. On the other hand, the gap also depends on the dc
current density J: the value of for which the gap is very
small increases if we go from from negative to positive
value of the dc current density. For a positive value of
J, we also observe the gap between the hybridized modes
slowly increases as increases and becomes maximum for
a particular value of , and then decreases if we further
increase the value of , as shown in the inset of Fig 4(f).
For the low regime, the anticrossing gap remains nearly
the same for di erent orders of magnitude and directions
of current density, as is shown in Figs. S2, S3 and S4
of the SM. However, for the high regime, we observe
a level repulsion in the real part and a level crossing in
the imaginary part of the eigenvalues for di erent orders
of magnitude and directions of the current density, as is
shown in Figs. S5, S6 and S7 of the SM. For a di erent
coupling strength ( KFKA), we observe a similar trend.
A positive current density is needed to increase the gap
between the two hybridized modes as is shown in Fig. S8
(SM).
In the following discussion, we chose Gilbert damping
parameters of = 2:2710 3and = 2:510 3for
di erent orders of magnitude of Jwhere a very small
anticrossing gap for zero current density is seen, as illus-
trated in Fig. 4. For large (= 410 3) and for very low
(= 510 5), the hybridized mode frequency ( ! !c)
and the linewidth ( !) plotted as a function of the eld
detuning (!m !c) are shown in the Fig. S1 and Fig.
S2 of the SM. Figure 5 shows the variation of the magni-
tude of the gap between the two hybridized modes with
respect to eld detuning ( !m !) for = 2:2710 3
and = 2:510 3. For a pure eld-like torque e ect,
there is a horizontal shift [as shown in Figs. 5(a) and
5(d)] of the gap between the hybridized modes towards
the positive value of eld detuning as we go from negative
to positive values of the current density J. However, for
a damping-like torque e ect, there is a vertical shift [as
shown in Figs. 5(b) and 5(e)] of the minimum gap (an-
ticrossing gap): the anticrossing gap increases if we go
from negative to positive values of J. For the combined
eld-like and damping-like torques e ect, there are both
horizontal and vertical shifts as can be seen in Figs. 5(c)
and 5(f). However, the horizontal shift due to the e ect of
eld-like torque, vertical shift due to damping-like torque
and the combined shift due to both eld and damping-
like torques are more pronounced for higher magnitudes6
FIG. 5. Variation of the gap between two hybridized modes with respect to the eld detunings ( !m !c) for = 2:2710 3
and continuous low Jfrom 5105A=cm2to 5105A=cm2[(a),(b) and (c)] and for = 2:510 3and pulsed high J
from 5106A=cm2to 5106A=cm2[(d),(e) and (f)]. There is (a),(d) a horizontal shift in the location of the gap for
a= 0 andb= 0:05 (pure eld-like torque e ect), (b), (e) vertical shift in the location of gap for a= 0:2 andb= 0
and (pure damping-like torque e ect), and (c),(f) horizontal and vertical shifts for a= 0:2 andb= 0:05 (combined e ect of
damping-like and eld-like torques).
ofJleading to an unusual behavior as is shown in panel
(f).
Finally, we note that introducing the eld-like and
damping-like torques in the Landau-Lifshitz-Gilbert
equation and coupling it with cavity mode through
combined Amp ere and Faraday e ects do not produce
level attraction. The coupling term in our analysis
is not a ected by the term [see Eq. (10)], which is
the parameter governed by spin torque. This means
that transitioning the system from strong coupling to
dissipative coupling and vice versa cannot be achieved
by spin-transfer torques.
IV. SUMMARY
By coupling of the generalized LLG equation with the
RLC equation of the cavity, we revealed the coupling be-
tween magnon and photon modes under the in
uence of
damping and eld-like torques. Our results indicate that
the magnitude of the level repulsion (manifested by the
frequency gap of the hybridized modes) and, hence, themagnon-photon coupling strength can eciently be con-
trolled by varying the magnitude and the direction of dc
current density Jfor realistic parameters of the magnetic
properties. Our model suggests that an on-demand ma-
nipulation of the magnon-photon coupling strength can
be achieved for current densities of the order as small
as 105A/cm2and an intermediate Gilbert damping of
the order 10 3. Higher values of Jcan de nitely en-
hance the e ect of damping and eld-like torques on the
magnon-photon coupling provided we use pulses of dc
current to reduce possible heating e ects. Therefore, the
experimental realization of the proposed magnon-photon
control mechanism should be feasible in YIG/Pt hybrid
structures.
ACKNOWLEDGMENT
We acknowledge fruitful discussions with Dr. J. Skle-
nar (Wayne State University) and Dr. J. Q. Xiao (Uni-
versity of Delaware). Research supported by the U.S.
Department of Energy, Oce of Basic Energy Sciences,
Division of Materials Sciences and Engineering under7
Award DE-SC0020308.
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1605.08698v1.A_reduced_model_for_precessional_switching_of_thin_film_nanomagnets_under_the_influence_of_spin_torque.pdf | A reduced model for precessional switching of thin- lm nanomagnets under the
in
uence of spin-torque
Ross G. Lund1, Gabriel D. Chaves-O'Flynn2, Andrew D. Kent2, Cyrill B. Muratov1
1Department of Mathematical Sciences, New Jersey Institute of Technology , University Heights, Newark, NJ 07102, USA
2Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA
(Dated: May 25, 2022)
We study the magnetization dynamics of thin- lm magnetic elements with in-plane magnetization
subject to a spin-current
owing perpendicular to the lm plane. We derive a reduced partial
di erential equation for the in-plane magnetization angle in a weakly damped regime. We then apply
this model to study the experimentally relevant problem of switching of an elliptical element when
the spin-polarization has a component perpendicular to the lm plane, restricting the reduced model
to a macrospin approximation. The macrospin ordinary di erential equation is treated analytically
as a weakly damped Hamiltonian system, and an orbit-averaging method is used to understand
transitions in solution behaviors in terms of a discrete dynamical system. The predictions of our
reduced model are compared to those of the full Landau{Lifshitz{Gilbert{Slonczewski equation for
a macrospin.
I. INTRODUCTION
Magnetization dynamics in the presence of spin-
transfer torques is a very active area of research with ap-
plications to magnetic memory devices and oscillators1{3.
Some basic questions relate to the types of magnetiza-
tion dynamics that can be excited and the time scales on
which the dynamics occurs. Many of the experimental
studies of spin-transfer torques are on thin lm magnetic
elements patterned into asymmetric shapes (e.g. an el-
lipse) in which the demagnetizing eld strongly con nes
the magnetization to the lm plane. Analytic models
that capture the resulting nearly in-plane magnetization
dynamics (see e.g.4{8) can lead to new insights and guide
experimental studies and device design. A macrospin
model that treats the entire magnetization of the ele-
ment as a single vector of xed length is a starting point
for most analyses.
The focus of this paper is on a thin- lm magnetic el-
ement excited by a spin-polarized current that has an
out-of-plane component. This out-of-plane component
of spin-polarization can lead to magnetization precession
about the lm normal or magnetization reversal. The for-
mer dynamics would be desired for a spin-transfer torque
oscillator, while the latter dynamics would be essential in
a magnetic memory device. A device in which a perpen-
dicular component of spin-polarization is applied to an
in-plane magnetized element was proposed in Ref. [9] and
has been studied experimentally10{12. There have also
been a number of models that have considered the in
u-
ence of thermal noise on the resulting dynamics, e.g., on
the rate of switching and the dephasing of the oscillator
motion13{15.
Here we consider a weakly damped asymptotic regime
of the Landau{Lifshitz{Gilbert{Slonczewski (LLGS)
equation for a thin- lm ferromagnet, in which the oscil-
latory nature of the in-plane dynamics is highlighted. In
this regime, we derive a reduced partial di erential equa-
tion (PDE) for the in-plane magnetization dynamics un-
der applied spin-torque, which is a generalization of theunderdamped wave-like model due to Capella, Melcher
and Otto8. We then analyze the solutions of this equa-
tion under the macrospin (spatially uniform) approxima-
tion, and discuss the predictions of such a model in the
context of previous numerical studies of the full LLGS
equation16.
The rest of this article is organized as follows. In Sec.
II, we perform an asymptotic derivation of the reduced
underdamped equation for the in-plane magnetization
dynamics in a thin- lm element of arbitrary cross sec-
tion, by rst making a thin- lm approximation to the
LLGS equation, then a weak-damping approximation. In
Sec. III, we then further reduce to a macrospin ordinary
di erential equation (ODE) by spatial averaging of the
underdamped PDE, and restrict to the particular case of
a soft elliptical element. A brief parametric study of the
ODE solutions is then presented, varying the spin-current
parameters. In Sec. IV, we make an analytical study of
the macrospin equation using an orbit-averaging method
to reduce to a discrete dynamical system, and compare
its predictions to the full ODE solutions. In Sec. V, we
seek to understand transitions between the di erent so-
lution trajectories (and thus predict current-parameter
values when the system will either switch or precess) by
studying the discrete dynamical system derived in Sec.
IV. Finally, we summarize our ndings in Sec. VI.
II. REDUCED MODEL
We consider a domain
R3occupied by a ferromag-
netic lm with cross-section DR2and thickness d, i.e.,
=D(0;d). Under the in
uence of a spin-polarized
electric current applied perpendicular to the lm plane,
the magnetization vector m=m(r;t), withjmj= 1 in
and 0 outside, satis es the LLGS equation (in SI units)
@m
@t=
0mHe + m@m
@t+ STT (1)arXiv:1605.08698v1 [cond-mat.mes-hall] 27 May 20162
in
, with@m=@n= (nr)m= 0 on@
, where nis the
outward unit normal to @
. In the above, > 0 is the
Gilbert damping parameter,
is the gyromagnetic ratio,
0is the permeability of free space, He = 1
0MsE
mis
the e ective magnetic eld,
E(m) =Z
Ajrmj2+K(m) 0MsHextm)
d3r
+0M2
sZ
R3Z
R3rm(r)rm(r0)
8jr r0jd3rd3r0(2)
is the micromagnetic energy with exchange constant A,
anisotropy constant K, crystalline anisotropy function
, external magnetic eld Hext, and saturation magne-
tizationMs. Additionally, the Slonczewski spin-transfer
torque STTis given by
STT=
~j
2deMsmmp; (3)
wherejis the density of current passing perpendicularly
through the lm, eis the elementary charge (positive),
pis the spin-polarization direction, and 2(0;1] is the
spin-polarization eciency.
We now seek to nondimensionalize the above system.
Let
`=s
2A
0M2s; Q =2K
0M2s;hext=Hext
Ms:(4)
We then rescale space and time as
r!`r; t!t
0Ms; (5)
obtaining the nondimensional form
@m
@t= mhe + m@m
@t mmp;(6)
where he =He =Ms, and
=~j
2de0M2s(7)
is the dimensionless spin-torque strength.
Since we are interested in thin lms, we now assume
thatmis independent of the lm thickness. Then, after
rescaling
E!0M2
sd`2E; (8)
we have he ' E
m, whereEis given by a local energy
functional de ned on the (rescaled) two-dimensional do-
mainD(see, e.g., Ref. [17]):
E(m)'1
2Z
D
jrmj2+Q(m) 2hextm
d2r
+1
2Z
Dm2
?d2r+1
4jlnjZ
@D(mn)2ds;(9)in which now m:D!S2,m?is its out-of-plane com-
ponent,=d=`is the dimensionless lm thickness, and
=d=L 1 (whereLis the lateral size of the lm) is the
lm's aspect ratio. The e ective eld is given explicitly
by
he = m Q
2rm(m) m?ez+hext; (10)
andmsatis es equation (6) in Dwith the boundary
condition
@m
@n= 1
2jlnj(mn)(n (mn)m) (11)
on@D.
We now parametrize min terms of spherical angles as
m= ( sincos ;coscos ;sin ); (12)
and the current polarization direction pin terms of an
in-plane angle and its out-of-plane component p?as
p=1p
1 +p2
?( sin ;cos ;p?): (13)
Writing = =p
1 +p2
?, after some algebra, one may
then write equation (6) as the system
@
@t= 1
cos he m+ cos @
@t
+ (p?cos sin cos( ));(14)
cos @
@t= he m + @
@t+ sin( );(15)
where m=@m=@andm =@m=@ formgiven by
(12). Again, since we are working in a soft thin lm, we
assume 1 and that the out-of-plane component of
the e ective eld in equation (10) is dominated by the
termhe ez' m?= sin . Note that this assumes
that the crystalline anisotropy and external eld terms
in the out-of-plane directions are relatively small, so we
assume the external eld is only in plane, though it is still
possible to include a perpendicular anisotropy simply by
renormalizing the constant in front of the m?term in
he . We then linearize the above system in , yielding
@
@t=E
+ @
@t+ (p? cos( )); (16)
@
@t= + sin( )
+ ( hxsin+hycos) + @
@t:(17)
wherehx=he exandhy=he ey, andE() isE(m)
evaluated at = 0.3
We now note that the last two terms in (17) are neg-
ligible relative to wheneverjhxj;jhyjand are small,
which is true of typical clean thin- lm samples of su-
ciently large lateral extent. Neglecting these terms, one
has
@
@t=E
+ @
@t+ (p? cos( )); (18)
@
@t= sin( ) + : (19)
Then, di erentiating (19) with respect to tand using the
result along with (19) to eliminate and@
@tfrom (18),
we nd a second-order in time equation for :
0 =@2
@t2+@
@t( + 2 cos( )) +E
+ p?+ 2
sin( ) cos( );(20)
where, explicitly, one has
E
= +Q
2~0() +hext(cos;sin); (21)
and~() = ( m()). In turn, from the boundary condi-
tion on min (11), we can derive the boundary condition
foras
nr=1
2jlnjsin( ') cos( '); (22)
where'is the angle parametrizing the normal nto@D
vian= ( sin';cos').
The model comprised of (20){(22) is a damped-driven
wave-like PDE for , which coincides with the reduced
model of Ref. [8] for vanishing spin-current density in
an in nite sample. This constitutes our reduced PDE
model for magnetization dynamics in thin- lm elements
under the in
uence of out-of-plane spin currents. It is
easy to see that all of the terms in (20) balance when the
parameters are chosen so as to satisfy
p? Q1=2jhextj1=2`
Ljlnj:(23)
This shows that it should be possible to rigorously obtain
the reduced model in (20){(22) in the asymptotic limit
ofL!1 and ; ;p?;Q;jhextj;!0 jointly, so that
(23) holds.
III. MACROSPIN SWITCHING
In this section we study the behavior of the reduced
model (20){(22) in the approximation that the magneti-
zation is spatially uniform on an elliptical domain, and
compare the solution phenomenology to that found by
simulating the LLGS equation in the same physical situ-
ation, as studied in Ref. [16].A. Derivation of macrospin model
Integrating equation (20) over the domain Dand using
the boundary condition (22), we have
Z
D@2
@t2+@
@t( + 2 cos( ))
+ p?+ 2
sin( ) cos( )
+Q
2~0() +hext(cos;sin)
d2r
=1
2jlnjZ
@Dsin( ') cos( ') ds:(24)
Assume now that does not vary appreciably across the
domainD, which makes sense in magnetic elements that
are not too large. This allows us to replace (r;t) by
its spatial average (t) =1
jDjR
D(r;t) d2r, wherejDj
stands for the area of Din the units of `2. Denoting
time derivatives by overdots, and omitting the bar on
for notational simplicity, this spatial averaging leads to
the following ODE for (t):
+_( + 2 cos( )) + 2
sin( ) cos( )
+ p?+Q
2~0() +hext(cos;sin)
=jlnj
4jDjsin 2Z
@Dcos(2') ds
jlnj
4jDjcos 2Z
@Dsin(2') ds:(25)
Next, we consider a particular physical situation in
which to study the macrospin equation, motivated by
previous work10,11. As in Refs. [14{16], we consider an
elliptical thin- lm element (recall that lengths are now
measured in the units of `):
D=
(x;y) :x2
a2+y2
b2<1
; (26)
with no in-plane crystalline anisotropy, Q= 0, and no
external eld, hext= 0. We take the long axis of the
ellipse to be aligned with the ey-direction, i.e. b > a ,
with the in-plane component of current polarization also
aligned along this direction, i.e., taking = 0. One can
then compute the integral over the boundary in equation
(25) explicitly, leading to the equation
+_( + cos) + sincos
+ 2
sincos+ p?= 0;(27)
where we introduced the geometric parameter 0 < 1
obtained by an explicit integration:
=jlnj
22abZ2
0b2cos2 a2sin2 p
b2cos2 +a2sin2 d : (28)4
(d)
(c)
(a)
(b)
FIG. 1: Solutions of macrospin equation (30) for = 0:01, = 0:1. In (a),p?= 0:2,= 0:03: decaying solution; in (b),
p?= 0:2,= 0:06: limit cycle solution (the initial conditions in (a) and (b) are (0) = 3:5, to better visualize the behavior).
In (c),p?= 0:3,= 0:08: switching solution; in (d), p?= 0:6,= 0:1: precessing solution.
This may be computed in terms of elliptic integrals,
though the expression is cumbersome so we omit it here.
Importantly, up to a factor depending only on the eccen-
tricity the value of is given by
d
LlnL
d: (29)
For example, for an elliptical nanomagnet with dimen-
sions 100302:5 nm (similar to those considered in
Ref. [16]), this yields '0:1.
It is convenient to rescale time byp
and divide
through by , yielding
+1p
_( + 2 cos) + sincos
+p?+2 sincos= 0;(30)
where we introduced = =. We then apply this
ODE to model the problem of switching of the thin- lm
elements, taking the initial in-plane magnetization direc-tion to be static and aligned along the easy axis, an-
tiparallel to the in-plane component of the spin-current
polarization. Thus, we take
(0) =; _(0) = 0; (31)
and study the resulting initial value problem.
B. Solution phenomenology
Let us brie
y investigate the solution phenomenology
as the dimensionless spin-current parameters andp?
are varied, with the material parameters, and , xed.
We take all parameters to be constant in time for simplic-
ity. We nd, by numerical integration, 4 types of solution
to the initial value problem de ned above. The sample
solution curves are displayed in Fig. 1 below. The rst
(panel (a)) occurs for small values of , and consists sim-
ply of oscillations of around a xed point close to the
long axis of the ellipse, which decay in amplitude towards
the xed point, without switching.5
Secondly (panel (b)), still below the switching thresh-
old, the same oscillations about the xed point can reach
a nite xed amplitude and persist without switching.
This behavior corresponds to the onset of relatively small
amplitude limit-cycle oscillations around the xed point.
Thirdly (panel (c)), increasing either ;p?or both, we
obtain switching solutions. These have initial oscillations
inabout the xed point near , which increase in ampli-
tude, and eventually cross the short axis of the ellipse at
==2. Thenoscillates about the xed point near 0,
and the oscillations decay in amplitude toward the xed
point.
Finally (panel (d)), further increasing andp?we
obtain precessing solutions. Here, the initial oscillations
about the xed point near quickly grow to cross =2,
after which continues to decrease for all t, the magne-
tization making full precessions around the out-of-plane
axis.
IV. HALF-PERIOD ORBIT-AVERAGING
APPROACH
We now seek to gain some analytical insight into the
transitions between the solution types discussed above.
We do this by averaging over half-periods of the oscil-
lations observed in the solutions to generate a discrete
dynamical system which describes the evolution of the
energy of a solution (t) on half-period time intervals.
Firstly, we observe that in the relevant parameter
regimes the reduced equation (30) can be seen as a weakly
perturbed Hamiltonian system. We consider both and
small, with .p
, and assume = and
p?.1. The arguments below can be rigorously jus-
ti ed by considering, for example, the limit !0 while
assuming that =O() and that the values of and
p?are xed. This limit may be achieved in the origi-
nal model by sending jointly d!0 andL!1 , while
keeping17
Ld
`2lnL
d.1: (32)
The last condition ensures the consistency of the assump-
tion thatdoes not vary appreciably throughout D.
Introducing !(t) =_(t), (30) can be written to leading
order as
_=@H
@!;_!= @H
@; (33)
where we introduced
H=1
2!2+V(); V () =1
2sin2+p?: (34)
At the next order, the e ects of nite and appear
in the rst-derivative term in (30), while the other forc-
ing term is still higher order. The behavior of (30) is
therefore that of a weakly damped Hamiltonian system
with Hamiltonian H, with the e ects of andservingto slowly change the value of Has the system evolves.
Thus, we now employ the technique of orbit-averaging to
reduce the problem further to the discrete dynamics of
H(t), where the discrete time-steps are equal (to the lead-
ing order) to half-periods of the underlying Hamiltonian
dynamics (which thus vary with H).
Let us rst compute the continuous-in-time dynamics
ofH. From (34),
_H=!( _!+V0()); (35)
which vanishes to leading order. At the next order, from
(30), one has
_H= !2
p
( + 2 cos): (36)
We now seek to average this dynamics over the Hamil-
tonian orbits. The general nature of the Hamiltonian
orbits is either oscillations around a local minimum of
V() (limit cycles) or persistent precessions. If the lo-
cal minimum of Vis close to an even multiple of ,H
cannot increase, while if it is close to an odd multiple
thenHcan increase if is large enough. The switching
process involves moving from the oscillatory orbits close
to one of these odd minima, up the energy landscape,
then jumping to oscillatory orbits around the neighbor-
ing even minimum, and decreasing in energy towards the
new local xed point.
We focus rst on the oscillatory orbits. We may de ne
their half-periods as
T(H) =Z
+
d
_; (37)
where
and
+are the roots of the equation V() =
Hto the left and right of the local minimum of V()
about which (t) oscillates. To compute this integral, we
assume that (t) follows the Hamiltonian trajectory:
_=p
2(H V()): (38)
We then de ne the half-period average of a function
f((t)) as
hfi=1
T(H)Z
+
f() dp
2(H V()); (39)
which agrees with the time average over half-period to
the leading order. Note that this formula applies irre-
spectively of whether the trajectory connects
to
+
or
+to
. Applying this averaging to _H, we then have
D
_HE
= 1
T(H)Z
+
(;H) d; (40)
where we de ned
(;H) =( + 2 cos)p
2(H V())p
: (41)6
If the value ofHis such that either of the roots
no
longer exist, this indicates that the system is now on a
precessional trajectory. In order to account for this, we
can de ne the period on a precessional trajectory instead
as
T(H) =ZC
C d
_; (42)
whereCis a local maximum of V(). On the preces-
sional trajectories, we then have
D
_HE
= 1
T(H)ZC
C (;H) d: (43)
In order to approximate the ODE solutions, we now
decompose the dynamics of Hinto half-period time in-
tervals. We thus take, at the n'th timestep,Hn=H(tn),
tn+1=tn+T(Hn) and
Hn+1=Hn Z
+(Hn)
(Hn) (;Hn) d; (44)
ifHncorresponds to a limit cycle trajectory. The same
discrete map applies to precessional trajectories, but with
the integration limits replaced with C andC, re-
spectively.
A. Modelling switching with discrete map
In order to model switching starting from inside a well
ofV(), we can iterate the discrete map above, starting
from an initial energy H0. We chooseH0by choosing a
static initial condition (0) =0close to an odd multiple
of(let us assume without loss of generality that we are
close to), and computing H0=V(0).
On the oscillatory trajectories, the discrete map then
predicts the maximum amplitudes of oscillation (
(Hn))
at each timestep, by locally solving Hn=V() for each
n. After some number of iterations, the trajectory will
escape the local potential well, and one or both roots of
Hn=V() will not exist. Due to the positive average
slope ofV() the most likely direction for a trajectory to
escape the potential well is _<0 (`downhill'). Assuming
this to be the case, at some timestep tN, it will occur that
the equationHN=V() has only one root =
+>,
implying that the trajectory has escaped the potential
well, and will proceed on a precessional trajectory in a
negative direction past ==2 towards= 0.
To distinguish whether a trajectory results in switching
or precession, we then perform a single half-period step
on the precessional orbit from CtoC , and check
whetherH< V (C ): if this is the case, the tra-
jectory moves back to the oscillatory orbits around the
well close to = 0, and decreases in energy towards the
xed point near = 0, representing switching. If how-
everH> V(C ) after the precessional half-period,
the solution will continue to precess.In Fig. 2 below, we display the result of such an iter-
ated application of the discrete map, for the same param-
eters as the switching solution given in Fig. 1(c). In Fig.
2(a), the continuous curve represents the solution to (30),
and the points are the predicted peaks of the oscillations,
from the discrete map (44). Fig. 2(b) shows the energy
of the same solution as a function of . Again the blue
curve givesH(t) for the ODE solution, the green points
are the prediction of the iterated discrete map, and the
red curve is V(). The discrete map predicts the switch-
ing behavior quite well, only su ering some error near
the switching event, when the change of His signi cant
on a single period.
B. Modelling precession
Here we apply the discrete map to a precessional
solution|one in which the trajectory, once it escapes
the potential well near , does not get trapped in the
next well, and continues to rotate. Fig. 3(a) below dis-
plays such a solution (t) and its discrete approximation,
and Fig. 3(b) displays the energy of the same solution.
Again, the prediction of the discrete map is excellent.
V. TRANSITIONS IN TRAJECTORIES
In this section we seek to understand the transi-
tions between the trapping, switching, and precessional
regimes as the current parameters andp?are varied.
A. Escape Transition
Firstly, let us consider the transition from states which
are trapped in a single potential well, such as those in
Figs. 1(a,b), to states which can escape and either switch
or precess. E ectively, the absolute threshold for this
transition is for the value of Hto be able to increase for
some value close to the minimum of V() near. Thus,
we consider the equation of motion (36) for H, and wish
to nd parameter values such that _H>0 for somenear
. This requires that
!2
p
( + 2 cos)<0: (45)
Assuming that !6= 0, we can see that the optimal value
ofto hope to satisfy this condition is =, yield-
ing a theoretical minimum =sfor the dimensionless
current density for motion to be possible, with
s=
2: (46)
This is similar to the critical switching currents derived
in previous work14. We then require >sfor the possi-
bility of switching or precession. Note that this estimate
is independent of the value of p?.7
(b) (a)
FIG. 2: Switching solution (blue line) and its discrete approximation (green circles). Parameters: = 0:01, = 0:1,p?= 0:3,
= 0:08. Panel (a) shows the solution (t), and panel (b) shows the trajectory for this solution in the H plane. The red
line in (b) shows V().
B. Switching{Precessing Transition
We now consider the transition from switching to pre-
cessional states. This is rather sensitive and there is not
in general a sharp transition from switching to precession.
It is due to the fact that for certain parameters, the path
that the trajectory takes once it escapes the potential
well depends on how much energy it has as it does so. In
fact, for a xed ;, and values of > swe can sep-
arate the (;p?)-parameter space into three regions: (i)
after escaping the initial well, the trajectory always falls
into the next well, and thus switches; (ii) after escaping,
the trajectory may either switch or precess depending on
its energy as it does so (and thus depending on its initial
condition); (iii) after escaping, the trajectory completely
passes the next well, and thus begins to precess.
We can determine in which region of the parameter
space a given point ( ;p?) lies by studying the discrete
map (44) close to the peaks of V(). Assume that the
trajectory begins at (0) =, and is thus initially in
the potential well spanning the interval =23=2.
Denote byCthe point close to ==2 at which V()
has a local maximum. It is simple to compute
C=
2+1
2sin 1(2p?): (47)
Moreover, it is easy to see that all other local maxima of
V() are given by =C+k, fork2Z.
We now consider trajectories which escape the initial
well by crossing C. These trajectories have, for some
value of the timestep nwhile still con ned in the initial
well, an energy value Hnin the range
Htrap<Hn<V(C+); (48)
where we de neHtrapto be the value of Hnsuch that
the discrete map (44) gives Hn+1=V(C). We thushaveHn+1> V (C). In order to check whether the
trajectory switches or precesses, we then compute Hn+2
and compare it to V(C ). We may then classify the
trajectories as switching if Hn+2 V(C )<0, and
precessional ifHn+2 V(C )>0.
Figure 4 displays a plot of Hn V(C+) against
Hn+2 V(C ). The blue line shows the result of
applying the discrete map, while the red line is the iden-
tity line. Values of Hn V(C+) which are inside the
range speci ed in (48) are thus on the negative x-axis
here. We can classify switching trajectories as those for
which the blue line lies below the x-axis, and precessing
trajectories as those which lie above. In Fig. 4, the pa-
rameters are such that both of these trajectory types are
possible, depending on the initial value of Hn, and thus
this set of parameters are in region (ii) of the parameter
space. We note that, since the curve of blue points and
the identity line intersect for some large enough value of
H, this gure implies that if the trajectory has enough
energy to begin precessing, then after several precessions
the trajectory will converge to one which conserves en-
ergy on average over a precessional period (indicated by
the arrows). In region (i) of the parameter space, the
portion of the blue line for Hn V(C+)<0 would
haveHn+2 V(C )<0, while in region (iii), they
would all haveHn+2 V(C )>0.
We can classify the parameter regimes for which
switching in the opposite direction (i.e. switches from
to 2) is possible in a similar way. It is not possible
to have a precessional trajectory moving in this direction
(_>0), though.
We may then predict, for a given point ( ;p?) in pa-
rameter space, by computing relations similar to that in
Fig. 4, which region that point is in, and thus generate
a theoretical phase diagram.
In Fig. 5 below, we display the phase diagram in the8
(a) (b)
FIG. 3: Precessing solution (blue line) and its discrete approximation (green circles). Parameters: = 0:01, = 0:1,p?= 0:6,
= 0:1. Panel (a) shows the solution (t), and panel (b) shows the trajectory for this solution in the H plane. The red
line in (b) shows V().
(;p?)-parameter space, showing the end results of solv-
ing the ODE (30) as a background color, together with
predictions of the bounding curves of the three regions
of the space, made using the procedure described above.
The predictions of the discrete map, while not perfect,
are quite good, and provide useful estimates on the dif-
ferent regions of parameter space. In particular, we note
that the region where downhill switching reliably occurs
(the portion of region (i) above the dashed black line) is
estimated quite well. We would also note that we would
expect the predictions of the discrete map to improve if
the values of and were decreased.
VI. DISCUSSION
We have derived an underdamped PDE model for mag-
netization dynamics in thin lms subject to perpendic-
ular applied spin-polarized currents, valid in the asymp-
totic regime of small and , corresponding to weak
damping and strong penalty for out-of-plane magnetiza-
tions. We have examined the predictions of this model
applied to the case of an elliptical lm under a macrospin
approximation by using an orbit-averaging approach. We
found that they qualitatively agree quite well with pre-
vious simulations using full LLGS dynamics16.
The bene ts of our reduced model are that they should
faithfully reproduce the oscillatory nature of the in-
plane magnetization dynamics, reducing computational
expense compared to full micromagnetic simulations. In
particular, in suciently small and thin magnetic ele-
ments the problem further reduces to a single second-
order scalar equation.
The orbit-averaging approach taken here enables the
investigation of the transition from switching to preces-
sion via a simple discrete dynamical system. The regionsin parameter space where either switching or precession
are predicted, as well as an intermediate region where
the end result depends sensitively on initial conditions.
It may be possible to further probe this region by includ-
ing either spatial variations in the magnetization (which,
in an earlier study16were observed to simply `slow down'
the dynamics and increase the size of the switching re-
gion), or by including thermal noise, which could result
in instead a phase diagram predicting switching proba-
bilities at a given temperature, or both.
−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05−0.05−0.04−0.03−0.02−0.0100.010.020.030.040.05
Hn−V(θC+π)Hn+2−V(θC−π)Switch Precess
FIG. 4: Precession vs switching prediction from the discrete
map. Parameters: = 0:01, = 0:1,p?= 0:35,= 0:08.
Values ofHn V(C+) to the left of the dashed line switch
after the next period, the trajectory becoming trapped in the
well around = 0. Values to the right begin to precess, and
converge to a precessional xed point of the discrete map.9
σp⊥
0 0.05 0.1 0.15 0.2 0.25 0.300.10.20.30.40.50.60.70.80.91
(i)(ii)(iii)
FIG. 5: Macrospin solution phase diagram: = 0:01; = 0:1.
The background color indicates the result of solving the ODE
(30) with initial condition (31): the dark region to the left of
the gure indicates solutions which do not escape their initial
potential well, and the vertical dashed white line shows the
computed value of the minimum current required to escape,
s= =(2). The black band represents solutions which
decay, like in Fig. 1(a), while the dark grey band represents
solutions like in Fig. 1(b). In the rest of the gure, the
green points indicate switching in the negative direction like
in Fig. 1(c), grey indicate switching in the positive direction,
and white indicates precession like in Fig. 1(d). The solid
black curves are the predictions of boundaries of the regions
(as indicated in the gure) by using the discrete map, and
the dashed line is the prediction of the boundary below which
switching in the positive direction is possible.ACKNOWLEDGMENTS
Research at NJIT was supported in part by NSF via
Grant No. DMS-1313687. Research at NYU was sup-
ported in part by NSF via Grant No. DMR-1309202.
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Matter Phys. 1, 71 (2010).
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1503.01478v2.Critical_current_destabilizing_perpendicular_magnetization_by_the_spin_Hall_effect.pdf | arXiv:1503.01478v2 [cond-mat.mes-hall] 1 Aug 2015Critical current destabilizing perpendicular magnetizat ion by the spin Hall effect
Tomohiro Taniguchi1, Seiji Mitani2, and Masamitsu Hayashi2
1National Institute of Advanced Industrial Science and Tech nology (AIST),
Spintronics Research Center, Tsukuba 305-8568, Japan
2National Institute for Materials Science, Tsukuba 305-004 7, Japan
(Dated: July 5, 2018)
The critical current needed to destabilize the magnetizati on of a perpendicular ferromagnet via
the spin Hall effect is studied. Both the dampinglike and field like torques associated with the spin
current generated by the spin Hall effect is included in the La ndau-Lifshitz-Gilbert equation to
model the system. In the absence of the fieldlike torque, the c ritical current is independent of the
damping constant and is much larger than that of conventiona l spin torque switching of collinear
magnetic systems, as in magnetic tunnel junctions. With the fieldlike torque included, we find that
the critical current scales with the damping constant as α0(i.e., damping independent), α, and
α1/2depending on the sign of the fieldlike torque and other parame ters such as the external field.
Numerical and analytical results show that the critical cur rent can be significantly reduced when
the fieldlike torque possesses the appropriate sign, i.e. wh en the effective field associated with the
fieldlike torque is pointing opposite to the spin direction o f the incoming electrons. These results
provideapathwaytoreducingthecurrentneededtoswitch ma gnetization usingthespin Hall effect.
PACS numbers: 75.78.-n, 75.70.Tj, 75.76.+j, 75.40.Mg
I. INTRODUCTION
The spin Hall effect1–3(SHE) in a nonmagnetic heavy
metal generates pure spin current flowing along the di-
rection perpendicular to an electric current. The spin
current excites magnetization dynamics in a ferromagnet
attached to the nonmagnetic heavy metal by the spin-
transfer effect4,5. There have been a number of exper-
imental reports on magnetization switching and steady
precession induced by the spin Hall effect6–9. These dy-
namics have attracted great attention recently from the
viewpoints ofboth fundamental physicsand practicalap-
plications.
An important issue to be solved on the magnetization
dynamics triggered by the spin Hall effect is the reduc-
tion of the critical current density needed to destabilize
the magnetization from its equilibrium direction, which
determines the current needed to switch the magneti-
zation direction or to induce magnetization oscillation.
The reported critical current density for switching8,10–13
or precession9is relatively high, typically larger than 107
A/cm2. One of the reasons behind this may be related
to the recently predicted damping constant independent
critical current when SHE is used14,15. This is in con-
trast to spin-transfer-induced magnetization switching in
a typical giant magnetoresistance (GMR) or magnetic
tunnel junction (MTJ) device where the critical current
is expected to be proportional to the Gilbert damping
constant α. Here the magnetization dynamics is excited
as a result of the competition between the spin torque
and the damping torque16. Since the damping constant
for typical ferromagnet in GMR or MTJ devices is rela-
tively small ( α∼10−2−10−3)17,18, it can explain why
the critical current is larger for the SHE driven systems.
Thus in particular for device application purposes, it is
crucial to find experimental conditions in which the mag-netization dynamics can be excited with lower current.
Another factor that might contribute to the reduc-
tion of the critical current is the presence of the field
like torque19. In the GMR/MTJ systems, both the con-
ventional spin torque, often referred to as the damp-
inglike torque, and the fieldlike torque arise from the
spin transfer between the conduction electrons and the
magnetization4,19–23. Due to the short relaxation length
of the transverse spin of the conduction electrons24,25,
the damping like torque is typically larger than the field-
like torque. Indeed, the magnitude of the field like
torque experimentally found in GMR/MTJ systems has
been reported to be much smaller than the damping like
torque26–29. Because of its smallness, the fieldlike torque
had notbeen consideredin estimatingthe criticalcurrent
intheGMR/MTJsystems16,30–32,althoughitdoesplaya
keyrolein particularsystems33,34. In contrast, recentex-
periments found that the fieldlike torque associated with
the SHE is larger than the damping like torque35–40.
The physical origin of the large SHE-induced field-
like torque still remains unclear. Other possible sources
can be the Rashba effect36,41–44, bulk effect45, and the
out of plane spin orbit torque46. Interestingly, the field
like torque has been reported to show a large angu-
lar dependence36,37,47(the angle between the current
and the magnetization), which cannot be explained by
the conventional formalism of spin-transfer torque in
GMR/MTJsystems. Thefieldliketorqueactsasatorque
duetoanexternalfieldandmodifiestheenergylandscape
of the magnetization. As a result, a large fieldlike torque
can significantly influence the critical current. However,
the fieldlike torque had not been taken into account in
considering the current needed to destabilize the magne-
tization from its equilibrium direction and thus its role
is still unclear.
In this paper, we study the critical current needed to2
destabilize a perpendicular ferromagnet by the spin Hall
effect. The Landau-Lifshitz-Gilbert(LLG) equationwith
the dampinglike and fieldlike torques associated with the
spin Hall effect is solved both numerically and analyti-
cally. Wefindthatthecriticalcurrentcanbesignificantly
reduced when the fieldlike torque possesses the appropri-
ate sign with respect to the dampinglike torque. With
the fieldlike torque included, the critical current scales
with the damping constant as α0(i.e., damping indepen-
dent),α, andα1/2, depending on the sign of the field-
like torque and other parameters. Analytical formulas
of such damping-dependent critical current are derived
[Eqs. (19)-(21)], and they show good agreement with the
numerical calculations. From these results, we find con-
ditions in which the critical current can be significantly
reduced compared to the damping-independent thresh-
old, i.e., systems without the fieldlike torque.
The paper is organized as follows. In Sec. II, we
schematically describe the system under consideration.
We discuss the definition of the critical current in Sec.
III. Section IV summarizes the dependences of the crit-
ical current on the direction of the damping constant,
the in-plane field, and the fieldlike torque obtained by
the numerical simulation. The analytical formulas of the
critical current and their comparison to the numerical
simulations are discussed in Sec. V. The condition at
which damping-dependent critical current occurs is also
discussed in this section. The conclusion follows in Sec.
VI.
II. SYSTEM DESCRIPTION
The system we consider is schematically shown in Fig.
1, where an electric current flowing along the x-direction
injects a spin current into the ferromagnet by the spin
Hall effect. The magnetization dynamics in the ferro-
magnet is described by the LLG equation,
dm
dt=−γm×H+αm×dm
dt
−γHsm×(ey×m)−γβHsm×ey,(1)
whereγandαarethe gyromagneticratioandtheGilbert
damping constant, respectively. We assume that the
magnetization of the ferromagnet points along the film
normal (i.e., along the zaxis), and an external in-plane
magnetic field is applied along the xoryaxis. The total
magnetic field His given by
H=HapplnH+HKmzez, (2)
whereHapplis the external field directed along the xor
yaxis and HKis the uniaxial anisotropy field along the
zaxis.nHandeiare unit vectors that dictate the di-
rection of the uniaxial anisotropy field and the iaxis,
respectively. Here we call the external field along the x
andydirections the longitudinal and transverse fields,
respectively. The third and fourth terms on the right-
hand side of Eq. (1) are the damping like and fieldlikeHappl // y
Happl // x
m
currentxz
y
FIG. 1. Schematic view of the spin-Hall system. The x
axis is parallel to current, whereas the zaxis is normal to the
film plane. The spin direction of the electrons entering the
magnetic layer via the spin Hall effect points along the + yor
−ydirection.
torques associated with the spin Hall effect, respectively.
Thetorquestrength Hscanbeexpressedwiththecurrent
densityj, the spin Hall angle ϑ, the saturation magneti-
zationM, and the thickness of the ferromagnet d, i.e.,
Hs=/planckover2pi1ϑj
2eMd. (3)
The ratio of the fieldlike torque to the damping like
torque is represented by β. Recent experiments found
thatβis positive and is larger than 135–40.
The magnetization dynamics described by the LLG
equation can be regarded as a motion of a point particle
on a two-dimensional energy landscape. In the presence
of the fieldlike torque, the energy map is determined by
the energy density given by34
E=−M/integraldisplay
dm·H−βMHsm·ey.(4)
Then, the external field torque and the fieldlike torque,
which are the first and fourth terms on the right-hand-
side of Eq. (1), can be expressed as −γm×B, where the
effective field Bis
B=−∂E
∂Mm. (5)
The initial state of the numerical simulation is chosen to
be the direction corresponding to the minimum of the
effective energy density E. The explicit forms of the ini-
tial state for the longitudinal and the transverse external
fields are shown in Appendix A.
We emphasize for the latter discussion in Sec. V that,
using Eqs. (1), (4), and (5), the time change of the effec-
tive energy density is described as
dE
dt=dEs
dt+dEα
dt. (6)3
Here the first and second terms on the right-hand side
are the rates of the work done by the spin Hall torque
and the dissipation due to damping, respectively, which
are explicitly given by
dEs
dt=γMHs[ey·B−(m·ey)(m·B)],(7)
dEα
dt=−αγM/bracketleftBig
B2−(m·B)2/bracketrightBig
. (8)
The sign of Eq. (7) depends on the current direction
and the effective magnetic field, while that of Eq. (8) is
always negative.
The magnetic parameters used in this paper mimic the
conditions achieved in CoFeB/MgO heterostructures48;
M= 1500 emu/c.c., HK= 540 Oe, ϑ= 0.1,γ=
1.76×107rad/(Oe s), and d= 1.0 nm. The value of
βis varied from −2, 0, to 2. Note that we have used a
reducedHK(Refs.8,49) in ordertoobtain criticalcurrents
that are the same order of magnitude with that obtained
experimentally. We confirmed that the following discus-
sions are applicable for a large value of HK(∼1T).
III. DEFINITION OF CRITICAL CURRENT
In this section, we describe how we determine the crit-
ical current from the numerical simulations. In exper-
iments, the critical current is determined from the ob-
servation of the magnetization reversal8,12,41,46,48–50. As
mentioned in Sec. II, in this paper, the initial state for
calculation is chosen to be the minimum of the effective
energy density. Usually, there are two minimum points
above and below the xyplane because of the symmetry.
Throughout this paper, the initial state is chosen to be
the minimum point above the xyplane, i.e., mz(0)>0,
for convention.” It should be noted that, once the mag-
netization arrives at the xyplane during the current ap-
plication, it can move to the other hemisphere after the
current is turned off due to, for example, thermal fluc-
tuation. Therefore, here we define the critical current as
the minimum current satisfying the condition
lim
t→∞mz(t)< ǫ, (9)
where a small positive real number ǫis chosen to be
0.001. The duration of the simulations is fixed to 5 µs,
long enough such that all the transient effects due to the
current application are relaxed. Figures 2(a) and 2(b)
show examples of the magnetization dynamics close to
the critical current, which are obtained from the numer-
ical simulation of Eq. (1). As shown, the magnetization
stays near the initial state for j= 3.1×106A/cm2, while
it moves to the xyplane for j= 3.2×106A/cm2. Thus,
the critical current is determined as 3 .2×106A/cm2in
this case.
We note that the choice of the definition of the criti-
cal current has some arbitrariness. For comparison, weFIG. 2. Time evolution ofthe zcomponentof themagnetiza-
tionmzin the presence of the transverse field of Happl= 200
with (a) j= 3.1×106A/cm2and (b)j= 3.2×106A/cm2.
The value of βis zero.
show numerically evaluated critical current with a differ-
ent definition in Appendix B. The main results of this
paper, e.g., the dependence of the critical current on the
damping constant, are not affected by the definition.
We also point out that the critical current defined by
Eq. (9) focuses on the instability threshold, and does
not guarantee a deterministic reversal. For example,
in the case of Fig. 2(b), the reversal becomes prob-
abilistic because the magnetization, starting along + z,
stops its dynamics at the xyplane and can move back
to its original direction or rotate to a point along −z
resulting in magnetization reversal. Such probabilistic
reversal can be measured experimentally using transport
measurements8,12,41,46,49,50or by studying nucleation of
magnetic domains via magnetic imaging48. On the other
hand, it hasbeen reportedthat deterministicreversalcan
take place when a longitudinal in-plane field is applied
alongside the current41,49. It is difficult to determine the
critical current analytically for the deterministic switch-
ing for all conditions since, as in the case of Fig. 2(b),
the magnetization often stops at the xyplane during the
current application. This occurs especially in the pres-
ence of the transverse magnetic field because all torques
become zero at m=±eyand the dynamics stops. Here
we thus focus on the probabilistic reversal.4
FIG. 3. Numerically evaluated mzatt= 5µs for (a)-(c) the longitudinal ( nH=ex) and (d)-(f) the transverse ( nH=ey)
fields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0. The damping constant is α= 0.005. The color scale
indicates the zcomponent of the magnetization ( mz) att= 5µs. The red/white boundary indicates the critical current fo r
probabilistic switching, whereas the red/blue boundary gi ves the critical current for deterministic switching.
IV. NUMERICALLY ESTIMATED CRITICAL
CURRENT
In this section, we show numerically evaluated critical
current for different conditions. We solve Eq. (1) and
apply Eq. (9) to determine the critical current. Figure
3 shows the value of mzatt= 5µs in the presence of
(a)-(c) the longitudinal ( nH=ex) and (d)-(f) the trans-
verse (nH=ey) fields. The value of βis 0 for Figs.
3(a) and 3(d), 2 .0 for Figs. 3(b) and 3(e), and −2.0 for
Figs. 3(c) and 3(f), respectively. The damping constant
isα= 0.005. The red/white boundary indicates the crit-
ical current for the probabilistic switching, whereas the
red and blue ( mz=−1) boundary gives the critical cur-
rent for the deterministic switching. Using these results
and the definition of the critical current given by Eq. (9),
and performing similar calculations for different values of
α, wesummarizethedependenceofthecriticalcurrenton
the longitudinal and transverse magnetic fields in Fig. 4.
The damping constant is varied as the following in each
plot:α= 0.005, 0.01, and 0 .02. The solid lines in Fig. 4
represent the analytical formula derived in Sec. V.A. In the presence of longitudinal field
In the case of the longitudinal field and β= 0
shown in Fig. 4(a), the critical current is damping-
independent. Such damping-independent critical current
has been reported previously for deterministic magneti-
zation switching14,15. Similarly, in the case of the longi-
tudinal field and negative β(β=−2.0) shown in Fig.
4(c), the critical current is damping-independent. In
these cases, the magnitude of the critical current is rel-
atively high. In particular, near zero field, the critical
current exceeds ∼108A/cm2, which is close to the limit
of experimentally accessible value. These results indicate
that the useofthe longitudinal field with zeroornegative
βis ineffective for the reduction of the critical current.
On the other hand, when βis positive, the critical cur-
rent depends on the damping constant, as shown in Fig.
4(b). Note that positive βis reported for the torques
associated with the spin Hall effect or Rashba effect in
the heterostructures studied experimentally35–37,39. The
magnitude of the critical current, ∼10×106A/cm2, is
relatively small compared with the cases of zero or neg-
ativeβ. In this case, the use of a low damping material
is effective to reduce the critical current. Interestingly,
the critical current is not proportional to the damping
constant, while that previously calculated for a GMR or
MTJ system16is proportional to α. For example, the5
longitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)
0 50 100 150 2000
-10050
-150100150
β=0.0
-50
-100 -50 -150 -200: α=0.005: α=0.01: α=0.02
transverse magnetic field (Oe)critical current density (10 6 A/cm 2)
0 50 100 150 2000
-10050
-150100150
β=0.0
-50
-100 -50 -150 -200: α=0.005: α=0.01: α=0.02
transverse magnetic field (Oe)critical current density (10 6 A/cm 2)
0 50 100 150 2000
-4020
-6040 60
β=2.0
-20
-100 -50 -150 -200: α=0.005: α=0.01: α=0.02longitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)
0 50 100 150 2000
-4020
-5040 50
β=2.0
-20
-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02-30-1010 30
transverse magnetic field (Oe)critical current density (10 6 A/cm 2)
0 50 100 150 2000
-10050
-150100150
β=-2.0
-50
-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02(a) (b) (c)
(d) (e) (f)longitudinal magnetic field (Oe)0 50 100 150 200β=-2.0
-100 -50 -150 -200: α=0.005: α=0.01: α=0.02critical current density (10 6 A/cm 2)
0
-10050
-150100150
-50
FIG. 4. Numerically evaluated critical currents in the pres ence of (a)-(c) the longitudinal ( nH=ex) and (d)-(f) the transverse
(nH=ey) fields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0, respectively. The solid lines are analytically
estimated critical current in Sec. V.
critical current at zero longitudinal field in Fig. 4(b) is
12.3,17.2, and24 .0×106A/cm2forα= 0.005,0.01, and
0.02, respectively. These values indicate that the critical
current is proportional to α1/2. In fact, the analytical
formula derived in Sec. V shows that the critical current
is proportional to α1/2for positive β[see Eq. (19)].
To summarize the case of the longitudinal field, the
use of a heterostructure with positive β, which is found
experimentally, has the possibility to reduce the critical
current if a ferromagnet with low damping constant is
used. In this case, the critical current is proportional to
α1/2, which has not been found in previous works.
B. In the presence of transverse field
In the presence of the transverse field with β= 0,
the critical current shows a complex dependence on the
damping constant α, as shown in Fig. 4(d). When the
current and the transversefield areboth positive (or neg-
ative), the critical current is proportional to the damping
constant αexcept near zero field. The numerically cal-
culated critical current matches well with the analytical
result, Eq. (20), shown by the solid lines. In this case,
the use of the low damping material results in the reduc-
tion of the critical current. On the other hand, when the
current and the transversefield possessthe opposite sign,
thecriticalcurrentisdampingindependent. Moreover,in
this case, thecriticalcurrentisofthe orderof108A/cm2.
Thus, it is preferable to use the current and field having
the same sign for the reduction of the critical current. Itshould be noted that, in our definition, the same sign of
current and field corresponds to the case when the direc-
tion ofincoming electrons’spin (due to the SHE) and the
transverse field are opposite to each other. The reason
why the critical current becomes damping dependent in
this situation will be explained in Sec. V.
Whenβis positive the critical current depends on the
damping constant for the whole range of the transverse
field, as shown in Fig. 4(e). The critical current is
roughly proportional to α1/2, in particular, close to zero
field. The solid lines display the analytical formula, Eq.
(21), and showgood agreementwith the numericalcalcu-
lations. The damping dependence of the critical current
becomes complex when the magnitude of the transverse
field is increased [see Eq. (21)]. We note that the critical
currentfor the positive βin Fig. 4(e) is smallerthan that
forβ= 0 in Fig. 4(d) for the whole range of Happl.
On the other hand, when βis negative, the critical
current is almost independent of α, especially near zero
field. However, when the transverse field is increased,
there is a regime where the critical current depends on
the damping constant. Such transition of the critical
current with the transverse field is also predicted by the
analytical solution, Eq. (21).
Tosummarizethe caseofthe transversefield, the αde-
pendence of the critical current can be categorized into
the following: α0(damping independent), α,α1/2, or
other complex behavior. As with the case of the longi-
tudinal field, the use of a heterostructure with positive β
allowsreductionofthe criticalcurrentwhen lowdamping
ferromagnet is used. Overall, the most efficient condition6
to reduce the critical current is to use the transverse field
with heterostructures that possess low αand positive β.
In this case, the critical current is reduced to the order
of 106A/cm2.
V. ANALYTICAL FORMULA OF CRITICAL
CURRENT
In this section, we derive the analytical formula of the
critical current from the linearized LLG equation51. The
complex dependences ofthe critical currentonthe damp-
ing constant αdiscussed in Sec. IV are well explained by
the analytical formula. We also discuss the physical in-
sight obtained from the analytical formulas.
A. Derivation of the critical current
To derive the critical current, we consider the stable
condition of the magnetization near its equilibrium. It is
convenient to introduce a new coordinate XYZin which
theZaxis is parallel to the equilibrium direction. The
rotationfromthe xyz-coordinatetothe XYZcoordinate
is performed by the rotation matrix
R=
cosθ0−sinθ
0 1 0
sinθ0 cosθ
cosϕsinϕ0
−sinϕcosϕ0
0 0 1
,(10)
where (θ,ϕ) are the polar and azimuth angles of the
magnetization at equilibrium. The equilibrium magne-
tization direction under the longitudinal and transverse
magnetic field is given by Eqs. (A1) and (A2), respec-
tively. Since we are interested in small excitation of the
magnetization around its equilibrium, we assume that
the components of the magnetization in the XYZcoor-
dinate satisfy mZ≃1 and|mX|,|mY| ≪1. Then, the
LLG equation is linearized as
1
γd
dt/parenleftbigg
mX
mY/parenrightbigg
+M/parenleftbigg
mX
mY/parenrightbigg
=−Hs/parenleftbigg
cosθsinϕ
cosϕ/parenrightbigg
,(11)
where the components of the 2 ×2 matrix Mare
M1,1=αBX−Hssinθsinϕ, (12)
M1,2=BY, (13)
M2,1=BX (14)
M2,2=αBY−Hssinθsinϕ. (15)
Here,BXandBYare defined as
BX=Happlsinθcos(ϕ−ϕH)+βHssinθsinϕ+HKcos2θ,
(16)BY=Happlsinθcos(ϕ−ϕH)+βHssinθsinϕ+HKcos2θ,
(17)
whereϕHrepresents the direction of the external field
within the xyplane:ϕH= 0 for the longitudinal field
andπ/2 for the transverse field.
The solution of Eq. (11) is mX,mY∝
exp{γ[±i/radicalbig
det[M]−(Tr[M]/2)2−Tr[M]/2]t}, where
det[M] and Tr[ M] are the determinant and trace of
the matrix M, respectively. The imaginary part of the
exponent determines the oscillation frequency around
theZaxis, whereas the real part determines the time
evolution of the oscillation amplitude. The critical
current is defined as the current at which the real part
of the exponent is zero. Then, the condition Tr[ M] = 0
gives
α(BX+BY)−2Hssinθsinϕ= 0,(18)
For the longitudinal field, Eq. (18) gives
jLONG
c=±2e√αMd
/planckover2pi1ϑ/radicalBig
2H2
K−H2
appl/radicalbig
β(2+αβ),(19)
indicating that the critical current is roughly propor-
tional to α1/2. This formula works for positive βonly52
if we assume 0 <2+αβ≃2, which is satisfied for typical
ferromagnets. The critical current when the transverse
field is applied reads
jTRANS
c=2αeMd
/planckover2pi1ϑ(Happl/HK)HK/bracketleftBigg
1−1
2/parenleftbiggHappl
HK/parenrightbigg2/bracketrightBigg
,(20)
whenβ= 0, indicating that the critical current is pro-
portional to α. The critical current for finite βis
jTRANS
c=2eMd
/planckover2pi1ϑ
×−(1+αβ)Happl±/radicalBig
H2
appl+2αβ(2+αβ)H2
K
β(2+αβ).
(21)
Equation (21) works for the whole range of |Happl|(<
HK) for positive β, while it only works when |Happl|>
2αβ(2 +αβ)HKfor negative β. For example, when
β=−2.0, this condition is satisfied when |Happl|>108
Oe forα= 0.005 and |Happl|>152 Oe for α= 0.01.
However the condition is not satisfied for the present
range of Happlforα= 0.02. The solid lines in Fig. 4(f)
show where Equation (21) is applicable. The zero-field
limits of Eqs. (19) and (21) become identical,
lim
Happl→0jc=±2e√αMd
/planckover2pi1ϑ√
2HK/radicalbig
β(2+αβ),(22)
indicating that the critical current near zero field is pro-
portional to α1/2whenβ >0.7
FIG. 5. Magnetization dynamics under the conditions of (a)
nH=ey,Happl= 50 Oe, β= 0,α= 0.005, and j= 13.2×106
A/cm2, and (b) nH=ex,Happl= 50 Oe, β= 0,α= 0.005,
andj= 90×106A/cm2.
B. Discussions
The solid lines in Fig. 4(b), 4(d), 4(e), and 4(f) show
the analytical formulas, Eqs. (19), (20), and (21). As
evident, these formulas agree well with the numerical re-
sults in the regions where the critical currents depend on
the dampingconstant. In this section, we discussthe rea-
son why the critical current becomes damping dependent
or damping independent depending on the field direction
and the sign of β.
It is useful for the following discussion to first study
typical magnetization dynamics found in the numerical
calculations. Figure 5 shows the time evolution of the
x,yandzcomponents of the magnetization when the
critical current depends on [Fig. 5(a)] or is independent
of [Fig. 5(b)] the damping constant. For the former,
the instability is accompanied with a precession of the
magnetization. On the other hand, the latter shows that
the instability takes place without the precession.
We start with the case when the critical current be-
comes damping dependent. To provide an intuitive pic-
ture, we schematically show in Fig. 6(a) the torques ex-
erted on the magnetization during one precession period
when current is applied. The condition is the same with
that described in Fig. 5(a), i.e., the transverse magnetic
field is applied with β= 0. In Fig. 6(a), magnetization
is shown by the large black arrow, while the directions
of the spin Hall torque, the damping torque and the ex-ternal field torque are represented by the solid, dotted
and dashed lines, respectively (the external field torque
is tangent to the precession trajectory). As evident in
Fig. 5(a), the precession trajectory is tilted to the posi-
tiveydirection due to the transversefield. Depending on
the direction of the magnetization the spin Hall torque
has a component parallel, antiparallel, or normal to the
damping torque. This means that the work done by the
spin Hall torque, denoted by ∆ Esin Fig. 6 (a), is pos-
itive, negative, or zero at these positions. This can be
confirmed numerically when we calculate the work done
by the spin Hall torque using Eq. (7). For an infinites-
imal time ∆ t, the work done by the spin Hall torque
is equal to the rate of its work ( dEs/dt), given in Eq.
(7), times ∆ t, i.e. ∆Es= (dEs/dt)∆t. The solid line
in Fig. 6(b) shows an example of the calculated rate of
the work done by the spin Hall torque (solid line), dEs/dt
in Eq. (7). As shown, dEs/dtis positive, negative, and
zero, when the magnetization undergoes one precession
period. Similarly, the energy dissipated by the damping
torque,dEα/dt, can be calculated using Eq. (8) and is
shown by the dotted line in Fig. 6(b). The calculated
dissipation due to damping over a precession period is
always negative. Details of how the rates, shown in Fig.
6, are calculated are summarized in Appendix C.
Note that the strength of the spin Hall torque for
∆Es>0 is larger than that for ∆ Es<0 due to the an-
gular dependence of the spin Hall torque, |m×(ey×m)|.
Although it is difficult to see, thesolid line in Fig. 6(b) is
slightly shifted upward. Thus the total energy supplied
by the spin Hall torque during one precession, given by/contintegraltext
dt(dEs/dt), does not average to zero and becomes posi-
tive. When the current magnitude, |j|, is larger than |jc|
in Eq. (20), the energy supplied by the spin Hall torque
overcomes the dissipation due to the damping and con-
sequently the precession amplitude grows, which leads to
the magnetization instability shown in Fig. 5(a). The
same picture is applicable when both directions of field
and current are reversed. For this condition, the insta-
bility of the magnetization is induced by the competition
between the spin Hall torque and the damping torque.
Therefore, the critical current depends on the damping
constant α. When only the current direction is reversed
in Figs. 6(a) and 6(b) (i.e., the sign of the magnetic field
and current is opposite to each other), the sign of ∆ Esis
reversed and thus the total energy supplied by the spin
Hall torque becomes negative. This means that the spin
Hall torque cannot overcome the damping torque to in-
duce instability. Therefore, the critical current shown in
Eq. (20) only applies to the case when the sign of the
field and current is the same. As described in Sec. IV,
the same sign of the current and field in our definition
means that the incoming electrons’ spin direction, due
to the spin Hall effect, is opposite to the transverse field
direction.
Next, we consider the case when the critical current is
damping independent. Figure 6 (c) schematically shows
the precession trajectory when the applied field points to8
FIG. 6. (a) A schematic view of the precession trajectory
in the presence of the applied field in the positive y-direction.
The solid and dotted arrows indicate the directions of the
spin Hall torque and the damping torque, respectively. The
dashed line, which is the tangent line to the precession tra-
jectory, shows the field torque. The damping torque always
dissipates energy from the ferromagnet. On the other hand,
the spin Hall torque supplies energy (∆ Es>0) when its di-
rection is anti-parallel to the damping torque, and dissipa tes
energy (∆ Es<0) when the direction is parallel to the damp-
ing torque. When the direction of the spin Hall torque is
orthogonal to the damping torque, the spin Hall torque does
not change the energy (∆ Es= 0). (b) Typical temporal vari-
ation of the rates of the work done by the spin Hall torque,
Eq. (7), (solid) and the dissipation due to damping, Eq. (8)
(dotted) in the presence of the transverse field. The time is
normalized by the period given by Eq. (C7). (c), (d) Similar
figures with the longitudinal field.
thexdirection and β= 0. The corresponding rate of
work done by the spin Hall torque and the dissipation
rate due to the damping torque are shown in Fig. 6 (d).
Similar to the previous case, ∆ Escan be positive, nega-
tive, or zero during one precession period. However, the
total workdoneby the spin Hall torque,/contintegraltext
dt(dEs/dt), be-
comes zero in this case due to the symmetry of angular
dependence of the spin Hall torque. This means that the
spin Hall torque cannot compensate the damping torque,
and thus, a steady precession assumed in the linearized
LLG equation is not excited. This is evident in the nu-
merically calculated magnetization trajectory shown in
Fig. 5(b). For this case, the linearized LLG equation
gives|jc| → ∞, indicating that the spin Hall torque can-
not destabilize the magnetization. The same picture is
alsoapplicable, forexample, in the absenceofthe applied
field and β= 0.
However, an alternative mechanism can cause destabi-
lization of the magnetization. As schematically shown in
Figs. 6(a) and 6(c), there is a component of the damping
like spin Hall torque that is orthogonal to the damping
torque when ∆ Es= 0. The spin Hall torque at this pointis parallel or antiparallel to the field torque depending on
the position of the magnetization. When the spin Hall
torqueissufficientlylargerthanthefieldtorque,themag-
netization moves from its equilibrium position even if the
total energy supplied by the spin Hall torque is zero or
negative. This leads to an instability that occurs before
one precession finishes. In this case, it is expected that
the critical current is damping-independent because the
instability is induced as a competition between the spin
Hall torque and the field torque, not the damping torque.
The time evolution of the magnetization shown in Fig.
5 (b) represents such instability. The work reported in
Refs.14,49discusses a similar instability condition.
The above physical picture is also applicable in the
presence of the fieldlike torque. The fieldlike torque,
which acts like a torque due to the transversefield, modi-
fies the equilibrium direction ofthe ferromagnetand thus
the precession trajectory. Consequently, the amount of
energy supplied by the spin Hall torque and the dissipa-
tion due to damping is changed when the fieldlike torque
is present. Depending on the sign of β, the amount of the
work done by the spin Hall torque increases or decreases
compared to the case with β= 0. In our definition, posi-
tiveβcontributes to the increase of the supplied energy,
resulting in the reduction of the critical current. The
complex dependence of the critical current on αarises
when the fieldlike torque is present.
To summarize the discussion, the critical current be-
comes damping dependent when the energy supplied by
the spin Hall torque during a precession around the equi-
librium is positive. The condition that meets this criteria
depends on the relative direction of the spin Hall torque
and the damping torque, as briefly discussed above. To
derive an analytical formula that describes the condition
atwhichthe criticalcurrentbecomesdamping dependent
is not an easy task except for some limited cases53.
VI. CONCLUSION
In summary, we have studied the critical current
needed to destabilize a perpendicularly magnetized fer-
romagnet by the spin Hall effect. The Landau-Lifshitz-
Gilbert (LLG) equation that includes both the damping-
like and fieldlike torques associated with the spin Hall
effect is solved numerically and analytically. The criti-
cal current is found to have different dependence on the
damping constant, i.e., the critical current scales with α0
(damping-independent), α, andα1/2depending on the
sign of the fieldlike torque. The analytical formulas of
the damping-dependent critical current, Eqs. (19), (20),
and (21), are derived from the linearized LLG equation,
which explain well the numerical results. We find that
systems with fieldlike torque having the appropriate sign
(β >0 in our definition) are the most efficient way to re-
duce the criticalcurrent. Fortypicalmaterialparameters
found in experiment, the critical current can be reduced
to the order of 106A/cm2when ferromagnets with rea-9
sonable parameters are used.
ACKNOWLEDGMENTS
The authorsacknowledgeT. Yorozu, Y. Shiota, and H.
Kubota in AIST for valuable discussion sthey had with
us. This workwassupported by JSPS KAKENHIGrant-
in-AidforYoungScientists(B),GrantNo. 25790044,and
MEXT R & D Next-Generation Information Technology.
Appendix A: Initial state of the numerical
simulation
We assume that the magnetization in the absence of
the applied field points to the positive zdirection. In
the presence of the field, the equilibrium direction moves
from the zaxis to the xyplane. Let us denote the zenith
and azimuth angles of the initial state m(t= 0) asθand
ϕ, i.e.,m(t= 0) = (sin θcosϕ,sinθsinϕ,cosθ). When
the applied field points to the x-direction ( nH=ex), the
initial state is
/parenleftbigg
θ
ϕ/parenrightbigg
nH=ex=/parenleftBigg
sin−1[/radicalBig
H2
appl+(βHs)2/HK]
tan−1(βHs/Happl)/parenrightBigg
,(A1)
where the value of ϕis 0< ϕ < π/ 2 forHappl>0 and
βHs>0,π/2< ϕ < π forHappl<0 andβHs>0,π <
ϕ <3π/2forHappl<0andβHs<0,and3π/2< ϕ <2π
forHappl>0 andβHs<0. On the other hand, when
the applied field points to the y-direction ( nH=ey), the
initial state is
/parenleftbigg
θ
ϕ/parenrightbigg
nH=ey=/parenleftbigg
sin−1[(Happl+βHs)/HK]
π/2/parenrightbigg
,(A2)
where the range of the inverse sine function is −π/2≤
sin−1x≤π/2. We note that the choice of the initial
state does not affect the evaluation of the critical cur-
rent significantly, especially in the small field and current
regimes.
Appendix B: Numerically evaluated critical current
with different definition
As mentioned in Sec. III, the definition of the critical
current has arbitrariness. As an example, we show the
time evolution of mzunder the conditions of nH=ex,
Happl=−30 Oe,β= 0, and j= 110×106A/cm2in
Fig. 7. In this case, the magnetization initially starts at
mz= cos[sin−1(Happl/HK)]≃0.99, and finally moves to
a pointmz→0.12. Since the final state does not satisfy
Eq. (9), this current, j= 110×106A/cm2, should be
regarded as the current smaller than the critical current
in Sec. IV. However, from the analytical point of view,
this current can be regarded as the current larger than
magnetization 01
-1 -0.50.5
j=110×10 6 A/cm2
time (ns)0 2 4 6 8 10 Happl=-30 Oe
FIG. 7. Time evolution of the zcomponent of the mag-
netization mzin the presence of the longitudinal field with
Happl=−30 Oe,β= 0, and j= 110×106A/cm2. The
dotted line is a guide showing mz= 0.
the critical current because the final state of the magne-
tization is far away from the initial equilibrium.
Regarding this point, we show the numerically eval-
uated critical current with a different definition. The
magnetic state can be regarded as unstable when it fi-
nally arrives at a point far away from the initial state54.
Thus, for example, one can define the critical current as
a minimum current satisfying
lim
t→∞|mz(t)−mz(0)|> δ, (B1)
where a small positive real number δis chosen to be
0.1 here. Figure 8 summarizes the numerically evalu-
ated critical current with the definition of Eq. (B1). The
analytical formulas, Eqs. (19)-(21), still fit well with the
numerical results. The absolute values of the damping-
dependent critical current are slightly changed when the
definition of the critical current is changed. This is be-
cause Eq. (B1) is more easily satisfied than Eq. (9),
and thus the critical current in Fig. 8 is smaller than
that shown in Fig. 4. However, the main results of this
paper, such as the damping dependence of the critical
current, are not changed by changing the definition of
the critical current in the numerical simulations.
Appendix C: Energy change during a precession
As described in Sec. V, the linearized LLG equation
assumes a steady precession of the magnetization due to
the field torque when the current magnitude is close to
the critical current. This is because the spin Hall torque
compensates with the damping torque. Thus, Figs. 6(b)
and 6(d) are obtained by substituting the solution of m
precessing a constant energy curve of Einto Eqs. (7) and
(8).
When the transverse field is applied and β= 0, i.e.,
E=E, whereE=−M/integraltext
dm·H, the precession trajec-
tory on the constant energy curve of Eis given by55
mx(E) = (r2−r3)sn(u,k)cn(u,k),(C1)10
transverse magnetic field (Oe)critical current density (10 6 A/cm 2)
0 50 100 150 2000
-4020
-6040 60
β=2.0
-20
-100 -50 -150 -200: α=0.005: α=0.01: α=0.02
transverse magnetic field (Oe)critical current density (10 6 A/cm 2)
0 50 100 150 2000
-10050
-150100150
β=-2.0
-50
-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02(a) (b) (c)
(d) (e) (f)longitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)
0 50 100 150 2000
-4020
-5040 50
β=2.0
-20
-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02-30-1010 30
longitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)
0 50 100 150 2000
-10050
-150100150
β=0.0
-50
-100 -50 -150 -200: α=0.005: α=0.01: α=0.02
longitudinal magnetic field (Oe)0 50 100 150 200β=-2.0
-100 -50 -150 -200: α=0.005: α=0.01: α=0.02
transverse magnetic field (Oe)critical current density (10 6 A/cm 2)
0 50 100 150 2000
-10050
-150100150
β=0.0
-50
-100 -50 -150 -200: α=0.005: α=0.01: α=0.02
critical current density (10 6 A/cm 2)
0
-10050
-150100150
-50
FIG. 8. Numerically evaluated critical currents with a diffe rent definition, Eq. (B1), in the presence of (a)-(c) the long itudinal
(nH=ex) and (d)-(f) the transverse ( nH=ey) fields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0. The
solid lines are the analytically estimated critical curren t described in Sec. V.
my(E) =r3+(r2−r3)sn2(u,k),(C2)
mz(E) =/radicalBig
1−r2
3−(r2
2−r2
3)sn2(u,k),(C3)
whereu=γ/radicalbig
HtHK/2√r1−r3t, andrℓare given by
r1(E) =−E
MHappl, (C4)
r2(E) =Happl
HK+/radicalBigg
1+/parenleftbiggHappl
HK/parenrightbigg2
+2E
MHK,(C5)
r3(E) =Happl
HK−/radicalBigg
1+/parenleftbiggHappl
HK/parenrightbigg2
+2E
MHK.(C6)The modulus of Jacobi elliptic functions is k=/radicalbig
(r2−r3)/(r1−r3). The precession period is
τ(E) =2K(k)
γ/radicalbig
HapplHK/2√r1−r3,(C7)
whereK(k) is the first kind of complete elliptic inte-
gral. The initial state is chosen to be my(0) =r3. Fig-
ure 6(b) is obtained by substituting Eqs. (C1), (C2),
and (C3) into Eqs. (7) and (8). We note that Eqs.
(C1), (C2), and (C3) are functions of the energy den-
sityE. Since we are interested in the instability thresh-
old near the equilibrium, the value of Eis chosen close
to the minimum energy Emin. In Fig. 6 (b), we use
E=Emin+ (Emax−Emin)/NwithN= 100, where
the minimum energy Eminand the maximum energy
EmaxareEmin=−(MHK/2)[1 + (Happl/HK)2] and
Emax=−MHappl, respectively. The value of Happlis
chosen to be 50 Oe. Figure 6(d) is obtained in a similar
way.
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1810.10595v4.Nearly_isotropic_spin_pumping_related_Gilbert_damping_in_Pt_Ni___81__Fe___19___Pt.pdf | Nearly isotropic spin-pumping related Gilbert damping in Pt/Ni 81Fe19/Pt
W. Cao,1,L. Yang,1S. Au ret,2and W.E. Bailey1, 2,y
1Materials Science and Engineering, Department of Applied Physics and Applied Mathematics,
Columbia University, New York, New York 10027, USA
2SPINTEC, Universit eGrenoble Alpes/CEA/CNRS, F-38000 Grenoble, France
(Dated: July 12, 2021)
A recent theory by Chen and Zhang [Phys. Rev. Lett. 114, 126602 (2015)] predicts strongly
anisotropic damping due to interfacial spin-orbit coupling in ultrathin magnetic lms. Interfacial
Gilbert-type relaxation, due to the spin pumping e ect, is predicted to be signi cantly larger for
magnetization oriented parallel to compared with perpendicular to the lm plane. Here, we have
measured the anisotropy in the Pt/Ni 81Fe19/Pt system via variable-frequency, swept- eld ferromag-
netic resonance (FMR). We nd a very small anisotropy of enhanced Gilbert damping with sign
opposite to the prediction from the Rashba e ect at the FM/Pt interface. The results are contrary
to the predicted anisotropy and suggest that a mechanism separate from Rashba spin-orbit coupling
causes the rapid onset of spin-current absorption in Pt.
INTRODUCTION
The spin-transport properties of Pt have been studied
intensively. Pt exhibits ecient, reciprocal conversion
of charge to spin currents through the spin Hall e ect
(SHE)[1{4]. It is typically used as detection layer for
spin current evaluated in novel con gurations[5{7]. Even
so, consensus has not yet been reached on the experi-
mental parameters which characterize its spin transport.
The spin Hall angle of Pt, the spin di usion length of Pt,
and the spin mixing conductance of Pt at di erent inter-
faces di er by as much as an order of magnitude when
evaluated by di erent techniques[2, 3, 8{12].
Recently, Chen and Zhang [13, 14] (hereafter CZ) have
proposed that interfacial spin-orbit coupling (SOC) is
a missing ingredient which can bring the measurements
into greater agreement with each other. Measurements of
spin-pumping-related damping, particularly, report spin
di usion lengths which are much shorter than those es-
timated through other techniques[15, 16]. The introduc-
tion of Rashba SOC at the FM/Pt interface leads to
interfacial spin-memory loss, with discontinuous loss of
spin current incident to the FM/Pt interface. The model
suggests that the small saturation length of damping en-
hancement re
ects an interfacial discontinuity, while the
inverse spin Hall e ect (ISHE) measurements re
ect the
bulk absorption in the Pt layer[15, 16].
The CZ model predicts a strong anisotropy of the en-
hanced damping due to spin pumping, as measured in
ferromagnetic resonance (FMR). The damping enhance-
ment for time-averaged magnetization lying in the lm
plane ( pc-FMR, or parallel condition) is predicted to be
signi cantly larger than that for magnetization oriented
normal to the lm plane ( nc-FMR, or normal condition).
The predicted anisotropy can be as large as 30%, with
pc-FMR damping exceeding nc-FMR damping, as will be
shown shortly.
In this paper, we have measured the anisotropy of the
enhanced damping due to the addition of Pt in symmet-ric Pt/Ni 81Fe19(Py)/Pt structures. We nd that the
anisotropy is very weak, less than 5%, and with the op-
posite sign from that predicted in [13].
THEORY
We rst quantify the CZ-model prediction for
anisotropic damping due to the Rashba e ect at the
FM/Pt interface. In the theory, the spin-memory loss
for spin current polarized perpendicular to the interfa-
cial plane is always larger than that for spin current po-
larized in the interfacial plane. The pumped spin po-
larization=m_mis always perpendicular to the
time-averaged or static magnetization hmit'm. For
nc-FMR, the polarization of pumped spin current is
always in the interfacial plane, but for pc-FMR, is nearly
equally in-plane and out-of-plane. A greater damping
enhancement is predicted in the pccondition than in the
nccondition, pc> nc:
nc=Kh1 + 4(tPt)
1 +(tPt)i
(1)
pc=Kh1 + 6(tPt)
1 +(tPt)+
2[1 +(tPt)]2i
(2)
(tPt) =(1)coth(tPt=sd) (3)
where the constant of proportionality K is the same for
both conditions and the dimensionless parameters, and
, are always real and positive. The Rashba parameter
= ( RkF=EF)2(4)
is proportional to the square of the Rashba coecient
R, de ned as the strength of the Rashba potential,arXiv:1810.10595v4 [cond-mat.mtrl-sci] 22 Feb 20192
FIG. 1. Frequency-dependent half-power FMR linewidth
H1=2(!) of the reference sample Py(5 nm) (black) and sym-
metric trilayer samples Pt(t)/Py(5 nm)/Pt(t) (colored). (a)
pc-FMR measurements. (b) nc-FMR measurements. Solid
lines are linear ts to extract Gilbert damping . (Inset):
inhomogeneous broadening H0inpc-FMR (blue) and nc-
FMR (red).
V(r) = R(z)(^k^z), where(z) is a delta function
localizing the e ect to the interface at z= 0 ( lm plane
isxy),kFis the Fermi wavenumber, and EFis the Fermi
energy. The back
ow factor is a function of Pt layer
thickness, where the back
ow fraction at in nitely large
Pt thickness de ned as =(1)=[1 +(1)].= 0 (1)
refers to zero (complete) back
ow of spin current across
the interface. sdis the spin di usion length in the Pt
layer.
To quantify the anisotropy of the damping, we de ne
Q:
Q( pc nc)= nc (5)
as an anisotropy factor , the fractional di erence be-
tween the enhanced damping in pc and nc conditions.
Positive Q (Q >0) is predicted by the CZ model. A
spin-memory loss factor of 0.90.1, corresponding
to nearly complete relaxation of spin current at the in-
terface with Pt, was measured through current perpen-
dicular to plane-magnetoresistance (CPP-GMR)[8] Ac-
cording to the theory[13, 14], the spin-memory loss can
be related to the Rashba parameter by = 2, so we
take0:45. The e ect of variable < 0:45 will be
shown in Figure 3. To evaluate the thickness dependent
back
ow(tPt), we assume Pt
sd= 14 nm, which is asso-
ciated with the absorption of the spin current in the bulk
of Pt layer, as found from CPP-GMR measurements[8]
and cited in [13]. Note that this Pt
sdis longer than that
used sometimes to t FMR data[15, 16]; Rashba interfa-
cial coupling in the CZ model brings the onset thickness
down. The calculated anisotropy factor Q should then
FIG. 2. Pt thickness dependence of Gilbert damping =
(tPt) inpc-FMR (blue) and nc-FMR (red). 0refers to the
reference sample ( tPt= 0). (Inset): Damping enhancement
(tPt) = (tPt) 0due to the addition of Pt layers in
pc-FMR (blue) and nc-FMR (red). Dashed lines refer to cal-
culated ncusing Equation 1 by assuming Pt
sd= 14 nm
and= 10%. The red dashed line ( = 0:15) shows a similar
curvature with experiments; The black dashed line ( 0:25)
shows a curvature with the opposite sign.
be as large as 0.3, indicating that pcis 30% greater
than nc(see Results for details).
EXPERIMENT
In this paper, we present measurements of the
anisotropy of damping in the symmetric Pt( tPt)/Py(5
nm)/Pt(tPt) system, where \Py"=Ni 81Fe19. Because
the Py thickness is much thicker than its spin coher-
ence length[17], we expect that spin-pumping-related
damping at the two Py/Pt interfaces will sum. The
full deposited stack is Ta(5 nm)/Cu(5 nm)/Pt( tPt)/Py(5
nm)/Pt(tPt)/Al 2O3(3 nm),tPt= 1{10 nm, deposited
via DC magnetron sputtering under computer control on
ion-cleaned Si/SiO 2substrates at ambient temperature.
The deposition rates were 0.14 nm/s for Py and 0.07
nm/s for Pt. Heterostructures deposited identically, in
the same deposition chamber, have been shown to exhibit
both robust spin pumping e ects, as measured through
FMR linewidth[18, 19], and robust Rashba e ects (in
Co/Pt), as measured through Kerr microscopy[20, 21].
The stack without Pt layers was also deposited as the ref-
erence sample. The lms were characterized using vari-
able frequency FMR on a coplanar waveguide (CPW)
with center conductor width of 300 m. The bias mag-
netic eld was applied both in the lm plane ( pc) and
perpendicular to the plane ( nc), as previously shown in
[22]. The nc-FMR measurements require precise align-
ment of the eld with respect to the lm normal. Here,3
FIG. 3. Anisotropy factor Q for spin-pumping enhanced damping, de ned in Equation 5. Solid lines are calculations using the
CZ theory[13], Equations 1{3, for variable Rashba parameter 0 :010:45.Pt
sdis set to be 14 nm. Back
ow fraction is
set to be 10% in (a) and 40% in (b). Black triangles, duplicate in (a) and (b), show the experimental values from Figure 2.
samples were aligned by rotation on two axes to maxi-
mize the resonance eld at 3 GHz.
RESULTS AND ANALYSIS
Figure 1 shows frequency-dependent half-power
linewidth H1=2(!) in pc- and nc-FMR. The measure-
ments were taken at frequencies from 3 GHz to a cut-o
frequency above which the signal-to-noise ratio becomes
too small for reliable measurement of linewidth. The
cuto ranged from 12{14 GHz for the samples with Pt
(linewidth200{300 G) to above 20 GHz for tPt= 0.
Solid lines stand for linear regression of the variable-
frequency FMR linewidth H1=2= H0+2 !=
, where
H1=2is the full-width at half-maximum, H0is the in-
homogeneous broadening, is the Gilbert damping, !
is the resonance frequency and
is the gyromagnetic ra-
tio. The ts show good linearity with frequency !=2for
all experimental linewidths H1=2(!). The inset sum-
marizes inhomogeneous broadening H0inpc- and nc-
FMR; its errorbar is 2 Oe.
In Figure 2, we plot Pt thickness dependence of damp-
ing parameters (tPt) extracted from the linear ts in
Figure 1, for both pc-FMR and nc-FMR measurements.
Standard deviation errors in the ts for are310 4.
The Gilbert damping saturates quickly as a function
oftPtin both pc and nc conditions, with 90% of the ef-
fect realized with Pt(3 nm). The inset shows the damp-
ing enhancement due to the addition of Pt layers = 0, normalized to the Gilbert damping 0of
the reference sample without Pt layers. The Pt thickness
dependence of matches our previous study on Py/Pt
heterostructures[19] reasonably; the saturation value of
Pt=Py=Pt is 1.7x larger than that measured for the
single interface Py=Pt [19] (2x expected). The dashed
lines in the inset refer to calculated ncusing Equation
1 (assuming Pt
sd= 14 nm and = 10%).= 0:25 shows
a threshold of Pt thickness dependence. When >0:25,
the curvature of (tPt) will have the opposite sign to
that observed in experiments, so = 0:25 is the maxi-
mum which can qualitatively reproduce the Pt thickness
dependence of the damping.
As shown in Figure 2 inset, the damping enhancement
due to the addition of Pt layers is slightly larger in the
ncgeometry than in the pcgeometry: nc> pc.
This is opposite to the prediction of the model in [13].
The anisotropy factor Q( pc nc)= ncfor the
model (Q>0) and the experiment (Q <0) are shown to-
gether in Figure 3 (a) and (b). The magnitude of Q
for the experiment is also quite small, with -0.05 <Q<0.
This very weak anisotropy, or near isotropy, of the spin-
pumping damping is contrary to the prediction in [13],
and is the central result of our paper.
The two panels (a) and (b), which present the same
experimental data (triangles), consider di erent model
parameters, corresponding to negligible back
ow ( =
0:1, panel a) and moderate back
ow ( = 0:4, panel b)
for a range of Rashba couplings 0 :010:45. A spin
di usion length sd= 14 nm for Pt[8] was assumed in all4
cases.
The choice of back
ow fraction = 0:1 or 0:4 and the
choice of spin di usion length of Pt sd= 14 nm follow
the CZ paper[13] for better evaluation of their theory.
For good spin sinks like Pt, the back
ow fraction is usu-
ally quite small. If = 0, then there will be no spin
back
ow. In this limit, pc, ncand the Q factor
will be independent of Pt thickness.
In the case of a short spin di usion length of Pt, e.g.,
sd= 3 nm, the anisotropy Q as a function of Pt thick-
ness decreases more quickly for ultrathin Pt, closer to
our experimental observations. However, we note that
the CZ theory requires a long spin di usion length in or-
der to reconcile di erent experiments, particularly CPP-
GMR with spin pumping, and is not relevant to evaluate
the theory in this limit.
Leaving apart the question of the sign of Q, we can see
that the observed absolute magnitude is lower than that
predicted for = 0:05 for small back
ow and 0.01 for
moderate back
ow. According to ref [13], a minimum
level for the theory to describe the system with strong
interfacial SOC is = 0:3.
DISCUSSION
Here, we discuss extrinsic e ects which may result in
a discrepancy between the CZ model (Q +0.3) and our
experimental result (-0.05 <Q<0). A possible role of two-
magnon scattering[23, 24], known to be an anisotropic
contribution to linewidth H1=2, must be considered.
Two-magnon scattering is present for pc-FMR and nearly
absent for nc-FMR. This mechanism does not seem to
play an important role in the results presented. It is
dicult to locate a two-magnon scattering contribution
to linewidth in the pure Py lm: Figure 1 shows highly
linear H1=2(!), without o set, over the full range to
!=2= 20 GHz, thereby re
ecting Gilbert-type damp-
ing. The damping for this lm is much smaller than
that added by the Pt layers. If the introduction of Pt
adds some two-magnon linewidth, eventually mistaken
for intrinsic Gilbert damping , this could only produce
a measurement of Q >0, which was not observed.
One may also ask whether the samples are appropriate
to test the theory. The rst question regards sample qual-
ity. The Rashba Hamiltonian models a very abrupt inter-
face. Samples deposited identically, in the same deposi-
tion chamber, have exhibited strong Rashba e ects, so we
expect the samples to be generally appropriate in terms
of quality. Intermixing of Pt in Ni 81Fe19(Py)/Pt[25] may
play a greater role than it does in Co/Pt[26], although
defocused TEM images have shown fairly well-de ned in-
terfaces for our samples[27].
A second question might be about the magnitude of
the Rashba parameter in the materials systems of in-
terest. Our observation of nearly isotropic damping isconsistent with the theory, within experimental error and
apart from the opposite sign, if the Rashba parameter is
very low and the back
ow fraction is very low. Ab-initio
calculations for (epitaxial) Co/Pt in the ref[28] have in-
dicated= 0.02{0.03, lower than the values of 0.45
assumed in [13, 14] to treat interfacial spin-memory loss.
The origin of the small, negative Q observed here is un-
clear. A recent paper has reported that pcis smaller
than ncin the YIG/Pt system via single-frequency,
variable-angle measurements[7], which is contrary to the
CZ model prediction as well. It is also possible that a
few monolayers of Pt next to the Py/Pt interfaces are
magnetized in the samples[19], and this may have an un-
known e ect on the sign, not taken into account in the
theory.
CONCLUSIONS
In summary, we have experimentally demonstrated
that in Pt/Py/Pt trilayers the interfacial damping at-
tributed to spin pumping is nearly isotropic, with an
anisotropy between lm-parallel and lm-normal mea-
surements of <5%. The nearly isotropic character of the
e ect is more compatible with conventional descriptions
of spin pumping than with the Rashba spin-memory loss
model predicted in [13].
ACKNOWLEDGEMENTS
We acknowledge support from the US NSF-DMR-
1411160 and the Nanosciences Foundation, Grenoble.
wc2476@columbia.edu
yweb54@columbia.edu
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1605.08965v3.Damped_Infinite_Energy_Solutions_of_the_3D_Euler_and_Boussinesq_Equations.pdf | DAMPED INFINITE ENERGY SOLUTIONS OF THE 3D EULER
AND BOUSSINESQ EQUATIONS
WILLIAM CHEN AND ALEJANDRO SARRIA
Abstract. We revisit a family of in nite-energy solutions of the 3D incompressible Euler
equations proposed by Gibbon et al. [9] and shown to blowup in nite time by Constantin
[6]. By adding a damping term to the momentum equation we examine how the damping
coecient can arrest this blowup. Further, we show that similar in nite-energy solutions of
the inviscid 3D Boussinesq system with damping can develop a singularity in nite time as
long as the damping e ects are insucient to arrest the (undamped) 3D Euler blowup in
the associated damped 3D Euler system.
1.Introduction
We consider a family of exact in nite-energy solutions of two three-dimensional (3D)
uid
models with a damping term, the incompressible Euler equations
(1)(
ut+uru+ u= rp;
divu= 0;
and the inviscid Boussinesq system
(2)8
><
>:ut+uru+ u= rp+e3;
t+ur= 0;
divu= 0:
In (1)-(2), urepresents the
uid velocity, pis the scalar pressure, is the scalar temperature
in the context of thermal convection or the density in the modeling of geophysical
uids,
e3= (0;0;1)T, and 2R+is a real parameter. For = 0, (1) reduces to the standard 3D
Euler equations describing the motion of an ideal, incompressible homogeneous
uid, while
(2) becomes the standard 3D inviscid Boussinesq system modeling large scale atmospheric
dynamics and oceanic
ows [11, 16, 19]. If 0, (2) reduces to (1). When uin (1) is
replaced by the di usion term u, we obtain the classical 3D Navier-Stokes equations.
The global (in time) regularity problem for the aforementioned 3D models are long-standing
open problems in mathematical
uid dynamics. See Constantin [7] for a history and survey
of results on the 3D Euler regularity problem and Fe erman [8] for a more precise account
of the Navier-Stokes regularity problem. In general, the main obstacle in obtaining global
existence of smooth solutions of these 3D models for general initial data is controlling non-
linear growth due to vortex stretching [14, 12]. To gain insight into this challenging problem,
2010 Mathematics Subject Classi cation. 35B44, 35B65, 35Q31, 35Q35.
Key words and phrases. 3D Euler; 3D Boussinesq; Blowup; Damping; In nite-energy solutions.
1arXiv:1605.08965v3 [math.AP] 30 Jun 20162 WILLIAM CHEN AND ALEJANDRO SARRIA
many researchers have turned their e orts to the 2D viscous Boussinesq equations
(3)8
><
>:ut+uru= rp+u+e2;
t+ur=;
divu= 0:
System (3) can be shown to be formally identical to the 3D Euler or Navier-Stokes equations
for axisymmetric swirling
ows and retains key features of the 3D models such as the vortex
stretching mechanism (see, e.g., [17]). The global regularity issue for (3) has been settled in
the armative under various degrees of viscosity and dissipation: with full viscosity >0
and >0, partial viscosity >0 and= 0, or= 0 and >0, for anisotropic models
[1, 4, 13, 15], and with fractional Laplacian dissipation (see [23] and references therein).
In contrast, the question of global regularity for (3) in the inviscid case == 0 remains
open; it is not apparent how to control vortex stretching when there is no dissipation ( = 0)
and no thermal di usion ( = 0). Using a somewhat di erent approach, Adhikaria et al.
[2] replaced uandin (3) with damping terms uand for > 0 and > 0
real parameters. Although the resulting damping e ects are insucient to control vortex
stretching for general initial data, the authors showed that a local (in time) solution will
persist globally in time if the initial data is small enough in some homogeneous Besov space.
The aim of this work is to examine how damping a ects the global regularity of a particular
class of in nite-energy solutions of (1) and (2) which, in the absence of damping ( = 0),
blowup in nite-time from smooth initial data. More particularly, the
uid velocity and
temperature considered here have the form
(4) u(x;z;t) = (u(x;t);v(x;t);z
(x;t)); (x;z;t) =z(x;t)
for (x;z) = (x;y;z ). Our spatial domain will be the semi-bounded 3D channel
(5) f(x;z)2QRg
of rectangular periodic cross-section Q[0;1]2withuandboth periodic in the xandy
variables with period one. Note that the unbounded geometry of (5) in the vertical direction
endows the
uid under consideration with, at best, locally nite kinetic energy.
Solutions of the 3D incompressible Euler equations
(6)(
ut+uru= rp
divu= 0
of the form (4)i) were proposed by Gibbon et al. [9], and then shown to blowup in nite
time numerically by Ohkitani and Gibbon [18], and analytically by Constantin [6]. See also
[5, 22, 20, 21] for blowup results of other similar in nite-energy solutions of the Euler and
Boussinesq equations in two and three dimensions. For convenience of the reader, we now
summarize Constantin's blowup result.
The set-up used by Constantin is e ectively the same as the one considered here. Imposing
a velocity eld of the form (4)i) on (6) subject to periodicity in the xandyvariables of period
one and with spatial domain (5), it follows that the vertical component of the velocity eld,
z
, satis es the vertical component of the momentum equation (6)i) if the mean-zero functionDAMPED INFINITE ENERGY SOLUTIONS OF THE 3D EULER AND BOUSSINESQ EQUATIONS 3
= (ux+vy) solves the nonlocal two-dimensional equation
(7)
t+u0r
=
2+ 2Z
Q
2dx
withu0= (u;v). For ( a;t)(a1;a2;t),a2Q, Constantin constructed the solution formula
(Y(a;t);t) = 0(t)
(t)(
1
1 +
0(a) (t) Z
Qda
1 +
0(a) (t) 1Z
Qda
(1 +
0(a) (t))2)
(8)
for
(a;0) =
0(a), satisfying the initial value problem (IVP)
(9) 0(t) =Z
Qda
1 +
0(a) (t) 2
; (0) = 0;
anda!Y(a;t) the 2D
ow-map de ned by
dY
dt=u0(Y(a;t);t); Y(a;0) =a:
By comparing the blowup rates of the time integrals in (8) against the local term, Constantin
proved the following blowup result for a large class of smooth initial data
0.
Theorem 1.1 ([6]).Consider the initial boundary value problem for (7)with smooth mean-
zero initial data
0and periodic boundary conditions. Suppose
0attains a negative minimum
m0at a nite number of locations a02Qand, near these locations,
0has non-vanishing
second-order derivatives. Set = 1
m0and let
(10) tE( ) =Z
0Z
Qda
1 +
0(a)2
d:
Then there exists a nite time TE>0, given by
TElim
% tE( );
such that both the maximum and minimum values of
diverge to positive and respectively
negative in nity as t%TE.
The outline for the remainder of the paper is as follows. In Section 2 we introduce the
damped two-dimensional equations (12)-(14) and summarize the main results of the paper.
Then in Section 3 we derive the solution formulae (33)-(37), which we use in Section 4 to
prove the Theorems.
2.Preliminaries
2.1.The Damped Two-dimensional Equations.
As stated in the previous section, we are interested in the global regularity of solutions of
(1) and (2) of the form (4) subject to periodic boundary conditions in the xandyvariables
(period one), and with spatial domain (5). First note that, from incompressibility and
periodicity,
= (ux+vy) satis es the mean-zero condition
(11)Z
Q
(x;t)dx= 0:4 WILLIAM CHEN AND ALEJANDRO SARRIA
Then imposing the ansatz (4) on the damped Boussinesq system (2), it is easy to check that
the vertical component of the velocity eld, z
, and the scalar temperature, z, satisfy the
vertical component of (2)i) and equation (2)ii) if
andsolve the nonlocal 2D system
(12)(
t+u0r
=
2
+I(t);x2Q; t> 0;
t+u0r=
; x2Q; t> 0
withu0= (u;v), >0 a real parameter, and
(13) I(t) = 2Z
Q
2(x;t)dx Z
Q(x;t)dx:
For0, (12)-(13) reduces to
(14)(
t+u0r
=
2
+I(t);x2Q; t> 0;
I(t) = 2R
Q
2(x;t)dx;
which is just the associated 2D equation obtained from the vertical component of the damped
3D Euler system (1).
For simplicity we will refer to equations (7) and (14), and the system (12)-(13), as the
undamped Euler equation ,the damped Euler equation , and the damped Boussinesq system ,
respectively.
Before summarizing the main results of the paper, we de ne some notation that will be
helpful in di erentiating among solutions of the various equations under consideration.
2.2.Notation.
For > 0, we denote by (
B
; ) and
E
the solution of the damped Boussinesq sys-
tem (12)-(13) and the damped Euler equation (14), respectively. The undamped ( = 0)
counterparts of (
B
; ) and
E
will be denoted by dropping the subscript, i.e., (
B;) and
respectively
E, with the latter given (along characteristics) by formula (8). Other notation
will be introduced in later sections in a similar manner. Lastly, by Cwe mean a generic
positive constant that may change in value from line to line.
2.3.Summary of Results.
For a smooth initial condition
0satisfying the conditions of Theorem 1.1, we determine
in Theorem 2.1 below \how much" damping is required for the solution
E
of the damped
Euler equation (14) to persist globally in time, or alternatively, for the nite-time blowup of
the solution
Eof the undamped Euler equation (7) to be suppressed.
Theorem 2.1. Consider the damped Euler equation (14) with smooth mean-zero initial data
0and periodic boundary conditions. Suppose
0satis es the conditions of Theorem 1.1, so
the solution
Eof the undamped Euler equation (7)blows up at a nite time TE. Then for
1=TE, the solution
E
of the damped Euler equation (14) exists globally in time. More
particularly, for = 1=TE,
E
converges to a non-trivial steady state as t!+1, whereas,
for > 1=TE, convergence is to the trivial steady state. In contrast, if 0< < 1=TE,
then there exists a nite time TE
> TEsuch that the maximum and minimum values of
E
diverge to positive and respectively negative in nity as t%TE
. More particularly, let
= 1=m 0form0the negative minimum of
0attained at a nite number of points in Q,DAMPED INFINITE ENERGY SOLUTIONS OF THE 3D EULER AND BOUSSINESQ EQUATIONS 5
and settE
( ) = 1
ln 1 tE( ) fortE( )the undamped Euler time variable (10). Then
for0< < 1=TE, the nite blowup time is TE
lim % tE
( ).
Before summarizing our next result, we note that Gibbon and Ohkitani [10] established a
regularity criterion of BKM [3] type whereby a solution
Eof (7) blows up at a nite time
TEif and only if
(15)ZTE
0k
E(x;s)k1ds= +1
forkk1the supremum norm. Let ( u;p) be a solution of the damped Euler system (1) for
uas in (4)i). Since the vertical component !=vx uyof the vorticity != curl usatis es
!t+u0r!= (
E
)!;
we may refer to
E
as the vorticity stretching rate. In the spirit of (15), a similar BKM-type
criterion can be established for a solution of the damped Euler equation (14) where the
blowup time TE
is de ned as the smallest time at which
ZTE
0k
E
(x;s)k1ds= +1:
Additionally, the argument used to prove Theorem 1.1 in [21], together with Theorem 2.2
below, shows that the same regularity criterion1is true for a solution of the damped Boussi-
nesq system (12)-(13). Part one of our next Theorem shows that for smooth
0satisfying
the conditions of Theorem 1.1 and 00, the existence of a nite blowup time TEfor the
undamped Euler equation (7) leads to nite-time blowup of the damped Boussinesq system
(12)-(13) if the damping coecient satis es 0 < < 1=TEand there is at least one point
a12Qsuch that
0(a1) =m0and0(a1) = 0. In particular, we nd that the time inte-
gral of
B
diverges to negative in nity fora=a1, in agreement with the aforementioned
BKM-type criterion. The second part of Theorem 2.2 shows that nite-time blowup is not
restricted to nonnegative 0, although the blowup mechanism we uncover for nonpositive 0
is of a di erent nature and the singularity is e ectively one dimensional. Brie
y, we show
the existence of smooth
0attaining its negative minimum m0at in nitely many points
a02Q, and00 satisfying 0(a0)6= 0, such that for TB>0 the nite blowup time of the
associated undamped Boussinesq system (i.e. (12)-(13) with = 0), if 0< < 1=TB, then
the time integral of
B
diverges to positive in nity fora6=a0.
Theorem 2.2. Consider the damped Bousinesq system (12)-(13) with periodic boundary
conditions.
(1)Suppose
0satis es the conditions of Theorem 1.1, so the solution
Eof the undamped
Euler equation (7)blows up at a nite time TE. Further, let 0< < 1=TE, so that
by Theorem 2.1 the solution
E
of the damped Euler equation (14) blows up at a
nite time TE
. Assume
0attains its negative minimum m0at nitely many points
ai2Q,1in, and00is smooth with 0(aj) = 0 for some 1jn. Then
there exists a nite time TB
, satisfying 0<TB
<TE
, such that
J(aj;t) = exp
Zt
0
B
(X(aj;s);s)ds
!+1
1With
E
replaced by
B
6 WILLIAM CHEN AND ALEJANDRO SARRIA
ast%TB
forJ= det@X
@a
the Jacobian of the 2D
ow-map (see (16)).
(2)There exist smooth mean-zero initial data
0attaining its negative minimum m0at
in nitely many points a02Q, and smooth 00with0(a0)6= 0, such that for
TB>0the nite blowup time of the associated undamped Boussinesq system, if
0< < 1=TBanda6=a0, then
J(a;t) = exp
Zt
0
B
(X(a;s);s)ds
!0
ast%TB
= 1
ln
1 TB
>TB.
3.Solution along characteristics
In this section we derive the representation formula (33)-(34) for the general solution
(
B
(x;t); (x;t)) (along characteristics) of the damped Boussinesq system (12)-(13). Our
approach is a natural generalization to higher dimensions of the arguments in [20, 21], and
is somewhat more direct than the argument used in [6] to derive (8)-(9).
For (a;t)(a1;a2;t),a2Q, and u0= (u;v), de ne the 2D
ow-map a!X(a;t) as the
solution of the IVP
(16)dX
dt=u0(X(a;t);t); X(a;0) =a:
Integrating equation (12)ii) along the
ow-map yields
(17) (X(a;t);t) =0(a) exp
Zt
0
B
(X(a;s);s)ds
:
Note that
(18) (X(a;t);t) =0(a)J(a;t)
forJ(a;t) = det@X
@a
the Jacobian determinant of Xsatisfying
(19) J(a;t) = exp
Zt
0
B
(X(a;s);s)ds
:
Formula (18) follows directly from (17) and the fact that the Jacobian satis es
(20) Jt(a;t) = J(a;t)
B
(X(a;t);t); J (a;0)1:
Next we use the above formulas to derive a second-order linear ODE for =J 1.
From (12)i), (18), and (20), it follows that
@
@t(
B
(X(a;t);t)) =0(a)J(a;t) Jt(a;t)
J(a;t)2
+ Jt(a;t)
J(a;t)+I(t): (21)
Then di erentiating (20) with respect to t, and using (20) and (21) on the resulting expression
leads to
Jtt=
2J2
t
J2 0J Jt
J I(t)
J: (22)
Setting=J 1in (22) yields, after some rearranging,
(23) tt(a;t) + t(a;t) I(t)(a;t) =0(a);DAMPED INFINITE ENERGY SOLUTIONS OF THE 3D EULER AND BOUSSINESQ EQUATIONS 7
a second-order linear ODE parametrized by a2Q. Fixa2Qand setf(t) =(a;t). Then
by (20) and (23), fsolves the IVP
(24) f00(t) + f0(t) I(t)f(t) =0; f (0) = 1; f0(0) =
0(a)
for0d
dt. To nd a general solution of (24) we follow a standard variation of parameters
argument. Consider the associated homogeneous equation
(25) f00
h(t) + f0
h(t) I(t)fh(t) = 0:
Let 1(t) and 2(t) be two linearly independent solutions of (25) satisfying 1(0) = 0
2(0) = 1
and 0
1(0) = 2(0) = 0. Then by reduction of order, the general solution of (25) takes the
formfh(t) = 1(t)(c1(a) +c2(a) (t)) for
(t) =Zt
0e s
2
1(s)ds:
Now consider a particular solution fp(t) =v1(a;t) 1(t) +v2(a;t) 2(t) of (24) for 2(t) =
(t) 1(t) andv1andv2to be determined. After some standard computations, we can write
the general solution of (24) as
(26) (a;t) = 1(t) [ (a;t) 0(a)(t)]
for
(a;t) = 1 +
0(a) (t) (27)
and
(t) =Zt
0e s (s) 1(s)ds (t)Zt
0e s 1(s)ds
= Zt
0Zt
s 1(s)
2
1(z)e (s z)dzds:(28)
The Jacobian is now obtained from (26) and =J 1as
(29) J(a;t) = 1
1(t)
(a;t) 0(a)(t):
To nd 1note that for xed a2Qandc2Z2, the IVP
(30) Z0(t) =u0(Z(t);t); Z(0) =a+c
has a unique solution as long as u0= (u;v) stays smooth. Then by periodicity of u0and (16),
Z1(t) =X(a;t) +candZ2(t) =X(a+c;t) both solve (30) with the same initial condition.
Thus X(a;t) +c=X(a+c;t), which implies that
(31)Z
QJ(a;t)da1:
Integrating (29) now yields
(32) 1(t) =Z
Qda
(a;t) 0(a)(t):
Fori2Z+, set
Ki(a;t) =
0(a)
( (a;t) 0(a)(t))i; Ki(t) =Z
Q
0(a)
( (a;t) 0(a)(t))ida8 WILLIAM CHEN AND ALEJANDRO SARRIA
and
Li(a;t) =0(a)
( (a;t) 0(a)(t))i; Li(t) =Z
Q0(a)
( (a;t) 0(a)(t))ida:
Then using (18), (20), (29) and (32) we obtain, after a lengthy but straight-forward compu-
tation, the solution formula
B
(X(a;t);t) = 0
(t)
K1(a;t) K2(t)
1(t)
+
L1(a;t) L2(t)
1(t)Zt
0e s 1(s)ds (33)
for (t) satisfying
(34) 0
(t) =e t
2
1(t); (0) = 0:
Moreover, the Jacobian (29) becomes
(35) J(a;t) =1
(a;t) 0(a)(t)Z
Qda
(a;t) 0(a)(t) 1
:
Since (18) implies that if 00, then 0 for as long as the solution is de ned, setting
00 in (33)-(34) yields, after some rearranging, the general solution of the damped Euler
equation (14) as
E
(X(a;t);t) = 0
(t)
(t)(
1
1 +
0(a) (t) Z
Qda
1 +
0(a) (t) 1Z
Qda
(1 +
0(a) (t))2)(36)
for
(37) 0
(t) =e tZ
Qda
1 +
0(a) (t) 2
; (0) = 0:
4.Proof of the Main Theorems
The blowup result in Theorem 1.1 for the solution
Eof the undamped Euler equation (7)
is established by estimating blowup rates for the integral terms in (8) under the assumption
that the smooth initial data
0behaves quadratically near the points where its minimum is
attained [6]. Since we are interested in how damping can arrest this blowup, we consider
the same class of initial data. In particular, this means that the blowup rates derived in [6]
for the integral terms also hold here; however, and for convenience of the reader, we outline
how to obtain these estimates in the proof of Theorem 2.1 below.
Proof of Theorem 2.1 .Suppose the mean-zero initial data
0(a) is smooth and attains
its negative minimum m0at a nite number of locations a02Q. Then the spatial term
(38)1
1 +
0(a) (t)
in (36) diverges to positive in nity when a=a0as approaches
= 1
m0;DAMPED INFINITE ENERGY SOLUTIONS OF THE 3D EULER AND BOUSSINESQ EQUATIONS 9
and remains nite and positive for all 0 anda6=a0. Suppose
0has nonzero
second-order partials near a0, so that locally,
m0+1
22ja a0j2
0(a)m0+1
21ja a0j2
for 0ja a0jr,r>0 small, and 1> 2>0 the eigenvalues of the Hessian matrix of
0ata0.2This implies that
(39) C1ln
1 +1r2
2
Z
Dda
+
0(a) m0C2ln
1 +2r2
2
for >0 small,Ci=2
i>0,i= 1;2, andDthe disk centered at a0of radiusr. Setting
=1
+m0in (39), it follows that
(40)Z
Qda
1 +
0(a) Cln( )
for >0 small. By a similar argument,
(41)Z
Qda
(1 +
0(a) )2C
:
Next we need the corresponding behavior of the exponential term
(42) e t= exp
tE
( )
in (36)-(37) as % . In (42) we have introduced the notation t=tE
to di erentiate the
time variable in (36)-(37) from that in (33)-(34). For > 0, we will refer to tE
andtB
as
the damped Euler time variable and respectively the damped Boussinesq time variable .
Since
0satis es the conditions of Theorem 1.1, the limit TElim ! tE( ), for
(43) tE( ) =Z
0Z
Qda
1 +
0(a)2
d
the associated undamped Euler time variable (10), is positive and nite, and represents the
blowup time for
Ein Theorem 1.1. That TEis nite follows from (40) and (43) which
imply, for >0 small, the asymptotic relation
(44) TE tE( )(ln( ) 1)2
whose right-hand side vanishes as % .
Since (37) implies that the damped Euler time variable satis es
(45) tE
( ) = 1
ln 1 tE( ) ;
it follows that the behavior of the exponential (42) is determined by the limit
(46) TE
lim
% tE
( );
which in turn depends on whether 0 < < 1=TEor 1=TE.
2If1=2=, then near a0,
0m0+1
2ja a0j2and the blowup rates (40)- (41) still hold.10 WILLIAM CHEN AND ALEJANDRO SARRIA
Case 1 - Finite-time blowup for 0< < 1=TE.
For 0< < 1=TE, the argument of the logarithm in (45) satis es 0 <1 tE( )1
for all 0 . This implies that the limits
lim
% tE
( ) =TE
; lim
% exp
tE
( )
= exp
TE
are both positive and nite. Using estimates (40) and (41) on (36), it follows that for a=a0
the spatial term dominates and
E
diverges to negative in nity,
E
(X(a0;t);t) C
( ) ln2( )! 1
ast%TE
. If instead a6=a0, the second term in the bracket of (36) now dominates and
the blowup is to positive in nity,
E
(X(a;t);t) C
( ) ln3( )!+1:
From (45) note that the blowup time TE
approaches the undamped blowup time TEas the
damping coecient vanishes. Further, for >0 and 0, we have that tE
( )tE( )
with equality only at = 0. ThusTE
>TE.
Case 2 - Convergence to a nontrivial steady state for = 1=TE.
For = 1=TE,
(47) lim
% tE
( ) = TElim
% 1 tE( )
TE = +1
and the exponential (42) vanishes,
lim
% exp
tE
( )
= 0:
To determine how fast we use (44) and (45) to obtain
(48) exp
tE
( )
=1
TE(TE tE( ))C( )(ln( ) 1)2
for >0 small. Using (40)-(41) and (47)-(48) on (36), we see that
E
converges to a
mean-zero nontrivial steady state as t!+1,
lim
t!+1
E
(X(a;t);t) =(
C; a=a0;
0; a6=a0:
Case 3 - Convergence to the trivial steady state for >1=TE.
For > 1=TE, (45) implies the existence of 0 < 1< such thattE( )%1
< TEas
% 1. ThentE
( )!+1as % 1, and
lim
% 1exp
tE
( )
= 0:
Since the spatial and integral terms in (36) stay positive and nite for 0 1, it follows
that
E
(x;t)!0 ast!+1for all x2Q.
DAMPED INFINITE ENERGY SOLUTIONS OF THE 3D EULER AND BOUSSINESQ EQUATIONS 11
Before proving Theorem 2.2 we establish the following Lemma.
Lemma 4.1. Let0(x)0and set = 1=m 0. If0 (t)< for allt2[0;T),
0<T+1, then 0< 1(t)<+1on.
Proof. Suppose0(a)0 and 0 (t)< for allt2[0;T), 0<T+1. The latter
and (27) imply that ( a;t) = 1 +
0(a) (t)>0 for all ( a;t)2Q. Since 1(t) satis es
1(0)>0, there exists, by continuity, a positive t12 such that 0 < 1(t)<+1for all
t2[0;t1). Note that 1(t) cannot diverge to positive in nity at t1. Indeed, suppose
(49) lim
t%t1 1(t) = +1:
Since0(a)0, (a;t)>0 for all ( a;t)2Q[0;t1], and(t)0 on [0;t1) (by (28)ii)),
(a;t) 0(a)(t) (a;t)>0
for all ( a;t)2Q[0;t1), and thus
(50) 1
1(t)Z
Qda
(a;t) 1
>0
fort2[0;t1). From (49) and (50) it follows that
lim
t%t1Z
Qda
(a;t)= +1;
contradicting ( a;t)>0 for all ( a;t)2Q.
Now suppose there exists t22 such that 1(t) vanishes as t%t2, namely,
(51) lim
t%t2Z
Qda
(a;t) 0(a)(t)= 0:
Since0(a)0 and 0< 1(t)<+1on [0;t2),(t)0 is continuous on [0 ;t2). Then
boundedness of on [0;t2] and ( a;t) 0(a)(t)>0 for all ( a;t)2Q[0;t2) imply that,
for (51) to hold, (t)! 1 ast%t2. From this and (28)i) it follows that
lim
t%t2
(t)Zt
0e s 1(s)ds
= +1;
or since 0 1<+1on [0;t2], (t)!+1ast%t2, contradicting 0 < on .
Proof of Theorem 2.2 .Suppose
0(x) satis es the conditions in Theorem 1.1 and 0(x)
0 for all x2Q. Further, and without loss of generality, assume there is one location a12Q
such that0(a1) = 0 and
0(a1) =m0. Set = 1=m 0. Then Lemma 4.1 and formulas
(28) and (35) imply that, as long as 0 < ,
(52) J(a;t)1
1 +
0(a) (t) 0(a)(t)Z
Qda
1 +
0(a) (t) 1
for all a2Q. Setting a=a1on (52) and using (40), it follows that
J(a1;t)1
1 +m0 (t)Z
Qda
1 +
0(a) (t) 1
C
( ) ln( )12 WILLIAM CHEN AND ALEJANDRO SARRIA
for >0 small, and therefore
J(a1;t)!+1
as % . Next, for 0 < < 1=TEandTE>0 the nite blowup time of the undamped
Euler equation (7), we prove the existence of a nite TB
>0 such that t%TB
as % .
From (32), (34) and Lemma 4.1,
(53) e tdtZ
Qda
1 +
0(a) 2
d
for 0 < . Denote the damped Boussinesq time variable t=tB
( ). Then integrating
(53) between 0 and tB
yields the upper-bound
(54) tB
( ) 1
ln 1 tE( ) =tE
( )
fortB
in terms of the damped Euler time variable tE
, which in turn depends on the undamped
Euler time variable tEin (10) and the damping coecient > 0. Since
0satis es the
conditions of Theorem 1.1, the limit TElim % tE( ) is positive and nite. Suppose
0< < 1=TE. Then Theorem 2.1 implies that TE
lim % tE
( ) is also positive and
nite. Thus letting % in (54), we see that the blowup time TB
>0 is nite and
satis es
TB
lim
% tB
( )TE
:
This nishes the proof of the rst part of the Theorem. For the second part we adapt an
argument used in [5, 21] to construct blowup and respectively global in nite-energy solutions
of the 2D Euler and inviscid Boussinesq equations.
Let0(x) = sin2(2x). Then we look for a solution of (23) satisfying (x;0)1 and
t(x;0) =
0(x). Suppose 1(t) and2(t) solve
(55) 00
1+ 0
1 I(t)1= 0; 00
2+ 0
2 I(t)2= 1
with1(0) =0
1(0) = 1 and 2(0) = 0,0
2(0)6= 0. Then
(56) (x;t) =1(t) +0(x)2(t)
satis es
tt+ t I(t)=0; (x;0) = 1:
Note that0
2(0)6= 0 must be such that
0(x) =t(x;0) = 1 +0(x)0
2(0) has mean zero over
Q, as required by (11). Now, since Jsatis es (31) it follows that
1 =Z
Qdx
1 sin2(2x)2;
which yields the relation
(57) 2=1 1
1:
Using (57) we see that
0(x) = cos(4x), which satis es the mean-zero condition (11).
Next, using (57) on (55)ii) to eliminate the nonlocal term I(t) in (55)i) gives, after some
simpli cation,
d
dt0
1
1
+ 0
1
1=1
2:DAMPED INFINITE ENERGY SOLUTIONS OF THE 3D EULER AND BOUSSINESQ EQUATIONS 13
Dividing both sides by 1, di erentiating the resulting equation, and then setting N(t) =
0
1=1now leads to
(58)d
dt
N0 1
2N2+ N
= N2:
SinceN(0) = 1 and N0(0) =1
2 , we integrate (58) and use a standard Gronwall-type
argument to obtain
(59) z01
2e tz2; z (0) = 1
forz(t) =e tN(t). Set
(60) TB
= 1
ln (1 2 )
for 0< < 1=2. Note that TB
is positive and nite. Then solving (59) for t2[0;TB
) gives
z(t)2
e t (1 2 );
or sincez(t) =e tN(t) =e td
dt(ln1(t)),
(61) 1(t)4 2
(e t (1 2 ))2:
From (61) it follows that 1!+1ast%TB
. Then by (56) and (57),
1
J(x;t)cos2(2x)1(t) +sin2(2x)
1(t);
which implies that J(x;t)!0 ast%TB
forx2Bf (x;y)2Qjx =2f1=4;3=4gg. Note
thatBare precisely the points where
0(x) = cos(4x) does not equal its negative minimum.
Lastly, by setting = 0 in (58) it is easy to see that the Jacobian of the associated
undamped Boussinesq system vanishes as t%2, which agrees with TB
!2 in (60) as
!0. Thus we may write 0 < < 1=2 as 0< < 1=TBforTB= 2 the blowup time for
the solution of the undamped Boussinesq system with the same initial data.
Note that the blowup result in part 2 of Theorem 2.2 is e ectively one dimensional in the
sense that the initial data depends only on one of the two coordinate variables. Currently
we do not know if this particular blowup is suppressed for 1=TB, or if a solution of
the damped Boussinesq system (12)-(13) persist globally in time for 1=TEwhen the
initial data satis es the conditions of part one of Theorem 2.2 andis such that the associated
undamped solution blows up in nite time.
5.Acknowledgments
This work was partially supported by the Div III & P Research Funding Committee at
Williams College.14 WILLIAM CHEN AND ALEJANDRO SARRIA
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Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267
E-mail address :wyc1@williams.edu
Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267
E-mail address :Alejandro.Sarria@williams.edu |
1504.06042v1.Magnetization_damping_in_noncollinear_spin_valves_with_antiferromagnetic_interlayer_couplings.pdf | arXiv:1504.06042v1 [cond-mat.mes-hall] 23 Apr 2015Magnetizationdamping innoncollinearspinvalveswithant iferromagnetic interlayer couplings
Takahiro Chiba1, Gerrit E. W. Bauer1,2,3, and Saburo Takahashi1
1Institute for Materials Research, Tohoku University, Send ai, Miyagi 980-8577, Japan
2WPI-AIMR, Tohoku University, Sendai, Miyagi 980-8577, Jap an and
3Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands
(Dated: October 29, 2018)
We study the magnetic damping in the simplest of synthetic an tiferromagnets, i.e. antiferromagnetically
exchange-coupled spin valves in which applied magnetic fiel ds tune the magnetic configuration to become
noncollinear. We formulate the dynamic exchange of spin cur rents in a noncollinear texture based on the spin-
diffusiontheorywithquantum mechanicalboundaryconditionsa ttheferrromagnet|normal-metal interfacesand
derive the Landau-Lifshitz-Gilbert equations coupled by t he static interlayer non-local and the dynamic ex-
change interactions. We predict non-collinearity-induce d additional damping that can be sensitively modulated
byanapplied magnetic field. The theoretical results compar e favorablywithpublished experiments.
I. INTRODUCTION
Antiferromagnets (AFMs) boast many of the functionali-
tiesofferromagnets(FM)thatareusefulinspintroniccirc uits
anddevices: Anisotropicmagnetoresistance(AMR),1tunnel-
ing anisotropicmagnetoresistance(TAMR),2current-induced
spintransfertorque,3–8andspincurrenttransmission9–11have
all been found in or with AFMs. This is of interest because
AFMshaveadditionalfeaturespotentiallyattractivefora ppli-
cations. InAFMsthetotalmagneticmomentis(almost)com-
pletely compensated on an atomic length scale. The AFM
order parameter is, hence, robust against perturbations su ch
as external magnetic fields and do not generate stray fields
themselveseither. AspintronictechnologybasedonAFM el-
ementsisthereforeveryattractive.12,13Drawbacksarethedif-
ficulty to controlAFMs by magnetic fields and much higher
(THz)resonancefrequencies,14–16whicharedifficulttomatch
with conventional electronic circuits. Man-made magnetic
multilayers in which the layer magnetizations in the ground
state isorderedin anantiparallelfashion,17i.e. so-calledsyn-
thetic antiferromagnets,donot su ffer fromthis drawbackand
have therefore been a fruitful laboratory to study and modu-
late antiferromagnetic couplings and its consequences,18but
also found applications as magnetic field sensors.19Trans-
port in these multilayers including the giant magnetoresis -
tance (GMR)20,21are now well understood in terms of spin
and charge diffusive transport. Current-induced magnetiza-
tionswitchinginF|N|Fspinvalvesandtunneljunctions,22has
been a game-changer for devices such as magnetic random
access memories(MRAM).23A keyparameterof magnetiza-
tiondynamicsisthemagneticdamping;asmalldampinglow-
ersthethresholdofcurrent-drivenmagnetizationswitchi ng,24
whereasalargedampingsuppresses“ringing”oftheswitche d
magnetization.25
Magnetization dynamics in multilayers generates “spin
pumping”, i.e. spin current injection from the ferromagnet
into metallic contacts. It is associated with a loss of an-
gular momentum and an additional interface-related magne-
tization damping.26,27In spin valves, the additional damp-
ing is suppressed when the two magnetizations precess in-
phase, while it is enhanced for a phase di fference ofπ(out-
of-phase).27–30This phenomenon is explained in terms of a
“dynamic exchange interaction”, i.e. the mutual exchange o fnon-equilibriumspin currents,which shouldbe distinguis hed
from(butcoexistswith)theoscillatingequilibriumexcha nge-
coupling mediated by the Ruderman-Kittel-Kasuya-Yosida
(RKKY) interaction. The equilibriumcoupling is suppresse d
whenthespacerthicknessisthickerthantheelasticmean-f ree
path,31,32while the dynamiccouplingise ffective onthe scale
oftheusuallymuchlargerspin-flipdi ffusionlength.
Antiparallel spin valves provide a unique opportunity to
study and control the dynamic exchange interaction between
ferromagnets through a metallic interlayer for tunable mag -
netic configurations.33,34An originallyantiparallel configura-
tionisforcedbyrelativelyweakexternalmagneticfieldsin toa
non-collinearconfigurationwith a ferromagneticcomponen t.
Ferromagneticresonance(FMR)andBrillouinlightscatter ing
(BLS) are two useful experimentalmethodsto investigateth e
nature and magnitude of exchange interactions and magnetic
damping in multilayers.35Both methods observe two reso-
nances, i.e. acoustic (A) and optical (O) modes, which are
characterizedbytheirfrequenciesandlinewidths.36,37
Timopheev etal.observedaneffectoftheinterlayerRKKY
coupling on the FMR and found the linewidth to be a ffected
by the dynamic exchange coupling in spin valves with one
layerfixed by the exchange-biasof an inert AFM substrate.38
They measured the FMR spectrum of the free layer by tun-
ing the interlayer coupling (thickness) and reported a broa d-
ening of the linewidth by the dynamic exchange interaction.
Taniguchi et al.addressed theoretically the enhancement of
the Gilbert damping constant due to spin pumping in non-
collinear F|N|F trilayer systems, in which one of the magne-
tizations is excited by FMR while the other is o ff-resonant,
butadoptaroleasspinsink.39Thedynamicsofcoupledspin
valvesinwhichbothlayermagnetizationsarefreetomoveha s
beencomputedby oneof us29and bySkarsvåg et al.33,49but
only for collinear (parallel and antiparallel) configurati ons.
Current-induced high-frequency oscillations without app lied
magnetic field in ferromagnetically coupled spin valves has
beenpredicted.40
Inthepresentpaper,wemodelthemagnetizationdynamics
of the simplest of synthetic antiferromagnets, i.e. the ant i-
ferromagnetically exchange-coupled spin valve in which th e
(in-plane) ground state magnetizations are for certain spa cer
thicknesses ordered in an antiparallel fashion by the RKKY
interlayercoupling.41We focusonthecoupledmagnetization2
modes in symmetric spin valves in which in contrast to pre-
vious studies, both magnetizations are free to move. We in-
clude static magnetic fields in the film plane that deform the
antiparallelconfigurationintoacantedone. Microwaveswi th
longitudinal and transverse polarizations with respect to an
externalmagneticfieldthenexciteAandOresonancemodes,
respectively.31,42–46We develop the theory for magnetization
dynamics and damping based on the Landau-Lifshitz-Gilbert
equationwithmutualpumpingofspincurrentsandspintrans -
fer torques based on the spin di ffusion model with quantum
mechanical boundary conditions.27,47,48We confirm28,49that
the additional damping of O modes is larger than that of the
A modes. We report that a noncollinear magnetization con-
figurationinducesadditionaldampingtorquesthat to the be st
of ourknowledgehavenotbeen discussedin magneticmulti-
layers before.50The external magnetic field strongly a ffects
the dynamics by modulating the phase of the dynamic ex-
change interaction. We compute FMR linewidths as a func-
tion of applied magnetic fields and find good agreement with
experimental FMR spectra on spin valves.31,32The dynam-
ics of magnetic multilayers as measured by ac spin trans-
fer torque excitation30reveals a relative broadening of the O
modes linewidths that is well reproduced by our spin valve
model.
In Sec. II we present our model for noncollinear spin
valves based on spin-di ffusiontheory with quantum mechan-
ical boundary conditions. In Sec. III, we consider the mag-
netization dynamics in antiferromagnetically coupled non -
collinear spin valves as shown in Fig. 1(b). We derive the
linearized magnetization dynamics, resonance frequencie s,
and lifetimes of the acoustic and optical resonance modes in
Sec. IV. We discuss the role of dynamicspin torqueson non-
collinear magnetization configurations in relation to exte rnal
magnetic field dependence of the linewidth. In Sec. V, we
compare the calculated microwave absorption and linewidth
with published experiments. We summarize the results and
endwiththeconclusionsinSec. VI.
II. SPINDIFFUSIONTRANSPORTMODEL
We consider F1|N|F2 spin valves as shown in Fig. 1(a), in
whichthemagnetizations MjoftheferromagnetsF j(j=1,2)
are coupled by a antiparallel interlayer exchange interact ion
and tilted towards the direction of an external magnetic fiel d.
Applied microwaves with transverse polarizations with re-
spect to an external magnetic field cause dynamics and, via
spinpumping,spincurrentsandaccumulationsinthenormal -
metal (NM) spacer. The longitudinal component of the spin
accumulation diffuses into and generates spin accumulations
inF thatwe showtobesmall later,butdisregardinitially. L et
usdenotethepumpedspincurrent JP
j,whileJB
jisthediffusion
(back-flow)spin currentdensity inducedby a spin accumula-NM zy
x
F1 F2 0 dN
J1PJ2P
J2BJ1Bµsθθ
(c) Acoustic mode (d) Optical mode (a)
m1 m2
H
*OUFSMBZFSDPVQMJOH
(b)
(b) H
hx hy
FIG.1. (a) Sketch of the sample withinterlayer exchange-co uplings
illustrating the spin pumping and backflow currents. (b) Mag netic
resonance in an antiferromagnetically exchange-coupled s pin valve
with a normal-metal (NM) film sandwiched by two ferromagnets
(F1,F2)subjecttoamicrowave magneticfield h. Themagnetization
vectors (m1,m2) are tilted by an angle θin a static in-plane mag-
neticfield Happliedalongthe y-axis. Thevectors mandnrepresent
thesum anddifference ofthe twolayer magnetizations, respectively.
(c) and (d): Precession-phase relations for the acoustic an d optical
modes.
tionµsjin NM,bothat theinterfaceF j, with27,51
JP
j=Gr
emj×/planckover2pi1∂tmj, (1a)
JB
j=Gr
e/bracketleftBig/parenleftBig
mj·µsj/parenrightBig
mj−µsj/bracketrightBig
, (1b)
wheremj=Mj//vextendsingle/vextendsingle/vextendsingleMj/vextendsingle/vextendsingle/vextendsingleis the unit vector along the magnetic
moment of F j(j=1,2). The spin current througha FM |NM
interface is governed by the complex spin-mixing conduc-
tance (per unit area of the interface) G↑↓=Gr+iGi.27The
real component Grparameterized one vector component of
the transverse spin-currentspumped and absorbed by the fer -
romagnets. The imaginary part Gican be interpreted as an
effective exchange field between magnetization and spin ac-
cumulation, which in the absence of spin-orbit interaction is
usuallymuchsmallerthantherealpart,forconductingaswe ll
asinsultingmagnets.523
Thediffusionspin-currentdensityin NMreads
Js,z(z)=−σ
2e∂zµs(z), (2)
whereσ=ρ−1is the electrical conductivity and µs(z)=
Ae−z/λ+Bez/λthe spin accumulationvectorthat is a solution
ofthespindiffusionequation∂2
zµs=µs/λ2,whereλ=√Dτsf
is the spin-diffusion length, Dthe diffusion constant, and τsf
the spin-flip relaxation time. The vectors AandBare de-termined by the boundary conditions at the F1 |NM (z=0)
and F2|NM (z=dN) interfaces: Js,z(0)=JP
1+JB
1≡Js1and
Js,z(dN)=−JP
2−JB
2≡−Js2. Theresultingspin accumulation
inN reads
µs(z)=2eλρ
sinh/parenleftBigdN
λ/parenrightBig/bracketleftBigg
Js1cosh/parenleftBiggz−dN
λ/parenrightBigg
+Js2cosh/parenleftbiggz
λ/parenrightbigg/bracketrightBigg
,(3)
withinterfacespin currents
Js1=ηS
1−η2δJP
1+η2/parenleftBig
m2·δJP
1/parenrightBig
1−η2(m1·m2)2m1×(m1×m2), (4a)
Js2=−ηS
1−η2δJP
2+η2/parenleftBig
m1·δJP
2/parenrightBig
1−η2(m1·m2)2m2×(m2×m1). (4b)
Here
δJP
1=JP
1+ηm1×(m1×JP
2), (5a)
δJP
2=JP
2+ηm2×(m2×JP
1), (5b)
S=sinh(dN/λ)/grandη=gr/[sinh(dN/λ)+grcosh(dN/λ)]
are the efficiency of the back flow spin currents, and gr=
2λρGris dimensionless. The first terms in Eqs. (4a) and (4b)
represent the mutual pumping of spin currents while the sec-
ondtermsmaybeinterpretedasa spincurrentinducedbythe
noncollinear magnetization configuration, including the b ack
flowfromthe NMinterlayer.
III. MAGNETIZATIONDYNAMICSWITH DYNAMIC
SPINTORQUES
We consider the magnetic resonance in the non-collinear
spin valve shown in Fig. 1. The magnetization dynamics are
describedbytheLandau-Lifshitz-Gilbert(LLG)equation,
∂tm1=−γm1×Heff1+α0m1×∂tm1+τ1,(6a)
∂tm2=−γm2×Heff2+α0m2×∂tm2−τ2.(6b)
The first term in Eqs. (6a) and (6b) represents the torque in-
ducedbytheeffectivemagneticfield
Heff1(2)=H+h(t)−4πMsm1(2)zˆz+Jex
MsdFm2(1),(7)
which consists of an in-plane applied magnetic field H,
a microwave field h(t), and the demagnetization field
−4πMsm1(2)zˆzwith saturation magnetization Ms. The inter-
layer exchange field is Jex/(MsdF)m2(1)with areal density of
theinterlayerexchangeenergy Jex<0(forantiferromagnetic-
coupling) and F layer thickness dF. The second term is the
Gilbert dampingtorque that governsthe relaxationcharact er-
ized byα0itowards an equilibrium direction. The third term,τ1(2)=γ/planckover2pi1/(2eMsdF)Js1(2), is the spin-transfertorque induced
by the absorption of the transverse spin currents of Eqs. (4a )
and (4b), andγandα0are the gyromagnetic ratio and the
Gilbert dampingconstant of the isolated ferromagneticfilm s,
respectively. SometechnicaldetailsofthecoupledLLGequ a-
tionsarediscussedinAppendixA.Introducingthetotalmag -
netizationdirection m=(m1+m2)/2andthedifferencevector
n=(m1−m2)/2,theLLG equationscanbewritten
∂tm=−γm×(H+h)
+2πγMs(mzm+nzn)׈z
+α0(m×∂tm+n×∂tn)+τm, (8a)
∂tn=−γn×/parenleftBigg
H+h+Jex
MsdFm/parenrightBigg
+2πγMs(nzm+mzn)׈z
+α0(m×∂tn+n×∂tm)+τn,(8b)
where the spin-transfer torques τm=(τ1+τ2)/2 andτn=
(τ1−τ2)/2become
τm/αm=m×∂tm+n×∂tn
+2ηm·(n×∂tn)
1−ηCm+2ηn·(m×∂tm)
1+ηCn,(9a)
τn/αn=m×∂tn+n×∂tm
−2ηm·(n×∂tm)
1+ηCm−2ηn·(m×∂tn)
1−ηCn,(9b)
andC=m2−n2,while
αm=α1gr
1+grcoth(dN/2λ), (10a)
αn=α1gr
1+grtanh(dN/2λ), (10b)
withα1=γ/planckover2pi12/(4e2λρMsdF).4
IV. CALCULATIONANDRESULTS
We consider the magnetization dynamics excited by lin-
early polarized microwaves with a frequency ωand in-plane
magnetic field h(t)=(hx,hy,0)eiωtthat is much smaller than
the saturation magnetization. For small angle magnetizati on
precession the total magnetization and di fference vector may
be separated into a static equilibrium and a dynamic com-
ponent as m=m0+δmandn=n0+δn, respectively,
wherem0=(0,sinθ,0),n0=(cosθ,0,0),C=−cos2θ,
ands=−ˆzsin2θ. The equilibrium (zero torque) conditions
m0×H=0 andn0×(H+Jex/(MsdF)m0)=0 lead to the
relation
sinθ=H/Hs, (11)
whereHs=−Jex/(MsdF)=|Jex|/(MsdF) is the saturation
field. TheLLGequationsread
∂tδm=−γδm×H−γm0×h
+2πγMs(δmzm0+δnzn0)׈z
+α0(m0×∂tδm+n0×∂tδn)+δτm,(12a)
∂tδn=−γδn×H−γn0×h
+2πγMs(δnzm0+δmzn0)׈z
−γHs(m0×δn−n0×δm)
+α0(m0×∂tδn+n0×∂tδm)+δτn,(12b)
withlinearizedspin-transfertorques
δτm/αm=m0×∂tδm+n0×∂tδn
−ηsin2θ
1+ηcos2θ∂tδnzm0+ηsin2θ
1−ηcos2θ∂tδmzn0,(13a)
δτn/αn=m0×∂tδn+n0×∂tδm
+ηsin2θ
1−ηcos2θδmzm0−ηsin2θ
1+ηcos2θδnzn0,(13b)
To leading order in the small precessing components δmand
δn,theLLG equationsinfrequencyspace become
δmx=γhxγ(Hs+4πMs)+iω/parenleftBig
α0+αm(1+η)
1−ηcos2θ/parenrightBig
ω2−ω2
A−i∆Aωsin2θ,(14a)
δny=−γhxγ(Hs+4πMs)+iω/parenleftBig
α0+αn(1−η)
1−ηcos2θ/parenrightBig
ω2−ω2
A−i∆Aωcosθsinθ,
(14b)
δmz=−γhxiω
ω2−ω2
A−i∆Aωsinθ, (14c)
δnx=−γhy4πγMs+iω/parenleftBig
α0+αn(1−η)
1+ηcos2θ/parenrightBig
ω2−ω2
O−i∆Oωcosθsinθ,(15a)
δmy=γhy4πγMs+iω/parenleftBig
α0+αm(1+η)
1−ηcos2θ/parenrightBig
ω2−ω2
O−i∆Oωcos2θ, (15b)
δnz=γhyiω
ω2−ω2
O−i∆Oωcosθ. (15c)The A modes (δmx,δny,δmz) are excited by hx, while the O
modes (δnx,δmy,δnz) couple to hy. The poles inδm(ω)and
δn(ω)define the resonance frequencies and linewidths that
do not depend on the magnetic field since we disregard
anisotropyandexchange-bias.
A. Acoustic andOpticalmodes
An antiferromagnetically exchange-coupled spin valves
generallyhave non-collinearmagnetizationconfiguration sby
thepresenceofexternalmagneticfields. For H<Hs(0<θ<
π/2),theacousticmode:
ωA=γH/radicalbig
1+(4πMs/Hs), (16)
∆A=α0γ/parenleftBig
Hs+4πMs+Hssin2θ/parenrightBig
+αmγ(Hs+4πMs)+αA(θ)γHs,(17)
andthe opticalmode:
ωO=γ/radicalBig
(4πMs/Hs)(H2s−H2), (18)
∆O=α0γ/parenleftBig
4πMs+Hscos2θ/parenrightBig
+αn4πγMs+αO(θ)γHs, (19)
where
αA(θ)=α1grsin2θ
1+grtanh(dN/2λ)+2grsin2θ/sinh(dN/λ),(20)
αO(θ)=α1grcos2θ
1+grtanh(dN/2λ)+2grcos2θ/sinh(dN/λ).(21)
The additional broadeningin ∆Ais proportional toαmand
αAwhile that in∆Oscales withαnandαOin. Figure 2 (a)
showsαmandαnas a function of spacer layer thickness, in-
dicating thatαnis always larger than αm, and thatαn(αm)
strongly increases (decreases) with decreasing N layer thi ck-
ness, especially for dN<λand large gr. Figure 2(b) shows
thedependenceofαAandαOonthetiltedangleθfordifferent
valuesof dN. Asθincreases,αAincreasesfrom0to αmwhile
αOdecreasesαmto 0. The additional damping can be ex-
plained by the dynamic exchange. When two magnetizations
in spin valves precess in-phase, each magnet receives a spin
current that compensates the pumped one, thereby reducing
the interface damping. When the magnetizations precess out
of phase, theπphase difference between both spin currents
means that the moduli have to be added, thereby enhancing
thedamping.
When the magnetizations are tilted by an angle θas
sketched in Fig. 1, we predict an additional damping torque
expressedbythesecondtermsofEqs.(4a)and(4b). Figure3
shows the ratiosαA(θ)/αmandαO(θ)/αnas a function ofθ
andgrfor different values ofλ/dN, thereby emphasizing the
additional damping in the presence of noncollinear magneti -
zations. In Fig. 3(a,b)with λ/dN=1, i.e. for a spin-di ffusive
interlayer,the additionaldampingof both A- and O-modesis
significant in a large region of parameter space. On the other5
0.1 1 10 100λ/d N012345αm/α1, αn/α1/α1 /α12
1
0.5
0 30 60 90
θ (degree)0.00.10.20.30.40.5αA, αOαO αAλ/d N=1
2
4
10(b)(a)
αnαmgr=5
FIG.2. (a)αm(dashed line)andαn(solidline) as a functionof λ/dN
for different values of the dimensionless mixing conductance gr. (b)
αAC(dashed line)andαOP(solidline)as afunction oftiltangle θfor
gr=5and different values ofλ/dN.
hand, in Fig. 3(c,d) with λ/dN=10, i.e. for an almost spin-
ballistic interlayer, the additional damping is more impor tant
fortheA-mode,whiletheO-modeisa ffectedonlyclosetothe
collinear magnetization. In the latter case the intrinsic d amp-
ingα0dominates,however.
B. In-phaseandOut-of-phasemodes
Whentheappliedmagneticfieldislargerthanthesaturation
field (H>Hs), both magnetizations point in the ˆydirection,
and theδm(A) andδn(O) modes morph into in-phase and
180◦out-of-phase (antiphase) oscillations of δm1andδm2,
respectively. The resonance frequency53and linewidth of the
in-phasemodefor H>Hs(θ=π/2)are
ωA=γ/radicalbig
H(H+4πMs), (22)
∆A=2(α0+αm)γ(H+2πMs), (23)
whilethoseoftheout-of-phasemodeare
ωO=γ/radicalbig
(H−Hs)(H−Hs+4πMs),(24)
∆O=2(α0+αn)γ(H−Hs+2πMs).(25)
B
C
D
E
HS
HS
θ (degree) θ (degree)
θ (degree)
θ (degree)
HS
HS0.1 0.3 0.5 0.7 0.9
0.1 0.2 0.3 0.4 0.5 0.6
0.55 0.65 0.75 0.85 0.95
0.01 0.02 0.03 0.04 0.05
FIG. 3. (a,c)αA(θ)/αmand (b,d)αO(θ)/αnas a function ofθandgr
for different values ofλ/dN. (a,b) withλ/dN=1, (c,d) withλ/dN=
10
Figure 4(a) shows the calculated resonance frequencies of
the A andO modesas a functionof an appliedmagneticfield
Hwhile 4(b) displays the linewidths for α1/α0=1, which
is representative for ferromagnetic metals, such as permal -
loy (Py) with an intrinsic magnetic damping of the order of
α0=0.01andacomparableadditionaldamping α1duetospin
pumping. A value gr=4/5 corresponds toλ=20/200nm,
ρ=10/2.5µΩcmfor N=Ru/Cu,54,55Gr=2/1×1015Ω−1m−2
for the N|Co(Py) interface56,57, anddF=1nm, for example.
Thecolorsinthefigurerepresentdi fferentrelativelayerthick-
nessesdN/λ. The linewidth of the A mode in Fig. 4(b) in-
creases with increasing H, while that of the O mode starts
to decrease until a minimum at the saturation field H=Hs.
Figure 4(c) shows the linewidths for α1/α0=10, which de-
scribes ferromagnetic materials with low intrinsic dampin g,
such as Heusler alloys58and magnetic garnets.59In this case,
the linewidth of the O modeis much largerthanthat of the A
mode,especiallyforsmall dN/λ.
In the limit of dN/λ→0 is easily established experimen-
tally. The expressions of the linewidth in Eqs. (17) and (19)
arethengreatlysimplifiedto ∆A=γ(Hs+4πMs+Hssin2θ)α0
and∆O=γ/parenleftBig
4πMs+Hscos2θ/parenrightBig
α0+(4πγMs)grα1,and∆A≪
∆Owhengrα1≫α0. The additional damping, Eq. (10b) re-
duces toαm→0 andαn→2[γ/planckover2pi1/(4πMsdF)(h/e2)Gr] when
themagnetizationsarecollinearandintheballisticspint rans-
port limit.27In contrast to the acoustic mode, the dynamic
exchange interaction enhances damping of the optical mode.
∆O≫∆AhasbeenobservedinPy |Ru|Pytrilayerspinvalves32
andCo|Cu multilayers30, consistentwiththepresentresults.
For spin valves with ferromagnetic metals,
the interface backflow spin-current [(1b)] reads
JB
j=(Gr/e)/bracketleftBig
ξF/parenleftBig
mj·µsj/parenrightBig
mj−µsj/bracketrightBig
,whereξF=6
012ω/(4 πγ Ms)ωAωO
02468∆/(4 πMsα0) dN/λ=
0.3
1(a)
(b)
(c)
0 0.5 1 1.5
H/H s050dN/λ=0.01
0.1
0.3
1∆/(4 πγ Msα0)0.01γ
A mode O mode 0.1
FIG. 4. (a) Resonance frequencies of the A and O modes as a func -
tion of magnetic field for Hs/(4πMs)=1. (b), (c) Linewidths of the
A (dashed line) and the O (solid line) modes for Hs/(4πMs)=1,
gr=5, and different values of dN/λ. (b)α1/α0=1 and (c)
α1/α0=10.
1−(G/2Gr)(1−p2)(1−ηF) (0≤ξF≤1),Gis the
N|F interface conductance per unit area, and pthe conduc-
tance spin polarization.51Here the spin diffusion efficiency
is
1
ηF=1+σF
GλFtanh(dF/λF)
cosh(dF/λF), (26)
whereσF,λF, anddFare the conductivity, the spin-flip dif-
fusion length, and the layer thickness of the ferromagnets,
respectively. For the material parameters of a typical fer-
romagnet with dF=1 nm, the resistivity ρF=10µΩcm,
G=2Gr=1015Ω−1m−2,λF=10nm,and p=0.7,ξF=0.95,
whichjustifiesdisregardingthiscontributionfromtheout set.
V. COMPARISONWITH EXPERIMENTS
FMRexperimentsyieldtheresonantabsorptionspectraofa
microwavefield ofa ferromagnet. Themicrowaveabsorptiont
dP/dH ϕ=90 o
20 o
0o×5
×5ϕHh(t)(a)
5
H (kOe) 22 4 6 0 0 0.4 0.8 1.2 H/H s
Co(3)|Ru(1)|Co(3) Co(3)|Ru(1)|Co(3)
Experiment (Ref. 11) Calculation
0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 ∆/(4 πγ Msα₀)H/H s
A mode
O mode
0 2000 4000 6000
Field (Oe) (b)
Experiment (Ref. 10): [Co(1)|Cu(1)] 10 Co(1) Calculation: Co(1)|Cu(1)|Co(1)
FIG.5. (a)Derivativeofthemicrowaveabsorptionspectrum dP/dH
at frequencyω/(2π)=9.22 GHz for different anglesϕbetween the
microwavefieldandtheexternalmagneticfieldfor Hs/(4πMs)=0.5,
ω/(4πγMs)=0.35,dN/λ=0.1,dF/λ=0.3α0=α1=0.02, and
gr=4. The experimental data have been adopted from Ref. 31.
(b) Computed linewidths of the A and O modes of a Co |Cu|Co spin
valve (dashed line) compared with experiments on a Co |Cu multi-
layer (solidline).30
powerP=2/angbracketlefth(t)·∂tm(t)/angbracketrightbecomesinourmodel
P=1
4γ2Ms(Hs+4πMs)∆A
(ω−ωA)2+(∆A/2)2h2
xsin2θ
+1
4γ2Ms(4πMs)∆O
(ω−ωO)2+(∆O/2)2h2
ycos2θ. (27)
Pdepends sensitively on the character of the resonance, the
polarization of the microwave, and the strength of the ap-
plied magnetic field. In Figure 5(a) we plot the normalized
derivative of the microwave absorption spectra dP/(P0dH)
at different anglesϕbetween the microwave field h(t) and
the external magnetic field H, where P0=γMsh2and
h(t)=h(sinϕ,cosϕ,0)eiωt. Here we use the experimen-
tal values Hs=5kOe, 4πMs=10kOe,dN=1nm,
dF=3nm, and microwave frequency ω/(2π)=9.22GHz
as found for a symmetric Co |Ru|Co trilayer.31λ=20nm for7
Ru,α0=α1=0.02, andgr=4 is adopted (correspond-
ing toGr=2×1015Ω−1m−2).Whenh(t) is perpendicularto
H(ϕ=90◦), only the A mode is excited by the transverse
(δmx,δmz) component. When h(t) is parallel to H(ϕ=0◦),
the O mode couples to the microwave field by the longitudi-
nalδmycomponent. For intermediate angles ( ϕ=20◦), both
modes are excited at resonance. We observe that the opti-
cal mode signal is broader than the acoustic one, as calcu-
lated. The theoretical resonance linewidths of the A and O
modes as well as the absorption power as a function of mi-
crowave polarization reproduce the experimental results f or
Co(3.2nm)|Ru(0.95nm)|Co(3.2nm)well.31
Figure 5(b) shows the calculated linewidths of A and
O modes as a function of an applied magnetic field for a
Co(1nm)|Cu(1nm)|Co(1nm) spin valve. The experimental
valuesλ=200nm andρ=2.5µΩcm for Cu,α0=0.01
and 4πMs=15kOe for Co, and gr=5 (corresponding to
Gr=1015Ω−1m−2) for the interface have been adopted.57
We partially reproduce the experimental data for magnetic
multilayers; for the weak-field broadenings of the observed
linewidthsagreementisevenquantitative. Theremainingd is-
crepanciesintheappliedmagneticfielddependencemightre -
flect exchange-dipolar49and/or multilayer30spin waves be-
yondourspinvalvemodelinthe macrospinapproximation.
VI. CONCLUSIONS
In summary, we modelled the magnetization dynamics
in antiferromagnetically exchange-coupled spin valves as a
model for synthetic antiferromagnets. We derivethe Landau -
Lifshitz-Gilbert equations for the coupled magnetization s in-
cluding the spin transfer torques by spin pumping based on
the spin diffusion model with quantum mechanical boundary
conditions. We obtain analytic expressionsfor the linewid ths
of magnetic resonance modes for magnetizations canted byapplied magneticfields and achieve goodagreementwith ex-
periments. We findthatthelinewidthsstronglydependonthe
type of resonance mode (acoustic and optical) as well as the
strength of magnetic fields. The magnetic resonance spectra
reveal complex magnetization dynamics far beyond a simple
precessionevenin the linear responseregime. Our calculat ed
results compare favorably with experiments, thereby provi ng
theimportanceofdynamicspincurrentsinthesedevices. Ou r
model calculation paves the way for the theoretical design o f
syntheticAFMmaterialthatisexpectedtoplayaroleinnext -
generationspin-baseddata-storageandinformationtechn olo-
gies.
VII. ACKNOWLEDGMENTS
TheauthorsthanksK.Tanaka,T.Moriyama,T.Ono,T.Ya-
mamoto, T. Seki, and K. Takanashi for valuable discussions
and collaborations. This work was supported by Grants-in-
AidforScientificResearch(GrantNos. 22540346,25247056,
25220910,268063)fromtheJSPS,FOM(StichtingvoorFun-
damenteel Onderzoek der Materie), the ICC-IMR, EU-FET
Grant InSpin 612759, and DFG Priority Programme 1538
“Spin-CaloricTransport”(BA 2954 /2).
AppendixA: CoupledLandau-Lifshitz-Gilbertequationsin
noncollinearspinvalves
Both magnets and interfaces in our NM |F|NM spin valves
are assumed to be identical with saturation magnetization Ms
andGrthe real part of the spin-mixing conductance per unit
area (vanishing imaginary part). When both magnetizations
are allowed to precess as sketched in Fig. 1 (a), the LLG
equationsexpandedtoincludeadditionalspin-pumpandspi n-
transfertorquesread
∂mi
∂t=−γmi×Heffi+α0imi×∂mi
∂t
+αSPi/bracketleftBigg
mi×∂mi
∂t−ηmj×∂mj
∂t+η/parenleftBigg
mi·mj×∂mj
∂t/parenrightBigg
mi/bracketrightBigg
+αnc
SPi(ϕ)mi×/parenleftBig
mi×mj/parenrightBig
, (A1)
αnc
SPi(ϕ)=αSPiη2
1−η2(mi·mj)2/bracketleftBigg
mj·mi×∂mi
∂t+η/parenleftBigg
mi·mj×∂mj
∂t/parenrightBigg
(mj·mi)/bracketrightBigg
, (A2)
whereγandα0iare the gyromagnetic ratio and the Gilbert
damping constant of the isolated ferromagnetic films labele d
byiand thickness dFi. Asymmetric spin valves due to the
thickness differencedFisuppress the cancellation of mutual
spin-pumpinA-mode,whichmaybeadvantagetodetectboth
modesintheexperiment. Thee ffectivemagneticfield
Heffi=Hi+h(t)+Hdii(t)+Hexj(t) (A3)consistsoftheZeemanfield Hi,amicrowavefield h(t),thedy-
namic demagnetization field Hdii(t), and interlayer exchange
fieldHexj(t). The Gilbert damping torque parameterized
byα0igoverns the relaxation towards an equilibrium direc-
tion. The third term in Eq. (A1) represents the mutual spin
pumping-induced damping-like torques in terms of damping
parameter8
αSPi=γ/planckover2pi12Gr
2e2MsdFiηS
1−η2, (A4)
where
η=gr
sinh(dN/λ)+grcosh(dN/λ)(A5)
andgr=2λρGrisdimensionless. ThefourthterminEq. (A1)
is the damping Eq. (A2) that depends on the relative angleϕbetween the magnetizations. When mjis fixed along the
Hidirection, i.e. a spin-sink limit, Eq. (A1) reduces to the
dynamicstiffnessin spinvalveswithoutanelectricalbias.60
When the magnetizations are noncollinear as in Fig. 1, we
have to take into account the additional damping torques de-
scribedbythe secondtermsin Eqs.(4a) and(4b ,). Inthe bal-
listic limit dN/λ→0 and collinear magnetizations, Eq. (A1)
reduces to the well known LLG equation with dynamic ex-
changeinteraction.27,28,38
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2109.12071v1.Damping_in_yttrium_iron_garnet_film_with_an_interface.pdf | arXiv:2109.12071v1 [cond-mat.mtrl-sci] 24 Sep 2021Damping in yttrium iron garnet film with an interface
Ravinder Kumar,1,2,∗B. Samantaray,1Shubhankar Das,3Kishori Lal,1D. Samal,2,4and Z. Hossain1,5,†
1Department of Physics, Indian Institute of Technology, Kan pur 208016, India.
2Institute of Physics, Bhubaneswar 751005, India.
3Institute of Physics, Johannes Gutenberg-University Main z, 55099 Mainz, Germany.
4Homi Bhabha National Institute, Anushakti Nagar, Mumbai 40 0085, India.
5Institute of Low Temperature and Structure Research, 50-42 2 Wroclaw, Poland.
(Dated: September 27, 2021)
We report strong damping enhancement in a 200 nm thick yttriu m iron garnet (YIG) film due
to spin inhomogeneity at the interface. The growth-induced thin interfacial gadolinium iron garnet
(GdIG) layer antiferromagnetically (AFM) exchange couple s with the rest of the YIG layer. The
out-of-plane angular variation of ferromagnetic resonanc e (FMR) linewidth ∆ Hreflects a large in-
homogeneous distribution of effective magnetization ∆4 πMeffdue to the presence of an exchange
springlike moments arrangement in YIG. We probe the spin inh omogeneity at the YIG-GdIG inter-
face by performing an in-plane angular variation of resonan ce fieldHr, leading to a unidirectional
feature. The large extrinsic ∆4 πMeffcontribution, apart from the inherent intrinsic Gilbert co n-
tribution, manifests enhanced precessional damping in YIG film.
I. INTRODUCTION
The viability of spintronics demands novel magnetic
materials and YIG is a potential candidate as it ex-
hibits ultra-low precessional damping, α∼3×10−5[1].
The magnetic properties of YIG thin films epitaxially
grown on top of Gd 3Ga5O12(GGG) vary significantly
due to growth tuning[ 2,3], film thickness[ 4], heavy met-
als substitution[ 5–7] and coupling with thin metallic
layers[8–10]. The growth processes may also induce the
formation of a thin interfacial-GdIG layer at the YIG-
GGG interface[ 11–13]. The YIG-GdIG heterostructure
derived out of monolithic YIG film growth on GGG ex-
hibits interestingphenomenasuchasall-insulatingequiv-
alent of a synthetic antiferromagnet[ 12] and hysteresis
loop inversion governed by positive exchange-bias [ 13].
The radio frequency magnetization dynamics on YIG-
GdIG heterostructure still remains unexplored and need
a detailed FMR study.
The relaxation of magnetic excitation towards equi-
librium is governed by intrinsic and extrinsic mecha-
nisms, leading to a finite ∆ H[14,15]. The former mech-
anism dictates Gilbert type relaxation, a consequence of
direct energy transfer to the lattice governed by both
spin-orbit coupling and exchange interaction in all mag-
netic materials[ 14,15]. Whereas, the latter mechanism
is a non-Gilbert-type relaxation, divided mainly into two
categories[ 14,15]- (i) the magnetic inhomogeneity in-
duced broadening: inhomogeneity in the internal static
magnetic field, and the crystallographic axis orienta-
tion; (ii) two-magnon scattering: the energy dissipates in
the spin subsystem by virtue of magnon scattering with
nonzero wave vector, k∝negationslash= 0, where, the uniform reso-
nance mode couples with the degeneratespin waves. The
∗ravindk@iitk.ac.in
†zakir@iitk.ac.inangular variation of Hrprovides information about the
presence of different magnetic anisotropies[ 4,6]. Most
attention has been paid towards the angular dependence
ofHr[4,6], whereas, the angular variation of the ∆ H
is sparsely investigated. The studies involving angular
dependence of ∆ Hmay help to probe different contribu-
tions to the precessional damping.
Inthispaper, theeffectsofintrinsicandextrinsicrelax-
ation mechanisms on precessionaldamping of YIG film is
studied extensively using FMR technique. An enhanced
value of α∼1.2×10−3is realized, which is almost two
orders of magnitude higher than what is usually seen in
YIGthinfilms, ∼6×10−5[1,2]. Theout-of-planeangular
variation of ∆ Hshows an unusual behaviour where spin
inhomogeneity at the interface plays significant role in
defining the ∆ Hbroadening and enhanced α. In-plane
angular variation showing a unidirectional feature, de-
mandstheincorporationofanexchangeanisotropytothe
free energy density, evidence of the presence of an AFM
exchangecoupling at the YIG-GdIG interface. The AFM
exchange coupling leads to a Bloch domain-wall-like spi-
ralmoments arrangementin YIG and givesrise to a large
∆4πMeff. This extrinsic ∆4 πMeffcontribution due to
spin inhomogeneity at the interface adds up to the inher-
ent Gilbert contribution, which may lead to a significant
enhancement in precessional damping.
II. SAMPLE AND MEASUREMENT SETUPS
We deposit a ∼200 nm thick epitaxial YIG film on
GGG(111)-substrate by employing a KrF Excimer laser
(Lambda Physik COMPex Pro, λ= 248 nm) of 20 ns
pulse width. A solid state synthesized Y3Fe5O12target
is ablated using an areal energy of 2.12 J.cm−2with a
repetition frequency of 10 Hz. The GGG(111) substrate
is placed 50 mm away from the target. The film is grown
at 800oC temperature and in-situpost annealed at the
same temperature for 60 minutes in pure oxygen envi-ronment. The θ−2θX-ray diffraction pattern shows epi-
taxial growth with trails of Laue oscillations (Fig. 3(a)
of ref[3]). FMR measurements are performed using a
Bruker EMX EPR spectrometer and a broadband copla-
nar waveguide (CPW) setup. The former technique uses
a cavity mode frequency f≈9.60 GHz, and enables us
to perform FMR spectra for various θHandφHangu-
lar variations. The latter technique enables us to mea-
sure frequency dependent FMR spectra. We define the
configurations Hparallel ( θH= 90o) and perpendicular
(θH= 0o) to the film plane for rf frequency and angu-
lar dependent measurements. The resultant spectra are
obtained as the derivative of microwave absorption w.r.t.
the applied field H.
III. RESULTS AND DISCUSSION
A. Broadband FMR
Fig.1(a) shows typical broadband FMR spectra in
a frequency frange of 1.5 to 13 GHz for 200 nm thick
YIG film at temperature T= 300 K and θH= 90o.
The mode appearing at a lower field value is the main
mode, whereas the one at higher field value represents
surface mode. We discuss all these features in detail in
the succeeding subsection IIIB. We determine the res-
onance field Hrand linewidth (peak-to-peak linewidth)
∆Hfrom the first derivative of the absorption spectra.
Fig.1(b) shows the rf frequency dependence of Hrat
θH= 90oand 0o. We use the Kittel equation for fitting
the frequency vs. Hrdata from the resonance condi-
tion expressed as[ 10],f=γ[Hr(Hr+4πMeff)]1/2/(2π)
forθH= 90oandf=γ(Hr−4πMeff)/(2π) for
θH= 0o. Where, γ=gµB/ℏis the gyromagnetic ratio,
4πMeff= 4πMS−Haniis the effective magnetization
consisting of 4 πMSsaturation magnetization (calculated
using M(H)) and Hanianisotropy field parametrizing cu-
bic and out-of-plane uniaxial anisotropies. The fitting
gives 4πMeff≈2000 Oe, which is used to calculate the
Hani≈ −370 Oe.
Fig.1(c) shows the frequency dependence of ∆ Hat
θH= 90o. The intrinsic and extrinsic damping contri-
butions are responsible for a finite width of the FMR
signal. The intrinsic damping ∆ Hintarises due to the
Gilbert damping of the precessing moments. Whereas,
the extrinsic damping ∆ Hextexists due to different non-
Gilbert-type relaxations such as inhomogeneity due to
the distribution of magnetic anisotropy ∆ Hinhom, or
two-magnon scattering (TMS) ∆ HTMS. The intrinsic
Gilbert damping coefficient ( α) can be determined using
the Landau-Liftshitz-Gilbert equation expressed as[ 10],
∆H= ∆Hin+ ∆Hinhom= (4πα/√
3γ)f+ ∆Hinhom.
Considering the above equation where ∆ Hobeys lin-
earfdependence, the slope determines the value of α,
and ∆Hinhomcorresponds to the intercept on the ver-
tical axis. We observe a very weak non-linearity in the
fdependence of ∆ H, which is believed to be due to thecontribution of TMS to the linewidth ∆ HTMS. The non-
linearfdependence of ∆ Hin Fig.1(c) can be described
in terms of TMS, assuming ∆ H= ∆Hin+ ∆Hinhom+
∆HTMS. We put a factor of 1 /√
3 to ∆Hdue to the
peak-to-peak linewidth value extraction[ 14]. The TMS
induces non-linear slope at low frequencies, whereas a
saturation is expected at high frequencies. TMS is in-
duced by scattering centers and surface defects in the
sample. The defects with size comparable to the wave-
length of spin waves are supposed to act as scattering
centres. The TMS term at θH= 90ocan be expressed
as[16]-
∆HTMS(ω) = Γsin−1/radicalBigg/radicalbig
ω2+(ω0/2)2−ω0/2/radicalbig
ω2+(ω0/2)2+ω0/2,(1)
withω= 2πfandω0=γ4πMeff. The prefactor Γ
defines the strength of TMS. The extracted values are
as follows: α= 1.2×10−3, ∆H0= 13 Oe and Γ = 2 .5
Oe. The Gilbert damping for even very thin YIG film
is extremely low, ∼6×10−5. Whereas, the value we
achieved is higher than the reported in the literature for
YIG thin films[ 2]. Also, the value of Γ is insignificant,
implying negligible contribution to the damping.
B. Cavity FMR
Fig.2(a) shows typical T= 300 K cavity-FMR
(f≈9.6 GHz) spectra for YIG film performed at dif-
ferentθH. The FMR spectra exhibit some universal
features: (i) Spin-Wave resonance (SWR) spectrum for
θH= 0o; (ii) rotating the Haway from the θH= 0o,
the SWR modes successively start diminishing, and at
certain critical angle θc(falls in a range of 30 −35o;
shaded region in Fig. 2(b)), all the modes vanish except
a single mode (uniform FMR mode). Further rotation
ofHforθH> θc, the SWR modes start re-emerging.
We observe that the SWR mode appearing at the higher
field side for θH> θc, represents an exchange-dominated
non-propagating surface mode[ 17–19]. The above dis-
cussed complexity in HrvsθHbehaviour has already
been realized in some material systems[ 19], including a
µ-thick YIG film[ 18]. The localized mode or surface
spin-wave mode appears for H∝bardblbut not⊥to the film-
plane[17–19]. WeassigntheSWR modesforthesequence
n= 1,2,3,...., as it provides the best correspondence
toHex∝n2, where, Hex=Hr(n)−Hr(0) defines ex-
change field[ 20]. The exchange stiffness can be obtained
by considering the modified Schreiber and Frait classical
approach using the mode number n2dependence of res-
onance field (inset Fig. 3(c))[ 20]. For a fixed frequency,
the exchange field Hexof thickness modes is determined
by subtracting the highest field resonance mode ( n= 1)
from the higher modes ( n∝negationslash= 1). In modified Schreiber
and Frait equation, the Hexshows direct dependency on
the exchange stiffness D:µ0Hex=Dπ2
d2n2(wheredis
2/s40/s99/s41/s40/s98/s41/s40/s97/s41
FIG. 1. Room temperature frequency dependent FMR measureme nts. (a) Representative FMR derivative spectra for differen t
frequencies at θH= 90o. (b) Resonance field vs. frequency data for θH= 90oandθH= 0oare represented using red and
blue data points, respectively. The fitting to both the data a re shown using black lines. (c) Linewidth vs. frequency data at
θH= 90o. The solid red circles represent experimental data, wherea s the solid black line represents ∆ Hfitting. Inhomogeneous
(∆Hinhom), Gilbert (∆ Hα) and two-magnon scattering (∆ HTMS) contributions to ∆ Hare shown using dashed green, solid
yellow and blue lines, respectively.
the film thickness). The linear fit of data shown in the
inset of Fig. 2(b) gives D= 3.15×10−17T.m2. The ex-
change stiffness constant Acan be determined using the
relationA=D MS/2. The calculated value is A= 2.05
pJ.m−1, which is comparable to the value calculated for
YIG,A= 3.7 pJ.m−1[20].
YIG thin films with in-plane easy magnetization ex-
hibit extrinsic uniaxial magnetic and intrinsic magne-
tocrystallinecubic anisotropies[ 21]. The total free energy
density for YIG(111) is given by[ 21,22]:
F=−HMS/bracketleftbigg
sinθHsinθMcos(φH−φM)
+cosθHcosθM/bracketrightbigg
+2πM2
Scos2θM−Kucos2θM
+K1
12/parenleftbigg7sin4θM−8sin2θM+4−
4√
2sin3θMcosθMcos3φM/parenrightbigg
+K2
108
−24sin6θM+45sin4θM−24sin2θM+4
−2√
2sin3θMcosθM/parenleftbig
5sin2θM−2/parenrightbig
cos3φM
+sin6θMcos6φM
(2)
The Eq. 2consists of the following different energy
terms; the first term is the Zeeman energy, the second
term is the demagnetization energy, the third term is
the out-of-plane uniaxial magnetocrystalline anisotropy
energyKu, and the last two terms are the first and sec-
ond order cubic magnetocrystalline anisotropy energies
(K1andK2), respectively. The total free energy density
equation is minimized by taking partial derivatives w.r.t.
toθMandφMtoobtaintheequilibriumorientationofthe
magnetization vector M(H), i.e.,∂F/∂θ M=∂F/∂φ M=
0. Theresonancefrequencyofuniformprecessionatequi-
librium condition is expressed as[ 21,23,24]:
ωres=γ
MSsinθM/bracketleftBigg
∂2F
∂θ2
M∂2F
∂φ2
M−/parenleftbigg∂2F
∂θM∂φM/parenrightbigg2/bracketrightBigg1/2
(3)
Mathematica is used to numerically solve the reso-nance condition described by Eq. 3for the energy den-
sity given by Eq. 2. The solution for a fixed frequency
is used to fit the angle dependent resonance data ( Hr
vs.θH) shown in Fig. 2(b). The main mode data
simulation is shown using a black line. The parame-
ters obtained from the simulation are Ku=−1.45×104
erg.cm−3,K1= 1.50×103erg.cm−3, andK2= 0.13×103
erg.cm−3. The calculated uniaxial anisotropy field value
isHu∼ −223 Oe.
The ∆Hmanifests the spin dynamics and related re-
laxation mechanisms in a magnetic system. The intrinsic
contribution to ∆ Harises due to Gilbert term ∆ Hint≈
∆Hα, whereas, the extrinsic contribution ∆ Hextconsists
of line broadening due to ∆ Hinhomand ∆HTMS. The
terms representing the precessional damping due to in-
trinsic and extrinsic contributions can be expressed in
different phenomenologicalforms. Figure 2(c) shows∆ H
as a function of θH. TheθHvariation of ∆ Hshows
distinct signatures due to different origins of magnetic
damping. We consider both ∆ Hintand ∆Hextmag-
netic damping contributions to the broadening of ∆ H,
∆H= ∆Hα+∆Hinhom+∆HTMS. The first term can
be expressed as[ 14]-
∆Hα=α
MS/bracketleftbigg∂2F
∂θ2
M+1
sin2θM∂2F
∂φ2
M/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂(2πf
γ)
∂Hr/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1
.(4)
The second term ∆ Hinhomhas a form[ 14]-
∆Hinhom=/vextendsingle/vextendsingle/vextendsingle/vextendsingledHr
d4πMeff/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆4πMeff+/vextendsingle/vextendsingle/vextendsingle/vextendsingledHr
dθH/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆θH.(5)
Where, the dispersion of magnitude and direction of
the 4πMeffare represented by ∆4 πMeffand ∆θH, re-
spectively. The ∆ Hinhomcontribution arises due to a
small spread of the sample parameters such as thickness,
internal fields, or orientation of crystallites within the
thin film. The third term ∆ HTMScan be written as[ 25]-
3/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53
/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s49/s50/s51/s32/s72
/s101/s120/s40/s107/s79/s101/s41
/s110/s50/s67/s114/s105/s116/s105/s99/s97/s108/s32/s97/s110/s103/s108/s101/s54/s53/s52/s51/s72
/s114/s32/s40/s32/s107/s79/s101/s32/s41/s50/s110/s32/s61/s32/s49
/s72/s32/s40/s32/s68/s101/s103/s114/s101/s101/s32/s41/s50 /s51 /s52 /s53 /s54/s68/s101/s114/s105/s118/s97/s116/s105/s118/s101/s32/s97/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41
/s72 /s32/s40/s107/s79/s101/s41/s48 /s49/s53 /s51/s48 /s52/s53 /s54/s48 /s55/s53 /s57/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48
/s32/s32
/s32/s69/s120/s112/s116/s46/s32/s68/s97/s116/s97
/s32 /s72
/s32 /s72
/s32 /s77
/s101/s102/s102
/s32
/s32 /s72
/s84/s77/s83/s72 /s32/s40/s79/s101/s41/s32
/s40/s100/s101/s103/s114/s101/s101 /s41/s40/s99/s41/s40/s98/s41/s40/s97/s41
FIG. 2. Room temperature out-of-plane angular θHdependence of FMR. (a) Derivative FMR spectra shown for diffe rentθH
performed at ≈9.6 GHz. (b) θHvariation of uniform mode and SWR modes of resonance field Hr. Inset: Exchange field
(Hex) vs mode number square ( n2). (c)θHvariation of the linewidth (∆ H), where, the experimental and simulated data are
represented by solid yellow circles and black line, respect ively. The different contributions ∆ Hα, ∆4πMeff, ∆θHand ∆HTMS
are represented by gray, purple, green and red lines, respec tively.
∆HTMS=/summationtext
i=1Γout
ifi(φH)
µ0γΦsin−1/radicalbigg√
ω2+(ω0/2)2−ω0/2√
ω2+(ω0/2)2+ω0/2,
Γout
i= Γ0
iΦA(θ−π/4)dHr(θH)
dω(θH)/slashbigg
dHr(θH=0)
dω(θH=0)
(6)
The prefactor Γout
idefines the TMS strength and has
aθHdependency in this case. The type and size of the
defects responsible for TMS is difficult to characterize
which makes it non-trivial to express the exact form of
Γout
i. Although, it mayhaveasimplified expressiongiven
in Eq.6, where, Γ0
iis a constant; A(θ−π/4), a step
function which makes sure that the TMS is deactivated
forθH< π/4; anddHr(θH)/dω(θH), a normalization
factor responsible for the θHdependence of the Γout
i.
In fig.2(c)the solid dark yellow circles and black solid
line represent the experimental and simulated ∆ HvsθH
data, respectively. We also plot contributions of different
terms such as ∆ Hα(blue color line), ∆4 πMeff(purple
colorline), ∆ θH(greencolorline) and ∆ HTMS(red color
line). The fitting provides following extracted parame-
ters,α= 1.3×10−3, ∆4πMeff= 58 Oe, ∆ θH= 0.29o
and Γ0
i= 1.3 Oe. The precessional damping calculated
fromthe∆ Hvs.θHcorroboratewiththevalueextracted
from the frequency dependence of ∆ Hdata (shown in
Fig.1(c));α= 1.2×10−3. The ∆ Hbroadening and
the overwhelmingly enhanced precessional damping are
thedirectconsequenceofcontributionsfromintrinsicand
extrinsic damping. Usually, the Gilbert term and the
inhomogeneity due to sample quality contribute to the
broadening of ∆ Hand enhanced αin YIG thin films. If
we interpret the ∆ HvsθHdata, it is clear that damping
enhancement in YIG is arising from the extrinsic mag-
netic inhomogeneity.The role of an interface in YIG coupled with metals
or insulators leading to the increments in ∆ Handαhas
been vastly explored. Wang et. al. [ 9] studied a variety
of insulating spacers between YIG and Pt to probe the
effect on spin pumping efficiency. Their results suggest
the generation of magnetic excitations in the adjacent
insulating layers due to the precessing magnetization in
YIG at resonance. This happens either due to fluctu-
ating correlated moments or antiferromagnetic ordering,
via interfacial exchange coupling, leading to ∆ Hbroad-
ening and enhanced precessional damping of the YIG[ 9].
Theimpurityrelaxationmechanismisalsoresponsiblefor
∆HbroadeningandenhancedmagneticdampinginYIG,
but is prominent only at low temperatures[ 16]. Strong
enhancement in magnetic damping of YIG capped with
Pt has been observed by Sun et. al. [ 8]. They suggest
ferromagneticorderingin an atomically thin Pt layerdue
to proximity with YIG at the YIG-Pt interface, dynam-
ically exchange couples to the spins in YIG[ 8]. In recent
years, some research groups have reported the presence
of a thin interfacial layer at the YIG-GGG interface[ 11–
13]. The 200 nm film we used in this study is of high
quality with a trails of sharp Laue oscillations [see Fig
3(a) in ref.[ 3]]. Thus it is quite clear that the observed
∆Hbroadening and enhanced αis not a consequence of
sample inhomogeneity. The formation of an interfacial
GdIG layer at the YIG-GGG interface, which exchange
couples with the YIG film may lead to ∆ Hbroadening
and increased α. Considering the above experimental ev-
idences leading to ∆ Hbroadening and enhanced Gilbert
dampingdueto couplingwithmetals andinsulators[ 8,9],
it is safe to assume that the interfacial GdIG layer at the
interface AFM exchange couples with the YIG[ 11–13],
and responsible for enhanced ∆ Handα.
Fig.3shows in-plane φHangular variation of Hr. We
4/s32/s68/s97/s116/s97
/s32/s84/s111/s116/s97/s108
/s32/s69/s120/s99/s104/s97/s110/s103/s101/s72
/s114/s32/s40/s79/s101/s41/s50/s48/s48/s32/s110/s109/s40/s97/s41
/s40/s98/s41
/s49/s48/s48/s32/s110/s109
/s72/s32/s40/s68/s101/s103/s114/s101/s101/s41
FIG. 3. (a) In-plane angular φHvariation of Hr. The exper-
imental data are represented by solid grey circles. Whereas ,
the simulated data for total and exchange (unidirectional)
anisotropy are represented by black and red solid lines, re-
spectively. (a) 200 nm thick YIG sample. (b) 100 nm thick
YIG sample.
simulate the in-plane HrvsφHangular variation using
the free energy densities provided in ref. [ 26] and an
additional term, −KEA.sinθM.cosφM, representing the
exchange anisotropy ( KEA). Even though φHvaria-
tion ofHrshown in Fig. 3(a) is not so appreciable
as the film is 200 nm thick, a very weak unidirectional
anisotropy trend is visible, suggesting an AFM exchange
coupling between the interface and YIG. It has been
shown that the large inhomogeneous 4 πMeffis a direct
consequence of the AFM exchange coupling at the inter-
face of LSMO and a growth induced interfacial layer[ 27].
The YIG thin film system due to the presence of a hard
ferrimagnetic GdIG interfacial layer possesses AFM ex-
change coupling[ 11–13]. A Bloch domain-wall-like spiral
moments arrangement takes place due to the AFM ex-
change coupling acrossthe interfacial GdIG and top bulk
YIG layer[ 11–13]. An exchange springlike characteris-
tic is found in YIG film due to the spiral arrangement
of the magnetic moments [ 11–13]. The FMR measure-
ment and the extracted value of ∆4 πMeffreflect inho-
mogeneous distribution of 4 πMeffin YIG-GdIG bilayer
system. The argument of Bloch domain-wall-like spiral
arrangement of moments is conceivable, as this arrange-
ment between the adjacent layers lowers the exchange
interaction energy[ 27]. To further substantiate the pres-ence of an interfacial AFM exchange coupling leading
to spin inhomogeneity at YIG-GdIG interface, we per-
formed in-plane φHvariation of Hron a relatively thin
YIG film ( ∼100 nm with growth conditions leading to
the formation of a GdIG interfacial layer[ 13]). Fig.3(b)
shows prominent feature of unidirectional anisotropydue
to AFM exchange coupling in 100 nm thick film. It is
evident that the interfacial layer exchange couples with
the rest of the YIG film and leads to a unidirectional
anisotropy. We observethatthe interfacialexchangecou-
pling may cause ∆ Hbroadening and enhanced αdue to
spin inhomogeneity at the YIG-GdIG interface, even in
a 200 nm thick YIG film.
IV. CONCLUSIONS
The effects of spin inhomogeneity at the YIG and
growth-induced GdIG interface on the magnetization dy-
namics of a 200 nm thick YIG film is studied extensively
using ferromagnetic resonance technique. The Gilbert
damping is almost two orders of magnitude larger
(∼1.2×10−3) than usually reported in YIG thin films.
The out-of-plane angular dependence of ∆ Hshows
an unusual behaviour which can only be justified after
considering extrinsic mechanism in combination with the
Gilbert contribution. The extracted parameters from
the ∆HvsθHsimulation are, (i) α= 1.3×10−3from
Gilbert term; (ii) ∆4 πMeff= 58 Oe and ∆ θH= 0.29o
from the inhomogeneity in effective magnetization and
anisotropy axes, respectively; (iii) Γ0
i= 1.3 Oe from
TMS. The TMS strength Γ is not so appreciable,
indicating high quality thin film with insignificant defect
sites. The AFM exchange coupling between YIG and
the interfacial GdIG layer causes exchange springlike
behaviour of the magnetic moments in YIG, leading to a
large ∆4 πMeff. The presence of large ∆4 πMeffimpels
the quick dragging of the precessional motion towards
equilibrium. A unidirectional behaviour is observed in
the in-plane angular variation of resonance field due to
the presence of an exchange anisotropy. This further
reinforces the spin inhomogeneity at the YIG-GdIG
interface due to the AFM exchange coupling.
ACKNOWLEDGEMENTS
We gratefully acknowledge the research support from
IIT Kanpur and SERB, Government of India (Grant
No. CRG/2018/000220). RK and DS acknowledge the
financial support from Max-Planck partner group. ZH
acknowledges financial support from Polish National
Agency for Academic Exchange under Ulam Fellowship.
The authors thank Veena Singh for her help with the
angular dependent FMR measurements.
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6 |
1110.3387v2.Atomistic_spin_dynamic_method_with_both_damping_and_moment_of_inertia_effects_included_from_first_principles.pdf | Atomistic spin dynamic method with both damping and moment of inertia e ects
included from rst principles
S. Bhattacharjee, L. Nordstr om, and J. Fransson
Department of Physics and Astronomy, Box 516, 75120, Uppsala University, Uppsala, Sweden
(Dated: October 28, 2018)
We consider spin dynamics for implementation in an atomistic framework and we address the
feasibility of capturing processes in the femtosecond regime by inclusion of moment of inertia. In the
spirit of an s-d-like interaction between the magnetization and electron spin, we derive a generalized
equation of motion for the magnetization dynamics in the semi-classical limit, which is non-local in
both space and time. Using this result we retain a generalized Landau-Lifshitz-Gilbert equation,
also including the moment of inertia, and demonstrate how the exchange interaction, damping, and
moment of inertia, all can be calculated from rst principles.
PACS numbers: 72.25.Rb, 71.70.Ej, 75.78.-n
In recent years there has been a huge increase in the in-
terest in fast magnetization processes on a femto-second
scale, which has been initialized by important develop-
ments in experimental techniques [1{5], as well as po-
tential technological applications [6]. From a theoretical
side, the otherwise trustworthy spin dynamical (SD) sim-
ulation method fails to treat this fast dynamics due to
the short time and length scales involved. Attempts have
been made to generalize the mesoscopic SD method to an
atomistic SD, in which the dynamics of each individual
atomic magnetic moment is treated [7, 8]. While this
approach should in principle be well suited to simulate
the fast dynamics observed in experiments, it has not
yet reached full predictive power as it has inherited phe-
nomenological parameters, e.g. Gilbert damping, from
the mesoscopic SD. The Gilbert damping parameter is
well established in the latter regime but it is not totally
clear how it should be transferred to the atomic regime.
In addition, very recently it was pointed out that the mo-
ment of inertia, which typically is neglected, plays an im-
portant role for fast processes [9]. In this Letter we derive
the foundations for an atomistic SD where all the rele-
vant parameters, such as the exchange coupling, Gilbert
damping, and moment of inertia, can be calculated from
rst principles electronic structure methods.
Usually the spin dynamics is described by the phe-
nomenological Landau-Lifshitz-Gilbert (LLG) equation
[10, 11] which is composed of precessional and damping
terms driving the dynamics to an equilibrium. By in-
cluding the moment of inertia, we arrive at a generalized
LLG equation
_M=M(
B+^G_M+^IM) (1)
where ^Gand^Iare the Gilbert damping and the moment
of inertia tensors, respectively. In this equation the e ec-
tive eld Bincludes both the external and internal elds,
of which the latter includes the exchange coupling and
anisotropy e ects. Here, we will for convenience include
the anisotropy arising from the classical dipole-dipole in-
teraction responsible for the shape anisotropy as a partof the external eld. The damping term in the LLG
equation usually consists of a single damping parame-
ter, which essentially means that the time scales of the
magnetization variables and the environmental variables
arewell separated . This separation naturally brings a
limitation to the LLG equation concerning its time scale
which is restricting it to the mesoscopic regime.
The addition of a moment of inertia term to the LLG
equation can be justi es as follows. A general process
of a moment Munder the in
uence of a eld Fis al-
ways endowed with inertial e ects at higher frequencies
[12]. The eld Fand moment Mcan, for example, be
stress and strain for mechanical relaxation, electric eld
and electric dipole moment in the case of dielectric re-
laxation, or magnetic eld and magnetic moment in the
case of magnetic relaxation. In this Letter we focus on
the latter case | the origin of the moment of inertia in
SD. The moment of inertia leads to nutations of the mag-
netic moments, see Fig. 1. Its wobbling variation of the
azimuthal angle has a crucial role in fast SD, such as fast
magnetization reversal processes.
In the case of dielectric relaxation the inertial e ects
are quite thoroughly mentioned in the literature [13, 14],
especially in the case of ferroelectric relaxors. Co ey et
al.[14] have proposed inertia corrected Debye's theory of
dielectric relaxation and showed that by including inertial
B
precessionĜ
dampingÎ
nutation
FIG. 1: The three contributions in Eq. (1), the bare preces-
sion arising from the e ective magnetic eld, and the super-
imposed e ects from the Gilbert damping and the moment of
inertia, respectively.arXiv:1110.3387v2 [cond-mat.stat-mech] 1 Feb 20122
e ects, the unphysical high frequency divergence of the
absorption co-ecient is removed.
Very recently Ciornei et al [9] have extended the LLG
equation to include the inertial e ects through a mag-
netic retardation term in addition to precessional and
damping terms. They considered a collection of uni-
formly magnetized particles and treated the total angular
momentum Las faster variable. They obtained Eq. (1)
from a Fokker-Plank equation where the number den-
sity of magnetized particles were calculated by integrat-
ing a non-equilibrium distribution function over faster
variables such that faster degrees of freedom appear as
parameter in the calculation.
The authors showed that at very short time scales the
inertial e ects become important as the precessional mo-
tion of magnetic moment gets superimposed with nuta-
tion loops due to inertial e ects. It is pointed out that
the existence of inertia driven magnetization dynamics
open up a pathway for ultrafast magnetic switching [15]
beyond the limitation [16] of the precessional switching.
In practice, to perform atomistic spin dynamics simu-
lations the knowledge of ^Gand^Iis necessary. There are
recent proposals [17, 18] of how to calculate the Gilbert
damping factor from rst principles in terms of Kubo-
Greenwood like formulas. Here, we show that similar
techniques may by employed to calculate the moment of
inertia tensor ^I. Finally, we present a microscopical jus-
ti cation of Eq. (1), considering a collective magnetiza-
tion density interacting locally with electrons constitut-
ing spin moments. Such a description would in principle
be consistent with the study of magnetization dynam-
ics where the exchange parameters are extracted from
rst-principles electronic structure calculations, e.g den-
sity functional theory (DFT) methods. We nd that in
an atomistic limit Eq. (1) actually has to be general-
ized slightly as both the damping and inertia tensors are
naturally non-local in the same way as the exchange cou-
pling included in the e ective magnetic eld B. From
our study it is clear that both the damping and the mo-
ment of inertia e ects naturally arise from the retarded
exchange interaction.
We begin by considering the magnetic energy E=M
B. Using that its time derivative is _E=M_B+_MB
along with Eq. (1), we write
_E=M_B+1
_M
^G_M+^IM
: (2)
Relating the rate of change of the total energy to the
HamiltonianH, through _E=hdH=dti, and expanding
theHlinearly around its static magnetization M0, with
M(t) =M0+(t), we can writeHH 0+(t)rH0,
whereH0=H(M0). Then the rate of change of the total
energy equals _E= _hrHito the rst order. Following
Ref. [19] and assuming suciently slow dynamics such
that(t0) =(t) _(t) + 2(t)=2, =t t0, we canwrite the rate of change of the magnetic energy as
_E= lim
!!0_i[ ij(!)j+i@! ij(!) _j @2
! ij(!)j=2]:
(3)
Here, ij(!) =R
( i)( )h[@iH0(t);@jH0(t0)]iei! dt0,
=t t0, is the (generalized) exchange interaction
tensor out of which the damping and moments of in-
ertia can be extracted. Summation over repeated in-
dices (i;j=x;y;z ) is assumed in Eq. (3). Equat-
ing Eqs. (2) and (3) results in an internal contribu-
tion to the e ective eld about which the magnetiza-
tion precesses Bint=lim!!0 (!), the damping term
^G=
lim!!0i@! (!) as well as the moment of inertia
^I=
lim!!0@2
! (!)=2.
For a simple order of magnitude estimate of the damp-
ing and inertial coecients, ^Gand^I, respectively, we
may assume for a state close to a ferromagnetic state
that the spin resolved density of electron states (")
corresponding to the static magnetization con guration
H0is slowly varying with energy. At low temperatures
we, then, nd
^G2
sp [h@iH0ih@jH0i]"="F; (4)
in agreement with previous results [19]. Here, sp denotes
the trace over spin 1/2 space. By the same token, the
moment of inertia is estimated as
^I (
=D) sp [h@iH0ih@jH0i]"="F; (5)
where 2Dis the band width of density of electron states
of the host material. Typically, for metallic systems the
band width 2 D1|10 eV, which sets the time-scale
of the inertial contribution to the femto second (10 15
s) regime. It, therefore, de nes magnetization dynamics
on a time-scale that is one or more orders of magnitude
shorter compared to e.g. the precessional dynamics of the
magnetic moment.
Next, we consider the physics leading to the LLG equa-
tion given in Eq. (1). As there is hardly any microscopical
derivation of the LLG equation in the literature, we in-
clude here, for completeness the arguments that leads to
the equation for the spin-dynamics from a quantum eld
theory perspective.
In the atomic limit the spin degrees of freedom are
deeply intertwined with the electronic degrees of free-
dom, and hence the main environmental coupling is the
one to the electrons. In this study we are mainly con-
cerned with a mean eld description of the electron
structure, as in the spirit of the DFT. Then a natural
and quite general description of the magnetic interac-
tion due to electron-electron interactions on the atomic
site around rwithin the material is captured by the s-d-
like modelHint= R
J(r;r0)M(r;t)s(r0;t)drdr0, where
J(r;r0) represents the interaction between the magneti-
zation density Mand the electron spin s. From a DFT3
perspective the interaction parameter J(r;r0) is related
to the e ective spin dependent exchange-correlation func-
tionalBxc[M(r0)](r). For generality we assume a fully
relativistic treatment of the electrons, i.e. including the
spin-orbit coupling. In this interaction the dichotomy of
the electrons is displayed, they both form the magnetic
moments and provide the interaction among them.
Owing to the general non-equilibrium conditions in the
system, we de ne the action variable
S=I
CHintdt+SZ+SWZWN (6)
on the Keldysh contour [20{22]. Here, the ac-
tionSZ=
H
CR
Bext(r;t)M(r;t)dtdrrepresents
the Zeeman coupling to the external eld Bext(r;t),
whereas the Wess-Zumino-Witten-Novikov (WZWN)
termSWZWN =RH
CR1
0M(r;t; )[@ M(r;t; )
@tM(r;t; )]d dtjM(r)j 2drdescribes the Berry phase
accumulated by the magnetization.
In order to acquire an e ective model for the magne-
tization density M(r;t), we make a second order [23] ex-
pansion of the partition function Z[M(r;t)]trTCeiS,
and take the partial trace over the electronic degrees of
freedom in the action variable. The e ective action SM
for the magnetization dynamics arising from the mag-
netic interactions described in terms of Hint, can, thus,
be written
SM= I Z
M(r;t)D(r;r0;t;t0)M(r0;t0)drdr0dtdt0;
(7)
whereD(r;r0;t;t0) =R
J(r;r1)( i)hTs(r1;t)s(r2;t0)i
J(r2;r0)dr1dr2is a dyadic which describes the electron
mediated exchange interaction.
Conversion of the Keldysh contour integrations into
real time integrals on the interval ( 1;1) results in
S=Z
M(fast)(r;t)[M(r;t)_M(r;t)]dtjM(r)j 2dr
+Z
M(fast)(r;t)Dr(r;r0;t;t0)M(r0;t0)drdr0dtdt0
Z
Bext(r;t)M(fast)(r;t)dtdr; (8)
withM(fast)(r;t) =Mu(r;t) Ml(r;t) and M(r;t) =
[Mu(r;t) +Ml(r;t)]=2 which de ne fast and slow vari-
ables, respectively. Here, Mu(l)is the magnetization den-
sity de ned on the upper (lower) branch of the Keldysh
contour. Notice that upon conversion into the real time
domain, the contour ordered propagator Dis replaced by
its retarded counterpart Dr.
We obtain the equation of motion for the (slow) mag-
netization variable M(r;t) in the classical limit by mini-
mizing the action with respect to M(fast)(r;t), cross mul-
tiplying by M(r;t) under the assumption that the totalmoment is kept constant. We, thus, nd
_M(r;t) =M(r;t)
Bext(r;t)
+Z
Dr(r;r0;t;t0)M(r0;t0)dt0dr0
:(9)
Eq. (9) provides a generalized description of the semi-
classical magnetization dynamics compared to the LLG
Eq. (1) in the sense that it is non-local in both time and
space. The dynamics of the magnetization at some point
rdepends not only on the magnetization locally at r,
but also in a non-trivial way on the surrounding magne-
tization. The coupling of the magnetization at di erent
positions in space is mediated via the electrons in the
host material. Moreover, the magnetization dynamics is,
in general, a truly non-adiabatic process in which the
information about the past is crucial.
However, in order to make connection to the magne-
tization dynamics as described by e.g. the LLG equa-
tion as well as Eq. (1) above, we make the following
consideration. Assuming that the magnetization dy-
namics is slow compared to the electronic processes in-
volved in the time-non-local eld D(r;r0;t;t0), we ex-
pand the magnetization in time according to M(r0;t0)
M(r0;t) _M(r0;t) + 2M(r0;t)=2. Then for the inte-
grand in Eq. (9), we get
Dr(r;r0;t;t0)M(r0;t0) =
Dr(r;r0;t;t0)[M(r0;t) _M(r0;t) + 2
2M(r0;t)]:(10)
Here, we observe that as the exchange coupling for the
magnetization is non-local and mediated through D, this
is also true for the damping (second term) and the inertia
(third term).
In order to obtain an equation of the exact same
form as LLG in Eq. (1) we further have to assume
that the magnetization is close to a uniform ferromag-
netic state, then we can justify the approximations
_M(r0;t)_M(r;t) and M(r0;t)M(r;t). When
Bint= R
D(r;r0;t;t0)M(r0;t)dr0dt0=
is included in
the total e ective magnetic eld B, the tensors of Eq. (1)
^Gand^Ican be identi ed with R
D(r;r0;t;t0)dr0dt0
andR
2D(r;r0;t;t0)dr0dt0=2, respectively. From a rst
principles model of the host materials we have, thus, de-
rived the equation for the magnetization dynamics dis-
cussed in Ref. 9, where it was considered from purely
classical grounds. However it is clear that for a treatment
of atomistic SD that allows for all kinds of magnetic or-
ders, not only ferromagnetic, Eq. (1) is not sucient and
the more general LLG equation of Eq. (9) together with
Eq. (10) has to be used.
We nally describe how the parameters of Eq. (1)
can be calculated from a rst principles point of view.
Within the conditions de ned by the DFT system, the
interaction tensor Dris time local which allows us to4
write lim "!0i@"Dr(r;r0;") =R
Dr(r;r0;t;t0)dt0and
lim"!0@2
"Dr(r;r0;") = R
2Dr(r;r0;t;t0)dt0, where
Dr(r;r0;") = 4 spZ
JrJ0r0f(!) f(!0)
" !+!0+i
ImGr
0(!)ImGr
0(!0)d!
2d!0
2dd0:(11)
Here,Jrr0J(r;r0) whereas Gr
rr0(!)Gr(r;r0;!) is
the retarded GF, represented as a 2 2-matrix in spin-
spaces. We notice that the above result presents a general
expression for frequency dependent exchange interaction.
Using Kramers-Kr onig's relations in the limit "!0, it
is easy to see that Eq. (11) leads to
Dr(r;r0; 0) = 1
sp ImZ
JrJ0r0f(!)
Gr
0(!)Gr
0(!)d!dd0;(12)
in agreement with e.g. Ref. [24]. We can make connection
with previous results, e.g. Refs. 25, 26, and observe that
Eq. (11) contains the isotropic Heisenberg, anisotropic
Ising, and Dzyaloshinsky-Moriya exchange interactions
between the magnetization densities at di erent points
in space [22], as well as the onsite contribution to the
magnetic anisotropy.
Using the result in Eq. (11), we nd that the damping
tensor is naturally non-local and can be reduced to
^G(r;r0) =1
spZ
JrJ0r0f0(!)
ImGr
0(!)ImGr
0(!)d!dd0;(13)
which besides the non-locality is in good accordance with
the results in Refs. [17, 25], and is closely connected to
the so-called torque-torque correlation model [27]. With
inclusion of the the spin-orbit coupling in Gr, it has been
demonstrated that Eq. (13) leads to a local Gilbert damp-
ing of the correct order of magnitude for the case of fer-
romagnetic permalloys [17].
Another application of Kramers-Kr onig's relations
leads, after some algebra, to the moment of inertia tensor
^I(r;r0) = spZ
JrJ0r0f(!)[ImGr
0(!)@2
!ReGr
0(!)
+ ImGr
0(!)@2
!ReGr
0(!)]d!
2dd0;(14)
where we notice that the moment of inertia is not sim-
ply a Fermi surface e ect but depends on the electronic
structure as a whole of the host material. Although the
structure of this expression is in line with the exchange
coupling in Eq. (12) and the damping of Eq. (13), it is
a little more cumbersome to compute due the presence
of the derivatives of the Green's functions. Note that it
is not possible to get completely rid of the derivatives
through partial integration. These derivatives also makethe moment of inertia very sensitive to details of the elec-
tronic structure, which has a few implications. Firstly the
moment of inertia can take large values for narrow band
magnetic materials, such as strongly correlated electron
systems, where these derivatives are substantial. For
such systems the action of moment of inertia can be im-
portant for longer time scales too, as indicated by Eq. (5).
Secondly, the moment of inertia may be strongly depen-
dent on the reference magnetic ordering for which it is
calculated. It is well known that already the exchange
tensor parameters may depend on the magnetic order.
It is the task of future studies to determine how trans-
ferable the moment of inertia tensor as well as damping
tensor are in-between di erent magnetic ordering.
In conclusion, we have derived a method for atomistic
spin dynamics which would be applicable for ultrafast
(femtosecond) processes. Using a general s-d-like interac-
tion between the magnetization density and electron spin,
we show that magnetization couples to the surrounding
in a non-adiabatic fashion, something which will allow for
studies of general magnetic orders on an atomistic level,
not only ferromagnetic. By showing that our method
capture previous formulas for the exchange interaction
and damping tensor parameter, we also derive a formula
for calculating the moment of inertia from rst principles.
In addition our results point out that all parameters are
non-local as they enter naturally as bilinear sums in the
same fashion as the well established exchange coupling.
Our results are straight-forward to implement in existing
atomistic SD codes, so we look on with anticipation to
the rst applications of the presented theory which would
be fully parameter-free and hence can take a large step
towards simulations with predictive capacity.
Support from the Swedish Research Council is ac-
knowledged. We are grateful for fruitful and encouraging
discussions with A. Bergman, L. Bergqvist, O. Eriksson,
C. Etz, B. Sanyal, and A. Taroni. J.F. also acknowledges
discussions with J. -X. Zhu.
Electronic address: Jonas.Fransson@physics.uu.se
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0708.0463v1.Strong_spin_orbit_induced_Gilbert_damping_and_g_shift_in_iron_platinum_nanoparticles.pdf | arXiv:0708.0463v1 [cond-mat.other] 3 Aug 2007Strong spin-orbit induced Gilbert damping and g-shift in ir on-platinum nanoparticles
J¨ urgen K¨ otzler, Detlef G¨ orlitz, and Frank Wiekhorst
Institut f¨ ur Angewandte Physik und Zentrum f¨ ur Mikrostruk turforschung,
Universit¨ at Hamburg, Jungiusstrasse 11, D-20355 Hamburg , Germany
(Dated: October 30, 2018)
The shape of ferromagnetic resonance spectra of highly disp ersed, chemically disordered
Fe0.2Pt0.8nanospheres is perfectly described by the solution of the La ndau-Lifshitz-Gilbert (LLG)
equation excluding effects by crystalline anisotropy and su perparamagnetic fluctuations. Upon
decreasing temperature , the LLG damping α(T) and a negative g-shift,g(T)−g0, increase pro-
portional to the particle magnetic moments determined from the Langevin analysis of the magneti-
zation isotherms. These novel features are explained by the scattering of the q→0 magnon from
an electron-hole (e/h) pair mediated by the spin-orbit coup ling, while the sd-exchange can be ruled
out. The large saturation values, α(0) = 0.76 andg(0)/g0−1 =−0.37, indicate the dominance of
an overdamped 1 meV e/h-pair which seems to originate from th e discrete levels of the itinerant
electrons in the dp= 3nmnanoparticles.
PACS numbers: 76.50.+g, 78.67.Bf, 76.30.-v, 76.60.Es
I. INTRODUCTION
Theongoingdownscalingofmagneto-electronicdevices
maintainstheyetintenseresearchofspindynamicsinfer-
romagnetic structures with restricted dimensions. The
effect of surfaces, interfaces, and disorder in ultrathin
films1, multilayers, and nanowires2has been examined
and discussed in great detail. On structures confined to
the nm-scale in all three dimensions, like ferromagnetic
nanoparticles, the impact of anisotropy3and particle-
particle interactions4on the Ne´ el-Brown type dynamics,
which controls the switching of the longitudinal magneti-
zation, is now also well understood. On the other hand,
the dynamics of the transverse magnetization, which e.g.
determines the externally induced, ultrafast magnetic
switching in ferromagnetic nanoparticles, is still a top-
ical issue. Such fast switching requires a large, i.e. a
critical value of the LLG damping parameter α5. This
damping has been studied by conventional6,7and, more
recently, by advanced8ferromagnetic resonance (FMR)
techniques, revealing enhanced values of αup to the or-
der of one.
By now, the LLG damping of bulkferromagnets is al-
most quantitatively explained by the scattering of the
q= 0 magnon by conduction electron-hole (e/h) pairs
due to the spin-orbit coupling Ω so9. According to recent
ab initio bandstructure calculations10, the rather small
values for α≈Ω2
soτresult from the small (Drude) re-
laxation time τof the electrons. For nanoparticles, the
Drude scattering and also the wave-vector conservation
are ill-defined, and ab initio many-body approaches to
the spin dynamics should be more appropriate. Numeri-
cal workby Cehovin et al.11considersthe modification of
the FMR spectrum by the discrete level structure of the
itinerant electrons in the particle. However, the effect of
the resulting electron-hole excitation, ǫp∼v−1
p, wherevp
is the nanoparticle volume on the intrinsic damping has
not yet been considered.
Here we present FMR-spectra recorded atω/2π=9.1 GHz on Fe0.2Pt0.8nanospheres, the struc-
tural and magnetic properties of which are summarized
in Sect. II. In Sect. III the measured FMR-shapes will be
examined by solutions of the LLG-equation of motion for
the particle moments. Several effects, in particular those
predicted for crystalline anisotropy12and superparam-
agnetic (SPM) fluctuations of the particle moments13
will be considered. In Sect. IV, the central results of this
study, i.e. the LLG-damping α(T) reaching values of
almost one and a large g-shift,g(T)−g0, are presented.
Since both α(T) and g(T) increase proportional to the
particle magnetization, they can be related to spin-orbit
damping torques, which, due to the large values of αand
∆gare rather strongly correlated. It will be discussed
which features of the e/h-excitations are responsible for
these correlations in a nanoparticle. A summary and the
conclusions are given in the final section.
II. NANOPARTICLE CHARACTERIZATION
The nanoparticleassemblyhasbeen prepared14follow-
ing the wet-chemical route by Sun et al.15. In order
to minimize the effect of particle-particle interactions,
the nanoparticles were highly dispersed14. Transmis-
sion electron microscopy (TEM) revealed nearly spher-
ical shapes with mean diameter dp= 3.1nmand a
rather small width of the log-normal size distribution,
σd= 0.17(3). Wide angle X-ray diffraction provided the
chemically disordered fccstructure with a lattice con-
stant a 0=0.3861 nm14.
Themeanmagneticmomentsofthenanospheres µp(T)
have been extracted from fits of the magnetization
isotherms M(H,T), measured by a SQUID magnetome-
ter (QUANTUM DESIGN, MPMS2) in units emu/g=
1.1·1020µB/g, to
ML(H,T) =Npµp(T)L(µpH
kBT). (1)2
Here are L(y) = coth( y)−y−1withy=µp(T)H/kBT
the Langevin function and Npthe number of nanoparti-
cles per gram. The fits shown in Fig.1(a) demanded for
a small paramagnetic background, M−ML=χb(T)H,
with a strong Curie-liketemperature variationof the sus-
ceptibility χb, signalizing the presence of paramagnetic
impurities. According to the inset of Fig. 1(b) this 1 /T-
law turns out to agree with the temperature dependence
of the intensity of a weak, narrow magnetic resonance
withgi= 4.3 depicted in Figs. 2 and 3. Such narrow
line with the same g-factor has been observed by Berger
et al.16on partially crystallized iron-containing borate
glass and could be traced to isolated Fe3+ions.
The results for µp(T) depicted in Fig.1(b) show the
moments to saturate at µp(T→0) = (910 ±30)µB.
This yields a mean moment per atom in the fccunit
cell ofµ(0) =µpa3
0/4vp= 0.7µBcorresponding to a
spontaneous magnetization Ms(0) = 5.5kOe. Accord-
ing to previous work by Menshikov et al.17on chemi-
cally disordered FexPt1−xthis corresponds to an iron-
concentration of x=0.20. Upon rising temperature the
moments decrease rapidly, which above 40 K can be
rather well parameterized by the empirical power law,
µp(T≥40K)∼(1−T/TC)βrevealing β= 2 and for
the Curie temperature TC= (320±20)K. This is con-
sistent with TC= (310±10)KforFe0.2Pt0.8emerging
from a slight extrapolation of results for TC(x≥0.25)
ofFexPt1−x18. No quantitative argument is at hand
for the exponent β= 2, which is much larger than the
mean field value βMF= 1/2. We believe that β= 2
may arise from a reduced thermal stability of the magne-
tization due to strong fluctuations of the ferromagnetic
exchange between FeandPtin the disordered struc-
ture and also to additional effects of the antiferromag-
neticFe−FeandPt−Ptexchange interactions. In
this context, it may be interesting to note that for low
Feconcentrations, x≤0.3, bulkFexPt1−xexhibits fer-
romagnetism only in the disordered structure17, while
structural ordering leads to para- or antiferromagnetism.
Recent first-principle calculations of the electronic struc-
ture produced clear evidence for the stabilizing effect of
disorder on the ferromagnetism in FePt19.
From the Langevin fits in Fig.1(a) we obtain for the
particle density Np= 3.5·1017g−1. Basing on the well
known mass densities of Fe0.2Pt0.8and the organic ma-
trix, we find by a little calculation20for the volume con-
centration of the particles cp= 0.013 and, hence, for the
mean inter-particle distance, dpp=dp/c1/3
p= 13.5nm.
This implies forthe maximum (i.e. T=0) dipolar interac-
tion between nearest particles, µ2
p(0)/4πµ0d3
pp= 0.20K,
so that at the present temperatures, T≥20K, the sam-
ple should act as an ensemble of independent ferromag-
netic nanospheres. Since also the blocking temperature,
Tb= 9K, as determined from the maximum of the ac-
susceptibility at 0.1 Hz in zero magnetic field20, turned
out to be low, the Langevin-analysis in Fig.1(a) is valid.FIG. 1: Fig. 1. (a) Magnetization isotherms of the
nanospheres fitted to the Langevin model plus a small para-
magnetic background χb·H; (b) temperature dependences
of the magnetic moments of the nanoparticles µpand of the
background susceptibility χb( inset units are emu/g kOe )
fitted to the indicated relations with TC= (320±20)K. The
inset shows also data from the intensity of the paramagnetic
resonance at 1.45 kOe, see Figs. 2 and 3.
III. RESONANCE SHAPE
Magnetic resonances at a fixed X-band frequency of
9.095 GHz have been recorded by a home-made mi-
crowave reflectometer equipped with field modulation to
enhancethesensitivity. Adouble-walledquartztubecon-
taining the sample powder has been inserted to a mul-
tipurpose, gold-plated VARIAN cavity (model V-4531).
Keeping the cavity at room temperature, the sample
couldbeeithercooledbymeansofacontinuousflowcryo-
stat (Oxford Instruments, model ESR 900) down to 15 K
orheatedupto500Kbyanexternal Pt-resistancewire20.
At all temperatures, the incident microwave power was
varied in order to assure the linear response.
Someexamplesofthespectrarecordedbelowthe Curie3
FIG. 2: (a) Derivative of the microwave (f=9.095 GHz) ab-
sorption spectrum recorded at T=52 K i.e. close to magnetic
saturation of the nanospheres. The solid and dashed curves
are based on fits to Eqs.(4) and (8), respectively, which both
ignore SPM fluctuations and assume either a g-shift and zero
anisotropy field HA(’∆g-FM’) or ∆ g= 0 and a randomly
distributed HA=0.5 kOe (’a-FM’), Eq. (9). Also shown are
fits to predictions by Eq.(11), which account for SPM fluctu-
ations, with HA=0.5 kOe and ∆ g= 0 (’a-SPM’) and, using
Eq.(11), for ∆ g/negationslash= 0 and HA=0 (’∆g-SPM’). The weak, nar-
row resonance at 1.45 kOe is attributed to the paramagnetic
background with g= 4.3±0.1 indicating Fe3+ 16impurities.
temperature are shown in Figs.2 and 3. The spectra have
been measured from -9.5 kOe to +9.5 kOe and proved to
be independent of the sign of H and free of any hystere-
sis. Thiscanbeexpectedduetothecompletelyreversible
behavior of the magnetization isotherms above 20 K and
the low blocking temperature of the particles. Lower-
ing the temperature, we observe a downward shift of the
mainresonanceaccompaniedbyastrongbroadening. On
the other hand, the position and width of the weak nar-
row line at (1 .50±0.05)kOecorresponding to gi∼=4.3
remain independent of temperature. This can be at-
tributed to the previously detected paramagnetic Fe3+-
impurities16and is supported by the integrated intensity
of this impurity resonance Ii(T) evaluated from the am-
plitudedifferenceofthederivativepeaks. Sincetheinten-
sity of a paramagnetic resonance is given by the param-
agnetic susceptibility, Ii(T)∼/integraltext
dH χ′′
xx(H,T)∼χi(T)
can be compared directly to the background suscepti-
bilityχb(T), see inset to Fig.1(b). The good agree-
ment between both temperature dependencies suggests
to attribute χbto these Fe3+-impurities. An analysis of
the fitted Curie-constant, Ci= 5emuK/g kOe , yields
Ni= 164.1017g−1for theFe3+- concentration, which
corresponds to 50 Fe3+per 1150 atoms of a nanosphere.
With regard to the main intensity, we want to extract
a maximum possible information, in particular, on theFIG. 3: Fig. 3. Derivative spectra at some representative
temperatures and fits to Eq.(4). The LLG-damping, g-shift,
and intensities are depicted in Fig. 4.
intrinsic magnetic damping in nanoparticles. Unlike the
conventional analysis of resonance fields and linewidths,
as applied e.g. to Ni6and Co7nanoparticles, our ob-
jective is a complete shape analysis in order to disen-
tangle effects by the crystalline anisotropy12, by SPM
fluctuations13, by an electronic g-shift21, and by differ-
ent forms ofthe damping torque /vectorR22. Additional difficul-
ties may enter the analysis due to non-spherical particle
shapes, size distributions and particle interaction, all of
which, however, can be safely excluded for the present
nanoparticle assembly.
The starting point of most FMR analyses is the phe-
nomenological equation of motion for a particle moment
( see e.g. Ref. 13 )
d
dt/vector µp=γ/vectorHeff×/vector µp−/vectorR , (2)
using either the original Landau-Lifshitz (LL) damping
with damping frequency λL
/vectorRL=λL
Ms(/vectorHeff×/vector µp)×/vector sp ,(3)
ortheGilbert-dampingwiththe Gilbertdampingparam-
eterαG,
/vectorRG=αGd/vector µp
dt×/vector sp , (4)
where/vector sp=/vector µp/µpdenotes the direction of the particle
moment. In Eq.(2), the gyromagneticratio, γ=g0µB/¯h,
is determined by the regular g-factorg0of the precessing
moments. It shouldperhapsbe notedthat the validityof
the micromagnetic approximationunderlying Eq. (2) has
been questioned23for volumes smaller than (2 λsw(T))3,4
whereλsw= 2a0TC/Tis the smallest wavelengthof ther-
mally excited spin waves. For the present particles, this
estimate leads to a fairly large temperature of ∼0.7TC
up to which micromagnetics should hold.
At first, we ignore the anisotropy being small in cubic
FexPt1−x24,25, so that for the present nanospheres the
effective field is identical to the applied field, /vectorHeff=
/vectorH. Then, the solutions of Eq. (2) for the susceptibility
of the two normal, i.e. circularly polarized modes, of
Npindependent nanoparticles per gram take the simple
forms
χL
±(H) =Npµpγ1∓iα
γH(1∓iα)∓ω(5)
for/vectorR=/vectorRLwithα=λL/γMsand for the Gilbert torque
/vectorRG
χG
±(H) =Npµpγ1
γH∓ω(1+iαG).(6)
For the LL damping, the experimental, transverse sus-
ceptibility, χxx=1
2(χ++χ−), takes the form
χL
xx(H) =NpµpγγH(1+α2)−iαω
(γH)2(1+α2)−ω2−2iαωγH.
(7)
As thesameshape isobtainedforthe Gilbert torquewith
α=αG, the damping is frequently denoted as LLG pa-
rameter. However, the gyromagnetic ratio in Eq.(7) has
tobereplacedby γ/(1+α2), whichonlyfor α≪1implies
also the same resonance field Hr. Upon increasing the
damping up to α≈0.7 (the regime of interest here), the
resonance field HrofχG
xx(H) , determined by dχ′′/dH=
0,remains constant, HG
r≈ω/γ, whileHL
rdecreases
rapidly. After renormalization γ/(1+α2) the resonance
fields and also the shapes of χL(H) andχG(H) become
identical. This effect should be observed when determin-
ing theg-factor from the resonance fields of broad lines.
It becomes even more important if the downward shift of
Hris attributed to anisotropy, as done recently for the
ratherbroadFMR absorptionof FexPt1−xnanoparticles
with larger Fe-content, x≥0.326.
In order to check here for both damping torques,
we selected the shape measured at a low temperature,
T=52 K, where the linewidth proved to be large (see
Fig. 2) and the magnetic moment µp(T) was close to sat-
uration (Fig. 1(b)). None of both damping terms could
explain both, the observed resonance field Hrand the
linewidth ∆ H=αω/γ, and, hence, the lineshape. By
using/vectorRG, the shift of Hrfromω/γ= 3.00kOewas not
reproduced by HG
r=ω/γ, while for /vectorRLthe resonance
fieldHL
r, demanded by the line width, became signifi-
cantly smaller than Hr.
This result suggested to consider as next the effect of a
crystallineanisotropyfield /vectorHAonthetransversesuscepti-
bility, which has been calculated by Netzelmann from thefree energy of a ferromagnetic grain12. Specializing his
general ansatz to a uniaxial /vectorHAoriented at angles ( θ,φ)
withrespecttothedc-field /vectorH||/vector ezandthemicrowavefield,
one obtains by minimizing
F(θ,φ,ϑ,ϕ) =−µp[Hcosϑ+
1
2HA(sinϑsinθ−cos(ϕ−φ)+cosθcosϑ)2] (8)
the equilibrium orientation ( ϑ0,ϕ0) of the moment /vector µpof
a spherical grain. After performing the trivial average
overφ, one finds for the transverse susceptibility of a
particle with orientation θ
χL
xx(θ,H) =γµp
2×
(Fϑ0ϑ0+Fϕ0ϕ0/tan2ϑ0)(1+α2)−iαµpω(1+cos2ϑ0)
(1+α2)(γHeff)2−ω2−iαωγ∆H.
(9)
HereH2
eff= (Fϑ0ϑ0Fϕ0ϕ0−F2
ϑ0ϕ0/(µpsinϑ0)2) and
∆H= (Fϑ0ϑ0+Fϕ0ϕ0/sin2ϑ0)/µpare given by the sec-
ond derivatives of F at the equilibrium orientation of /vector µp.
For the randomly distributed /vectorHAofNpindependent par-
ticles per gram one has
χL
xx(H) =/integraldisplayπ/2
0d(cosθ)χL
xx(θ,H).(10)
In a strict sense, this result should be valid at fields
larger than the so called thermal fluctuation field HT=
kBT/µp(T), see e.g. Ref. 13, which for the present case
amounts to HT= 1.0kOe. Hence, in Fig. 2 we fit-
ted the data starting at high fields, reaching there an
almost perfect agreement with the curve a-FM. The fit
yields a rather small HA= 0.5kOewhich implies a small
anisotropyenergyper atom, EA=1
2µp(0)HA= 1.0µeV.
This number is smaller than the calculated value for bulk
fcc FePt ,EA= 4.0µeV25, most probably due to the
lowerFe-concentration (x=0.20) and the strong struc-
tural disorder in our nanospheres. We emphasize, that
the main defect of this a-FM fit curve arises from the
finite value of dχ′′
xx/dHatH= 0. By means of Eq. (9)
one finds χ′′
xx(H→0,θ)∼HAH/ω2, which remains fi-
nite even after averaging over all orientations according
toθ(Eq. (10)).
Thefinite value ofthe derivativeof χ′′
xx(H→0)should
disappear if superparamagnetic (SPM) fluctuations of
the particles are taken into account. Classical work27
predicted the anisotropy field to be reduced by SPM,
HA(y) =HA·(1/L(y)−3/y), which for y=H/HT≪1
impliesHA(y) =HA·y/5 and, therefore, χ′′
xx(H→0)∼
H2. A statistical theory for χL
xx(H,T) which considers
the effect of SPM fluctuations exists only to first order in
HA/H21. The result of this linear model (LM) in HA/H
which generalizes Eq. (4), can be cast in the form
χLM
±(θ,H) =NpµpL(y)γ(1+A∓iαA)
γ(1+B∓iαB)H∓ω.
(11)5
The additional parameters are given in Ref. 21 and con-
tain, depending on the symmetry of HA, higher-order
Langevin functions Lj(y) and their derivatives. Observ-
ing the validity of the LM for H≫HA= 0.5kOe, we
fitted the data in Fig. 2 to Eq. 11 with χLM
xx(θ,H) =
1
2(χLM
++χLM
−) at larger fields. There one has also H≫
HT= 1.0kOeand the fit, denoted as a-SPM, agrees
with the ferromagnetic result (a-FM). However, increas-
ing deviations appear below fields of 4 kOe. By varying
HAandα, we tried to improve the fit near the resonance
Hr= 2.3kOeand obtained unsatisfying results. For low
anisotropy, HA≤3kOe, the resonance field could be
reproduced only by significantly lower values of α, which
are inconsistent with the measured width and shape. For
HA>3kOe, a small shift of Hroccurs, but at the
same time the lineshape became distorted, tending to a
two-peak structure also found in previous simulations13.
Even at the lowest temperature, T= 22K, where the
thermal field drops to HT= 0.4kOe, no signatures of
such inhomogeneous broadening appear (see Fig. 3). Fi-
nally, it should be mentioned that all above attempts to
incorporate the anisotropy in the discussion of the line-
shape were based on the simplest non-trivial, i.e. uni-
axial symmetry, which for FePtwas also considered by
the theory25. For cubic anisotropy, the same qualitative
discrepancies were found in our simulations20. This in-
sensitivitywith respecttothe symmetry of HAoriginates
from the orientational averaging in the range of the HA-
values of relevance here.
As a finite anisotropy failed to reproduce Hr,∆H, and
also the shape, we tried a novel ansatz for the magnetic
resonanceofnanoparticlesbyintroducingacomplexLLG
parameter,
ˆα(T) =α(T)−i β(T). (12)
According to Eq. (4) this is equivalent to a negative g-
shift,g(T)−g0=−β(T)g0, which is intended to com-
pensate the too large downward shift of HL
rdemanded
byχL
xx(H) due to the large linewidth. In fact, insert-
ing this ansatz in Eq. (5), the fit, denoted as ∆ g-FM
in Fig. 2, provides a convincing description of the line-
shape down to zero magnetic field. It may be interest-
ing to note that the resulting parameters, α= 0.56 and
β= 0.27, revealed the same shape as obtained by using
the Gilbert-susceptibilities, Eq. (6).
In spite of the agreement of the ∆ g-FM model with
the data, we also tried to include here SPM fluctuations
by using ˆ α(T,H) = ˆα(T)(1/L(y)−1/y)21forHA= 0.
The result, designated as ∆ g-SPM in Fig. 2 agrees with
the ∆g-FM curve for H≫HTwhere ˆα(T,H) = ˆα(T),
but again significant deviations occur at lower fields.
They indicate that SPM fluctuations do not play any
role here, and this conclusion is also confirmed by the
results at higher temperatures. There, the thermal fluc-
tuation field, HT=kBT/µp(T), increases to values
larger than the maximum measuring field, H= 10kOe,
so that SPM fluctuations should cause a strong ther-/s48 /s51/s48/s48/s32/s75/s48/s55/s46/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s56/s49/s46/s48/s32
/s32
/s32/s97/s41
/s48/s46/s49/s56/s32/s43/s32/s48/s46/s53/s56/s32/s40/s32/s49/s45/s84/s47/s84
/s67/s32/s41/s32/s50
/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s53/s48/s48/s48/s46/s48/s48/s46/s50
/s48/s46/s51/s57/s32/s40/s32/s49/s45/s84/s47/s84
/s67/s32/s41/s32/s50
/s32/s84/s32/s40/s32/s75/s32/s41/s103/s32/s47/s32/s103
/s48
/s32/s32
/s98/s41/s32
/s32/s73/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s126/s32/s40/s32/s49/s45/s84/s47/s84
/s67/s32/s41/s32/s50
FIG. 4: Temperature variation (a) of the LLG-damping αand
(b) of the relative g-shifts with g0= 2.16 (following from the
resonance fields at T > T C). Within the error margins, α(T)
and ∆g(T) and also the fitted intensity of the LLG-shape
(see inset) display the same temperature dependence as the
particle moments in Fig. 1(b).
mal, homogeneous broadening of the resonance due to
ˆα(H≫HT) = ˆα·2HT/H. However, upon increasing
temperature, the fitted linewidths, (Fig. 3) and damping
parameters (Fig. 4) display the reverse behavior.
IV. COMPLEX DAMPING
In order to shed more light on the magnetization dy-
namics of the nanospheres we examined the tempera-
ture variation of the FMR spectra. Figure 3 shows
some examples recorded below the Curie temperature,
TC= 320K, together with fits to the a−FM model
outlined in the last section. Above TC, the resonance
fields and the linewidths are temperature independent
revealing a mean g−factor,g0= 2.16±0.02, and a
damping parameter ∆ H/Hr=α0= 0.18±0.01. Since
g0is consistent with a recent report on g-values of
FexPt1−xforx≥0.4321, we suspect that this resonance
arises from small FexPt1−x-clusters in the inhomoge-
neousFe0.2Pt0.8structure. Fluctuations of g0and of
local fields may be responsible for the rather large width.6
This interpretation is supported by the observation that
aboveTCthe lineshape is closer to a Gaussian than to
the Lorentzian following from Eq.(7) for small α.
The temperature variation for both components of the
complex damping, obtained from the fits below TCto
Eq. (7), are shown in Fig. 4. Clearly, they obey the same
powerlaw as the moments, µp(T), displayed in Fig. 1(b),
which implies
ˆα(T) = (α−i β)ms(T)+α0.(13)
Herems(T)=µp(T)/µp(0),α=0.58, and β=-∆g(0)/g0=
0.39 denote the reduced spontaneous magnetization and
the saturation values for the complex damping, respec-
tively. It should be emphasized that the fitted inten-
sityI(T) of the spectra, shown by the inset to Fig. 4(b),
exhibits the same temperature variation I(T)∼µp(T).
This behavior is predicted by the ferromagnetic model,
Eq. (7), and is a further indication for the absence of
SPM effects on the magnetic resonance. If the reso-
nance were dominated by SPM fluctuations, the inten-
sity should decrease like the SPM Curie-susceptibility,
ISPM(T)∼µ2
p(T)/T, following from Eq. (11), being
much stronger than the observed I(T).
At the beginning of a physical discussion of ˆ α(T), we
should point out that the almost perfect fits of the line-
shape to Eq. (7) indicate that the complex damping is
related to an intrinsic mechanism and that eventual in-
homogeneous effects by distributions of particle sizes and
shapes in the assembly, as well as by structural disorder
areratherunlikely. Sinceageneraltheoryofthemagneti-
zation dynamics in nanoparticles is not yet available, we
start with the current knowledge on the LLG-damping
inbulkand thin film ferromagnets, as recently reviewed
by B. Heinrich5. Based on experimental work on the
archetypal metallic ferromagnets and on recent ab initio
band structure calculations10there is now ratherfirm ev-
idence that the damping ofthe q=0-magnonis associated
with the torques /vectorTso=/vector ms×/summationtext
j(ξj/vectorLj×/vectorS) on the spin /vectorS
due to the spin-orbit interaction ξjat the lattice sites j.
The action of the torque is limited by the finite lifetime
τof an e/h excitation, the finite energy ǫof which may
cause a phase, i.e. a g-shift. As a result of this magnon
- e/h-pair scattering, the temperature dependent part of
the LLG damping parameter becomes
ˆα(T)−α0=λL(T)
γMs(T)
=(Ωso·ms(T))2
τ−1+i ǫ/¯h·1
γMs(T). (14)
Forintraband scattering, ǫ≪¯h/τ, the aforementioned
numerical work10revealed Ω so= 0.8·1011s−1and 0.3·
1011s−1as effective spin-orbit coupling in fcc Niand
bcc Fe, respectively. Hence, the narrow unshifted (∆ g=
0)bulkFMR lines in pure crystals, where α≤10−2,
are related to intraband scattering with ǫ≪¯h/τand toelectronic (momentum) relaxation times τsmaller than
10−13s.
Basing on Eq. (14), we discuss at first the temperature
variation,whichimpliesalineardependence, ˆ α(T)−α0∼
ms(T). Obviously, both, the real and imaginary part of
ˆα(T)−α0, agree perfectly with the fits to the data in
Fig. 4, if the relaxation time τremains constant. It may
be interestingtonote herethat the observedtemperature
variation of the complex damping λL(T) is not predicted
by the classical model28incorporating the sd-exchange
coupling Jsd. According to this model, which has been
advanced recently to ferromagnets with small spin-orbit
interaction29and ferromagnetic multilayers30,Jsdtrans-
fers spin from the localized 3d-moments to the delocal-
ized s-electron spins within their spin-flip time τsf. From
the mean field treatment of their equations of motion by
Turov31, we find a form analogous to Eq. (14)
αsd(T) =Ω2
sdχs
τ−1
sf+i/tildewideΩsd·1
γMs(T)(15)
where Ω sd=Jsd/¯his the exchange frequency , χsthe
Pauli-susceptibility of the s-electrons and /tildewideΩsd/Ωsd=
(1 + Ω sdχs/γMd). The same form follows from more
detailed considerations of the involved scattering process
(see e.g. Ref. 5). As a matter of fact, the LLG-damping
αsd=λsd/γMdcannot account for the observed tem-
perature dependence, because Ω sdandχsare constants.
The variation of the spin-torques with the spontaneous
magnetization ms(T) drops out in this model, since the
sd-scattering involves transitions between the 3d spin-up
and -down bands due to the splitting by the exchange
fieldJsdms(T).
By passing from the bulk to the nanoparticle ferro-
magnet, we use Eq. (14) to discuss our results for the
complex ˆ α(T), Eq. (13). Recently, for Conanoparticles
with diameters 1-4 nm, the existence of a discrete level
structure near ǫFhas been evidenced32, which suggests
to associate the e/h-energy ǫwith the level difference ǫp
at the Fermi energy. From Eqs. (13),(14) we obtain rela-
tions between ǫand the lifetime of the e/h-pair and the
experimental parameters αandβ:
τ−1=α
βǫ
¯h, (16a)
ǫ
¯h=β
α2+β2Ω2
so
γMs(0). (16b)
Due toα/β= 1.5, Eq. (16a) reveals a strongly over-
damped excitation, which is a rather well-founded con-
clusion. The evaluation of ǫ, on the other hand, de-
pends on an estimate for the effective spin-orbit cou-
pling, Ω so=ηLχ1/2
eξso/¯hwhereηLrepresents the ma-
trix element of the orbital angular momentum between7
the e/h states5. The spin-orbit coupling of the minor-
ityFe-spins in FePthas been calculated by Sakuma24,
ξso= 45meV, while the density of states D(ǫF)≈1/(eV
atom)24,33yields a rather high susceptibility of the elec-
trons,ηLχe=µ2
BD(ǫF) = 4.5·10−5. Assuming ηL=1,
both results lead to Ω so≈3.5·1011s−1, which is by
one order of magnitude larger than the values for Fe
andNimentioned above. One reason for this enhance-
ment and for a large matrix element, ηL=1, may be the
strong hybridization between 3 dand 4d−Ptorbitals24in
FexPt1−x. By inserting this result into Eq. (16b) we find
ǫ= 0.8 meV. In fact, this value is comparable to an esti-
mate for the level difference at ǫF32,ǫp= (D(ǫF)·Np)−1
which for ourparticles with Np= (2π/3)(dp/a0)3= 1060
atoms yields ǫp= 0.9 meV. Regarding the several in-
volved approximations, we believe that this good agree-
ment between the two results on the energy of the e/h
excitation, ǫ≈ǫp, maybe accidental. However,wethink,
that this analysisprovidesa fairlystrongevidence for the
magnon-scattering by this excitation, i.e. for the gap in
the electronic states due to confinement of the itinerant
electrons to the nanoparticle.
V. SUMMARY AND CONCLUSIONS
The analysis of magnetization isotherms explored
the mean magnetic moments of Fe0.2Pt0.8nanospheres
(dp= 3.1nm) suspended in an organic matrix, their
temperature variation up to the Curie temperature TC,
the large mean particle-particle distance Dpp≫dpand
the presence of Fe3+impurities. Above TC, the res-
onance field Hrof the 9.1GHzmicrowave absorption
yielded a temperature independent mean g-factor,g0=
2.16, consistent with a previous report21for paramag-
neticFexPt1−xclusters. There, the lineshape proved
to be closer to a gaussian with rather large linewidth,
∆H/Hr= 0.18, which may be associated with fluctua-
tions of g0and local fields both due to the chemically
disordered fccstructure of the nanospheres.
Below the Curie temperature, a detailed discussion of
the shape of the magnetic resonance spectra revealed a
number of novel and unexpected features.
(i) Starting at zero magnetic field, the shapes could be
described almost perfectly up to highest field of 10 kOe
by the solution of the LLG equation of motion for inde-
pendentferromagneticsphereswithnegligibleanisotropy.
Signatures of SPM fluctuations on the damping, which
have been predicted to occur below the thermal field
HT=kBT/µp(T), could not be realized.
(ii) Upon decreasing temperature, the LLG damping in-
creases proportional to µp(T), i.e. to the spontaneous
magnetization of the particles, reaching a rather largevalueα= 0.7 forT≪TC. We suspect that this high
intrinsic damping may be responsible for the absence of
the predicted SPM effects on the FMR, since the under-
lying statistical theory13has been developed for α≪1.
This conjecturemayfurther be based onthe fact that the
large intrinsic damping field ∆ H=α·ω/γ= 2.1kOe
causes a rapid relaxation of the transverse magnetization
(q= 0 magnon) as compared to the effect of statistical
fluctuations of HTadded to Heffin the equation of mo-
tion, Eq.(2)13.
(iii) Along with the strong damping, the lineshape analy-
sis revealed a significant reduction of the g-factor, which
also proved to be proportional to µp(T). Any attempts
to account for this shift by introducing uniaxial or cubic
anisotropy fields failed, since low values of /vectorHAhad no
effects on the resonance field due to the orientational av-
eraging. On the other hand, larger /vectorHA’s, by which some
small shifts of Hrcould be obtained, produced severe
distortions of the calculated lineshape.
The central results of this work are the temperature
variation and the large magnitudes of both α(T) and
∆g(T). They were discussed by using the model of the
spin-orbit induced scattering of the q= 0 magnon by
an e/h excitation ǫ, well established for bulk ferromag-
nets, where strong intraband scattering with ǫ≪¯h/τ
proved to dominate5. In nanoparticles, the continuous
ǫ(/vectork)-spectrum of a bulk ferromagnet is expected to be
split into discrete levels due to the finite number of lat-
tice sites creating an e/h excitation ǫp. According to
the measured ratio between damping and g-shift, this
e/h pair proved to be overdamped, ¯ h/τp= 1.5ǫp. Based
on the free electron approximation for ǫp32and the den-
sity of states D(ǫF) from band-structure calculations
forFexPt1−x24,33, one obtains a rough estimate ǫp≈
0.9 meV for the present nanoparticles. Using a reason-
able estimate of the effective spin-orbit coupling to the
minority Fe-spins, this value could be well reproduced
by the measured LLG damping, α= 0.59. Therefore we
conclude that the noveland unexpected results of the dy-
namics of the transverse magnetization reported here are
due to the presence of a broade/h excitation with energy
ǫp≈1meV. Deeper quantitative conclusions, however,
must await more detailed information on the real elec-
tronic structure of nanoparticles near ǫF, which are also
required to explain the overdamping of the e/h-pairs, as
it is inferred from our data.
The authors are indebted to E. Shevchenko and H.
Weller (Hamburg) for the synthesis and the structural
characterizationof the nanoparticles. One of the authors
(J. K.) thanks B. Heinrich (Burnaby) and M. F¨ ahnle
(Stuttgart) for illuminating discussions.
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2002.03418v2.Fujita_modified_exponent_for_scale_invariant_damped_semilinear_wave_equations.pdf | arXiv:2002.03418v2 [math.AP] 9 Jan 2021Noname manuscript No.
(will be inserted by the editor)
Fujita modified exponent for scale invariant damped semiline ar
wave equations.
Felisia Angela Chiarello ·Giovanni Girardi ·
Sandra Lucente
Received: date / Accepted: date
Abstract The aim of this paper is to prove a blow up result of the solution for a se milinear
scale invariant damped wave equation under a suitable decay conditio n on radial initial data. The
admissible range for the power of the nonlinear term depends both o n the damping coefficient and
on the pointwise decay order of the initial data. In addition we give an upper bound estimate for
the lifespan of the solution. It depends not only on the exponent of the nonlinear term, not only
on the damping coefficient but also on the size of the decay rate of th e initial data.
Mathematics Subject Classification (2010) Primary 35B33 ·Secondary 35L70
1 Introduction
In the recent years, the following Cauchy problem for the wave equ ation with scale invariant
damping spreads a new line of research on variable coefficient type eq uations. More precisely, we
are dealing with
vtt(t,x)−∆v(t,x)+µ
1+tvt(t,x)+ν
(1+t)2v(t,x) =|v(t,x)|p, t≥0, x∈Rn,
v(0,x) = 0,
vt(0,x) =εg(x),(1)
withn≥2,µ, ν∈R,p >1 andga radial smooth function. In [DLR], [DL] and [Pe] and [Po]
some results on the global existence of a solution for (1) with non co mpactly initial data appeared
assuming a suitable decay behavior for g. Many other results concern blow-up and global existence
for this equation, see [PR] for a summary of this problem. The main po int is to find a critical
exponent, fixed a suitable space of data. More precisely, a level ¯ pis critical if for p >¯pone can
prove that for ε >0 sufficiently small and for any gchosen in the fixed space there exists a unique
global (in time) solution of the problem, and conversely if p∈(1,¯p), for any ε >0 there exist
somegin this space such that the local solution cannot be prolonged over a finite time. Coming
Felisia Angela Chiarello
Department of Mathematical Sciences “G. L. Lagrange”, Poli tecnico di Torino, Corso Duca degli Abruzzi 24, 10129
Torino, Italy. Supported by “Compagnia di San Paolo” (Torin o, Italy)
E-mail: felisia.chiarello@polito.it
Giovanni Girardi
Department of Mathematics, University of Bari, Via Orabona n. 4, Italy.
E-mail: giovanni.girardi@uniba.it
Sandra Lucente
Dipartimento Interateneo di Fisica, University of Bari, Vi a Orabona n. 4, Italy. Partially supported by PRIN 2017-
linea Sud ”Qualitative and quantitative aspects of nonline ar PDEs.”
E-mail: sandra.lucente@uniba.it2 Felisia Angela Chiarello et al.
back to (1), in dependence on µ, νandn,a competition between two critical exponents appeared.
In some cases the Strauss exponent is dominant; it is given by the wa ve equation theory, it will
be denoted by pS(d) and it is the positive root of the quadratic equation
(d−1)p2−(d+1)p−2 = 0.
For other assumptions, the equation goes to an heat equation and a Fujita-type exponent pF(h) :=
1+2
happears. In all known results, the quantities d >1,h >0 depend on ν, µandn. Changing
the space of data, a change of critical exponent may appear. The novelty of our result consists
in showing that if one takes into account the decay rate of the initial data then the Fujita type
exponent depends also on such decay rate. In addition we give an up per bound estimate for the
lifespan of the solution, in terms of the power of the nonlinear term, the size and the growth of
the initial data. Let us recall that the lifespan of the solution is a fun ction of εwhich gives the
maximal existence time:
T(ε) := sup{T >0 such that the local solution uto (1) is defined on [0 ,T)×Rn}.
Finally, we will prove the following.
Theorem 1 Letn≥2. Letε >0andga radial smooth function satisfying
g(|x|)≥M
(1+|x|)¯k+1,with¯k >−1, (2)
for some M >0and for any x∈Rn. Assuming in addition that
¯k+µ
2>0,µ
2/parenleftBigµ
2−1/parenrightBig
≥ν
and
1< p < p F/parenleftBig
¯k+µ
2/parenrightBig
,
then the classical solution of (1)blows up. More precisely, the lifespan of the solution T(ε)>0is
finite and satisfies
T(ε)≤Cε−2(p−1)
4−(µ+2¯k)(p−1), (3)
withC >0, independent of ε.
Remark 1 Recently Ikeda, Tanaka, Wakasa in [ITW] consider a similar question f or cubic convo-
lution nonlinearity and a critical decay appears.
Remark 2 In [GL] we will also considera variantof problem (1), in which the nonline arity depends
onv,t,vtcombined in a suitable way.
Remark 3 The lifespan estimate for the same equation with compactly support ed data and ν/\e}atio\slash=
µ
2/parenleftbigµ
2−1/parenrightbig
has been considered in [PT]. If ν≤µ
2/parenleftbigµ
2−1/parenrightbig
the lifespan estimate is different from (3)
due to the compactness of the support of the initial data.
The paper is organized as follows: in Section 2 we give an overview of th e known results and
we state an auxiliary theorem; in Section 3 we prove the main results.Fujita modified exponent for scale invariant damped semilin ear wave equations. 3
2 Motivations
2.1 The case µ= 2,ν= 0
Let us start with a quite simple case
vtt−∆v+2
1+tvt=|v|p, t≥0, x∈Rn,
v(0,x) = 0,
vt(0,x) =εg(x).(4)
The global existence of small data solutions for this problem was firs t solved in [D] for a suitable
range of nandp. Some non-existence results were also established for p < p F(n) := 1 +2
n.
Except for the one-dimensional case a gap between this value and t he admissible exponents in
[D] appeared. In [DLR] for dimension n= 2,3,this gap was covered with an unexpected result.
Indeed, in that paper the Strauss exponent came into play. After wards, the global existence of
small data solutions to (4) has been proved for any p > pS(n+2) also in odd dimension n≥5 in
[DL] and in even dimension n≥4 in [Pe].
Moreover,we know that the exponent p2(n) := max {pS(n+2),pF(n)}is optimal; in fact, in [DLR]
the authors prove the blow up of solutions of (4) for each 1 < p≤p2(n) in each dimension n∈N.
In [DL,DLR,Pe], the authors provea global existence result not ne cessarilywhen the initial datum
g=g(x) has compact support. More precisely, let n≥3, given a radial initial datum g(x) =g(|x|)
withg∈C1(R), for any p > pS(n+ 2) it is possible to choose ¯k >0 andε0>0 such that (4)
admits a radial global solution u∈C([0,∞)×Rn)∩C2([0,∞)×Rn\{0}) provided
|g(h)(r)| ≤ε/a\}bracketle{tr/a\}bracketri}ht−(¯k+1+h)forh= 0,1. (5)
In the present paper we discuss the dependence of ¯kfromnandp. In (5), the exponent ¯khas
to belong to a suitable interval [ k1(n,p),k2(n,p)]. It is interesting to investigate the case of ¯k/\e}atio\slash∈
[k1(n,p),k2(n,p)]. In the sequel we will see that the bound k2(n,p) can be easily improved (see
Remark 4). On the contrary if k < k1(n,p) then a new result appears. The known situation is the
following:
-k1(3,p) = max/braceleftbig3−p
p−1,1
p−1/bracerightbig
andk2(3,p) = 2(p−1), see [DLR].
-k1(n,p) = max/braceleftbig3−p
p−1,n−1
2/bracerightbig
andk2(n,p) = min/braceleftbig(n+1)p
2−2,n2−2n+13
2(n−3)/bracerightbig
ifn≥5 odd, see [DL].
-k1(n,p) = max/braceleftbig3−p
p−1,n−1
2/bracerightbig
andk2(n,p) = min/braceleftBig
(n+1)p
2−2,n−1/bracerightBig
ifn≥4, see [Pe].
We can write in a different way the previous conditions. Firstly we conc entrate on the case n= 3.
Forp∈(1,2) we have ¯k≥3−p
p−1that is equivalent to
p≥1+2
¯k+1=pF(¯k+1).
From above we have ¯k≤2(p−1) that is
p≥¯k
2+1.
The intersection of p=pF(¯k+1) and p= 1+¯k/2 is exactly in ¯k=−1+√
17
2andp=pS(5). We
summarize the situation in Figure 1. In the following graphs we denote in blue the zone of the
known global existence results, in red the zone of the known blow-u p results. In this paper we
want to cover the white zones.4 Felisia Angela Chiarello et al.
p
¯k12
13
3+√
17
4
−1+√
17
2
Fig. 1:n= 3, µ= 2, ν= 0
Reading [DL] we see that the same situation appears for any odd n≥5. The critical curve
p=pF(¯k+1)
intersect the line
p=2(¯k+2)
n+1
in the Strauss couple
/parenleftbigg
¯k0,2(¯k0+2)
n+1/parenrightbigg
=/parenleftBigg
n−5+√
n2+14n+17
4,pS(n+2)/parenrightBigg
.
The only difference with the case n= 3 is that, in the global existence zone, a bound from above
appears for pand this has some influence on k2(n,p). More precisely one can take
p≤n+1
n−3,¯k≤n2−2n+13
2(n−3)ifn≥7,
andp≤2,¯k≤3 ifn= 5. Hence, the result of such paper can be represented as in Figur e 2. Our
aim is to prove blow up in the white zone below the Fujita curve.
p
¯k1n+5
n+1n+1
n−3
n−1
2pS(n+2)
¯k0
Fig. 2:n≥5 odd,µ= 2, ν= 0
Even dimension is more delicate. In [Pe] the global existence result is e stablished in the blue
zone below the line p=n+5
n+1except on the curve p=pF(¯k+1). For convenience of the reader, we
precise that in the notation of [Pe] the role of ¯kis taken by the quantity k+n+1
2.Fujita modified exponent for scale invariant damped semilin ear wave equations. 5
p
¯k1n+5
n+1
n−1
2 n+1pS(n+2)
¯k0
Fig. 3:n≥4 even,µ= 2, ν= 0
2.2 The case µ >2 andν=µ
2(µ
2−1)
In [Pe] and [Po] the author considers the Cauchy problem (1) for th e semilinear wave equation
with scale invariant damping and mass terms, that is ν=µ
2(µ
2−1)≥0. We see that for µ= 2,it
reduces to (4). Global existence of solutions to (1) holds under th e conditions
µ∈[2,M(n)], M(n) =n−1
2/parenleftBigg
1+/radicalbigg
n+7
n−1/parenrightBigg
.
In the even case [Pe], the initial data satisfies (5) for ¯k∈(k1(n,p,µ),k2(n,p,µ)] such that
k1(n,p,µ) = max/braceleftBign−1
2,2
p−1−µ
2/bracerightBig
; (6)
k2(n,p,µ) = min/braceleftBig
n−1,n+µ−1
2p−µ+2
2/bracerightBig
. (7)
Rewriting these conditions in term of p, we find that
p > pF/parenleftBig
¯k+µ
2/parenrightBig
, p≥2¯k+µ+2
n+µ−1.
The intersection of the curves those define the global existence z one gives p=pS(n+µ). Hence,
the condition p > pS(n+µ) appears. Moreover, another bound from above appears:
p <¯p:= min/braceleftbigg
pF(µ),pF/parenleftbiggn+µ−1
2/parenrightbigg/bracerightbigg
.
This means that different results for large µand small µhold. This influences the positions of k1
andk2. For our purpose it is sufficient to say that for µ/\e}atio\slash= 2 and even nthe situation is similar
to Figure 3. More precisely, in Figure 4 pS(n+µ) appears. The blow up result is indeed given in
[NPR]. The zone between p=pS(n+µ) andp=pF/parenleftbig¯k+µ
2/parenrightbig
is not covered by any known result.
p
¯k1¯p
k1 k2pS(n+µ)
¯k06 Felisia Angela Chiarello et al.
Fig. 4:n≥4 even,ν=µ
2(µ
2−1)≥0
The corresponding global existence result for the Cauchy problem (1) in odd space dimension
n≥1isstudiedin[Po]forradialandsmalldata,assumingcondition(5)wit h¯k∈[k1(n,p,µ),k2(n,p,µ)]
wherek2satisfies (7) and it holds:
k1(3,p,µ) = max/braceleftBig
1,2
p−1−µ
2,1
p−1/bracerightBig
;
k1(n,p,µ) = max/braceleftBign−1
2,2
p−1−µ
2/bracerightBig
, n≥5µ∈[2,n−1];
k1(n,p,µ) = max/braceleftBign−1
2,2
p−1−µ
2,1
p−1,/bracerightBig
, n≥5µ∈(n−1,M(n)].
In anycase the condition p > pF(¯k+µ
2) appears.Hence in odd space dimension n≥5 the situation
is not different from Figure 4.
Reading Theorem 1 in the case ν=µ
2(µ
2−1), it is clear that the aim of this paper is to find
blowing-up solutions to (1) even for p > pS(n+µ) by considering initial data with slow decay.
More precisely, let us consider
g(x)≃M
(1+|x|)¯k+1,forn−1
2<¯k <¯k0, (8)
where¯k0is such that
pF/parenleftBig
¯k0+µ
2/parenrightBig
=pS(n+µ).
We will prove the blow up result in the left white side zones in Figures 1, 2 , 3, 4 where ¯k <¯k0,
p > pS(n+µ) andp < pF(¯k0+µ
2). Under the same assumption on g, the quoted results assure
that for p≥pF(¯k+µ
2) andp > pS(n+µ) there is global existence. Hence, p=pF(¯k+µ
2) is a
critical curve for the Cauchy problem (1), provided ν=µ
2(µ
2−1)≥0.
Remark 4 Still fixing ν=ν
2(ν
2−1)≥0, let us consider ¯k >¯k0andp > pS(n+µ). As discussed,
the global existence results in the previous literature require pabove a line which depends on ¯k,
because of a restriction of type ¯k≤k2(n,p,µ) which everytime appears. Actually, this restriction
can be avoided; indeed, if the initial datum satisfies (8) with ¯k > k2(n,p,µ), then we can say that
the initial datum also satisfies (5) with ¯k=k2(n,p,µ). Hence, the global existence of a solution
to (1) follows from the known results.
Remark 5 Forν=µ
2(µ
2−1)≥0,Theorem 1 provides some new information about the solution
of (1) also when pbelongs to the red zone of Figure 1, 2, 3, 4, 5. In fact, for
p <min/braceleftBig
pS(n+µ),pF/parenleftBig
¯k+µ
2/parenrightBig/bracerightBig
by the previous literature we know that the solution blows up in finite t ime, whereas Theorem 1
gives a life-span estimate in the case of radial initial data with non com pact support, relating this
estimate with the decay rate of the data.
2.3 The case µ= 0 and ν= 0
In Figure 5 we summarize the wave equation case µ=ν= 0. The red blow-up zone was coveredby
many authors, see [S] and the reference therein for the whole list o f blow up results. For µ=ν= 0
the global existence result has been completely solved in [GLS], wher e the interested reader can
find a long bibliography of previous contributes. In particular the blu e zone, for radial solution
without compact support assumption for the initial data has been e xploited by Kubo, see for
example [K] and [KK]. Before these papers, Takamura obtained a blow up result in the green
zone. In [T] the point is to find a critical decay level k0=2
p−1,equivalently p≤1 +2
k0.We
underline that this a Fujita-type exponent.Fujita modified exponent for scale invariant damped semilin ear wave equations. 7
p
¯k13
1pS(n)
2
pS(n)−1
Fig. 5:µ=ν= 0
In Theorem 1, we generalize Takamura’s result when µ/\e}atio\slash= 0 and ν≤µ
2(µ
2−1). To this aim, it
is sufficient to consider a peculiar wave equation with nonlinear term ha ving a decaying time-
dependent variable coefficient. This means that we will deduce Theor em 1 from the following
result.
Theorem 2 Letn≥2. Given a smooth function g=g(|x|)withx∈Rn, we set r=|x|and we
consider g=g(r)satisfying
g(r)≥M
(1+r)¯k+1,with¯k >−1, (9)
for some M >0. Letu=u(t,r)be the radial local solution to
utt−urr−n−1
rur= (1+t)−µ
2(p−1)|u|p, r >0,
u(0,r) = 0,
ut(0,r) =εg(r).(10)
withp >1andp < pF(µ
2−1)ifµ >2. Assume in addition that
−1<¯k <2
p−1−µ
2. (11)
Then, given ε >0, the lifespan T(ε)>0of classical solutions to (10)satisfies
T(ε)≤Cε−2(p−1)
4−(µ+2¯k)(p−1), (12)
withC >0, independent of ε.
Remark 6 The assumption p < pF(µ
2−1) ifµ >2 guarantees that the range of admissible ¯kin
(11) is not empty.
In the case µ= 0 Theorem 2 coincides with Takamura’s result in [T]. In the proof of T heorem
2, we will follows the same approach of that paper.
3 Proof of the main results
3.1 Proof of Theorem 2
We recall the crucial lemma of [T].8 Felisia Angela Chiarello et al.
Lemma 1 Letn≥2andm= [n/2]. Given a smooth function g=g(|x|)withx∈Rn, we set
r=|x|and we consider g=g(r). Let us denote by u0(t,r)the solution of the free wave problem
/braceleftBigg
/squareu0= 0 ( t,r)∈[0,∞)×[0,∞)
u0(0,r) = 0, u0
t(0,r) =g(r).
Letu=u(t,r)be a solution to
utt−urr−n−1
rur=F(t,u) (13)
with the initial condition
u(0,r) = 0, ut(0,r) =εg(r), r∈[0,∞). (14)
IfFis nonnegative, there exists a constant δm>0such that
u(t,r)≥εu0(t,r)+1
8rm/integraldisplayt
0dτ/integraldisplayr+t+τ
r−t+τλmF(t,u(t,λ))dλ, (15)
u0(t,r)≥1
8rm/integraldisplayr+t
r−tλmg(λ)dλ, (16)
provided
r−t≥2
δmt >0.
The constant δmin the previous lemma is described in [T, Lemma 2.5]; it depends on the spa ce
dimension, in particular it changes accordingly with the different repr esentations of the free wave
solution in odd and even dimension.
We are ready to prove that if (9) holds, then the solution of (10) blo ws up in finite time even
for small ε.
Let us fix δ >0; we define a blow-up set,
Σδ=/braceleftBig
(t,r)∈(0,∞)2:r−t≥max/braceleftbigg2
δmt,δ/bracerightbigg/bracerightBig
, (17)
whereδm>0 is the constant given in Lemma 1. Combining the assumption (10) with the formulas
(15) and (16), for any ( t,r)∈Σδ,it holds
u(t,r)≥εu0(t,r)≥ε
8rm/integraldisplayr+t
r−tλmg(λ)dλ≥Mε
8rm/integraldisplayr+t
r−tλm(1+λ)−(¯k+1)dλ.
Then, (17) implies that
u(t,r)≥Mε
8rm/parenleftbigg1+δ
δ/parenrightbigg−(¯k+1)/integraldisplayr+t
r−tλm−(¯k+1)dλ
≥Mε
8rm/parenleftbigg1+δ
δ/parenrightbigg−(¯k+1)
(r+t)−(¯k+1)/integraldisplayr+t
r−tλmdλ≥Mε
8/parenleftbigg1+δ
δ/parenrightbigg−(¯k+1)(r−t)m2t
rm(r+t)¯k+1.
Since (t,r)∈Σδ, we have
u(t,r)≥C0tm+1
rm(r+t)¯k+1,
where we set
C0=ε2m−2M
δmm/parenleftbiggδ
1+δ/parenrightbigg¯k+1
>0. (18)Fujita modified exponent for scale invariant damped semilin ear wave equations. 9
Now we assume an estimate of the form
u(t,r)≥Cta
rm(r+t)bfor (t,r)∈Σδ, (19)
wherea,b, andCare positive constant. In particular, (19) holds true for a=m+1,b=¯k+1
andC=C0.
Beingg≥0, from (16) we deduce u0≥0. Combining (15) and (19), for ( t,r)∈Σδ, we get
u(t,r)≥1
8rm/integraldisplayt
0dτ/integraldisplayr+t−τ
r−t+τλm
(1+τ)µ
2(p−1)|u(τ,λ)|pdλ (20)
≥Cp
8rm/integraldisplayt
0τpa
(1+τ)µ
2(p−1)dτ/integraldisplayr+t−τ
r−t+τλm(1−p)(λ+τ)−pbdλ
≥Cp
8rm(r+t)pb+m(p−1)/integraldisplayt
0τpa
(1+τ)µ
2(p−1)dτ/integraldisplayr+t−τ
r−t+τdλ
≥Cp
4rm(r+t)pb+m(p−1)/integraldisplayt
0(t−τ)
(1+τ)µ
2(p−1)τpadτ.
By means of integration by parts, we obtain
/integraldisplayt
0(t−τ)τpa
(1+τ)µ
2(p−1)dτ≥1
(1+t)µ
2(p−1)/integraldisplayt
0(t−τ)τpadτ≥1
(1+t)µ
2(p−1)tpa+2
(pa+1)(pa+2).
While searching a finite lifespan of a solution, it is not restrictive to ass umet >1. We have
/integraldisplayt
0(t−τ)τpa
(1+τ)p−1dτ≥tp(a−µ
2)+2+µ
2
2p−1(pa+1)(pa+2). (21)
Let (t,r)∈Σδ, from (19)-(21), we can conclude
u(t,r)≥C∗ta∗
rm(r+t)b∗for (t,r)∈Σδ, (22)
with
a∗=p/parenleftBig
a−µ
2/parenrightBig
+2+µ
2, b∗=pb+m(p−1), C∗=(C/2)p
2(pa+2)2.
Let us define the sequences {ak},{bk},{Ck}fork∈Nby
ak+1=p/parenleftBig
ak−µ
2/parenrightBig
+2+µ
2, a 1=m+1, (23)
bk+1=pbk+m(p−1), b1=¯k+1, (24)
Ck+1=(Ck/2)p
2(pak+2)2, C1=C0, (25)
whereC0is defined by (18). Hence, we have
ak+1=pk/parenleftbigg
m+1−µ
2+2
p−1/parenrightbigg
+µ
2−2
p−1, (26)
bk+1=pk(¯k+1+m)−m, (27)
Ck+1≥KCp
k
p2k(28)10 Felisia Angela Chiarello et al.
for some constant K=K(p,µ,m)>0 independent of k. The relation (28) implies that for any
k≥1 it holds
Ck+1≥exp/parenleftbig
pk(log(C0)−Sp(k))/parenrightbig
, (29)
Sp(k) =Σk
j=0dj, (30)
d0= 0 and dj=jlog(p2)−logK
pjforj≥1. (31)
We note that dj>0 for sufficiently large j.Since lim j→∞dj+1/dj= 1/p,the sequence Sp(k)
converges for p >1 by using the ratio criterion for series with positive terms. Hence, t here is a
positive constant Sp,K≥Sp(k) for any k∈N, so that
Ck+1≥exp(pk(log(C0)−Sp,K)). (32)
Therefore, by (22), (26)- (29), we obtain
u(r,t)≥(r+t)m
rmt−µ
2+2
p−1exp(pkJ(t,r)), (33)
where
J(t,r) := log(C0)−Sp,K+/parenleftBig
m+1−µ
2+2
p−1/parenrightBig
logt−(¯k+1+m)log(r+t).
Thus if we prove that there exists ( t0,r0)∈Σδsuch that J(t0,r0)>0, then we can conclude that
the solution to (10) blows up in finite time, in fact
u(t0,r0)→ ∞fork→ ∞.
By the definition of J=J(t,r), we find that J(t,r)>0 if
/parenleftBig2
p−1−µ
2−¯k/parenrightBig
logt >log/parenleftBigeSp,K
C0/parenleftBig
2+r−t
t/parenrightBig¯k+1+m/parenrightBig
.
In particular, we can take ( t,r) = (t,t+max{2t
δm,δ})∈Σδ; then, it is enough to prove that
/parenleftBig2
p−1−µ
2−¯k/parenrightBig
logt >log/parenleftBigeSp,K
C0/parenleftBig
2+2
δm/parenrightBig¯k+1+m/parenrightBig
.
Now, the crucial assumption (9) comes into play. The coefficient in th e left side is positive and
by using (18) we find that J(t,r)>0 provided
t > Cε−(2
p−1−µ
2−¯k)−1
, (34)
where
C=/parenleftBigeSp,Kδm
m
2m−2M/parenleftBig1+δ
δ/parenrightBig¯k+1/parenleftBig
2+2
δm/parenrightBig1+¯k+m/parenrightBig 1
2
p−1−µ
2−¯k,
which is positive. As by-product, the inequality (34) gives the lifespa n estimate (12) and conclude
the proof of Theorem 2.Fujita modified exponent for scale invariant damped semilin ear wave equations. 11
3.2 Proof of Theorem 1
We start rewriting the Cauchy problem (1) as a nonlinear wave equat ion with a time dependent
potential. Let v=v(t,x) be a solution of (1); we define
u(t,x) := (1+ t)µ
2v(t,x).
Then the function u=u(t,x) is a solution of the Cauchy problem
utt−∆u= (1+t)−µ
2(p−1)|u|p+(µ
2(µ
2−1)−ν)u
(1+t)2, t≥0, x∈Rn,
u(0,x) = 0,
ut(0,x) =εg(x).(35)
Ifgis radial, then uis radial and it satisfies equations (13) and (14) with
F(t,u) = (1+ t)−µ
2(p−1)|u|p+/parenleftBigµ
2/parenleftBigµ
2−1/parenrightBig
−ν/parenrightBigu
(1+t)2.
Let us fix δ >0 and use the same notation of the proof od Theorem 2. Since we are assuming
µ
2(µ
2−1)−ν≥0, by comparison lemma, see [T, Lemma 2.9], we deduce u >0 inΣδ.Then it
holds
F(t,u)≥(1+t)−µ
2(p−1)|u|p;
hence, by formula (15) in Lemma 1 we still derive the estimate (20). T hus, the proof of Theorem
2 guarantees the result of Theorem 1.
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1807.11808v3.Comparative_study_of_methodologies_to_compute_the_intrinsic_Gilbert_damping__interrelations__validity_and_physical_consequences.pdf | Comparative study of methodologies to compute the intrinsic Gilbert damping:
interrelations, validity and physical consequences
Filipe S. M. Guimar~ aes,J. R. Suckert, Jonathan Chico, Juba Bouaziz, Manuel dos Santos Dias, and Samir Lounis
Peter Gr unberg Institut and Institute for Advanced Simulation,
Forschungszentrum J ulich & JARA, 52425 J ulich, Germany
(Dated: December 20, 2018)
Relaxation e ects are of primary importance in the description of magnetic excitations, leading
to a myriad of methods addressing the phenomenological damping parameters. In this work, we
consider several well-established forms of calculating the intrinsic Gilbert damping within a uni ed
theoretical framework, mapping out their connections and the approximations required to derive
each formula. This scheme enables a direct comparison of the di erent methods on the same footing
and a consistent evaluation of their range of validity. Most methods lead to very similar results for
the bulk ferromagnets Fe, Co and Ni, due to the low spin-orbit interaction strength and the absence
of the spin pumping mechanism. The e ects of inhomogeneities, temperature and other sources of
nite electronic lifetime are often accounted for by an empirical broadening of the electronic energy
levels. We show that the contribution to the damping introduced by this broadening is additive, and
so can be extracted by comparing the results of the calculations performed with and without spin-
orbit interaction. Starting from simulated ferromagnetic resonance spectra based on the underlying
electronic structure, we unambiguously demonstrate that the damping parameter obtained within
the constant broadening approximation diverges for three-dimensional bulk magnets in the clean
limit, while it remains nite for monolayers. Our work puts into perspective the several methods
available to describe and compute the Gilbert damping, building a solid foundation for future
investigations of magnetic relaxation e ects in any kind of material.
I. INTRODUCTION
Dynamical processes lie at the core of magnetic manip-
ulation. From the torques acting on the magnetic mo-
ments to how fast they relax back to their equilibrium
orientations, a material-speci c time-dependent theory
is essential to describe and predict their behavior. In
most cases, the description of the time evolution of the
magnetization is done via micromagnetics1or atomistic
spin dynamics (ASD)2,3approaches, in which the mag-
netization is considered either as a classical continuous
vector eld or as individual 3D vectors on a discrete
lattice, respectively. They have been successfully used
to describe a plethora of magnetic phenomena, ranging
from spin waves in low dimensional magnets4, domain
walls5and skyrmion6dynamics to thermal stability of
magnetic textures7. These approaches model the mag-
netization dynamics via a phenomenological equation of
motion that contains both precessional and relaxation
terms.
A rst attempt to address these processes was per-
formed by Landau and Lifshitz (LL), by considering
a Larmor-like precessional torque and adding to it a
(weaker) damping term of relativistic origin8. Since its
phenomenological inception in 1935, the precise nature
of the relaxation processes has been a source of intense
debate. In particular, the original LL formulation was
found to not properly describe situations in which the
damping was large. This problem was addressed by
Gilbert, who introduced a Rayleigh-like dissipation term
into the magnetic Lagrangian, thus obtaining the now-ubiquitous Landau-Lifshitz-Gilbert (LLG) equation9,
dM
dt=
MB+
MMdM
dt
= e
MB e
MM(MB):(1)
where
>0 is the gyromagnetic factor, Mis the (spin)
magnetic moment, Bis the time-dependent e ective
magnetic eld acting on M, and is the scalar damping
parameter named after Gilbert. The upper form of the
LLG equation is due to Gilbert, and the lower one shows
that it is equivalent to a LL equation with a renormalized
gyromagnetic factor, e
=
=(1 + 2). The rst term in
the right-hand side of Eq. (1) describes the precession of
the magnetic moments around the e ective eld, while
the second term is the Gilbert damping one, that de-
scribes the relaxation of the magnetic moments towards
B. This equation corrects the previously mentioned issue
for large values of , for which the original LL equation
is expected to fail10,11.
The ferromagnetic resonance (FMR) technique is one
of the most common procedures to probe magnetiza-
tion dynamics12, in which the damping parameter is re-
lated to the linewidth of the obtained spectra13. Al-
though many measurements have been carried out in bulk
materials12,14{18, their description at low temperatures is
still controversial19{22. This can be attributed to the dif-
ferent intrinsic and extrinsic mechanisms that can con-
tribute to the relaxation processes23{36. When varying
the temperature, two distinct regimes could be identi-
ed in the measured relaxation parameters37. For high
temperatures, a proportionality between the linewidth
and the temperature was observed in most of the exper-arXiv:1807.11808v3 [cond-mat.mes-hall] 19 Dec 20182
iments. It was called resistivity-like, due to the simi-
larity with the temperature dependence of this quantity.
A conductivity-like regime (linewidth inversely propor-
tional to the temperature) was identi ed at low temper-
atures for certain materials such as Ni15,17, but not for
Fe18,38. It was also seen that di erent concentrations
of impurities a ected this low-temperature regime, even
suppressing it altogether16.
From the theoretical point-of-view, the calculation of
the Gilbert parameter is a challenging problem due to
the many di erent mechanisms that might be at play for
a given material39,40. Perhaps this is why most of the
theoretical approaches have focused on contributions to
the damping from electronic origin. The ultimate goal
then becomes the development of a predictive theory of
the Gilbert damping parameter, based on the knowledge
of a realistic electronic structure of the target magnetic
material. The ongoing e orts to complete this quest
have resulted in the development of a myriad of tech-
niques21,22,37,41{43. Comparisons between a few of these
approaches are available44,45, including experimental val-
idation of some methods24,46, but a complete picture is
still lacking.
We clarify this subject by addressing most of the well-
established methods to calculate the Gilbert damping
from rst principles. First, we connect the many dif-
ferent formulas, highlighting the approximations made
in each step of their derivations, determining what con-
tributions to the damping they contain, and establish-
ing their range of validity. These are schematically illus-
trated in Fig. 1. Second, we select a few approaches and
evaluate the Gilbert damping within a uni ed and con-
sistent framework, making use of a multi-orbital tight-
binding theory based on rst-principles electronic struc-
ture calculations. FMR simulations and the mapping of
the slope of the inverse susceptibility are used to bench-
mark the torque correlation methods based on the ex-
change and spin-orbit torques. We apply these di erent
techniques to bulk and monolayers of transition metals
(Fe, Ni and Co), for which the spin pumping mecha-
nism is not present and only the spin-orbit interaction
(SOI) contributes to the relaxation. Disorder and tem-
perature e ects are included by an empirical broadening
of the electronic energy levels37,43,47,48. Third, we engage
a longstanding question regarding the behavior of the
damping in the low-temperature and low-disorder limits:
should the intrinsic contribution to the Gilbert damping
diverge for clean systems? Our results using the con-
stant broadening model demonstrate that the divergence
is present in the clean limit of 3D systems but not of
the 2D ones49, which we attest by eliminating the pos-
sibility of them being caused by numerical convergence
issues or di erent anisotropy elds. Our results also in-
dicate that the limit !!0 is not responsible for the
divergence of the intrinsic damping, as it is commonly
attributed19,37,43,50. Finally, we propose a new way to
obtain the spin-orbit contribution that excludes the c-
titious temperature/disorder contribution caused by thearti cial broadening51,52: they can be discounted by sub-
tracting the values of damping calculated without SOI.
For bulk systems, this yields the total damping, while in
layered materials this method should also discount part
of the spin pumping contribution. In Ref. 20, where tem-
perature and disorder are included via a CPA analogy, a
similar arti cial increase of for high temperatures was
removed by including vertex corrections.
This work is organized as follows. We start, in Sec. II,
with a brief overview of the di erent methods proposed
in the literature. In Sec. III, we explain the theory used
to calculate the response functions. We then turn to
the distinct theoretical forms of calculating the damping:
In Sec. IV, we analyze the di erent approaches related
to the spin-spin responses, while in Sec. V, the torque
methods are explored. We then discuss the obtained re-
sults and conclude in Sec. VII. The Hamiltonian used in
the microscopic theory is given in Appendix A, while the
anisotropy elds for the 3D and 2D systems together with
the transverse dynamical magnetic susceptibility given
by the LLG equation are given in Appendix B.
II. OVERVIEW OF METHODS ADDRESSING
INTRINSIC GILBERT DAMPING
We now focus on the di erent methods to describe
the microscopic contributions to the Gilbert parameter,
which encompasses e ects that transfers energy and an-
gular momentum out of the magnetic system. Within
these mechanisms, the relativistic SOI comes to the fore.
This is often referred to as the intrinsic contribution to
the damping, and was rst identi ed by Landau and Lif-
shitz8. The origin of this damping mechanism lies in
the non-hermiticity of the relativistic corrections to the
spin Hamiltonian when the magnetization precesses26,27.
The elementary magnetic excitations, called magnons,
can also be damped via Stoner excitations (electron-hole
pairs with opposite spins)33,34,53. Alternatively, the con-
duction electrons can carry spin angular momentum even
in absence of the SOI. This leads to damping via the spin-
pumping mechanism32,54{56.
Early models proposed to describe these processes al-
ready argued that the interaction between the magnetic
moments and the conduction electrons is a key ingre-
dient57. This led to the so-called breathing Fermi sur-
face model, where the shape of the Fermi surface de-
pends on the orientation of the magnetization through
the SOI41. This approach, however, could only capture
the conductivity-like regime, which diverges at low tem-
peratures. The decay of magnons into Stoner excitations
was also considered early on39, describing well the exper-
imental behavior of Ni but also missing the increase at
larger temperatures of other materials.
An important progress was made by Kambersky us-
ing the spin-orbit torque correlation function to calculate
the damping parameter37. This approach captures both
conductivity- and resistivity-like behaviors, which were3
↵ ↵noSOI
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Inversion
Inversion for low frequencies + sum ruleDyson equation of susceptibilityEquation of motion of susceptibility
Spectral representation at T=0KSpectral representation at T=0KLow SOI
Low spin pumpingNo spin pumpingComputational costsFull SOI Spin pumpingFMR linewidthSlope of inverse susceptibility
Slope of inverse mean-field susceptibility
Exchange torque correlation at Fermi surfaceSlope of mean-field SO-torque susceptibility with SOIProduct of spectral functions of opposite spinsSlope of mean-field susceptibilitySpin correlation at Fermi surfaceSO-torque correlation at Fermi surface with SOISpin response methodsTorque response methods
Equation of motion of susceptibility + perturbation theory
Dyson equation for Green function + Orbital quenchingSlope of SO-torque susceptibility without SOI
Spectral representation at T=0KSlope of mean-field SO-torque susceptibility without SOISO-torque correlation at Fermi surface without SOILarge broadening
Figure 1. Diagram exhibiting the di erent methods investigated in this work, their connections and range of validity. Two
groups are identi ed: one related to the spin susceptibility (spin response methods), including the ferromagnetic resonance
and the slope of the inverse susceptibility that involves a direct mapping of this quantity to the LLG equation; and the other
associated with torque responses, for which approximations need to be taken. The steps indicated by solid lines represent
exact connections, while dashed arrows involve some kind of approximation. The arrow on the left points from the methods
that require less computational power (lower part) to the more demanding ones (upper part). Boxes are hyper-linked with the
respective equations and sections.
shown to originate from the intra- and interband transi-
tions, respectively58. Recently, this so-called torque cor-
relation method was re-obtained using a di erent per-
turbative approach19, spurring discussions about the va-
lidity of the obtained results, specially the divergence
caused by the intraband transitions22. A similar method
also based on torque correlation functions was developed
using a scattering theory approach42involving the ex-
change torque operator instead of the spin-orbit torque
one. Results obtained in this way also present diverg-
ing behavior in the clean limit of 3D structures20. A
similar scattering framework was used to explain the
enhancement of the Gilbert damping due to the spin
pumping in thin lms32. Yet another method relating
the Gilbert damping to the spin-spin response was pro-
posed and related to the existing spin-orbit torque cor-
relation method43. It also presented diverging intraband
contributions when the parameter used to broaden the
delta functions (which mimics the e ect of disorder or
temperature) was taken to zero59. The vertex correc-
tions proposed in Ref. 59 did not remove this diver-
gence. More recently, Costa and Muniz21showed that
the damping parameters of layered structures remain -nite in the zero broadening limit, when extracted directly
from the linewidth of the dynamical magnetic suscepti-
bility (within the random phase approximation).
Several of these methods have been implemented
for material-speci c calculations20,47,49,58,60{62, and some
approaches were compared and related43{45,63. In this
work, we start our analysis with the uniform frequency-
dependent spin-spin susceptibility, which is measured ex-
perimentally in FMR setups, to derive the other expres-
sions for the damping parameter based on the spin- and
torque-correlation methods.
III. MICROSCOPIC THEORY
We begin by setting the grounds of the theory we use
to evaluate the di erent formulas of the Gilbert damping
on equal footing. The electronic structure of the system
is described by the mean- eld Hamiltonian
^H=^H0+^Hxc+^HSOI+^Hext: (2)
The paramagnetic band structure is described by ^H0
within a multi-orbital tight-binding parametrization. An4
e ective local electron-electron interaction within the
mean- eld approximation is included in ^Hxc, which is re-
sponsible for ferromagnetism. We also account for spin-
orbit interaction through ^HSOI, and the interaction with
external static magnetic elds via ^Hext. The explicit
forms of all the terms are given in Appendix A.
In this work, we investigate the di erent methods to
compute the intrinsic Gilbert damping utilizing the pro-
totypical bulk magnets Fe (bcc), Co (fcc) and Ni (fcc),
and also square lattices corresponding to the (001) planes
of those materials, with the same nearest-neighbor dis-
tances as in its bulk forms.
For simplicity, we consider the spin-orbit interaction
and the local e ective Coulomb interaction only on the
dorbitals, with U= 1 eV64{66for all systems, and the
spin-orbit strengths Fe
SOI= 54 meV67,Co
SOI= 70 meV68,
andNi
SOI= 133 meV68. The magnetic ground state is
found by self-consistently enforcing charge neutrality for
the bulk materials69. For the monolayer cases, the total
number of electrons in the atomic plane is decreased to
n= 7:3 (Fe),n= 8:1 (Co) and n= 9:0 (Ni), as the re-
maining charge spills into the vacuum (which we are not
explicitly taking into account within the model). The
ground-state properties (spin moment M, orbital mo-
mentM`and magnetic anisotropy energy K) obtained
within this framework are listed in Table I. The easy axis
for all the bulk systems and the monolayers were found
to be along the (001) direction. We emphasize that our
goal is not to achieve the most realistic description of the
electronic structure of these materials, but rather to de-
ne a concrete set of cases that allow us to compare the
di erent methods to compute the Gilbert damping.
The magnetic excitations are described using linear re-
sponse theory, where the transverse magnetic response
M(t) due to an oscillatory magnetic eld B(t) is given
by70
M (t) =Z
dt0 (t t0)B (t0); (3)
where the convention to sum over repeated indices of
the components =fx;y;zgis used. This approach
captures the orbitally-averaged part of the response. The
bulk monolayer
bcc Fe fcc Co fcc Ni Fe Co Ni
M(B) 2.32 1.48 0.43 2.90 1.90 0.96
M`(B) 0.072 0.079 0.055 0.28 0.22 0.20
K(meV) 0.19 0.26 0.084 1.7 1.8 1.9
Table I. Ground state properties of the investigated systems.
MandM`denotes the spin and orbital magnetic moments,
respectively. Values obtained for = 1:36 meV. The mag-
netic anisotropy constant Kis obtained from the anisotropy
elds given by Eq. (B3).magnetic susceptibility is given by
(t t0) = 4
^S (t);^S (t0)
= 4i
^S (t);^S (t0) ; (4)
in atomic units. ^S (t) is the -component of the spin op-
erator. In the rst line of the equation above, we reprise
the double-bracket notation of Zubarev for the spin-spin
retarded Green function71. This notation is convenient
for the derivations of Sec. V.
For the crystal symmetries of the systems we are in-
terested in, it is convenient to work in the circular ba-
sis^S=^Sxi^Sy, which diagonalizes the susceptibil-
ity matrix with components +(t) and + (t). The
frequency- and wave vector-dependent transverse suscep-
tibility +(q;!) is obtained within the random phase
approximation (RPA), which captures the collective spin
wave modes21,72, as well as the possible decay into
particle-hole excitations (Stoner modes) described by the
single-particle response function +
0(!). Considering
matrices that take into account the orbital dependency,
the two susceptibilities are related by
[ +] 1= [ +
0] 1 1
4U: (5)
Here,U=Uis a matrix with the e ective lo-
cal Coulomb interaction strength within the dorbitals.
It plays a similar role to the exchange-correlation ker-
nel in the adiabatic local-density approximation of time-
dependent DFT calculations73. We de ne the transverse
magnetic response of the system by summing the suscep-
tibility matrix over all the dorbitals.
The uniform single particle transverse susceptibility
+
0(!) = +
0(q= 0;!), obtained within the mean-
eld approximation, is expressed in terms of the single-
particle Green functions as
+
0;(!) =1
NX
kZF
d"
G""
(k;"+!) ImG##
(k;")
+ ImG""
(k;")
G##
(k;" !)o
:
(6)
The sum is over the wave vectors in the rst Brillouin
zone, with Ntheir number. The indices ;represent
orbitals and Fis the Fermi level.
In the spirit of many preceding works37,43,47,48, the
e ect of temperature and disorder is modeled by in-
troducing a constant band broadening on the en-
ergy levels, such that G(!)!G(!+ i). The imag-
inary part of the Green function is then de ned as
ImG(!) =1
2ifG(!+ i) G(! i)g. This ap-
proach attempts to capture all the intrinsic e ects origi-
nated from the electronic structure of the system.
The imaginary part of the susceptibility is related
to the energy dissipation of the system74, encoding
the relaxation mechanism of the magnetization towards
equilibrium. The damping parameter is then obtained5
by mapping the transverse magnetic susceptibility ob-
tained from the quantum mechanical calculation de-
scribed above to the phenomenological form provided by
the LLG, Eq. (1). On the following sections, we present
di erent mapping procedures involving several approx-
imations and explore their range of validity when the
broadening is taken to zero (clean limit).
IV. SPIN RESPONSE METHODS
A. Ferromagnetic resonance
Magnetic excitations can be investigated by applying
time-dependent perturbations. This is done in FMR ex-
periments where the magnetic sample is subjected to a
static magnetic eld and an oscillatory radio-frequency
one. By varying either the strength of the static compo-
nent or the frequency of the oscillatory eld, the system
can be driven through magnetic resonance. This setup
yields the uniform mode of the transverse magnetic sus-
ceptibility. As the Gilbert parameter describes the relax-
ation mechanisms of the magnetization, it is related to
the linewidth of the resonance peak21,75.
We simulate this kind of experiments by calculating
the transverse magnetic response relying on the linear
response theory discussed in Sec. III, and mapping the
imaginary part of the susceptibility into the result ob-
tained from the LLG equation (see Appendix B),
Im +(!) = 2
!M
[!
(Bext+Ban)]2+ ( !)2:(7)
When xing the frequency and varying Bext;z, this func-
tion presents a resonance at Bres= (!
Ban;z)=
with linewidth given by the full width at half maxi-
mum B= 2 !=
. On the other hand, when the
eld is kept xed and the frequency is varied, the res-
onance is located at !res=
(Bext;z+Ban;z)=p
1 + 2
with full width at half maximum approximately given by
!2
jBext;z+Ban;zj, in the limit 175.
The Gilbert parameter can then be obtained either by
tting Eq. (7) or through the ratio between the linewidth
and the resonance position. In this sense, a divergence of
the damping when !0 seems counter-intuitive, since
this would imply that either the resonance position ( Bres
or!res) goes to zero or that the corresponding linewidth
increases drastically. In the presence of SOI, the SU(2)
rotational symmetry is broken and the anisotropy eld
Ban;zshifts the resonance position to a nite value | it
costs a nite amount of energy to set the magnetization
into precession76. Therefore, the divergence of the damp-
ing parameter can only happen if the linewidth increases
and goes to in nity.
To verify this claim, we simulate FMR experiments
in fcc Co bulk by calculating the imaginary part of the
transverse magnetic susceptibility as a function of the
frequency!, in the presence of the spin-orbit interac-
tion. In Fig. 2a, we present the obtained spectra fordi erent values of the broadening . When a relatively
large value of the broadening is used, = 13:6 meV (solid
curve), the spectra displays a broad resonance peak,
which can be characterized by a value = 1:310 2,
obtained by tting the linear response data with Eq. (7).
When the broadening of the energy levels is decreased
to= 4:1 meV (dashed curve), the peak shifts and be-
comes sharper ( = 3:810 3), as one intuitively ex-
pects when disorder and/or temperature decreases. No-
tice that most of the change in is due to the change
in the peak width, while the resonance shift is relatively
small. This can be viewed as a consequence of the smaller
energy overlap between the bands, which decrease possi-
ble interband transitions58. Surprisingly, by further de-
creasing the broadening to = 0:41 meV (dotted curve),
the peak becomes broader when compared to the pre-
vious case, with = 5:610 3. This counter-intuitive
result represents a shorter lifetime of the magnetic excita-
tion when the electronic lifetime (mean time between two
successive scattering events) = 1becomes longer.
Obtaining the damping from the FMR curves is com-
putationally demanding, though. The response function
must be calculated for many frequencies (or magnetic
elds) to resolve the peak. For the case of low broaden-
ings that require many k-points in the Brillouin zone for
a converged result, this task becomes prohibitive. In the
next section, we provide alternative methods to obtain
the Gilbert parameter based on the static limit of the
susceptibility, and compare their outcomes with the ones
obtained using the resonance approach.
B. Inverse Susceptibility Method
We proceed now to investigate a di erent mapping
of the microscopic transverse susceptibility to the LLG
equation and possible approximations to simplify the cal-
culation of the Gilbert damping. From Eq. (B4), one can
see that de nes the slope of the imaginary part of the
inverse susceptibility43, i.e.,
= 2
Mlim
!!0Im[ +(!)] 1
!: (8)
We will refer to this as the inverse susceptibility method
(ISM). The mapping to the LLG model of the slope at
small frequencies has a great advantage over the FMR
one since it only requires a single frequency-point calcula-
tion, instead of a full sweep over frequencies or magnetic
elds for the tting procedure.
In Fig. 2b, we display the damping parameter for bcc
Fe, fcc Co and fcc Ni bulk systems calculated as a func-
tion of the electronic energy broadening. We also include
the results obtained from the FMR approach (solid sym-
bols), which compare well with the ISM given in Eq. (8).
Note that although Eq. (8) has an explicit linear depen-
dence on the spin moment M, the susceptibility implic-
itly depends on its value. The obtained curves are in-
versely related to M: highest for Ni ( M0:45B), low-6
0.5 0.55 0.6 0.6502468·105
Frequencyω(meV)−Imχ−+(ω) (states/eV)η1= 13.6 meV→α= 1.3·10−2
η2= 4.1 meV→α= 3.8·10−3
η3= 0.41 meV→α= 5.6·10−3
1 10 10010−210−1100101102
Broadening η(meV)Gilbert damping αXC-TCM ISM
Fe
Fe 5·λSOI
Fe 10·λSOI
1 10 10010−310−210−1100101
Broadening η(meV)Gilbert damping αISM Fe
ISM Co
ISM Co no SOI
FMR Co
ISM Ni
1 10 10010−310−210−1100101
Broadening η(meV)Gilbert damping αFe monolayer
Co monolayer
Ni monolayer10 100 1,000Temperature (K)
10 100 1,000Temperature (K)
10 100 1,000Temperature (K)(a)
(b)(c)
(d)
Figure 2. Characteristics of the Gilbert damping in 3D and 2D systems in presence and absence of SOI. (a) Ferromagnetic
resonance spectra for fcc Co, in presence of spin-orbit interaction and no external eld, calculated for three di erent decreasing
broadenings 1= 13:6 meV (solid), 2= 4:1 meV (dashed) and 3= 0:41 meV (dotted). The values of the Gilbert damping
given in the legend box, obtained by tting to Eq. (7), decrease from the rst case to the second, but increases again when is
further decreased. (b) Gilbert damping in presence of spin-orbit interaction for bcc Fe (blue triangles), fcc Co (red circles, solid
line) and fcc Ni (green squares) as a function of the broadening, obtained from the slope of the inverse susceptibility, Eq. (8).
All values were computed with 108k-points in the full Brillouin zone. Solid red circles are the values obtained from the FMR
spectra in (a), while the open red circles connected by dashed lines represent the damping parameter for fcc Co when SOI is
not included in the calculations. (c) Damping parameter for bcc Fe for di erent SOI strenghts: SOI= 54:4 meV, 5 SOI,
and 10 SOI. (d) Gilbert damping of Fe, Co and Ni monolayers in the presence of SOI. No increase in the Gilbert damping
is seen when the broadening is decreased.
est for Fe (M2:3B) and Co in-between ( M1:5B).
This trend is con rmed by setting the SOI strength SOI
to the same values for all the elements (not shown). The
position of the minimum value of is connected with
SOI, which determines when the intraband or interband
transitions become more important58. To substantiate
this claim, we employed the technique of arti cially scal-
ing theSOI, as previously done in connection to the
magnetic anisotropy energy77. The results are shown in
Fig. 2c, where the SOI strength SOIof Fe bulk is mag-
ni ed by factors of 5 and 10. Indeed, the minimum can
clearly be seen to shift to larger values of .An important aspect to be considered is the conver-
gence of Eq. (6) | failing to achieve numerical precision
may give rise to spurious results49,78. This can be partly
solved using sophisticated schemes to perform those cal-
culations79,80. When the broadening is lowered, the con-
vergence of the wave vector summation is a ected by
the increasingly dominant role of the poles of the Green
functions in the vicinity of the Fermi energy. For that
reason, to capture the intricacies of the electronic states
| in particular, the important contributions from the
small gaps opened by the weak SOI |, we calculated
the slope of the response function using a very ne in-7
tegration mesh on the Brillouin zone reaching up to 109
k-points. The results in Fig. 2c also demonstrate that the
divergence is not an issue of numerical convergence, since
this behavior is shifted to larger values of broadenings,
for which the convergence is more easily achieved.
Nevertheless, such diverging e ect only occurs in the
presence of spin-orbit interaction. In Fig. 2b we also dis-
play the values of for Co fcc obtained using the ISM
when the SOI is not included in the calculations (cir-
cles connected by dashed lines). In this case, noSOI lin-
early goes to zero when the broadening is decreased21.
The non-vanishing damping when SOI is not present
can be interpreted as originating from the nite elec-
tronic lifetimes introduced by the constant broadening
parameter. As it stands, represents the coupling to a
ctitious reservoir51,52providing dissipation mechanisms
that physically should originate from disorder or temper-
ature, for example.
Obtaining the damping from the FMR spectra when
SOI is not present requires an applied magnetic eld,
such that the resonance frequency becomes nite and
avoiding an in nite response at zero frequency (repre-
senting no cost of energy due to the rotational symmetry,
i.e., the Goldstone mode). Nevertheless, the results pre-
sented in Fig. 2d were obtained using the ISM without
any applied eld. Calculations with an applied magnetic
eld shifting the peak to the original anisotropy energy
were indistinguishable from those values (with variations
smaller than 3%). This is accordance to the phenomeno-
logical expectations expressed through Eq. (B4), where
the slope is independent of the magnetic eld.
One can put our results for bulk ferromagnets into
perspective by comparing with low dimensional systems.
We investigated this case within our linear response ap-
proach, using monolayers of Fe, Co and Ni. The calcu-
lations follow the same procedure, except that the sum
overkvectors in Eq. (6) is restricted to the 2D Brillouin
zone. The results are presented as triangles (Fe), cir-
cles (Co) and squares (Ni) connected by dotted lines in
Fig. 2d, and once again exhibit a monotonous decay with
the decrease of . We note that previous calculations of
the damping parameter in thin lms also did not nd it
to increase rapidly for decreasing broadening21,49.
Besides the dimensionality, another main di erence
from the bulk to the layered case is the larger anisotropy
elds of the latter (see Table I). Nevertheless, this can-
not explain the non-diverging behavior in the monolay-
ers. We have already shown that by arti cially increas-
ing the SOI strength of the bulk | and, consequently, its
anisotropy eld |, the conductivity-like behavior of the
damping occurs at even larger broadenings (see Fig. 2c).
On the other hand, to rule out a possible divergence hap-
pening at lower broadenings ( <0:1 meV, not reachable
in our calculations), we have also scaled up SOIof the
monolayers by one order of magnitude. This resulted in
larger dampings, nonetheless, the same decreasing be-
haviour with !0 was observed (not shown). There-
fore, the divergence can only be attributed to the three-dimensionality of the ferromagnet.
C. Approximate static limit methods
We now look back to Fig. 1 and proceed to perform
approximations on Eq. (8) in order to simplify the calcu-
lations of the damping parameter. Here we follow Ref. 43.
First, we use Eq. (5) that relates the RPA susceptibility
matrix to the mean- eld response matrix 0, such that
Im 1Im 1
0. Although Uis a real matrix, the sum
over orbitals ( =P
) ends up mixing the real
and imaginary parts of the matrix elements. Only when
Re 1
0=U=4 the relation above becomes an equality.
This means that, within our model with Uacting only on
thedorbitals, must also be de ned by summing over
those orbitals only. Under the previous assumption, we
obtain
2
Mlim
!!0Im[ +
0(!)] 1
!: (9)
This relation is only valid when +
0is decoupled from
the other types of susceptibilities (transverse and longi-
tudinal), as in the systems we investigate in this work.
The damping parameter can therefore be obtained from
the single-particle transverse susceptibility 0.
For frequencies !in the meV range (where the col-
lective spin excitations are located), +
0has a simple
!-dependence81:
+
0(!)Re 0(0) + i!Im 0
0(0): (10)
where 0
0(0) =d +
0
d!
!=0. These results are valid also in
the presence of spin-orbit coupling. Using Eq. (10), the
Gilbert damping can be written as
2
M
Re +
0(0) 2lim
!!0Im +
0(!)
!:(11)
Although the expansion of the susceptibility for low fre-
quencies was used, no extra approximation is employed,
since Eq. (9) is calculated in the limit !!0. Re +
0(0)
can be obtained using the sum rule that relates the
static susceptibility with the magnetic moments76. For
3dtransition metals, the external and the spin-orbit
elds are three orders of magnitude smaller than U, and
so the static susceptibility of the bulk systems reads
Re +
0(0)4=U. Thus,
MU2
8lim
!!0Im +
0(!)
!: (12)
Finally, from Eq. (6) it is possible to show that Eq. (12)8
simpli es as
MU2
2NX
k;TrfImG(k;F)^S ImG(k;F)^S+g
=
2MNX
k;TrfImG(k;F)^T
xcImG(k;F)^T+
xcg
=
MU2
8NX
k;n#
(k;F)n"
(k;F):
(13)
wheren
(k;F) = 1
ImG
(k;F) is the matrix el-
ement of the spectral function of spin calculated at
kandF. The second equation is written in terms
of the \exchange-correlation torque operator", T
xc=
i^S;^Hxc
=iUM^S. This result is equivalent
to the one obtained in Ref. 42, which we reference as
theexchange torque correlation method (XC-TCM) |
although, in reality, it relates with the spin-spin re-
sponse. The last step in Eq. (13) connects the damping
with the product of spectral functions of opposite spins
at the Fermi level, as shown theoretically in Ref. 81 and
con rmed experimentally in Ref. 46.
In Fig. 2c, we compare the results obtained with this
approximated method with the ISM described before, for
the di erent values of SOI scalings. For the bulk tran-
sition metals we investigate, the approximation is very
good, since the SOI is relatively small. In fact, even
when the SOI is scaled one order of magnitude higher,
the results of the XC-TCM are still very good.
The formulas in Eq. (13) show that we have arrived
at the bottom of the triangle in Fig. 1. These forms
do not involve an integral over energy, which simpli es
substantially the calculation of . For that reason, they
are suitable for rst-principles approaches (e.g., Refs. 20
and 62). This concludes our investigations of the spin
response methods. In the next section, we take a di erent
path to calculate the Gilbert damping.
V. TORQUE RESPONSE METHODS
Despite the simplicity of the methods based on the spin
susceptibility discussed in the previous section, seminal
work was based on a di erent type of response function.
The main idea, rst proposed by Kambersky37, is to di-
rectly relate to the spin-orbit interaction. Here, our
aim is twofold. First, we connect the spin susceptibility
with the spin-orbit torque response via the equation of
motion, clarifying the damping mechanisms captured by
this formalism. Second, we compare the results obtained
with both types of methods.
We start with the equation of motion for the spin-spin
susceptibility. Its time-Fourier transform can be written
as19
!
^S ;^S+
!=M+
^S ;^H
;^S+
!; (14)whereM= 2
^Sz
. From the Hamiltonian given in
Eq. (2), the commutator [ ^S ;^H
has four contributions:
kinetic (spin currents, from ^H0), exchange torque (from
^Hxc), external torque (from ^Hext) and spin-orbit torque
(from ^HSOI). In presence of SOI, the total spin magnetic
moment is not a conserved quantity and spin angular
momentum can be transferred to the orbital degrees of
freedom. For bulk systems subjected to static external
elds and in the present approximation for the electron-
electron interaction, the only two non-vanishing torques
are due to the external eld and the spin-orbit interac-
tion. It also follows from these assumptions that the
mechanisms that contribute to the relaxation arises then
from the spin-orbit torques ^T
SOI= i^S;^HSOI
and
from the broadening of the energy levels .
It can be shown19that the inverse of the susceptibility
+(!) =
^S ;^S+
!is given by
+(!) 1=
+
noSOI (!) 1
1 + +
noSOI (!) (!) 1
+
noSOI (!) 1 (!):
(15)
Here, +
noSOI (!) is the susceptibility calculated excluding
the SOI contribution to the Hamiltonian. The connection
between the two susceptibilities in Eq. (15) is provided
by the quantity
M2 (!) = i
^T
SOI;^S+
+
^T
SOI;^T+
SOI
!:(16)
Using Eq. (8), and noticing that the rst term on the
right-hand side of the equation above does not contribute
to the imaginary part, we nd
= noSOI 2
Mlim
!!0Im
^T
SOI;^T+
SOI
!
!: (17)
noSOI is the contribution obtained by inputting
+
noSOI (!) into Eq. (8), which is nite due to the broad-
ening.
Kambersky37 rst obtained this same result following a
di erent approach. In our framework, this would involve
starting from Eq. (5) and exploiting the consequences of
the fact that the collective spin excitations ( !meV)
have low frequencies when compared to the exchange en-
ergy (UeV). On the other hand, Hankiewicz et al.19
described the same expansion for low SOI, and justi ed
its use for !.
Bext. Finally, Edwards22shows that
this formula is equivalent to a perturbation theory cor-
rect to2
SOI(compared to
Bext !). For that rea-
son, he suggests that the states used in the calculation
of
^T
SOI;^T+
SOI
!should not include SOI, since the op-
erator ^T
SOI/SOI. Due to the orbital quenching in the
states without SOI, this leads to the absence of intra-
band contributions and, consequently, of the divergent
behavior for !082.
In this approach, temperature and disorder e ects are
included in noSOI (shown in Fig. 2d for Co), while the9
spin-orbit intrinsic broadening is calculated by the sec-
ond term in Eq. (17), which can also be obtained as
noSOI . An extra advantage of calculating the damp-
ing as the aforementioned di erence is that one explic-
itly subtracts the contributions introduced by , provid-
ing similar results to those obtained with vertex correc-
tions20. Considering the torque-torque response within
the mean- eld approximation (an exact result in the per-
turbative approach22), we obtain, similarly to Eq. (13),
noSOI =
2
MNX
kTrfImG(k;F)^T
SOIImG(k;F)^T+
SOIg:
(18)
In this formula, the involved quantities are matrices in
spin and orbital indices and the trace runs over both.
This is known as Kambersy's formula, commonly used in
the literature43,44,47,49,58, which we refer to as spin-orbit
torque correlation method (SO-TCM). As in Eq. (13), it
relates the damping to Fermi level quantities only. When
the SOI is not included in the calculation of the Green
functionsG(k;F) and enters only through the torque
operators, we name it perturbative SO-TCM22. These
methods are placed at the bottom right of Fig. 1, with
the main approximations required indicated by the long
dashed arrows.
We now proceed to compare these approaches with the
ISM explained in Sec. IV B. Fig. 3 presents the calcula-
tions of the SOI contribution to the damping parameter
of bulk Fe (a), Co (b) and Ni (c) using the SO-TCM ob-
tained in Eq. (18), when no external eld is applied. Both
approaches, including SOI (red curve with squares) in
the Green functions or not (green curve with triangles),
are shown. For a meaningful comparison, we compute
noSOI within the ISM.
We rst note that the perturbative approach suggested
by Edwards22describes reasonably well the large broad-
ening range (i.e., mostly given by the interband transi-
tions), but deviates from the other approaches for low .
This is an expected behaviour since it does not include
the intraband transitions that display the 1behav-
ior within the constant broadening model. In the clean
limit, the Gilbert damping computed from the pertur-
bative SO-TCM approaches zero for all elements, in a
very monotonic way for Co and Ni, but not for Fe. This
method is thus found to be in agreement with the other
ones only when SOI . The SO-TCM formula in-
cluding the SOI in the states (i.e., Kambersky's formula)
matches very well obtained within ISM in the whole
range of broadenings.
Finally, after demonstrating that the SO-TCM pro-
vides very similar results to the ISM, we can use it to
resolve the wave-vector-dependent contributions to the
Gilbert parameter by planes in the reciprocal space, as
(kmax
z) =kmax
zX
jkzj (kz); (19)
1 10 10010−310−2Gilbert damping α(a)Fe
ISM ( α−αnoSOI )
SO-TCM
perturbative SO-TCM10 100 1,000Temperature [K]
1 10 10010−410−310−2Gilbert damping α(b)Co
10 100 1,000
1 10 10010−410−310−210−1100
Broadening η[meV]Gilbert damping α(c)Ni
10 100 1,000Figure 3. Comparison between noSOI for (a) Fe bcc, (b)
Co fcc and (c) Ni fcc, obtained using the inverse susceptibility
method (ISM) with the spin-orbit-torque correlation method
(SO-TCM) with and without SOI in the states (perturbative
SO-TCM). All the points were computed with 108k-points in
the full Brillouin zone.
where (kz) is given by the right-hand side of Eq. (18)
summed over kx;ky. The result, displayed in Fig. 4, uses
100 million k-points for all curves and shows the expected
divergence in presence of SOI and a decrease with when
this interaction is absent. In every case, most of the con-
tribution arises from the rst half ( kmax
z<0:4). Note
that when the broadening of the energy levels is low, the
integrated alpha without SOI (Fig. 4b) displays step-
like contributions, while when SOI is present, they are
smoother. This is a consequence of the damping being10
01234Gilbert damping α(·10−2)(a) With SOI
η(meV):
0.14
0.41
0.68
1.1
1.4
4.1
6.8
10.9
13.6
40.8
0 0.2 0.4 0.6 0.8 100.20.40.60.81
kmax
zGilbert damping α(·10−3)(b) Without SOIDecreasingη
Decreasingη
Figure 4. Integrated Gilbert damping for fcc Co as a function
ofkzplotted against the maximum value, kmax
z(see Eq. (19)),
with SOI (a) and without SOI (b). The curves were obtained
using the SO-TCM given in Eq. (18). Colors represent di er-
ent values of the broadening (in units of meV). The value of
forkmax
z= 0 (i.e., a single value of kzin the sum) represents
a two-dimensional system, whilst for kmax
z= 1 the sum covers
the whole 3D Brillouin zone. In the latter case, the damping
decreases when is decreased without SOI, while it increases
drastically when SOI is present. For 2D systems, the
caused by interband transitions in the former and intra-
band in the latter.
The convergence of the previous results for the smallest
including SOI were tested with respect to the total
number of k-points in the Brillouin zone in Fig. 5. By
going from 10 million to 10 billion k-points, the results
vary20%. However, compared with the result shown
in Fig. 4a, the damping gets even larger, corroborating
once more the divergent results.
VI. DISCUSSIONS
In this section, we make a few nal remarks on the pre-
viously obtained results and we go beyond bulk systems
to comment on the approximations taken and additional
physical mechanisms that may come into play in other
0 0.2 0.4 0.6 0.8 1024
kmax
zGilbert damping α(·10−2)
k-points:
107
108
109
1010Figure 5. Integrated Gilbert damping for fcc Co as a func-
tion ofkzplotted against the maximum value, kmax
z, for
= 0:14 meV and di erent amount of k-points (up to 10
billion) in the Brillouin zone.
materials. We also provide a new analytical explanation
for the divergence of the damping parameter within the
constant broadening model.
Our rst comment regards the application of static
magnetic elds B. As described in Refs. 19 and 22, the
approximations done in Eq. (15) to derive an expression
for involves comparisons between the excitation en-
ergy andB. However, all the results we have presented
here were obtained in absence of static elds. We also
performed calculations including external magnetic elds
up toB7 T, and the computed damping parameter is
weakly in
uenced by their presence. We conclude that
the validity of the SO-TCM formula given in Eq. (18)
does not hinge on having a magnetic eld, supporting
the arguments already given in Ref. 19.
A further remark concerns the approximations made
to obtain the mean- eld result in Eq. (12). We assumed
that SOI is weak when using the magnetic sum rule.
This approximation may break down when this is not
the case. The spin pumping also a ects the magnetic
sum rule, which may worsen the agreement with the ISM
results. Although this contribution is not present in the
investigated (bulk-like) systems, it plays an important
role in magnetic multilayers. This e ect enhances the
damping factor32,54,55. Furthermore, the SO-TCM ex-
plicitly excludes spin pumping, as this is described by
^I
S= i^S ;^H0
, dropped from the equation of mo-
tion. These validity conditions are indicated in Fig. 1 by
the large blue rectangle (low SOI), red triangle (low spin
pumping) and green rectangle (no spin pumping).
Another mechanism that opens new spin relaxation
channels is the coupling between transverse and longi-
tudinal excitations induced by the SOI. This was one of
the reasons raised in Ref. 21 to explain the divergence of
the damping parameter. However, this is absent not only11
when the system has full spin rotational symmetry83, but
also when rotational symmetry is broken by the SOI in
2D and 3D systems for the symmetries and materials we
investigated. Even though the damping is nite in the
rst two cases (as shown in Fig. 2d), the divergence is
still present in the latter (Fig. 2b).
We can also recognize that the mathematical expres-
sion for in terms of the mean- eld susceptibility given
in Eq. (12) is similar to the conductivity one (i.e., the
slope of a response function)84| which leads to the same
issues when approaching the clean limit ( !0). How-
ever, the physical meaning is the exact opposite: While
the divergence of the conductivity represents an in nite
acceleration of an ideal clean system, in nite damping
denotes a magnetic moment that is instantly relaxed in
whichever direction it points (as d M=dt!0 for !1 )
| i.e., no dynamics10,11. This means that a clean 3D
spin system is in nitely viscous. Within the constant
broadening model, the divergence of the Gilbert damp-
ing can also be seen analytically by comparing Eq. (12)
with the calculations of the torkance done in Ref. 48.
By replacing the torque operator and the current density
by the spin lowering and raising operators, respectively,
the even contribution (in the magnetization) to the re-
sponse function vanishes and only the odd one remains.
In this approximation, it is also seen that only the Fermi
surface quantities are left, while the Fermi sea does not
contribute85. In the limit of low broadenings, this con-
tribution is shown to diverge as 1. This divergence
arises from intraband transitions which are still present
in the clean limit, and originate from the nite electronic
lifetimes introduced by the constant broadening approx-
imation.
The static limit ( !!0) is another reason that many
authors considered to be behind the divergent damping
behavior19,37,43,50. This limit is taken in Eq. (8) in or-
der to eliminate the contribution of terms nonlinear in
frequency from the inverse susceptibility (e.g., inertia ef-
fects68,86). They can be present in the full microscopic
calculation of the susceptibility but are not included in
the phenomenological model discussed in Appendix B.
Adding the quadratic term in frequency leads to an in-
verse susceptibility given by
Im[ +(!)] 1= !
2
M( !I)
whereIis the o -diagonal element of the moment of iner-
tia tensor86. The t to the expression linear in frequency
then yields an e ective e (!). In the vicinity of the res-
onance frequency, e (!res) = !resI, which is clearly
reduced in comparison to the one obtained in the static
limit, e (0) = . According to Ref. 68, I =, which
explains the discrepancy between the FMR and the ISM
seen in Fig. 2b as !0. We can then conclude that the
static limit is not the culprit behind the divergence of
in the clean limit.VII. CONCLUSIONS
In this work, we presented a study of di erent meth-
ods to calculate the intrinsic Gilbert damping , o ering
a panorama of how the approaches are related and their
range of validity (see Fig. 1). They can be grouped into
three main categories: the methods that directly employ
the results of full microscopic calculations of the dynam-
ical magnetic susceptibility (!) (FMR and ISM); the
exchange-torque method (XC-TCM), which is also based
on (!) but making use of the mean- eld approximation;
and the spin-orbit torque-correlation method (SO-TCM),
obtained from the (spin-orbit) torque-torque response via
an equation of motion for (!). While the FMR, ISM
and XC-TCM include all the contributions to the mag-
netic relaxation, the SO-TCM provides only the intrinsic
contribution due to the angular momentum transfer to
the orbital degrees of freedom (not including, for exam-
ple, the spin pumping mechanism). The XC- and SO-
TCM, given by Eqs. (13) and (18), are predominant in
the literature due to their simplicity in obtaining in
terms of Fermi level quantities. It is important to note,
however, that they rely on approximations that may not
always be full lled21.
In order to implement and compare the di erent meth-
ods, we constructed a uni ed underlying framework
based on a multi-orbital tight-binding Hamiltonian using
as case studies the prototypical bulk 3D systems: bcc Fe,
fcc Co and fcc Ni. For this set of materials, the di erent
methods lead to similar results for , showing that the
corresponding approximations are well-founded. Even
when the SOI strength is scaled up by one order of mag-
nitude, this excellent agreement remains, as we explic-
itly veri ed for bcc Fe. We found one method that falls
out-of-line with the others in the clean limit, namely the
perturbative form of the SO-TCM formula22,82. In this
case, although the equation is identical to the well-known
Kambersky formula, Eq. (18), the electronic states used
to evaluate it do not include SOI. By comparison with
the other methods, we conclude that the results obtained
by the perturbative SO-TCM are only valid in the large
broadening regime (compared to the SOI strength). Cen-
tral to our analysis was a careful study of the convergence
of our results with respect to the number of k-points,
reaching up to 1010k-points in the full Brillouin zone.
The behavior of is intimately connected with the con-
stant broadening approximation for the electronic life-
times. For high temperatures, the Gilbert damping in-
creases with increasing temperature ( ), while for
low temperatures it diverges for 3D ferromagnets (
1=), but not for 2D (ferromagnetic monolayers). Our
calculations revealed that the high temperature values
of arise mostly from the broadening of the electronic
states. In Ref. 20, the strongly increasing behaviour of
for high temperatures was found to be spurious, and
cured employing a more realistic treatment of disorder
and temperature, and the so-called vertex corrections.
We found that the contribution of the intrinsic SOI to 12
is additive to the one arising from the broadening, and
can be easily extracted by performing a calculation of
without SOI and subtracting this result from the SOI
one, noSOI . Combined with the ISM, this provides
a relatively simple and accurate way to obtain the in-
trinsic damping, which discounts contributions from the
additional broadening . This establishes an alternative
way of accessing the high temperature regime of .
The low-temperature divergence of when approach-
ing the clean limit for 3D ferromagnets has also been the
subject of much discussion. The rst diculty is in es-
tablishing numerically whether this quantity actually di-
verges or not. Our results consistently show an increase
of with decreasing , down to the smallest achievable
value of= 0:14 meV (Fig. 5), with no hints of a plateau
being reached, but only when accounting for SOI. This
divergence arises from the intraband contributions to ,
as discussed in Ref. 58. Refs. 22 and 82 used pertur-
bation theory arguments to claim that such intraband
contributions should be excluded. However, as we dis-
cussed in Sec. IV B, adapting the formalism of Ref. 48 to
the calculation of shows that these intraband terms are
enabled by the constant broadening approximation, and
so should be included in the calculations. Contrary to
the high temperature regime, works that employ a more
realistic treatment of disorder and temperature still nd
the diverging behavior of 20,52.
In real experiments, any kind of material disturbance
such as disorder or temperature e ects leads to a nite
value of the damping. Besides that, a non-uniform com-
ponent of the oscillatory magnetic eld (either from the
apparatus itself or due to its limited penetration into the
sample) induces excitations with nite wave vectors and
nite linewidths39,87. A di erent way to determine the
damping parameter is using the time-resolved Magneto-
Optic Kerr E ect (TR-MOKE)40,88. It has the advan-
tage that, as it accesses a smaller length scale ( 1µm)
than FMR experiments (which probe the whole magnetic
volume), the measured magnetic properties are more ho-
mogeneous and thus the e ect of linewidth broadening
may be weaker. The magnetic excitations in nanomag-
nets can also be probed by recent re nements of FMR
experimental setups89,90.
Although the methods we described here are gen-
eral, we did not explicitly addressed non-local sources
of damping such as the spin-pumping32. As a future
project, we plan to ascertain whether our conclusions
have to be modi ed for systems where this mechanism
is present. Systems that combine strong magnetic el-
ements with heavy ones possessing strong SOI are ex-
pected to have anisotropic properties, as well-known for
the magnetic interactions91. It is then natural to explore
when the Gilbert damping can also display signi cant
anisotropy, becoming a tensor instead of a scalar quan-
tity47,78. Indeed, this has been observed experimentally
in magnetic thin lms92,93. As the SOI, magnetic non-
collinearity can also lead to other forms of damping in do-
main walls and skyrmions50,94{98. From the microscopicpoint of view, the potential coupling between transverse
and longitudinal degrees of freedom allowed by the non-
collinear alignment should also be considered. Lastly,
higher order terms in frequency, such as the moment of
inertia68,86,99{101, might also become important in the
dynamical magnetic susceptibility for large frequencies
or for antiferromagnets, for instance.
The description of magnetization dynamics of real ma-
terials helps to design new spintronic devices able to con-
trol the
ow of information. Our work sheds light on fun-
damental questions about the main relaxation descrip-
tions used in the literature and sets ground for future
theoretical predictions.
Appendix A: Ground-state Hamiltonian
In this Appendix, we give the explicit forms of the
terms in the Hamiltonian written in Eq. 2. As the inves-
tigated systems only have one atom in the unit cell, the
site indices are omitted.
The electronic hoppings in the lattice are described by
^H0=1
NX
kX
t(k)cy
(k)c(k); (A1)
withcy
(k) andc(k) being the creation and annihila-
tion operators of electrons with spin and wave vector
kin the orbitals and, respectively. The tight-binding
parameters t(k) were obtained by tting paramagnetic
band structures from rst-principles calculations up to
second nearest neighbors102, within the two-center ap-
proximation103.
The electron-electron interaction is characterized by
a local Hubbard-like104interaction within the Lowde-
Windsor approximation105, resulting in the mean- eld
exchange-correlation term
^Hxc= X
2d
U
2(
M
0+X
2dn(20 0))
cy
(k)c0(k):
(A2)
Here,Uis the local e ective Coulomb interaction, M
and are the -component of the magnetic moment
vector (summed over the dorbitals) and of the Pauli
matrix, respectively. nis the change in the occupation
of orbitalcompared to the DFT calculations included
in Eq. A1. M andnare determined self-consistently.
The atomic SOI is described by
^HSOI=X
0^L
^S
0cy
(k)c0(k);(A3)
whereL andS are the components of the orbital and
spin vector operators, respectively. The strength of the
SOI,, is also obtained from rst-principles calculations.13
The interaction with a static magnetic eld Bextis
described by
^Hext=B
extX
0(^L
0+0)cy
(k)c0(k);
(A4)
whereBis absorbed to B
extand we used gL= 1 and
gS= 2 as the Land e factors for the orbital and spin
angular momentum.
Appendix B: Phenomenology of FMR
The semi-classical description of the magnetization is
obtained using the Landau-Lifshitz-Gilbert (LLG) equa-
tion (1)9. The e ective eld acting on the magnetic mo-
ment is obtained from the energy functional of the system
asBe (t) = @E=@M. For the symmetries we investi-
gate, the model energy106for the 3D cubic cases77can
be written as
E3D(M) =K4
M4(M2
xM2
y+M2
yM2
z+M2
xM2
z) MBext;
(B1)
while for 2D systems,
E2D(M) = K2
M2M2
z MBext: (B2)
Positive values of K4andK2yield easy magnetization
direction along the (001) direction.
We consider magnetic moments pointing along the easy
axis, which de nes the ^ zdirection. Static magnetic elds
are applied along the same orientation. The magnetic
moment is set into small angle precession, M=M^ z+Mx(t)^ x+My(t)^ y, by an oscillatory eld in the trans-
verse plane, i.e., Bext(t) =Bext^ z+Bext(t). In this form,
the e ective eld (linear in the transverse components of
the magnetization) is given by Be (t) =Ban(t)+Bext(t),
with
B3D
an(t) = 2K4
M2(Mx^ x+My^ y) , and B2D
an=2K2
M^ z
(B3)
being the anisotropy elds for 3D and 2D systems, respec-
tively. In the following expressions, K4andK2appear
in the same way, so they are denoted by K.
The Fourier transform of the linearized equation of mo-
tion can be written using the circular components M=
MxiMy. Within this convention, M =B =
+=2 and
+(!) = 2
M
[!
(Bext+Ban)] i !; (B4)
whereBan= 2K=M .
ACKNOWLEDGMENTS
We are very grateful to R. B. Muniz, A. T. Costa
and D. M. Edwards for fruitful discussions. The authors
also gratefully acknowledge the computing time granted
through JARA-HPC on the supercomputers JURECA
and JUQUEEN at Forschungszentrum J ulich, and the
computing resources granted by RWTH Aachen Univer-
sity under project jara0175. This work is supported by
the European Research Council (ERC) under the Eu-
ropean Union's Horizon 2020 research and innovation
programme (ERC-consolidator grant 681405 { DYNA-
SORE).
f.guimaraes@fz-juelich.de
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2205.14166v2.Scalar_field_damping_at_high_temperatures.pdf | arXiv:2205.14166v2 [hep-ph] 27 Sep 2022Scalar field damping at high temperatures
Dietrich B¨ odeker1, Jan Nienaber2
Fakult¨ at f¨ ur Physik, Universit¨ at Bielefeld, 33615 Biel efeld, Germany
Abstract
The motion of a scalar field that interacts with a hot plasma, l ike the inflaton
during reheating, is damped, which is a dissipative process . At high tempera-
tures the damping can be described by a local term in the effecti ve equation of
motion. The damping coefficient is sensitive to multiple scat tering. In the loop
expansion its computation would require an all-order resum mation. Instead we
solve an effective Boltzmann equation, similarly to the compu tation of transport
coefficients. For an interaction with another scalar field we o btain a simple re-
lation between the damping coefficient and the bulk viscosity , so that one can
make use of known results for the latter. The numerical prefa ctor of the damping
coefficient turns out to be rather large, of order 104.
1bodeker@physik.uni-bielefeld.de
2jan.nienaber@uni-bielefeld.de
11 Introduction
Scalar fields may play an important role in the early Universe. They can drive cosmic
inflation, andtheirquantumfluctuationscanprovidetheseedofga laxyformation, they
can cause phase transitions [ 1] and generate the baryon asymmetry of the Universe [ 2].
They can also be part or all of dark matter [ 3], or be responsible for today’s dark
energy [4].
We consider a scalar field ϕwhich is (approximately) constant in space and which
evolves in time. Important examples are the inflaton field which drives inflation or
the axion field which can be dark matter. To be specific we will consider ϕbeing the
inflaton, keeping in mind that our discussion applies to many other situ ations as well.
When inflation ends, the inflaton field ϕstarts oscillating coherently around the
minimum of its potential. It interacts with other fields leading to an ene rgy transfer
thus creating a plasma and (re-)heating the Universe. At the same time, the motion
ofϕgets damped. The plasma makes up an increasing fraction of the tot al energy
density. For sufficiently strong interaction the plasma thermalizes. Eventually this
thermal plasma dominates the energy density; the corresponding temperature is called
reheat temperature TRH. This is, however, not the largest temperature of the plasma,
which rises early during the reheating process and then decreases before it reaches
TRH[5,6].
An oscillating inflaton field with frequency ω=mϕ, can be viewed as a state with
high occupancy of inflaton particles with zero momentum and mass mϕ. When there
are only few decay products present, the damping is dominated by in flaton decay into
lighter particles [ 7]. If many particles have already been produced such that their oc-
cupation numbers are of order one or larger, other effects come in to play. Parametric
resonance can lead to very efficient particle production [ 8]. Then the decay products
thermalize and acquire a thermal mass. At high temperature, ther mal masses can
become larger than the inflaton mass such that the decay of an infla ton into plasma
particlesiskinematically forbidden[ 9]. Thenotherprocessesthatinvolve multiplescat-
terings become unsuppressed and open new channels for the ener gy transfer [ 10,11].
In this paper we consider the damping rate in the high-temperature regime where T
is much larger than mϕand the mass of the plasma particles. We assume that the
characteristic frequency ω∼˙ϕ/ϕand the damping rate γofϕare small compared to
the thermalization rate of the plasma. Then the inflaton interacts n early adiabatically
with an almost thermal plasma. In particular, there is no non-pertu rbative particle
production through parametric resonance [ 8]. When the plasma is approximately ther-
mal, its properties are fully specified by the temperature and by the instantaneous
2value ofϕ. Therefore the plasma ‘forgets’ about its past, and its effect on t he inflaton
dynamics can be described by local terms. The effective equation of motion (without
Hubble expansion) for the zero-momentum mode of ϕtakes the form [ 12]
¨ϕ+V′
eff+γ˙ϕ= 0, (1)
where the prime denotes a derivative with respect to ϕ. The effective potential Veff
and the damping coefficient γonly depend on the value of ϕand the temperature. For
sufficiently slow evolution, higher derivative terms in Eq. ( 1) can be neglected. Note
that the form of Eq. ( 1) follows from the separation of timescales alone.
Ifωand the damping rate γare small compared to the thermalization rate, then
γcan be obtained from a finite-temperature real-time correlation fu nction, evaluated
in the the zero-momentum, zero-frequency limit [ 13,14]. E.g., the damping coefficient
for the axion field is proportional to the Chern-Simons diffusion rate in QCD [ 13], the
so-called strong sphaleron rate, which is non-perturbative and ha s been calculated on
the lattice [ 15,16]. In many cases the required correlation functions can in principle
be calculated perturbatively in thermal field theory. They are, how ever, sensitive to
timescales much larger than the mean free time of the plasma particle s, so that one
has to take into account multiple scatterings. This requires the res ummation of an
infinite set of diagrams. Several authors have applied 1- or 2-loop a pproximations with
resummed propagators containing a finite width (see e.g. [ 17–20]), which gives rise to a
nonzero damping rate. However, proper treatment of the multiple interaction requires
the resummation of a much larger class of diagrams [ 21,22].
Similar complications arise in the computation of transport coefficient s, such as
bulk viscosity, which can be written as the zero-momentum zero-fr equency limit of
a stress-tensor correlation function [ 23,24]. For viscosities, the required summation
of diagrams has been performed. It was shown that this is equivalen t to solving an
effective Boltzmann equation [ 23].
The physics behind bulk viscosity ζis closely related to the one of the damping
coefficient γ. Both describe small deviations from thermal equilibrium. In the cas e of
ζit is due to a uniform expansion of the system. The deviation of the tr ace of the
stress tensor from the ideal-fluid form is proportional to ζ. When the field ϕchanges
with time and interacts with the plasma particles, it changes their par ameters such
as masses or couplings, driving the plasma out of equilibrium. Since ϕis spatially
constant, this deviation from equilibrium is homogeneous and isotrop ic as well.
For the model considered in Ref. [ 14] the damping coefficient could be related to
a correlation function of the stress tensor. Thus there is a simple r elation between γ
and the bulk viscosity ζof the thermal plasma, so that one can use the known result
3forζ. In this work we consider interactions of ϕwith another scalar field χthrough
operators that cannot be expressed in terms the stress tensor , but which still allow
for a perturbative treatment. We can proceed similarly to the comp utation of the
bulk viscosity, for which the required resummation of diagrams is equ ivalent to solving
an effective Boltzmann equation [ 23]. Damping coefficients have been computed from
a Boltzmann equation long ago using a relaxation time approximation [ 12].3This
approximation, however, does not give the correct result for the bulk viscosity in scalar
theory [23,24]. Here we carefully treat the collision term as well as thermal effects
by employing the effective Boltzmann equations which were used to pe rturbatively
compute bulk viscosities in scalar theories [ 23,24], and in gauge theories [ 25,26]. This
allows us to obtain the correct dependence on the coupling constan ts and explicitly
compute γat leading order in perturbation theory.
This paper is organized as follows. In Sec. 2we obtain the effective equation of
motion for ϕand an expression for the damping coefficient in terms of the plasma-
particle occupancy. The latter is computed in Sec. 3from an effective Boltzmann
equation. In Sec. 4the solution to the Boltzmann equation is inserted into the effective
equation of motion for ϕ, and the damping coefficients is expressed in terms of the
known bulk viscosity of the plasma. Section 5contains conclusions and a brief outlook.
Appendix Adeals with the thermodynamics of the plasma particles, and Appendix B
describes the solution of the Boltzmann equation.
2 Effective equation of motion
In this section, largely following Ref. [ 12], we obtain the effective equation of motion
(1) from quantum field theory andrelate the coefficients therein to mic roscopic physics.
We consider a scalar field Φ is coupled to another scalar field χthrough the interaction
LΦχ=−A(Φ)χ2. (2)
Restricting ourselves to renormalizable interactions we can have
A(Φ) =µ
2Φ+λ
4Φ2(3)
with coupling constants µandλ. Without Hubble expansion the equation of motion
for Φ reads
¨Φ−∆Φ+V′(Φ)+A′(Φ)χ2= 0, (4)
3The damping coefficients computed in Refs. [ 18–20] are of the same form as in Ref. [ 12].
4whereVis the part of the tree-level potential that depends only on Φ. Eq. (4) is still
an equation for field operators. We want to write an equation of mot ion for the zero-
momentum mode ϕof Φ, and we assume that ϕcan be approximated by a classical
field. We write
Φ =ϕ+ˆΦ (5)
whereˆΦ contains the non-zero momentum modes of Φ. Through the intera ction,χ
particles are produced. Once the χparticles are created, they can also produce Φ par-
ticles which are represented by ˆΦ. Thus the production of Φ particles also contributes
to the damping of ϕ. This effect is discussed in [ 27], where it was found that this
contribution is subdominant unless the energy density in ϕis small compared to the
energy density of the χparticles. In the context of reheating after inflation this would
already be during radiation domination. We assume that ϕstill dominates the energy
density and neglect this contribution. Then we can replace the fort h term in Eq. ( 4)
byA′(ϕ)χ2.
We assume that χinteracts rapidly with itself or other fields, so that it thermalizes
on timescales which are short compared to the period of ϕoscillations. Furthermore,
we assume that the interactions of χare weak enough, so that the typical mean free
path ofχparticles is much larger than their typical de Broglie wavelength. The n
χis made up of weakly interacting particles which can be described by th eir phase
space density, or occupancy f(t,p). Since we consider a homogeneous system, it only
depends on time tand on the particle momentum p. We may then replace χ2by its
expectation value computed from the occupancy using the free-fi eld expression
/angbracketleftbig
χ2/angbracketrightbig
=/integraldisplayd3p
(2π)3f(t,p)
E, (6)
whereEis the one-particle energy (see below). Thus we arrive at the effect iveclassical
equation of motion
¨ϕ+V′(ϕ)+A′(ϕ)∝angbracketleftχ2∝angbracketright= 0. (7)
The deviations from equilibrium are assumed to be small, so that the oc cupancy in
Eq. (6) can be written as
f(t,p) =feq(t,p)+δf(t,p) (8)
with the local equilibrium distribution
feq(t,p) =1
exp/parenleftbig
E/T)−1. (9)
5andδf≪feq. The temperature Tin Eq. (9) varies slowly with time. The mass of the
χparticles depends on the value of ϕ,
m2=m2
0+2A(ϕ) (10)
wherem0is the zero-temperature mass at vanishing ϕ. Throughout this paper we
assume that mis small compared to the temperature.4The mass appearing in the one-
particle energy Ein Eqs. ( 6) and (9) also receives a thermal contribution m2
th∝T2,
so thatE= (p2+m2
eff)1/2with
m2
eff=m2+m2
th. (11)
To avoid a tachyonic instability [ 29],m2
effmust be positive.
With the help of Eq. ( 8), the expectation value in Eq. ( 6) becomes
∝angbracketleftχ2∝angbracketright=∝angbracketleftχ2∝angbracketrighteq+δ∝angbracketleftχ2∝angbracketright. (12)
The first term inEq. ( 12)is nondissipative. It gives a thermal correction intheeffective
potential in Eq. ( 1) [12],
V′
eff=V′+A′∝angbracketleftχ2∝angbracketrighteq, (13)
which is precisely the leading term in the high-temperature limit of the 1 -loop effective
potential (see, e.g., [ 30]). The second term in Eq. ( 12) is dissipative and will give rise
to the damping term in Eq. ( 1).
3 Boltzmann equation
The occupancy of χparticles in Eq. ( 6) can be computed by solving a Boltzmann
equation, because the timescale on which their mass changes is of or der 1/ωwhich
is much larger than their typical de Broglie wavelength of order 1 /T. Due to the
homogeneity, spatial momentum is conserved. Thus the Boltzmann equation takes the
form
∂tf=C (14)
whereCis the collision term. Now we insert Eq. ( 8) on the left-hand side of Eq. ( 14).
We neglect ∂tδfbecause it is quadratic in small quantities, so that
∂tf≃ −feq(1+feq)∂t(E/T). (15)
4The opposite limit m≫Tis consideredin Ref. [ 28] with additional light degreesoffreedom. Then
χcan be integrated out giving rise to an effective interaction of ϕwith the light plasma particles.
6The zero-momentum mode ϕdepends on time and changes the mass of the plasma
particles through the interaction ( 2). If the oscillation is much slower than the ther-
malization of the plasma, this is an adiabatic process that changes th e temperature in
Eq. (9) at constant volume.5Thus the time dependence of the temperature is deter-
mined by
∂tT=/parenleftbigg∂T
∂m2/parenrightbigg
S,V∂tm2. (16)
In the limit T≫mwe obtain (see Appendix A)
/parenleftbigg∂T
∂m2/parenrightbigg
S,V=T
4ρ/angbracketleftbig
χ2/angbracketrightbig
eq, (17)
whereρistheenergydensityofthethermalplasma. Theone-particleener gyEdepends
on time through the effective mass. We thus have
∂t(E/T) =1
2TE/bracketleftBigg
1−∝angbracketleftχ2∝angbracketrighteq
2ρ/parenleftbigg
E2−T2∂m2
eff
∂T2/parenrightbigg/bracketrightBigg
∂tm2. (18)
The third term in the square bracket is small compared to the first, both for hard
(|p| ∼T) and for soft ( |p| ∼meff) momenta, and can be neglected, so that
∂tf≃ −feq(1+feq)Q
2T∂tm2(19)
with
Q(p)≡1
E−∝angbracketleftχ2∝angbracketrighteq
2ρE. (20)
Now we insert Eq. ( 8) into the collision term. Since Cvanishes in equilibrium, its
expansion in δfstarts at linear order,
C≃/hatwideCδf. (21)
Here we have neglected the contribution of Φ particles because the corresponding colli-
sion term is quadratic in the Φ- χcouplings which we assume to be much smaller than
the self-coupling of χentering /hatwideC.
It is convenient to write the deviation from equilibrium as
δf=−feq(1+feq)X. (22)
5This does not apply to the case ω>∼Twhich is considered in Refs. [ 31–33].
7Similarly we write the linearized collision term as
/hatwideCδf=feq(1+feq)/tildewideCX, (23)
with the convolution
[/tildewideCX](p)≡/integraldisplayd3p′
(2π)3/tildewideC(p,p′)feq(p′)/parenleftBig
1+feq(p′)/parenrightBig
X(p′). (24)
Then the kernel /tildewideCis symmetric [ 34],
/tildewideC(p,p′) =/tildewideC(p′,p). (25)
The Boltzmann equation thus turns into an equation for X,
−∂tm2
2TQ=/tildewideCX. (26)
Since the collision term vanishes in equilibrium for any temperature, th e linearized
collision term has a zero mode X=X1associated with a shift of the temperature,
which is given by X1(p) =E. Due to the symmetry of /tildewideCthe right-hand side of
Eq. (26) is orthogonal to X1. For Eq. ( 26) to be consistent, the left-hand side must be
orthogonal to X1as well. This is indeed the case when the second term in Eq. ( 20) is
taken into account, which can be easily checked.
Due to the zero mode the linear operator /tildewideCcannot be inverted. However, it can be
inverted on the subspace orthogonal to the zero mode,6where orthogonality is defined
with respect to the inner product
(X,X′)≡/integraldisplayd3p
(2π)3feq(1+feq)X(p)X′(p). (27)
We can then write the solution as
X=−∂tm2
2T/tildewideC−1Q, (28)
and we finally obtain
δf=feq(1+feq)∂tm2
2T/tildewideC−1Q. (29)
6This is equivalent to imposing the Landau-Lifshitz condition on the ene rgy density δρ=
(2π)−3/integraltext
d3pEδf= 0.
84 Damping coefficient and bulk viscosity
Coming back to the effective equation of motion ( 7), we insert the solution ( 29) into
Eq. (6) to obtain the second term in Eq. ( 12) as
δ∝angbracketleftχ2∝angbracketright=1
2T/parenleftBig
Q′,/tildewideC−1Q/parenrightBig
∂tm2. (30)
Here we have introduced Q′(p)≡1/E. The factor ∂tm2is proportional to ˙ ϕ. Com-
paring Eqs. ( 1) and (7) we see that the second term in Eq. ( 12) is indeed responsible
for the damping,
γ˙ϕ=A′δ∝angbracketleftχ2∝angbracketright. (31)
Inserting Afrom Eq. ( 3) we obtain
γ=1
4T/parenleftBig
Q′,/tildewideC−1Q/parenrightBig
(µ+λϕ)2(32)
which is our main result.
The computation of /tildewideC−1Qis described in Appendix B. However, at this point we do
not need it explicitly,7because, as we shall see in a moment, the coefficient ( Q′,/tildewideC−1Q)
also appears in the computation of the bulk viscosity ζof theχplasma. Therefore it
can be read off directly from known results for ζ. To see this, we first recall that /tildewideC−1Q
is orthogonal to X1=E, i.e., (E,/tildewideC−1Q) = 0. In Eq. ( 32) we may therefore replace
Q′=Q+(∝angbracketleftχ2∝angbracketrighteq/2ρ)EbyQwithout changing our result for γ, which then reads
γ=1
4T/parenleftBig
Q,/tildewideC−1Q/parenrightBig
(µ+λϕ)2. (33)
Letusnowrecallsomepropertiesofthebulkviscosity, asdescribe d, e.g., inRef.[ 25].
When a plasma is uniformly compressed or rarified it leaves equilibrium, u nless this
happens infinitely slowly. The pressure of the plasma then differs fro m the value it
would have in the equilibrium state with the same energy density. This d eviation of
the pressure from equilibrium is proportional to the bulk viscosity.
In a plasma with scale invariance the bulk viscosity vanishes, for two d ifferent
reasons. The first one is that a uniform expansion or rarefaction is a dilatation which is
asymmetry transformationinascaleinvariant theory. Therefore suchatransformation
does not take the system out of equilibrium. The second is that in a sc ale invariant
theory the trace of the energy-momentum tensor Tµνalways vanishes. Therefore the
pressure P=Tmm/3 equals ρ/3 even out of equilibrium.
7It will be usefull later, when we estimate the size of δfin order to check the accuracy of our
approximations.
9Scale invariance is broken by zero-temperature masses and by the trace anomaly,
i.e., by quantum effects. The bulk viscosity is then quadratic in the mea sure which
controls the breaking of scale invariance.
The bulk viscosity ζof the thermal plasma of scalar particles with mass mwas
computed for scalar theory in Refs. [ 23,24]. Like in QCD [ 25] it can be written as
ζ=1
T/parenleftBig
q,/tildewideC−1q/parenrightBig
, (34)
with
q(p) =−1
E/bracketleftbigg/parenleftbigg
c2
s−1
3/parenrightbigg
p2+c2
sm2
sub/bracketrightbigg
. (35)
Herecswithc2
s=∂P/∂ρis the speed of sound. In a scale invariant theory c2
sequals
1/3, so that the first term in the square bracket in Eq. ( 35) vanishes. Furthermore,
msubwith
m2
sub≡m2
eff−T2∂m2
eff
∂T2(36)
is the so-called subtracted mass. In the massless limit m= 0,m2
effequalsT2times a
function of the coupling constants (see Eqs. ( 10), (11)). Then the only contribution to
the subtracted mass is from the running of the couplings renormaliz ed at the scale T.
The subtracted mass is thus a measure of the deviation from scale in variance as well,
because it vanishes when m= 0 and the couplings do not run. Since qappears twice
in Eq. (34), the bulk viscosity is indeed quadratic in the measure of scale-invar iance
violation.
Now we replace p2byE2−m2
effin Eq. (35) which turns it into
q(p) =/bracketleftbigg/parenleftbigg
c2
s−1
3/parenrightbigg
m2
eff−c2
sm2
sub/bracketrightbigg1
E−/parenleftbigg
c2
s−1
3/parenrightbigg
E. (37)
Comparing Eqs. ( 20) and (37) we see that both Qandqconsist of a term proportional
to 1/E, and one proportional to E. Furthermore, qappears on the left-hand side of a
Boltzmann equation precisely like Qin Eq. (19),8and is thus orthogonal to X1as well.
Therefore qmust be proportional to Q. Here we are interested in the limit T≫min
which [23]
|c2
s−1/3|=O(m2
sub/T2)≪1 (T≫m), (38)
8See Eq. (3.7) of Ref. [ 25].
10andalsom2
eff≪T2. Thereforewecanapproximatethesquarebracketin( 37)by−m2
sub/3.
This gives us the approximate factor of proportionality, so that q≃ −(m2
sub/3)Q. Then
we obtain the following simple relation
γ(ϕ,T) =9
4ζ
m4
sub(µ+λϕ)2(39)
of the damping coefficient in the effective equation of motion ( 1) and the bulk viscosity
of theχplasma. Like in Ref. [ 14] the nontrivial dependence on the interaction of the
plasma particles, on thermal masses, etc., is precisely the same for both quantities.
Note that m4
subin the denominator of Eq. ( 39) removes the factors related to the
breaking of scale invariance from ζ, which can also be seen explicitly in Eqs. ( 41) and
(43) below. Thus, despite its similarity to the bulk viscosity, the damping c oefficient
is not related to the breaking of scale invariance.
The bulk viscosity for a self-interacting scalar field was computed in R ef. [23]. For
the quartic self-interaction
Lχχ=−g2
4!χ4(40)
andT≫mthe leading-order result reads
ζ=b
4m4
subm2
eff
g8T3ln2/parenleftbiggκ2m2
eff
T2/parenrightbigg
(41)
withb= 5.5×104andκ= 1.25. The effective mass for the χparticles is given by
m2
eff=m2+g2
24T2, m2=m2
0+λ
2ϕ2. (42)
Inserting Eq. ( 41) into Eq. ( 39) we obtain the damping coefficient
γ(ϕ,T) =am2
eff
g8T3ln2/parenleftbiggκ2m2
eff
T2/parenrightbigg
(µ+λϕ)2(43)
with the remarkably large numerical prefactor
a= 3.1×104. (44)
In the temperature range T≫m/g2the form ( 41) and thus Eq. ( 43) remain valid
when a cubic self-interaction is included in ( 40), while in the intermediate regime m≪
T≪m/g2the bulk viscosity depends nontrivially on the relative strengths of c ubic
and quartic χself-couplings [ 23]. It is obvious from the dependence on the coupling
constant gthat the result ( 43) cannot be obtained from a one-loop approximation to a
11correlation function, as anticipated in Refs. [ 22,28]. Instead, by solving the Boltzmann
equation we have summed an infinite set of diagrams which all contribu te at leading
order ing.
We can compare our result with the one obtained in Ref. [ 12] for a single scalar
field by putting χ=ϕ,µ= 0, and, up to an O(1) factor, g2=λ. In Ref. [ 12] the
Boltzmann equation was solved in the collision-time approximation, i.e., b y replacing
the linearized collision term on the right-hand side of Eq. ( 21) by a constant times
δf. Such an approximation does not take into account the zero eigenv alues and the
hierarchy of nonzero eigenvalues of /tildewideC. In Ref. [ 12] the collision time is determined by
2→2 scattering which changes momenta but not particle numbers. Bulk viscosity
and the damping coefficient γare, however, determined by the slowest equilibration
process, corresponding to the smallest eigenvalue of the linear collis ion operator, since
it is the inverse of the collision operator that appears in Eqs. ( 32) and (34). In scalar
field theory the slowest process is particle number equilibration. The refore the com-
putation of Ref. [ 12] does not give the correct dependence on the coupling constant
and underestimates the values of γandζ. Similarly, in Ref. [ 35] the rate for elastic
scattering was used to estimate the damping coefficient.
The importance of particle number changing processes for the bulk viscosity is well
known. The reason why they are also important for the damping coe fficient is the
following. When ϕevolves in time, it changes the mass of the χparticles, but not their
momenta. This, in turn, changes the energy density of χparticles but leaves their
number density unaffected. In order to relax to equilibrium, the χparticle number has
to adjust to the equilibrium value corresponding to their new energy density.
Let as finally discuss the range of validity of the effective equation of motion (1).
There the damping term is linear in ˙ ϕ. This is related to the linearization of the
Boltzmann equation, which requires that δf≪feq. In Appendix Bwe show that
δf/feq∼meff
g8T4∂tm2(45)
for the interaction ( 40). When ϕoscillates with frequency ωand amplitude /tildewideϕ, the time
derivative ∂tm2can be estimated as λω/tildewideϕ2. ForgT≫mwe thus need
λω/tildewideϕ2≪g7T3(46)
in order to be able to linearize the Boltzmann equation.
We may also apply the condition ( 46) to a model with a single scalar field ϕwhich
was considered in Ref. [ 12] by putting µ= 0 and λ∼g2. Then ( 46) turns into
ω/tildewideϕ2≪g5T3. The energy density in ϕwould be ρϕ∼ω2/tildewideϕ2≪(ω/T)g5T4. Due to
12ω≪T, the energy carried by ϕwould be only a tiny fraction of the plasma energy
densityρ∼T4. For the more interesting case that we have several fields, ϕcan give
the dominant contribution to the total energy without violating the condition ( 46).
5 Conclusion
A slowly moving homogeneous scalar field ϕinteracting with a thermal plasma drives
it slightly out of equilibrium, giving rise to dissipation and damping. In the high-
temperature regime the damping coefficient in the effective equation of motion for ϕ
canbeefficientlycomputedbysolvinganappropriateBoltzmannequa tion, seeEq.( 32).
We have considered a plasma made of a single species of scalar particle s. In this case
we obtained a simple relation of the damping coefficient to the bulk visco sity of the
plasma, Eq. ( 39). This extends a result [ 14] which was obtained for a scalar field
with derivative interaction. Like in the computation of viscosity, the solution of the
Boltzmann equation is dominated by the slowest process required fo r equilibration.
This can be easily generalized to multicomponent plasmas, where again one has to
identify the slowest process to solve the Boltzmann equation and th en use the resulting
phase spacedensity tocomputethedissipative terms intheeffectiv e equationofmotion
for the scalar field.
Acknowledgements
We thank Mikko Laine and Simona Procacci for comments and discuss ions. D.B. ac-
knowledges support by the Deutsche Forschungsgemeinschaft ( DFG, German Research
Foundation) through the CRC-TR 211 ’Strong-interaction matter under extreme con-
ditions’– project number 315477589 – TRR 211.
A Mass dependence of the temperature
The free energy of an ideal gas has the high-temperature expans ion
F(T,V,m2) =V(−aT4+bT2m2+···) (A.1)
with positive constants aandb;mis the mass of one particle species. The coefficient
acan also contain the contributions from other light species. At leadin g order our
expansion is related to the energy density by ρ= 3aT4. For a scalar
∂F
∂m2=V
2∝angbracketleftχ2∝angbracketrighteq (A.2)
13which gives b=∝angbracketleftχ2∝angbracketrighteq/(2T2). The entropy is
S=−∂F
∂T=V(4aT3−2bTm2+···). (A.3)
This can be inverted to obtain the expansion for the temperature, T=T0+T2+···,
for which we obtain
T0=/parenleftbiggS
4aV/parenrightbigg1/3
, (A.4)
T2=b
6am2
T0. (A.5)
Differentiating T2with respect to m2then gives Eq. ( 17).
B SolvingtheBoltzmannequationandestimating δf
The linearization of the Boltzmann equation is only possible if the deviat ion from
equilibrium is small, δf≪feq. This condition restricts the allowed values of the
couplings and the amplitude of the zero-momentum mode ϕ.
To estimate the size of δfin Eq. (29) we derive its explicit form, closely following
Ref. [24]. For a self-interacting scalar field one has to include two contributio ns in the
collision term,
/tildewideC=/tildewideCel+/tildewideCinel. (B.1)
/tildewideCeldescribes elastic 2 →2scattering which conserves particlenumber. Therefore ithas
the additional zero mode X0= 1, associated with a shift of the chemical potential, and
cannot be inverted on the subspace orthogonal to X1=E. One also has to include an
inelastic contribution /tildewideCineldescribing particle number changing processes, even though
its matrix element is higher order. /tildewideChas a single small eigenvalue con the subspace
orthogonal to X1, with the approximate eigenvector
X0⊥=X0−αX1 (B.2)
whereα= (X1,X0)/(X1,X1). The small eigenvalue is approximately
c=/parenleftbig
X0⊥,/tildewideCinelX0⊥/parenrightbig
(X0⊥,X0⊥)(B.3)
while the other nonvanishing eigenvalues areof order /tildewideCel. Inthe numerator ofEq. ( B.3)
we may replace X0⊥byX0because /tildewideCinelX1vanishes. This eigenvalue gives the leading
14contribution to /tildewideC−1, so that
/tildewideC−1Q≃(X0⊥,Q)
(X0,/tildewideCinelX0)X0⊥. (B.4)
In the numerator of Eq. ( B.4) we can replace X0⊥byX0, because X1is orthogonal to
Q. We insert this into Eq. ( 29), which finally gives
δf(p) =feq(p)/bracketleftbig
1+feq(p)/bracketrightbig∂tm2
2T(X0,Q)
(X0,/tildewideCinelX0)X0⊥(p). (B.5)
We now use this to estimate the size of δf. We will encounter the integrals
In≡/integraldisplayd3p
(2π)3feq(1+feq)En(B.6)
forn=−1,0, and 1. For n≥0 these are saturated at |p| ∼T, givingIn∼T3+n.
Sincefeq≃T/EforE≪T, the integral I−1, is logarithmically infrared divergent in
the massless limit and is cut off by meff. ThusI−1receives leading order contributions
both from |p| ∼Tand from |p| ∼meff≪T, with the result
I−1=T2
2π2ln/parenleftbigg2T
meff/parenrightbigg
. (B.7)
The factor ( X0,Q) in the numerator of Eq. ( B.5) contains I−1andI1and is of order T2
modulo logarithms, because ∝angbracketleftχ2∝angbracketrighteq/ρ∼T−2. The denominator depends on the type of
interaction (see below).
Since the size of δf(p) depends on |p|, we need to know which values of |p|give the
dominant contributionsto( Q′,/tildewideC−1Q)∝(Q′,X0⊥)whichentersthedampingcoefficient
in Eq. (32). Using Eq. ( B.2) we find that the integrals ( B.6) appear in the combination
I−1−αI0. The factor αis of order 1 /T. Thus|p| ∼Tand|p| ∼meffare equally
important. In both regions X0⊥∼1. Due to the Bose factors in Eq. ( 22) the ratio
δf/feqincreases with decreasing |p|, so that it takes its largest value when |p| ∼meff.
Putting|p| ∼meff, collecting all factors and ignoring logarithms we obtain
δf/feq∼T2∂tm2
meff(X0,/tildewideCinelX0). (B.8)
Fortheχself-interaction( 40)andthe χ-ϕinteraction ( 2) withµ= 0,/tildewideCineldescribes
scattering involving 6 particles. The corresponding squared matrix is proportional
tog8. The momentum integral which enters the denominator in Eq. ( B.4) is saturated
by soft momenta ( T∼meff)) [23]. It contains up to six Bose distributions, which for
soft momenta satisfy feq(p)≃T/E, giving rise to a factor T6. By dimensional analysis
one then finds ( X0,/tildewideCinelX0)∼g8T6/m2
eff, which yields the estimate ( 45).
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18 |
1604.07053v3.Coupled_Spin_Light_dynamics_in_Cavity_Optomagnonics.pdf | Coupled Spin-Light dynamics in Cavity Optomagnonics
Silvia Viola Kusminskiy,1Hong X. Tang,2and Florian Marquardt1, 3
1Institute for Theoretical Physics, University Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
2Department of Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA
3Max Planck Institute for the Science of Light, Günther-Scharowsky-Straße 1, 91058 Erlangen, Germany
Experiments during the past two years have shown strong resonant photon-magnon coupling in
microwave cavities, while coupling in the optical regime was demonstrated very recently for the first
time. Unlike with microwaves, the coupling in optical cavities is parametric, akin to optomechanical
systems. This line of research promises to evolve into a new field of optomagnonics, aimed at
the coherent manipulation of elementary magnetic excitations by optical means. In this work we
derive the microscopic optomagnonic Hamiltonian. In the linear regime the system reduces to the
well-known optomechanical case, with remarkably large coupling. Going beyond that, we study the
optically induced nonlinear classical dynamics of a macrospin. In the fast cavity regime we obtain
an effective equation of motion for the spin and show that the light field induces a dissipative term
reminiscent of Gilbert damping. The induced dissipation coefficient however can change sign on
the Bloch sphere, giving rise to self-sustained oscillations. When the full dynamics of the system is
considered, the system can enter a chaotic regime by successive period doubling of the oscillations.
I. INTRODUCTION
The ability to manipulate magnetism has played his-
torically an important role in the development of infor-
mation technologies, using the magnetization of materi-
als to encode information. Today’s research focuses on
controlling individual spins and spin currents, as well as
spin ensembles, with the aim of incorporating these sys-
tems as part of quantum information processing devices.
[1–4]. In particular the control of elementary excitations
of magnetically ordered systems –denominated magnons
or spin waves, is highly desirable since their frequency is
broadly tunable (ranging from MHz to THz) [2, 5] while
theycanhaveverylonglifetimes, especiallyforinsulating
materials like the ferrimagnet yttrium iron garnet (YIG)
[6]. The collective character of the magnetic excitations
moreoverrendertheserobustagainstlocalperturbations.
Whereas the good magnetic properties of YIG have
beenknownsincethe60s, itisonlyrecentlythatcoupling
andcontrollingspinwaveswithelectromagneticradiation
in solid-state systems has started to be explored. Pump-
probe experiments have shown ultrafast magnetization
switching with light [7–9], and strong photon-magnon
coupling has been demonstrated in microwave cavity ex-
periments [10–18] –including the photon-mediated cou-
pling between a superconducting qubit and a magnon
mode [19]. Going beyond microwaves, this points to the
tantalizing possibility of realizing optomagnonics : the
coupled dynamics of magnons and photons in the op-
tical regime, which can lead to coherent manipulation
of magnons with light. The coupling between magnons
and photons in the optical regime differs from that of
the microwave regime, where resonant matching of fre-
quencies allows for a linear coupling: one magnon can be
convertedintoaphoton, andviceversa[20–22]. Intheop-
tical case instead, the coupling is a three-particle process.
This accounts for the frequency mismatch and is gener-
z
yzx
optical mode
optical shift GSx0magnonmodeopticalmodeGˆ~SˆaabcFigure 1. (Color online) Schematic configuration of the model
considered. (a)Optomagnoniccavitywithhomogeneousmag-
netization along the z-axis and a localized optical mode with
circular polarization in the y-z-plane. (b) The homogeneous
magnon mode couples to the optical mode with strength G.
(c) Representation of the magnon mode as a macroscopic spin
on the Bloch sphere, whose dynamics is controlled by the cou-
pling to the driven optical mode.
ally called parametric coupling. The mechanism behind
the optomagnonic coupling is the Faraday effect, where
the angle of polarization of the light changes as it prop-
agates through a magnetic material. Very recent first
experiments in this regime show that this is a promising
route, by demonstrating coupling between optical modes
and magnons, and advances in this field are expected to
develop rapidly [23–27].
In this work we derive and analyze the basic op-
tomagnonic Hamiltonian that allows for the study of
solid-state cavity optomagnonics. The parametric op-
tomagnonic coupling is reminiscent of optomechanicalarXiv:1604.07053v3 [cond-mat.mes-hall] 19 Sep 20162
models. In the magnetic case however, the relevant oper-
atorthatcouplestotheopticalfieldisthespin, insteadof
the usual bosonic field representing a mechanical degree
of freedom. Whereas at small magnon numbers the spin
can be replaced by a harmonic oscillator and the ideas of
optomechanics [28] carry over directly, for general trajec-
tories of the spin this is not possible. This gives rise to
rich non-linear dynamics which is the focus of the present
work. Parametric spin-photon coupling has been studied
previously in atomic ensembles [29, 30]. In this work we
focus on solid-state systems with magnetic order and de-
rive the corresponding optomagnonic Hamiltonian. After
obtaining the general Hamiltonian, we consider a simple
model which consists of one optical mode coupled to a
homogeneous Kittel magnon mode [31]. We study the
classical dynamics of the magnetic degrees of freedom
and find magnetization switching, self-sustained oscilla-
tions, and chaos, tunable by the light field intensity.
The manuscript is ordered as follows. In Sec. (II) we
present the model and the optomagnonic Hamiltonian
which is the basis of our work. In Sec. (IIA) we discuss
briefly the connection of the optomagnonic Hamiltonian
derived in this work and the one used in optomechanic
systems. In Sec. (IIB) we derive the optomagnonic
Hamiltonian from microscopics, and give an expression
for the optomagnonic coupling constant in term of ma-
terial constants. In Sec. (III) we derive the classical
coupled equations of motion of spin and light for a ho-
mogeneous magnon mode, in which the spin degrees of
freedomcanbetreatedasamacrospin. InSec. (IIIA)we
obtain the effective equation of motion for the macrospin
in the fast-cavity limit, and show the system presents
magnetization switching and self oscillations. We treat
the full (beyond the fast-cavity limit) optically induced
nonlinear dynamics of the macrospin in Sec. (IIIB), and
follow the route to chaotic dynamics. In Sec. (IV) we
sketch a qualitative phase diagram of the system as a
function of coupling and light intensity, and discuss the
experimental feasibility of the different regimes. An out-
look and conclusions are found in Sec. (V). In the Ap-
pendix we give details of some of the calculations in the
main text, present more examples of nonlinear dynamics
asafunctionofdifferenttuningparameters, andcompare
optomagnonic vs.optomechanic attractors.
II. MODEL
Further below, we derive the optomagnonic Hamilto-
nian which forms the basis of our work:
H= ~^ay^a ~
^Sz+~G^Sx^ay^a; (1)
where ^ay(^a) is the creation (annihilation) operator for a
cavity mode photon. We work in a frame rotating at the
laser frequency !las, and =!las !cavis the detuning
with respect to the optical cavity frequency !cav. Eq. (1)assumes a magnetically ordered system with (dimension-
less) macrospin S= (Sx;Sy;Sz)with magnetization axis
along ^ z, and a precession frequency
which can be con-
trolled by an external magnetic field [32]. The coupling
between the optical field and the spin is given by the
last term in Eq. (1), where we assumed (see below) that
light couples only to the x component of the spin as
shown in Fig. (1). The coefficient Gdenotes the para-
metric optomagnonic coupling. We will derive it in terms
of the Faraday rotation, which is a material-dependent
constant.
A. Relation to optomechanics
Close to the ground state, for deviations such that
S S(withS=jSj), we can treat the spin in the
usual way as a harmonic oscillator, ^Sxp
S=2(^b+^by),
withh
^b;^byi
= 1. Then the optomagnonic interaction
~G^Sx^ay^a~Gp
S=2^ay^a(^b+^by)becomes formally equiv-
alent to the well-known opto mechanical interaction [28],
with bare coupling constant g0=Gp
S=2. All the phe-
nomena of optomechanics apply, including the “optical
spring” (here: light-induced changes of the magnon pre-
cession frequency) and optomagnonic cooling at a rate
opt, and the formulas (as reviewed in Ref. [28]) can be
taken over directly. All these effects depend on the light-
enhanced coupling g=g0 , where =pnphotis the
cavity light amplitude. For example, in the sideband-
resolved regime (
, whereis the optical cavity
decay rate) one would have opt= 4g2=. Ifg > ,
one enters the strong-coupling regime, where the magnon
mode and the optical mode hybridize and where coher-
ent state transfer is possible. A Hamiltonian of the form
of Eq. (1) is also encountered for light-matter interaction
in atomic ensembles [29], and its explicit connection to
optomechanics in this case was discussed previously in
Ref. [30]. In contrast to such non-interacting spin en-
sembles, the confined magnon mode assumed here can
be frequency-separated from other magnon modes.
B. Microscopic magneto-optical coupling G
In this section we derive the Hamiltonian presented in
Eq. (1) starting from the microscopic magneto-optical
effect in Faraday-active materials. The Faraday effect is
captured by an effective permittivity tensor that depends
on the magnetization Min the sample. We restrict our
analysis to non-dispersive isotropic media and linear re-
sponseinthemagnetization, andrelegatemagneticlinear
birefringence effects which are quadratic in M(denomi-
nated the Cotton-Mouton or Voigt effect) for future work
[5, 33]. In this case, the permittivity tensor acquires an
antisymmetric imaginary component and can be written3
as"ij(M)="0("ij ifP
kijkMk), where"0(") is the
vacuum (relative) permittivity, ijkthe Levi-Civita ten-
sorandfamaterial-dependentconstant[33](hereandin
what follows, Latin indices indicate spatial components).
The Faraday rotation per unit length
F=!fMs
2cp"; (2)
depends on the frequency !, the vacuum speed of light
c, and the saturation magnetization Ms. The magneto-
optical coupling is derived from the time-averaged energy
U=1
4
drP
ijE
i(r;t)"ijEj(r;t), using the complex
representation of the electric field, (E+E)=2. Note
that Uis real since "ijis hermitean [5, 33]. The magneto-
optical contribution is
UMO= i
4"0f
dr M(r)[E(r)E(r)]:(3)
This couples the magnetization to the spin angular mo-
mentum density of the light field. Quantization of this
expression leads to the optomagnonic coupling Hamilto-
nian. A similar Hamiltonian is obtained in atomic en-
semble systems when considering the electric dipolar in-
teraction between the light field and multilevel atoms,
where the spin degree of freedom (associated with M(r)
in our case) is represented by the atomic hyperfine struc-
ture [29]. The exact form of the optomagnonic Hamil-
tonian will depend on the magnon and optical modes.
In photonic crystals, it has been demonstrated that opti-
cal modes can be engineered by nanostructure patterning
[34], and magnonic-crystals design is a matter of intense
current research [3]. The electric field is easily quantized,
^E(+)(r;t) =P
E (r)^a (t), where E (r)indicates the
theigenmode of the electric field (eigenmodes are indi-
cated with Greek letters in what follows). The magne-
tization requires more careful consideration, since M(r)
dependsonthelocalspinoperatorwhich, ingeneral, can-
not be written as a linear combination of bosonic modes.
There are however two simple cases: (i) small deviations
of the spins, for which the Holstein-Primakoff representa-
tion is linear in the bosonic magnon operators, and (ii) a
homogeneous Kittel mode M(r) =Mwith macrospin S.
In the following we treat the homogeneous case, to cap-
ture nonlinear dynamics. From Eq. (3) we then obtain
the coupling Hamiltonian ^HMO =~P
j
^SjGj
^ay
^a
with
Gj
= i"0fMs
4~SX
mnjmn
drE
m(r)E
n(r);(4)
where we replaced Mj=Ms=^Sj=S, withSthe extensive
total spin (scaling like the mode volume). One can diago-
nalize the hermitean matrices Gj, though generically not
simultaneously. In the present work, we treat the con-
ceptually simplest case of a strictly diagonal coupling tosome optical eigenmodes ( Gj
6= 0butGj
= 0). This is
precludedonlyiftheopticalmodesarebothtime-reversal
invariant ( E real-valued) and non-degenerate. In all the
other cases, a basis can be found in which this is valid.
For example, a strong static Faraday effect will turn op-
tical circular polarization modes into eigenmodes. Al-
ternatively, degeneracy between linearly polarized modes
implies we can choose a circular basis.
Consider circular polarization (R/L) in the y z-plane,
such thatGxis diagonal while Gy=Gz= 0. Then we
find
Gx
LL= Gx
RR=G=1
ScF
4p"; (5)
where we used Eq. (2) to express the coupling via the
Faraday rotation F, and where is a dimensionless over-
lap factor that reduces to 1if we are dealing with plane
waves (see App. A). Thus, we obtain the coupling Hamil-
tonianHMO=~G^Sx(^ay
L^aL ^ay
R^aR). This reduces to
Eq. (1) if the incoming laser drives only one of the two
circular polarizations.
The coupling Ggives the magnon precession frequency
shift perphoton. It decreases for larger magnon mode
volume, in contrast to GS, which describes the overall
opticalshift for saturated spin ( Sx=S). For YIG,
with"5andF200ocm 1[5, 35], we obtain
GS1010Hz(taking= 1), which can easily become
comparable to the precession frequency
. The ultimate
limit for the magnon mode volume is set by the optical
wavelength,(1m)3, which yields S1010. There-
foreG1Hz, whereas the coupling to a single magnon
would be remarkably large: g0=Gp
S=20:1MHz.
This provides a strong incentive for designing small mag-
netic structures, by analogy to the scaling of piezoelectri-
cal resonators [36]. Conversely, for a macroscopic volume
of(1mm)3, withS1019, this reduces to G10 9Hz
andg010Hz.
III. SPIN DYNAMICS
The coupled Heisenberg equations of motion are ob-
tainedfromtheHamiltonianinEq. (1)byusing
^a;^ay
=
1,h
^Si;^Sji
=iijk^Sk. Wenextfocusontheclassicallimit,
where we replace the operators by their expectation val-
ues:
_a= i(GSx )a
2(a max)
_S= (Gaaex
ez)S+G
S(_SS):(6)
Here we introduced the laser amplitude maxand the in-
trinsic spin Gilbert-damping [37], characterized by G,
due to phonons and defects ( G10 4for YIG [38]).
After rescaling the fields (see App.. B), we see that the4
classical dynamics is controlled by only five dimension-
less parameters:GS
;G 2
max
;
;
; G. These are inde-
pendent of ~as expected for classical dynamics.
In the following we study the nonlinear classical dy-
namics of the spin, and in particular we treat cases where
the spin can take values on the whole Bloch sphere and
therefore differs significantly from a harmonic oscilla-
tor, deviating from the optomechanics paradigm valid
forS S. The optically induced tilt of the spin
can be estimated from Eq. (6) as S=S =Gjaj2=
G 2
max=
=B max=
, whereB max=G 2
maxis an op-
tically induced effective magnetic field. We would ex-
pect therefore unique optomagnonic behavior (beyond
optomechanics) for large enough light intensities, such
thatB maxis of the order of or larger than the preces-
sion frequency
. We will show however that, in the case
of blue detuning, even small light intensity can destabi-
lize the original magnetic equilibrium of the uncoupled
system, provided the intrinsic Gilbert damping is small.
A. Fast cavity regime
As a first step we study a spin which is slow compared
to the cavity, where G_Sx 2. In that case we can
abyzx-0.10-0.0500.050.10
-0.2-0.100.10.2
Figure 2. (Color online) Spin dynamics (fast cavity limit)
at blue detuning =
and fixedGS=
= 2,=
= 5,
G= 0. The left column depicts the trajectory (green full
line) of a spin (initially pointing near the north pole) on the
Bloch sphere. The color scale indicates the optical damping
opt. The right column shows a stereographic projection of
the spin’s trajectory (red full line). The black dotted line
indicates the equator (invariant under the mapping), while
the north pole is mapped to infinity. The stream lines of the
spin flow are also depicted (blue arrows). (a) Magnetization
switching behavior for light intensity G 2
max=
= 0:36. (b)
Limit cycle attractor for larger light intensity G 2
max=
=
0:64.expand the field a(t)in powers of _Sxand obtain an ef-
fective equation of motion for the spin by integrating out
the light field. We write a(t) =a0(t) +a1(t) +:::, where
the subscript indicates the order in _Sx. From the equa-
tion fora(t), we find that a0fulfills the instantaneous
equilibrium condition
a0(t) =
2 max1
2 i( GSx(t));(7)
from which we obtain the correction a1:
a1(t) = 1
2 i( GSx)@a0
@Sx_Sx:(8)
To derive the effective equation of motion for the spin,
we replacejaj2ja0j2+a
1a0+a
0a1in Eq. (6) which
leads to
_S=Be S+opt
S(_SxexS) +G
S(_SS):(9)
HereBe =
ez+Bopt, where Bopt(Sx) =Gja0j2ex
acts as an optically induced magnetic field. The second
term is reminiscent of Gilbert damping, but with spin-
velocity component only along ex. Both the induced field
Boptand dissipation coefficient optdepend explicitly on
the instantaneous value of Sx(t):
Bopt=G
[(
2)2+ ( GSx)2]
2 max2
ex(10)
opt= 2GSjBoptj( GSx)
[(
2)2+ ( GSx)2]2:(11)
This completes the microscopic derivation of the optical
Landau-Lifshitz-Gilbert equation for the spin, an impor-
tant tool to analyze effective spin dynamics in different
contexts [39]. We consider the nonlinear adiabatic dy-
namics of the spin governed by Eq. (9) below. Two
distinct solutions can be found: generation of new sta-
ble fixed points (magnetic switching) and optomagnonic
limit cycles (self oscillations).
Given our Hamiltonian (Eq. (1)), the north pole is sta-
ble in the absence of optomagnonic coupling – the se-
lection of this state is ensured by the intrinsic damping
G>0. By driving the system this can change due to
the energy pumped to (or absorbed from) the spin, and
the new equilibrium is determined by Be andopt, when
optdominates over G. Magnetic switching refers to the
rotation of the macroscopic magnetization by , to a
new fixed point near the south pole in our model. This
can be obtained for blue detuning >0, in which case
optis negative either on the whole Bloch sphere (when
> GS) or on a certain region, as shown in Fig. (2)a.
Similar results were obtained in the case of spin opto-
dynamics for cold atoms systems [30]. The possibility of
switching the magnetization direction in a controlled way
is of great interest for information processing with mag-
netic memory devices, in which magnetic domains serve5
as information bits [7–9]. Remarkably, we find that for
blue detuning, magnetic switching can be achieved for
arbitrary small light intensities in the case of G= 0.
This is due to runaway solutions near the north pole for
>0, as discussed in detail in App. C. In physical sys-
tems, the threshold of light intensity for magnetization
switching will be determined by the extrinsic dissipation
channels.
For higher intensities of the light field, limit cycle at-
tractors can be found for jj<GS, where the optically
induced dissipation optcan change sign on the Bloch
sphere (Fig. (2)b). The combination of strong nonlinear-
ity and a dissipative term which changes sign, leads the
system into self sustained oscillations. The crossover be-
tween fixed point solutions and limit cycle attractors is
determined by a balance between the detuning and the
light intensity, as discussed in App. C. Limit cycle at-
tractors require B max=
>jj=GS(note that from (11)
BoptB maxif ( GS)).
We note that for both examples shown in Fig. (2), for
the chosen parameters we have opt Gin the case of
YIG, and hence taking G= 0is a very good approx-
imation. More generally, from Eqs. (10) we estimate
optGSB opt=3and therefore we can safely neglect
Gfor( maxG)2S G3. The qualitative results (limit
cycle, switching) survive up to opt&G, although quan-
titatively modified as Gis increased: for example, the
size of the limit cycle would change, and there would be
a threshold intensity for switching.
B. Full nonlinear dynamics
The nonlinear system of Eq. (6) presents even richer
behavior when we leave the fast cavity regime. For limit
cycles near the north pole, when S S, the spin is
well approximated by a harmonic oscillator, and the dy-
namics is governed by the attractor diagram established
for optomechanics [40]. In contrast, larger limit cycles
will display novel features unique to optomagnonics, on
which we focus here.
Beyond the fast cavity limit, we can no longer give
analytical expressions for the optically induced magnetic
field and dissipation. Moreover, we can not define a coef-
ficientoptsince an expansion in _Sxis not justified. We
therefore resort to numerical analysis of the dynamics.
Fig. (3) shows a route to chaos by successive period dou-
bling, upon decreasingthe cavity decay . This route can
be followed in detail as a function of any selected param-
eter by plotting the respective bifurcation diagram. This
is depicted in Fig. (4). The plot shows the evolution
of the attractors of the system as the light intensity is
increased. The figure shows the creation and expansion
of a limit cycle from a fixed point near the south pole,
followed by successive period doubling events and finally
entering into a chaotic region. At high intensities, a limit
t⌦
GSz⌦
yzxabc
deIncreasing period of the limit cycle
Chaos2⇡ 3 0.5
13 2.52
3 2 1Figure 3. (Color Online) Full non-linear spin dynamics and
route to chaos for GS=
= 3andG 2
max=
= 1(G= 0).
The system is blue detuned by =
and the dynamics,
after a transient, takes place in the southern hemisphere. The
solid red curves represent the spin trajectory after the initial
transient, on the Bloch sphere for (a) =
= 3, (b)=
= 2,
(c)=
= 0:9, (d)=
= 0:5. (f)Szprojection as a function
of time for the chaotic case =
= 0:5.
spin projectionGSz/⌦chaoslimit cycleperioddoublingcoexistence
1.0laser amplitudepG|↵max|2/⌦210-1-21.5
Figure 4. Bifurcation density plot for GS=
= 3and=
= 1
at =
(G= 0), as a function of light intensity. We plot
theSzvalues attained at the turning points ( _Sz= 0). For
other possible choices ( eg. _Sx= 0) the overall shape of
the bifurcation diagram is changed, but the bifurcations and
chaotic regimes remain at the same light intensities. For the
plot, 30 different random initial conditions were taken.
cycle can coexist with a chaotic attractor. For even big-
ger light intensities, the chaotic attractor disappears and
thesystemprecessesaroundthe exaxis, asaconsequence
of the strong optically induced magnetic field. Similar bi-
furcation diagrams are obtained by varying either GS=
or the detuning =
(see App. D).6
11
2
xy-plane limit cycles"optomechanics"chaosoptomagnonic limit cyclesswitching
chaos yz plane limit cycles⌦GS
xy-plane limit cyclesB↵max⌦
Figure 5. Phase diagram for blue detuning with =
, as a
function of the inverse coupling strength
=GSand the op-
tically induced field B max=
=G 2
max=
. Boundaries are
qualitative. Switching, in white, refers to a fixed point solu-
tion with the spin pointing near the south pole. Limit cycles
in thexyplane are shaded in blue, and they follow the op-
tomechanical attractor diagram discussed in Ref. [40]. For
higherB max, chaos can ensue. Orange denotes the param-
eter space in which limit cycles deviate markedly from op-
tomechanical predictions. These are not in the xyplane and
also undergo period doubling leading to chaos. In red is de-
picted the area where pockets of chaos can be found. For
largeB max=
, the limit cycles are in the yzplane. In the
case of red detuning =
, the phase diagram remains as
is, except that instead of switching there is a fixed point near
the north pole.
IV. DISCUSSION
We can now construct a qualitative phase diagram for
our system. Specifically, we have explored the qualitative
behavior (fixed points, limit cycles, chaos etc.) as a func-
tion of optomagnonic coupling and light intensity. These
parameters can be conveniently rescaled to make them
dimensionless. We chose to consider the ratio of magnon
precessionfrequencytocoupling, intheform
=GS. Fur-
thermore, we express the light intensity via the maxi-
mal optically induced magnetic field B max=G 2
max.
The dimensionless coupling strength, once the material
of choice is fixed, can be tuned viaan external magnetic
fieldwhichcontrolstheprecessionfrequency
. Thelight
intensity can be controlled by the laser.
We start by considering blue detuning, this is shown
in Fig. (5). The “phase diagram” is drawn for =
,
and we set =
andG= 0. We note that some of the
transitions are rather crossovers (“optomechanical limit
cycles” vs.“optomagnonic limit cycles”). In addition, the
other “phase boundaries” are only approximate, obtained
from direct inspection of numerical simulations. These
are not intended to be exact, and are qualitatively validfor departures of the set parameters, if not too drastic
– for example, increasing will lead eventually to the
disappearance of the chaotic region.
As the diagram shows, there is a large range of pa-
rameters that lead to magnetic switching, depicted in
white. This area is approximately bounded by the con-
ditionB max=
.=GS, which in Fig. (5) corresponds
to the diagonal since we took =
. This condition
is approximate since it was derived in the fast cavity
regime, see App. C. As discussed in Sec. III, magnetic
switching should be observable in experiments even for
small light intensity in the case of blue detuning, pro-
vided that all non-optical dissipation channels are small.
The caveat of low intensity is a slow switching time. For
B max=
&=GS, the system can go into self oscilla-
tions and even chaos. For optically induced fields much
smaller than the external magnetic field, B max
we
expect trajectories of the spin in the xyplane, precessing
around the external magnetic field along ezand therefore
the spin dynamics (after a transient) is effectively two-
dimensional. This is depicted by the blue-shaded area
in Fig.(5). These limit cycles are governed by the op-
tomechanical attractor diagram presented in Ref. [40],
as we show in App. E. There is large parameter region
in which the optomagnonic limit cycles deviate from the
optomechanical attractors. This is marked by orange in
Fig.(5). As the light intensity is increased, for
=GS 1
the limit cycles remain approximately confined to the xy
plane but exhibit deviations from optomechanics. This
approximate confinement of the trajectories to the xy
plane at large B max=
(B max=
&0:5for =
)
can be understood qualitatively by looking at the ex-
pression of the induced magnetic field Boptdeduced in
the fast cavity limit, Eq. (10). Since we consider =
,
=GS 1impliesGS . In this limit, Bopt=
can
become very small and the spin precession is around the
ezaxis. For moderate B max=
and
=GS, the limit cy-
cles are tilted and precessing around an axis determined
by the effective magnetic field, a combination of the opti-
calinducedfieldandtheexternalmagneticfield. Bluede-
tuning causes these limit cycles to occur in the southern
hemisphere. Period doubling leads eventually to chaos.
The region where pockets of chaos can be found is rep-
resented by red in the phase diagram. For large light
intensity, such that B max
, the optical field domi-
nates and the effective magnetic field is essentially along
theexaxis. The limit cycle is a precession of the spin
around this axis.
According to our results optomagnonic chaos is at-
tained for values of the dimensionless coupling GS=
1 10and light intensities G 2
max=
0:1 1. This
implies a number of circulating photons similar to the
number of locked spins in the material, which scales with
the cavity volume. This therefore imposes a condition
on the minimum circulating photon density in the cavity.
For YIG with characteristic frequencies
1 10GHz,7
theconditiononthecouplingiseasilyfulfilled(remember
GS= 10GHz as calculated above). However the condi-
tion on the light intensity implies a circulating photon
density of108 109photons/m3which is outside
of the current experimental capabilities, limited by the
power a typical microcavity can support (around 105
photons/m3). On the other hand, magnetic switching
and self-sustained oscillations of the optomechanical type
(but taking place in the southern hemisphere) can be at-
tained for low powers, assuming all external dissipation
channels are kept small. While self-sustained oscillations
and switching can be reached in the fast-cavity regime,
morecomplexnonlinearbehaviorsuchasperioddoubling
and chaos requires approaching sideband resolution. For
YIG the examples in Figs. 3, 4 correspond to a preces-
sion frequency
3109Hz(App. D), whereas can be
estimated to be1010Hz, taking into account the light
absorption factor for YIG ( 0:3cm 1) [35].
For red detuning <0, the regions in the phase dia-
gram remain the same, except that instead of magnetic
switching, the solutions in this parameter range are fixed
points near the north pole. This can be seen by the sym-
metry of the problem: exchanging ! together
withex! exandez! ezleaves the problem un-
changed. The limit cycles and trajectories follow also
this symmetry, and in particular the limit cycles in the
xyplane remain invariant.
V. OUTLOOK
The observation of the spin dynamics predicted here
will be a sensitive probe of the basic cavity optomagnonic
model, beyond the linear regime. Our analysis of the op-
tomagnonic nonlinear Gilbert damping could be general-
ized to more advanced settings, leading to optomagnonic
reservoir engineering (e.g. two optical modes connected
by a magnon transition). Although the nonlinear dy-
namics presented here requires light intensities outside of
the current experimental capabilities for YIG, it should
be kept in mind that our model is the simplest case for
which highly non-linear phenomena is present. Increas-
ing the model complexity, for example by allowing for
multiple-mode coupling, could result in a decreased light
intensity requirement. Materials with a higher Faraday
constantwouldbealsobeneficial. Inthisworkwefocused
on the homogeneous Kittel mode. It will be an interest-
ing challenge to study the coupling to magnon modes at
finite wavevector, responsible for magnon-induced dissi-
pation and nonlinearities under specific conditions [41–
43]. The limit cycle oscillations can be seen as “opto-
magnonic lasing”, analogous to the functioning principle
of a laser where energy is pumped and the system set-
tles in a steady state with a characteristic frequency, and
also discussed in the context of mechanics (“cantilaser”
[44]). These oscillations could serve as a novel sourceof traveling spin waves in suitable geometries, and the
synchronization of such oscillators might be employed to
improve their frequency stability. We may see the de-
sign of optomagnonic crystals and investigation of opto-
magnonic polaritons in arrays. In addition, future cav-
ity optomagnonics experiments will allow to address the
completely novel regime of cavity-assisted coherent op-
tical manipulation of nonlinear magnetic textures, like
domain walls, vortices or skyrmions, or even nonlinear
spatiotemporal light-magnon patterns. In the quantum
regime, prime future opportunities will be the conversion
of magnons to photons or phonons, the entanglement be-
tween these subsystems, and their applications to quan-
tum communication and sensitive measurements.
We note that different aspects of optomagnonic sys-
tems have been investigated in a related work done
simultaneously [45]. Our work was supported by an
ERC-StG OPTOMECH and ITN cQOM. H.T. ac-
knowledges support by the Defense Advanced Research
ProjectsAgency(DARPA)MicrosystemsTechnologyOf-
fice/Mesodynamic Architectures program (N66001-11-1-
4114) and an Air Force Office of Scientific Research
(AFOSR) Multidisciplinary University Research Initia-
tive grant (FA9550-15-1-0029).
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Appendix A: Optomagnonic coupling Gfor plane waves
In this section we calculate explicitly the optomagnonic coupling presented in Eq.. (5) for the case of plane
waves mode functions for the electric field. We choose for definiteness the magnetization axis along the ^ zaxis, and
consider the case Gx
6= 0. The Hamiltonian HMOis then diagonal in the the basis of circularly polarized waves,
eR=L=1p
2(eyiez). The rationale behind choosing the coupling direction perpendicular to the magnetization axis,
is to maximize the coupling to the magnon mode, that is to the deviations of the magnetization with respect to the
magnetization axis. The relevant spin operator is therefore ^Sx, which represents the flipping of a spin. In the case of
plane waves, we quantize the electric field according to ^E+( )(r;t) = +( )iP
jejq
~!j
2"0"V^a(y)
j(t)e+( )ikjr;whereV
is the volume of the cavity, kjthe wave vector of mode jand we have identified the positive and negative frequency
components of the field as E!^E+,E!^E . The factor of "0"in the denominator ensures the normalization
~!j="0"hjj
d3rjE(r)j2jji "0"h0j
d3rjE(r)j2j0i, which corresponds to the energy of a photon in state jjiabove
the vacuumj0i. For two degenerate (R/L) modes at frequency !, using Eq. (2) we see that the frequency dependence
cancels out and we obtain the simple form for the optomagnonic Hamiltonian HMO=~G^Sx(^ay
L^aL ^ay
R^aR)with
G=1
ScF
4p". Therefore the overlap factor = 1in this case.9
Appendix B: Rescaled fields and linearized dynamics
To analyze Eq. (6) it is convenient to re-scale the fields such that a= maxa0,S=SS0and measure all times and
frequencies in
. We obtain the rescaled equations of motion (time-derivatives are now with respect to t0=
t)
_a0= i(GS
S0
x
)a0
2
(a0 1) (B1)
_S0=G 2
max
ja0j2ex ez
S0+G
S
_S0S0
(B2)
If we linearize the spin-dynamics (around the north-pole, e.g.), we should recover the optomechanics behavior. In
this section we ignore the intrinsic Gilbert damping term. We set approximately S0(S0
x;S0
y;1)Tand from Eq. (B1)
we obtain
_S0
x=S0
y (B3)
_S0
y= G 2
max
ja0j2 S0
x (B4)
We can now choose to rescale further, via S0
x=
max=p
S
S00
xand likewise for S0
y. We obtain the following
spin-linearized equations of motion:
_S00
x=S00
y (B5)
_S00
y= Gp
S max
ja0j2 S00
x (B6)
_a0= i(Gp
S max
S00
x
)a0
2
(a0 1) (B7)
This means that the number of dimensionless parameters has been reduced by one, since the two parameters initially
involving G, S, and maxhave all been combined into
Gp
S max
(B8)
In other words, for S0
x;y=Sx;y=S 1, the dynamics should only depend on this combination, consistent with the
optomechanicalanalogyvalidinthisregimeasdiscussedinthemaintext(wherewearguedbasedontheHamiltonian).
Appendix C: Switching in the fast cavity limit
From Eq. (9) in the weak dissipation limit ( G 1) we obtain
_Sx=
Sy
_Sy= SzBopt
Sx opt
S_SxSz;
from where we obtain an equation of motion for Sx. We are interested in studying the stability of the north pole once
the driving is turned on. Hence we set Sz=S,
Sx=
SBopt
2Sx opt
_Sx;
and we consider small deviations SxofSxfrom the equilibrium position that satisfies S0
x= SBopt=
, whereBopt
is evaluated at S0
x. To linear order we obtain
Sx=
+S@Bopt
@Sx
Sx+ 2GS
Bopt( +GSB opt=
)
h
(=
)2+ ( +GSB opt=
)2i2_Sx:
We see that the dissipation coefficient for blue detuning ( >0) is always negative, giving rise to runaway solutions.
Therefore the solutions near the north pole are always unstable under blue detuning, independent of the light intensity.10
These trajectories run to a fixed point near the south pole, which accepts stable solutions for >0(switching) or to
a limit cycle. Near the south pole, Sz= S,S0
x=SBopt=
and
Sx=
S@Bopt
@Sx
Sx 2GS
Bopt( GSB opt=
)
h
(=
)2+ ( GSB opt=
)2i2_Sx:
Therefore for > GSB opt=
there are stable fixed points, while in the opposite case there are also runaway
solutions that are caught in a limit cycle. For red detuning, ! and the roles of south and north pole are
interchanged.
Appendix D: Nonlinear dynamics
In this section we give more details on the full nonlinear dynamics described in the main text. In Figs. 3 and (4) of
the main text we chose a relative coupling GS=
= 3, around which a chaotic attractor is found. With our estimated
GS1010Hzfor YIG, this implies a precession frequency
3109Hz. In Fig. (3) the chaotic regime is reached at
=2withG 2
max=
= 1, which implies 2
maxS=3, that is, a number of photons circulating in the (unperturbed)
2.53
-0.5
-1Spin projectionSz/S
GS/⌦Normalized coupling
Figure 6. (Color online) Bifurcation density plot for G 2
max=
= 1and=
= 1at =
(G= 0), as a function of the
relative coupling strength GS=
. The dotted blue line indicates GS=
= 3, for comparison with Fig. (4). As in the main
text, the points (obtained after the transient) are given by plotting the values of Szattained whenever the trajectory fulfills
the turning point condition _Sz= 0, for 20 different random initial conditions.11
1.01.50-1-2Spin projectionGSz/⌦
/⌦Detuning
Figure 7. (Color online) Bifurcation density plot for GS=
= 3,G 2
max=
= 1and=
= 1(G= 0), as a function of the
detuning =
. The dotted blue line indicates =
= 1, for comparison with Fig. (4).
cavity of the order of the number of locked spins and hence scaling with the cavity volume. Bigger values of the cavity
decay rate are allowed for attaining chaos at the same frequency, at the expense of more photons in the cavity, as can
be deduced from Fig. (4) where we took =
. On the other hand we can think of varying the precession frequency
by an applied external magnetic field and explore the nonlinearities by tuning GS=
in this way (note that GSis
a material constant). This is done in Fig. (6). Alternatively, the nonlinear behavior can be controlled by varying the
detuning , as shown in Fig. (7).
Appendix E: Relation to the optomechanical attractors
In this appendix we show that the optomagnonic system includes the higher order nonlinear attractors found in
optomechanics as a subset in parameter space.
In optomechanics, the high order nonlinear attractors are self sustained oscillations with amplitudes Asuch that
the optomechanical frequency shift GAis a multiple of the mechanical frequency
. Translating to our case, this
meansGSn
. SinceS=SGj maxj2=
=B max=
we obtain the condition
GS
B max
n (E1)
for observing these attractors. We can vary B maxaccording to Eq. (E1). For
=GS 1we are in the limit of small
B max=
and we expect limit cycles precessing along ezas discussed in Sec. (IV). In Fig. 8 the attractor diagram12
5101520
2015105GS/⌦GSx/⌦
10302020301040GS/⌦GSx/⌦
Figure 8. Attractor diagram for = 1:5
and=
= 1with condition G2Sj maxj2=n
2. Top:n= 1, bottomn= 10. We
plot theSxvalues attained at the turning points ( _Sx= 0) forSx>0. The diagram is symmetric for Sx<0as expected for
a limit cycle on the Bloch sphere. The diagram at the left coincides to a high degree of approximation with the predictions
obtained for optomechanical systems (i.e. replacing the spin by a harmonic oscillator). In contrast, this is no longer the case
for the diagram on the right, which involves higher light intensities.13
obtained by imposing condition (E1) is plotted. Since the trajectories are in the xyplane, we plot the inflection point
of the coordinate Sx. We expect GSx=
evaluated at the inflection point, which gives the amplitude of the limit
cycle, to coincide with the optomechanic attractors for small B max=
and hence flat lines at the expected amplitudes
(as calculated in Ref. [40]) as GS=
increases. Relative evenly spaced limit cycles increasing in number as larger
values ofGS=
are considered are observed, in agreement with Ref. [40]. Remarkable, these limit cycles attractors
are found on the whole Bloch sphere, and not only near the north pole where the harmonic approximation is strictly
valid. These attractors are reached by allowing initial conditions on the whole Bloch sphere. For n= 1, (Fig. 8, top),
switching is observed up to GS=
4and then perfect optomechanic behavior. For higher values of n, deviations
from the optomechanical behavior are observed for small GS=
(implying large B max=
according to Eq. (E1)) and
large amplitude limit cycles, as compared to the size of the Bloch sphere. An example is shown in Fig. 8, bottom,
forn= 10. |
0807.2901v1.Current_induced_dynamics_of_spiral_magnet.pdf | arXiv:0807.2901v1 [cond-mat.str-el] 18 Jul 2008Current-induced dynamics of spiral magnet
Kohei Goto,1,∗Hosho Katsura,1,†and Naoto Nagaosa1,2,‡
1Department of Applied Physics, The University of Tokyo,
7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
2Cross-Correlated Materials Research Group (CMRG), ASI, RI KEN, Wako 351-0198, Japan
We study the dynamics of the spiral magnet under the charge cu rrent by solving the Landau-
Lifshitz-Gilbert equation numerically. In the steady stat e, the current /vectorjinduces (i) the parallel shift
of the spiral pattern with velocity v= (β/α)j(α,β: the Gilbert damping coefficients), (ii) the
uniform magnetization Mparallel or anti-parallel to the current depending on the ch irality of the
spiral and the ratio β/α, and (iii) the change in the wavenumber kof the spiral. These are ana-
lyzed by the continuum effective theory using the scaling arg ument, and the various nonequilibrium
phenomena such as the chaotic behavior and current-induced annealing are also discussed.
PACS numbers: 72.25.Ba, 71.70.Ej, 71.20.Be, 72.15.Gd
The current-induced dynamics of the magnetic struc-
tureisattractingintensiveinterestsfromtheviewpointof
the spintronics. A representative example is the current-
driven motion of the magnetic domain wall (DW) in fer-
romagnets [1, 2]. This phenomenon can be understood
from the conservation of the spin angular momentum,
i.e., spin torque transfer mechanism [3, 4, 5, 6, 7]. The
memorydevicesusing this current-inducedmagneticDW
motion is now seriously considered [8]. Another example
is the motion of the vortex structure on the disk of a
ferromagnet, where the circulating motion of the vortex
core is sometimes accompanied with the inversion of the
magnetization at the core perpendicular to the disk [9].
Therefore, the dynamics of the magnetic structure in-
duced by the current is an important and fundamental
issue universal in the metallic magnetic systems. On the
otherhand, thereareseveralmetallicspiralmagnetswith
the frustrated exchange interactions such as Ho metal
[10, 11], and with the Dzyaloshinskii-Moriya(DM) inter-
action suchas MnSi [12, 13, 14], (Fe,Co)Si [15], and FeGe
[16]. Thequantumdisorderingunderpressureorthenon-
trivial magnetic textures have been discussed for the lat-
ter class of materials. An important feature is that the
direction of the wavevector is one of the degrees of free-
dom in addition to the phase of the screw spins. Also the
non-collinear nature of the spin configuration make it an
interesting arena for the study of Berry phase effect [17],
whichappearsmostclearlyinthecouplingtothecurrent.
However, the studies on the current-induced dynamics of
the magnetic structures with finite wavenumber, e.g., an-
tiferromagnet and spiral magnet, are rather limited com-
pared with those on the ferromagnetic materials. One
reason is that the observation of the magnetic DW has
been difficult in the case of antiferromagnets or spiral
magnets. Recently, the direct space-time observation of
the spiral structure by Lorentz microscope becomes pos-
sible for the DM induced spiralmagnets [15, 16] since the
wavelength of the spiral is long ( ∼100nm). Therefore,
the current-induced dynamics of spiral magnets is now
an interesting problem of experimental relevance.In this paper, we study the current-induced dynamics
of the spiral magnet with the DM interaction as an ex-
plicit example. One may consider that the spiral magnet
can be regarded as the periodic array of the DW’s in fer-
romagnet,butithasmanynontrivialfeaturesunexpected
from this naive picture as shown below.
The Hamiltonian we consider is given by [18]
H=/integraldisplay
d/vector r/bracketleftBigJ
2(/vector∇/vectorS)2+γ/vectorS·(/vector∇×/vectorS)/bracketrightBig
,(1)
whereJ >0 is the exchange coupling constant and γis
the strength of the DM interaction. The ground state of
His realized when /vectorS(/vector r) is a proper screw state such that
/vectorS(/vector r) =S(/vector n1cos/vectork·/vector r+/vector n2sin/vectork·/vector r), (2)
where the wavenumber /vectork=/vector n3|γ|/J, and/vector ni(i= 1,2,3)
form the orthonormal vector sets. The ground state en-
ergy is given by −VS2γ2/2JwhereVis the volume of
the system. The sign of γis equal to that of( /vector n1×/vector n2)·/vector n3,
determining the chirality of the spiral.
The equation of motion of the spin under the current
is written as
˙/vectorS=gµB
¯h/vectorBeff×/vectorS−a3
2eS(/vectorj·/vector∇)/vectorS+a3
2eSβ/vectorS×(/vectorj·/vector∇)/vectorS+α
S/vectorS×˙/vectorS
(3)
where/vectorBeff=−δH/δ/vectorSis the effective magnetic field and
α,βare the Gilbert damping constants introduced phe-
nomenologically [19, 20].
We discretize the Hamiltonian Eq.(1) and the equation
of motion Eq.(3) by putting spins on the chain or the
square lattice with the lattice constant a, and replacing
the derivative by the difference. The length of the spin
|/vectorSi|is a constant of motion at each site i, and we can
easily derive ˙H=δH
δ/vectorS·˙/vectorS=−α|˙/vectorS|2from Eq.(3), i.e., the
energy continues to decrease as the time evolution.
We start with the one-dimensional case along x-axis
as shown in Fig.1. The discretization means replacing
∂x/vectorS(x) by (/vectorSi+1−/vectorSi−1)/2a, and∂2
x/vectorS(x) by (/vectorSi+1−2/vectorSi+2
/vectorSi−1)/a2. We note that the wavenumber which mini-
mizes Eq.(1) is k=k0= arcsin( γ/J) on the discretized
one-dimensional lattice. Numerical study of Eq.(3) have
been done with gµB/¯h= 1, 2e= 1,S= 1,a= 1
J= 2, and γ= 1.2. In this condition, the wavelength
of the spiral λ= 2π/k0≈11.6 is long compared with
the lattice constant a= 1, and we choose the time scale
∆t/(1 +α2) = 10−2. We have confirmed that the re-
sults do not depend on ∆ teven if it is reduced by the
factor 10−1or 10−2. The sample size Lis 104with the
open boundary condition. As we will show later, the
typical value of the current is j∼2γand in the real
situation with the wavelength λ[nm], the exchange cou-
pling constant J[eV] and the lattice constant a[nm], it is
j≈3.2×1015J/(λa)[A/m2]. Substituting J= 0.02,
λ= 100,a= 0.5 into above estimatation, the typi-
cal current is 1012[A/m2], and the unit of the time is
∆t=J/¯h≈30[ps].
The Gilbert damping coefficients α,βare typically
10−3∼10−1in the realistic systems. In most of the cal-
culations, however,wetake α= 5.0toaccelaratethecon-
vergence to the steady state. The obtained steady state
depends only on the ratio β/αexcept the spin configura-
tions near the boundaries as confirmed by the simlations
withα= 0.1. We employ the two types of initial condi-
tion, i.e., the ideal proper screw state with the wavenum-
berk0, and the random spin configurations. The differ-
ence of the dynamics in these two cases are limited only
in the early stage ( t <5000∆t).
Now we consider the steady state with the constant
velocity for the shift of the spiral pattern obtained after
the time of the order of 105∆t. One important issue here
is the current-dependence of the velocity, which has been
discussed intensivelyfor the DW motion in ferromagnets.
In the latter case, there appears the intrinsic pinning in
the case of β= 0 [4], while the highly nonlinear behavior
forβ/α/negationslash= 0 [20]. In the special case of β=α, the trivial
solution corresponding to the parallel shift of the ground
state configuration of Eq.(1) with the velocity v=jis
considered to be realized [5]. Figure 2 shows the results
for the velocity, the induced uniform magnetization Sx
alongx-axis, and the wavevector kof the spiral in the
steady state. The current-dependence of the velocity for
the cases of β= 0.1,0.5α,αand 2αis shown in Fig.
2(a). Figure 2(b) shows the β/α-dependence of the ve-
locity for the fixed current j= 1.2. It is seen that the
velocity is almost proportional to both the current jand
the ratio β/α. Therefore, we conclude that the velocity
v= (β/α)jwithout nonlinear behavior up to the current
j∼2γ, which is in sharp contrast to the case of the DW
motion in ferromagnets. The unit of the velocity is given
bya/∆t, which is of the order of 20[m /s] fora≈5[˚A]
and ∆t≈30[ps]. In Fig. 2(c) shown the wavevector k
of the spiral under the current j= 1.2 for various values
ofβ/α. It shows a non-monotonous behavior with the
maximum at β/α≈0.2, and is always smaller than the
FIG. 1: Spin configurations in the spiral magnet (a) in
the equilibrium state, and (b) under the current. Under
the current /vectorj, the uniform magnetization Sxalong the spi-
ral axis/current direction is induced together with the ro-
tation of the spin, i.e., the parallel shift of the spiral pat -
tern with the velocity v. Note that the magnetization
is anti-parallel/parallel to the current direction with po si-
tive/negative γforβ < α, while it is reversed for β > α,
and the wavenumber kchanges from the equilibrium value.
0 0.5 1 1.5 2 2.5 3 3.5 4
0 0.5 1 1.5 2 2.5v
jβ/α=2.0
β/α=1.0
β/α=0.5
β/α=0.1 (a)
0 0.5 1 1.5 2
0 0.5 1 1.5 2 v / j
β / α(b)
0 0.2 0.4 0.6 0.8 1
0 0.5 1 1.5 2 k
β / α(c)
-1 -0.5 0 0.5 1
0 0.5 1 1.5 2 Sx
β / α(d)
FIG. 2: For the case of γ= 1.2, the numerical result is shown.
(a) The steady state velocity vas a function of the current
jwith the fixed values of β= 0.1,0.5α,α, and 2α. (b) The
velocityvas a function of β/αfor a fixed value of the current
j= 1.2.visalmost proportional to β/α. (c)Thewavenumber
kas a function of β/α. The dotted line shows k0in the
equilibrium. (d) The uniform magnetization Sxalong the
current direction as a function of β/αfor a fixed value of
j= 1.2.
wavenumber k0in the equilibrium shown in the dotted
line. Namely, the period of the spiral is elongated by the
current. As shown in Fig. 2(d), there appears the uni-
form magnetization Sxalong the x-direction. Sxis zero
and changes the sign at β/α= 1. With the positive γ
(as in the case of Fig. 2(d)), Sxis anti-parallel to the
currentj//xwithβ < αand changes its direction for
β > α. For the negative γ, the sign of Sxis reversed. As
for the velocity /vector v, on the other hand, it is always parallel
to the current /vectorj.3
Now we present the analysis of the above results in
terms of the continuum theory and a scaling argument.
For one-dimensional case, the modified LLG Eq.(3) can
be recast in the following form:
˙/vectorS=−J/vectorS×∂2
x/vectorS−(2γSx+j)∂x/vectorS+/vectorS×(α˙/vectorS+βj∂x/vectorS).(4)
It is convenientto introduceamovingcoordinates ˆξ(x,t),
ˆη, andˆζ(x,t) (see Fig.1) [21]. They are explicitly defined
through ˆ x, ˆyand ˆzas
ˆζ(x,t) = cos( k(x−vt)+φ)ˆy+sin(k(x−vt)+φ)ˆz,
ˆξ(x,t) =−sin(k(x−vt)+φ)ˆy+cos(k(x−vt)+φ)ˆz,
and ˆη= ˆx. We restrict ourselves to the following ansatz:
/vectorS(x,t) =Sxˆη+/radicalbig
1−S2xˆζ(x,t), (5)
whereSxis assumed to be constant.
By substituting Eq.(5) into Eq.(4), we obtain vas
v=β
αj, (6)
by requiring that there is no force along ˆ η- andˆζ-
directions acting on each spin. In contrast to the DW
motioninferromagnets,thevelocity vbecomeszerowhen
β→0 even for large value of the current. The numerical
results in Fig. 2(a), (b) show good agreement with this
prediction Eq.(6).
On the other hand, the magnetization Sxalongx-axis
is given by
Sx=β/α−1
2γ−Jkj, (7)
once the wavevector kis known. Here we note that the
above solution is degenerate with respect to k, which
needs to be determined by the numerical solution. From
the dimensional analysis, the spiral wavenumber kis
given by the scaling form, k=k0g(j/(2γ),β/α) with
the dimensionless function g(x,y) and also is Sxthrough
Eq.(7).
Motivated by the analysis above, we study the γ-
dependence of the steady state properties. In Fig.3,
we show the numerical results for k/k0andSxas the
functions of j/2γin the cases of β/α= 0.1,0.5 and
2. Roughly speaking, the degeneracies of the data are
obtained approximately for each color points (the same
β/αvalue) with different γvalues. The deviation from
the scaling behavior is due to the discrete nature of the
lattice model, which is relevant to the realistic situation.
Forβ/α= 0.1(black points in Fig.3), kremainsconstant
andSxis induced almost proportional to the current up
j/2γ≈0.4, where the abrupt change of koccurs. For
β/α= 0.5 (blue points) and β/α= 2.0 (red points), the
changes of kandSxare more smooth. A remarkable
result is that the spin Sion the lattice point iis well 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 k / k 0
j / 2γβ/α=2.0, γ=1.2
β/α=2.0, γ=1.0
β/α=2.0, γ=0.8
β/α=2.0, γ=0.6
β/α=0.5, γ=1.2
β/α=0.5, γ=1.0
β/α=0.5, γ=0.8
β/α=0.5, γ=0.6
β/α=0.1, γ=1.2
β/α=0.1, γ=1.0
β/α=0.1, γ=0.8
β/α=0.1, γ=0.6 (a)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Sx
j / 2γβ/α=2.0, γ=1.2
β/α=2.0, γ=1.0
β/α=2.0, γ=0.8
β/α=2.0, γ=0.6
β/α=0.5, γ=1.2
β/α=0.5, γ=1.0
β/α=0.5, γ=0.8
β/α=0.5, γ=0.6
β/α=0.1, γ=1.2
β/α=0.1, γ=1.0
β/α=0.1, γ=0.8
β/α=0.1, γ=0.6 (b)
FIG.3: Thescalingplotfor (a) k/k0wherek0isthewavenum-
ber in the equilibrium without the current, and (b) Sxin the
steady state as the function of j/2γ. The black, blue, and
red color points correspond to β/α= 0.1, 0.5 and 2.0, respec-
tively. The curves in (b) indicate Eq.(7) calculated from th e
kvalues in (a), showing the good agreement with the data
points.
described by Eq.(5) at x=xi, and hence the relation
Eq.(7) is well satisfied as shown by the curves in Fig.
3(b), even though the scaling relation is violated to some
extent. For larger values of jbeyond the data points,
i.e.,j/2γ >0.75 forβ/α= 0.1,j/2γ >1.5 forβ/α= 0.5
andj/2γ >0.9 forβ/α= 2.0, the spin configuration is
disordered from harmonic spiral characterized by a sin-
gle wavenumber k. The spins are the chaotic funtion of
both space and time in this state analogousto the turbu-
lance. This instability is triggered by the saturated spin
Sx=±1, occuring near the edge of the sample.
Next, we turn to the simulations on the two-
dimensional square lattice in the xy-plane. In this case,
the direction of the spiral wavevector becomes another
important variable because the degeneracy of the ground
state energy occurs.
Starting with the random spin configuration, we sim-
ulate the time evolution of the system without and with
the current as shown in Fig.4. Calculation has been done
with the same parameters as in the one-dimensional case4
(a)/vectorj= 0
(b)/vectorj= (0.3,0)
(c)/vectorj= (0.3/√
2,0.3/√
2)
color box of Sz(r)
color box of |Sz(k)|2
FIG. 4: The time evolution of the zcomponent Szof the
spin from the random initial configuration of the 102×102
section in the middle of the sample is shown in the case of
(a)j= 0, (b) j= 0.3 along the x-axis, (c) j= 0.3 along
the (1,1)-direction. From the left, t= 102∆t, 1.7×103∆t,
5×103∆t. The rightmost panels show the spectral intensity
|Sz(/vectork,5×103∆t)|2from the whole sample of the size 210×210
in the momentum space /vectork= (kx,ky).
whereγ= 1.2,β= 0, and the system size is 210×210.
In the absence of the current, the relaxation of the
spins into the spiral state is very slow, and many dislo-
cations remain even after a long time. Correspondingly,
the energy does not decrease to the ground state value
but approaches to the higher value with the power-law
like long-time tail. The momentum-resolved intensity is
circularly distributed with the broad width as shown in
Fig. 4(a) corresponding to the disordered direction of
/vectork. This glassy behavior is distinct from the relaxation
dynamics of the ferromagnet where the large domain for-
mation occurs even though the DW’s remain. Now we
put the current along the ˆ x(Fig. 4(b)) and (ˆ x+ˆy) (Fig.
4(c)) directions. It is seen that the direction of /vectorkis con-
trolled by the current also with the radial distribution
in the momentum space being narrower than that in the
absence of j(Fig. 4(a)). This result suggests that the
currentjwith the density ∼1012[A/m2] of the time du-
ration∼0.1[µsec] can anneal the directional disorder of
the spiral magnet. After the alignment of /vectorkis achieved,
the simulations on the one-dimensional model described
above are relevant to the long-time behavior.To summarize, we have studied the dynamics of the
spiral magnet with DM interaction under the current j
by solving the Landau-Lifshitz-Gilbert equation numeri-
cally. In the steady state under the charge current j, the
velocityvis given by ( β/α)j(α,β: the Gilbert-damping
coefficients), the uniform magnetization is induced par-
allel or anti-parallel to the current direction, and period
of the spiral is elongated. The annealing effect especially
on the direction of the spiral wavevector /vectorkis also demon-
strated.
TheauthorsaregratefultoN.FurukawaandY.Tokura
for fruitful discussions. This work was supported in part
by Grant-in-Aids (Grant No. 15104006, No. 16076205,
and No. 17105002) and NAREGI Nanoscience Project
fromtheMinistryofEducation, Culture, Sports, Science,
andTechnology. HK wassupported bythe JapanSociety
for the Promotion of Science.
∗Electronic address: goto@appi.t.u-tokyo.ac.jp
†Electronic address: katsura@appi.t.u-tokyo.ac.jp
‡Electronic address: nagaosa@appi.t.u-tokyo.ac.jp
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1805.01815v2.Effective_damping_enhancement_in_noncollinear_spin_structures.pdf | Effective damping enhancement in noncollinear spin structures
Levente Rózsa,1,∗Julian Hagemeister,1Elena Y. Vedmedenko,1and Roland Wiesendanger1
1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany
(Dated: August 29, 2021)
Damping mechanisms in magnetic systems determine the lifetime, diffusion and transport prop-
erties of magnons, domain walls, magnetic vortices, and skyrmions. Based on the phenomenological
Landau–Lifshitz–Gilbert equation, here the effective damping parameter in noncollinear magnetic
systems is determined describing the linewidth in resonance experiments or the decay parameter
in time-resolved measurements. It is shown how the effective damping can be calculated from the
elliptic polarization of magnons, arising due to the noncollinear spin arrangement. It is concluded
that the effective damping is larger than the Gilbert damping, and it may significantly differ be-
tween excitation modes. Numerical results for the effective damping are presented for the localized
magnons in isolated skyrmions, with parameters based on the Pd/Fe/Ir(111) model-type system.
Spinwaves(SW)ormagnonsaselementaryexcitations
of magnetically ordered materials have attracted signifi-
cant research attention lately. The field of magnonics[1]
concerns the creation, propagation and dissipation of
SWs in nanostructured magnetic materials, where the
dispersion relations can be adjusted by the system ge-
ometry. A possible alternative for engineering the prop-
erties of magnons is offered by noncollinear (NC) spin
structures[2] instead of collinear ferro- (FM) or antifer-
romagnets (AFM). SWs are envisaged to act as informa-
tion carriers, where one can take advantage of their low
wavelengths compared to electromagnetic waves possess-
ing similar frequencies[3]. Increasing the lifetime and the
stability of magnons, primarily determined by the relax-
ation processes, is of crucial importance in such applica-
tions.
The Landau–Lifshitz–Gilbert (LLG) equation[4, 5] is
commonly applied for the quasiclassical description of
SWs, where relaxation is encapsulated in the dimen-
sionless Gilbert damping (GD) parameter α. The life-
time of excitations can be identified with the resonance
linewidth in frequency-domain measurements such as fer-
romagnetic resonance (FMR)[6], Brillouin light scatter-
ing (BLS)[7] or broadband microwave response[8], and
with the decay speed of the oscillations in time-resolved
(TR) experiments including magneto-optical Kerr effect
microscopy (TR-MOKE)[9] and scanning transmission x-
ray microscopy (TR-STXM)[10]. Since the linewidth is
knowntobeproportionaltothefrequencyofthemagnon,
measuring the ratio of these quantities is a widely ap-
plied method for determining the GD in FMs[3, 6]. An
advantage of AFMs in magnonics applications[11, 12] is
their significantly enhanced SW frequencies due to the
exchange interactions, typically in the THz regime, com-
pared to FMs with GHz frequency excitations. However,
it is known that the linewidth in AFM resonance is typ-
ically very wide because it scales with a larger effective
damping parameter αeffthan the GD α[13].
The tuning of the GD can be achieved in magnonic
crystals by combining materials with different values of
α. It was demonstrated in Refs. [14–16] that this leadsto a strongly frequency- and band-dependent αeff, based
on the relative weights of the magnon wave functions in
the different materials.
Magnetic vortices are two-dimensional NC spin config-
urations in easy-plane FMs with an out-of-plane magne-
tized core, constrained by nanostructuring them in dot-
orpillar-shapedmagneticsamples. Theexcitationmodes
ofvortices, particularlytheirtranslationalandgyrotropic
modes, havebeeninvestigatedusingcollective-coordinate
models[17] based on the Thiele equation[18], linearized
SW dynamics[19, 20], numerical simulations[21] and ex-
perimental techniques[22–24]. It was demonstrated theo-
retically in Ref. [21] that the rotational motion of a rigid
vortex excited by spin-polarized current displays a larger
αeffthan the GD; a similar result was obtained based on
calculating the energy dissipation[25]. However, due to
the unbounded size of vortices, the frequencies as well
as the relaxation rates sensitively depend on the sample
preparation, particularly because they are governed by
the magnetostatic dipolar interaction.
In magnetic skyrmions[26], the magnetic moment di-
rections wrap the whole unit sphere. In contrast to vor-
tices, isolated skyrmions need not be confined for stabi-
lization, and are generally less susceptible to demagneti-
zation effects[3, 27]. The SW excitations of the skyrmion
lattice phase have been investigated theoretically[28–30]
and subsequently measured in bulk systems[3, 8, 31]. It
was calculated recently[32] that the magnon resonances
measured via electron scattering in the skyrmion lattice
phase should broaden due to the NC structure. Calcula-
tions predicted the presence of different localized modes
concentrated on the skyrmion for isolated skyrmions
on a collinear background magnetization[33–35] and for
skyrmions in confined geometries[20, 36, 37]. From the
experimental side, the motion of magnetic bubbles in a
nanodisk was investigated in Ref. [38], and it was pro-
posed recently that the gyration frequencies measured in
Ir/Fe/Co/Pt multilayer films is characteristic of a dilute
array of isolated skyrmions rather than a well-ordered
skyrmion lattice[6]. However, the lifetime of magnons in
skyrmionic systems based on the LLG equation is appar-arXiv:1805.01815v2 [cond-mat.mtrl-sci] 15 Oct 20182
ently less explored.
It is known that NC spin structures may influence the
GD via emergent electromagnetic fields[29, 39, 40] or via
the modified electronic structure[41, 42]. Besides deter-
mining the SW relaxation process, the GD also plays
a crucial role in the motion of domain walls[43–45] and
skyrmions[46–48] driven by electric or thermal gradients,
both in the Thiele equation where the skyrmions are
assumed to be rigid and when internal deformations of
the structure are considered. Finally, damping and de-
formations are also closely connected to the switching
mechanisms of superparamagnetic particles[49, 50] and
vortices[51], as well as the lifetime of skyrmions[52–54].
Theαeffin FMs depends on the sample geometry due
to the shape anisotropy[13, 55, 56]. It was demonstrated
in Ref. [56] that αeffis determined by a factor describing
the ellipticity of the magnon polarization caused by the
shape anisotropy. Elliptic precession and GD were also
investigated by considering the excitations of magnetic
adatomsonanonmagneticsubstrate[57]. Thecalculation
of the eigenmodes in NC systems, e.g. in Refs. [6, 20, 35],
also enables the evaluation of the ellipticity of magnons,
but this property apparently has not been connected to
the damping so far.
Although different theoretical methods for calculating
αeffhave been applied to various systems, a general de-
scriptionapplicabletoallNCstructuresseemstobelack-
ing. Here it is demonstrated within a phenomenological
description of the linearized LLG equation how magnons
in NC spin structures relax with a higher effective damp-
ing parameter αeffthan the GD. A connection between
αeffand the ellipticity of magnon polarization forced by
the NC spin arrangement is established. The method
is illustrated by calculating the excitation frequencies
of isolated skyrmions, considering experimentally deter-
mined material parameters for the Pd/Fe/Ir(111) model
system[58]. It is demonstrated that the different local-
ized modes display different effective damping parame-
ters, with the breathing mode possessing the highest one.
The LLG equation reads
∂tS=−γ/primeS×Beff−αγ/primeS×/parenleftBig
S×Beff/parenrightBig
,(1)
withS=S(r)the unit-length vector field describing
the spin directions in the system, αthe GD and γ/prime=
1
1+α2ge
2mthe modified gyromagnetic ratio (with gbeing
theg-factor of the electrons, ethe elementary charge and
mthe electron mass). Equation (1) describes the time
evolution of the spins governed by the effective magnetic
fieldBeff=−1
MδH
δS, withHthe Hamiltonian or free
energy of the system in the continuum description and
Mthe saturation magnetization.
The spins will follow a damped precession relaxing
to a local minimum S0ofH, given by the condition
S0×Beff=0. Note that generally the Hamiltonian rep-
resents a rugged landscape with several local energy min-
ima, corresponding to e.g. FM, spin spiral and skyrmionlattice phases, or single objects such as vortices or iso-
lated skyrmions. The excitations can be determined by
switching to a local coordinate system[20, 34, 47] with
the spins along the zdirection in the local minimum,
˜S0= (0,0,1), and expanding the Hamiltonian in the
variablesβ±=˜Sx±i˜Sy, introduced analogously to spin
raising and lowering or bosonic creation and annihila-
tion operators in the quantum mechanical description of
magnons[59–61]. The lowest-order approximation is the
linearized form of the LLG Eq. (1),
∂tβ+=γ/prime
M(i−α)/bracketleftbig
(D0+Dnr)β++Daβ−/bracketrightbig
,(2)
∂tβ−=γ/prime
M(−i−α)/bracketleftbig
D†
aβ++ (D0−Dnr)β−/bracketrightbig
.(3)
For details of the derivation see the Supplemental
Material[62]. The term Dnrin Eqs. (2)-(3) is respon-
sible for the nonreciprocity of the SW spectrum[2]. It
accounts for the energy difference between magnons
propagating in opposite directions in in-plane oriented
ultrathin FM films[63, 64] with Dzyaloshinsky–Moriya
interaction[65, 66] and the splitting between clockwise
and counterclockwise modes of a single skyrmion[20].
Here we will focus on the effects of the anomalous
term[34]Da, which couples Eqs. (2)-(3) together. Equa-
tions (2)-(3) may be rewritten as eigenvalue equations by
assuming the time dependence
β±(r,t) =e−iωktβ±
k(r). (4)
Forα= 0, the spins will precess around their equilib-
riumdirection ˜S0. Iftheequationsareuncoupled, the ˜Sx
and ˜Syvariables describe circular polarization, similarly
to the Larmor precession of a single spin in an exter-
nal magnetic field. However, the spins are forced on an
elliptic path due to the presence of the anomalous terms.
The effective damping parameter of mode kis defined
as
αk,eff=/vextendsingle/vextendsingle/vextendsingle/vextendsingleImωk
Reωk/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (5)
which is the inverse of the figure of merit introduced in
Ref. [15]. Equation (5) expresses the fact that Im ωk,
the linewidth in resonance experiments or decay coeffi-
cient in time-resolved measurements, is proportional to
the excitation frequency Re ωk.
Interestingly, there is a simple analytic expression con-
nectingαk,effto the elliptic polarization of the modes at
α= 0. Forα/lessmuch1, the effective damping may be ex-
pressed as
αk,eff
α≈/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)
k(r)/vextendsingle/vextendsingle/vextendsingle2
+/vextendsingle/vextendsingle/vextendsingleβ+(0)
k(r)/vextendsingle/vextendsingle/vextendsingle2
dr/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)
k(r)/vextendsingle/vextendsingle/vextendsingle2
−/vextendsingle/vextendsingle/vextendsingleβ+(0)
k(r)/vextendsingle/vextendsingle/vextendsingle2
dr=/integraltext
a2
k(r) +b2
k(r)dr/integraltext
2ak(r)bk(r)dr,
(6)3
0.0 0.2 0.4 0.6 0.8 1.00246810
FIG. 1. Effective damping parameter αk,effas a function of
inverseaspectratio bk/akofthepolarizationellipse, assuming
constantakandbkfunctions in Eq. (6). Insets illustrate the
precession for different values of bk/ak.
where the (0)superscript denotes that the eigenvectors
β±
k(r)defined in Eq. (4) were calculated for α= 0, while
ak(r)andbk(r)denote the semimajor and semiminor
axes of the ellipse the spin variables ˜Sx(r)and ˜Sy(r)
are precessing on in mode k. Details of the derivation
are given in the Supplemental Material[62]. Note that
an analogous expression for the uniform precession mode
in FMs was derived in Ref. [56]. The main conclusion
from Eq. (6) is that αk,effwill depend on the considered
SW mode and it is always at least as high as the GD
α. Although Eq. (6) was obtained in the limit of low
α, numerical calculations indicate that the αk,eff/αratio
tends to increase for increasing values of α; see the Sup-
plementalMaterial[62]foranexample. Theenhancement
of the damping from Eq. (6) is shown in Fig. 1, with the
space-dependent ak(r)andbk(r)replaced by constants
for simplicity. It can be seen that for more distorted po-
larization ellipses the spins get closer to the equilibrium
directionafterthesamenumberofprecessions, indicating
a faster relaxation.
Since the appearance of the anomalous terms Dain
Eqs. (2)-(3) forces the spins to precess on an elliptic
path, it expresses that the system is not axially sym-
metric around the local spin directions in the equilib-
rium state denoted by S0. Such a symmetry breaking
naturally occurs in any NC spin structure, implying a
mode-dependent enhancement of the effective damping
parameter in NC systems even within the phenomeno-
logical description of the LLG equation. Note that the
NC structure also influences the electronic properties of
the system, which can lead to a modification of the GD
itself, see e.g. Ref. [42].
In order to illustrate the enhanced and mode-
dependent αk,eff, we calculate the magnons in isolated
chiralskyrmionsinatwo-dimensionalultrathinfilm. Thedensity of the Hamiltonian Hreads[67]
h=/summationdisplay
α=x,y,z/bracketleftBig
A(∇Sα)2/bracketrightBig
+K(Sz)2−MBSz
+D(Sz∂xSx−Sx∂xSz+Sz∂ySy−Sy∂ySz),(7)
withAthe exchange stiffness, Dthe Dzyaloshinsky–
Moriya interaction, Kthe anisotropy coefficient, and B
the external field.
In the following we will assume D>0andB≥0
without the loss of generality, see the Supplemental
Material[62] for discussion. Using cylindrical coordi-
nates (r,ϕ)in real space and spherical coordinates S=
(sin Θ cos Φ ,sin Θ sin Φ,cos Θ)in spin space, the equi-
librium profile of the isolated skyrmion will correspond
to the cylindrically symmetric configuration Θ0(r,ϕ) =
Θ0(r)andΦ0(r,ϕ) =ϕ, the former satisfying
A/parenleftbigg
∂2
rΘ0+1
r∂rΘ0−1
r2sin Θ 0cos Θ 0/parenrightbigg
+D1
rsin2Θ0
+Ksin Θ 0cos Θ 0−1
2MBsin Θ 0= 0 (8)
with the boundary conditions Θ0(0) =π,Θ0(∞) = 0.
The operators in Eqs. (2)-(3) take the form (cf.
Refs. [34, 35, 47] and the Supplemental Material[62])
D0=−2A/braceleftBigg
∇2+1
2/bracketleftbigg
(∂rΘ0)2−1
r2/parenleftbig
3 cos2Θ0−1/parenrightbig
(∂ϕΦ0)2/bracketrightbigg/bracerightBigg
−D/parenleftbigg
∂rΘ0+1
r3 sin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg
−K/parenleftbig
3 cos2Θ0−1/parenrightbig
+MBcos Θ 0, (9)
Dnr=/parenleftbigg
4A1
r2cos Θ 0∂ϕΦ0−2D1
rsin Θ 0/parenrightbigg
(−i∂ϕ), (10)
Da=A/bracketleftbigg
(∂rΘ0)2−1
r2sin2Θ0(∂ϕΦ0)2/bracketrightbigg
+D/parenleftbigg
∂rΘ0−1
rsin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg
+Ksin2Θ0.(11)
Equation (11) demonstrates that the anomalous terms
Daresponsible for the enhancement of the effective
damping can be attributed primarily to the NC arrange-
ment (∂rΘ0and∂ϕΦ0≡1) and secondarily to the
spins becoming canted with respect to the global out-
of-plane symmetry axis ( Θ0∈ {0,π}) of the system.
TheDnrtermintroducesanonreciprocitybetweenmodes
with positive and negative values of the azimuthal quan-
tum number (−i∂ϕ)→m, preferring clockwise rotat-
ing modes ( m < 0) over counterclockwise rotating ones
(m > 0) following the sign convention of Refs. [20, 34].
BecauseD0andDnrdepend onmbutDadoes not, it is
expected that the distortion of the SW polarization el-
lipse and consequently the effective damping will be more
enhanced for smaller values of |m|.
The different modes as a function of external field
are shown in Fig. 2(a), for the material parameters de-
scribing the Pd/Fe/Ir(111) system. The FMR mode at4
0.7 0.8 0.9 1.0 1.1 1.20255075100125150175
(a)
0.7 0.8 0.9 1.0 1.1 1.21.01.52.02.53.0
(b)
FIG. 2. Localized magnons in the isolated skyrmion, with the
interaction parameters corresponding to the Pd/Fe/Ir(111)
system[58]:A = 2.0pJ/m,D =−3.9mJ/m2,K =
−2.5MJ/m3,M= 1.1MA/m. (a) Magnon frequencies f=
ω/2πforα= 0. Illustrations display the shapes of the excita-
tion modes visualized on the triangular lattice of Fe magnetic
moments, with red and blue colors corresponding to positive
and negative out-of-plane spin components, respectively. (b)
Effective damping coefficients αm,eff, calculated from Eq. (6).
ωFMR =γ
M(MB−2K), describing a collective in-phase
precession of the magnetization of the whole sample, sep-
arates the continuum and discrete parts of the spectrum,
with the localized excitations of the isolated skyrmion
located below the FMR frequency[34, 35]. We found a
single localized mode for each m∈{0,1,−2,−3,−4,−5}
value, so in the following we will denote the excita-
tion modes with the azimuthal quantum number. The
m=−1mode corresponds to the translation of the
skyrmion on the field-polarized background, which is a
zero-frequency Goldstone mode of the system and not
shown in the figure. The m=−2mode tends to zero
aroundB= 0.65T, indicating that isolated skyrmions
become susceptible to elliptic deformations and subse-
quently cannot be stabilized at lower field values[68].
The values of αm,effcalculated from Eq. (6) for the
different modes are summarized in Fig. 2(b). It is impor-
tant to note that for a skyrmion stabilized at a selected
0 20 40 60 80 100-0.04-0.020.000.020.040.06
-0.03 0.00 0.03-0.030.000.03FIG. 3. Precession of a single spin in the skyrmion in the
Pd/Fe/Ir(111) system in the m= 0andm=−3modes at
B= 0.75T, from numerical simulations performed at α=
0.1. Inset shows the elliptic precession paths. From fitting
the oscillations with Eq. (4), we obtained |Reωm=0|/2π=
39.22GHz,|Imωm=0|= 0.0608ps−1,αm=0,eff= 0.25and
|Reωm=−3|/2π= 40.31GHz,|Imωm=−3|= 0.0276ps−1,
αm=−3,eff= 0.11.
field value, the modes display widely different αm,effval-
ues, with the breathing mode m= 0being typically
damped twice as strongly as the FMR mode. The ef-
fective damping tends to increase for lower field values,
and decrease for increasing values of |m|, the latter prop-
erty expected from the m-dependence of Eqs. (9)-(11)
as discussed above. It is worth noting that the αm,eff
parameters are not directly related to the skyrmion size.
Wealsoperformedthecalculationsfortheparametersde-
scribing Ir|Co|Pt multilayers[69], and for the significantly
largerskyrmionsinthatsystemweobtainedconsiderably
smaller excitation frequencies, but quantitatively similar
effective damping parameters; details are given in the
Supplemental Material[62].
The different effective damping parameters could pos-
sibly be determined experimentally by comparing the
linewidths of the different excitation modes at a selected
field value, or investigating the magnon decay over time.
An example for the latter case is shown in Fig. 3, dis-
playing the precession of a single spin in the skyrmion,
obtained from the numerical solution of the LLG Eq. (1)
withα= 0.1. AtB= 0.75T, the frequencies of the
m= 0breathing and m=−3triangular modes are close
to each other (cf. Fig. 2), but the former decays much
faster. Because in the breathing mode the spin is follow-
ing a significantly more distorted elliptic path (inset of
Fig. 3) than in the triangular mode, the different effective
damping is also indicated by Eq. (6).
In summary, it was demonstrated within the phe-
nomenological description of the LLG equation that the
effective damping parameter αeffdepends on the consid-
ered magnon mode. The αeffassumes larger values if5
the polarization ellipse is strongly distorted as expressed
by Eq. (6). Since NC magnetic structures provide an
anisotropic environment for the spins, leading to a dis-
tortion of the precession path, they provide a natural
choice for realizing different αeffvalues within a single
system. The results of the theory were demonstrated for
isolated skyrmions with material parameters describing
the Pd/Fe/Ir(111) system. The results presented here
may stimulate further experimental or theoretical work
on the effective damping in skyrmions, vortices, domain
walls or spin spirals.
The authors would like to thank U. Atxitia and G.
Meier for fruitful discussions. Financial support by the
Alexander von Humboldt Foundation, by the Deutsche
Forschungsgemeinschaft via SFB 668, by the European
Union via the Horizon 2020 research and innovation pro-
gram under Grant Agreement No. 665095 (MAGicSky),
and by the National Research, Development and Inno-
vation Office of Hungary under Project No. K115575 is
gratefully acknowledged.
∗rozsa.levente@physnet.uni-hamburg.de
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[72] J. Hagemeister, A. Siemens, L. Rózsa, E. Y. Vedme-
denko, and R. Wiesendanger, Phys. Rev. B 97, 174436
(2018).Supplemental Material to
Effective damping enhancement in noncollinear spin structures
Levente Rózsa,1,∗Julian Hagemeister,1Elena Y. Vedmedenko,1and Roland Wiesendanger1
1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany
(Dated: August 29, 2021)
In the Supplemental Material the derivation of the linearized equations of motion and the effective
damping parameter are discussed. Details of the numerical determination of the magnon modes in
the continuum model and in atomistic spin dynamics simulations are also given.
S.I. LINEARIZED
LANDAU–LIFSHITZ–GILBERT EQUATION
Here we will derive the linearized form of the Landau–
Lifshitz–Gilbert equation given in Eqs. (2)-(3) of the
maintextanddiscussthepropertiesofthesolutions. The
calculation is similar to the undamped case, discussed in
detail in e.g. Refs. [1–3]. Given a spin configuration sat-
isfying the equilibrium condition
S0×Beff=0, (S.1)
the local coordinate system with ˜S0= (0,0,1)may be
introduced, andtheHamiltonianbeexpandedinthevari-
ables ˜Sxand˜Sy. Thelineartermmustdisappearbecause
the expansion is carried out around an equilibrium state.
The lowest-order nontrivial term is quadratic in the vari-
ables and will be designated as the spin wave Hamilto-
nian,
HSW=/integraldisplay
hSWdr, (S.2)
hSW=1
2/bracketleftbig˜Sx˜Sy/bracketrightbig/bracketleftbiggA1A2
A†
2A3/bracketrightbigg/bracketleftbigg˜Sx
˜Sy/bracketrightbigg
=1
2/parenleftBig
˜S⊥/parenrightBigT
HSW˜S⊥. (S.3)
The operator HSWis self-adjoint for arbitrary equi-
librium states. Here we will only consider cases where
the equilibrium state is a local energy minimum, mean-
ing thatHSW≥0; the magnon spectrum will only be
well-defined in this case. Since hSWis obtained as an
expansion of a real-valued energy density around the
equilibrium state, and the spin variables are also real-
valued, fromtheconjugateofEq.(S.3)onegets A1=A∗
1,
A2=A∗
2, andA3=A∗
3.
The form of the Landau–Lifshitz–Gilbert Eq. (1) in
the main text may be rewritten in the local coordinates
by simply replacing Sby˜S0everywhere, including the
definitionoftheeffectivefield Beff. TheharmonicHamil-
tonianHSWin Eq. (S.2) leads to the linearized equation
of motion
∂t˜S⊥=γ/prime
M(−iσy−α)HSW˜S⊥,(S.4)
∗rozsa.levente@physnet.uni-hamburg.dewithσy=/bracketleftbigg
0−i
i0/bracketrightbigg
the Pauli matrix.
By replacing ˜S⊥(r,t)→˜S⊥
k(r)e−iωktas usual, for
α= 0the eigenvalue equation
ωk˜S⊥
k=γ
MσyHSW˜S⊥
k (S.5)
is obtained. If HSWhas a strictly positive spectrum,
thenH−1
2
SWexists, and σyHSWhas the same eigenvalues
asH1
2
SWσyH1
2
SW. Since the latter is a self-adjoint ma-
trix with respect to the standard scalar product on the
Hilbert space, it has a real spectrum, consequently all ωk
eigenvalues are real. Note that the zero modes of HSW,
which commonly occur in the form of Goldstone modes
due to the ground state breaking a continuous symme-
try of the Hamiltonian, have to be treated separately.
Finally, we mention that if the spin wave expansion is
performed around an equilibrium state which is not a
local energy minimum, the ωkeigenvalues may become
imaginary, meaning that the linearized Landau–Lifshitz–
Gilbert equation will describe a divergence from the un-
stable equilibrium state instead of a precession around
it.
Equations (2)-(3) in the main text may be obtained
by introducing the variables β±=˜Sx±i˜Syas described
there. The connection between HSWand the operators
D0,Dnr, andDais given by
D0=1
2(A1+A3), (S.6)
Dnr=1
2i/parenleftBig
A†
2−A2/parenrightBig
, (S.7)
Da=1
2/bracketleftBig
A1−A3+i/parenleftBig
A†
2+A2/parenrightBig/bracketrightBig
.(S.8)
An important symmetry property of Eqs. (2)-(3) in
the main text is that if (β+,β−) =/parenleftbig
β+
ke−iωkt,β−
ke−iωkt/parenrightbig
is an eigenmode of the equations, then (β+,β−) =/parenleftBig/parenleftbig
β−
k/parenrightbig∗eiω∗
kt,/parenleftbig
β+
k/parenrightbig∗eiω∗
kt/parenrightBig
is another solution. Following
Refs. [1, 3], this can be attributed to the particle-hole
symmetry of the Hamiltonian, which also holds in the
presence of the damping term. From these two solutions
mentioned above, the real-valued time evolution of the
variables ˜Sx,˜Symay be expressed as
˜Sx
k=eImωktcos (ϕ+,k−Reωkt)/vextendsingle/vextendsingleβ+
k+β−
k/vextendsingle/vextendsingle,(S.9)
˜Sy
k=eImωktsin (ϕ−,k−Reωkt)/vextendsingle/vextendsingleβ+
k−β−
k/vextendsingle/vextendsingle,(S.10)arXiv:1805.01815v2 [cond-mat.mtrl-sci] 15 Oct 20182
withϕ±,k= arg/parenleftbig
β+
k±β−
k/parenrightbig
. As mentioned above, the
Imωkterms are zero in the absence of damping close to
a local energy minimum, and Im ωk<0is implied by
the fact that the Landau–Lifshitz–Gilbert equation de-
scribes energy dissipation, which in the linearized case
corresponds to relaxation towards the local energy min-
imum. In the absence of damping, the spins will precess
on an ellipse defined by the equation
/parenleftBig
˜Sx
k/parenrightBig2
/vextendsingle/vextendsingle/vextendsingleβ+(0)
k+β−(0)
k/vextendsingle/vextendsingle/vextendsingle2
cos2(ϕ+,k−ϕ−,k)
+2˜Sx
k˜Sy
ksin (ϕ+,k−ϕ−,k)/vextendsingle/vextendsingle/vextendsingleβ+(0)
k−β−(0)
k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ+(0)
k+β−(0)
k/vextendsingle/vextendsingle/vextendsinglecos2(ϕ+,k−ϕ−,k)
+/parenleftBig
˜Sy
k/parenrightBig2
/vextendsingle/vextendsingle/vextendsingleβ+(0)
k−β−(0)
k/vextendsingle/vextendsingle/vextendsingle2
cos2(ϕ+,k−ϕ−,k)= 1,(S.11)
where the superscript (0)indicatesα= 0. The semima-
jor and semiminor axes of the ellipse akandbkmay be
expressed from Eq. (S.11) as
akbk=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ−(0)
k/vextendsingle/vextendsingle/vextendsingle2
−/vextendsingle/vextendsingle/vextendsingleβ+(0)
k/vextendsingle/vextendsingle/vextendsingle2/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (S.12)
a2
k+b2
k= 2/parenleftbigg/vextendsingle/vextendsingle/vextendsingleβ−(0)
k/vextendsingle/vextendsingle/vextendsingle2
+/vextendsingle/vextendsingle/vextendsingleβ+(0)
k/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg
.(S.13)
Note thatβ+
kandβ−
k, consequently the parameters of
the precessional ellipse akandbk, are functions of the
spatial position r.
S.II. CALCULATION OF THE EFFECTIVE
DAMPING PARAMETER FROM
PERTURBATION THEORY
Here we derive the expression for the effective damping
parameter αeffgiven in Eq. (6) of the main text. By
introducingβk=/parenleftbig
β+
k,−β−
k/parenrightbig
,
D=/bracketleftbiggD0+Dnr−Da
−D†
aD0−Dnr/bracketrightbigg
,(S.14)
and using the Pauli matrix σz=/bracketleftbigg
1 0
0−1/bracketrightbigg
, Eqs. (2)-(3)
in the main text may be rewritten as
−ωkσzβk=γ/prime
M(D+iασzD)βk(S.15)
in the frequency domain. Following standard perturba-
tion theory, we expand the eigenvalues ωkand the eigen-
vectorsβkin the parameter α/lessmuch1. For the zeroth-order
terms one gets
−ω(0)
kσzβ(0)
k=γ
MDβ(0)
k, (S.16)
0.0 0.1 0.2 0.3 0.4 0.50.00.51.01.52.02.5FIG. S1. Effective damping coefficients αm,effof the isolated
skyrmion in the Pd/Fe/Ir(111) system at B= 1T, calcu-
lated from the numerical solution of the linearized Landau–
Lifshitz–Gilbert equation (S.15), as a function of the Gilbert
damping parameter α.
with realω(0)
keigenvalues as discussed in Sec. S.I. The
first-order terms read
−ω(0)
k/angbracketleftBig
β(0)
k/vextendsingle/vextendsingle/vextendsingleσz/vextendsingle/vextendsingle/vextendsingleβ(1)
k/angbracketrightBig
−ω(1)
k/angbracketleftBig
β(0)
k/vextendsingle/vextendsingle/vextendsingleσz/vextendsingle/vextendsingle/vextendsingleβ(0)
k/angbracketrightBig
=γ
M/angbracketleftBig
β(0)
k/vextendsingle/vextendsingle/vextendsingleD/vextendsingle/vextendsingle/vextendsingleβ(1)
k/angbracketrightBig
+iαγ
M/angbracketleftBig
β(0)
k/vextendsingle/vextendsingle/vextendsingleσzD/vextendsingle/vextendsingle/vextendsingleβ(0)
k/angbracketrightBig
,
(S.17)
after taking the scalar product with β(0)
k. The first terms
on both sides cancel by letting Dact to the left, then
using Eq. (S.16) and the fact that the ω(0)
kare real. By
applying Eq. (S.16) to the remaining term on the right-
hand side one obtains
ω(1)
k=−iαω(0)
k/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)
k/vextendsingle/vextendsingle/vextendsingle2
+/vextendsingle/vextendsingle/vextendsingleβ+(0)
k/vextendsingle/vextendsingle/vextendsingle2
dr/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)
k/vextendsingle/vextendsingle/vextendsingle2
−/vextendsingle/vextendsingle/vextendsingleβ+(0)
k/vextendsingle/vextendsingle/vextendsingle2
dr,(S.18)
by writing in the definition of the scalar product. By
using the definition αk,eff=|Imωk/Reωk|≈/vextendsingle/vextendsingle/vextendsingleω(1)
k/ω(0)
k/vextendsingle/vextendsingle/vextendsingle
and substituting Eqs. (S.12)-(S.13) into Eq. (S.18), one
arrives at Eq. (6) in the main text as long as/vextendsingle/vextendsingle/vextendsingleβ−(0)
k/vextendsingle/vextendsingle/vextendsingle2
−
/vextendsingle/vextendsingle/vextendsingleβ+(0)
k/vextendsingle/vextendsingle/vextendsingle2
does not change sign under the integral.
It is worthwhile to investigate for which values of α
does first-order perturbation theory give a good estimate
forαk,effcalculated from the exact solution of the lin-
earized equations of motion, Eq. (S.15). In the materials
where the excitations of isolated skyrmions or skyrmion
lattices were investigated, significantly different values of
αhave been found. For example, intrinsic Gilbert damp-
ing parameters of α= 0.02-0.04were determined experi-
mentallyforbulkchiralmagnetsMnSiandCu 2OSeO 3[4],
α= 0.28was deduced for FeGe[5], and a total damp-
ing ofαtot= 0.105was obtained for Ir/Fe/Co/Pt mag-
netic multilayers[6], where the latter value also includes3
various effects beyond the Landau–Lifshitz–Gilbert de-
scription. Figure S1 displays the dependence of αm,eff
onαfor the eigenmodes of the isolated skyrmion in the
Pd/Fe/Ir(111) system, shown in Fig. 2 of the main text.
Most of the modes show a linear correspondence between
the two quantities with different slopes in the displayed
parameter range, in agreement with Eq. (6) in the main
text. For the breathing mode m= 0the convex shape
of the curve indicates that the effective damping param-
eter becomes relatively even larger than the perturbative
expression Eq. (6) as αis increased.
S.III. EIGENMODES OF THE ISOLATED
SKYRMION
Here we discuss the derivation of the skyrmion profile
Eq. (8) and the operators in Eqs. (9)-(11) of the main
text. The energy density Eq. (7) in polar coordinates
reads
h=A/bracketleftbigg
(∂rΘ)2+ sin2Θ (∂rΦ)2+1
r2(∂ϕΘ)2
+1
r2sin2Θ (∂ϕΦ)2/bracketrightbigg
+D/bracketleftbigg
cos (ϕ−Φ)∂rΘ
−1
rsin (ϕ−Φ)∂ϕΘ + sin Θ cos Θ sin ( ϕ−Φ)∂rΦ
+1
rsin Θ cos Θ cos ( ϕ−Φ)∂ϕΦ/bracketrightbigg
+Kcos2Θ−MBcos Θ.
(S.19)
The Landau–Lifshitz–Gilbert Eq. (1) may be rewritten
as
sin Θ∂tΘ =γ/primeBΦ+αγ/primesin ΘBΘ,(S.20)
sin Θ∂tΦ =−γ/primeBΘ+αγ/prime1
sin ΘBΦ,(S.21)
with
Bχ=−1
MδH
δχ
=−1
M/bracketleftbigg
−1
r∂r/parenleftbigg
r∂h
∂(∂rχ)/parenrightbigg
−∂ϕ∂h
∂(∂ϕχ)+∂h
∂χ/bracketrightbigg
,
(S.22)
whereχstands for ΘorΦ. Note that in this form it is
common to redefine BΦto include the 1/sin Θfactor in
Eq. (S.21)[7]. The first variations of Hfrom Eq. (S.19)may be expressed as
δH
δΘ=−2A/braceleftbigg
∇2Θ−sin Θ cos Θ/bracketleftbigg
(∂rΦ)2+1
r2(∂ϕΦ)2/bracketrightbigg/bracerightbigg
−2Ksin Θ cos Θ +MBsin Θ
−2Dsin2Θ/bracketleftbigg
sin (ϕ−Φ)∂rΦ + cos (ϕ−Φ)1
r∂ϕΦ/bracketrightbigg
,
(S.23)
δH
δΦ=−2A/braceleftbigg
sin2Θ∇2Φ + sin 2Θ/bracketleftbigg
∂rΘ∂rΦ +1
r2∂ϕΘ∂ϕΦ/bracketrightbigg/bracerightbigg
+ 2Dsin2Θ/bracketleftbigg
sin (ϕ−Φ)∂rΘ + cos (ϕ−Φ)1
r∂ϕΘ/bracketrightbigg
,
(S.24)
TheequilibriumconditionEq.(8)inthemaintextmay
be obtained by setting ∂tΘ =∂tΦ = 0in Eqs. (S.20)-
(S.21) and assuming cylindrical symmetry, Θ0(r,ϕ) =
Θ0(r)and Φ0(r,ϕ) =ϕ. In the main text D>0
andB≥0were assumed. Choosing D<0switches
the helicity of the structure to Φ0=ϕ+π, in which
caseDshould be replaced by |D|in Eq. (8). For the
background magnetization pointing in the opposite di-
rectionB≤0, one obtains the time-reversed solutions
with Θ0→π−Θ0,Φ0→Φ0+π,B→−B. Time rever-
sal also reverses clockwise and counterclockwise rotating
eigenmodes; however, the above transformations do not
influence the magnitudes of the excitation frequencies.
Finally, we note that the frequencies remain unchanged
even if the form of the Dzyaloshinsky–Moriya interaction
in Eq. (S.19), describing Néel-type skyrmions common in
ultrathin films and multilayers, is replaced by an expres-
sion that prefers Bloch-type skyrmions occurring in bulk
helimagnets – see Ref. [3] for details.
Fordeterminingthelinearizedequationsofmotion,one
can proceed by switching to the local coordinate system
as discussed in Sec. S.I and Refs. [1, 3]. Alternatively,
they can also directly be derived from Eqs. (S.20)-(S.21)
by introducing Θ = Θ 0+˜Sx,Φ = Φ 0+1
sin Θ 0˜Syand
expanding around the skyrmion profile from Eq. (8) up
to first order in ˜Sx,˜Sy– see also Ref. [2]. The operators
in Eq. (S.3) read
A1=−2A/parenleftbigg
∇2−1
r2cos 2Θ 0(∂ϕΦ0)2/parenrightbigg
−2D1
rsin 2Θ 0∂ϕΦ0−2Kcos 2Θ 0+MBcos Θ 0,
(S.25)
A2=4A1
r2cos Θ 0∂ϕΦ0∂ϕ−2D1
rsin Θ 0∂ϕ,(S.26)
A3=−2A/braceleftbigg
∇2+/bracketleftbigg
(∂rΘ0)2−1
r2cos2Θ0(∂ϕΦ0)2/bracketrightbigg/bracerightbigg
−2D/parenleftbigg
∂rΘ0+1
rsin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg
−2Kcos2Θ0+MBcos Θ 0, (S.27)4
which leads directly to Eqs. (9)-(11) in the main text via
Eqs. (S.6)-(S.8).
The excitation frequencies of the ferromagnetic state
may be determined by setting Θ0≡0in Eqs. (9)-(11) in
the main text. In this case, the eigenvalues and eigenvec-
tors can be calculated analytically[1],
ωk,m=γ/prime
M(1−iα)/bracketleftbig
2Ak2−2K+MB/bracketrightbig
,(S.28)
/parenleftBig
β+
k,m(r),β−
k,m(r)/parenrightBig
= (0,Jm−1(kr)),(S.29)
withJm−1theBesselfunction ofthefirstkind, appearing
due to the solutions being regular at the origin. Equa-
tion (S.28) demonstrates that the lowest-frequency exci-
tation of the background is the ferromagnetic resonance
frequencyωFMR =γ
M(MB−2K)atα= 0. Since the
anomalous term Dadisappears in the out-of-plane mag-
netized ferromagnetic state, all spin waves will be circu-
larly polarized, see Eq. (S.29), and the effective damping
parameterwillalwayscoincidewiththeGilbertdamping.
Regarding the excitations of the isolated skyrmion, for
α= 0the linearized equations of motion in Eq. (S.15)
are real-valued; consequently, β±
k,m(r)can be chosen to
be real-valued. In this case Eqs. (S.9)-(S.10) take the
form
˜Sx
k,m= cos (mϕ−ωk,mt)/parenleftBig
β+
k,m(r) +β−
k,m(r)/parenrightBig
,(S.30)
˜Sy
k,m= sin (mϕ−ωk,mt)/parenleftBig
β+
k,m(r)−β−
k,m(r)/parenrightBig
.(S.31)
This means that modes with ωk,m>0form> 0will
rotate counterclockwise, that is, the contours with con-
stant ˜Sx
k,mand ˜Sy
k,mwill move towards higher values of
ϕastis increased, while the modes with ωk,m>0for
m < 0will rotate clockwise. Modes with m= 0corre-
spond to breathing excitations. This sign convention for
mwas used when designating the localized modes of the
isolated skyrmion in the main text, and the kindex was
dropped since only a single mode could be observed be-
low the ferromagnetic resonance frequency for each value
ofm.
S.IV. NUMERICAL SOLUTION OF THE
EIGENVALUE EQUATIONS
The linearized Landau–Lifshitz–Gilbert equation for
the isolated skyrmion, Eqs. (2)-(3) with the operators
Eqs.(9)-(11)inthemaintext, weresolvednumericallyby
a finite-difference method. First the equilibrium profile
was determined from Eq. (8) using the shooting method
for an initial approximation, then obtaining the solution
on a finer grid via finite differences. For the calculationswe used dimensionless parameters (cf. Ref. [8]),
Adl= 1, (S.32)
Ddl= 1, (S.33)
Kdl=KA
D2, (S.34)
(MB)dl=MBA
D2, (S.35)
rdl=|D|
Ar, (S.36)
ωdl=MA
γD2ω. (S.37)
The equations were solved in a finite interval for
rdl∈[0,R], with the boundary conditions Θ0(0) =
π,Θ0(R) = 0. For the results presented in Fig. 2 in the
main text the value of R= 30was used. It was confirmed
bymodifying Rthattheskyrmionshapeandthefrequen-
cies of the localized modes were not significantly affected
by the boundary conditions. However, the frequencies of
the modes above the ferromagnetic resonance frequency
ωFMR =γ
M(MB−2K)did change as a function of
R, since these modes are extended over the ferromag-
netic background – see Eqs. (S.28)-(S.29). Furthermore,
in the infinitely extended system the equations of mo-
tion include a Goldstone mode with/parenleftbig
β+
m=−1,β−
m=−1/parenrightbig
=/parenleftbig
−1
rsin Θ 0−∂rΘ0,1
rsin Θ 0−∂rΘ0/parenrightbig
, corresponding to
the translation of the skyrmion on the collinear
background[1]. This mode obtains a finite frequency in
the numerical calculations due to the finite value of R
and describes a slow clockwise gyration of the skyrmion.
However, this frequency is not shown in Fig. 3 of the
main text because it is only created by boundary effects.
In order to investigate the dependence of the effective
damping on the dimensionless parameters, we also per-
formed the calculations for the parameters describing the
Ir|Co|Pt multilayer system[9]. The results are summa-
rized in Fig. S2. The Ir|Co|Pt system has a larger di-
mensionless anisotropy value ( −KIr|Co|Pt
dl = 0.40) than
the Pd/Fe/Ir(111) system ( −KPd/Fe/Ir(111)
dl = 0.33). Al-
though the same localized modes are found in both cases,
the frequencies belonging to the m= 0,1,−3,−4,−5
modes in Fig. S2 are relatively smaller than in Fig. 2
compared to the ferromagnetic resonance frequency at
the elliptic instability field where ωm=−2= 0. This
agrees with the two limiting cases discussed in the lit-
erature: it was shown in Ref. [1] that for Kdl= 0the
m= 1,−4,−5modes are still above the ferromagnetic
resonance frequency at the elliptic instability field, while
in Ref. [2] it was investigated that all modes become soft
withfrequenciesgoingtozeroat (MB)dl= 0inthepoint
−Kdl=π2
16≈0.62,belowwhichaspinspiralgroundstate
is formed in the system. Figure S2(b) demonstrates that
the effective damping parameters αm,effare higher at the
ellipticinstabilityfieldinIr|Co|PtthaninPd/Fe/Ir(111),
showing an opposite trend compared to the frequencies.
Regarding the physical units, the stronger exchange
stiffness combined with the weaker Dzyaloshinsky–5
0.03 0.04 0.05 0.06 0.07 0.080246810
(a)
0.03 0.04 0.05 0.06 0.07 0.081.01.52.02.53.03.5
(b)
FIG. S2. Localized magnons in the isolated skyrmion, with
the interaction parameters corresponding to the Ir|Co|Pt
multilayer system from Ref. [9]: A= 10.0pJ/m,D=
1.9mJ/m2,K=−0.143MJ/m3,M = 0.96MA/m. The
anisotropy reflects an effective value including the dipolar in-
teractions as a demagnetizing term, −K =−K 0−1
2µ0M2
withK0=−0.717MJ/m3. (a) Magnon frequencies f=ω/2π
forα= 0. Illustrations display the shapes of the excitation
modes visualized as the contour plot of the out-of-plane spin
componentsona 1×1nm2grid,withredandbluecolorscorre-
sponding to positive and negative Szvalues, respectively. (b)
Effective damping coefficients αm,eff, calculated from Eq. (6)
in the main text.Moriya interaction and anisotropy in the multilayer sys-
tem leads to larger skyrmions stabilized at lower field val-
ues and displaying lower excitation frequencies. We note
that demagnetization effects were only considered here
as a shape anisotropy term included in K; it is expected
that this should be a relatively good approximation for
the Pd/Fe/Ir(111) system with only a monolayer of mag-
netic material, but it was suggested recently[6] that the
dipolar interaction can significantly influence the excita-
tion frequencies of isolated skyrmions in magnetic multi-
layers.
S.V. SPIN DYNAMICS SIMULATIONS
For the spin dynamics simulations displayed in Fig. 3
in the main text we used an atomistic model Hamiltonian
on a single-layer triangular lattice,
H=−1
2/summationdisplay
/angbracketlefti,j/angbracketrightJSiSj−1
2/summationdisplay
/angbracketlefti,j/angbracketrightDij(Si×Sj)−/summationdisplay
iK(Sz
i)2
−/summationdisplay
iµBSz
i, (S.38)
with the parameters J= 5.72meV for the Heisenberg
exchange,D=|Dij|= 1.52meV for the Dzyaloshinsky–
Moriya interaction, K= 0.4meV for the anisotropy,
µ= 3µBfor the magnetic moment, and a= 0.271nm
for the lattice constant. For the transformation be-
tween the lattice and continuum parameters in the
Pd/Fe/Ir(111) system see, e.g., Ref. [10]. The simula-
tionswereperformedbynumericallysolvingtheLandau–
Lifshitz–Gilbert equation on an 128×128lattice with
periodic boundary conditions, which was considerably
larger than the equilibrium skyrmion size to minimize
boundary effects. The initial configuration was deter-
mined by calculating the eigenvectors in the continuum
model and discretizing it on the lattice, as shown in the
insets of Fig. 2 in the main text. It was found that such
a configuration was very close to the corresponding exci-
tation mode of the lattice Hamiltonian Eq. (S.38), simi-
larly to the agreement between the continuum and lattice
equilibrium skyrmion profiles[10].
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2307.02352v2.Optimal_damping_of_vibrating_systems__dependence_on_initial_conditions.pdf | Optimal damping of vibrating systems: dependence on initial conditions
K. Lelasa,∗, I. Nakićb
aFaculty of Textile Technology, University of Zagreb, Croatia
bDepartment of Mathematics, Faculty of Science, University of Zagreb, Croatia
Abstract
Common criteria used for measuring performance of vibrating systems have one thing in common: they
do not depend on initial conditions of the system. In some cases it is assumed that the system has zero
initial conditions, or some kind of averaging is used to get rid of initial conditions. The aim of this paper
is to initiate rigorous study of the dependence of vibrating systems on initial conditions in the setting of
optimal damping problems. We show that, based on the type of initial conditions, especially on the ratio of
potential and kinetic energy of the initial conditions, the vibrating system will have quite different behavior
and correspondingly the optimal damping coefficients will be quite different. More precisely, for single degree
of freedom systems and the initial conditions with mostly potential energy, the optimal damping coefficient
willbeintheunder-dampedregime, whileinthecaseofthepredominantkineticenergytheoptimaldamping
coefficient will be in the over-damped regime. In fact, in the case of pure kinetic initial energy, the optimal
damping coefficient is +∞! Qualitatively, we found the same behavior in multi degree of freedom systems
with mass proportional damping. We also introduce a new method for determining the optimal damping of
vibrating systems, which takes into account the peculiarities of initial conditions and the fact that, although
in theory these systems asymptotically approach equilibrium and never reach it exactly, in nature and in
experiments they effectively reach equilibrium in some finite time.
Keywords: viscous damping, optimal damping, multi-degree of freedom, initial conditions
1. Introduction
If we have an multi-degree of freedom (MDOF) linear vibrating system, i.e. a system of coupled damped
oscillators, how to determine damping coefficients that ensure optimal evanescence of free vibrations? In
the literature one finds several different criteria, typically based on frequency domain analysis of the system,
although there are also approaches based on time domain analysis [1]. The tools used for designing the
criteria include modal analysis [2], transfer functions [3], H2andH∞norms coming from systems theory
[4, 5] and spectral techniques [6]. A general overview of the optimization tools for structures analysis can
be found in e.g. [7]. Another optimization criterion used is to take as optimal the damping coefficients that
minimize the (zero to infinity) time integral of the energy of the system, averaged over all possible initial
conditions corresponding to the same initial energy [8]. This criterion was investigated widely, mostly by
mathematicians in the last two decades, more details can be found, e.g., in references [8, 9, 10, 11, 12].
However, what is common to all these criteria is that they implicitly or explicitly ignore the dependence
of the dynamics of the system on the initial conditions. Sometimes this is suitable, e.g. for systems with
continuous excitation, but in some cases it make sense to study the free vibrations of the system with non-
zero initial conditions. A prominent example where this is the case is the vibration control of buildings
subjected to earthquake excitation [13, 14]. Indeed, depending on the initial conditions, MDOF systems can
∗Corresponding author
Email addresses: klelas@ttf.unizg.hr (K. Lelas), nakic@math.hr (I. Nakić)
Preprint submitted to Journal of Sound and Vibration January 31, 2024arXiv:2307.02352v2 [physics.class-ph] 30 Jan 2024exhibit oscillatory or non-oscillatory response [15], so it is clear that initial conditions can play an important
role in the overall dynamics of the system.
Implicitly, dependence of the behavior of system on initial conditions has been investigated in the context
of time-optimal vibrations reduction [16] and transient response [17] in terms of computationally efficient
methods for the calculation of the system response. Our aim with this paper is to start more systematic
investigation of the role of initial conditions in the study of linear vibrating systems. Specifically, the
dependence of the energy integral on the initial conditions has not been investigated, as far as we are aware,
and therefore it is not clear how much information about the behavior of vibrating systems is lost by taking
the average over all initial conditions or by assuming zero initial conditions and it is not clear how well the
optimal damping obtained in this way works for a specific choice of initial conditions, e.g. for an experiment
with initial conditions such that the initial energy consists only of potential energy, etc. We have chosen to
study the particular criterion of minimizing time integral of the energy as in this case it is straightforward to
modify it to take into account the initial conditions: instead of averaging over all possible initial conditions,
we study the dependence of the time integral of the energy of the system on initial conditions. Specifically,
for criteria based on frequency domain approach, which are designed for forced vibrations, it is not clear
how to take into account the non-zero initial conditions in a systematic way.
We will explore this dependence by considering free vibrations of single degree of freedom (SDOF),
two-degree of freedom (2-DOF) and MDOF vibrating systems with mass proportional damping (MPD). In
particular, for a SDOF, averaging over all initial conditions gives the critical damping as optimal [8, 10], and
we show, by considering the minimization of the energy integral without averaging over initial conditions,
that damping coefficients approximately 30%less than critical to infinite are obtained as optimal, depending
on the initial conditions. We systematize all our results with respect to the relationship between initial
potential and initial kinetic energy, e.g., for initial conditions with initial potential energy grater than initial
kinetic energy the optimal damping coefficient is in the under-damped regime, while for initial conditions
with initial kinetic energy grater than initial potential energy we find the optimal damping deep in the
over-damped regime. We also consider the minimization of the energy integral averaged over a subset of
initial conditions and obtain a significant dependence of the optimal damping coefficient on the selected
subset. Qualitatively, we find the same behavior in 2-DOF and MDOF systems as well.
Furthermore, we show that the minimization of the energy integral for certain types of initial conditions
does not give a satisfactory optimal damping coefficient. Specifically, for SDOF systems, the obtained
optimal damping coefficient does not distinguish between two initial states with the same magnitude of
initial displacement and initial velocity, but which differ in the relative sign of initial displacement and
initial velocity. These initial conditions differ significantly in the rate of energy dissipation as a function of
the damping coefficient, i.e. it is not realistic for one damping coefficient to be optimal for both of these
initial conditions. The same is true for each individual mode of MDOF systems with respect to the signs of
initial displacements and velocities, expressed via modal coordinates. Another disadvantage of this criterion
isthat, forinitialconditionswithpurelykineticinitialenergy, itgivesaninfiniteoptimaldampingcoefficient,
which is not practical for experiments. Also, the energy integral is calculated over the entire time, due to
the fact that these systems asymptotically approach equilibrium and never reach it exactly, but in nature
and experiments they effectively reach equilibrium in some finite time.
We introduce a new method for determining the optimal damping of MDOF systems, which practically
solves the aforementioned problems and gives optimal damping coefficients that take into account the pecu-
liarities of each initial condition and the fact that these systems effectively reach equilibrium in some finite
time. We take that the system has effectively reached equilibrium when its energy drops to some small
fraction of the initial energy, e.g., to the energy resolution of the measuring device with which we observe
the system. Our method is based on the determination of the damping coefficient for which the energy of
the system drops to that desired energy level the fastest.
In this paper we focus on mass proportional damping so that we could analytically perform a modal
analysis and present ideas in the simplest possible way, but, as we briefly comment at the end of the paper,
everything we have done can be done in a similar fashion analytically for the case of Rayleigh damping [18]
as well as for tuned mass damper [19, 20]. Also, it is possible to carry out this kind of analysis numerically
2for systems with damping that does not allow analytical treatment. This will be the subject of our further
research.
The rest of the paper is organized as follows: Section 2 is devoted to SDOF systems, in particular
minimization of the energy integral and optimal damping is studied for the chosen set of initial conditions.
In Section 3 we analyze 2-DOF systems with MPD. MDOF systems with MPD are the subject of Section
4. In Section 5 we propose a new optimization criterion and analyze its properties. Section 6 summarizes
important findings of the paper.
2. SDOF systems
Free vibrations of a SDOF linear vibrating system can be described by the equation
¨x(t) + 2γ˙x(t) +ω2
0x(t) = 0 , x(0) = x0,˙x(0) = v0, (1)
where x(t)denotes the displacement from the equilibrium position (set to x= 0) as a function of time,
the dots denote time derivatives, γ > 0is the damping coefficient, ω0stands for the undamped oscillator
angular frequency (sometimes called the natural frequency of the oscillator) and (x0, v0)encode the initial
conditions [21, 22]. The physical units of the displacement x(t)depend on the system being considered. For
example, for a mass on a spring (or a pendulum) in viscous fluid, when it is usually called elongation , it is
measured in [m], while for an RLC circuit it could either be voltage, or current, or charge. In contrast, the
units of γandω0are[s−1]for all systems described with the SDOF model. The form of the solution to
the equation (1) depends on the relationship between γandω0, producing three possible regimes [21, 22]:
under-damped ( γ < ω 0), critically damped ( γ=ω0) and over-damped ( γ > ω 0) regime.
Here we would like to point out that, although it is natural to classify the solution of SDOF into three
regimes depending on the value of γ, we can actually take one form of the solution as a unique solution valid
for all values of γ >0,γ̸=ω0,
x(t) =e−γt
x0cos(ωt) +v0+γx0
ωsin(ωt)
, (2)
where ω=p
ω2
0−γ2is the (complex valued) damped angular frequency. In order to describe the critically
damped regime, one can take the limit γ→ω0of the solution (2) to obtain the general solution of the
critically damped regime
xc(t) =e−ω0t(x0+ (v0+ω0x0)t). (3)
Therefore, in order to calculate the energy and the time integral of the energy, we do not need to perform
separate calculations for all three regimes, but a single calculation using the displacement given by (2) and
the velocity given by
˙x(t) =e−γt
v0cos(ωt)−γv0+ω2
0x0
ωsin(ωt)
. (4)
For simplicity, in this section we will refer to the quantity
E(t) = ˙x(t)2+ω2
0x(t)2(5)
as the energyof the system, and to the quantities EK(t) = ˙x(t)2andEP(t) =ω2
0x(t)2as the kinetic energy
andpotential energy of the system respectively. The connection of the quantity (5) to the usual expressions
for the energy is straightforward, e.g., for a mass mon a spring in viscous fluid
E(t) =m
2E(t), (6)
and similarly for other systems described with the SDOF model. Using (2) and (4) in (5), we obtain
E(t) =e−2γt
E0cos2(ωt) +γ
ω2
0x2
0−v2
0sin(2ωt)
ω+
E0(ω2
0+γ2) + 4ω2
0γx0v0sin2(ωt)
ω2
(7)
3for the energy of the system, where E0=v2
0+ω2
0x2
0is the initial energy given to the system at t= 0.
Accordingly, E0K=v2
0istheinitialkineticenergyand E0P=ω2
0x2
0istheinitialpotentialenergy. Expression
(7) is valid for both under-damped and over-damped regimes, and to obtain the energy of the critically
damped regime we take the γ→ω0limit of the energy (7), and obtain
Ec(t) =e−2ω0t
E0+ 2ω0
ω2
0x2
0−v2
0
t+ 2ω2
0(E0+ 2ω0x0v0)t2
. (8)
2.1. Minimization of the energy integral and optimal damping in dependence of initial conditions
We consider the SDOF system with initially energy E0. All possible initial conditions that give this
energy can be expressed in polar coordinates with constant radius r=√E0and angle θ= arctan
v0
ω0x0
,
i.e. we have
ω0x0=rcosθ
v0=rsinθ .(9)
In Fig. 1 we sketch the circle given by (9), i.e. given by all possible initial conditions with the same energy
E0. For clarity of the exposition, here we comment on a few characteristic points of the circle presented in
Fig. 1:
•Initial conditions ω0x0=±√E0andv0= 0, i.e. with purely potential initial energy (and zero initial
kinetic energy), correspond to two points on the circle with θ={0, π}.
•Initial conditions ω0x0=±p
E0/2andv0=±p
E0/2, i.e. with initial potential energy equal to initial
kinetic energy, correspond to four points on the circle with θ={π/4,3π/4,5π/4,7π/4}.
•Initial conditions ω0x0= 0andv0=±√E0, i.e. with purely kinetic initial energy (and zero initial
potential energy), correspond to two points on the circle with θ={π/2,3π/2}.
θ r=√E0(ω0x0, v0)
ω0x0v0
Figure 1: Sketch of all possible initial conditions with the same initial energy E0in the (ω0x0, v0)coordinate system. Square
of the coordinates corresponds to initial potential energy E0P=ω2
0x2
0and initial kinetic energy E0K=v2
0respectively. This
representation gives us a useful visualization, e.g.: all initial conditions with E0P> E 0Kare represented by two arcs, i.e.
points with θ∈(−π/4, π/4)∪(3π/4,5π/4)(blue dotted arcs); initial conditions with E0K=E0andE0P= 0are represented
by two points on a circle with θ={π/2,3π/2}(two red filled circles); etc.
Using (9) in (7) and (8), we obtain the energy of the under-damped and over-damped regime
E(t, θ) =E0e−2γt
cos2(ωt) +γcos 2θsin(2ωt)
ω+
ω2
0+γ2+ 2ω0γsin 2θsin2(ωt)
ω2
,(10)
and the energy of the critically damped regime
Ec(t, θ) =E0e−2ω0t
1 + 2 ω0(cos 2 θ)t+ 2ω2
0(1 + sin 2 θ)t2
, (11)
40.1 0.71 1 2 3012345
γ/ω 0I(γ, θ)ω0/E0θ= 0
θ=π/4
θ=π/2
Figure 2: Integral (13) for three initial conditions θ={0, π/4, π/2}.
as functions of θ, instead of x0andv0. Now we integrate energy (10) over all time, i.e.
I(γ, θ) =Z∞
0E(t)dt , (12)
and obtain
I(γ, θ) =E0
2ω0ω2
0+γ2
γω0+γ
ω0cos 2θ+ sin 2 θ
. (13)
Integral (13) is valid for all three regimes, i.e. for any γ >0.
We note here that the energy (see (7) and (8)) is invariant to a simultaneous change of the signs of the
initial conditions, i.e. to the change (x0, v0)→(−x0,−v0)(but not to x0→ −x0orv0→ −v0separately).
This change of signs corresponds to the change in angle θ→θ+π, therefore, functions (10), (11) and (13)
are all periodic in θwith period π.
In Fig. 2 we show the integral (13) for γ∈[0.1ω0,3ω0]for three different initial conditions, i.e. for
θ={0, π/4, π/2}. We can see that I(γ, θ= 0)(red solid curve), with purely potential initial energy and
zero initial kinetic energy, attains minimum for γ= 0.707ω0(rounded to three decimal places), i.e. well in
the under-damped regime. For the initial condition with equal potential and kinetic energy, I(γ, θ=π/4)
(black dotted curve) attains minimum for γ=ω0, i.e. at the critical damping condition. Interestingly,
for the initial condition with purely kinetic energy and zero potential energy, I(γ, θ=π/2)(blue dashed
curve) has no minimum in the displayed range of damping coefficients, therefore here we explicitly show this
function
I(γ, θ=π/2) =E0
2γ, (14)
and it is clear that (14) has no minimum. This is easy to understand from a physical point of view, i.e. if
all the initial energy is kinetic, the higher the damping coefficient, the faster the energy dissipation will be.
If we consider the optimal damping as the one for which the integral (13) is minimal, we can easily
determine the optimal damping coefficient γopt(θ)from the condition
∂I(γ, θ)
∂γ
γopt= 0, (15)
and we obtain
γopt(θ) =r
1
2 cos2θω0. (16)
In Fig. 3 we show the optimal damping coefficient (16) for θ∈[0,2π](function (16) has a period π, but
here we choose this interval for completeness), and here we comment on the shown results with respect to
the relationship between initial potential energy ( E0P=ω2
0x2
0) and initial kinetic energy ( E0K=v2
0) for any
given initial condition, i.e. for any θ:
50 0.25 0.50.75 1 1.25 1.51.75 201234
γopt=ω0
θ/πγopt(θ)/ω0
Figure 3: Optimal damping coefficient (16) (solid red curve) as a function of all possible initial conditions, i.e. for θ∈[0,2π].
Below the dashed horizontal line, optimal damping coefficients are in the under-damped regime, above the line in the over-
damped regime, and in the critically damped regime at the crossing points of the line and the solid red curve.
•Initial conditions with E0P> E 0Kcorrespond to the set θ∈(−π/4, π/4)∪(3π/4,5π/4). For these
initial conditions, optimal damping coefficients (16) are in the under-damped regime, i.e. γopt∈√
2ω0/2, ω0
, with the minimum value γopt=√
2ω0/2 = 0 .707ω0(rounded to three decimal places)
obtained for θ={0, π}, i.e. for two initial conditions with E0=E0PandE0K= 0.
•Initial conditions with E0P=E0Kcorrespond to four points θ={π/4,3π/4,5π/4,7π/4}with optimal
damping coefficient (16) equal to critical damping, i.e. γopt=ω0.
•Initial conditions with E0K> E 0Pcorrespond to the set θ∈(π/4,3π/4)∪(5π/4,7π/4). For these
initial conditions, optimal damping coefficients (16) are in the over-damped regime, i.e. γopt∈(ω0,∞),
where γoptdiverges for θ={π/2,3π/2}, i.e. for two initial conditions with E0K=E0andE0P= 0.
Before closing this subsection, we would like to point out two more ways in which we can write relation
(16) that will prove useful when dealing with MDOF systems. The ratio of the initial potential energy to
the initial total energy is
β=E0P
E0= cos2θ , (17)
where we used first of the relations (9) and E0P=ω2
0x2
0. Using (17), optimal damping coefficient (16) can
be written as a function of the fraction of potential energy in the initial total energy, i.e.
γopt(β) =r1
2βω0. (18)
Thus, from (18) one can simply see that γoptis in the under-damped regime for β∈(1/2,1], in the critically
damped regime for β= 1/2and in the over-damped regime for β∈[0,1/2). Using β=ω2
0x2
0/E0in (18) we
can express the optimal damping coefficient in yet another way, as a function of the initial displacement x0,
i.e.
γopt(x0) =s
E0
2x2
0=s
v2
0+ω2
0x2
0
2x2
0, (19)
where x0∈[−√E0/ω0,√E0/ω0]and for v0the condition v2
0=E0−ω2
0x2
0holds. One of the benefits of
relation (19) is that it can be seen most directly that the optimal damping coefficient does not distinguish
initial conditions (±x0,±v0)and(±x0,∓v0), which is a shortcoming of this optimization criterion, because
the energy as a function of time is not the same for those two types of initial conditions (see (7) and (8))
and the energy decay may differ significantly depending on which of those initial conditions is in question.
We will deal with these and other issues of energy integral minimization as an optimal damping criterion in
the subsection 4.2.
60.1 0.781 1.66 2 3012345
γ/ω 0I(γ, ϕ1, ϕ2)ω0/E0ϕ1=−π/4,ϕ2=π/4
ϕ1=π/4,ϕ2= 3π/4
ϕ1=−π/4,ϕ2= 3π/4
Figure 4: Averaged integral (21) for three sets of initial conditions.
2.2. Minimization of the energy integral averaged over a set of initial conditions and optimal damping in
dependence of the chosen set
Now we calculate the average of the integral (12) over a set of initial conditions with θ∈[ϕ1, ϕ2], i.e.
I(γ, ϕ1, ϕ2) =1
ϕ2−ϕ1Zϕ2
ϕ1I(γ, θ)dθ , (20)
and we obtain
I(γ, ϕ1, ϕ2) =E0
2ω0ω2
0+γ2
γω0+γ
2ω0(ϕ2−ϕ1)(sin 2 ϕ2−sin 2ϕ1) +1
2(ϕ2−ϕ1)(cos 2 ϕ1−cos 2ϕ2)
.(21)
In Fig. 4 we show averaged integral (21) for three different sets of initial conditions. For the set of initial
conditions with ϕ1=−π/4andϕ2=π/4, i.e. with E0P≥E0K(where the equality holds only at the end
points of the set), minimum of the averaged integral (solid red curve) is at γ= 0.781ω0(rounded to three
decimal places). For the set of initial conditions with ϕ1=π/4andϕ2= 3π/4, i.e. with E0K≥E0P(where
the equality holds only at the end points of the set), minimum of the averaged integral (dashed blue curve)
is atγ= 1.658ω0(rounded to three decimal places). For the set of mixed initial conditions with ϕ1=−π/4
andϕ2= 3π/4, i.e. with E0P> E 0KandE0K> E 0Ppoints equally present in the set, minimum of the
averaged integral (dotted black curve) is at the critical damping condition γ=ω0.
If we consider the optimal damping as the one for which the averaged integral (21) is minimal, we can
easily determine the optimal damping coefficient γopt(ϕ1, ϕ2)form the condition
∂I(γ, ϕ1, ϕ2)
∂γ
γopt= 0, (22)
and we obtain
γopt(ϕ1, ϕ2) =s
2(ϕ2−ϕ1)
2(ϕ2−ϕ1) + sin 2 ϕ2−sin 2ϕ1ω0. (23)
We note here that averaged integral (21) and optimal damping coefficient (23) are not periodic functions in
variables ϕ1andϕ2, if we keep one variable fixed and change the other. But they are periodic, with period
π, if we change both variables simultaneously.
In Fig. 5 we show the optimal damping coefficient (23) as a function of ϕ2with fixed ϕ1= 0, and the
results shown can be summarized as follows:
•Forϕ1= 0andϕ2∈[0, π/2)∪(π,3π/2), the optimal damping coefficient (23) is in the under-damped
regime. In this case, integral (20) is averaged over sets that have more points corresponding to initial
conditions with E0P> E 0K, in comparison to the points corresponding to initial conditions with
E0K> E 0P.
70 0.25 0.50.75 1 1.25 1.51.75 20.70.80.911.1
γopt=ω0
ϕ2/πγopt(0, ϕ2)/ω0
Figure 5: Optimal damping coefficient (23) (solid red curve) as a function of ϕ2∈[0,2π]for fixed ϕ1= 0. Below the dashed
horizontal line, optimal damping coefficients are in the under-damped regime, above the line in the over-damped regime, and
in the critically damped regime at the crossing points of the line and the solid red curve.
•Forϕ1= 0andϕ2={π/2, π,3π/2,2π}, the optimal damping coefficient (23) is equal to critical damp-
ing. In this case, integral (20) is averaged over sets that have equal amount of points corresponding
to initial conditions with E0P> E 0Kand initial conditions with E0K> E 0P.
•Forϕ1= 0andϕ2∈(π/2, π)∪(3π/2,2π), the optimal damping coefficient (23) is in the over-damped
regime. In this case, integral (20) is averaged over sets that have more points corresponding to initial
conditions with E0K> E 0P, in comparison to the points corresponding to initial conditions with
E0P> E 0K.
3. 2-DOF systems with MPD
Figure 6: Schematic figure of a 2-DOF system.
Here we consider 2-DOF system shown schematically in Fig. 6. The corresponding equations of motion
are
m1¨x1(t) =−c1˙x1(t)−k1x1(t)−k2(x1(t)−x2(t)),
m2¨x2(t) =−c2˙x2(t)−k3x2(t) +k2(x1(t)−x2(t)).(24)
We will consider MPD [23], i.e. masses {m1, m2}, spring constants {k1, k2, k3}, and dampers {c1, c2}can
in general be mutually different but the condition c1/m1=c2/m2holds. In this case we can use modal
analysis [2, 21] and the system of equations (24) can be written via modal coordinates [21] as
¨q1(t) + 2γ˙q1(t) +ω2
01q1(t) = 0
¨q2(t) + 2γ˙q2(t) +ω2
02q2(t) = 0 ,(25)
where qi(t)andω0i, with i={1,2}, denote the modal coordinates and undamped modal frequencies of
the two modes, while γ=ci/2miis the damping coefficient. In the analysis that we will carry out in this
8subsection, we will not need the explicit connection of modal coordinates qi(t)and mass coordinates, i.e.
displacements xi(t), and we will deal with this in the next subsection when considering a specific example
with given masses, springs and dampers. Similarly as in Section 2 (see (2)), the general solution for the i-th
mode can be written as
qi(t) =e−γt
q0icos(ωit) +˙q0i+γq0i
ωisin(ωit)
, (26)
where ωi=p
ω2
0i−γ2is the damped modal frequency, and qi(0)≡q0iand ˙qi(0)≡˙q0iare the initial condi-
tions of the i-th mode. Thus, the reasoning and the results presented in Section 2, with some adjustments,
can by applied for the analysis of the 2-DOF system we are considering here.
The energy of the system is
E(t) =2X
i=1mi˙xi(t)2
2+k1x1(t)2
2+k3x2(t)2
2+k2(x1(t)−x2(t))2
2, (27)
and we take that the modal coordinates are normalised so that (27) can be written as
E(t) =2X
i=1Ei(t) =2X
i=1
˙qi(t)2+ω2
0iqi(t)2
(28)
where Ei(t)in (28) denotes the energy of the i-th mode. Total energy at t= 0, i.e. the initial energy, is
given by
E0=2X
i=1E0i=2X
i=1(E0Ki+E0Pi) =2X
i=1
˙q2
0i+ω2
0iq2
0i
, (29)
where E0idenotes the initial energy of the i-th mode, E0Ki= ˙q2
0iandE0Pi=ω2
0iq2
0idenote initial kinetic
and initial potential energy of the i-th mode.
All possible initial conditions with the same initial energy (29) can be expressed similarly as in the SDOF
case (see (9) and Fig. 1) but with two pairs of polar coordinates, one pair for each mode. For the i-th mode
we have radius ri=√E0iand angle θi= arctan
˙q0i
ω0iq0i
, i.e. we can write
ω0iq0i=ricosθi
˙q0i=risinθi.(30)
Thus, each initial condition with energy E0=E01+E02can be represented by points on two circles with
radii r1=√E01andr2=√E02, for which condition r2
1+r2
2=E0holds, and with angles θ1andθ2that tell
us how initial potential and initial kinetic energy are distributed within the modes. Using relation (10) for
SDOF systems, we can write the energy of the i-th mode in polar coordinates (30) as
Ei(t) =E0ie−2γt
cos2(ωit) +γcos 2θisin(2ωit)
ωi+
ω2
0i+γ2+ 2ω0iγsin 2θisin2(ωit)
ω2
i
(31)
for the under-damped ( γ < ω 0i) and over-damped ( γ > ω 0i) regime, and the energy of the i-th mode in the
critically damped regime is obtained analogously using the relation (11).
Consequently, the integral of the energy (28) over the entire time, for some arbitrary initial condition, is
simply calculated using relation (13) for each individual mode, we obtain
I(γ,{E0i},{θi}) =2X
i=1Z∞
0Ei(t)dt=2X
i=1E0i
2ω0iω2
0i+γ2
γω0i+γ
ω0icos 2θi+ sin 2 θi
. (32)
Furthermore, initial energy of the i-th mode can be written as E0i=a2
iE0, where coefficient a2
i∈[0,1]
denotes the fraction of the initial energy of the i-th mode in the total initial energy. Coefficients of the two
9modes satisfy a2
1+a2
2= 1and therefore can be parameterized as
a1= cos ψ
a2= sin ψ,(33)
where ψ∈[0, π/2]. Taking (33) into account, we can write (32) as
I(γ, ψ, θ 1, θ2) =E02X
i=1a2
i
2ω0iω2
0i+γ2
γω0i+γ
ω0icos 2θi+ sin 2 θi
. (34)
If we consider the optimal damping coefficient as the one for which the integral (34) is minimal, we can
easily determine the optimal damping coefficient form the condition
∂I(γ, ψ, θ 1, θ2)
∂γ
γopt= 0, (35)
and we obtain
γopt(ψ, θ1, θ2) =s
ω2
01ω2
02
2ω2
02cos2ψcos2θ1+ 2ω2
01sin2ψcos2θ2. (36)
It is easy to see that, for any fixed ψ, the function (36) has smallest magnitude for cos2θ1= cos2θ2= 1,
which corresponds to the initial conditions with initial energy comprised only of potential energy distributed
within the two modes, i.e E0=E0P1+E0P2. In that case we can write the denominator of (36) as
f(ψ) =q
2ω2
02cos2ψ+ 2ω2
01sin2ψ=q
2(ω2
02−ω2
01) cos2ψ+ 2ω2
01, (37)
where we used sin2ψ= 1−cos2ψ. Since ω01< ω 02, the function (37) has maximum for ψ= 0. Thus,
the minimum value of the optimal damping coefficient (36) is√
2ω01/2, and it is obtained for ψ= 0and
θ1={0, π}, which corresponds to the initial conditions with initial energy comprised only of potential energy
in the first mode, i.e. E0=E0P1. On the other hand, for any fixed ψ, the function (36) has singularities
forcos2θ1= cos2θ2= 0, which corresponds to the initial conditions with initial energy comprised only of
kinetic energy. Thus, the range of the optimal damping coefficient (36) is
γopt∈h√
2ω01/2,+∞
. (38)
Now we calculate the average of the integral (34) over a set of all initial conditions, we obtain
I(γ) =1
2π3Zπ/2
0dψZ2π
0dθ1Z2π
0dθ2I(γ, ψ, θ 1, θ2) =E0
42X
i=1ω2
0i+γ2
γω2
0i
, (39)
and from the condition
∂I(γ)
∂γ
γopt= 0, (40)
we find that the optimal damping coefficient with respect to the averaged integral (39) is given by
γopt=s
2ω2
01ω2
02
ω2
01+ω2
02. (41)
Inordertomoreeasilyanalyzethebehaviorofthedampingcoefficient(36)withregardtothedistribution
of the initial potential energy within the modes and its relationship with the damping coefficient (41),
10similarly as in subsection 2.1 (see (17) and (18)), we define the ratio of the initial potential energy of the
i-th mode and the total initial energy, i.e.
βi=E0Pi
E0. (42)
Since the initial potential energy satisfies E0P=E0P1+E0P2≤E0, we have βi∈[0,1]and the condition
0≤(β1+β2)≤1holds. Taking E0Pi=E0icos2θi(see (30)) and E0i=a2
iE0with (33) into account, we
have
β1= cos2ψcos2θ1
β2= sin2ψcos2θ2.(43)
Using (43), relation (36) can be written as
γopt(β1, β2) =s
ω2
01ω2
02
2ω2
02β1+ 2ω2
01β2. (44)
For clarity, we will repeat briefly, the minimum value of (44) is√
2ω01/2, obtained for β1= 1andβ2= 0
(or in terms of the angles in (36), for ψ= 0andθ1={0, π}), while γopt→+∞forβ1=β2= 0(or in
terms of the angles in (36), for any ψwith θ1={π/2,3π/2}andθ2={π/2,3π/2}). The benefit of relation
(44) is that we expressed (36) through two variables instead of three, i.e. this way we lost information about
the signs of the initial conditions and about distribution of initial kinetic energy within the modes, but the
optimal damping coefficient (36) does not depend on those signs anyway, due to the squares of trigonometric
functions in variables θ1andθ2, and, for a fixed distribution of initial potential energy within the modes,
the optimal damping coefficient (36) is constant for different distributions of initial kinetic energy within
the modes. By looking at relations (44) and (41), it is immediately clear that γopt(β1, β2) =γoptfor
ω2
02β1+ω2
01β2=ω2
01+ω2
02
4, (45)
while γopt(β1, β2)<γoptif the left hand side of relation (45) is greater than the right hand side, and
γopt(β1, β2)>γoptif the left hand side of relation (45) is smaller than the right hand side.
Again, similarly as in subsection 2.1 (see (19)), using βi=ω2
0iq2
0i/E0we can express the optimal damping
coefficient (44) as a function of the initial modal coordinates as well, i.e.
γopt(q01, q02) =s
E0
2q2
01+ 2q2
02, (46)
where q0i∈[−√E0/ω0i,√E0/ω0i]and the condition 0≤(ω2
01q2
01+ω2
02q2
02)≤E0holds. We can express
condition (45) in terms of initial modal coordinates, i.e. γopt({q0i}) =γoptfor
q2
01+q2
02
E0=ω2
01+ω2
02
4ω2
01ω2
02, (47)
while γopt({q0i})<γoptif the left hand side of relation (47) is greater than the right hand side, and
γopt({q0i})>γoptif the left hand side of relation (47) is smaller than the right hand side.
We note here that we did not use explicit values of the undamped modal frequencies ω01andω02in
the analysis so far, and relations presented so far are valid for any 2-DOF system with MPD. In the next
subsection, we provide a more detailed quantitative analysis using an example with specific values of modal
frequencies.
113.1. Quantitative example
Here we consider the 2-DOF system as the one shown schematically in Fig. 6, but with m1=m2=m,
k1=k2=k3=kandc1=c2=c. The corresponding equations of motion are
m¨x1(t) =−c˙x1(t)−kx1(t)−k(x1(t)−x2(t)),
m¨x2(t) =−c˙x2(t)−kx2(t) +k(x1(t)−x2(t)).(48)
For completeness, we will investigate here the behavior of the optimal damping coefficient given by the
minimization of the energy integral for different initial conditions, and its relationship with the optimal
dampingcoefficientgivenbytheminimizationoftheaveragedenergyintegral, inallthreecoordinatesystems
that we introduced in the previous subsection and additionally in the coordinate system defined by the initial
displacements of the masses. System of equations (48) can be easily recast to the form (25) with the modal
coordinates
q1(t) =rm
4(x1(t) +x2(t))
q2(t) =rm
4(x1(t)−x2(t)),(49)
and with the natural (undamped) frequencies of the modes ω01=ω0andω02=√
3ω0, where ω0=p
k/m.
Normalisation factorsp
m/4in (49) ensure that our expression (28) for the energy of the system corresponds
to energy expressed over the displacements and velocities of the masses, i.e.
E(t) =2X
i=1
˙qi(t)2+ω2
0iqi(t)2
=2X
i=1m˙xi(t)2
2+kxi(t)2
2
+k(x1(t)−x2(t))2
2. (50)
Using the specific values of undamped modal frequencies of this system, relations (36), (41) and (44)
become
γopt(ψ, θ1, θ2) =s
3
6 cos2ψcos2θ1+ 2 sin2ψcos2θ2ω0, (51)
γopt=√
6
2ω0, (52)
γopt(β1, β2) =r3
6β1+ 2β2ω0. (53)
Since ω01=ω0, the range of (53) is γopt∈[√
2ω0/2,+∞)(see (38)).
As examples of the behavior of the damping coefficient (51) as a function of the angles {ψ, θ1, θ2}and
its relationship with the damping coefficient (52), in Fig. 7 we show γopt(ψ, θ1, θ2)/γoptforψ={π/3, π/6}
andθi∈[0, π]. In Fig. 8 we show ratio of the damping coefficient (53) and the damping coefficient (52), i.e.
γopt(β1, β2)/γopt.
If the initial energy is comprised only of potential energy, in terms of initial modal coordinates we have
E0=ω2
0q2
01+ 3ω2
0q2
02, thus, the initial modal coordinates satisfy
q01s
ω2
0
E0∈[−1,1],
q02s
ω2
0
E0∈h
−√
3/3,√
3/3i
,
0≤ω2
0
E0
q2
01+ 3q2
02
≤1.(54)
12Figure 7: Ratio γopt(ψ, θ1, θ2)/γoptof the optimal damping coefficients (51) and (52) for ψ={π/3, π/6}andθi∈[0, π]. (a)
Forψ=π/3, the total initial energy E0is distributed within the modes as E01=E0/4andE02= 3E0/4. (b) For ψ=π/6,
the total initial energy is distributed within the modes as E01= 3E0/4andE02=E0/4. Singularities for θ1=θ2=π/2
are indicated by infinity symbols, and the points around singularities for which γopt(ψ, θ1, θ2)/γopt>3are removed on both
figures (central white areas). Black lines, on both figures, indicate the points for which γopt(ψ, θ1, θ2)/γopt= 1. On both
figures, ratio attains minimum for the corner points, i.e. for (θ1, θ2) ={(0,0),(0, π),(π,0),(π, π)}.
Figure 8: Ratio γopt(β1, β2)/γoptof the optimal damping coefficients (53) and (52) for β1∈[0,1],β2∈[0,1]and the constraint
0≤(β1+β2)≤1. Singularity for β1=β2= 0is indicated by the filled red circle, and the points near singularity, for which
γopt(β1, β2)/γopt>3, are removed, thus, a small white triangle is formed with the right angle at the origin. Black line indicates
the points for which γopt(β1, β2)/γopt= 1. The minimum value of the ratio is at the point (β1, β2) = (1 ,0).
Furthermore, we can write the optimal damping coefficient (46) as
γopt(q01, q02) =s
E0
2ω2
0(q2
01+q2
02)ω0, (55)
and the condition (47) as
ω2
0
E0
q2
01+q2
02
=1
3. (56)
13In Fig. 9(a) we show the ratio of (55) and (52), i.e. γopt(q01, q02)/γopt. The domain of this function
consists of points inside and on the ellipse, i.e. it is given by (54). Similarly as before, singularity at
(q01, q02) = (0 ,0)is indicated by the infinity symbol, and the points for which γopt(q01, q02)/γopt>3are
removed. For points inside the circle we have γopt(q01, q02)/γopt>1, and for points outside the circle
we have γopt(q01, q02)/γopt<1. Minimum values of this ratio are√
3/3≈0.58, obtained for the points
(q01, q02) ={(−√E0/ω0,0),(√E0/ω0,0)}.
Figure 9: (a) Ratio γopt(q01, q02)/γoptof the optimal damping coefficients (55) and (52). (b) Ratio γopt(x01, x02)/γoptof the
optimal damping coefficients (57) and (52). Singularities, at points (0,0)on the both figures, are denoted by infinity symbols,
and the points near singularities, for which γopt/γopt>3, are removed. Black circles on both figures indicate the points for
which γopt/γopt= 1.
Using (49) we can write the optimal damping coefficient (55) in terms of initial displacements xi(0)≡x0i
as
γopt(x01, x02) =s
E0
m(x2
01+x2
02)=s
E0
mω2
0(x2
01+x2
02)ω0. (57)
If the initial energy is comprised only of potential energy, in terms of initial displacements we have E0=
mω2
0(x2
01+x2
02−x01x02), thus, the initial displacements of the masses satisfy
x0is
mω2
0
E0∈[−1,1],
0≤mω2
0
E0
x2
01+x2
02−x01x02
≤1,(58)
and the condition (56) is now
mω2
0
E0
x2
01+x2
02
=2
3. (59)
In Fig. 9(b) we show the ratio of (57) and (52), i.e. γopt(x01, x02)/γopt. The domain of this function
consists of points given by (58). Similarly as before, singularity at (x01, x02) = (0 ,0)is indicated by the
infinity symbol, and the points for which γopt(x01, x02)/γopt>3are removed. For points inside the circle
γopt(x01, x02)/γopt>1, and for points outside the circle γopt(x01, x02)/γopt<1. Minimum values of this
ratio are√
3/3≈0.58, obtained for the points (x01, x02) =
±q
E0
mω2
0,±q
E0
mω2
0
.
14Figure 10: Schematic figure of a MDOF system with Ndegrees of freedom.
4. MDOF systems with MPD
Here we consider the MDOF system with Ndegrees of freedom shown schematically in Fig. 10. As in
the Section 3, we will consider MPD, i.e. masses {m1, m2, ..., m N}, spring constants {k1, k2, ..., k N+1}, and
dampers {c1, c2, ..., c N}can in general be mutually different but the condition ci/mi= 2γholds for any
i={1, ..., N}, where γis the damping coefficient. Therefore, the reasoning we presented in Section 3 can
be applied here, with the main difference that now the system has Nmodes instead of two. Again, we can
write each initial condition over polar coordinates, as in the 2-DOF case (see (30)), only now we have N
pairs of polar coordinates instead of two.
The energy of each mode is given by (31), and consequently, the integral of the total energy over the
entire time, for some arbitrary initial condition, is simply calculated similarly as in (32), i.e.
I(γ,{E0i},{θi}) =NX
i=1Z∞
0Ei(t)dt=NX
i=1E0i
2ω0iω2
0i+γ2
γω0i+γ
ω0icos 2θi+ sin 2 θi
, (60)
where, again, Ei(t)is the energy of the i-th mode, E0iis the initial energy of the i-th mode. Thus, each
initial condition with energy E0=PN
i=1E0iis represented by points on Ncircles with radii ri=√E0i, for
which conditionPN
i=1r2
i=E0holds, and with angles θithat tell us how initial potential and initial kinetic
energy are distributed within the modes.
Similarly as before, initial energy of the i-th mode can be written as E0i=a2
iE0, where coefficient
a2
i∈[0,1]denotes the fraction of the initial energy of the i-th mode in the total initial energy E0, and the
condition
NX
i=1a2
i= 1 (61)
holds. Relation (61) defines a sphere embedded in N-dimensional space and we can express the coefficients
aiover N-dimensional spherical coordinates ( N−1independent coordinates, i.e. angles, since the radius
is equal to one), but for the sake of simplicity we will not do that here and we will stick to writing the
expressions as a functions of the coefficients ai. Thus, we can write (60) as
I(γ,{ai},{θi}) =NX
i=1Z∞
0Ei(t)dt=E0NX
i=1a2
i
2ω0iω2
0i+γ2
γω0i+γ
ω0icos 2θi+ sin 2 θi
.(62)
We differentiate relation (62) by γand equate it to zero and get
γopt({ai},{θi}) = NX
i=12a2
icos2θi
ω2
0i!−1/2
(63)
as the optimal damping coefficient for which integral (62) is minimal. For any fixed set of coefficients
{ai}, the smallest magnitude of the function (63) is obtained for cos2θi= 1∀i, which corresponds to the
15initial conditions with initial energy comprised only of potential energy distributed within the modes, i.e
E0=PN
i=1E0Pi. In that case the denominator of (63) is
f({ai}) = NX
i=12a2
i
ω2
0i!1/2
(64)
and using a2
1= 1−PN
i=2a2
i(see (61)) we can write (64) as
f({ai}) =
2
ω2
01+NX
i=22a2
i1
ω2
0i−1
ω2
01!1/2
. (65)
Since ω01< ω 0ifor any i≥2, each term in the sum of relation (65) is negative, and we can conclude that
the function (65) has maximum for the set {ai}={1,0, ...,0}. Thus, the minimum value of the optimal
damping coefficient (63) is√
2ω01/2, and it is obtained for a1= 1andθ1={0, π}, which corresponds to the
initial conditions with initial energy comprised only of potential energy in the first mode, i.e. E0=E0P1.
On the other hand, for any fixed set {ai}, the function (63) has singularities for cos2θi= 0∀i. Thus, the
range of the optimal damping coefficient (63) is
γopt∈h√
2ω01/2,+∞
. (66)
In Appendix A we have calculated the average of the integral (62) over a set of all initial conditions and
obtained
I(γ) =E0
2NNX
i=1ω2
0i+γ2
γω2
0i
. (67)
We differentiate relation (67) by γand equate it to zero and obtain
γopt=N1/2 NX
i=11
ω2
0i!−1/2
(68)
as the optimal damping coefficient with respect to the averaged integral (67).
Since the ratio of the initial potential energy of the i-th mode and the total initial energy is
βi=E0Pi
E0=a2
icos2θi, (69)
where βi∈[0,1]and the condition 0≤PN
i=1βi≤1holds, we can write (63) as a function of the distribution
of the initial potential energy over the modes, i.e.
γopt({βi}) = NX
i=12βi
ω2
0i!−1/2
. (70)
The minimum value of (70) is√
2ω01/2, obtained for β1= 1andβi= 0fori≥2, while γopt→+∞for
βi= 0∀i. Using βi=ω2
0iq2
0i/E0, we can write (63) as a function of initial modal coordinates as well, i.e.
γopt({q0i}) =s
E0
2PN
i=1q2
0i, (71)
where q0i∈[−√E0/ω0i,√E0/ω0i]and the condition 0≤PN
i=1ω2
0iq2
0i≤E0holds.
164.1. Quantitative example
Here we consider the MDOF system as the one shown schematically in Fig. 10 but with mi=m,ci=c
fori={1, ..., N}, and with ki=kfori={1, ..., N + 1}. Such a system without damping, i.e with ci= 0∀i,
is a standard part of the undergraduate physics/mechanics courses [22]. Therefore, for the MDOF system
with Ndegrees of freedom we are considering here, the undamped modal frequencies are [21, 22]
ω0i= 2ω0siniπ
2(N+ 1)
,with i={1, ..., N}, (72)
and where ω0=p
k/m. In Fig. 11(a) we show undamped modal frequencies ω01,ω0Nand damping
coefficient γopt, i.e. (68), calculated with (72), as functions of N. We clearly see that the coefficient γoptis
in the over-damped regime from the perspective of the first mode, and in the under-damped regime from
the perspective of highest mode, for any N > 1, and in the case N= 1all three values match. In Fig. 11(b)
we show ratios γopt/ω01andω0N/γoptand we see that both ratios increase with increasing N.
Figure 11: (a) Undamped modal frequencies ω01(blue circles), ω0N(red x’s) and the damping coefficient γopt(black squares)
as functions of the number of the masses N. (b) Ratios γopt/ω01(blue line) and ω0N/γopt(red line), shown as solid lines due
to the high density of the shown points.
We show in Appendix B that the following limits hold
lim
N→+∞γopt(N) = 0 , (73)
lim
N→+∞γopt(N)
ω01(N)= +∞, (74)
lim
N→+∞ω0N(N)
γopt(N)= +∞. (75)
We note here that these limit values do not correspond to the transition from a discrete to a continuous
system, but simply tell us the behavior of these quantities with respect to the increase in the number of
masses, i.e. with respect to the increase in the size of the discrete system.
From everything that has been said so far, it is clear that the damping coefficient γopt, obtained by
minimizing the energy integral averaged over all initial conditions that correspond to the same initial energy,
cannot be considered generally as optimal and that, by itself, it says nothing about optimal damping of the
system whose dynamics started with some specific initial condition. Damping coefficient (63), which is given
by the minimization of the energy integral for a specific initial condition, is of course a better choice for
optimal damping of an MDOF system, than the damping coefficient γopt, if we want to consider how the
system dissipates energy the fastest for a particular initial condition, but, as we argue in the subsection 4.2,
this damping coefficient also has some obvious deficiencies.
174.2. Issues with the minimum of the energy integral as a criterion for optimal damping
We can ask, for example, whether in an experiment, with known initial conditions, in which an MDOF
system is excited to oscillate, a damping coefficient (63) would be the best choice if we want that the system
settles down in equilibrium as soon as possible? Here, in three points, we explain why we think the answer
to that question is negative:
•From relation (62), we see that, due to the term sin 2θi, the energy integral is sensitive to changes
θi→ −θiandθi→π−θi, which correspond to changes of initial conditions (q0i,˙q0i)→(q0i,−˙q0i)and
(q0i,˙q0i)→(−q0i,˙q0i). When we differentiate (62) to determine γfor which the energy integral has a
minimum, the term sin 2θicancels and as a result the coefficient (63) is not sensitive to this change
in initial conditions. Such changes in the initial conditions lead to significantly different situations.
For example, if q0i>0and ˙q0i>0, the i-th mode in the critical and over-damped regime (i.e. for
γ≥ω0i) will never reach the equilibrium position, while for q0i>0and ˙q0i<0, and i-th mode
initial kinetic energy grater than initial potential energy, it can go through the equilibrium position
once, depending on the magnitude of the damping coefficient, and there will be the smallest damping
coefficient in the over-damped regime for which no crossing occurs and for which the solution converges
to equilibrium faster than for any other damping coefficient [24]. Therefore, the damping coefficient
considered optimal would have to be sensitive to this change in initial conditions.
•Damping coefficient (63) has singularities for cosθi= 0∀i, i.e. for initial conditions for which all
initial energy is kinetic. For such initial conditions, the higher the damping coefficient, the higher and
faster the dissipation. In other words, the higher the damping coefficient, the faster the energy integral
decreases. Therefore, coefficient (63) diverges for that type of initial conditions. This would actually
mean that, for this initial conditions, it is optimal to take the damping coefficient as high as possible,
but in principle this corresponds to a situation in which all modes are highly over-damped, i.e. all
masses reach their maximum displacements in a very short time and afterwards they begin to return
to the equilibrium position almost infinitely slowly. Figuratively speaking, it is as if we immersed the
system in concrete. This issue has recently been addressed in the context of free vibrations of SDOF
[24] and was already noticed in [25]. Therefore, simply taking the highest possible damping coefficient,
as suggested by relation (63) for this type of initial conditions, is not a good option.
•The damping coefficient (63) is determined on the basis of the energy integral over the entire time and
therefore it does not take into account that in nature and experiments these systems effectively return
to the equilibrium state for some finite time.
Because of the above points, in the next section we provide a new approach to determine the optimal
damping of MDOF systems.
5. Optimal damping of an MDOF system: a new perspective
From a theoretical perspective, systems with viscous damping asymptotically approach the equilibrium
state and never reach it exactly. In nature and in experiments, these systems reach the equilibrium state
which is not an exact zero energy state, but rather a state in which the energy of the system has decreased
to the level of the energy imparted to the system by the surrounding noise, or to the energy resolution of
the measuring apparatus. Following this line of thought, we will define a system to be in equilibrium for
times t > τsuch that
E(τ)
E0= 10−δ, (76)
where E(τ)is the energy of the system at t=τ,E0is the initial energy, and δ > 0is a dimensionless
parameter that defines what fraction of the initial energy is left in the system. This line of thought has
recently been used to determine the optimal damping of SDOF systems [24], and here we extend it to
MDOF systems. Therefore, in what follows, we will consider as optimal the damping coefficient for which
the systems energy drops to some energy level of interest, e.g. to the energy resolution of the experiment,
the fastest and we will denote it with ˜γ.
185.1. Optimal damping of the i-th mode of a MDOF system with MPD
Here we will consider the behavior of the energy of the i-th mode of the MDOF system with MPD and
determine the optimal damping coefficient ˜γiof the i-th mode with respect to criterion (76). For any MDOF
system with N≥1degrees of freedom with MPD, each mode behaves as a SDOF system studied in Section
2, with the damping coefficient γand the undamped (natural) frequency ω0i. Thus (see relation (31)), the
ratio of the energy of the i-th mode, Ei(γ, t), and initial energy of the i-th mode, E0i, is given by
Ei(γ, t)
E0i=e−2γt
cos2(ωit) +γcos 2θisin(2ωit)
ωi+
ω2
0i+γ2+ 2ω0iγsin 2θisin2(ωit)
ω2
i
(77)
for the under-damped ( γ < ω oi) and over-damped ( γ > ω 0i) regime. We will repeat here briefly for clarity,
ωi=p
ω2
0i−γ2is the damped angular frequency and θiis the polar angle which determines the initial
conditions q0iand ˙q0iof the i-th mode and the distribution of the initial energy within the mode, i.e.
initial potential and initial kinetic energy of the i-th mode are E0Pi=E0icos2θiandE0Ki=E0isin2θi
respectively. Energy to initial energy ratio for the i-th mode in the critically damped regime ( γ=ω0i) is
simply obtained by taking γ→ω0ilimit of the relation (77), and we obtain
Ei(γ=ω0i, t)
E0i=e−2ω0it
1 + 2 ω0i(cos 2 θi)t+ 2ω2
0i(1 + sin 2 θi)t2
. (78)
In relations (77) and (78), we explicitly show that the energy depends on the damping coefficient and time,
because in what follows we will plot these quantities as functions of these two variables for fixed initial
conditions, i.e. fixed θi. We will investigate the behavior for several types of initial conditions, which of
course will not cover all possible types of initial conditions, but will give us a sufficiently clear picture of the
determinationandbehavioroftheoptimaldampingwithrespecttotheinitialconditionsandtheequilibrium
state defined with condition (76).
5.1.1. Initial energy of the i-th mode comprised only of potential energy
In Fig. 12 we show the base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition
θi= 0, which corresponds to the initial energy of the i-th mode comprised only of potential energy. Four
black contour lines denote points with Ei(γ, t)/E0i={10−3,10−4,10−5,10−6}respectively, as indicated by
the numbers placed to the left of each contour line. Each contour line has a unique point closest to the γ
axis, i.e. corresponding to the damping coefficient ˜γifor which that energy level is reached the fastest. As an
example, we draw arrow in Fig. 12 that points to the coordinates (γ, t) = (0 .840ω0i,5.15ω−1
0i), i.e. to the tip
of the contour line with points corresponding to Ei(γ, t) = 10−4E0i. Thus, for the initial condition θi= 0,
˜γi= 0.840ω0iis the optimal damping coefficient for the i-th mode to reach this energy level the fastest, and
it does so at the instant τi= 5.15ω−1
0i. In Table 1 we show results for other energy levels corresponding
to contour lines shown in Fig. 12. Here, and in the rest of the paper, we have rounded the results for the
damping coefficient to three decimal places, and for the time to two decimal places.
Ei(γ, t)/E0i˜γi[ω0i]τi[ω−1
0i]
10−30.769 4.18
10−40.840 5.15
10−50.885 6.16
10−60.915 7.20
Table 1: Optimal damping coefficient ˜γifor which the energy of the i-th mode drops to the level 10−δE0ithe fastest, with the
initial condition θi= 0.
Consider now, for example, a thought experiment in which we excite a MDOF system so that it vibrates
only in the first mode and that all initial energy was potential, i.e. E01=E0andθ1= 0. Furthermore,
19Figure 12: The base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi= 0. For this initial
condition, initial energy of the i-th mode is comprised only of potential energy. Four black contour lines denote points with
Ei(γ, t)/E0i={10−3,10−4,10−5,10−6}respectively, as indicated by the numbers placed to the left of each contour line. As
an example of determining the optimal damping for which the system reaches the desired energy level the fastest, i.e. with
respect to the condition (76), we draw the arrow that points to the coordinates (γ, t) = (0 .840ω0i,5.15ω−1
0i)for which the i-th
mode reaches the level Ei(γ, t)/E0i= 10−4the fastest. Thus, ˜γi= 0.840ω0iis the optimal damping coefficient to reach this
energy level the fastest. Optimal values for other energy levels, denoted with contour lines, are given in Table 1.
suppose that the system has effectively returned to equilibrium when its energy drops below 10−6E0, due
to the resolution of the measuring apparatus. It is clear form the Table 1 that ˜γ1= 0.915ω01would be
optimal in such a scenario. In the same scenario, optimal damping coefficient given by the minimization of
the energy integral, i.e. (63), would be γopt=√
2ω01/2 = 0 .707ω01, thus, a very bad choice in the sense
that this damping coefficient would not be optimal even in an experiment with a significantly poorer energy
resolution (see Table 1). This simple example illustrates that, from a practical point of view, one has to take
into account both the initial conditions and the resolution of the measuring apparatus in order to determine
the optimal damping coefficient.
5.1.2. Initial energy of the i-th mode comprised only of kinetic energy
In Fig. 13(a) and (b) we show the base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial
condition θi=π/2, which corresponds to the initial energy of the i-th mode comprised only of kinetic
energy. In Fig. 13(b) we show results for larger data span than in Fig. 13(a), and only contour line for points
corresponding to Ei(γ, t) = 10−3E0i. The left arrow in Fig. 13(b) indicates the same coordinates as the
arrow in Fig. 13(a), and the right arrow in Fig. 13(b) points to the coordinates (γ, t) = (13 .316ω0i,4.66ω−1
0i)
with Ei(γ, t) = 10−3E0i. Thus, for γ >13.316ω0ithe system comes sooner to the energy level 10−3E0ithan
forγ= 0.722ω0i, but these highly over-damped damping coefficients would correspond to restricting the
system to infinitesimal displacements from equilibrium, after which the system returns to the equilibrium
state practically infinitely slowly [24]. Thus, for this initial condition we take the damping coefficient in the
under-damped regime, i.e. ˜γi= 0.722ω0i, as optimal for reaching the level Ei(γ, t) = 10−3E0ithe fastest.
For all energy levels the behaviour is qualitatively the same, and the results are given in Table 2.
Consider now, for example, a thought experiment in which we excite a MDOF system so that it vibrates
only in the first mode and that all initial energy was kinetic, i.e. E01=E0andθ1=π/2. Furthermore,
suppose that the system has effectively returned to equilibrium when its energy drops below 10−6E0, due
to the resolution of the measuring apparatus. It is clear form the Table 2 that ˜γ1= 0.892ω01would be
optimal in such a scenario. In the same scenario, optimal damping coefficient given by the minimization of
the energy integral, i.e. (63), would be γopt= +∞.
20Figure 13: The base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi=π/2. For this initial
condition, initial energy of the i-th mode is comprised only of kinetic energy. (a) Four black contour lines denote points with
Ei(γ, t)/E0i={10−3,10−4,10−5,10−6}respectively, andthearrowpointstothecoordinates (γ, t) = (0 .722ω0i,4.66ω−1
0i), with
Ei(γ, t)/E0i= 10−3, for which this level of energy is reached in shortest time for the shown data span. (b) Contour line for
points with Ei(γ, t) = 10−3E0iis shown for larger data span, left arrow points to the coordinates (γ, t) = (0 .722ω0i,4.66ω−1
0i),
and the right arrow to the coordinates (γ, t) = (13 .316ω0i,4.66ω−1
0i), both with Ei(γ, t)/E0i= 10−3. Thus, for γ >13.316ω0i
energy level 10−3E0iis reached faster than for γ= 0.7223ω0i. See text for details.
Ei(γ, t)/E0i˜γi[ω0i]τi[ω−1
0i]
10−30.722 4.66
10−40.794 5.50
10−50.852 6.42
10−60.892 7.40
Table 2: Optimal damping coefficient ˜γifor which the energy of the i-th mode drops to the level 10−δE0ithe fastest, with the
initial condition θi=π/2.
Here we note that if in such an experiment we can set the damping coefficient to be in the over-damped
regime in the first part of the motion, i.e. when the system is moving from the equilibrium position to
the maximum displacement, and in the under-damped regime in the second part of the motion, i.e. when
the system moves from the position of maximum displacement back towards the equilibrium position, then
the fastest way to achieve equilibrium would be to take the largest experimentally available over-damped
coefficient in the first part of the motion, and the under-damped coefficient optimised like in 5.1.1 in the
second part of the motion, with the fact that we have to carry out the optimization with respect to the
energy left in the system at the moment when the system reached the maximum displacement and with
respect to the energy resolution of the experiment.
5.1.3. Initial energy of the i-th mode comprised of potential and kinetic energy
In Fig. 14(a) we show the base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition
θi=π/3, whichcorrespondstothe initialenergyofthe i-thmodecomprisedofkineticenergy E0Ki= 3E0i/4
and potential energy E0Pi=E0i/4, with both initial normal coordinate and velocity positive, i.e. with
q0i>0and ˙q0i>0. The results for optimal damping are obtained by the same method as in 5.1.1 and are
given in Table 3 for data shown in Fig. 14(a), and in Table 4 for data shown in Fig. 14(b). We see that
the energy dissipation strongly depends on the relative sign between q0iand ˙q0i. It was recently shown,
21for free vibrations of SDOF, that for an initial condition with initial kinetic energy greater than initial
potential energy and opposite signs between x0andv0, an optimal damping coefficient can be found in the
over-damped regime [24], thus, the same is true when we consider any mode of a MDOF system with MPD.
Figure 14: The base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), (a) for initial condition θi=π/3, and (b) for initial
condition θi=−π/3. For both initial conditions, initial energy of the i-th mode is comprised of kinetic energy E0Ki= 3E0i/4
and potential energy E0Pi=E0i/4. For θi=π/3initial normal coordinate and velocity are of the same signs, i.e. q0i>0and
˙q0i>0. For θi=−π/3initial normal coordinate and velocity are of the opposite signs, i.e. q0i>0and ˙q0i<0.
Ei(γ, t)/E0i˜γi[ω0i]τi[ω−1
0i]
10−30.751 4.66
10−40.825 5.58
10−50.875 6.55
10−60.908 7.58
Table 3: Optimal damping coefficient ˜γifor which the energy
of the i-th mode drops to the level 10−δE0ithe fastest, with
the initial condition θi=π/3.Ei(γ, t)/E0i˜γi[ω0i]τi[ω−1
0i]
10−31.075 1.87
10−41.112 2.42
10−51.135 3.02
10−61.145 3.64
Table 4: Optimal damping coefficient ˜γifor which the energy
of the i-th mode drops to the level 10−δE0ithe fastest, with
the initial condition θi=−π/3.
Consider now, for example, a thought experiment in which we excite a MDOF system so that it vibrates
only in the first mode and that 75%of initial energy was kinetic and 25%of initial energy was potential,
and with q01>0and ˙q01>0, i.e. E01=E0andθ1=π/3. Furthermore, suppose that the system
has effectively returned to equilibrium when its energy drops below 10−6E0, due to the resolution of the
measuring apparatus. It is clear form the Table 3 that ˜γ1= 0.908ω01would be optimal in such a scenario. In
the same scenario, but with q01>0and ˙q01<0, i.e. for θ1=−π/3, we see from Table 4 that ˜γ1= 1.145ω01
would be optimal. Optimal damping coefficient given by the minimization of the energy integral, i.e. (63),
is insensitive to the change of the sign of ˙q01, and it would be γopt=√
2ω01= 1.414ω01in both cases.
We note here, that for the initial conditions of the i-th mode with initial kinetic energy much grater than
initial potential energy, i.e. E0Ki>> E 0Pi, and with opposite signs of initial displacement and velocity,
i.e.sgn(q0i)̸= sgn( ˙ q0i), the optimal damping coefficient is going to be deep in the over-damped regime and
dissipation of initial energy will happen in a very short time. If, for any reason, this is not desirable in
some particular application, one can always find damping coefficient in the under-damped regime, with that
same initial condition, which can serve as an alternative. As an example of such a situation, in Fig. 15 we
show the base 10 logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi=−9π/20, which
corresponds to the initial energy of the i-th mode comprised of kinetic energy E0Ki≈0.98E0iand potential
energy E0Pi≈0.02E0i, with q0i>0and ˙q0i<0. In Fig. 15 we see that the i-th mode will reach the energy
22level 10−6E0ithe fastest for γ= 3.222ω0i, and in case, e.g., that such damping coefficient is difficult to
achieve experimentally, another choice for the optimal damping coefficient can be γ= 0.883ω0i.
Figure 15: The base 10logarithm of the ratio (77), i.e. log (Ei(γ, t)/E0i), for initial condition θi=−9π/20. Black contour
line denotes the points with Ei(γ, t) = 10−6E0i. Left arrow points to the coordinates (γ, t) = (0 .883ω0i,7.30ω−1
0i)for which
level 10−6E0iis reached the fastest in the under-damped regime, and the right arrow points to the coordinates (γ, t) =
(3.222ω0i,0.87ω−1
0i)for which the same level is reached the fastest in the over-damped regime.
5.2. Optimal damping of a MDOF system with MPD
If all modes of a MDOF system with Ndegrees of freedom are excited, the ratio of the energy of the
system, E(γ, t), and initial energy of the system, E0, is given by
E(γ, t)
E0=NX
i=1E0i
E0e−2γt
cos2(ωit) +γcos 2θisin(2ωit)
ωi+
ω2
0i+γ2+ 2ω0iγsin 2θisin2(ωit)
ω2
i
,(79)
where the set of all initial energies of the modes, i.e. {E0i}, and the set of all polar angles, i.e. {θi},
determines the initial condition of the whole system. Since for MPD the damping of the system as a whole
is determined by only one damping coefficient γ, we can calculate the base 10logarithm of the ratio (79),
but using a unique units for γ,tandω0ifor all modes, and from these data determine the optimal damping
coefficient ˜γ, for which the system will come to equilibrium in the sense of the condition (76) the fastest,
in the same way as in subsubsections 5.1.1-5.1.3 where we showed how to determine the optimal damping
of individual modes. One practical choice for the units might be ω01forγand for ω0i∀i, and ω−1
01fort.
This way, we have the easiest insight into the relationship between the first mode and the optimal damping
coefficient that we want to determine, in the sense that we can easily see whether the first mode is under-
damped, over-damped or critically damped in relation to it, which is important since the first mode is often
the dominant mode. If we apply this to the 2-DOF system studied in 3.1, we obtain
E(γ, t)
E0=2X
i=1E0i
E0e−2γt
cos2(ωit) +γcos 2θisin(2ωit)
ωi+
ω2
0i+γ2+ 2ω0iγsin 2θisin2(ωit)
ω2
i
,(80)
where ω01=ω0,ω02=√
3ω0,ω1=p
ω2
0−γ2,ω2=p
3ω2
0−γ2and we take that the damping coefficient
is in ω0units, while the time is in ω−1
0units. We are now in a position to determine the optimal damping
of this 2-DOF system for different initial conditions. Again, we will not investigate all possible types of the
23initial conditions, but two qualitatively different ones, one with initial energy comprised only of potential
energy, and the other with initial energy comprised only of kinetic energy. These two examples will give us
a picture of the procedure for determining the optimal damping coefficient ˜γfor this 2-DOF system. The
same procedure for determining the optimal damping can be in principle carried out for any MDOF system
with MPD, with any initial condition.
5.2.1. Optimal damping of the 2-DOF system with initial energy comprised only of potential energy
Herewechooseinitialconditionwith E01=E02=E0/2andθ1=θ2= 0, i.e. withinitialpotentialenergy
distributed equally between the two modes and zero initial kinetic energy. In Fig. 16 we show the base 10
logarithm of the ratio (80), i.e. log (E(γ, t)/E0), for the chosen initial condition. In Table 5 we show results
for other energy levels corresponding to contour lines shown in Fig. 16. For this initial condition, optimal
damping coefficient given by the minimization of the energy integral, i.e. (63), is γopt=p
3/4ω0= 0.866ω0.
Figure16: Thebase 10logarithmoftheratio(80), i.e. log (E(γ, t)/E0), forinitialcondition E01=E02=E0/2andθ1=θ2= 0.
For this initial condition, initial energy of the 2-DOF system is comprised only of potential energy distributed equally between
the modes. Four black contour lines denote points with E(γ, t)/E0={10−3,10−4,10−5,10−6}respectively, as indicated by
the numbers placed to the left of each contour line. As an example of determining the optimal damping for which the system
reaches the desired energy level the fastest, i.e. with respect to the condition (76), we draw the arrow that points to the
coordinates (γ, t) = (0 .859ω0,5.37ω−1
0)for which the energy of the system reaches the level E(γ, t)/E0= 10−4the fastest.
Thus, ˜γ= 0.859ω0is the optimal damping coefficient to reach this energy level the fastest.
E(γ, t)/E0˜γ[ω0]τ[ω−1
0]
10−30.817 4.36
10−40.859 5.37
10−50.893 6.27
10−60.924 7.55
Table 5: Optimal damping coefficient ˜γfor which the energy of the system drops to the level 10−δE0the fastest, with the
initial condition E01=E02=E0/2andθ1=θ2= 0.
245.2.2. Optimal damping of the 2-DOF system with initial energy comprised only of kinetic energy
Here we choose initial condition with E01=E02=E0/2andθ1=θ2=π/2, i.e. with initial kinetic
energy distributed equally between the two modes and zero initial potential energy. In Fig. 17(a) and (b)
we show the base 10logarithm of the ratio (80), i.e. log (E(γ, t)/E0), for the chosen initial condition. In
Table 6 we show results for other energy levels corresponding to contour lines shown in Fig. 17(a). For this
initial condition, optimal damping coefficient given by the minimization of the energy integral, i.e. (63), is
γopt= +∞.
Figure 17: The base 10logarithm of the ratio (80), i.e. log (E(γ, t)/E0), for initial condition E01=E02=E0/2andθ1=
θ2=π/2. For this initial condition, initial energy of the 2-DOF system is comprised only of kinetic energy distributed equally
between the modes. (a) Four black contour lines denote points with E(γ, t)/E0={10−3,10−4,10−5,10−6}respectively, and
thearrowpointstothecoordinates (γ, t) = (0 .783ω0,4.60ω−1
0), with E(γ, t)/E0= 10−3, forwhichthislevelofenergyisreached
in shortest time for the shown data span. (b) Contour line for points with E(γ, t) = 10−3E0is shown for larger data span, left
arrow points to the coordinates (γ, t) = (0 .783ω0,4.60ω−1
0), and the right arrow to the coordinates (γ, t) = (15 .927ω0,4.60ω−1
0),
both with Ei(γ, t)/E0i= 10−3. Thus, for γ >15.927ω0energy level 10−3E0is reached faster than for γ= 0.783ω0.
E(γ, t)/E0˜γ[ω0]τ[ω−1
0]
10−30.783 4.60
10−40.838 5.72
10−50.861 6.47
10−60.909 7.78
Table 6: Optimal damping coefficient ˜γfor which the energy of the system drops to the level 10−δE0the fastest, with the
initial condition E01=E02=E0/2andθ1=θ2=π/2.
6. Conclusion and outlook
The main message of this paper is that the dissipation of the initial energy in vibrating systems signifi-
cantly depends on the initial conditions with which the dynamics of the system started, and ideally it would
be optimal to always adjust the damping to the initial conditions. We took one of the known criteria for
optimal damping, the criterion of minimizing the (zero to infinity) time integral of the energy of the system,
averaged over all possible initial conditions corresponding to the same initial energy, and modified it to take
into account the initial conditions, i.e. instead of averaging over of all possible initial conditions, we studied
25the dependence of the time integral of the energy of the system on initial conditions and determined the
optimal damping as a function of the initial conditions. We found that the thus obtained optimal damping
coefficients take on an infinite range of values depending on the distribution of initial potential energy and
initial kinetic energy within the modes. We also pointed out the shortcomings of the thus obtained optimal
damping coefficients and introduced a new method for determining optimal damping. Our method is based
on the determination of the damping coefficients for which the energy of the system drops the fastest be-
low some energy threshold (e.g. below the energy resolution of the experiment). We have shown that our
method gives, both quantitatively and qualitatively, different results from the energy integral minimization
method. In particular, the energy integral minimization method gives infinite optimal damping for initial
conditions with purely kinetic energy, i.e. this method overlooks the region of underdamped coefficients
for which strong energy dissipation occurs with this type of initial conditions, while this region is clearly
seen and taken into account if one looks the energy behaviour directly, as we did. Furthermore, the energy
integral minimization method gives the optimal damping which does not depend on the signs of the initial
conditions, and we have shown that energy dissipation can strongly depend on them, which is taken into
account in our method.
Although the paper is dedicated to the case of mass-proportional damping, the new method we propose
for determining the optimal damping can be applied to the types of damping we did not study in this paper.
For example, in the case of a system with Rayleigh damping, the energy can be determined analytically
using modal analysis, and based on that analytical expression, it can be numerically investigated for which
values of the mass and stiffness proportionality constants the energy of the system drops the fastest below
some energy threshold which effectively corresponds to the equilibrium state. In the case of a system with
damping that does not allow analytical treatment, energy, as a function of time and magnitudes of individual
dampers, can be determined numerically, e.g. by studying the vibrating system as a first order ordinary
differential equation with matrix coefficients and using modern numerical methods for finding a solution of
such an equation. This approach allows one to numerically solve systems with many degrees of freedom.
Thus, we can numerically analyze the time evolution of the energy and find a set of damping parameters for
which the energy drops to a desired energy threshold the fastest. Of course, this approach can be applied
only for systems with a moderate number of degrees of freedom and a small number of dampers, due to
the rapid growth of the parameter space that needs to be searched. Despite these limitations, we believe
that our approach to optimal damping can be useful because, as we have shown, it can provide insights
that other approaches overlook. Therefore, in future work we will investigate in detail the application of
our approach to systems with damping that does not allow modal analysis. Furthermore, real systems can
respond to many different initial conditions in operating conditions. We envision that our approach can be
used to provide an overall optimal damping with respect to all initial conditions or with respect to some
expected range of initial conditions. For this purpose, one could consider the energy averaged over the
initial conditions and find the damping for which this averaged energy drops to a desired energy threshold
the fastest. This will be the topic of our next work.
7. Acknowledgments
We are grateful to Bojan Lončar for making schematic figures of 2-DOF and MDOF systems, i.e. Fig.
6 and 10, according to our sketches. This work was supported by the QuantiXLie Center of Excellence, a
projectco-financedbytheCroatianGovernmentandEuropeanUnionthroughtheEuropeanRegionalDevel-
opment Fund, the Competitiveness and Cohesion Operational Programme (Grant No. KK.01.1.1.01.0004).
The authors have no conflicts to disclose.
Appendix A. Average of the integral (62)over a set of all initial conditions
For reader’s convenience, we will repeat the integral (62) here
I(γ,{ai},{θi}) =E0NX
i=1a2
i
2ω0iω2
0i+γ2
γω0i+γ
ω0icos 2θi+ sin 2 θi
. (A.1)
26In order to calculate the average of (A.1) over a set of all initial conditions, one has to integrate (A.1)
over all coefficients ai, which satisfyPN
i=1a2
i= 1andai∈[−1,1], and over all angles θi∈[0,2π]. Due toR2π
0cos 2θidθi=R2π
0sin 2θidθi= 0, terms with sine and cosine functions don’t contribute to the average of
(A.1). Integration over all possible coefficients aiamounts to calculating the average of a2
iover a sphere
of radius one embedded in Ndimensional space. If we were to calculate the average of the equation of a
spherePN
i=1a2
i= 1over a sphere defined by that equation, we would get
NX
i=1a2
i= 1, (A.2)
where a2
idenotes the average of a2
iover a sphere. Due to the symmetry of the sphere and the fact that we
are integrating over the whole sphere, contribution of each a2
iin the sum (A.2) has to be the same, so we
can easily conclude that
a2
i=1
N, (A.3)
for any i. Thus, the average of (A.1) over all possible initial conditions is
I(γ) =E0
2NNX
i=1ω2
0i+γ2
γω2
0i
. (A.4)
Appendix B. Limit values (73),(74)and(75)
For reader’s convenience, we repeat here (68) and (72)
γopt=N1/2 NX
i=11
ω2
0i!−1/2
(B.1)
ω0i= 2ω0siniπ
2(N+ 1)
,with i={1, ..., N}. (B.2)
Using (B.2), we can write (B.1) as
γopt= 2ω0N1/2 NX
i=11
sin2ζi!−1/2
, (B.3)
where ζi=iπ
2(N+1). Using the fact that sinx < xfor0< x < π/ 2, we obtain
γopt<2ω0N1/2
4(N+ 1)2
π2NX
i=11
i2!−1/2
= 2ω0N1/2 π
2(N+ 1) NX
i=11
i2!−1/2
.
Now taking N→ ∞and using the well-known formulaP∞
i=11
i2=π2
6, we obtain (73).
Now we focus on the limit (74). We will use the following well-known inequality sinx > x/ 2for0< x <
π/2(this can be easily seen by, e.g. using the fact that sinis a concave function on [0, π/2]). From (B.3) it
follows
γopt
ω01=N1/2(sinζ1)−1 NX
i=11
sin2ζi!−1/2
> N1/2ζ−1
1·1
2 NX
i=11
ζ2
i!−1/2
=1
2N1/2 NX
i=11
i2!−1/2
,
hence we obtain (74).
27The limit (75) is also easy to prove. Since
lim
N→+∞ω0N= lim
N→+∞2ω0sinNπ
2(N+ 1)
= 2ω0 (B.4)
and we already showed (73), it is easy to conclude that
lim
N→+∞ω0N
γopt= +∞, (B.5)
i.e. the limit (75) holds.
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28 |
2312.07116v2.Sliding_Dynamics_of_Current_Driven_Skyrmion_Crystal_and_Helix_in_Chiral_Magnets.pdf | Sliding Dynamics of Current-Driven Skyrmion Crystal and Helix in Chiral Magnets
Ying-Ming Xie,1Yizhou Liu,1and Naoto Nagaosa1,∗
1RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
(Dated: December 14, 2023)
The skyrmion crystal (SkX) and helix (HL) phases, present in typical chiral magnets, can each be
considered as forms of density waves but with distinct topologies. The SkX exhibits gyrodynamics
analogous to electrons under a magnetic field, while the HL state resembles topological trivial
spin density waves. However, unlike the charge density waves, the theoretical analysis of the sliding
motion of SkX and HL remains unclear, especially regarding the similarities and differences in sliding
dynamics between these two spin density waves. In this work, we systematically explore the sliding
dynamics of SkX and HL in chiral magnets in the limit of large current density. We demonstrate
that the sliding dynamics of both SkX and HL can be unified within the same theoretical framework
as density waves, despite their distinct microscopic orders. Furthermore, we highlight the significant
role of gyrotropic sliding induced by impurity effects in the SkX state, underscoring the impact of
nontrivial topology on the sliding motion of density waves. Our theoretical analysis shows that
the effect of impurity pinning is much stronger in HL compared with SkX, i.e., χSkX/χHL∼α2
(χSkX,χHL: susceptibility to the impurity potential, α(≪1) is the Gilbert damping). Moreover,
the velocity correction is mostly in the transverse direction to the current in SkX. These results are
further substantiated by realistic Landau-Lifshitz-Gilbert simulations.
Introduction.— Density waves in solids represent a
prevalent phenomenon, particularly in low-dimensional
systems [1, 2]. They break the translational symmetry
of the crystal, leading to the emergence of Goldstone
bosons, i.e., phasons, which remain gapless when the pe-
riod of density waves is incommensurate with the crystal
periodicity. The sliding motion of density waves under an
electric field Ehas been extensively studied. In this con-
text, the impurity pinning of phasons results in a finite
threshold field [1, 2]. In general, exploring the dynam-
ics of pinning and depinning offers valuable insights into
understanding the behavior of density waves.
The skyrmion crystal (SkX) and helix (HL) phases
in chiral magnets can be recognized as periodic density
waves of spins, as depicted in Figs. 1 (a) and (b). The
HL phase is stabilized in chiral magnet at small mag-
netic field regions, with spins of neighboring magnetic
moments arranging themselves in a helical pattern. SkX
is a superposition of three phase-locked HL and comprises
arrays of magnetic skyrmions, nanoscale vortex-like spin
textures characterized by a non-zero skyrmion number
Nsk=1
4πR R
d2rs·(∂xs×∂ys) (sbeing the unit vec-
tor of spin). Theoretically proposed magnetic skyrmions
[3–5] were initially observed in the chiral magnet MnSi
under magnetic fields [6–8], wherein the skyrmion lattice
structure produces a six-fold neutron scattering pattern.
Since then, the chiral magnetic states encompassing SkX
and HL states have been the focus of extensive research
[9–13].
The dynamics of SkX in a random environment, specif-
ically the pinning effects from impurities, are manifested
through the topological Hall effect. The current depen-
dence of topological Hall resistivity ρxywas initially ex-
plored theoretically by Zang et al [14] and experimentally
∗nagaosa@riken.jpby Schulz et al. [15]. To illustrate, a schematic plot is
presented in Fig. 1(c). Typically, there are three distinct
regions characterizing the dynamics of SkX: the pinned,
creep, and flow regions. The topological Hall resistiv-
ity decreases when SkX is depinned because the motion
of SkX induces temporal changes in the emerging mag-
netic fields Be, subsequently generating emergent elec-
tric fields Eeand an opposing Hall contribution. Theo-
retically, the pinning problem of both SkX and HL was
investigated in terms of replica symmetry breaking [16],
revealing a distinct difference in glassy states between
SkX and HL. The key factor lies in the nontrivial topol-
ogy of SkX, contrasting with the trivial topology in HL
and most density wave states. However, this difference
has not been theoretically explored in the context of slid-
ing/moving density wave states for chiral magnets.
In this work, we systemically study the current-driven
sliding dynamics of the SkX and HL in chiral magnets.
We employ the methodology proposed by Sneddon et al.
[17] in their investigation of charge density waves and
apply it to magnetic materials. This method allows us
to investigate the current-driven dynamics of SkX and
HL, considering both deformation and impurity pinning
effects. Through this method, we reveal that the drift ve-
locity correction ∆ vddue to the impurity pinning effects
versus the current density jsin the flow region, follows
∆vd∝(vd0)d−2
2(−e∥+G
αDe⊥) for the SkX phase, while
∆vd∝ −(vd0)d−2
2e∥for the HL phase with the spatial di-
mension denoted as d. Here, e∥represents the direction
of the intrinsic drift velocity vd0(the magnitude of vd0is
proportional to the current density jsdue to the univer-
sal linear current-velocity relation [18]), G= 4πNsk,Dis
a form factor at the order of unity, α≪1 is the Gilbert
damping parameter so that G/αD ≫1. Although the
scaling relation ( vd0)d−2
2applies to both SkX and HL, we
can see that the gyrodynamics of the SkX state inducedarXiv:2312.07116v2 [cond-mat.mes-hall] 13 Dec 20232
(a)
(c)
vd
vs(b)
(d)
HL SkX
ρxy
vdCreep Flow pinned
js
FIG. 1. (a), (b) The current-driven motion of the SkX and
HL, respectively. (c) Schematic of the Hall resistivity ρxy
and drift velocity vdversus current density jswith pinned
(yellow), creeping (green), and flowing (purple) highlighted.
(d) The collective flow motion of the SkX, where the center
of each skyrmion (red dots) and the impurities (black crosses)
are highlighted.
by its nontrivial topology results in its sliding dynamics
more robust than in HL and mostly in the transverse di-
rection. Finally, we explicitly conduct the micromagnetic
simulations on both the SkX and HL systems, aligning
well with our theoretical expectations.
Our work demonstrates the unification of sliding dy-
namics between spin density waves and charge density
waves within the same theoretical framework. Our re-
sults also vividly illuminate both the similarities and dif-
ferences in the sliding dynamics between SkX and HL
phases. This insight significantly enhances our under-
standing of the sliding dynamics associated with topo-
logical density wave phenomena, which possesses pos-
sible applications in areas, such as skyrmion-based de-
vices [19–21], depinning dynamics [22–27], Hall responses
[14, 15, 28], and current-driven motion of Wigner crystals
under out-of-plane magnetic fields [29–31].
Sliding dynamics for skyrmion crystals.— The current-
driven motion of SkX is described by the Thiele equation
assuming that its shape does not change [18, 32, 33]:
G×(vs−vd) +D(βvs−αvd) +F= 0. (1)
Here, the first term on the left represents the Magnus
force, the second term is the dissipative force, and the
last term arises from deformation and impurity-pinning
effects. Here, vsis the velocity of conduction electrons,
αis the damping constant of the magnetic system, and β
describes the non-adiabatic effects of the spin-polarized
current. The gyromagnetic coupling vector is denoted as
G= (0,0,4πNsk), and the dissipation matrix Dij=δijD
where i, j∈ {x, y}. It is noteworthy that the Thieleequation respects out-of-plane rational symmetry [Sup-
plementary Material (SM) Sec. IA [34]].
To obtain the equation of motion of SkX, the displace-
ment vector field of skyrmions is defined as u(r, t) so
that the drift velocity vd=∂u(r,t)
∂t, where ris the posi-
tion vector, tis the time. The force Fcan be expressed
withu(r, t) asF(r, t) =Fimp+Fde, where the impurity
pinning force Fimp=−P
i∇U(r+u(r, t)−ri)ρ(r) =
fimp(r+u(r, t))ρ(r) and the deformation force Fde=R
dr′D(r−r′)u(r′, t′). Here, U(r−ri) is the impu-
rity potential around site ri,D(r−r′) characterizes the
restoration strength after deformation, and ρ(r) is the
skyrmion density. Based on these definitions, the Thiele
equation can be expressed as an equation of motion:
∂u(r, t)
∂t=ˆM0vs+ˆM1Z
dr′D(r−r′)u(r′, t)
+ˆM1fimp(r+u(r, t))ρ(r). (2)
where ˆM0=1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
and
ˆM1=1
G2+α2D2
αD G
−G αD
. Note that each skyrmion is
now considered as a center-of-mass particle, and these
skyrmions form a triangular lattice and move collec-
tively with scatterings from impurities, as illustrated in
Fig. 1(d).
The displacement vector can be expanded around the
uniform motion,
u(r, t) =vdt+˜u(r, t). (3)
Here,vdis the dominant uniform skyrmion motion veloc-
ity,˜u(r, t) characterizes a small non-uniform part. Using
the Green’s function approach to solve the differential
equation Eq. (2), ˜u(r, t) can be obtained as [17, 29, 34]
˜u(r, t) =Z
dr′Z
dt′G(r−r′, t−t′){vd0−vd
+M1fimp(r′+vdt′+˜u(r′, t′))ρ(r′)},(4)
where the intrinsic drift velocity vd0=ˆM0vs, the Fourier
component of the Green’s function Gis given by
G−1(k, ω) =−iω−ˆM1D(k). (5)
Here,D(k) arises from the Fourier transformation of de-
formation D(k) =R
dd(r)D(r)e−ik·r(the spatial dimen-
sion is denoted as d).
In the flow region, ˜u(r, t) in Eq. (4) can be solved
perturbatively. Up to the second order, ˜u(r, t)≈
˜u0(r, t) +˜u1(r, t) +˜u2(r, t), which, respectively, are ob-
tained by replacing the terms in the brackets of Eq. (4) as
M0vs−vd, M1fimp(r′+vdt′)ρ(r′), M1∇fimp(r′+vdt′)·
˜u1(r, t)ρ(r′). Based on this approximation and making
use of ⟨˜u(r,t)
∂t⟩= 0, the self-consistent equation for the3
velocity reads (for details see SM Sec. IB [34])
vd=vd0+X
gZddq
(2π)d|ρ(g)|2Λ(q)ˆM1×
q2
xqxqy
qxqyq2
y
Im [G(q−g,−q·vd)]ˆM1
qx
qy
.(6)
where ρ(g) is the Fourier component of ρ(r) with gas the
reciprocal skyrmion lattice vectors, and Λ( q) arises from
the impurity average U(q1)U(q2) = (2 π)dΛ(q2)δ(q1+
q2). Note that the crucial aspects for the above method
to be valid are (i) the impurity strength is weak, (ii) the
drift velocity is large compared to the impurity effects
and the SkX remains elastic, (iii) the deformation within
each skyrmion is negligible so that each skyrmion can be
regareded as a point object.
To proceed further, we adopt the following approxi-
mations. The current-driven distortion is expected to be
weak so that D(k) would be dominant by the long-wave
limit. In this case, D(k) can be expanded as Kxk2
x+Kyk2
y
for the 2D case and as Kxk2
x+Kyk2
y+Kzk2
zfor the 3D
case. On the other hand, the characterized frequency
that enters into the Green’s function is q·vd∼vd/a.
Using a reasonable parameter vd= 10 m/s, the skyrmion
lattice constant a= 25 nm, we estimate vd/a∼0.4 GHz.
This frequency is much smaller compared with the one of
Kj, which is roughly the scale of exchange energy J∼1
meV∼240 GHz [14, 18]. As a result, the dominant
contribution to the integral is given by the elastic modes
vdgj≈ωk≈ D(k)/√
G2+α2D2withk=q−g→0,
around which the imaginary part of Green’s function is
the largest.
With the above approximations, we perform the in-
tegral in Eq. (6) and sum over the smallest gvectors:
gj=√
3κ0(sin(j−1)π
3,cos(j−1)π
3) with jas integers from
1 to 6 and κ0=4π
3a. Since the Thiele equation exhibits
out-of-plane rotational symmetry, without loss of gen-
erality, we set vd0along x-direction here. After some
simplifications (for details see SM. Sec IB), we find the
correction (∆ vd=vd−vd0) on the drift velocity due to
the impurity and deformation are given by
∆vd≈χSkX
d(vd0)d−2
2(−e∥+G
αDe⊥), (7)
where the susceptibility to the impurity potential
χSkX
d =9κ3
0|ρ1|2Λ0αD
4√
KxKy(G2+α2D2)ford= 2, while χSkX
d =
9√
3κ7/2
0Γ(G2+α2D2
4α2D2)|ρ1|2Λ0(αD)3/2
π2√
KxKyKz(G2+α2D2)ford= 3 (the function
Γ(a) =R+∞
0dxx6
(x4−a)2+x4). Note that we have replaced
ρ(gj) =ρ1,Λ(gj) = Λ 0given the six-fold rotational sym-
metry of the skyrmion lattice.
The first important aspect in Eq. (7) is that the cor-
rection ∆ vdis insensitive to vd0in 2D limit but follows
a square-root scaling: ( vd0)1/2in 3D limit. Similar to
many scaling phenomena, the dimension plays a critical
0 0.005 0.01-1.5-1-0.50
0 0.005 0.0100.510 0.005 0.01-1.5-1-0.50
0 0.005 0.0100.511.5
0 0.005 0.01-1.5-1-0.50
Skyrmion 3DSkyrmion 2D, G < 0
i=i=||Helix 2D
Skyrmion 2D, G > 0(a) (b)
(c) (d)
(e) (f)-10 -8 -6 -4-2.5-2-1.5-1-0.50
Log(vd0/K)Log(|∆vd|) Slope ≈ 0.5
0°30°60°90°
120°
150°
180°
210°
240°
270°300°330°00.20.40.60.81
i=i=||
i=i=||
Helix 3DFIG. 2. (a) and (b) The correction ∆ vdversus vd0(in units
ofK) for the 2D and 3D HL state, respectively. (c) and (d)
The correction ∆ vdversus vd0(in units of K) for the 2D
SkX with G= 4πandG=−4π, respectively, where the
longitudinal (transverse) component is in blue (red). The 3D
SkX case is plotted in (f) with G= 4π. (e) The angular-
dependence of |∆vd(θ)|, where θis the angle of vd0. All the
of ∆vdin these plots has been normalized. The parameters
D= 5.577π,α= 0.04.
role here. The second important aspect is that the cor-
rection along the transverse direction directly reflects the
skyrmion topological number Gwith the ratio compared
to the longitudinal one as G/αD . These interesting as-
pects embedded in the Eq. (7) will be further highlighted
later.
Helix case.— It is straightforward to generalize the
above treatment to the helical spin order. The Thiele
equation is reduced to one dimension:
D(βvs−αvd) +F= 0. (8)
The essential difference here is the absence of gyrotropic
coupling ( G= 0). Following the same procedure [SM
Sec. II], the self-consistent equation for the drift velocity
is given by
vd=vd0+Zddq
(2π)dX
g|ρ(g)|2
α2D2Λ(q)q3
xIm[G(q−g,−qxvd)].
(9)
Here, vd0=β
αvs, the flow direction of the HL is defined
asx-direction. After adopting the approximation in the4
previous section, the analytical expression of the correc-
tion ∆ vdof the helical magnetic state is
∆vd≈ −χHL
d(vd0)d−2
2 (10)
where χHL
d =(KxKy)−1/2|ρ1|2Λ0g3
0
4αD,ford= 2
(KxKyKz)−1/2|ρ1|2Λ0g7/2
0
2√
2π(αD)1/2 ,ford= 3 with g0=π
a. Despite
different magnetic state nature, the ∆ vdas a function
vd0in Eq. (10) for the HL displays a consistent scaling
behavior as the one of SkX shown in Eq. (7).
Numerical evaluation.— To further justify our analyt-
ical results, we calculate the ∆ vdnumerically according
to Eqs. (6) and (9). For simplicity, we set the elastic
coefficient Kjas isotropic with K≡Kj. Figs. 2(a) and
(b) display the correction ∆ vdas a function of vd0of
HL. Note that the zero drify velocity limit should be ig-
nored since the impurity pinning effect would be dom-
inant in practice. When the vdis beyond this limit so
that the pinning effect can be treated as a perturbation,
which is true for the flow region, the plots clearly indi-
cate ∆ vd∝(vd0)d−2
2. The square root behavior in 3D
(d= 3) is explicitly checked with the log-log plot (inset
of Fig. 2(b)).
Figs. 2(c) and (d) show the longitudinal component
(blue) and transverse component (red) of the correction
for the SkX case with positive Gand negative G, respec-
tively. It is consistent with Eq. (7) that the transverse
component is odd with respect to Gand is much larger
than the longitudinal component as G/αD ≫1. This
gyrotropic type correction is inherited from the Magnus
forces in the Thiele equation, and this correction also
implies that there exists a net change on the skyrmion
Hall angle due to the impurities. Moreover, the angu-
lar dependence of the total correction |∆vd|is shown in
Figs. 2(e), where the anisotropy is very small. The ∆ vd
as a function vd0also displays distinct scaling behavior
between 2D [Figs. 2(c),(d)] and 3D case [Fig. 2(f)].
Overall, the scaling behavior of skyrmion similar to
that of the HL in Fig. 2, as expected from our theoretical
analysis. Moreover, the intrinsic drift velocity vd0is lin-
early proportional to the current density jsfor both SkX
and HL ( vd0∝js) at large js. As a result, we can replace
vd0with jsin the scaling relation, i.e., ∆ vd∝(js)d−2
2. It
is worth noting that the charge density wave also respects
this scaling relation [17], despite its distinct microscopic
nature.
Physical interpretation.— Now we provide a physical
interpretation of the observed scaling behavior: ∆ vd∝
(vd0)d−2
2. As we mentioned earlier, the dominant con-
tribution to the drift velocity correction arises from the
excitation of elastic modes. Hence, we expect the correc-
tion to be proportional to the number of excited elastic
modes at a fixed vd0. For the SkX case, these modes
follow the dispersion: vd0|gj|=D(k)/√
G2+α2D2,
which can be rewritten as vd0=kd/2m′with m′=
2π√
G2+α2D2/(√
3aK). Next, the problem is mapped
to evaluate the density of states of free fermions with
0.050.10.150.20.251234
0.10.20.30.40.52D HL2D SkX0102030051015202530vd,x clean
js (1010 A/m2)vd (m/s)2D SkX, weak impuritiesvd,x weak impurities vd,y defect vd,y weak impurities
0102030051015202530cleanweak impurities2D HL, weak impuritiesvd (m/s)
js (1010 A/m2)(a)(b)
(d)∆vd (m/s)α(c)0102030012345|Δvd/<Δv∥>||Δv∥/<Δv∥>||Δv⊥/<Δv∥>|G/αD⃗vd⃗y⃗xΔ⃗v⊥Δ⃗v∥⃗vd0finalColumn 6Column 5|Δvd|FIG. 3. Simulation results of LLG equation. (a) and (c)
The drift velocity vdversus current density js(in units of
1010A/m2) of the 2D SkX and HL at the clean and impurity
case. (b) The magnitdue of longitutinal (∆ d,∥) and trans-
verse (∆ d,⊥) drift velocity correction versus the current den-
sity (normalized the avegare value of ∆ d,∥), where the G/αD
ratio is highlighted (red dashed line). The coordinate relation
between different vectors are shown in the inset. (d) the drift
velocity correction as a function of the damping parameter α
atjs= 2×1011A/m2(only vd,xis used for the SkX). For (a)
to (c), α= 0.2 and β= 0.5αare employed in the simulations.
an energy vd0. Recall that the density of states of
free fermion N(E)∝Ed−2
2at energy E. Hence, it is
expected that the correction follows the same scaling:
∆vd∝(vd0)d−2
2according to this argument. We em-
phasize that the microscopic nature of the density waves
in this argument are not essential, which mainly stems
from the long-wave characteristic of elastic modes. This
explains why the HL and charge density wave also follow
the same scaling behavior.
Micromagnetic simulation.— We now further validate
our theory through solving the Landau–Lifshitz–Gilbert
(LLG) equation with the spin transfer torque effect [35–
39] (for details see SM). The calculated drift velocity vd
versus current density jscurves are shown in Fig. 3 for
both the SkX and HL. For simplicity, we mainly focus
on 2D SkX and HL with weak impurities here, where
our analytical expressions from perturbation theory are
applicable.
Figs. 3(a) and (c) show vdin the clean and disordered
case with α= 0.2. The correction between these two
cases at both SkX and HL is indeed insensitive to the
current density within the flow limit. It is noteworthy
that due to the gyrodynamcs, the SkX exhibits a much
smaller depinning critical current density. Fig. 3(b) is to
show that the correction along the transverse direction is
obviously larger than the longitudinal one with the ratio5
∼G/αD , being consistent with Eq. (7). Interestingly,
the longitudinal correction versus the damping parame-
terαof 2D SkX and HL show a positive and negative
correlation, respectively [Fig. 3(d)], which is also consis-
tent with our analytical expressions [see χSkX
dandχHL
d
in Eqs. (7) and (10)]. It can also be seen that the im-
purity correction along longitudinal direction is typically
much stronger in HL than in SkX as χSkX
d/χHL
d∼α2.
These distinct features between SkX and HL highlight
the importance of the nontrivial topology in the sliding
dynamics of density waves.
Discussion.— We have provided a thorough analy-
sis of the sliding dynamics exhibited by the SkX and
HL phases, highlighting both their similarities and dif-
ferences in terms of density waves sliding with distinct
topologies. Our theory could have broader applica-
tions. For instance, one can explore the relationship be-
tween the topological Hall effect and the current den-
sity in the flow region. In the clean limit, the uni-
versal linear current-velocity relation vd0∝jsimplies
that the topological Hall resistivity ρxy, proportional to
|(vs−vd0)×Be|/|vs|[15], is expected to exhibit a plateau
in the flow region, as illustrated in Fig. 1(c). In thepresence of impurities, the topological Hall resistivity is
modified to ρxy∝ |(vs−vd0−∆vd)×Be|/|vs|. Consid-
ering that ∆ vd∝(vd0)(d−2)/2, we anticipate a modified
relation ρxy=a+bj−2+d/2
s , where aandbremain inde-
pendent of the current magnitude js. The second term,
bj−2+d/2
s , represents the correction from impurities. Con-
sequently, we expect that the ρxy-jsplateau in the flow
region will gradually diminish with increasing disorder.
Our theory can also be applied to investigate the slid-
ing dynamics of a 2D Wigner crystal under out-of-plane
magnetic fields [29–31]. The crucial distinction lies in the
fact that the Lorentz force is typically much smaller than
the damping force, whereas in the SkX, the Magnus force
dominates over the damping force. In essence, the SkX
represents an extreme limit of gyrodynamics.
Acknowledgment.— We thank Max Birch and Yoshi-
nori Tokura for presenting us their Hall measurement
data on the SkX, which motivated this study. N.N.
was supported by JSTCREST Grants No.JMPJCR1874.
Y.M.X. and Y.L. acknowledge financial support from
the RIKEN Special Postdoctoral Researcher(SPDR) Pro-
gram.
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Supplementary Material for “ Sliding Dynamics of Current-Driven Skyrmion Crystal
and Helix in Chiral Magnets ”
Ying-Ming Xie,1Yizhou Liu,1Nato Nagaosa,1
1RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
I. THE SLIDING DYNAMICS OF SKYRMION CRYSTAL WITH PINING AND DEFORMATION
EFFECTS
A. The skyrmion dynamics and Thiele equation
From the Landau-Lifshitz-Gilbert equation, it was obtained that the current-driven skyrmion dynamics are captured
by the Thiele equation:
G×(vs−vd) +D(βvs−αvd) +F= 0. (S1)
One can rewrite the equation as
−G(vsy−vdy) +D(βvsx−αvdx) +Fx= 0, (S2)
G(vsx−vdx) +D(βvsy−αvdy) +Fy= 0. (S3)
In the matrix form:
G αD
αD−G
vdx
vdy
=
G βD
βD−G
vsx
vsy
+
Fy
Fx
. (S4)
Then,
vdx
vdy
=
G αD
αD−G−1
G βD
βD−G
vsx
vsy
+
G αD
αD−G−1
Fy
Fx
. (S5)
vdx
vdy
=1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
+
G αD
αD−G
0 1
1 0
Fx
Fy
,
=1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
+
αD G
−G αD
Fx
Fy
(S6)
Without loss of generality, we can choose the current direction to be x-direction: vs= (vs,0). When the pinning
force is set to be F= 0, one can solve
vd∥,0=G2+αβD2
G2+α2D2vs,vd⊥,0=(α−β)GD
G2+α2D2ˆz×vs. (S7)
Therefore, the longitudinal drift velocity vdxis proportional to the electric current when the force Fis neglectable.
In a general direction, we can write the intrinsic drift velocity as
vd0=1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
(S8)
=s
1 +β2γ2
1 +α2γ2
cosθSkH−sinθSkH
sinθSkH cosθSkH
vscosθs
vssinθs
(S9)
=vd0
cos(θs+θSkH)
sin(θs+θSkH)
. (S10)
Here, the angle θsis to characterize the applied current direction, the skyrmion Hall angle θSkH= atanγ(α−β)
1+αβγ2with
γ=D
G, and the magnitude of drift velocity vd0=|vd0|=vsq
1+β2γ2
1+α2γ2.2
Now we show that the Thiele equation respects rotational symmetry with the principal axis along z-direction from
Eq. (S6). The rotational operator is defined as Rz=
cosϕ−sin(ϕ)
sin(ϕ) cos( ϕ)
with ϕas the rotational angle. Under this
rotational operation, Eq. (S6) becomes
Rz
vdx
vdy
=1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
+
G αD
αD−G
0 1
1 0
Fx
Fy
,
=1
G2+α2D2
Rz
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
R−1
zRz
vsx
vsy
+Rz
αD G
−G αD
R−1
zRz
Fx
Fy
(S11)
It is easy to show
Rz
A B
−B A
R−1
z=
A B
−B A
(S12)
with AandBas constant. The Eq. (S13) is simplified as
Rz
vdx
vdy
=1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
+
G αD
αD−G
0 1
1 0
Fx
Fy
,
=1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
Rz
vsx
vsy
+
αD G
−G αD
Rz
Fx
Fy
(S13)
Hence, we have shown that the Thiele equation respects out-of-plane rotational symmetry.
B. The correction on the drifted velocity due to the pining and deformation effects
Let us define the displacement of skyrmion lattice as u(r, t) so that the drift velocity vd(r, t) =∂u(r,t)
∂t. The force
is given by
F(r, t) = Fimp+Fde (S14)
Fimp =−X
i∇U(r+u(r, t)−ri)ρ(r) =fimp(r+u(r, t))ρ(r) (S15)
Fde=Z
dr′D(r−r′)u(r′, t′), (S16)
where Fimdescribes the pining effect from impurities and Fdearises from the deformation of skyrmion lattice, U(r−ri)
is the impurity potential around the site ri.ρ(r) is the skyrmion densities.
∂u(r, t)
∂t=1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
+
αD G
−G αDZ
dr′D(r−r′)
ux(r′, t)
uy(r′, t)
+1
G2+α2D2
αD G
−G αD
fimp(r+u(r, t))ρ(r). (S17)
The displacement vector can be expanded around the uniform motion,
u(r, t) =vdt+˜u(r, t). (S18)
Here, vdis the dominant uniform skyrmion motion velocity, ˜u(r, t) characterizes a small non-uniform motion. Then
the equation of motion is written as
∂
∂t−1
G2+α2D2
αD G
−G αDZ
dr′D(r−r′)
˜ux(r′, t)
˜uy(r′, t)
=1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
−vd
+1
G2+α2D2
αD G
−G αD
fimp(r+vdt+˜u(r, t))ρ(r).
(S19)3
Here, we have used D(q) = 0 in the long wave limit ( q→0) so thatR
dr′D(r−r′) = 0.
Let us try to solve the Green’s function of the operator at the left-hand side, which is given by
∂
∂t−1
G2+α2D2
αD G
−G αDZ
dr′D(r−r′)
G(r′, t) =δ(t)δ(r)
1 0
0 1
(S20)
It is more economical to work in the momentum space with
G(r, t) =Zdω
2πZddk
(2π)de−iωt+ik·rG(k, ω). (S21)
Let us define D(r) =Rddq
(2π)deiq·rD(q), and then
Z
dr′Z
D(r−r′)G(r′, t) =Z
dr′Zddq
(2π)deiq·(r−r′)D(q)Zddk
(2π)dZdω
2πG(k, ω)ei(k·r′−ωt)
=Zddk
(2π)dZdω
(2π)G(k, ω)D(k)ei(k·r−ωt). (S22)
In the momentum space, we find
G−1(k, ω) =−iω−1
G2+ (Dα)2
αD G
−G αD
D(k) (S23)
Therefore, Eq. (S19) can be rewritten as
˜u(r, t) =Z
dr′Z
dt′G(r−r′, t−t′){1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
−vd (S24)
+1
G2+α2D2
αD G
−G αD
fimp(r′+vdt′+˜u(r′, t′))ρ(r′)}.
In the flow limit, the perturbation from the deformation and impurity can be regarded as small in comparison with
the leading order term. As a result, the displacement vector can be expanded as
˜u0(r, t) =Z
dr′Z
dt′G(r−r′, t−t′)1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
−vd
, (S25)
˜u1(r, t) =1
G2+α2D2Z
dr′Z
dt′G(r−r′, t−t′)
αD G
−G αD
fimp(r′+vdt′)ρ(r′), (S26)
˜u2(r, t) =1
G2+α2D2Z
dr′Z
dt′G(r−r′, t−t′)
αD G
−G αD
∇fimp(r′+vdt′)·˜u1(r′, t′)ρ(r′).(S27)
Here, u0,u1, and u2are the leading, first, and second order terms, respectively. Next, let us evaluate the volume-
average velocity
∂˜u(r, t)
∂t
= ∂˜u0(r, t)
∂t
+ ∂˜u2(r, t)
∂t
(S28)
Note the fact that under the impurity average fimp(r′+vt′) = 0 has been used so that u1(r, t) would not contribute
directly. Since non-uniform motion must vanish over the volume average, we can obtain a self-consistent equation for
the velocity vd. Next, let us work out the self-consistent equation for vd.
The leading order
∂u0(r, t)
∂t=Z
dr′Z
dt′G(r−r′, t−t′)
∂t1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
−vd
=Z
dr′Z
dt′Zddk
(2π)dZdω
2πeik·(r−r′)−iω(t−t′)(−iω)G(k, ω)1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
−vd
= lim
ω→0−iωG(k= 0, ω)1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
−vd
=1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
−vd
(S29)4
As mentioned theD
∂˜u1(r,t)
∂tE
would not contribute, now let us show it explicitly. Recall that
˜u1(r, t) =1
G2+α2D2Z
dr′Z
dt′G(r−r′, t−t′)
αD G
−G αD
fimp(r′+vdt′)ρ(r′) (S30)
Then,
∂˜u1(r, t)
∂t=Z
dr′Z
dt′ 1
G2+α2D2G(r−r′, t−t′)
αD G
−G αD
fimp(r′+vdt′)ρ(r′) (S31)
=−1
G2+α2D2Z
dr′Z
dt′Zddk
(2π)dZdω
2πeik·(r−r′)−iω(t−t′)G(k, ω)
αD G
−G αD
× (S32)
Zddq
(2π)d
iqx
iqy
Uqeiq·(r′+vdt′)ρ(r′) (S33)
= 0, (S34)
because after averaging over the impurity configurations, Uk= 0.
Now let us look at the second-order term
∂˜u2(r, t)
∂t=1
G2+α2D2Z
dr′Z
dt′∂G(r−r′, t−t′)
∂t
αD G
−G αD
∇fimp(r′+vdt′)·˜u1(r′, t′)ρ(r′).(S35)
Note that
∇fimp(r′+vdt′)·˜u1(r′, t′) =−Zddq
(2π)dU(q)eiq·(r′+vdt′)q2
xqxqy
qxqyq2
y
˜u1x(r′, t′)
˜u1y(r′, t′)
(S36)
Substitute the form of ˜u1(r′, t),
∂˜u2(r, t)
∂t=−1
(G2+α2D2)2Z
dr′Z
dt′∂G(r−r′, t−t′)
∂t
αD G
−G αD
ρ(r′)×
Zddq1
(2π)dU(q1)eiq1(·r′+vdt′)q2
1xq1xq1y
q1xq1yq2
1y
×
Z
dt′′Z
dr′′G(r′−r′′, t′−t′′)
αD G
−G αD
ρ(r′′)Zddq2
(2π)d
iq2x
iq2y
U(q2)eiq2·(r′′+vdt′′)(S37)
Then write the terms at the right-hand side of the equation with their Fourier components,
∂˜u2(r, t)
∂t=−1
(G2+α2D2)2Z
dt′Z
dr′Zddk
(2π)dZdω
2πG(k, ω)(−iω)eik·(r−r′)−iω(t−t′)
αD G
−G αD
×
Zddq1
(2π)dq2
1xq1xq1y
q1xq1yq2
1y
U(q1)eiq1·(r′+vdt′)X
g1ρ(g1)eig1·r′×
Z
dt′′Z
dr′′Zddk′
(2π)dZdω′
2πG(k′, ω′)eik′·(r′−r′′)−iω′(t′−t′′)
αD G
−G αDX
g2ρ(g2)eig2·r′′×
Zddq2
(2π)d
iq2x
iq2y
U(q2)eiq2·(r′′+vdt′′)(S38)
We can take integrals with respect to the space and time, and take the average over disorders, several delta functions5
would appear on the right-hand side:
U(q1)U(q2) = (2 π)dΛ(q2)δ(q1+q2), (S39)Z
dt′eiωt′eiq1·vdt′e−iω′t′= 2πδ(ω−ω′+q1·vd), (S40)
Z
dr′e−ik·r′eiq1·r′eig1·r′eik′·r′= (2π)dδ(k′−k+q1+g1), (S41)
Z
dt′′eiω′t′′eiq2·vdt′′= 2πδ(ω′+q2·vd), (S42)
Z
dr′′e−ik′·r′′eig2·r′′eiq2·r′′= (2π)dδ(g2+q2−k′). (S43)
Take the volume average, and consider constraints from delta functions: q2=−q1=q,g1=−g2=g,ω′=−q·vd,
we find
∂˜u2(r, t)
∂t
=1
(G2+α2D2)2X
gZddq
(2π)d|ρ(g)|2Λ(q)
αD G
−G αDq2
xqxqy
qxqyq2
y
×
Im [G(q−g,−q·vd)]
αD G
−G αD
qx
qy
. (S44)
Therefore,
∂˜u(r, t)
∂t
=1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
−vd
+
1
(G2+α2D2)2X
gZddq
(2π)d|ρ(g)|2Λ(q)
G αD
αD−Gq2
xqxqy
qxqyq2
y
×
Im [G(q−g,−q·vd)]
G αD
αD−G
qx
qy
(S45)
SetD
∂˜u(r,t)
∂tE
= 0, the self-consistent equation for the velocity is
vd=1
G2+α2D2
G2+αβD2GD(β−α)
GD(α−β)G2+αβD2
vsx
vsy
+
1
(G2+α2D2)2X
gZddq
(2π)d|ρ(g)|2Λ(q)
αD G
−G αDq2
xqxqy
qxqyq2
y
×
Im [G(q−g,−q·vd)]
αD G
−G αD
qx
qy
. (S46)
As argued in the main text, the largest imaginary part is contributed by k=q−gin long wave limit ( kis small).
To further proceed, let us evaluate Im[ G(k, ω)].
G(k, ω) =1
−iω−1
G2+(Dα)2
αD G
−G αD
D(k)=−G2+α2D2
λ(k)iω−D(k)
λ(k)
αD−G
G αD
, (S47)
where
λ(k) = [D(k) +ω(iαD+G)][D(k) +ω(iαD−G)] =D(k)2−(G2+α2D2)ω2+ 2iαDωD(k). (S48)
The imaginary part of Green’s function is given by
Im[G(k, ω)] = −(G2+α2D2)[D2(k)−(G2+α2D2)ω2]ω
[D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k)
+2αDωD2(k)
[D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k)
αD−G
G αD
. (S49)6
The largest imaginary part is given by the real mode ωk=D(k)/√
G2+α2D2. As a result, the first term in Im[ G(k, ω)]
can be negligible. In 2D, we can expand
D(k) =Kxk2
x+Kyk2
y. (S50)
Now we can show that
Zd2k
(2π)22αDωD2(k)
[D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k)
= (KxKy)−1/2Z+∞
0dk′2πk′
(2π)22αDωk′4
(k′4−(G2+α2D2)ω2)2+ 4α2D2ω2k′4
=(KxKy)−1/2sgn(ω)
4. (S51)
where the integralR+∞
0dtt2
(t2−√
G2+α2D2
2αD)2+t2=π
2is used with t=k′2
2αD|ω|. Note that αD̸= 0 is taken. The
multiplications between matrices give
αD G
−G αDq2
xqxqy
qxqyq2
y
αD−G
G αD
αD G
−G αD
qx
qy
= (G2+α2D2)(q2
x+q2
y)
Gqy+αDq x
−Gqx+αDq y
.(S52)
Finally, we obtain
δvd=(KxKy)−1/2
4(G2+α2D2)X
g|ρ(g)|2Λ(g)sgn(−g·vd0)|g|2
Ggy+αDg x
−Ggx+αDg y
(S53)
We have shown that δvdrespects out-of-plane rotational symmetry in the main text, which is inherited from the
Theiele equation. Without loss of generality, let us set vd0to be along x-direction. In this case, after summing over
the six smallest gvectors: gj=√
3κ0(sin(j−1)π
3,cos(j−1)π
3) with κ0=4π
3a,jare integers from 1 to 6, we find
δvd=9κ3
0|ρ1|2Λ0αD
4p
KxKy(G2+α2D2)−1
G
αD
(S54)
where ρ1=ρ(gj),Λ0= Λ(gj). In the 3D case,
D(k) =Kxk2
x+Kyk2
y+Kzk2
z. (S55)
Then,
Zd3k
(2π)32αDωD2(k)
[D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k)
= (KxKyKz)−1/2Z+∞
04πk′2dk′
(2π)32αDωk′4
(k′4−(G2+α2D2)ω2)2+ 4α2D2ω2k′4
=(KxKyKz)−1/2Γ(G2+α2D2
4α2D2)sgn( ω)p
2αD|ω|
2π2. (S56)
where Γ( a) =R+∞
0dxx6
(x4−a)2+x4. Similarly, we can obtain
δvd=(2αD)1/2(KxKyKz)−1/2Γ(G2+α2D2
4α2D2)
2π2(G2+α2D2)X
g|ρ(g)|2Λ(g)sgn(−g·vd0)p
|g·vd0||g|2
Ggy+αDg x
−Ggx+αDg y
.(S57)
After summing over the six smallest gvectors, we find
δvd=9√
3κ7/2
0Γ(G2+α2D2
4α2D2)|ρ1|2Λ0αD√αDv d0
π2p
KxKyKz(G2+α2D2)−1
G
αD
. (S58)7
II. HELICAL SPIN ORDER CASE
In this section, we consider the helical spin order case. Without loss of generality, we denote the helical spin order
has a variation along x-direction. The Thiele equation for the helical spin order would be
D(βvs−αvd) +F= 0. (S59)
For simplicity, we have omitted the index xin the following. The equation of motion becomes
∂u(r, t)
∂t=β
αvs+1
αDZ
dr′D(r−r′)u(r′, t′) +1
αDfimp(r+u(r, t))ρ(r). (S60)
Similarly, by defining u(r, r) =vdt+ ˜u(r, t), the equation of motion can be rewritten as
[∂
∂t−1
αDZ
dr′D(r−r′)]˜u(r′, t) =β
αvs+1
αDfimp(r+u(r, t))ρ(r). (S61)
Let us define the Green’s function G(r, t) so that
[∂
∂t−1
αDZ
dr′D(r−r′)]˜u(r′, t)G(r′, t) =δ(r)δ(t). (S62)
It is easy to obtain
G−1(k, ω) =−iω−D(k)
αD. (S63)
The displacement ˜ u(r, t) is given by
˜u(r, t) =Z
dr′Z
dt′G(r−r′, t−t′)[β
αvs−vd+1
αDfimp(r′+u(r′, t′))ρ(r′)]. (S64)
Then up to the second order,
˜u0(r, t) =Z
dr′dt′G(r−r′, t−t′)[β
αvs−vd], (S65)
˜u1(r, t) =1
αDZ
dr′Z
dt′G(r−r′, t−t′)fimp(r′+vdt′)ρ(r′), (S66)
˜u2(r, t) =1
αDZ
dr′Z
dt′G(r−r′, t−t′)∂xfimp(r′+vdt′)˜u1(r′, t′)ρ(r′). (S67)
Following a similar procedure in Sec. II, using ⟨∂u(r,t)
∂t⟩= 0, we obtain
vd=vd0+1
α2D2X
g|ρ(g)|2Zddq
(2π)dΛ(q)q3
xIm[G(q−g,−qxvd)]. (S68)
where the intrinsic drift velocity vd0=β
αvs. Consider the Im[ G(k, ω)] is dominant in the long wave limit ( k→0) and
expand D(k) =Kxk2
x+Kyk2
yind= 2, we find
vd≈vd0−(KxKy)−1/2
8αDX
g|ρ(g)|2Λ(g)|gx|3(S69)
and similarly in d= 3 case, D(k) =Kxk2
x+Kyk2
y+Kzk2
z, we obtain
vd≈vd0−(KxKyKz)−1/2
4√
2π(αD)1/2X
g|ρ(g)|2Λ(g)|gx|3p
|gxvd0|. (S70)
After summing over the smallest reciprocal lattice vectors for the helix: g= (±1,0)g0with g0=π
a, the correction
∆vd=β
αvs−χHL
d(vd0)d−2
2 (S71)
where
χHL
d=
(KxKy)−1/2|ρ1|2Λ0g3
0
4αD, ford= 2
(KxKyKz)−1/2|ρ1|2Λ0g7/2
0
2√
2π(αD)1/2 ,ford= 3(S72)
Here, we have set ρ(g0) =ρ1and Λ( g0) = Λ 0in this case.8
III. NUMERICAL METHOD DETAILS
A. Details for the main text Fig.2
The main text Fig.2 is obtained from the main text Eq. (6) and (9) by numerically integrating qand summing over
the smallest reciprocal lattice gvectors. For the SkX, the 4000 ×4000 in-plane momentum grids of qare taken with
a hexagonal boundary (the boundary length is 4 κ0); while for the HL, the 1000 ×1000 in-plane momentum grids are
taken with a square boundary (the boundary length is 4 g0). In the 3D case, the 1000 out-of-plane momentum points
ofqwithin [ −2g0,2g0] are used for both SkX and HL in evaluating the integral. Also, we set the elastic coefficients
Kj= 10000, the lattice constant aas a natural unit of one, the damping parameter α= 0.04, the dissipative coefficient
D= 5.577π, the additional parameter G=±4πfor the SkX.
B. Micromagnetic simulations
The micromagnetic simulations were performed using MuMax3 [35]. The Landau-Lifshitz-Gilbert (LLG) equation
is numerically solved
˙s=−|γ0|s×Heff+αs×˙s+pa3
2eS(js· ∇)s−pa3β
2eM2ss×(js· ∇)s, (S73)
where sis the unit vector of spin, γ0is the gyromagnetic constant, αis the Gilbert damping constant, Msis the
saturation magnetization, and Heff=−1
µ0MsδH
δSis the effective field. The spin transfer torque effect of the current
is described by the last two terms [36–38]. βdescribes the non-adiabaticity of the spin transfer torque effect. The
current is applied along the x-direction in the simulations.
A typical chiral magnet can be described by the following Hamiltonian density
H=A(∇S)2−DS·(∇ ×S)−µ0MsB·S (S74)
The corresponding parameters and their values employed in the simulations are: the saturation magnetization
Ms= 111 kA/m, the exchange stiffness A= 3.645 pJ/m, and the Dzyaloshinskii-Moriya interaction strength
D= 0.567 mJ/m2. An external magnetic field B= 0.3 T (with its direction perpendicular to the skyrmion plane) is
used in the simulations to stabilize the skyrmions. The simulations for helical state are performed at zero-field. The
cell size is 1 nm ×1 nm×1 nm.
We consider magnetic impurities with uniaxial magnetic anisotropy Himp=−KimpS2
z, where the easy-axis is
perpendicular to the skyrmion 2D plane. For the weak impurity case, an impurity concentration x= 0.1% and
impurity strength Kimp= 0.2A/l2(lis the cell size) were used. For the strong impurity case, an impurity concentration
x= 0.5% and impurity strength Kimp= 0.6A/l2were used. The simulation results are averaged over 100 impurity
distributions. The skyrmion velocity is extracted by using the emergent electric field method [39]. For each current
density, the emergent electric field is also averaged over 100 time steps in order to get the skyrmion velocity. For the
transverse correction, the damping value α= 0.2 is employed in the main text Figs. 3(a) and (b) for computational
efficiency, as using smaller damping values results in significantly longer simulation times to obtain a reasonable
correction along the transverse direction. |
1308.3787v1.Thickness_and_power_dependence_of_the_spin_pumping_effect_in_Y3Fe5O12_Pt_heterostructures_measured_by_the_inverse_spin_Hall_effect.pdf | arXiv:1308.3787v1 [cond-mat.mes-hall] 17 Aug 2013Thickness and power dependence of the spin-pumping effect in Y3Fe5O12/Pt
heterostructures measured by the inverse spin Hall effect
M. B. Jungfleisch,1,∗A. V. Chumak,1A. Kehlberger,2V. Lauer,1
D. H. Kim,3M. C. Onbasli,3C. A. Ross,3M. Kl¨ aui,2and B. Hillebrands1
1Fachbereich Physik and Landesforschungszentrum OPTIMAS,
Technische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany
2Institute of Physics, Johannes Gutenberg-University Main z, 55099 Mainz, Germany
3Department of Materials Science and Engineering, MIT, Camb ridge, MA 02139, USA
(Dated: September 17, 2018)
The dependence of the spin-pumpingeffect on the yttrium iron garnet (Y 3Fe5O12, YIG) thickness
detected by the inverse spin Hall effect (ISHE)has been inves tigated quantitatively. Due to the spin-
pumping effect driven by the magnetization precession in the ferrimagnetic insulator Y 3Fe5O12film
a spin-polarized electron current is injected into the Pt la yer. This spin current is transformed into
electrical charge current by means of the ISHE. An increase o f the ISHE-voltage with increasing film
thickness is observed and compared to the theoretically exp ected behavior. The effective damping
parameter of the YIG/Pt samples is found to be enhanced with d ecreasing Y 3Fe5O12film thickness.
The investigated samples exhibit a spin mixing conductance ofg↑↓
eff= (7.43±0.36)×1018m−2and
a spin Hall angle of θISHE= 0.009±0.0008. Furthermore, the influence of nonlinear effects on the
generated voltage and on the Gilbert damping parameter at hi gh excitation powers are revealed. It
is shown that for small YIG film thicknesses a broadening of th e linewidth due to nonlinear effects at
highexcitation powers is suppressedbecause of alack ofnon linear multi-magnon scatteringchannels.
We have found that the variation of the spin-pumping efficienc y for thick YIG samples exhibiting
pronounced nonlinear effects is much smaller than the nonlin ear enhancement of the damping.
I. INTRODUCTION
The generation and detection of spin currents have at-
tracted much attention in the field of spintronics.1,2An
effective method for detecting magnonic spin currents is
the combination of spin pumping and the inverse spin
Hall effect (ISHE). Spin pumping refers to the genera-
tion of spin-polarized electron currents in a normal metal
from the magnetization precession in an attached mag-
netic material.3,4These spin-polarized electron currents
are transformed into conventional charge currents by the
ISHE, which allows for a convenient electric detection of
spin-wave spin currents.5–7
After the discovery of the spin-pumping effect in fer-
rimagnetic insulator (yttrium iron garnet, Y 3Fe5O12,
YIG)/non-magnetic metal (platinum, Pt) heterosystems
by Kajiwara et al.7, there was rapidly emerging inter-
est in the investigation of these structures.6–13Since
Y3Fe5O12is an insulator with a bandgap of 2.85 eV14no
direct injection of a spin-polarized electron current into
the Pt layer is possible. Thus, spin pumping in YIG/Pt
structures can only be realized by exchange interaction
between conduction electrons in the Pt layer and local-
ized electrons in the YIG film.
Spin pumping into the Pt layer transfers spin angular
momentum from the YIG film thus reducing the mag-
netization in the YIG. This angular momentum transfer
results in turn in an enhancement of the Gilbert damp-
ing of the magnetization precession. The magnitude of
the transfer of angular momentum is independent of the
ferromagnetic film thickness since spin pumping is an in-
terface effect. However, with decreasing film thickness,
the ratio between surface to volume increases and, thus,the interface character of the spin-pumping effect comes
into play: the deprivation of spin angular momentum be-
comes notable with respect to the precession ofthe entire
magnetizationinthe ferromagneticlayer. Thus, theaver-
age damping for the whole film increases with decreasing
film thicknesses. It is predicted theoretically3and shown
experimentallyinferromagneticmetal/normalmetalhet-
erostructures (Ni 81Fe19/Pt) that the damping enhance-
ment due to spin pumping is inversely proportional to
the thickness of the ferromagnet.15,16
Since the direct injection of electrons from the insula-
tor YIG into the Pt layer is not possible and spin pump-
ing is an interface effect, an optimal interface quality is
required in order to obtain a high spin- to charge cur-
rent conversion efficiency.17,18Furthermore, Tashiro et
al. have experimentally demonstrated that the spin mix-
ing conductance is independent of the YIG thickness in
YIG/Pt structures.11Recently, Castel et al. reported
on the YIG thickness and frequency dependence of the
spin-pumpingprocess.19Incontrasttoourinvestigations,
they concentrate on rather thick ( >200 nm) YIG films,
which are much thicker than the exchange correlation
length in YIG20–22and thicker than the Pt thickness.
Thus, the YIG film thickness dependence in the nanome-
ter range is still not addressed till now.
In this paper, we report systematic measurements of
the spin- to charge-current conversion in YIG/Pt struc-
tures as a function of the YIG film thickness from 20 nm
to 300 nm. The Pt thickness is kept constant at 8.5 nm
for all samples. We determine the effective damping as
well as the ISHE-voltage as a function of YIG thickness
and find that the thickness plays a key role. From these
characteristics the spin mixing conductance and the spin2
FIG. 1. (Color online) (a) Schematic illustration of the ex-
perimental setup. (b) Dimensions of the structured Pt layer
on the YIG films. The Pt layer was patterned by means of
optical lithography and ion etching. (c) Scheme of combined
spin-pumping process and inverse spin Hall effect.
Hall angle are estimated. The second part of this paper
addresses microwave power dependent measurements of
the ISHE-induced voltage UISHEand the ferromagnetic
resonance linewidth for varying YIG film thicknesses.
The occurrence of nonlinear magnon-magnon scattering
processesonthe widening ofthe linewidth aswell astheir
influence on the spin-pumping efficiency are discussed.
II. SAMPLE FABRICATION AND
EXPERIMENTAL DETAILS
In Fig.1(a) a schematic illustration of the investigated
samples is shown. Mono-crystalline Y 3Fe5O12samples of
20, 70, 130, 200 and 300 nm thickness were deposited by
means of pulsed laser deposition (PLD) from a stoichio-
metric target using a KrF excimer laser with a fluence of
2.6J/cm2andarepetition rateof10Hz.23In orderto en-
sure epitaxial growth of the films, single crystalline sub-
TABLE I. Variation of saturation magnetization MSand
Gilbert damping parameter α0as a function of the YIG film
thickness. Results are obtained using a VNA-FMR measure-
ment technique.
dYIG(nm)MS(kA/m) α0(×10−3)
20 161.7 ±0.2 2.169 ±0.069
70 176.4 ±0.1 0.489 ±0.007
130 175.1 ±0.2 0.430 ±0.015
200 176.4 ±0.1 0.162 ±0.008
300 176.5 ±0.1 0.093 ±0.007FIG. 2. Original Gilbert damping parameter α0measured by
VNA-FMR technique. The increased damping at low sample
thicknesses is explained by an enhanced ratio between surfa ce
to volume, which results in an increased number of scatterin g
centers and, thus, in an increased damping. The inset shows
the saturation magnetization as a function of the YIG film
thickness dYIG. The error bars are not visible in this scale.
stratesofgadoliniumgalliumgarnet(Gd 3Ga5O12, GGG)
in the (100) orientation were used. We achieved opti-
mal deposition conditions for a substrate temperature
of 650◦C±30◦C and an oxygen pressure of 6.67 ×10−3
mbar. Afterwards, each film was annealed ex-situ at
820◦C±30◦C by rapid thermal annealing for 300 s un-
der a steady flow of oxygen. This improves the crystal-
lographic order and reduces oxygen vacancies. We deter-
mined the YIG thickness by profilometer measurements
andthecrystallinequalitywascontrolledbyx-raydiffrac-
tion(XRD). InordertodepositPtontothesamples,they
were transferred at atmosphere leading to possible sur-
face adsorbates. Therefore, the YIG film surfaces were
cleaned in-situ by a low power ion etching before the
Pt deposition.17We used DC sputtering under an ar-
gon pressure of 1 ×10−2mbar at room temperature to
deposit the Pt layers. XRR measurements yielded a Pt
thickness of 8.5 nm, which is identical for every sample
due to the simultaneously performed Pt deposition. The
Pt layer was patterned by means of optical lithography
and ion etching. In order to isolate the Pt stripes from
the antenna we deposited a 300 nm thick square of SU-8
photoresist on the top. A sketch of the samples and the
experimental setup is shown in Fig. 1(a), the dimensions
of the structured Pt stripe are depicted in Fig. 1(b).
In order to corroborate the quality of the fabricated
YIG samples, we performed ferromagnetic resonance
(FMR) measurements using a vector network analyzer
(VNA).25Since the area deposited by Pt is small com-
pared to the entire sample size, we measure the damp-
ingα0of the bare YIG by VNA (this approach results
in a small overestimate of α0), whereas in the spin-
pumping measurement we detect the enhanced damp-
ingαeffof the Pt covered YIG films. The VNA-FMR
results are summarized in Tab. Iand in Fig. 2. Appar-
ently, the 20 nm sample features the largest damping3
ofα20nm
0= (2.169±0.069)×10−3. With increasing film
thickness α0decreasesto α300nm
0= (0.093±0.007)×10−3.
There might be two reasons for the observed behavior:
(1) The quality of the thinner YIG films might be worse
due to the fabrication process by PLD. (2) For smaller
YIG film thicknesses, the ratio between surface to vol-
ume increases. Thus, the two-magnon scattering pro-
cess at the interface is more pronounced for smaller film
thicknesses and gives rise to additional damping.26The
VNA-FMR technique yields the saturation magnetiza-
tionMSfor the YIG samples (see inset in Fig. 2and
Tab.I). The observed values for MSare larger than the
bulkvalue,27,28butinagreementwiththevaluesreported
for thin films.29The general trend of the film thickness
dependence of MSis in agreement with the one reported
in Ref.29,30and might be associated with a lower crystal
quality after the annealing.
Thespin-pumpingmeasurementsfordifferentYIGfilm
thicknesses were performed in the following way. The
samples were magnetized in the film plane by an exter-
nal magnetic field H, and the magnetization dynamics
was excited at a constant frequency of f= 6.8 GHz by
an Agilent E8257D microwave source. The microwave
signals with powers Pappliedof 1, 10, 20, 50, 100, 250 and
500mW wereapplied to a 600 µm wide 50Ohm-matched
Cumicrostripantenna. Whiletheexternalmagneticfield
was swept, the ISHE-voltage UISHEwas recorded at the
edges of the Pt stripe using a lock-in technique with an
amplitudemodulationatafrequencyof500Hz, aswellas
the absorbed microwave power Pabs. All measurements
were performed at room temperature.
III. THEORETICAL BACKGROUND
The equations describing the ferromagnetic resonance,
the spin pumping and the inverse spin Hall effect are
provided in the following and used in the experimental
part of this paper.
A. Ferromagnetic resonance
In equilibrium the magnetization Min aferromagnetic
material is aligned along the bias magnetic field H. Ap-
plying an alternating microwave magnetic field h∼per-
pendicularly to the external field Hresults in a torque
onMand causes the magnetic moments in the sample to
precess (see also Fig. 1(a)). In ferromagnetic resonance
(FMR)themagneticfield Handtheprecessionfrequency
ffulfill the Kittel equation31
f=µ0γ
2π/radicalbig
HFMR(HFMR+MS), (1)
whereµ0is the vacuum permeability, γis the gyromag-
netic ratio, HFMRis the ferromagnetic resonance field
andMSis the saturation magnetization (experimentallyobtained values of MSfor our samples can be found in
Tab.I).
The FMR linewidth ∆ H(full width at half maximum)
is related to the Gilbert damping parameter αas16,18,27
µ0∆H= 4πfα/γ. (2)
B. Spin pumping
By attaching a thin Pt layer to a ferromagnet, the
resonance linewidth is enhanced,3which accounts for an
injection of a spin current from the ferromagnet into the
normal metal due to the spin-pumping effect (see illus-
tration in Fig. 1(c)). In this process the magnetization
precession loses spin angular momentum, which gives
rise to additional damping and, thus, to an enhanced
linewidth. The effective Gilbert damping parameter αeff
for the YIG/Pt film is described as16
αeff=α0+∆α=α0+gµB
4πMSdYIGg↑↓
eff,(3)
whereα0is the intrinsic damping of the bare YIG film,
gis the g-factor, µBis the Bohr magneton, dYIGis the
YIG film thickness and g↑↓
effis the real part of the ef-
fective spin mixing conductance. The effective Gilbert
damping parameter αeffis inversely proportional to the
YIG film thickness dYIG: with decreasing YIG thickness
the linewidth and, thus, the effective damping parameter
increases.
When the system is resonantlydriven in the FMR con-
dition, a spin-polarized electron current is injected from
the magnetic material (YIG) into the normal metal (Pt).
Inaphenomenologicalspin-pumpingmodel, theDCcom-
ponent of the spin-current density jsat the interface, in-
jected in y-direction into the Pt layer (Fig. 1(c)), can be
described as15,16,32
js=f/integraldisplay1/f
0¯h
4πg↑↓
eff1
M2
S/braceleftBig
M(t)×dM(t)
dt/bracerightBig
zdt,(4)
whereM(t) is the magnetization. {M(t)×dM(t)
dt}zis the
z-component of {M(t)×dM(t)
dt}, which is directed along
the equilibrium axis of the magnetization (see Fig. 1(c)).
Due to spin relaxation in the normal metal (Pt) the
injected spin current jsdecays along the Pt thickness
(y-direction in Fig. 1(c)) as15,16
js(y) =sinhdPt−y
λ
sinhdPt
λj0
s, (5)
whereλis the spin-diffusion length in the Pt layer. From
Eq. (4) one can deduce the spin-current density at the
interface ( y= 0)15
j0
s=g↑↓
effγ2(µ0h∼)2¯h(µ0MSγ+/radicalbig
(µ0MSγ)2+16(πf)2)
8πα2
eff((µ0MSγ)2+16(πf)2).
(6)4
FIG. 3. (Color online) ISHE-induced voltage UISHEas a func-
tion of the magnetic field Hfor different YIG film thicknesses
dYIG. Applied microwave power Papplied= 10 mW, ISHE-
voltage for the 20 nm thick sample is multiplied by a factor
of 5.
Sincej0
sis inverselyproportionalto α2
effandαeffdepends
inversely on dYIG(Eq. (3)), the spin-current density at
the interface j0
sincreases with increasing YIG film thick-
nessdYIG.
C. Inverse spin Hall effect
The Pt layer acts as a spin-current detector and trans-
forms the spin-polarized electron current injected due to
the spin-pumping effect into an electrical charge current
by means of the ISHE (see Fig. 1(c)) as6,7,12,15,16
jc=θISHE2e
¯hjs×σ, (7)
whereθISHE,e,σdenote the spin Hall angle, the elec-
tron’s elementary charge and the spin-polarization vec-
tor, respectively. Averaging the charge-current density
over the Pt thickness and taking into account Eqs. ( 4) –
(7) yields
¯jc=1
dPt/integraldisplaydPt
0jc(y)dy=θISHEλ
dPt2e
¯htanh/parenleftbigdPt
2λ/parenrightbig
j0
s.(8)
Taking into account Eqs. ( 3), (6) and (8) we calcu-
late the theoretically expected behavior of IISHE=A¯jc,
whereAis the cross section of the Pt layer. Ohm’s law
connects the ISHE-voltage UISHEwith the ISHE-current
IISHEviaUISHE=IISHE·R, whereRis the electric resis-
tance of the Pt layer. Rvaries between 1450 Ω and 1850
Ω for the different samples.
IV. YIG FILM THICKNESS DEPENDENCE OF
THE SPIN-PUMPING EFFECT DETECTED BY
THE ISHE
In Fig.3the magnetic field dependence of the gener-
ated ISHE-voltage UISHEas a function of the YIG filmthickness is shown. Clearly, the maximal voltage UISHE
at the resonance field HFMRand the FMR linewidth ∆ H
vary with the YIG film thickness. The general trend
shows, that the thinner the sample the smaller is the
magnitude of the observed voltage UISHE. At the same
time the FMR linewidth increases with decreasing YIG
film thickness.
In the following the ISHE-voltage generated by spin
pumping is investigated as a function of the YIG film
thickness. For these investigations we have chosen a
rather small exciting microwave power of 1 mW. Thus,
nonlinear effects like the FMR linewidth broadening due
to nonlinear multi-magnon processes can be excluded
(such processes will be discussed in Sec. V). Sec.IVA
covers the YIG thickness dependent variation of the en-
hanced damping parameter αeff. From these measure-
ments the spin mixing conductance g↑↓
effis deduced. In
Sec.IVBwe focus on the maximal ISHE-voltage driven
by spin pumping as a function of the YIG film thickness.
Finally, the spin Hall angle θISHEis determined.
A. YIG film parameters as a function of the YIG
film thickness
As described in Sec. IIIB, the damping parameter is
enhanced when a Pt layeris deposited onto the YIG film.
ThisenhancementisinvestigatedasafunctionoftheYIG
film thickness: the effective Gilbert damping parameter
αeff(see Eq. ( 3)) is obtained from a Lorentzian fit to the
experimental data depicted in Fig. 3and Eq. ( 2). The
resultisshowninFig. 4. With decreasingYIG film thick-
ness the linewidth and, thus, the effective damping αeff
increases. This behavior is theoretically expected: ac-
cording to Eq. ( 3)αeffis inversely proportional to dYIG.
Since the Pt film is grown onto all YIG samples simulta-
FIG. 4. (Color online) Enhanced damping parameter αeff
of the YIG/Pt samples obtained by spin-pumping measure-
ments. The red solid curve shows a fit to Eq. (3) taking the
FMR measured values for MSand a constant value for g↑↓
eff
into account. Papplied= 1 mW. The error bars for the mea-
surement points at higher sample thicknesses are not visibl e
in this scale.5
FIG. 5. (Color online) (a) ISHE-voltage UISHEas a function
of the YIG film thickness dYIG. The black line is a linear in-
terpolation as a guide tothe eye. (b) Corresponding thickne ss
dependent charge current IISHE. The red curve shows a fit to
Eqs. (6), (7), (8) with theparameters g↑↓
eff=(7.43±0.36)×1018
m−2andθISHE= 0.009±0.0008. The applied microwave
power used is Papplied= 1 mW.
neously, the spin mixing conductance g↑↓
effat the interface
is considered to be constant for all samples.11Assum-
ingg↑↓
effas constant and taking the saturation magne-
tizationMSobtained by VNA-FMR measurements (see
Fig.2and Tab. I) into account, a fit to Eq. ( 3) yields
g↑↓
eff= (7.43±0.36)×1018m−2. The fit is depicted as a
red solid line in Fig. 4.
B. YIG thickness dependence of the ISHE-voltage
driven by spin pumping
Fig.5(a) shows the maximum voltage UISHEat the
resonance field HFMRas a function of the YIG film
thickness. UISHEincreases up to a YIG film thickness
of around 200 nm when it starts to saturate (in the
case of an applied microwave power of P applied= 1
mW). The corresponding charge current IISHEis shown
in Fig.5(b). The observed thickness dependent behavior
is in agreement with the one reported for Ni 81Fe19/Pt16
and for Y 3Fe5O12/Pt.11With increasing YIG film thick-
ness the generated ISHE-current increases and tends to
saturate at thicknesses near 200 nm (Fig. 5(b)). Accord-
ing to Eq. ( 3), (6) and (8) it isIISHE∝j0
s∝1/α2
eff∝
(α0+c/dYIG)−2, wherecis a constant. Therefore, theISHE-current IISHEincreases with increasing YIG film
thickness dYIGand goes into saturation at a certain YIG
thickness.
From Eqs. ( 3), (6) and (8) we determine the expected
behavior of IISHE=A¯jcand compare it with our ex-
perimental data. In order to do so, the measured values
forMS(see Tab. I), the original damping parameter α0
determined by VNA-FMR measurements at 1 mW (see
Tab.I) and the enhanced damping parameter αeffob-
tained by spin-pumping measurements at a microwave
power of 1 mW (see Fig. 4) are used. The Pt layer thick-
ness isdPt= 8.5 nm and the microwave magnetic field
is determined to be h∼= 3.2 A/m for an applied mi-
crowavepower of 1 mW using an analytical expression.24
The spin-diffusion length in Pt is taken from literature as
λ= 10 nm33,36and the damping parameter is assumed
to be constant as α0= 6.68×10−4, which is the aver-
age of the measured values of α0. The fit is shown as a
red solid line in Fig. 5(b). We find a spin Hall angle of
θISHE= 0.009±0.0008,which is in agreementwith litera-
ture values varying in a range of 0.0037- 0.08.33–35Using
FIG. 6. (Color online) (a) YIG thickness dependence of the
ISHE-voltage driven by spin pumping for microwave powers
in the range between 1 and 500 mW. The general thickness
dependent behavior is independent of the applied microwave
power. The error bars for the measurement at lower mi-
crowave powers are not visible in this scale. (b) Deviation
of the ISHE-voltage from the linear behavior with respect to
the measured voltage U500mW
ISHE. The inset shows experimental
data for a YIG film thickness dYIG= 20 nm and the theoret-
ically expected curve. The error bars of the 20 nm and the
70 nm samples are not visible in this scale.6
the fit we estimate the saturation value of the generated
current. Although we observe a transition to saturation
at sample thicknesses of 200 – 300 nm, we find that ac-
cording to our fit, 90% of the estimated saturation level
of 5 nA is reached at a sample thickness of 1.2 µm.
V. INFLUENCE OF NONLINEAR EFFECTS ON
THE SPIN-PUMPING PROCESS FOR VARYING
YIG FILM THICKNESSES
In order to investigate nonlinear effects on the spin-
pumping effect for varying YIG film thicknesses, we per-
formed microwavepower dependent measurements of the
ISHE-voltage UISHEas function of the film thickness
dYIG. For higher microwavepowersin the rangeof 1 mW
to 500 mW we observe the same thickness-dependent
behavior of the ISHE-voltage as in the linear case
(Papplied= 1 mW, discussed in Sec. IVB): Near 200 nm
UISHEstarts to saturate independently of the applied mi-
crowave power, as it is shown in Fig. 6(a). Furthermore,
it is clearly visible from Fig. 6(a) that for a constant
film thickness the spin pumping driven ISHE-voltage in-
creases with increasing applied microwave power. At
high microwave powers the voltage does not grow lin-
early and saturates. Fig. 6(b) shows the deviation of
the ISHE-voltage ∆ UISHEfrom the linear behavior with
respect to the measured value of U500mW
ISHEat the excita-
tion power Papplied= 500 mW. In order to obtain the
relation between UISHEandPappliedfor each YIG film
thickness dYIGthe low power regime up to 20 mW is
fitted by a linear curve and extrapolated to 500 mW.
Theinsetin Fig. 6(b) showsthe correspondingviewgraph
for the case of the 20 nm thick sample. As it is visible
from Fig. 6(b), the deviation from the linear behavior
is drastically enhanced for larger YIG thicknesses. For
the thin 20 nm and 70 nm samples we observe an almost
linear behavior between UISHEandPappliedover the en-
tire microwavepower range, whereas for the thicker sam-
ples the estimated linear behavior and the observed non-
linear behavior differ approximately by a factor of 2.5
(Fig.6(b)). We observe an increase of the ISHE-voltage
as well as an broadening of the FMR linewidth with in-
creasing microwave power. In Fig. 7(a) the normalized
ISHE-voltage UISHEas function of the external magnetic
fieldHis shown for different microwave powers Papplied
in the range of 1 mW to 500 mW (YIG film thickness
dYIG= 300 nm). The linewidth tends to be asymmet-
ric at higher microwave powers. The shoulder at lower
magnetic field is widened in comparison to the shoulder
at higher fields. The reason for this asymmetry might
be due to the formation of a foldover effect,37,38due to
nonlinear damping or a nonlinear frequency shift.39,40
The results of the damping parameter αeffobtained
by microwave power dependent spin-pumping measure-
ments are depicted in Fig. 7(b). It can be seen, that
with increasing excitation power the Gilbert damping for
thicker YIG films is drastically increased. To present
FIG. 7. (Color online) (a) Illustration of the linewidth bro ad-
ening at higher excitation powers. The normalized ISHE-
voltage spectra are shown as a function of the magnetic
fieldHfor different excitation powers. Sample thickness:
300 nm. (b) Power dependent measurement of the damp-
ing parameter αefffor different YIG film thicknesses dYIGob-
tained by a Lorentzian fit to the ISHE-voltage signal. The
error bars are omitted in order to provide a better readabil-
ity of the viewgraph. (c) Nonlinear damping enhancement
(α500mW
eff−α1mW
eff)/α1mW
effas a function of the YIG film thick-
nessdYIG. Due to a reduced number of scattering channels to
other spin-wave modes for film thicknesses below 70 nm, the
damping is only enhanced for thicker YIG films with increas-
ing applied microwave powers. The error bars of the 200 nm
and the 300 nm samples are not visible on this scale.
this result more clearly the nonlinear damping enhance-
ment (α500mW
eff−α1mW
eff)/α1mW
effis shown in Fig. 7(c). The
dampingparameterat asamplethicknessof20nm α20nm
eff7
FIG. 8. (Color online) Dispersion relations calculated for each sample thickness taking into account the measured valu es of the
saturation magnetization MS(see Tab. I). Backward volume magnetostatic spin-wave mode s as well as magnetostatic surface
spin-wave modes (in red) and the first perpendicular standin g thickness spin-wave modes are depicted (in black and gray) .
(a)–(e) show the dispersion relations for the investigated sample thicknesses of 20 nm – 300 nm.
is almost unaffected by a nonlinear broadening at high
microwave powers. With increasing film thickness the
original damping α1mW
effatPapplied= 1 mW increases by
a factor of around 3 at Papplied= 500 mW. This factor
is very close to the value of the deviation of the ISHE-
voltage from the linear behavior (Fig. 6(b)).
This behavior can be attributed to the enhanced prob-
ability of nonlinear multi-magnon processes at larger
sample thicknesses: In order to understand this, a fun-
damental understanding of the restrictions for multi-
magnon scattering processes can be derived from the en-
ergy and momentum conservation laws:
N/summationdisplay
i¯hωi=M/summationdisplay
j¯hωjandN/summationdisplay
i¯hki=M/summationdisplay
j¯hkj,(9)
where the left/right sum of the equations runs over the
initial/final magnons with indices i/j which exist be-
fore/after the scattering process, respectively.41–43The
most probable scattering mechanism in our case is the
four-magnon scattering process with N= 2 and M=
2.43In Eq. (9) the wavevector ki/jand the frequency ωi/j
are connected by the dispersion relation 2 πfi/j(ki/j) =
ωi/j(ki/j). The calculated dispersion relations are shown
in Fig.8(backward volume magnetostatic spin-wave
modes with a propagation angle /negationslash(H,k) = 0◦as well
as magnetostatic surface spin-wave modes /negationslash(H,k) =
90◦).44For this purpose, the measured values of MS
(Tab.I) for each sample are used. In the case of the
20 nm sample thickness, the first perpendicular standing
spin-wave mode (thickness mode) lies above 40 GHz, thesecond above 120 GHz. Thus, the nonlinear scattering
probability obeying the energy- and momentum conser-
vation is largely reduced. This means magnons cannot
find a proper scattering partner and, thus, multi-magnon
processes are prohibited or at least largely suppressed.
With increasing film thickness the number of standing
spin-wavemodesincreasesand, thus, thescatteringprob-
ability grows. As a result, the scattering of spin waves
from the initially excited uniform precession (FMR) to
other modes is allowed and the relaxation of the original
FMR mode is enhanced. Thus, the damping increases
and we observe a broadening of the linewidth, which is
equivalent to an enhanced Gilbert damping parameter
αeffat higher YIG film thicknesses (see Fig. 7).
In orderto investigatehowthe spin-pumping efficiency
is affected by the applied microwave power, we measure
simultaneously the generated ISHE-voltage UISHEand
the transmitted ( Ptrans), as well as the reflected ( Prefl)
microwave power, which enables us to determine the
absorbed microwave power Pabs=Papplied−(Ptrans+
Prefl).17Since the 300 nm sample exhibits a strong non-
linearity(largedeviationfromthelinearbehavior(Fig. 6)
and large nonlinear linewidth enhancement (Fig. 7)), we
analyze this sample thickness. In Fig. 9the normalized
absorbed microwave power Pnorm=Pabs/PPapplied=1mW
abs
and the normalized ISHE-voltage in resonance Unorm=
UISHE/UPapplied=1mW
ISHEare shown as a function of the ap-
plied power Papplied. Both curves tend to saturate at
high microwave powers above 100 mW. The absorbed
microwave power increases by a factor of 110 for applied
microwave powers in the range between 1 and 500 mW,8
FIG. 9. (Color online) Normalized absorbed power Pnorm=
Pabs/PPapplied=1mW
abs (black squares) and normalized ISHE-
voltageUnorm=UISHE/UPapplied=1mW
ISHE (red dots) for varying
microwave powers Papplied. The inset illustrates the indepen-
dence of the spin-pumping efficiency UISHE/PabsonPapplied.
YIG thickness illustrated: 300 nm. Error bars of the low
power measurements are not visible in this scale.
whereas the generated voltage increases by a factor of
80. The spin-pumping efficiency UISHE/Pabs(see inset
in Fig.9) varies within a range of 30% for the differ-
ent microwave powers Pappliedwithout clear trend. Since
the 300 nm thick film shows a nonlinear deviation of the
ISHE-voltageby afactorof2.3 (Fig. 6(b)) and the damp-
ing is enhanced by a factor of 3 in the same range of
Papplied(Fig.7(c)), we conclude that the spin-pumping
process is only weakly dependent on the magnitude of
the applied microwave power (see inset in Fig. 9). In our
previous studies reported in Ref.12,13we show that sec-
ondary magnons generated in a process of multi-magnon
scattering contribute to the spin-pumping process and,
thus, the spin-pumping efficiency does not depend on the
applied microwave power.
VI. SUMMARY
The Y 3Fe5O12thickness dependence of the spin-
pumping effect detected by the ISHE has been inves-
tigated quantitatively. It is shown that the effective
Gilbert damping parameter of the the YIG/Pt sam-
ples is enhanced for smaller YIG film thicknesses, which
is attributed to an increase of the ratio between sur-
face to volume and, thus, to the interface character of
the spin-pumping effect. We observe a theoretically
expected increase of the ISHE-voltage with increasing
YIG film thickness tending to saturate above thick-
nesses near 200 – 300 nm. The spin mixing conductance
g↑↓
eff= (7.43±0.36)×1018m−2as well as the spin Hall an-gleθISHE= 0.009±0.0008 are calculated and are found
to be in agreement with values reported in the literature
for our materials.
The microwave power dependent measurements reveal
the occurrence of nonlinear effects for the different YIG
film thicknesses: for low powers, the induced voltage
grows linearly with the power. At high powers, we ob-
serve a saturation of the ISHE-voltage UISHEand a de-
viation by a factor of 2.5 from the linear behavior. The
microwave power dependent investigations of the Gilbert
damping parameter by spin pumping show an enhance-
ment by a factor of 3 at high sample thicknesses due
to nonlinear effects. This enhancement of the damping
is due to nonlinear scattering processes representing an
additional damping channel which absorbs energy from
the originally excited FMR. We have shown that the
smaller the sample thickness, the less dense is the spin-
wave spectrum and, thus, the less nonlinear scattering
channels exist. Hence, the smallest investigated sample
thicknesses (20 and 70 nm) exhibit a small deviation of
the ISHE-voltage from the linear behavior and a largely
reduced enhancement of the damping parameter at high
excitation powers. Furthermore, we have found that the
variation of the spin-pumping efficiencies for thick YIG
samples which show strongly nonlinear effects is much
smaller than the nonlinear enhancement of the damping.
This is attributed to secondary magnons generated in a
processofmulti-magnonscatteringthatcontributetothe
spin pumping. It is shown, that even for thick samples
(300 nm) the spin-pumping efficiency is only weakly de-
pendent on the applied microwave power and varies only
within a range of 30% for the different microwave powers
without a clear trend.
Our findings provide a guideline to design and create
efficient magnon- to charge current converters. Further-
more, the results are also substantial for the reversed
effects: the excitation of spin waves in thin YIG/Pt bi-
layers by the direct spin Hall effect and the spin-transfer
torque effect.45
VII. ACKNOWLEDGMENTS
We thank G.E.W. Bauer and V.I. Vasyuchka for valu-
able discussions. Financial support by the Deutsche
Forschungsgemeinschaft within the project CH 1037/1-
1 are gratefully acknowledged. AK would like to thank
the Graduate School of Excellence Materials Science in
Mainz (MAINZ) GSC 266. CAR, MCO and DHK ac-
knowledge support from the National Science Founda-
tion. Shared experimental facilities supported by NSF
MRSEC award DMR-0819762 were used.
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1301.2114v1.First_principles_calculation_of_the_Gilbert_damping_parameter_via_the_linear_response_formalism_with_application_to_magnetic_transition_metals_and_alloys.pdf | arXiv:1301.2114v1 [cond-mat.other] 10 Jan 2013APS/123-QED
First-principles calculation of the Gilbert damping param eter via the linear response
formalism with application to magnetic transition-metals and alloys
S. Mankovsky1, D. K¨ odderitzsch1, G. Woltersdorf2, and H. Ebert1
1University of Munich, Department of Chemistry,
Butenandtstrasse 5-13, D-81377 Munich, Germany and
2Department of Physics, Universit¨ at Regensburg, 93040 Reg ensburg, Germany
(ΩDated: September 9, 2018)
A method for the calculations of the Gilbert damping paramet erαis presented, which based on
the linear response formalism, has been implemented within the fully relativistic Korringa-Kohn-
Rostoker band structure method in combination with the cohe rent potential approximation alloy
theory. To account for thermal displacements of atoms as a sc attering mechanism, an alloy-analogy
model is introduced. This allows the determination of αfor various types of materials, such as
elemental magnetic systems and ordered magnetic compounds at finite temperature, as well as for
disordered magnetic alloys at T= 0 K and above. The effects of spin-orbit coupling, chemical a nd
temperature induced structural disorder are analyzed. Cal culations have been performed for the
3dtransition-metals bcc Fe, hcp Co, and fcc Ni, their binary al loys bcc Fe 1−xCox, fcc Ni 1−xFex,
fcc Ni 1−xCoxand bcc Fe 1−xVx, and for 5 dimpurities in transition-metal alloys. All results are in
satisfying agreement with experiment.
PACS numbers: 72.25.Rb 71.20.Be 71.70.Ej 75.78.-n
I. INTRODUCTION
Duringthe lastdecadesdynamicalmagneticproperties
have attracted a lot of interest due to their importance in
the development of new devices for spintronics, in par-
ticular, concerning their miniaturization and fast time
scale applications. A distinctive property of such devices
is the magnetization relaxation rate characterizing the
time scale on which a system being deviated from the
equilibrium returns to it, or how fast the device can be
switched from one state to another. In the case of dy-
namics of a uniform magnetization /vecMthis property
is associated with the Gilbert damping parameter ˜G(M)
used first in the phenomenologicalLandau-Lifshitz (LL)1
and Landau-Lifshitz-Gilbert (LLG) theory2describing
the magnetization dynamics processes by means of the
equation:
1
γdM
dτ=−M×Heff+M×/bracketleftBigg˜G(M)
γ2M2sdM
dτ/bracketrightBigg
,(1)
whereMsis the saturation magnetization, γthe gy-
romagnetic ratio and Heff=−∂MF[M(r)] being the
effective magnetic field. Sometimes it is more conve-
nient to use a dimensionless Gilbert damping parame-
terαgiven byα=˜G/(γMs) (see, e.g.3–5). Safonov
has generalized the Landau-Lifshitz equation by intro-
ducing a tensorial form for the Gilbert damping parame-
ter with the diagonal terms characterising magnetization
dissipation6. Beingintroducedasaphenomenologicalpa-
rameter, the Gilbert damping is normally deduced from
experiment. In particular, it can be evaluated from the
resonant line width in ferromagnetic-resonance (FMR)
experiments. The difficulty of these measurements con-
sists in the problem that there exist several different
sources for the broadening of the line width, which havebeen discussed extensively in the literature7–13. The line
width that is observed in ferromagnetic resonance spec-
tra is usually caused by intrinsic and extrinsic relax-
ation effects. The extrinsic contributions are a conse-
quence of spatially fluctuating magnetic properties due
to sample imperfections. Short range fluctuations lead
to two magnon scattering while long range fluctuations
lead to an inhomogeneous line broadening due a super-
position of local resonances14. In order to separate the
intrinsic Gilbert damping from the extrinsic effects it is
necessary to measure the frequency and angular depen-
dence of the ferromagnetic resonance line width, e. g.
two magnon scattering can be avoided when the mag-
netization is aligned along the film normal11(perpen-
dicular configuration). Usually one finds a linear fre-
quency dependence with a zero frequency offset and one
can write ∆ H(ω) =αω
γ+∆H(0). When such measure-
ments are performed over a wide frequency range the
slope of ∆ Has a function of frequency can be used
to extract the intrinsic Gilbert damping constant. In
metallic ferromagnets Gilbert damping is mostly caused
by electron magnon scattering. In addition Gilbert-like
damping can be caused by eddy currents. The magni-
tude of the eddy current damping is proportional to d2,
wheredis the sample thickness10. In sufficiently thin
magnetic films ( d≤10 nm) the eddy current damping
can be neglected10. However, for very thin films relax-
ation mechanisms that occur at the interfaces can also
increase and even dominate the damping. Such effects
are spin pumping15,16and the modified electronic struc-
ture at the interfaces. In the present work spin pumping
and the modified interface electronic structure are not
considered and we assume that bulk-like Gilbert damp-
ing dominates.
Much understanding of dynamical magnetic properties
could in principle be obtained from the simulation of2
these processes utilizing time-dependent first-principles
electronic structure calculations, that in turn would pave
the way to developing and optimizing new materials for
spintronic devices. In spite of the progress in the de-
velopment of time-dependent density functional theory
(TD-DFT) during the last decades17that allows to study
variousdynamicalprocessesin atomsand molecules from
firstprinciples, applicationstosolidsarerare. Thisisdue
to a lack of universally applicable approximations to the
exchange-correlation kernel of TD-DFT for solids. Thus,
atthemoment, atractableapproachconsistsintheuseof
the classical LLG equations, and employing parameters
calculated within a microscopic approach. Note however
that this approach can fail dealing with ultrafast mag-
netization dynamics, which is discussed, for instance, in
Refs. [18 and 19], but is not considered in the present
work.
Most of the investigations on the magnetization dissi-
pation have been carried out within model studies. Here
one has to mention, in particular, the so-called s-dor
p-dexchange model20–23based on a separate considera-
tion of the localized ’magnetic’ d-electrons and delocal-
izeds- andp-electrons mediating the exchange interac-
tions between localized magnetic moments and responsi-
ble for the magnetization dissipation in the system. As
was pointed out by Skadsem et al.24, the dissipation pro-
cess in this case can be treated as an energy pumping
out of thed-electron subsystem into the s-electron bath
followed by its dissipation via spin-flip scattering pro-
cesses. This model gave a rather transparent qualita-
tive picture for the magnetization relaxation in diluted
alloys, e.g. magnetic semiconductors such as GaMnAs.
However, it fails to give quantitative agreement with ex-
periment in the case of itinerant metallic systems (e.g.
3d-metalalloys),wherethe d-statesareratherdelocalised
and strongly hybridized with the sp-electrons. As a con-
sequence the treatment of allvalence electrons on the
same footing is needed, which leads to the requirement
of first-principles calculations of the Gilbert damping go-
ing beyond a model-based evaluation.
Various such calculations of the Gilbert damping pa-
rameter are already present in the literature. They usu-
ally assume a certain dissipation mechanism, like Kam-
bersky’sbreathingFermisurface(BFS)25,26, ormoregen-
eral torque-correlation models (TCM)3,27. These models
include explicitly the spin-orbit coupling (SOC), high-
lighting its key role in the magnetization dissipation pro-
cesses. However, the latter methods used for electronic
structure calculations cannot take explicitly into account
disorder in the system that in turn is responsible for the
aforementioned spin-flip scattering process. Therefore,
this has to be simulated by using external parameters
characterizing the finite lifetime of the electronic states.
ThisweakpointwasrecentlyaddressedbyBrataas et al.4
who described the Gilbert damping by means of scatter-
ing theory. This development supplied the formal basis
for the first parameter-free investigations on disordered
alloys for which the dominant scattering mechanism ispotential scattering caused by chemical disorder5.
Theoretical investigations of the magnetization dissi-
pation by means of first-principles calculations of the
Gilbert damping parameter already brought much un-
derstanding of the physical mechanisms responsible for
this effect. First of all, key roles are played by two ef-
fects: the SOC of the atomic species contained in the
system and scattering on various imperfections, either
impurities or structural defects, phonons, etc. Account-
ing for the crucial role of scattering processes respon-
sible for the energy dissipation, different types of scat-
tering phenomena have to be considered. One can dis-
tinguish between the ordered-compound or pure-element
systems for which electron-phonon scattering is a very
important mechanism for relaxation, and disordered al-
loys with dominating scattering processes resulting from
randomly distributed atoms of different types. In the
first case, the Gilbert damping behavior is rather differ-
ent at low and high temperatures. At high temperature
atomic displacements create random potentials leading
toSOC-inducedspin-flipscattering. At lowtemperature,
where the magnetization dissipation is well described via
the BFS (Breathing Fermi-surface) mechanism25,26, the
spin-conserving electron-phonon scattering is required to
bring the electronic subsystem to the equilibrium at ev-
ery step of the magnetization rotation, i.e. to reoccupy
the modified electronic states.
In this contribution we describe a formalism for the
calculation of the Gilbert damping equivalent to that
of Brataas et al.4, however, based on the linear re-
sponce theory28as implemented in fully relativistic mul-
tiple scattering based Korringa-Kohn-Rostoker (KKR)
formalism. It will be demonstrated that this allows to
treat elegantly and efficiently the temperature depen-
dence ofαin pure crystals as well as disordered alloys.
II. THEORETICAL APPROACH
To have direct access to real materials and to obtain
a deeper understanding of the origin of the properties
observed experimentally, the phenomenological Gilbert
damping parameter has to be treated on a microscopic
level. This implies to deal with the electrons responsible
forthe energydissipation in the magnetic dynamicalpro-
cesses. Thus, one equates the corresponding expressions
for the dissipation rate obtained in the phenomenologi-
cal and microscopic approaches ˙Emag=˙Edis. Although
a temporal variation of the magnetization is a required
condition for the energy dissipation to occur, the Gilbert
damping parameter is defined in the limit ω→0 (see
e.g., Ref. [24]) and therefore can be calculated within the
adiabatic approximation.
In the phenomenological LLG theory the time depen-
dent magnetization M(t) is described by Eq. (1). Ac-
cordingly, the time derivative of the magnetic energy is3
given by:
˙Emag=Heff·dM
dτ=1
γ2(˙ˆm)T[˜G(M)˙ˆm](2)
whereˆm=M/Msdenotes the normalized magnetiza-
tion. To represent the Gilbert damping parameter in
terms of a microscopic theory, following Brataas et al.4,
the energy dissipation is associated with the electronic
subsystem. The dissipation rate upon the motion of the
magnetization ˙Edis=/angbracketleftBig
dˆH
dτ/angbracketrightBig
, is determined by the under-
lying Hamiltonian ˆH(τ). Assuming a small deviation of
the magnetic moment from the equilibrium the normal-
ized magnetization ˆm(τ) can be expanded around the
equilibrium magnetization ˆm0
ˆm(τ) =ˆm0+u(τ), (3)
resulting in the expression for the linearized time depen-
dent Hamiltonian for the system brought out of equilib-
rium:
ˆH=ˆH0(ˆm0)+/summationdisplay
µuµ∂
∂uµˆH(ˆm0).(4)
Due tothe smalldeviation from the equilibrium, ˙Ediscan
be obtained within the linearresponseformalism, leading
to the expression4:
˙Edis=−π/planckover2pi1/summationdisplay
ij/summationdisplay
µν˙uµ˙uν/angbracketleftBigg
ψi/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH
∂uµ/vextendsingle/vextendsingle/vextendsingle/vextendsingleψj/angbracketrightBigg/angbracketleftBigg
ψj/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH
∂uν/vextendsingle/vextendsingle/vextendsingle/vextendsingleψi/angbracketrightBigg
×δ(EF−Ei)δ(EF−Ej),(5)
whereEFis the Fermi energy and the sums run over
all eigenstates of the system. As Eq. (5) characterizes
the rate of the energy dissipation upon transition of the
system from the tilted state to the equilibrium, it can
be identified with the corresponding phenomenological
quantity in Eq. (2), ˙Emag=˙Edis. This leads to an ex-
plicit expression for the Gilbert damping tensor ˜Gor
equivalently for the damping parameter α=˜G/(γMs)
(Ref. [4]):
αµν=−/planckover2pi1γ
πMs/summationdisplay
ij/summationdisplay
µν/angbracketleftBigg
ψi/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH
∂uµ/vextendsingle/vextendsingle/vextendsingle/vextendsingleψj/angbracketrightBigg/angbracketleftBigg
ψj/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH
∂uν/vextendsingle/vextendsingle/vextendsingle/vextendsingleψi/angbracketrightBigg
×δ(EF−Ei)δ(EF−Ej),(6)
where the summation is running over all states at the
Fermi surface EF.
In full analogy to the problem of electric
conductivity29, the sum over eigenstates |ψi/angbracketrightmay be
expressed in terms of the retarded single-particle Green’s
function Im G+(EF) =−π/summationtext
i|ψi/angbracketright/angbracketleftψi|δ(EF−Ei). This
leads for the parameter αto a Kubo-Greenwood-like
equation:
αµν=−/planckover2pi1γ
πMsTrace
/angbracketleftBigg
∂ˆH
∂uµImG+(EF)∂ˆH
∂uνImG+(EF)/angbracketrightBigg
c(7)with/angbracketleft.../angbracketrightcindicating a configurational average in case of
a disordered system.
The most reliable way to account for spin-orbit cou-
pling as the source of Gilbert damping is to evaluate
Eq. (7) using a fully relativistic Hamiltonian within the
framework of local spin density formalism (LSDA)30:
ˆH=cα·p+βmc2+V(r)+βσ·ˆmB(r).(8)
Hereαiandβare the standard Dirac matrices, σde-
notes the vector of relativistic Pauli matrices, and pis
the relativistic momentum operator31. The functions
V(r) andB=σ·ˆmB(r) are the spin-averaged and
spin-dependent parts, respectively, of the LSDA poten-
tial. The spin density ms(r) as well as the effective ex-
change field B(r) are assummed to be collinear within
the unit cell and aligned along the z-direction in the
equilibrium (i. e. ms,0(r) =ms(r)ˆm0=ms(r)ezand
B0(r) =B(r)ˆm0=B(r)ez). Tilting of the magnetiza-
tion direction by the angle θaccording to Eq. (3), i.e.
ms(r) =ms(r)ˆm=ms(r)(sinθcosφ,sinθsinφ,cosθ)
andB(r) =B(r)ˆmleads to a perturbation term in the
Hamiltonian
∆V(r) =βσ·(ˆm−ˆm0)B(r) =βσ·uB(r),(9)
with (see Eq. (4))
∂
∂uµˆH(ˆm0) =βσµB(r). (10)
The Green’s function G+in Eq. (7) can be obtained in
a very efficient way by using the spin-polarized relativis-
tic version of multiple scattering theory30that allows us
to treat magnetic solids:
G+(r,r′,E) =/summationdisplay
ΛΛ′Zn
Λ(r,E)τnm
ΛΛ′(E)Zm×
Λ′(r′,E)
−δnm/summationdisplay
Λ/bracketleftbig
Zn
Λ(r,E)Jn×
Λ′(r′,E)Θ(r′
n−rn)
+Jn
Λ(r,E)Zn×
Λ′(r′,E)Θ(rn−r′
n)/bracketrightbig
.(11)
Herer,r′refer to site nandm, respectively, where
Zn
Λ(r,E) =ZΛ(rn,E) =ZΛ(r−Rn,E) is a function
centered at site Rn. The four-component wave functions
Zn
Λ(r,E) (Jn
Λ(r,E)) are regular (irregular) solutions to
the single-site Dirac equation labeled by the combined
quantum numbers Λ (Λ = ( κ,µ)), withκandµbeing
the spin-orbit and magnetic quantum numbers31. The
superscript ×indicates the left hand side solution of the
Dirac equation. τnm
ΛΛ′(E) is the so-called scattering path
operator that transfers an electronic wave coming in at
siteminto a wave going out from site nwith all possible
intermediate scattering events accounted for.
Using matrix notation with respect to Λ, this leads to
the following expression for the damping parameter:
αµµ=g
πµtot/summationdisplay
nTrace/angbracketleftbig
T0µ˜τ0nTnµ˜τn0/angbracketrightbig
c(12)
with the g-factor 2(1 + µorb/µspin) in terms of the spin
and orbital moments, µspinandµorb, respectively, the4
total magnetic moment µtot=µspin+µorb, ˜τ0n
ΛΛ′=
1
2i(τ0n
ΛΛ′−τ0n
Λ′Λ) and with the energy argument EFomit-
ted. The matrix elements in Eq. (12) are identical to
those occurring in the context of exchange coupling32:
Tnµ
Λ′Λ=/integraldisplay
d3rZn×
Λ′(r)/bracketleftbigg∂
∂uµˆH(ˆm0)/bracketrightbigg
Zn
Λ(r)
=/integraldisplay
d3rZn×
Λ′(r) [βσµBxc(r)]Zn
Λ(r).(13)
The expression in Eq. (12) for the Gilbert-damping
parameterαis essentially equivalent to the one obtained
within the torque correlation method (see e.g. Refs. [33–
35]). However, in contrast to the conventional TCM the
electronicstructureishererepresentedusingtheretarded
electronic Green function giving the present approach
much more flexibility. In particular, it does not rely on
a phenomenological relaxation time parameter.
The expression Eq. (12) can be applied straightfor-
wardly to disordered alloys. This can be done by de-
scribing in a first step the underlying electronic struc-
ture (forT= 0 K) on the basis of the coherent po-
tential approximation (CPA) alloy theory. In the next
step the configurational average in Eq. (12) is taken fol-
lowing the scheme worked out by Butler29when dealing
with the electrical conductivity at T= 0 K or residual
resistivity, respectively, of disordered alloys. This im-
plies in particular that so-called vertex corrections of the
type/angbracketleftTµImG+TνImG+/angbracketrightc− /angbracketleftTµImG+/angbracketrightc/angbracketleftTνImG+/angbracketrightcthat
account for scattering-in processes in the language of the
Boltzmann transport formalism are properly accounted
for.
One has to note that the factorg
µtotin Eq. (12) is sep-
arated from the configurational average /angbracketleft.../angbracketrightc, although
both values, gandµtot, have to represent the averageper
unit cell doing the calculations for compounds and al-
loys. This approximation is rather reasonable in the case
of compounds or alloys where the properties of the ele-
ments of the system are similar (e.g. 3 d-element alloys),
but can be questionable in the case of systems containing
elements exhibiting significant differences (3 d-5d-, 3d-4f-
compounds, etc), or in the case of non-uniform systems
as discussed by Nibarger et al36.
Thermal vibrations as a source of electron scattering
can in principle be accounted for by a generalization of
Eqs. (7) – (13) to finite temperatures and by including
the electron-phonon self-energy Σ el−phwhen calculating
the Green’s function G+. Here we restrict our considera-
tion to elastic scattering processes by using a quasi-static
representation of the thermal displacements of the atoms
from their equilibrium positions. The atom displaced
from the equilibrium position in the lattice results in a
corresponding variation ∆ tn=tn−tn
0of the single-site
scattering matrix in the global frame of reference37,38. A
single-site scattering matrix tn(the underline denotes a
matrix in an angular momentum representation Λ) for
the atomndisplaced by the value sn
νfrom the equilib-
rium position in the lattice can be obtained using thetransformation matrices37,39
Un
LL′(sν,E) = 4π/summationdisplay
L′′il′′+l−l′
×CLL′L′′jl′′(sn
ν√
E)YL′′(ˆsn
ν).(14)
Heremeis the electron mass, jla spherical Bessel func-
tion,CLL′L′′stands for the Gaunt coefficients, and a
non-relativistic angular momentum representation with
L= (l,ml) has been used. Performing a Clebsch-Gordon
transformation for the transformation matrix Un
LL′to
the relativistic Λ representation, the tmatrixtnfor the
shifted atom can be obtained from the non-shifted one
tn
0from the expression
tn
ν= (Un
ν)−1tn
0Un
ν. (15)
Treating for a discrete set of displacements sn
νeach
displacement as an alloy component, we introduce an
alloy-analogy model to average over the set sn
νthat
is chosen to reproduce the thermal root mean square
average displacement/radicalbig
/angbracketleftu2/angbracketrightTfor a given temperature
T. This in turn may be set according to /angbracketleftu2/angbracketrightT=
1
43h2
π2mkΘD[Φ(ΘD/T)
ΘD/T+1
4] with Φ(Θ D/T) the Debye func-
tion,hthe Planck constant, kthe Boltzmann constant
andΘ DtheDebyetemperature40. Ignoringthezerotem-
perature term 1 /4 and assuming a frozen potential for
the atoms, the situation can be dealt with in full analogy
to the treatment of disordered alloys on the basis of the
CPA (see above).
For small displacements the transformation Eq. (14)
can be expanded with respect to sn
ν(see Ref. [39]) re-
sulting in a linear dependence on sn
νfor non-vanishing
contributions with ∆ l=|l−l′|=±1. This leads, in
particular, in the presence of atomic displacements for
transition-metals (TM), for which an angular momen-
tum cut-off of lmax= 2 in the KKR multiple scattering
expansion is in general sufficient for an undistorted lat-
tice, to an angular momentum expansion up to at least
lmax= 3. However, this is correct only under the assump-
tion of very small displacements allowing linearisation of
the transformation Uwith respect to the displacement
amplitudes. Thus, since the temperature increase leads
to a monotonous increaseof s, the cut-off lmaxshould also
be increased.
III. MODEL CALCULATIONS
In the following we present results of calculations for
which single parameters have artificially been manipu-
lated in the first-principles calculations in order to sys-
tematically reveal their role for the Gilbert-damping.
This approach is used to disentangle competing influ-
ences on the Gilbert-damping parameter.5
A. Vertex corrections
The impact of vertex corrections is shown in Fig. 1
for two different cases: Fig. 1(a) represents the Gilbert
damping parameter for an Fe 1−xVxdisordered alloy as a
function of concentration, while Fig. 1(b) gives the cor-
responding value for pure Fe in the presence of temper-
ature induced disorder and plotted as a function of tem-
perature. Both figures show results calculated with and
without vertexcorrectionsallowingforcomparison. First
of all, a significant effect of the vertex corrections is no-
ticeable in both cases, although the variation depends on
increasingconcentrationofVinthebinaryFe 1−xVxalloy
and the temperature in the case of pure Fe, respectively.
Some differences in their behavior can be explained by
the differences of the systems under consideration. Deal-
ing with temperature effects via the alloy analogy model,
the system is considered as an effective alloy consisting
of a fixed number of components characterizing different
types of displacements. Thus, in this case the tempera-
ture effect is associated with the increase of disorder in
the system caused only by the increase of the displace-
ment amplitude, or, in other words – with the strength
of scattering potential experienced by the electrons rep-
resented by tn(T)−tn
0. In the case of a random alloy
theA1−xBxvariation of the scattering potential, as well
as the difference tn
A−tn
B, upon changing the concentra-
tions is less pronounced for small amounts of impurities
Bandtheconcentrationdependenceisdeterminedbythe
amount of scatterers of different types. However, when
the concentration of impurities increases, the potentials
of the components are also modified (this is reflected,
e.g., in the shift of electronic states with respect to the
Fermi level, that will be discussed below) and this can
lead to a change of the concentration dependence of the
vertex corrections. An important issue which one has to
stress that neglect of the vertex corrections may lead to
the unphysical result, α <0, as is shown in Fig. 1(a).
In terms of the Boltzmann transport formalism, this is
because of the neglect of the scattering-in term41lead-
ing obviously to an incomplete description of the energy
transfer processes.
B. Influence of spin-orbit coupling
Aswasalreadydiscussedabove,thespin-orbitcoupling
for the electrons of the atoms composing the system is
the main driving force for the magnetization relaxation,
resultingintheenergytransferfromthemagneticsubsys-
tem to the crystal lattice. Thus, the Gilbert damping pa-
rameter should approach zero upon decreasing the SOC
in the system. Fig. 2 shows the results for Py+15%Os,
where√αis plotted as a function of the scaling param-
eter of the spin-orbit coupling42applied to all atoms in
the alloy. As one can see,√αhas a nearly linear depen-
dence on SOC implying that αvaries in second order in
the strength of the spin-orbit coupling43.00.10.2 0.3 0.4 0.5
concentration xV02040α × 103without vertex corrections
with vertex correctionsFe1-xVx
(a)
0100200 300 400 500
temperature (K)051015202530α × 103without vertex corrections
with vertex correctionsbcc Fe
(b)
FIG. 1. The Gilbert damping parameter for (a) bcc Fe 1−xVx
(T= 0 K) as a function of V concentration and (b) for bcc-Fe
as a function of temperature. Full (open) symbols give resul ts
with (without) the vertex corrections.
0 0.5 1 1.5 2
SOC scaling parameter00.10.20.30.4 α1/2 Py+15%Os
FIG. 2. The Gilbert damping parameter for Py+15%Os as
a function of the scaling parameter of spin-orbit coupling a p-
plied to all atoms contained in the alloy. Red dashed line in
plot – linear fit. The values 0 and 1 for the SOC scaling pa-
rameter correspond to the scalar-relativistic Schr¨ oding er-like
and fully relativistic Dirac equations, respectively.6
IV. RESULTS AND DISCUSSIONS
A. 3dtransition-metals
We have mentioned above the crucial role of scatter-
ing processes for the energy dissipation in magnetiza-
tion dynamic processes. In pure metals, in the absence
of any impurity, the electron-phonon scattering mecha-
nism is of great importance, although it plays a different
role in the low- and high-temperature regimes. This was
demonstrated by Ebert et al.28using the alloy analogy
approach,aswellasbyLiu44et al.usingthe ’frozenther-
mal lattice disorder’ approach. In fact both approaches
are based on the quasi-static treatment of thermal dis-
placements. However, while the average is taken by the
CPA within the alloy analogy model the latter requires
a sequence of super-cell calculations for this purpose.
As a first example bcc Fe is considered here. The cal-
culations have been performed accounting for the tem-
perature induced atomic displacements from their equi-
librium positions, according to the alloy analogy scheme
described in section II. This leads, even for pure systems,
to a scattering process and in this way to a finite value
forα(see Fig. 3(a)). One can see that the experimen-
tal results available in the literature are rather different,
depending on the conditions of the experiment. In par-
ticular, the experimental results Expt. 2 (Ref. [45]) and
Expt. 3 (Ref. [46]) correspondto bulk while the measure-
ments Expt. 1 (Ref. [47]) have been done for an ultrathin
film with 2 .3 nm thickness. The Gilbert damping con-
stant obtained within the present calculations for bcc Fe
(circles,a= 5.44 a.u.) is compared in Fig. 3(a) with
the experiment exhibiting rather good agreement at the
temperature above 100 K despite a certain underestima-
tion. One can also see a rather pronounced increase of
the Gilbert damping observed in the experiment above
400 K (Fig. 3(a), Expt. 2 and Expt. 3), while the theo-
retical value shows only little temperature dependent be-
havior. Nevertheless, the increase of the Gilbert damp-
ing with temperature becomes more pronounced when
the temperature induced lattice expansion is taken into
account, that can be seen from the results obtained for
a= 5.45 a.u. (squares). Thick lines are used to stress
the temperature regions for which corresponding lattice
parameters are more appropriate. At low temperatures,
below 100 K, the calculated Gilbert damping parameter
goes up when the temperature decreases, that was ob-
served only in the recent experiment47. This behavior is
commonly denoted as a transition from low-temperature
conductivity-like to high-temperature resistivity-like be-
havior reflecting the dominance of intra- and inter-band
transitions, respectively3. The latter are related to the
increase of the smearing of electron energy bands caused
by the increase of scattering events with temperature.
Note that even a small amount of impurities reduces
strongly the conductivity-like behavior28,45, leading to
the more pronounced effect of impurity-scattering pro-
cesses due to the increase of scattering events caused bychemical disorder. Large discrepancies between the lat-
ter experimental data47and theoretical results of the α
calculationsforbcc Fe arerelatedto the verysmall thick-
ness of the film investigated experimentally, that leads to
an increase of spin-transfer channels for magnetization
dissipation as was discussed above.
Results for the temperature dependent Gilbert-
damping parameter αfor hcp Co are presented in Fig.
3(b) which shows, despite certain underestimation, a rea-
sonable agreement with the experimental results45. The
general trends at low and high temperatures are similar
to those seen in Fe.
The results for pure Ni are given in Fig. 3(c) that show
in full accordance with experiment a rapid decrease of α
with increasing temperature until a regime with a weak
variation of αwithTis reached.
Note that in the discussions above we have treated α
as a scalar instead of a tensorial quantity ignoring a pos-
sible anisotropy of the damping processes. This approx-
imation is reasonable for the systems considered above
with the magnetization directions along a three- or four-
fold symmetry axis (see, e.g., the discussions in Ref. [48
and 49]). For a more detailed discussion of this issue
the anisotropy of the Gilbert damping tensor α(M) has
been investigated for bcc Fe. To demonstrate the depen-
dence ofαon the magnetization direction M, the cal-
culations have been performed for M= ˆz|M|with the
ˆzaxis taken along the /angbracketleft001/angbracketright,/angbracketleft111/angbracketrightand/angbracketleft011/angbracketrightcrystallo-
graphic directions. Fig. 4 presents the temperature de-
pendence of the diagonal elements αxxandαyy. As to be
expected for symmetry reasons, αxxdiffers from αyyonly
in the case of ˆ z/bardbl/angbracketleft011/angbracketright. One can see that the anisotropic
behavior of the Gilbert damping is pronounced at low
temperatures. With an increase of the temperature the
anisotropy nearly disappears, because of the smearing of
the energy bands caused by thermal vibrations49. A sim-
ilar behavior is caused by impurities with a random dis-
tribution, aswasobservedforexamplefortheFe 0.95Si0.05
alloy system. The calculations of the diagonal elements
αxxandαyyfor two different magnetization directions
along/angbracketleft001/angbracketrightand/angbracketleft011/angbracketrightaxes giveαxx=αyy= 0.00123 in
the first case and αxx= 0.00123 and αyy= 0.00127 in
the second, i.e. the damping is nearly isotropic.
The damping parameter αincreases very rapidly with
decreasing temperature in the low temperature regime
(T≤100 K) for all pure ferromagnetic 3 dmetals, Fe,
Co, and Ni (see Fig. 3), leading to a significant discrep-
ancy between theoretical and experimental results in this
regime. The observed discrepancy between theory and
experiment can be related to the exact limit ω= 0 taken
intheexpressionfortheGilbertdampingparameter. Ko-
renmann and Prange13have analyzedthe magnon damp-
ing in the limit of small wave vector of magnons q→0,
assuming indirect transitions in the electron subsystem
and taking into account the finite lifetime τof the Bloch
states due to electron-phonon scattering. They discuss
the limiting cases of low and high temperatures showing
the analogy of the present problem with the problem of7
0 200 400 600
Temperature (K)0246810α × 103Theory: a = 5.42 a.u.
Theory: Fe+0.1% Vac.
Theory: a = 5.45 a.u.
Expt. 1
Expt. 2
Expt. 3bcc Fe
(a)
0100200 300 400 500600
Temperature (K)05101520α × 103Expt
Theory
Theory: Co + 0.03% Vac
Theory: Co + 0.1% Vachcp Co
(b)
0100200 300 400 500
Temperature (K)00.050.10.150.20.25 αExpt
Theory
Theory: Ni + 0.1%Vacfcc Ni
(c)
FIG. 3. Temperature variation of the Gilbert damping pa-
rameter of pure systems. Comparison of theoretical results
with experiment: (a) bcc-Fe: circles and squares show the re -
sults for ideal bcc Fe for two lattice parameters, a= 5.42 a.u.
anda= 5.45 a.u.; stars show theoretical results for bcc Fe
(a= 5.42 a.u.) with 0.1% of vacancies (Expt. 1 - Ref. [47],
Expt. 2 - Ref. [45], Expt. 3 - Ref. [46]); (b) hcp-Co: circles
show theoretical results for ideal hcp Co, stars - for Co with
0.03% of vacancies, and ’pluses’ - for Co with 0.1% of vacan-
cies (Expt. Ref. [45]); and (c) fcc-Ni (Expt. Ref. [45]).extreme cases for the conductivity leading to the normal
and anomalous skin effect. On the basis of their result,
the authors point out that the expression for the Gilbert
damping obtained by Kambersky25, withα∼τis cor-
rect in the limit of small lifetime (i.e. qvFτ≪1, in their
model consideration, where qis a magnon wave vector
andvFis a Fermi velocity of the electron). In the low-
temperature limit the lifetime τincreases with decreas-
ingTand one has to use the expression corresponding
to the ’anomalous’ skin effect for the conductivity, i.e.
α∼tan−1(qvFτ)/qvF, leading to a saturation of αupon
the increase of τ.
Another possible reason for the low-temperature be-
havior of the Gilbert damping observed experimentally
can be structural defects present in the material. To
simulate this effect, calculations have been performed for
fcc Ni and bcc Fe with 0.1% of vacancies and for hcp
Co with 0.1% and 0.03% of vacancies. Fig. 3(a)-(c)
shows the corresponding temperature dependence of the
Gilbert dampingparameterapproachingafinite value for
T→0. The remaining difference in the T-dependent be-
havior can be attributed to the non-linear dependence of
the scattering cross section at low temperatures as is dis-
cussed in the literature for transport properties of metals
and is not accounted for within the present approxima-
tion.
B. 3dTransition-metal alloys
As is mentioned above, the use of the linear response
formalism within multiple scattering theory for the elec-
tronicstructurecalculationsallowsustoperformthenec-
essary configurational averaging in a very efficient way
avoiding supercell calculations and to study with mod-
erate effort the influence of varying alloy composition
onα. The corresponding approach has been applied to
0100200 300 400 500
Temperature (K)051015202530α × 103Theory: z||<001>: αxx=αyy
Theory: z||<011>: αxx
Theory: z||<011>: αyy
Theory: z||<111>: αxx=αyybcc Fe
FIG. 4. Temperature variation of the αxxandαyycom-
ponents of the Gilbert damping tensor of bcc Fe with the
magnetization direction taken along different crystallogr aphic
directions: M= ˆz|M|/bardbl/angbracketleft001/angbracketright(circles), M/bardbl/angbracketleft011/angbracketright(squares),
M/bardbl/angbracketleft111/angbracketright(diamonds).8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
concentration xCo0123456 α × 103bcc, CPA
CsCl: CPA (partial ord)
bcc, NL CPA (ord)
bcc, NL CPA (disord)
n(EF)Fe1-xCox
n(EF)
n(EF) (sts./Ry)
102030405060
0
(a)
-8-6-4 -2 0 2
energy (eV)00.511.52n↑tot(E) (sts./eV)00.511.52n↓tot(E) (sts./eV)x = 0.01
x = 0.5EF
(b)
FIG. 5. (a) Theoretical results for the Gilbert damping pa-
rameter of bcc Fe 1−xCoxas a function of Co concentration:
CPA results for the bcc structure (full circles) describing
the random alloy system, results for the partially ordered
system (opened square) for x= 0.5 (i.e. for Fe 1−xCox
alloy with CsCl structure and alloy components randomly
distributed in two sublattices in the following proportion s:
(Fe0.9Co0.1)(Fe0.1Co0.9), the NL-CPA results for random al-
loy with bcc structure (opened circles) and the NL-CPA re-
sults for the the system with short-range order within the
first-neighbor shell (opened diamonds). The dashed line rep -
resents the DOS at the Fermi energy, EF, as a function of Co
concentration. (b) spin resolved DOS for bcc Fe 1−xCoxfor
x= 0.01 (dashed line) and x= 0.5 (solid line).
the ferromagnetic 3 d-transition-metal alloy systems bcc
Fe1−xCox, fcc Ni 1−xFex, fcc Ni 1−xCoxand bcc Fe 1−xVx.
Fig. 5(a) shows as an example results for the Gilbert
damping parameter α(x) calculated for bcc Fe 1−xCox
forT= 0 K at different conditions. Full circles rep-
resent the results of the single-cite CPA calculations
characterizing the random Fe-Co alloy. These results
are compared to those obtained employing the non-local
CPA52,53(NL-CPA) assuming no short-range order in
the system (opened circles). Dealing in both cases (CPA
and NL-CPA), with completely disordered system, the
NL-CPA maps the alloy problem on that of an impurity
cluster embedded in a translational invariant effective
medium determined selfconsistently, thereby accounting0 0.2 0.4 0.6
concentration xCo0246 α × 103Exp.
TheoryFe1-xCox
(a)
0 0.2 0.4 0.6
concentration xCo010203040 α × 103Exp.
Theory (Starikov et al.)
TheoryNi1-xCox
(b)
0 0.2 0.4 0.6
concentration xFe01020 α × 103Theory (Starikov et al.)
Theory
Expt 1
Expt 2Ni1-xFex
(c)
0 0.2 0.4 0.6
concentration xV012345 α × 103Expt.
Theory, T = 0 K
Theory, T = 300 KFe1-xVx
(d)
FIG. 6. The Gilbert damping parameter for Fe 1−xCox(a)
Ni1−xCox(b) and Ni 1−xFex(c) as a function of Co and Fe
concentration, respectively: present results within CPA ( full
circles), experimental data by Oogane50(full diamonds). (d)
Results for bcc Fe 1−xVxas a function of V concentration:
T= 0 K (full circles) and T= 300 K (open circles). Squares:
experimental data51. Open circles: theoretical results by
Starikov et al.5.9
for nonlocal correlations up to the range of the cluster
size. The present calculations have been performed for
the smallestNL-CPAclusterscontainingtwositesforbcc
based system, accountingfor the short-rangeorder in the
first-neighborshell. As onecan see, this results in a small
decrease of the αvalue in the region of concentrations
aroundx= 0.5 (opened diamonds), that is in agreement
with the results obtained for partially ordered system
(opened square) for x= 0.5. The latter have been calcu-
lated for the Fe 1−xCoxalloy having CsCl structure and
alloycomponentsrandomlydistributed intwosublattices
in the following proportions: (Fe 0.9Co0.1)(Fe0.1Co0.9).
Because the moments and spin-orbit coupling strength
do not differ very much for Fe and Co, the variation of
α(x) should be determined in the concentrated regime
essentially by the electronic structure at the Fermi en-
ergyEF. As Fig. 5(a) shows, there is indeed a close
correlation with the density of states n(EF) that may be
seen as a measure for the number of available relaxation
channels. The change of α(x) due to the increase of the
Co concentration is primarily determined by an appar-
ent shift of the Fermi energy also varying with concen-
tration (Fig. 5(b)). The alloy systems considered have
the common feature that the concentration dependence
ofαis governed by the concentration dependent density
of statesn(EF).
A comparison of theoretical αvalues with the experi-
mentforbccFe 1−xCoxisshowninFig. 6(a),demonstrat-
ing satisfying agreement. In the case of Ni 1−xFexand
Ni1−xCoxalloysshown in Fig. 6, (b) and (c), the Gilbert
dampingdecreasesmonotonouslywith the increaseofthe
Fe and Co concentration, in line with experimental data.
At all concentrations the experimental results are under-
estimated by theory approximately by a factor of 2. The
calculated dampingparameter α(x) is found in verygood
agreementwith theresultsbasedonthe scatteringtheory
approach5demonstrating numerically the equivalence of
the two approaches. An indispensable requirement to
achieve this agreement is to include the vertex correc-
tions mentioned above. As suggested by Eq. (12) the
variation of α(x) with concentration xmay also reflect
to some extent the variation of the averagemagnetic mo-
ment of the alloy, µtot.
ThepeculiarityoftheFe 1−xVxalloywhencomparedto
those discussed above is that V is a non-magnetic metal
and has only an induced spin magnetic moment. De-
spite that, the concentration dependence of the Gilbert
damping parameter at T= 0 K for small amounts of
V (see Fig. 6(d)) displays the same trend as the pre-
viously discussed alloys shown in Fig. 6(a)-(c). Taking
into account a finite temperature of T= 300 K changes
αvalue significantly at small V concentrations leading
to an improved agreement with experiment for pure Fe,
while it still compares poorly with the experimental data
atxV= 0.27. One should stress once more that the con-
centration dependent behavior of the Gilbert damping
parameter of the alloys discussed above is different for
an increased amount of impurities (more than 10%), as aresultofadifferentvariationoftheDOS n(EF)causedby
a concentration dependent modification of the electronic
states and shift of the Fermi level.
C. 5dimpurities in 3 dtransition metals
As discussed in our recent work28investigating the
temperature dependent Gilbert damping parameter for
pure Ni and for Ni with Cu impurities, αis primarily
determined by the thermal displacement in the regime of
small impurity concentrations. This behavior can also be
seen in Fig. 7, where the results forFe with 5 d-impurities
are shown. Solid lines represent results for T= 0 K for
an impurity concentration of 1% (full squares) and 5%
(full circles). As one can see, at smaller concentrations
the maximum of the Gilbert damping parameter occurs
for Pt. With increasing impurity content the αparame-
ter decreases in such a way that at the concentration of
5% a maximum is observed for Os.
The reason for this behavior lies in the rather weak
scattering efficiency of Pt atoms due to a small DOS
n(EF) of the Pt states when compared for example for
Osimpurities (see Fig. 9). This results in a slowdecrease
ofαat small Pt concentration when the BFS mechanism
is mostly responsible for the energy dissipation. A con-
sequence of this feature can be seen in the temperature
dependence of α(T= 300 K, opened squares): a most
pronounced temperature induced decrease of the αvalue
is observed for Pt and Au. When the concentration of
5d-impurities is increased up to 5%, the maximum in α
occurs for the element with the most efficient scatter-
ing potential resulting in spin-flip scattering processes
responsible for dissipation. The temperature effect at
this concentration is very small.
Considering in more detail the temperature dependent
behavior of the Gilbert damping parameter for Fe with
Os and Pt impurities, shown in Fig. 8, one can also ob-
serve the consequence of the features mentioned above.
At 1% of Pt impurities αdecreases much steeper upon
increasing the temperature, as compared to the case of
Os impurities. Therefore, in the first case the role of the
scattering processes due to atomic displacements is much
more pronounced than in the second case with rather
strong scattering on the Os impurities. When the con-
centration increases to 5% the dependence of αon the
temperature in both cases becomes less pronounced.
The previousresults can be comparedto the results for
the 5d-impurities in the permalloy Fe 80Ni20(Py), which
has been investigated also experimentally54. This system
shows some difference in the concentration dependence
when compared with pure Fe, because Py is a disordered
alloy with a finite value of the αparameter. Therefore,
a substitution of 5 dimpurities leads to a nearly linear
increase of the Gilbert damping with impurity content,
just as seen in experiment54.
The total damping for 10% of 5 d-impurities shown in
Fig.10(a)variesroughlyparabolicallyoverthe 5 dTMse-10
Ta W Re Os Ir Pt Au0246810 α × 103x = 0.05; T = 0 K
x = 0.05; T = 300 K
x = 0.01; T = 0 K
x = 0.01; T = 300 K
FIG. 7. Gilbert damping parameter for bcc Fe with 1%
(squares) and 5% (circles) of 5 dimpurities calculated for
T= 0K (full symbols) and for T= 300K (opened sysmbols).
ries. This variation of αwith the type of impurity corre-
lateswellwith the densityofstates n5d(EF) (Fig. 10(b)).
Again the trend of the experimental data is well repro-
duced by the calculated values that are however some-
what too low.
V. SUMMARY
In summary, aformulationforthe Gilbert dampingpa-
rameterαin terms of linear response theory was derived
thatledtoaKubo-Greenwood-likeequation. Thescheme
was implemented using the fully relativistic KKR band
structuremethod incombinationwiththe CPAalloythe-
ory. This allows to account for various types of scat-
tering mechanisms in a parameter-free way, that might
be either due to chemical disorder or to temperature-
induced structural disorder (i.e. electron-phonon scat-
tering effect). The latter has been described by using
the so-called alloy-analogy model with the thermal dis-
placement of atoms dealt with in a quasi-static manner.
Corresponding applications to pure metals (Fe, Co, Ni)
aswellastodisorderedtransition-metalalloysledto very
good agreement with results based on the scattering the-
ory approach of Brataas et al.4and well reproduces the
experimental results. The crucial role of vertex correc-
tions for the Gilbert damping is demonstrated both in
the case of chemical as well as structural disorder and
the accuracy of finite-temperature results is analyzed via
test calculations.
Furthermore, the flexibility and numerical efficiency
of the present scheme was demonstrated by a study
on metallic systems on a series of binary 3 d-alloys
(Fe1−xCox, Ni1−xFex, Ni1−xCoxand Fe 1−xVx), 3d−5d
TM systems, the permalloy-5 dTM systems. The agree-
ment between the present theoretical and experimental
results is quite satisfying, although one has to stress
a systematic underestimation of the Gilbert damping
by the numerical results. This disagreement could be
caused either by the idealized system considered theoret-
ically (e.g., the boundary effects are not accounted for0100200 300 400 500
temperature (K)12345α × 103Fe0.99Me0.01Pt
Os
(a)
0100200 300 400 500
temperature (K)22.533.54α × 103Fe0.95Me0.05
PtOs
(b)
FIG. 8. Gilbert damping parameter for bcc Fe 1−xMxwith
M= Pt (circles) and M= Os (squares) impurities as a func-
tion of temperature for 1% (a) and 5% (b) of the impurities.
-8-6-4 -2 0 2
energy (eV)00.81.62.43.2n↑(E) (sts./eV)00.81.62.43.2n↓(E) (sts./eV)Pt
OsEF
EF
FIG. 9. DOS for Pt in Fe 1−xPtx(full line) and Os in
Fe1−xOsx(dashed line) for x= 0.01.11
Ta W Re OsIr Pt Au02468α × 102Expt.
Theory
(a)
Ta W Re OsIr Pt Au051015n5d(EF) (Sts/Ry)
(b)
FIG. 10. (a) Gilbert damping parameter αfor Py/5d TM
systems with 10 % 5d TM content in comparison with
experiment54; (b) spin magnetic moment m5d
spinand density
of states n(EF) at the Fermi energy of the 5 dcomponent in
Py/5d TM systems with 10 % 5d TM content.in present calculations) or because of additional intrin-
sic dissipation mechanisms for bulk systems which have
to be taken into account. These could be, for instance,
effects of temperature induced spin disorder44.
ACKNOWLEDGMENTS
The authors would like to thank the DFG for finan-
cial support within the SFB 689 “Spinph¨ anomene in re-
duzierten Dimensionen” and within project EBE-154/23
for financial support.
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2308.08331v1.Discovery_and_regulation_of_chiral_magnetic_solitons__Exact_solution_from_Landau_Lifshitz_Gilbert_equation.pdf | Discovery and regulation of chiral magnetic solitons: Exact solution from Landau-Lifshitz-Gilbert
equation
Xin-Wei Jin,1, 2Zhan-Ying Yang,1, 2,∗Zhimin Liao,3Guangyin Jing,1, †and Wen-Li Yang2, 4
1School of Physics, Northwest University, Xi’an 710127, China
2Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China
3School of Physics, Peking University, Beijing, 100871,China
4Insititute of Physics, Northwest University, Xi’an 710127, China
(Dated: August 17, 2023)
The Landau-Lifshitz-Gilbert (LLG) equation has emerged as a fundamental and indispensable framework
within the realm of magnetism. However, solving the LLG equation, encompassing full nonlinearity amidst
intricate complexities, presents formidable challenges. Here, we develop a precise mapping through geometric
representation, establishing a direct linkage between the LLG equation and an integrable generalized nonlinear
Schr ¨odinger equation. This novel mapping provides accessibility towards acquiring a great number of exact
spatiotemporal solutions. Notably, exact chiral magnetic solitons, critical for stability and controllability in
propagation with and without damping effects are discovered. Our formulation provides exact solutions for the
long-standing fully nonlinear problem, facilitating practical control through spin current injection in magnetic
memory applications.
Introduction.— The seminal 1935 work by Landau and Lif-
shitz, which laid down the foundational dynamical equation
governing magnetization based on phenomenological insights
[1–3], and the subsequent introduction of a damping term by
Gilbert [4], the amalgamation of these concepts has given
rise to the renowned Landau-Lifshitz-Gilbert (LLG) equation.
Over the years, this equation has emerged as a fundamental
and indispensable framework within magnetism field. Its con-
temporary significance has been amplified through remarkable
advancements, most notably the incorporation of an additional
term that facilitates the explication of spin torque phenomena
in spintronics [5–10], spin waves [11–18], magnetic solitons
[19–28], spatio-temporal patterns [29, 30], and even chaotic
behavior [31]. Further advancements have paved the way
for applications in next-generation magnetic storage [32–34],
neural networks [35–37], and logic gates [38–42].
Despite its deceptively simple appearance, solving the LLG
equation poses an exceptional challenge [29, 30], rendering
it a persistently unresolved problem for nearly nine decades.
This complexity emanates from its intricate nature as a vector-
based highly nonlinear partial differential equation. In real-
world scenarios, the LLG equation encompasses a myriad of
complex interactions among the components of the magneti-
zation vector [4]. Consequently, solutions often necessitate
recourse to linearization, approximations, and asymptotic
techniques such as the Holstein-Primakoff (HP) transforma-
tion [43, 44], reductive perturbation scheme [45, 46], and long
wavelength approximation. Nonetheless, these techniques
prove utterly ineffectual in regions of large amplitudes or
strong nonlinearity. Therefore, exact solution of the LLG
equation emerges as a potent bridge, overcoming these gaps
and revealing profound revelations regarding magnetization
dynamics, thereby furnishing insightful understandings for
simulating and comprehending intricate magnetic systems.
In this Letter, through a geometric representation [47],
we establish an exact mapping of the LLG equation onto
an integrable generalized nonlinear Schr ¨odinger equation,
free of any approximation. This novel mapping provides
accessibility towards acquiring a great number of exact
spatiotemporal solutions of the original equation. Notably, we
unveil an analytical formulation for chiral magnetic solitons,encompassing a spectrum ranging from left-handed, neutral to
right-handed configurations, determined by a defined chirality
factor. The derived exact solution indicates the potential for
arbitrary manipulation of magnetic soliton motion through the
injection of spin current —a discovery that aligns seamlessly
with our numerical findings. To encapsulate the realism of
dissipative devices, we incorporate Gilbert damping into the
dynamics of these chiral magnetic solitons, thereby estimating
their dynamic propagation.
Modeling.— We consider an isotropic ferromagnetic
nanowire with spin-polarized current flowing along the axis
of nanowire as depicted in Fig. 1. A “nanowire” as defined
here is a planar ferromagnetic stripe of length Lx, width
Ly, and thickness Lzalong ˆx,ˆy, and ˆz, respectively, with
Lx≫Ly>Lz.
Figure 1: Schematic diagram of 1D ferromagnetic structure.
Magnetic soliton excitation driven by spin-polarized currents.
Here∆represents the width of magnetic solitons.
The magnetization dynamics is described by the famous
LLG equation
∂m
∂t=−γm×Heff+α/parenleftbigg
m×∂m
∂t/parenrightbigg
+τb, (1)
where m=M/Ms= (mx,my,mz)is the unit magnetization
vector with Msbeing the saturated magnetization. The first
term on the right-hand side represents the torque contributed
by the effective field Heff(including applied, demagnetizing,
anisotropy, and exchange fields), γis the gyromagnetic
constant. The second term describes the Gilbert damping
torque, parameterized by a dimensionless damping factor α.arXiv:2308.08331v1 [nlin.PS] 16 Aug 20232
Figure 2: Spatial structure and classification of chiral magnetic solitons. (a) Vertical views of the left-handed and right-handed
magnetic solitons. (b) Schematic plot of the chirality defined by the azimuth angle change. The pair of red arrows delineate the
azimuthal directional changes of the left and right chiral magnetic solitons across the distribution axis. Their discrepancies in
azimuthal variation are denoted by ∆ϕ′
LHand∆ϕ′
RH. (c)-(e) Spatial spin structures of left-handed magnetic soliton, neutral
magnetic soliton, and right-handed magnetic soliton. (f)-(h) illustrate the azimuthal, polar angle, and phase gradient flow of the
three kinds of chiral solitons.
The last term τbrepresents the spin-transfer torque (STT),
which comprises dual components that can be written as τb=
−bJ(ˆJ·∇)M+βbJM×(ˆJ·∇)M. Here ˆJis the unit vector in
the direction of the current. These two components are most
commonly termed adiabatic and non-adiabatic spin torques,
respectively, with bJ=P jeµB/(eMs)andβdefined as the non-
adiabatic torque coefficient. Wherein, Prepresents the spin
polarization of current, jeis the electric current density, µBis
the Bohr magneton, and eis the magnitude of electron charge.
In what follows, we take only adiabatic STT into consideration
for two reasons: one is that the most widely agreed upon
interaction between a spin-polarized current and a magnetic
soliton is adiabatic STT; and the other is that the magnitude
of the nonadiabatic spin torque is about 2 orders of magnitude
smaller than adiabatic torque (β≈10−2). Let us begin by
examining the most elementary effective field, encompassing
solely exchange fields, i.e. Heff= (2A/Ms)∇2m, where Ais
the exchange stiffness constant.
The spatiotemporal transformation τ=γµ0Mst/(1+α2)
andζ=λex·xare introduced to recast the LLG equation
into the dimensionless Landau-Lifshitz form (Note that λex=/radicalbig
2A/(µ0M2s)is the exchange length):
mτ=−m×mζζ−αm×/parenleftbig
m×mζζ/parenrightbig
+Qmζ, (2)
whereQ=bJ(1+αβ)//radicalbig
2Aγ2µ0, a dimensionless number
measuring the ratio of external spin current over exchange
interaction strength. This dimensionless STT-LLG model
(2) effectively describes the dynamics of nonlinear excita-
tions, such as magnetic solitons, occurring in ferromagnetic
nanowires upon spin injection. Moreover, it exhibits qualita-tive reproduction much of the behavior seen experimentally.
For a permalloy nanowire, the standard material parameters
are:γ=1.76×1011rad/s·T,Ms=8×105A/m,A=1.3×
10−11J/m,P=0.5. As a result, the units in time and space
after rescaling are 1 τ≈5.70 ps ,1ζ≈5.68 nm.
Chiral magnetic soliton.— The high nonlinearity of STT-
LLG model (2) presents a great challenge for comprehensive
analytical research and restricts the exploration of novel
spin textures to the realm of micromagnetic simulation or
weak nonlinearity. In this context, to obtain an analytical
depiction of large-amplitude magnetic textures, we exactly
map the STT-LLG equation (2) into a generalized nonlinear
Schr ¨odinger (GNLS) equation (See Supplementary Material
for further details on the spatial curve mapping procedure),
devoid of any approximations. For no damping, the GNLS
equation reads
iΨτ+Ψζζ+2|Ψ|2Ψ−iQΨζ=0. (3)
Exact cycloidal chiral magnetic soliton solutions can be
constructed by applying the Darboux transformation (DT) [48,
49]. Indeed, using the mapping relationship between equation
solutions, the specific expression of three components of
magnetization are obtained (See Supplementary Material for3
detailed calculations):
mx=2a
a2+b2[asinh(Ξ)sin(Γ)−bcosh(Ξ)cos(Γ)]·sech2(Ξ),
my=2a
a2+b2[asinh(Ξ)cos(Γ)+bcosh(Ξ)sin(Γ)]·sech2(Ξ),
mz=1−2a2
a2+b2sech2(Ξ),
(4)
with
Ξ=2a/bracketleftbigg
x+/integraldisplay
(Q+4b)dt/bracketrightbigg
,
Γ=2b/bracketleftbigg
x+/integraldisplay
(Q−2(a2−b2)/b)dt/bracketrightbigg
,
where aandbdescribe the wave number and the velocity of
the magnetic soliton.
Fig. 2(a) depicts the spin textures of the obtained
magnetic solitons. Evidently, these solitons showcase a mirror
symmetry relative to the wave vector axis, indicating their
inherent chirality. The chirality can be characterized by
variations of the azimuth angle. To clarify, we denote the polar
and azimuthal angles of mbyθandϕ, respectively (as shown
in Fig. 1), such that M+=mx+imy=M0sin(θ)exp(iϕ),mz=
M0cos(θ). Note that the azimuthal angle exhibits a periodic
background, which arises from the variation of magnetization
in space, as revealed by the solution (4). The oscillation
structures present in the azimuthal angle profiles are related
to the small oscillation of mν(ν=x,y), even though they
may not be readily visible in Fig. 2(a). By eliminating the
meaningless periodic background phase, the real phase jumps
of the chiral magnetic solitons are obtained by calculating the
intrinsic argument ϕ′(x) =argM′
+, where M′
+=M+exp(iΓ).
As a result, the azimuthal angles of both chiral solitons
are demonstrated in Fig. 2(b). Two red arrows span
between the blue dashed line representing negative infinity and
the corresponding positive infinity, delineating the azimuthal
evolution of magnetic solitons across the distribution axis. The
distinction in azimuthal variation for chiral magnetic solitons
are denoted as ∆ϕ′
LHand∆ϕ′
RH. Notably, the two classes
of chiral magnetic solitons exhibit opposite phase jumps,
corresponding to two distinct chiralities.
The total phase change is defined as ∆ϕ′=ϕ′(x→+∞)−
ϕ′(x→ −∞). In general, the phase change of arbitrary
magnetic solitons can be determined by integrating the phase
gradient flow. Insight can be gained from combining both
argument and the phase gradient flow ∇ϕ′(x). Starting from
M′
+that constructed from exact solutions, we obtain
∇ϕ′(x) =2b[sech(2Ξ)+1]/parenleftig
a2−b2
a2+b2/parenrightig
sech(2Ξ)−1. (5)
One can observe that the denominator of the aforementioned
expression is consistently a non-positive value, which indi-
cates that “ +” and “ −” families of phase gradient flow are
characterized by the opposite signs of b. Here, we define
a chirality factor C=sgn(b) =±1, which determines the
chirality of magnetic solitons. It is straightforward to verify
that the nonzero phase variation is characterized by a simple
Figure 3: Coupling between chiral magnetic solitons and spin
current injection. (a) Velocities of three distinct classes of
chiral magnetic solitons plotted against spin-polarized
currents ( j). (b)-(d) Controlled manipulation of right-handed
magnetic solitons under varying current strengths, enabling
forward, backward, and arrested motion.
expression: ∆ϕ′=2Carctan (|a/b|).Thus, the chirality of
the chiral magnetic soliton is entirely determined by this
chirality factor C. When b=0, a special case naturally
occurs, where the chirality factor cannot be defined, and
chirality disappears, corresponding to the neutral cycloidal
magnetic solitons. Finally, we can now classify the exact
solution (4) into three categories based on the chirality factor,
corresponding to neutral, left-handed, and right-handed chiral
magnetic solitons. Figs. 2(c)–2(h) depict the typical spin
textures, azimuthal angles, polar angles, and phase gradient
flow at t=0 when no spin current is applied.
Spin-current coupling and damping effect.— We now move
to study the coupling between chiral magnetic soliton and the
injection of spin current. It has been demonstrated that the
spin transfer torque is capable of driving the domain wall
or skyrmion [7, 26], eliciting their prompt movement at a
considerable velocity upon the application of spin current.
Here we report a comparable phenomenon on the chiral
magnetic soliton from both theoretical and simulation results.
The numerical simulation results depicting the relationship
between the velocities of three categories of chiral magnetic
solitons and the injected spin currents are illustrated in Fig.
3(a), and are in direct agreement with those obtained from
the analytical solutions (4). These linear correlations can be
realized from the equivalent GNLS equation (3), wherein the
spin current term can be normalized to resemble the “driving
velocity”, as supported by the dimensional analysis of Q.
Figs. 3(b)–3(d) exemplify the current manipulation for right-
handed magnetic soliton that comprise a series of transient
snapshots captured during the magnetization evolution process
(the model parameters are shown in the caption). The above4
Figure 4: Transmission of magnetic solitons in damped ferromagnetic nanowires and the anti-damping effect of non-adiabatic
STT. (a) Schematic diagram of magnetization dissipation under damping. (b) Propagation of right-handed magnetic solitons in
ferromagnetic nanowires with Gilbert-damping constant α=0.05. (c) Temporal evolution of the magnetization component mz
in the absence of non-adiabatic STT, where damping constant α=0.01. (d) Temporal evolution of the magnetization component
mzwith spin-polarized current j=3.7×107A·cm−2. (e) Gilbert-damping dependence of lifetime and moving distance.
results highlight two notable aspects. Firstly, the chiral
magnetic soliton possesses an inherent velocity linked to its
initial magnetization state. Secondly, the solitons’ motion can
be stimulated by a spin-polarized current, while preserving
their chirality. The external injection of spin current offers a
means to manipulate chiral magnetic solitons, granting control
over their forward, backward, and frozen motion.
Until now, our analysis is based on a perfect ferromagnetic
wire in the absence of damping. Strictly speaking, in
realistic nanowires, magnetic solitons cannot move over a
large distance due to Gilbert damping. The existence of
damping introduces a small torque field, which dissipates the
energy of the system during magnetization dynamics, and
leads to a helical precession of the magnetization towards
the direction of the effective field, i.e. the minimum energy
state (See Fig. 4(a)). To understand this damping effect in
greater detail we performed numerical simulations of single-
soliton dynamics. Fig. 4(b) shows the evolution of a right-
handed magnetic soliton in a nanowire with Gilbert-damping
constant α=0.05. It can be seen that the chiral magnetic
soliton degenerates to a homogeneous magnetized state after
propagating for about 223.8 ps. During the whole process, the
magnetic soliton undergoes continuous deformation. This has
two consequences: magnetic soliton spreading and slowing of
internal oscillations. In order to characterize the presence of a
chiral magnetic soliton, we define the soliton polarization Ps=
[1−min(mz)]
2. The soliton is deemed to have dissipated when its
polarization is lower than 5% in comparison to the maximum
magnetization. The movement of magnetic solitons within
ferromagnetic nanowires, subject to varying damping, resultsin distinct lifetimes. Fig. 4(e) depicts line graphs illustrating
the relationship between the damping coefficient, lifetime,
and moving distance. The dissipation of solitons due to
damping is a challenge to circumvent, and one approach is to
seek ferromagnetic materials with low damping coefficients.
Here, we explored the potential anti-damping effect of non-
adiabatic STT, as depicted in Figs. 4(c) and 4(d). In the
absence of external spin current, the mzcomponent of the
magnetic soliton diminishes during transmission. However,
upon injecting an appropriate spin current, the incorporation
of non-adiabatic STT enables the chiral magnetic soliton to
propagate uniformly in its original velocity, resulting in a
significantly extended lifetime.
Conclusions.— In this Letter, we have shown that the
dimensionless LLG equation containing STT is entirely
equivalent to the generalized nonlinear Schr ¨odinger equation
without any approximation. This remarkable integrable
system enables us to predict novel exact spatiotemporal
magnetic solitons. By applying the Darboux transformation,
we obtain exact solution of chiral magnetic solitons, emerging
within an isotropic ferromagnetic nanowire. Our analytical
formulation establishes a distinct correlation between chiral
magnetic solitons and the infusion of spin currents, corrobo-
rating our numerical findings. This interrelation underscores
the potential for arbitrary manipulation of magnetic soliton
motion through spin current injection. The inherent chirality
of the micromagnetic structure plays a pivotal role in soliton
motion: a reversal in chirality leads to a shift in motion
direction. To encapsulate the realism of dissipative devices,
we investigate the influence of Gilbert damping on the motion5
of chiral magnetic solitons. The results reveal that in
the presence of damping, chiral magnetic solitons gradually
evolve toward a uniformly magnetized state. This implies
a topological equivalence between the two magnetization
states. We propose selecting an appropriate spin current
intensity to introduce an anti-damping effect, thereby ensuring
the long-distance transmission of the chiral magnetic soliton.
These results present new possibilities for developing chiral
magnetic soliton-based racetrack memory.
The authors thank Prof. H. M. Yu, Prof. L. C. Zhao, Prof. J.
Liu and Prof. C. P. Liu for their helpful discussions. This work
was supported by the National Natural Science Foundation
of China (No.12275213, 12174306,12247103), and Natural
Science Basic Research Program of Shaanxi (2023-JC-JQ-02,
2021JCW-19).
∗zyyang@nwu.edu.cn
†jing@nwu.edu.cn
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Science and Numerical Simulation 32, 285 (2016).Supplementary Materials for “Discovery and regulation of chiral magnetic solitons: Exact solution
from Landau-Lifshitz-Gilbert equation”
Xin-Wei Jin and Zhan-Ying Yang∗
School of Physics, Northwest University, Xi’an 710127, China and
Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China
Zhimin Liao
School of Physics, Peking University, Beijing, 100871,China
Guangyin Jing†
School of Physics, Northwest University, Xi’an 710127, China
Wen-Li Yang
Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China and
Insititute of Physics, Northwest University, Xi’an 710127, ChinaarXiv:2308.08331v1 [nlin.PS] 16 Aug 20232
In this supplementary material, we will show more details on the exact geometric mapping between Landau-Lifshitz-Gilbert
and generalized nonlinear Schr ¨odinger equation, and calculation details of solving the magnetic solitons.
A. Exact mapping between LLG and GNLS equation: Geometric Representation
We identify the magnetization state of ferromagnetic nanowire at any instant of time with a moving space curve in Euclidean
three-dimensional space E3. This is achieved by mapping the unit magnetization vector m(x,t)on the unit tangent vector e1
associated with the curve. Thus the dimensionless STT-LLG equation (in the absence of damping) becomes
e1t=e1×e1xx+Qe1x. (A.1)
In the usual way, the normal and binormal vectors of the moving space curve are constructed by taking e2in the direction of e′1
ande3=e1×e2. The spatial variations of these orthogonal unit vectors is determined by the Serret-Frenet equations
e1
e2
e3
x=
0κ0
−κ0τ
0−τ0
e1
e2
e3
, (A.2)
where κ(x,t)andτ(x,t)are the curvature and torsion of the space curve. In view of (A.1) and (A.2) alongside the orthogonality
of the three unit vectors, it is easy to obtain
e1
e2
e3
t=
0 −κτ+Qκ κ x
κτ−Qκ 0 −τ2+Qτ+κ−1κxx
−κxτ2−Qτ−κ−1κxx 0
e1
e2
e3
. (A.3)
The compatibility conditions∂
∂t/parenleftig
∂ei
∂x/parenrightig
=∂
∂x/parenleftig
∂ei
∂t/parenrightig
,i=1,2,3, between Eqs. (A.2) and (A.3) lead to the following evolution
equations for κandτ
κt=−(κτx+2κxτ)−Qκx, (A.4a)
τt=/parenleftbig
κ−1κxx−τ2/parenrightbig
x+κκx−Qτx. (A.4b)
On making the complex transformation Ψ=1
2κexp/parenleftbig
i/integraltext
τdx/parenrightbig
, we finally arrive at a generalized nonlinear Schr ¨odinger (GNLS)
equation (it is easy to verify that the real and imaginary parts of (A.5) is equivalent to (A.4a) and (A.4b), respectively)
iΨτ+Ψζζ+2|Ψ|2Ψ−iQΨζ=0. (A.5)
Thus we have proved that the STT-LLG equation can be exactly mapped into the integrable GNLS equation.
B. Lax Representation and Darboux Transformation
We now turn to establish the connection between the solutions of the LLG equation and the GNLS equation. Using the Pauli
matrices ( σ1,σ2,σ3), the LLG equation can be rewriten into the matrix form
/hatwidemt=1
2i[/hatwidem,/hatwidemxx]+Q/hatwidemx, (B.1)
where/hatwidem=mxσ1+myσ2+mzσ3and[·,·]denotes the Lie bracket of the matrices. For this equation, the boundary condition is
given by lim
x→±∞/hatwidem=σ3, i.e., lim
x→±∞m= (0,0,1). Considering the Lax representation of the GNLS (A.5)
∂Φ
∂x=UΦ,∂Φ
∂t=VΦ, (B.2)
where U=U0+λU1,V=V0+λV1+λ2V2andλis the spectral parameter,
U0=/parenleftigg
0Ψ
−Ψ∗0/parenrightigg
,U1=−σ3,V0=/parenleftigg
i|Ψ|2iΨx+QΨ
iΨ∗
x−QΨ∗−i|Ψ|2/parenrightigg
,V1=−2iU0−Qσ3,V2=2iσ3. (B.3)3
Suppose Φ1(x,t,λ)andΦ2(x,t,λ)are two linear independence eigenvectors of Lax pair (B.2), then Ω= (Φ1,Φ2)also satisfies
Eq. (B.2). Let g(x,t) =Ω(x,t,λ)/vextendsingle/vextendsingle
λ=0, we have gx=U0g,gt=V0g. From transformation /hatwideΦ=g−1Φ, we obtain
∂/hatwideΦ
∂x=/hatwideU/hatwideΦ,∂/hatwideΦ
∂t=/hatwideV/hatwideΦ, (B.4)
where
/hatwideU=g−1Ug−g−1gx=g−1(U−U0)g=λg−1U1g,
/hatwideV=g−1Vg−g−1gt=g−1(V−V0)g=λg−1V1g+λ2g−1V2g,(B.5)
Let/hatwidem=−g−1σ3gbe a solution of Eq. (B.1), then
/hatwidem/hatwidemx=−g−1σ3U0σ3g+g−1σ2
3U0g=2g−1U0g, (B.6)
Substitute (B.3) into (B.5), together with the definition of /hatwidemand (B.6), matrices (B.5) of new Lax pair (B.2) can be rewritten as
/hatwideU=λ/hatwidem,/hatwideV=λQ/hatwidem−iλ/hatwidem/hatwidemx−2iλ2/hatwidem, (B.7)
Using the factor /hatwidem2=I, the compatibility condition /hatwideUt−/hatwideVx+ [/hatwideU,/hatwideV] =0 exactly yields the matrix form LLG equation (B.1).
Thus we proved the Lax gauge equivalence of the GNLS equation (A.5) and the dimensionless STT-LLG equation. Through the
established gauge equivalence detailed above, it becomes evident that given a non-zero solution Φof the GNLS equation (A.5),
the corresponding eigenfunctions can be derived via Lax pair (B.2). This process thereby elucidates the determination of the
invertible matrices Ω(x,t,λ)andg(x,t). Further through the transformation /hatwideΦ=g−1Φand/hatwidem=−g−1σ3g, we are enabled to
acquire the solution /hatwidemfor (B.1). Finally, the three components of magnetization, namely mx,my, and mz, can be obtained from
the definition of /hatwidem, constituting the non-trivial solution to the original STT-LLG equation.
To obtain the dynamical magnetic soliton in the ferromagnetic nanowire, we are going to construct the Darboux transformation
of (A.5). Let Φ[0]
1(x,t,λ)andΦ[0]
2(x,t,λ)be the eigenfunction of the Lax pair (B.2) corresponding to the zero solution of
the GNLS equation (A.5). Demonstrating the reciprocity of the Lax pair solution with respect to spectral parameters, it is
straightforward to establish that if/parenleftbig
Φ[0]
1(x,t,λ1),Φ[0]
2(x,t,λ1)/parenrightbigTrepresents the solution for Lax pair (B.2) corresponding to the
spectral parameter λ1, then/parenleftbig
Φ[0]∗
2(x,t,λ∗
1),−Φ[0]∗
1(x,t,λ∗
1)/parenrightbigTconstitutes the solution for the corresponding spectral parameter
λ∗
1. Denote
H1=/parenleftigg
Φ[0]
1Φ[0]∗
2
Φ[0]
2−Φ[0]∗
1/parenrightigg
,Λ1=/parenleftigg
λ10
0λ∗
1/parenrightigg
, (B.8)
where/parenleftbig
Φ[0]
1,Φ[0]
2/parenrightbigT=/parenleftbig
exp[−λ1x+2iλ2
1t−λ1/integraltext
Qdt],exp[λ1x−2iλ2
1t+λ1/integraltext
Qdt]/parenrightbigT. The Darboux matrix is acquired through
the standard procedure
T[1]=λI−H1Λ1H−1
1, (B.9)
leading to the solution Φ[1]=T[1]Φof new spectral problem. Therefore the Darboux transformation is written as
Ψ[1](x,t) =Ψ[0](x,t)−2(λ1+λ∗
1)Φ[0]
1Φ[0]∗
2
|Φ[0]
1|2+|Φ[0]
2|2. (B.10)
Taking λ1=a+ibwe get the soliton solution of GNLS equation
Ψ[1](x,t) =−2asech[2a(x+4bt+Qt)]exp/bracketleftbig
2i/bracketleftbig
(2a2−2b2−Q)t−bx/bracketrightbig/bracketrightbig
, (B.11)
and the corresponding eigenfunction Φ[1](x,t,λ) = (Φ[1]
1,Φ[1]
2)T. Substitute the above results into Ω(x,t,λ) =/parenleftigg
Φ[1]
1Φ[1]∗
2
Φ[1]
2−Φ[1]∗
1/parenrightigg
,
g(x,t) =Ω(x,t,λ)/vextendsingle/vextendsingle
λ=0,/hatwidem=−g−1σ3gin sequence, we finally obtain exact cycloidal chiral magnetic soliton solution in
ferromagnetic nanowires under the influence of spin current injection4
mx=/hatwidem12+/hatwidem21
2=2a
a2+b2[asinh(Ξ)sin(Γ)−bcosh(Ξ)cos(Γ)]·sech2(Ξ),
my=/hatwidem21−/hatwidem12
2i=2a
a2+b2[asinh(Ξ)cos(Γ)+bcosh(Ξ)sin(Γ)]·sech2(Ξ),
mz=/hatwidem11=1−2a2
a2+b2sech2(Ξ),(B.12)
with
Ξ=2a/bracketleftbigg
x+/integraldisplay
(Q+4b)dt/bracketrightbigg
,Γ=2b/bracketleftbigg
x+/integraldisplay
(Q−2(a2−b2)/b)dt/bracketrightbigg
,
where aandbdescribe the wave number and the velocity of the chiral soliton.
To gain deeper insight into the interaction dynamics between two chiral magnetic solitons, we continue to utilize gauge
transformation (2.2.67) to construct two-soliton solutions based on the above single soliton solutions. The second-order Darboux
matrix is expressed as follows
H2=/parenleftigg
Φ[1]
1Φ[1]∗
2
Φ[1]
2−Φ[1]∗
1/parenrightigg
,Λ2=/parenleftigg
λ20
0λ∗
2/parenrightigg
,T[2]=λI−H2Λ2H−1
2, (B.13)
and the specific Darboux transformation form of the two-soliton solution is subsequently obtained
Ψ[2](x,t) =Ψ[1](x,t)−2(λ2+λ∗
2)Φ[1]
1Φ[1]∗
2
|Φ[1]
1|2+|Φ[1]
2|2. (B.14)
Taking λ1=a1+ib1,λ2=a2+ib2, after tedious simplification, we get two-soliton solution of GNLS equation
Ψ[2](x,t) =4η1eiβ2cosh(α2)+η2eiβ1cosh(α1)+iη3/parenleftbig
eiβ1sinh(α1)−eiβ2sinh(α2)/parenrightbig
η4cosh(α1+α2)+η5cosh(α1−α2)+η6cos(β1−β2), (B.15)
where
α1=2a1(x+4b1t+/integraldisplay
Qdt),β1=4(a2−b2)t−2b2(x+/integraldisplay
Qdt),
α2=2a2(x+4b2t+/integraldisplay
Qdt),β2=4(a2−b2)t−2b2(x+/integraldisplay
Qdt),
η1= [(a2
2−a2
1)−(b2−b1)2]a1,η4=−(a2−a1)2−(b2−b1)2,η3=2a1a2(b2−b1),
η2= [(a2
1−a2
2)−(b1−b2)2]a2,η5=−(a2+a1)2−(b2−b1)2,η6=4a1a2.
Continuing with the same approach in the previous text, we are able to provide a precise expression for the three-component of
the magnetization mfor the dynamic chiral magnetic two-solitons. Owing to the complexity of its explicit expression, we opt
to omit it and solely showcase the corresponding figure. Two typical solutions for the interaction between two chiral magnetic
solitons are shown in Fig.(I).
C. Chirality of cycloidal chiral magnetic soliton in Bloch sphere
As shown in the main text, we denote the polar and azimuthal angles of the vector masθandϕ, respectively. This notation
allows us to express M+=mx+imy=M0sin(θ)exp(iϕ)andmz=M0cos(θ). By employing the three-component analytical
formulation, we can infer the inverse solution for θandϕ, which in turn can be mapped onto the Bloch unit sphere. This
approach yields a trajectory map delineating the movement of chiral magnetic solitons across the unit sphere. Consequently,
within the magnetization unit sphere, a chiral magnetic soliton traces a closed curve encompassing a single pole. The trajectories
of motion for the two distinct types of chiral magnetic soliton solutions on the Bloch spheres can indirectly manifest their
chirality. Commencing from negative infinity, which corresponds to the pole of ground state, the left and right-handed chiral5
(a)
(b)
Figure I: The interaction between two chiral magnetic solitons. (a) Interaction between left-handed and right-handed magnetic
soliton. (b) Bound states formed by two right-handed magnetic Solitons.
Figure II: Trajectories of chiral magnetic solitons on the Bloch sphere at time t=0. (a) Left-handed chiral magnetic soliton
a=1,b=1, (b) Right-handed chiral magnetic soliton a=1,b=−1.
magnetic solitons will give rise to enclosed paths, one proceeding in a clockwise direction and the other counterclockwise. This
motion pattern eventually in mirror-symmetrical trajectories.
∗zyyang@nwu.edu.cn
†jing@nwu.edu.cn |
2103.07008v1.Magnetoelastic_Gilbert_damping_in_magnetostrictive_Fe___0_7__Ga___0_3___thin_films.pdf | Magnetoelastic Gilbert damping in magnetostrictive Fe 0.7Ga 0.3thin films
W. K. Peria,1X. Wang,2H. Yu,2S. Lee,2, 3I. Takeuchi,2and P. A. Crowell1,∗
1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA
2Department of Materials Science and Engineering,
University of Maryland, College Park, Maryland 20742, USA
3Department of Physics, Pukyong National University, Busan 48513, South Korea
We report an enhanced magnetoelastic contribution to the Gilbert damping in highly magne-
tostrictive Fe 0.7Ga0.3thin films. This effect is mitigated for perpendicular-to-plane fields, leading
to a large anisotropy of the Gilbert damping in all of the films (up to a factor of 10 at room tem-
perature). These claims are supported by broadband measurements of the ferromagnetic resonance
linewidths over a range of temperatures (5 to 400 K), which serve to elucidate the effect of both the
magnetostriction andphonon relaxation on the magnetoelastic Gilbert damping.
Among the primary considerations in the design of
spintronics devices is Gilbert damping. However, a full
understanding of the mechanisms which cause damping
of magnetization dynamics in ferromagnets remains elu-
sive. Reports of anisotropy in the Gilbert damping have
proven to be useful tools in the understanding of the un-
derlying mechanisms involved [1–3], but there is much
that is yet unclear. Studies of the temperature depen-
dence also promise to be a uniquely powerful tool for a
complete physical understanding [4, 5], however, there
are few such reports in existence.
Recently, it has been shown that spins can be co-
herently coupled over large distances ( ∼1 mm) using
magnon-phonon coupling [6–8]. It is also well known
that magnetization dynamics can be excited elastically
through this phenomenon [9], but its effect on Gilbert
damping has been largely confined to theoretical calcu-
lations [10–13] and lacks clear experimental validation.
Furthermore, most studies have focused on yttrium iron
garnet (YIG), which is weakly magnetostrictive.
In this Letter, we observe a large and anisotropic mag-
netoelastic contribution to the Gilbert damping in highly
magnetostrictive Fe 0.7Ga0.3films through broadband
measurements of the ferromagnetic resonance (FMR)
linewidths over a wide range of temperatures. The
perpendicular-to-plane linewidths exhibit a relatively low
minimum in the Gilbert damping of approximately 0.004,
similar to that of bcc Fe [14]. At room temperature, the
Gilbert damping is as large as a factor of 10 greater with
field applied in plane relative to out of plane. In fact, for
any given sample and temperature, the anisotropy is, at
minimum, about a factor of 2. We argue this is due to
a mitigation of the magnetoelastic contribution for per-
pendicular magnetization, arising from finite-thickness
boundary conditions and weak elastic coupling to the
substrate. The nonmonotonic temperature dependence
of the Gilbert damping also shows the competing effects
of the magnetostriction, which increases at low tempera-
ture, and the phonon viscosity, which generally decreases
at low temperature.
The Fe 0.7Ga0.3films studied in this letter were de-
posited on SiO 2/Si wafers at room temperature by dcmagnetron sputtering of an Fe 0.7Ga0.3target. The base
pressure of the deposition chamber was 5 ×10−8torr,
and the working pressure was kept at 5 ×10−3torr
with Ar gas. The composition of the Fe 0.7Ga0.3films
was quantitatively analyzed by energy dispersive spec-
troscopy (EDS). Films were grown with thicknesses of
21 nm, 33 nm, 57 nm, and 70 nm (the 21 nm, 57 nm,
and 70 nm belong to the same growth series). An addi-
tional 33 nm film was grown at 200◦C. The 33 nm room
temperature deposition was etched using an ion mill to
obtain films with thicknesses of 17 nm and 26 nm. The
thicknesses of the films were measured using x-ray reflec-
tometry (see Supplemental Material).
The FMR linewidths were measured using a setup in-
volving a coplanar waveguide and modulation of the ap-
plied magnetic field for lock-in detection as described in
Ref. [15]. Measurements were done with the field applied
in the plane (IP) and perpendicular to the plane (PP) of
the film. The sample temperature was varied from 5 K
to 400 K for both IP and PP configurations [16] with
microwave excitation frequencies up to 52 GHz. The res-
onance fields and linewidths were isotropic in the plane,
and the absence of in-plane magnetic anisotropy was ver-
ified with vibrating sample magnetometry (see Supple-
mental Material). This is also consistent with the abun-
dance of grain boundaries observed with atomic force mi-
croscopy (AFM). In analyzing the FMR linewidths, we
consider three contributions: Gilbert damping 4 παf/γ
(αis the Gilbert damping coefficient, fis the microwave
frequency, and γis the gyromagnetic ratio), inhomo-
geneous broadening ∆ H0, and two-magnon scattering
∆HTMS (for IP fields). Eddy current damping and ra-
diative damping contributions [17] are neglected because
we expect them to be small ( <10−4) for these films.
Linewidths of the 70 nm film at 300 K for both con-
figurations of the applied field are shown in Fig. 1(a),
and the IP linewidths with individual contributions to
the linewidth plotted separately in Fig. 1(b). We fit the
IP linewidths using a model of two-magnon scattering
based on granular defects [15, 18, 19]. The fit for the
70 nm film is shown in Fig. 1(b), along with the two-
magnon contribution alone given by the magenta curve.arXiv:2103.07008v1 [cond-mat.mtrl-sci] 11 Mar 20212
01 02030405005001000150020000500100015002000/s61508
H0/s61508HTMS/s61508H (Oe)F
requency (GHz)H in-plane/s61508
HGilbertT = 300 K/s61537
= 0.0390 /s61617 0.0005/s61560
= 17 nm(a)(
b)/s61508 H (Oe)T = 300 K/s61537
IP = 0.0390 /s61617 0.0005/s61537
PP = 0.0035 /s61617 0.0001/s61508HIP/s61508
HPP70 nm film
FIG. 1. (a) FMR linewidths for IP (black squares) and
PP (red circles) configurations for the 70 nm film. The IP
linewidths are fit to a model of two-magnon scattering and
the PP linewidths are fit using the standard Gilbert damping
model. (b) Total linewidth (solid black), Gilbert linewidth
(dotted blue), two-magnon scattering linewidth (dashed ma-
genta), and inhomogeneous broadening (dashed/dotted red)
for the 70 nm film with IP field.
The fit parameters are the Gilbert damping α(indicated
on the figure) and the RMS inhomogeneity field H/prime. The
defect correlation length ξis fixed to 17 nm based on the
structural coherence length obtained with x-ray diffrac-
tion (XRD), which agrees well with the average grain di-
ameter observed with AFM (see Supplemental Material).
Furthermore, the high-frequency slope of the linewidths
approaches that of the Gilbert damping since the two-
magnon linewidth becomes constant at high frequencies
[see Fig. 1(b)].
We now compare the IP and PP linewidths of the
70 nm film shown in Fig. 1(a). The two-magnon scat-
tering mechanism is inactive with the magnetization per-
pendicular to the plane [20], and so the PP linewidths are
fit linearly to extract the Gilbert damping. We obtain a
value of 0.0035±0.0001 for PP fields and 0 .039±0.0005
for IP fields, corresponding to an anisotropy larger than
a factor of 10. Li et al. [3] recently reported a large
anisotropy (∼factor of 4) in epitaxial Co 50Fe50thin films.
First we discuss the dependence of the PP Gilbert
dampingαPPon temperature for all of the films, shown
in Fig. 2. We observe a significant temperature depen-
01 002 003 004 00024685
7 nm 17 nm2
6 nm3
3 nm (RT dep)/s61537 (×10-3)T
emperature (K)21 nm 3
3 nm (200 °C dep)H
perpendicular-to-plane70 nmFIG. 2. Gilbert damping αfor PP field shown as a func-
tion of temperature for the 17 nm (orange), 21 nm (blue),
26 nm (green), 33 nm room temperature deposition (ma-
genta), 33 nm 200◦C deposition (gold), 57 nm (red), and
70 nm (black) Fe 0.7Ga0.3films.
dence in all cases (with the exception of the 33 nm room
temperature deposition), characterized by a maximum
at around 50 K. Then, at the lowest temperatures (5 to
10 K),αPPapproaches the same value for all of the films
(/similarequal0.004).
Now we turn to the temperature dependence of the
IP Gilbert damping αIPshown in Fig. 3. The values
obtained here were obtained by fitting the linewidths lin-
early, but excluding the low-frequency points ( <∼20 GHz)
since the two-magnon scattering becomes constant at
high frequencies [21]. Here we note, upon comparison
with Fig. 2, that a large anisotropy of the Gilbert damp-
ing exists for all of the samples. In the 70 nm film, for
instance,αIPis more than a factor of 10 larger than αPP
at 300 K. In the temperature dependence of αIP, we ob-
serve behavior which is similar to that seen in αPP(Fig.
2), namely, a maximum at around 50 K (with the excep-
tion of the 21 nm film). Here, however, αIPdoes not
approach a common value at the lowest temperatures in
all of the samples as it does in the PP case.
The IP Gilbert damping is larger than the PP Gilbert
damping for all of the samples over the entire range of
temperatures measured. This anisotropy of the Gilbert
damping—along with the nonmonotonic temperature
dependence—in all seven samples implies a contribution
to the Gilbert damping in addition to Kambersk´ y damp-
ing. We have verified that the orientation of FeGa(110)3
planes is completely random with XRD for the 33 nm
(both depositions) and 70 nm films (see Supplemen-
tal Material), and it is therefore not possible that the
anisotropy is due to Kambersk´ y damping. Interface
anisotropy has reportedly led to anisotropic Kambersk´ y
damping in ultrathin ( ∼1 nm) films of Fe [2], but this
is highly unlikely in our case due to the relatively large
thicknesses of the films. In addition, the fact that the
damping anisotropy shows no clear correlation with film
thickness furthers the case that intrinsic effects, which
tend to show a larger anisotropy in thinner films [2],
cannot be the cause. The longitudinal resistivity ρxxof
the 33 nm (both depositions) and 70 nm films (see Sup-
plemental Material) shows very weak temperature de-
pendence. In the Kambersk´ y model, the temperature
dependence of the damping is primarily determined by
the electron momentum relaxation time τ, and we would
therefore not expect the Kambersk´ y damping to show
a significant temperature dependence for samples where
the residual resistivity ratio is approximately unity. It is
plausible that the Kambersk´ y damping would still show
a temperature dependence in situations where the spin
polarization is a strong function of temperature, due to
changes in the amount of interband spin-flip scattering.
This kind of damping, however, would be expected to
decrease at low temperature [22, 23]. The temperature
dependence we observe for both αPPandαIPis therefore
inconsistent with Kambersk´ y’s model, and the similarity
between the two cases in this regard suggests that the
enhanced Gilbert damping has a common cause that is
mitigated in the PP configuration.
It has been proposed that magnetoelastic coupling
can lead to Gilbertlike magnetization damping through
phonon relaxation processes [10, 12, 24]. Similar treat-
ments calculate the magnetoelastic energy loss through
interaction with the thermal population of phonons
[11, 25]. The Kambersk´ y mechanism is often assumed to
be the dominant Gilbert damping mechanism in metal-
lic samples, so magnetoelastic Gilbert damping is usually
studied in magnetic insulators, particularly yttrium iron
garnet (YIG). There is the possibility, however, for the
magnetoelastic damping to dominate in metallic samples
where the magnetostriction is large, such as in Fe-Ga al-
loys. Later we will discuss how magnetoelastic damping
can be mitigated in thin films by orienting the magneti-
zation perpendicular to the plane, and how the degree to
which it is mitigated depends on the boundary conditions
of the film.
Here we outline a theory of magnetoelastic damping,
which relies on the damping of magnetoelastic modes
through phonon relaxation mechanisms. Figure 4 illus-
trates the flow of energy through such a process. Analyt-
ically, the procedure is to equate the steady-state heating
rate due to Gilbert damping to the heating rate due to
crystal viscosity, and solve for the Gilbert damping α
in terms of the crystal shear viscosity ηand the mag-
01 002 003 004 000123453
3 nm (200 °C dep)17 nm2
6 nm/s61537 (×10-2)T
emperature (K)33 nm (RT dep)57 nm2
1 nmH in-plane7
0 nmFIG. 3. Gilbert damping αfor IP field shown as a func-
tion of temperature for the 17 nm (orange), 21 nm (blue),
26 nm (green), 33 nm room temperature deposition (ma-
genta), 33 nm 200◦C deposition (gold), 57 nm (red), and
70 nm (black) Fe 0.7Ga0.3films.
netostrictive coefficients λhkl. Shear strain uijresult-
ing from the magnetoelastic interaction can be expressed
asuij=λ111mimj[26], where mi≡Mi/Msare the
reduced magnetizations. The leading-order shears thus
have equations of motion given by ˙ uiz=λ111˙mi, where
i=xory, andzis the direction of the static magnetiza-
tion so that mz≈1. Longitudinal modes are quadratic
in the dynamical component of the magnetization [24]
and so will be neglected in this analysis.
The heating rate due to Gilbert damping can be writ-
ten as ˙Qα=Ms
γα( ˙m2
x+ ˙m2
y), and the heating rate due to
the damping of phonon modes as ˙Qη= 4η( ˙u2
xz+ ˙u2
yz) =
4ηλ2
111( ˙m2
x+ ˙m2
y) [12], with the factor of 4 accounting
for the symmetry of the strain tensor. Equating the two,
and solving for α(henceforward referred to as αme), we
obtain
αme=4γ
Msηλ2
111. (1)
We will restrict our attention to the case of isotropic mag-
netostriction, and set λ111=λ.
In order to use Eq. 1 to estimate αmein our films,
we first estimate the shear viscosity, given for transverse
phonons with frequency ωand relaxation time τas [27]
η=2ρc2
t
ω2τ, (2)4
(b)
u(t)
phonon pumpingM(t) H0
M(t)
u(t)(a)
kph
dH0
FIG. 4. (a) Depiction of magnetoelastic damping process
for magnetization in plane and (b) perpendicular to plane,
where M(t) is the magnetization vector and u(t) is the lattice
displacement. In panel (b), the magnon-phonon conversion
process is suppressed when d < π/k ph, wheredis the film
thickness and kphis the transverse phonon wavenumber at
the FMR frequency.
whereρis the mass density and ctis the transverse
speed of sound. Using ω/2π= 10 GHz, τ= 10−11s,
andct= 2.5 km/s, we obtain η≈2.3 Pa s. (The
estimate of the phonon relaxation time is based on a
phonon mean free path of the order of the grain size:
∼10 nm.) Furthermore, the magnetostriction of an equiv-
alent sample has been measured to be ∼100 ppm at room
temperature [28]. Then, with γ/2π= 29 GHz/T and
Ms= 1123 emu/cc (extracted from FMR data taken at
300 K), we estimate αme≈0.016. This estimate gives us
immediate cause to suspect that magnetoelastic Gilbert
damping is significant (or even dominant) in these films.
We now discuss why the magnetoelastic damping can
be much weaker for PP magnetization in sufficiently thin
films. We will start by assuming that there is no coupling
between the film and substrate, and later we will relax
this assumption. In this case the only phonons excited
by the magnetization, to leading-order in the magneti-
zations and strains, are transverse modes propagating in
the direction of the static magnetization [24]. One may
assume that the minimum allowable phonon wavenum-
ber is given by π/d, wheredis the film thickness, since
this corresponds to the minimum wavenumber for a sub-
strate having much lower acoustic impedance than the
film (requiring the phonons to have antinodes at the in-
terfaces) [13]. (We also assume an easy-axis magnetic
anisotropy at the interfaces, so that the dynamical mag-
netizations have antinodes at the interfaces.) We expect
then that the magnetoelastic damping will be suppressed
for cases where the phonon wavelength, at the frequencyof the precessing magnetization, is greater than twice the
film thickness [see Fig. 4(b)]. Thus, in sufficiently thin
films (with weakly-coupled substrates), the magnetoelas-
tic damping process can be suppressed when the mag-
netization is perpendicular to the plane. However, the
magnetoelastic damping can be active (albeit mitigated)
when there is nonnegligible or “intermediate” coupling
to the substrate.
Before moving on, we briefly note the implications of
Eq. (1) for the temperature dependence of the Gilbert
damping. On the basis of the magnetostriction alone,
αmewould be expected to increase monotonically as tem-
perature is decreased ( λhas been shown to increase by
nearly a factor of 2 from room temperature to 4 K in
bulk samples with similar compositions [29]). However,
the viscosity ηwould be expected to decrease at low tem-
perature, leading to the possibility of a local maximum
inαme. In polycrystalline samples where the grain size
is smaller than the phonon wavelength, viscous damping
of phonons due to thermal conduction caused by stress
inhomogeneities can be significant [27, 30]. (In our case
the phonon wavelengths are ∼100 nm and the grain
sizes are∼10 nm.) This effect scales with temperature
asη∼Tα2
T/Cχ [30], where αTis the thermal expansion
coefficient, Cis the specific heat at constant volume, and
χis the compressibility. At higher temperatures, αTand
Cwill approach constant values, and χwill always de-
pend weakly on temperature. We therefore expect that
the viscosity is approximately linear in T. In this case,
αmeis maximized where λ2(T) has an inflection point.
We proceed to explain our data in terms of the mecha-
nism described above, turning our attention again to the
PP Gilbert damping for all of the films shown in Fig.
2. We previously argued that the magnetoelastic damp-
ing mechanism will be suppressed for the case where the
acoustic impedances of the film and substrate are mis-
matched. However, the clear dependence on tempera-
ture, which we have already shown is inconsistent with
Kambersk´ y damping, appears to be consistent with the
magnetoelastic damping mechanism. We estimate that
the acoustic impedance of the film (defined as the product
of mass density ρand transverse speed of sound ct[13])
is about a factor of 2 larger than the substrate. This sug-
gests that the elastic coupling between the film and sub-
strate, albeit weak, may be nonnegligible. Furthermore,
experiments with YIG/GGG heterostructures (where the
acoustic match is good) have demonstrated magnetic ex-
citation of phononic standing waves that have boundary
conditions dictated by the combined thickness of the film
and substrate, rather than the film thickness alone (i.e.,
the wavelengths are much larger than the film thickness)
[6, 31]. In this case, the Gilbert damping may contain
some contribution from the magnetoelastic mechanism.
A final point is that αPPapproaches/similarequal0.004 at 5 to
10 K for all of the films. Both the magnetostriction and
the viscosity are quantities which could have significant5
01 0020030040002468/s61537me (×10-2)T
emperature (K)Ref. [29]2
1 nm70 nm5
7 nm0
.000.250.500.751.00/s61548
2(T)//s615482(0)010020030001/s61544 (arb. units)T
(K)
FIG. 5. Magnetoelastic Gilbert damping αmefor the 21 nm
(blue), 57 nm (red), and 70 nm (black) films (left ordinate)
andλ2(T)/λ2(0) from Clark et al. [29] (magenta; right or-
dinate) shown as a function of temperature. Inset shows the
ratio ofαmeandλ2(T)/λ2(0), labeled as η(T), along with lin-
ear fits for the 21 nm (blue), 57 nm (red), and 70 nm (black)
films.
variation between samples, leading to variations in αme.
However, the viscosity becomes small at low temperature,
which means that the Gilbert damping will approach the
Kambersk´ y “limit,” a property that is determined by the
electronic structure, implying that the Kambersk´ y damp-
ing is/similarequal0.004 in these films and that it is the primary
contribution to the Gilbert damping near T= 0.
Now we revisit the IP Gilbert damping shown in Fig.
3. In this configuration, there is a strong temperature
dependence of the Gilbert damping similar to that of
the PP case, again implying the presence of magnetoe-
lastic damping. However, the overall magnitude is much
higher. That is because in this case arbitrarily long wave-
length phonons can be excited regardless of the thick-
ness of the film. Although we cannot directly measure
the magnetostriction as a function of temperature, we
estimate the scaling behavior of λby interpolating the
data in Ref. [29] taken for bulk samples of similar com-
position. In order to demonstrate that αIPscales with
temperature as expected from the model, we have plot-
ted the quantities αmeandλ2(T)/λ2(0) as functions of
temperature in Fig. 5—where we define the quantity
αme≡αIP−0.004—for the 21 nm, 57 nm, and 70 nm
films (which are part of the same growth). The corre-
lation between the two quantities is not completely con-
vincing. There is, however, an additional temperature
dependence in αmebesidesλ2(T), namely, the viscosity
η(T). The inset of Fig. 5 shows the ratio of αmeand
λ2(T), which [from Eq. (1)] is proportional to η(T). The
linear fits provide strong evidence that the mechanism
behind the viscosity is indeed the thermal conduction
process that we have argued is approximately linear inT. It is noteworthy that the maximum in αme(∼50 to
75 K for all of the samples) coincides approximately with
the inflection point in λ2(T). This was a consquence of
our assumption that η(T) should be roughly linear. We
also obtain a significant value for the zero-temperature
viscosity, which is around 25 % of the value at 300 K. This
is likely due to boundary-scattering processes which will
preventαmefrom going to zero at low temperatures, par-
ticularly for in-plane magnetization where αmeis much
larger than 0.004 (our estimate for the Kambersk´ y damp-
ing). For the PP case, αmeis much smaller due to limita-
tions on the wavelengths of phonons that can be excited,
so the Gilbert damping of all the samples approaches the
Kambersk´ y limit of 0.004 near zero temperature. We also
found that η(T) was linear for the 33 nm (200◦C depo-
sition) film, but had a more complicated dependence on
Tfor the 17 nm, 26 nm, and 33 nm (room temperature
deposition) films (the latter three being notably of the
same growth). The viscosity near zero temperature is
within roughly a factor of 2 for all seven of the samples,
however.
Finally, we propose that this mechanism may be re-
sponsible for a Gilbert damping anisotropy of similar
magnitude reported in Ref. [3], observed in an epitax-
ial Co 0.5Fe0.5thin film. The authors attributed the
anisotropy to the Kambersk´ y mechanism [22, 23, 32, 33],
arising from tetragonal distortions of the lattice. The
magnetostriction is known to be highly anisotropic in
bulk Co 0.5Fe0.5,viz.,λ100= 150 ppm and λ111= 30 ppm
[34]. We therefore expect that the Gilbert damping aris-
ing from the mechanism we have described may be much
larger for M/bardbl(110) than M/bardbl(100), which is precisely
what the authors observed.
In summary, we observe large and anisotropic magne-
toelastic Gilbert damping in Fe 0.7Ga0.3polycrystalline
thin films (thicknesses ranging from 17 to 70 nm). At
300 K, the damping coefficient is more than a factor of
10 larger for field in plane than it is for field perpendicu-
lar to the plane in the 70 nm film. The large anisotropy
is caused by a mitigation of the magnetoelastic effect for
perpendicular-to-plane fields due to a dependence on the
elastic coupling of the film to the substrate, which in our
case is weak. Finally, there is a nonmonotonic tempera-
ture dependence of the Gilbert damping, which we show
is consistent with our model.
We acknowledge Rohit Pant and Dyland Kirsch for as-
sistance with thin film deposition and characterization.
This work was supported by SMART, a center funded
by nCORE, a Semiconductor Research Corporation pro-
gram sponsored by NIST. Parts of this work were carried
out in the Characterization Facility, University of Min-
nesota, which receives partial support from NSF through
the MRSEC program, and the Minnesota Nano Cen-
ter, which is supported by NSF through the National
Nano Coordinated Infrastructure Network, Award Num-
ber NNCI - 1542202.6
∗Author to whom correspondence should be addressed:
crowell@umn.edu
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Phys. 31, S157 (1960).Supplemental Material for
“Magnetoelastic Gilbert damping in magnetostrictive Fe 0.7Ga 0.3thin films”
W. K. Peria,1X. Wang,2H. Yu,2S. Lee,2, 3I. Takeuchi,2and P. A. Crowell1
1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA
2Department of Materials Science and Engineering,
University of Maryland, College Park, Maryland 20742, USA
3Department of Physics, Pukyong National University, Busan 48513, South Korea
CONTENTS
S1. Magnetization dynamics 1
S2. Ferromagnetic resonance linewidths of 70 nm film 2
S3. X-ray reflectivity 2
S4. X-ray diffraction 3
S5. Atomic force microscopy 3
S6. Vibrating sample magnetometry 3
S7. Longitudinal resistivity 3
References 4
S1. MAGNETIZATION DYNAMICS
The treatment of magnetization dynamics begins with the Landau-Lifshitz-Gilbert equation of motion
dM
dt=−γM×Heff+α
MsM×dM
dt(S1)
where the relaxation is characterized by the Gilbert damping parameter α. Upon linearizing this equation in the
dynamic component of the magnetization, one obtains for the ac magnetic susceptibility of the uniform q= 0 mode
χac(q= 0,ω)∝αω/γ
(H−HFMR )2+ (αω/γ )2(S2)
so that the field-swept full-width-at-half-maximum linewidth is given by ∆ HFWHM = 2αω/γ . Therefore, the Gilbert
damping parameter αis obtained by measuring ∆ HFWHM as a function of ω.
Relaxation of the uniform mode can include mechanisms which are not described by Gilbert damping. The most
common of these is inhomogeneous broadening, which results from inhomogeneities in the system and is constant
as a function of frequency. Another mechanism is two-magnon scattering, which is also extrinsic in nature. Two-
magnon scattering originates from the negative group velocity at low qof the backward volume mode magnons for
in-plane magnetization. The negative group velocity is due to a lowering of the magnetostatic surface charge energy
for increasing q. The existence of negative group velocity at low qleads to the appearance of a mode at nonzero q
that is degenerate with the uniform mode. Two-magnon scattering refers to the scattering of the uniform mode to
the nonuniform degenerate mode.
Much work has been done on the treatment of two-magnon scattering [S1–S3], and here we will simply give an
expression for the contribution of two-magnon scattering to the field-swept linewidth
∆HTMS =γ2ξ2H/prime2
dω/dH/integraldisplay
d2qΛ0q1
(1 + (qξ)2)3/21
πωα
(ωα)2+ (ω−ωFMR )2(S3)
withξthe defect correlation length, H/primethe RMS inhomogeneity field, and Λ 0qthe magnon-magnon coupling. In
general, this leads to a nonlinear dependence of the linewidth on frequency. Eq. S3 is used to fit the IP linewidths.arXiv:2103.07008v1 [cond-mat.mtrl-sci] 11 Mar 20212
0102030405005001000150020002500(b)
5 K
50 K
150 K
225 K
300 K
400 K/s61508H (Oe)F
requency (GHz)H in-plane
010203040500100200300400500 5 K
50 K
150 K
225 K
300 K
400 K/s61508H (Oe)F
requency (GHz)H perpendicular-to-plane(a)
FIG. S1. FMR linewidths of the 70 nm film with field PP (a) and IP (b) for sample temperatures of 5 K (blue), 50 K (gold),
150 K (black), 225 K (magenta), 300 K (red), and 400 K (orange). The solid lines are linear fits in both panels. In (b), the
vertical dashed line indicates the lower bound of the points included in the fit.
12345610-1100101102103104105106(c)Intensity (arb. units)/s61553
/2/s61553 (degrees)d = 56.6 nm
12345610-1100101102103104105106(b)Intensity (arb. units)/s61553
/2/s61553 (degrees)200 °C depositiond
= 33.1 nm
123410-1100101102103104105106(a)R
T depositiond
= 33.2 nm Intensity (arb. units)/s61553
/2/s61553 (degrees)
FIG. S2. X-ray reflectivity data (black) overlaid with fits (red) for the (a) 33 nm (room temperature deposition), (b) 33 nm
(200◦C deposition), and (c) 57 nm films. Thicknesses dobtained from the fits are indicated on the figure.
S2. FERROMAGNETIC RESONANCE LINEWIDTHS OF 70 nm FILM
The field-swept FMR linewidths of the 70 nm film are shown in Fig. S1 for field PP and IP. For the case of field
IP, the data above 23 GHz were fit linearly to obtain the Gilbert damping. (This value varied between different
samples since the characteristic roll-off frequency depends on both defect lengthscale and film thickness, but remained
in the range 20 to 25 GHz.) It is safe to do this provided there are no inhomogeneities at lengthscales smaller than a
few nm, which could cause the two-magnon scattering contribution to the linewidth to roll off at higher frequencies.
We believe that defects at such small lengthscales are highly unlikely given the characterization performed on these
samples.
S3. X-RAY REFLECTIVITY
In Fig. S2 we show x-ray reflectivity measurements at grazing incidence for 33 nm (room temperature and 200◦C
depositions) and 57 nm films. The measurements were taken using a Rigaku SmartLab diffractometer. The thicknesses
dyielded by the fits of the data are indicated on the figure.3
4243444546036912F
e0.7Ga0.3(110)(a)I
ntensity (arb. units)/s61553
/2/s61553 (degrees)33 nm (RT deposition)2
/s61553c= 44.14 /s61617 0.03 °F
WHM = 0.78 /s61617 0.09 °
4243444546036912F
e0.7Ga0.3(110)(b)I
ntensity (arb. units)/s61553
/2/s61553 (degrees)33 nm (200 °C deposition)2
/s61553c = 44.07 /s61617 0.05 °F
WHM = 1.18 /s61617 0.17 °
4243444546036912(c)F
e0.7Ga0.3(110)Intensity (arb. units)/s61553
/2/s61553 (degrees)2/s61553c = 44.36 /s61617 0.02 °F
WHM = 0.50 /s61617 0.07 °70 nm
FIG. S3. X-ray diffraction symmetric θ/2θscans for (a) 33 nm room temperature deposition, (b) 33 nm 200◦C deposition,
and (c) 70 nm films. Full width at half maxima (FWHM) and 2 θcenter positions are indicated on the figure.
S4. X-RAY DIFFRACTION
X-ray diffraction (XRD) measurements were performed in order to determine both the degree of orientation and
the structural coherence length of the films.
Symmetric θ/2θscans were taken with a Rigaku Smartlab diffractometer using Cu Kα 1(λ= 1.54˚A) radiation.
The data for both samples are shown in Fig. S3. The grain size was estimated using the Scherrer formula for spherical
grains [S4] as 13 nm, 9 nm, and 17 nm for the 33 nm (room temperature deposition), 33 nm (200◦C deposition), and
70 nm films respectively.
Two-dimensional images were collected with a Bruker D8 Discover diffractometer using Co Kα 1(λ= 1.79˚A)
radiation. Detector images showing the “ring” corresponding to the Fe 0.7Ga0.3(110) peak in four different samples
are shown in Fig. S4. The ring indicates that the Fe 0.7Ga0.3(110) planes are randomly oriented over the range of the
detector, which we take to be evidence that there is no texture over a macroscopic scale in these samples. Furthermore,
the films were grown directly on top of amorphous SiO 2layers, so we do not expect an epitaxial relationship between
the film and substrate. The Fe 0.7Ga0.3(110) peaks were the only measurable Bragg peaks since the structure factor
is highest for this case.
S5. ATOMIC FORCE MICROSCOPY
Atomic force microscopy data are shown in Fig. S5 for the 33 nm (room temperature and 200◦C depositions),
57 nm, and 70 nm films. The field-of-view is 250 nm for the 33 nm films and 500 nm for the 57 nm and 70 nm films.
The root-mean-square (RMS) roughness of the sample surfaces is 0.7 nm, 0.4 nm, 1.5 nm, and 1.3 nm for the 33 nm
(room temperature deposition), 33 nm (200◦C deposition), 57 nm, and 70 nm films, respectively .
S6. VIBRATING SAMPLE MAGNETOMETRY
Vibrating sample magnetometry (VSM) data for the 33 nm (room temperature and 200◦C depositions) and 70 nm
films are shown in Fig. S6. The magnetic field was applied in 3 different directions, with no discernible difference in the
hysteresis loops. We conclude that there is no in-plane magnetocrystalline anisotropy over macroscopic lengthscales,
which is consistent with the FMR measurements.
S7. LONGITUDINAL RESISTIVITY
Longitudinal resistivity ρxxwas measured as a function of temperature for the 33 nm (room temperature and 200◦C
depositions) and 70 nm films (Fig. S7) by patterning Hall bars and performing 4-wire resistance measurements.4
65 60 55 50 45 40 35
2/s61553(°)
0100200300400Intensity
(arb.units)
65 60 55 50 45 40 35
2/s61553(°)
0100200300400Intensity
(arb.units)
65 60 55 50 45 40 35
2/s61553(°)
020406080Intensity
(arb.units)
65 60 55 50 45 40 35
2/s61553(°)
020406080Intensity
(arb.units)(b) (a)
(c) (d)q
xyz
q
xyz33 nm (RT) 33 nm (200 °C)
57 nm 70 nm
FIG. S4. Two-dimensional detector images of the Fe 0.7Ga0.3(110) peak for (a) 33 nm (room temperature deposition), (b)
33 nm (200◦C deposition), (c) 57 nm, and (d) 70 nm films. The total scattering angle is 2 θand is shown on the abscissa. The
measurement is conducted such that the symmetric configuration corresponds to the center of the detector, which is to say
that the incident radiation is at an angle ω/similarequal26◦relative to the sample surface. In panel (a), the effect of moving vertically
from the center of the detector on the scattering vector qis shown ( qis canted into the y-zplane).
[S1] R. Arias and D. L. Mills, Extrinsic contributions to the ferromagnetic resonance response of ultrathin films, Phys. Rev. B
60, 7395 (1999).
[S2] R. D. McMichael and P. Krivosik, Classical Model of Extrinsic Ferromagnetic Resonance Linewidth in Ultrathin Films,
IEEE Trans. Magn. 40, 2 (2004).
[S3] P. Krivosik, N. Mo, S. Kalarickal, and C. E. Patton, Hamiltonian formalism for two magnon scattering microwave relaxation:
Theory and applications, J. Appl. Phys. 101, 083901 (2007).5
-2-1012Height(nm)(a)
-2-1012Height(nm)(b)
(c) (d)
-2-1012Height(nm)
-2-1012Height(nm)100 nm
100 nm100 nm
100 nm33 nm (RT)
57 nm33 nm (200 °C)
70 nm
FIG. S5. Atomic force microscopy for (a) 33 nm (room temperature deposition), (b) 33 nm (200◦C deposition), (c) 57 nm,
and (d) 70 nm films. RMS roughnesses are (a) 0.7 nm, (b) 0.4 nm, (c) 1.5 nm, and (d) 1.3 nm.
[S4] M. Birkholz, Thin Film Analysis by X-Ray Scattering (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2006).6
-2000-1000010002000-101M/MsF
ield (Oe) H || Si[100]
H || Si[110]
H || Si[010]70 nm(c)
-200-1000100200-101
H || Si[100]
H || Si[110]
H || Si[010]M/MsF
ield (Oe)33 nm (
200 °C)(b)
-50-2502550-101
H || Si[100]
H || Si[110]
H || Si[010]M/Ms F
ield (Oe)33 nm (RT)(a)
FIG. S6. Vibrating sample magnetometry of (a) 33 nm (room temperature deposition), (b) 33 nm (200◦C deposition), and
(c) 70 nm films for H/bardblSi[100] (black), H/bardblSi[110] (red), and H/bardblSi[010] (blue).
0100200300400020040060080010007
0 nm(c)/s61554 xx (µΩ cm)T
emperature (K)
0100200300400050100150200250/s61554xx (µΩ cm)T
emperature (K)33 nm (200 °C deposition)(b)
0100200300400050100150200250(a)/s61554xx (µΩ cm)T
emperature (K)33 nm (RT deposition)
FIG. S7. Longitudinal resistivity ρxxas a function of temperature for the (a) 33 nm (room temperature deposition), (b) 33 nm
(200◦C deposition), and (c) 70 nm films. |
1905.13042v1.Predicting_New_Iron_Garnet_Thin_Films_with_Perpendicular_Magnetic_Anisotropy.pdf | 1
Predicting New Iron Garnet Thin Film s with Perpendicular Magnetic Anisotropy
Saeedeh Mokarian Zanjani1, Mehmet C. Onbaşlı1,2,*
1 Graduate School of Materials Science and Engineering, Koç University , Sarıyer, 34450 Istanbul,
Turkey .
2 Department of Electrical and Electronics Engineering, Koç University, Sarıyer, 34450 Istanbul,
Turkey.
* Corresponding Author: monbasli@ku.edu.tr
Abstract:
Perpendicular magnetic anisotropy (PMA) is a necessary condition for many spintronic
applications like spin -orbit torques switching , logic and memory devices. An important class of
magnetic insulators with low Gilbert damping at room temperature are iron garnets, which only
have a few PMA types such as terbium and samarium iron garnet. More and stable PMA garnet
options are necessary for researchers to be able to investigate new spintronic phenomena. In this
study, we predict 20 new substrate/magnetic iron garnet film pairs with stable PMA at room
temperature. The effective anisotropy energies of 10 different garnet films that are lattice -matched
to 5 different commercially available garnet substrates have been calculated using shape,
magnetoelastic and magnetocrystalline anisotropy terms . Strain type, tensile o r compressive
depending on substrate choice, as well as the sign and the magnitude of the magnetostriction
constants of garnets determine if a garnet film may possess PMA. We show the conditions in which
Samarium, Gadolinium, Terbium, Holmium, Dysprosium a nd Thulium garnets may possess PMA
on the investigated garnet substrate types . Guidelines for obtaining garnet films with low damping
are presented. New PMA garnet film s with tunable saturation moment and field may improve spin -
orbit torque memory and compensated magnonic thin film devices.
2
Introduction
With the development of sputtering and pulsed laser deposition of high -quality iron garnet thin
films with ultralow Gilbert damping, researchers have been able t o investigate a wide variety of
magnetization switching and spin wave phenomena1-3. The key enabler in many of these studies
has been Yttrium iron garnet (Y 3Fe5O12, YIG)4 which has a very low Gilbert damping allowing
spin waves to propagate over multiple millimeters across chip. YIG thin films are useful for spin
wave device applications, but since their easy axes lie along film plane, their utility cannot b e
extended to different mechanisms such as spin -orbit torques, Rashba -Edelstein effect, logic
devices, forward volume magnetostatic spin waves1. At the same time, to have reliable and fast
response using low current densities as in spin -orbit torque switching, magnetization orientation
needs to be perpendicular to the surface plane5. The possibility of having Dzyaloshinskii –Moriya
interaction (DMI) in TmIG/GGG may enable stabilizing skyrmions and help drive skyrmion
motion with pure spin currents6.
There is a number of studies on tuning anisotropy or obtaining perpendicular magnetic anisotropy
in insulator thin films7-12. Among the materials studied, insulating magnetic garnet s whose
magnetic pr operties can be tuned have been a matter of interest over the past decades13-15 due to
their low damping and high magnetooptical Faraday rotation. In order to obtain perpendicular
magnetic anisotropy in magnetic garnets, one needs to engineer the anisotropy terms that give rise
to out -of-plane easy axis. Angular dependence of total m agnetization energy density is called
magnetic anisotropy energy and consists of contributions from shape anisotropy, strain -induced
(magnetoelastic) and magnetocrystalline anisotropy. A magnetic material preferentially relaxes its
magnetization vector tow ards its easy axis, which is the least energy axis , when there is no external
field bias . Such energy minimization process drives magnetic switching rates as well as the stability
of total magnetization v ector . Controlling magnetic anisotropy in thin film garnets not only offers
researchers different testbeds for experimenting new PMA -based switching phenomena, but also
allows the investigation of anisotropy -driven ultrafast dynamic magnetic response in thin film s and
nanostructures.
The most extensively studied garnet thin film is Yttrium Iron Garnet ( YIG). YIG films display in -
plane easy axis because of their large shape anisotropy and negligible magnetocrystalline
anisotropy3. Although PMA of ultrathin epitaxial YIG films has been reported16,17, the tolerance 3
for fabrication condition variations for PMA YIG is very limited and strain effects were found to
change magnetocrystalline anisotropy in YIG. Strain -controlled anisotr opy has been observed in
polycrystalline ultrathin YIG films17,18. In case of YIG thin film grown on Gadolinium Gallium
Garnet (GGG), only partial anisotropy control has been possible through significant change in
oxygen stoichiometry19, which increases damping. Since the fabrication of high -quality and highly
PMA YIG films is not easy for practical thicknesses on gadolinium gallium garnet substrates
(GGG), researchers have explored tuning magnetic anisotropy by substituting Yttrium sites with
other rare earth elements20,21. New garnet thin films that can exhibit PMA with different
coercivities, saturation fields, compensation points and tunable Gilbert damping values must be
developed in order to evaluate the effect of these p arameters on optimized spintronic insulator
devices .
Since the dominant anisotropy energy term is s hape anisotropy in thin film YIG , some studies focus
on reducing the shape anisotropy contribution by micro and nanopatterning22-24. Continuous YIG
films were etched to form rectangular nanostrips with nanometer -scale thickness es, as
schematic ally shown on Fig. 1(a) . Thus, least magnetic saturation field is needed along the longest
dimension of YIG nanostrips . By growing ultrathin YIG, magnetoelastic strain contributions lead
to a negative anisotropy field and thus PMA in YIG film s24. As the length -to-thickness ratio
decreases , the effect of shape anisotropy is reduced and in-plane easy axis is converted to PMA17.
While reducing the effect of shape anisotropy is necessary, one also needs to use m agnetoelastic
anisotropy contribut ion to reorient magnetic easy axis towards out of film plane , as schematically
shown on Fig. 1 (b). Strain-induced perpendicular magnetic anisotropy in rare earth (RE) iron
garnets, especially in YIG, has been demonstrated to overcome shape anisotropy16,17,25,26. If
magnetoelastic anisotropy term induced by crystal lattice mismatch is large r than shape anisotropy
and has opposite sign, then magnetoelastic anisotropy overcome s shape . Thus, the easy axis of the
film becomes perpendicular to the film plane and the hyster esis loop becomes square -shaped with
low saturation field8. One can also achieve PMA in other RE magnetic iron garnets due to their
lattice parameter mismatch with their substrates . PMA has previously been achieved using
Substituted Gadolinium Gallium Garnet (SGGG) as substrate and a Samarium Gallium Garnet
(SmGG) ultrathin film as buffer layer under (and on) YIG16. In case of thicker YIG films (40nm),
the magnetic easy axis becomes in -plane again. An important case shown by Kubota et.al19
indicates that increasing in-plane strain (ε ||) or anisotropy field (H a) helps achieve perpendicular 4
magnetic anisotropy. In ref.8,19, they reported that if magnetostriction coefficient (λ 111) is negative
and large eno ugh to overcome shape anisotropy, and tensile strain is introduced to the thin film
sample (ε ||>0), the easy axis becomes perpendicular to the sample plane as in the case of Thul ium
iron garnet (Tm 3Fe5O12, TmIG ).
A different form of magnetoelastic anisotropy effect can be induced by using porosity in garnet
thin films. Mesoporous Holmium Iron Garnet (Ho 3Fe5O12, HoIG) thin film on Si (001)27 exhibit s
PMA due to reduced shape anisotropy, increased magnetostrictive and growth -induced anisotropy
effects. Such combined effects lead to PMA in HoIG. In this porous thin film, the PMA was found
to be independent of the substrate used, because the mechanical s tress does not result from a lattice
or thermal expansion mismatch between the substrate and the film. Instead, the pore -solid
architecture itself imposes an intrinsic strain on the solution processed garnet film. This example
indicates that the film struc ture can be engineered in addition to the substrate choices in order to
overcome shape anisotropy in thin film iron garnets.
Another key method for controlling anisotropy is strain doping through substitutional elements and
using their growth -induced aniso tropy effects , as schematically shown on Fig. 1(c) . Bi-doped
yttrium iron garnet (Bi:YIG and Bi:GdIG) has been reported to possess perpendicular magnetic
anisotropy due to the chemical composition change as the result of increased annealing
temperature21,28. Another reason for PMA in these thin films is strain from GGG substrate29.
Doping of oxides by Helium implantation was shown to reversibly and locally tune magnetic
anisotropy30. For TbxY3-xFe5O12 (x=2.5, 2.0, 1.0, 0.37 ) samples grown by spontaneous nucleation
technique31, magnetic easy axis was found to change from [111] to [100] direction as Tb
concentration was decreased. The first -order anisotropy constant K 1 undergoes a change of sign
near 190K . Another temperature -dependent lattice distortion effect that caused anisotropy chang e
was also reported for YIG films32. These results indicate that temperature also plays an important
role in both magnetic compensation, lattice distortion and change in anisotropy.
In this study, we systematically calculate the anisotropy energ ies of 10 different types of lattice -
matched iron garnet compounds epitaxially -grown as thin films (X 3Fe5O12, X = Y, Tm, Dy, Ho,
Er, Yb, Tb, Gd, Sm, Eu) on commercial ly available (111) -oriented garnet substrates (Gd 3Ga5O12-
GGG , Y 3Al5O12-YAG , Gd 3Sc2Ga3O12 -SGGG, Tb 3Ga5O12 –TGG, Nd 3Ga5O12 -NGG). Out of the
50 different film/substrate pairs, we found that 20 cases are candidates for room temperature PMA.
Out of these 20 cases, 7 film/substrate pairs were experimentally tested and shown to exhibit 5
characteristics originating from PMA. The remaining 13 pairs, to the best of our knowledge, have
not been tested for PMA experimentally. We indicate through systematic anisotropy calculations
that large strain -induced magnetic anisotropy terms may overcome shape when the films are highly
strained . We use only the room temperature values of λ 11133 and only report predictions for room
temperature (300K) . Throughout the rest of this study, the films are labelled as XIG (X = Y, Tm,
Dy, Ho , Er, Yb, Tb, Gd, Sm, Eu), i.e. TbIG (Terbium iron garnet) or SmIG (Samarium iron garnet)
etc. to distinguish them based on the rare earth element. Our model could accurately predict the
magnetic easy axis in almost all experimentally tested garnet film/substrate cases provided that the
actual film properties are entered in the model and that the experimental film properties satisfy
cubic lattice mat ching condition to the substrate. 6
Figure 1. Methods to achieve perpendicular magnetic anisotropy in iron garnet thin films.
(a) Micro/nano -patterning reduces shape anisotropy and magnetoelastic anisotropies overcome
shape. (b) Large strain -induced anisotropy must over come shape anisotropy to yield out -of-plane
easy axis. (c) Substitutional doping in garnets overcome shape anisotropy by enhancing
magnetocrystalline, growth -induced or magnetoelastic terms.
Anisotropy energy den sity calculation s
Total anisotropy energy density contains three main contributions; according the Equation 1, shape
anisotropy ( Kshape ), first order cubic magnetocrystalline anisotropy ( K1), and strain -induced
7
(magnetoelastic) anisotropy ( Kindu) parameters determine the total effective anisotropy energy
density16.
Keff=Kindu +Kshape +K1 (1)
In case of garnet film magnetized along [111] direction (i.e. on a 111 substrate) , the magneto -elastic
anisotropy energy density, resulting from magneto -elastic coupling is calculated by Equation 2:
Kindu =−3
2λ111σ|| (2)
where λ111 is magnetostriction constant along [111] direction and it is usually negative at room
temperature34. In Eqn. 2, σ|| is the in -plane stress induced in the material as a result of lattice
mismatch between film and the substrate , and the in -plane stress is calculated from Equation 335:
σ||=Y
1−νε|| (3)
where Y is elastic mo dulus, and ν is P oisson’s ratio36.
For calculation of in -plane strain, lattice parameter values obtaine d from the XRD characterization
of the thin films are used . Equation 4 shows the strain relation as the lattice constant difference
between the bulk form of the film and that of the substrate divided by the lattice constant for the
bulk form of the film16.
ε||=afilm −abulk
afilm (4)
Assuming the lattice parameter of the thin film matches with that of the substrate, the lattice
constant of substrate can be used as the lattice constant of thin film for calculation of strain in
Equation 537:
ε||=asub−afilm
afilm (5)
The lattice constants used for the films and substrates examined for this study are presented on
Table 1.
Shape anisotropy energy density depends on the geometry and the intrinsic saturation magnetic
moment of the iron garnet material. Shape anisotropy has a demagnetizing effect on the total 8
anisotropy energy density. These significant anisotropy effects can be observed in magnetic
hysteresis loop s and FMR measurements24.
The most common anisotropies in magnetic materials are shape anisotropy and magneto -crystalline
anisotropy38,39. Considering that the film is continuous, the shape anisotropy is calculated as16
Kshape =2πMs2 (6)
By obtaining the values of M s for rare earth iron garnets as a function of temperature40,41, the value
for shape anisotropy energy density have been calculated using Equation 6.
Intrinsic magnetic anisotropy42, so called magnetocrystalline anisotropy, has the weakest
contribution to anisotropy energy densit y compared to shape, and strain -induced
anisotropies9,11,16,19. The values for the first order magnetocrystalline anisotropy is calculated and
reported previously for rare earth iron garnets at different temperatures43. A key consideration in
magnetic thin films is saturation field. In anisotropic magnetic thin films, the anisotropy fiel ds have
also been calculated using equation 7 as a measure of how much field the films need for magnetic
saturation along the easy axis:
HA=2Keff
Ms (7)
Table 1. List of magnetic iron garnet thin films and garnet substrates available off -the-shelf used
for this study. The fourth column shows the lattice constants used for calculating the magnetoelastic
anisotropy values of epitaxial garnets on the given substrates.
Garnet
material Chemical
formula Purpose Bulk
lattice
constant
(Å)
GGG Gd3Ga5O12 Substrate 12.383
YAG Y3Al5O12 Substrate 12.005
SGGG Gd3Sc2Ga3O12 Substrate 12.480
TGG Tb3Ga5O12 Substrate 12.355
NGG Nd3Ga5O12 Substrate 12.520
YIG Y3Fe5O12 Film 12.376
TmIG Tm 3Fe5O12 Film 12.324
DyIG Dy3Fe5O12 Film 12.440
HoIG Ho3Fe5O12 Film 12.400
ErIG Er3Fe5O12 Film 12.350
YbIG Yb3Fe5O12 Film 12.300 9
TbIG Tb3Fe5O12 Film 12.460
GdIG Gd3Fe5O12 Film 12.480
SmIG Sm 3Fe5O12 Film 12.530
EuIG Eu3Fe5O12 Film 12.500
Results and Discussion
Table s 1 and 2 list in detail the parameters used and the calculated anisotropy energy density terms
for magnetic rare earth iron garnets at 300K . These tables show only the cases predicted to be PMA
out of a total of 50 film/substrate pairs investigated . The extended version of Tables 1 and 2 for all
calculated anisotropy energy density terms for all combinations of the 50 film/subst rate pairs are
provided in the s upplementary tables. The tabulated values for saturation magnetization40,41 and
lattice parameters44 have been used for the calculations . In this study , we assumed the value of
Young’s modulus and Poisson ratio as 2.00×1012 dyne ·cm-2 and 0.29 for all garnet types ,
respectively, based on ref.36. We also assume that the saturation magnetization, used for calculation
of shape anisotropy, does not change with the film thickness. The saturation magnetization (Ms)
values and shape anisotropy for iron garnet films are presented in the third and fourth columns ,
respectively. The stress values for fully lattice -matched film s σ calculated using equation 3 and
magnetostriction constants of the films , λ111, are presented on column s 6 and 7. Magnetoelastic
anisotropy K indu, magnetocrystalline anisotropy energy density K 1, and the total magnetic
anisotropy energy density K eff are calculated and listed on columns 8 , 9 and 10, respectively . H A
on column 11 is the anisotropy field ( the fields required to saturate the film s).
Table 2 . Anisotropy energy density parameters calculation results. Rare earth iron garnets on GGG
(as=12.3 83Å), YAG ( Y3Al5O12, as=12.005Å) , SGGG (a s=12.48Å) and TGG (Tb 3Ga5O12,
as=12.355Å), and NGG ( Nd3Ga5O12, as=12.509Å) substrates, with K eff < 0, are presented.
Film Substr ate Ms
(emu·cm-
3) Kshape
(erg·cm-
3) (× 103) ε
(×
10-3) σ
(dyn·cm-
2)
(×1010) λ111
(×10-
6) Kindu
(erg·cm-
3) (×104) K1
(300K)
(erg·cm-
3)
(× 103) Keff
(erg·cm-
3)
(× 103) HA
(Oe)
(×
103)
DyIG GGG 31.85 6.37 -4.58 -1.29 -5.9 -11.4 -5.00 -113 -7.09
HoIG GGG 55.73 19.5 -1.37 -0.386 -4 -2.3 -5.00 -8.66 -0.311
GdIG GGG 7.962 0.398 -7.77 -2.19 -3.1 -10.2 -4.10 -106 -26.5
SmIG GGG 140 123 -11.7 -3.30 -8.6 -42.6 -17.4 -321 -4.58
YIG YAG 141.7 126 -30.0 -8.44 -2.4 -30.4 -6.10 -184 -2.60 10
TmIG YAG 110.9 77.2 -25.9 -7.29 -5.2 -56.9 -5.80 -497 -8.97
DyIG YAG 31.85 6.37 -35.0 -9.85 -5.9 -87.2 -5.00 -870 -54.7
HoIG YAG 55.73 19.5 -31.9 -8.97 -4 -53.8 -5.00 -524 -18.8
ErIG YAG 79.62 39.8 -27.9 -7.87 -4.9 -57.8 -6.00 -545 -13.7
YbIG YAG 127.3 102 -24.0 -6.76 -4.5 -45.6 -6.10 -360 -5.66
GdIG YAG 7.962 0.398 -38.1 -10.7 -3.1 -49.9 -4.10 -502 -126
SmIG YAG 140 123 -41.9 -11.8 -8.6 -152.3 -17.4 -1420 -20.2
TbIG SGGG 15.92 1.59 1.61 0.452 12 -8.14 -8.20 -88.0 -11.1
GdIG SGGG 7.962 0.398 0.00 0.00 -3.1 0.00 -4.10 -3.70 -0.930
SmIG SGGG 140 123 -3.99 -1.12 -8.6 -14.5 -17.4 -39.3 -0.562
DyIG TGG 31.85 6.37 -6.83 -1.92 -5.9 -17.0 -5.00 -169 -10.6
HoIG TGG 55.73 19.5 -3.63 -1.02 -4 -6.13 -5.00 -46.8 -1.68
GdIG TGG 7.962 0.398 -10.0 -2.82 -3.1 -13.1 -4.10 -135 -33.9
SmIG TGG 140 123 -14.0 -3.93 -8.6 -50.8 -17.4 -402 -5.74
TbIG NGG 15.92 1.59 3.93 1.11 12 -19.9 -8.20 -206 -25.9
In this study, we take the same sign convention as in ref. 16 and the film s exhibit PMA when Keff
< 0. So for obtaining PMA, negative and large values for anisotropy energy density are desired. As
all the garnets (except TbIG ) possess negative magnetostriction constant s at room temperature, the
sign of the strain (tensile or compressive) determines whether the induced anisotropy is negative
or positive . In the literature16,20,45,46 however , we observe that PMA was defined for either positive
or ne gative effective anisotropy energy density (K eff). This inconsistency may cause confusion
among researchers . Thermodynamically, a higher energy means an unstable state with respect to
lower energy cases. Easy axis, by definition, is the axis along which the magnetic material can be
saturated with lowest external field or lowest total energy. A magnetic material would thus
spontaneously minimize its energy and reorient it s magnetic moment along the easy axis. As a
result, we use here Keff < 0 for out-of-plane easy axis . Due to the thermodynamic arguments
mentioned above, we suggest researchers to use K eff < 0 definition for PMA.
Effect of Substrate on Anisotropy Energy Density
Changing the substrate alters the strain in the film, which also changes strain -induced anisotropy
in the film. Figure 2 show s the calculated anisotropy energy density of rare earth iron garnet thin
films grown on five commercially available differ ent substrates : Gadolinium Gallium Garnet
(Gd3Ga5O12, GGG), Yttrium Aluminum Garnet ( Y3Al5O12, YAG), Substituted Gadolinium 11
Gallium Garnet ( Gd3Sc2Ga3O12, SGGG), Terbium Gallium Garnet (Tb 3Ga5O12, TGG ), and
Neodymium Gallium Garnet (Nd 3Ga5O12, NGG) . As shown on Fig. 2(a), when gr own on GGG
substrate ; Dysprosium Iron Garnet (DyIG), Holmium Iron Garnet (HoIG), Gadolinium Iron Garnet
(GdIG), and Samarium Iron Garnet (SmIG) possess compressive strain (afilm>asubstrate ). Considering
the large negative magnetostriction constant (λ 111) for each case, the strain -induced anisotropy
energy densit ies are estimated to cause negative total effective anisotropy energy density . As a
result, DyIG, HoIG, GdIG, and SmIG on GGG are predicted to be PMA cases.
Based o n the shape, magnetoelastic and magnetocrystalline anisotropy terms (room temperature
K1), Thulium iron garnet (TmIG) on GGG (111) is estimated to be in -plane easy axis although
unambiguous experimental evidence indicates that TmIG grows with PMA on GGG (111) [1,19] .
The fact that only considering shape, magnetocrystalline and magnetoelastic anisotropy terms does
not verify this experimental result suggests that the PMA in TmIG/GGG (111) may originate from
a different anisotropy term such as surface anisotropy , growth -induced or stoichiometry -driven
anisotropy. Since the films used in the experiments are less than 10 nm or 5 -8 unit cells thick,
surface effects may become more significant and may require density functional theory -based
predictions to account for surface anisotropy effects . 12
Figure 2. Calculated effective anisotropy values for each rare earth iron garnet thin film when they
are epitaxially grown on ( a) GGG, ( b) YAG, ( c) SGGG, ( d) TGG, (e) NGG substrates . Note that
the scales of the axes are different in each part.
13
Yttrium Aluminum Garnet (YAG) is a substrate with smaller lattice parameter than all the rare
earth iron garnet films considered . With a substrate lattice parameter of as=12.005Å, YAG causes
significant and varying amounts of strain on YIG (af=12.376Å) , TmIG (af=12.324Å) , DyIG
(af=12.440Å) , HoIG (af=12.400Å) , ErIG (af=12.350Å) , YbIG (af=12.300Å) , TbIG (af=12.460Å) ,
GdIG (af=12.480Å) , SmIG (af=12.530Å) and EuIG (a f=12.500Å) . Strain from YAG substrate
yields negative strain -induced anisotropy energy density for these films . The strain -induced
anisotropy term overcomes the shape anisotropy in these garnets when they are grown on YAG.
Consequently , effective anisotropy energy densit ies become negative and these garnet films are
estimated to possess perpendicular magnetic anisotropy . In the e xceptional case s of Terbium Iron
Garnet (TbIG) and Europium Iron Garnet (EuIG) , compressive strain is not enough to induce
negative strain anisotropy because the magnetostriction coefficient s of TbIG and EuIG are positive.
So the strain -induced anisotropy term s are also positive for both TbIG (Kindu(TbIG) = 1.85×106
erg·cm-3) and EuIG (K indu(EuIG) = 3.01×105 erg·cm-3) and do not yield PMA.
Other potential PMA garnets as a film on SGGG substrate are GdIG , TbIG , and SmIG. TbIG and
GdIG cases are particularly interesting as growth conditions of these materials can be further
optimized to achieve room temperature compensation and zero saturation magnetization. This
property enables PMA garnet -based room temperature terahertz magnonics. The lattice parameters
of GdIG (af=12.480Å) and SGGG (as=12.48Å) match exactly, so the in -plane strain value is zero
and the effect of strain -induced anisotropy is eliminated completely. Consequently, due to small
value for saturation magnetizat ion of GdIG, shape anisotropy (3.98 ×102 erg·cm-3) cannot compete
with magnetocrystalline anisotropy ( -4.1×103 erg·cm-3). In other words, in this case, the influence
of magnetocrystalline anisotropy is not negligible compared to the other anisotropy terms.
Consequently , the anisotropy energy density is negative for GdIG when grown on SGGG due to
the influence of magnetocrystalline anisotropy energy density.
One other candidate for a PMA rare earth iron garnet on SGGG substrate is SmIG. Since the film
lattice parameter is greater than that of the substrate, compressive strain ( -3.99 ×10-3) is induced in
the film such that the resulting anisotropy energy density possesses a negative value of an order of
magnitude ( -1.45×105 erg·cm-3) comparable to the sh ape anisotropy energy density (1.23×105
erg·cm-3). With its relatively large magnetocrystalline anisotropy energy density ( -1.74×104
erg·cm-3), SmIG has a perpendicular magnetic anisotropy due to negative value for effective 14
anisotropy energy density (-4.80×105 erg·cm-3). TbIG film on SGGG substrate is a PMA candidate
with positive strain and this film was also recently experimentally demonstrated to have PMA47.
Since TbIG’s lattice constant is smaller than that of the substrate, the film becomes subject to tensile
strain ( +1.61 ×10-3). Since TbIG also has a positive λ 111 (in contrast to that of SmIG), the film’s
magnetoelastic anisotropy term becomes large and negative and overcomes the shape anisotropy.
In case of TbIG, magnetocrystalline anisotropy alone overcomes shape and renders the film PMA
on SGGG. With the additio nal magnetoelastic anisotropy contribution ( -8.14×104 erg·cm-3),
significant stability of PMA can be achieved.
Terbium Gallium Garnet (TGG) is a substrate with lattice parameter (as=12.355Å ) such that it can
induce tensile strain on TmIG, ErIG and YbIG and it induces compressive strain on the rest of the
rare earth iron garnets (YIG, DyIG, HoIG, TbIG, GdIG, SmIG, EuIG) . In none of the tensile -
strained cases, PMA can be achieved since the sign of the magnetoelastic anisotropy is positive
and has the same sign as the shape anisotropy. Among the compressively strained cases, YIG, TbIG
and EuIG are found to have weak magnetoelastic anisotropy terms which cannot overcome shape.
As a result, YIG, TbIG and EuIG on TGG substrate are expected to have in -plane easy axis. DyIG,
HoIG, GdIG and SmIG films on TGG achieve large and negative effective total anisotropy energy
densities due to their negative λ 111 values . In addition, since the materials have compressive strain,
the signs cancel and lead to large magnetoelastic anisotropy energy terms that can overcome shape
in these materials. So these cases are similar to the conditions explained for GGG substrate, on
which only DyIG, HoIG, GdIG, and SmIG films with compressive strain can gain a large negative
strain -induced anisotropy energy density which can overcome shape a nisotropy .
Neodymium gallium garnet (NGG) is a substrate 45 used for growing garnet thin films by pulsed
laser deposition method. NGG has large lattice constant compared with the rest of the bulk rare
earth garnets and yield compressive strain in all rare earth garnets investigated except for Samarium
iron garnet (SmIG). For all cases other than SmIG, the sign of the magnetoelastic a nisotropy term
is determined by the respective λ 111 for each rare earth iron garnet. YIG, TmIG, DyIG, HoIG, ErIG,
YbIG cases have positive magnetoelastic anisotropy terms, which lead to easy axes along the ir film
plane s. For SmIG and EuIG on NGG, magnetoel astic strain and anisotropy terms are not large
enough to overcome large shape anistropy. For GdIG, the weak tensile strain on NGG substrates
actually causes in -plane easy axes as magnetoelastic strain offsets the negative magnetocrystalline 15
anisotropy. The only rare earth iron garnet that can achieve PMA on NGG is Terbium Iron Garnet
(TbIG) due to its large negative strain -induced anisotropy energy density ( -2.44×105 erg·cm-3). Its
large and negative magnetoelastic anisotropy can offset shape (1.59×103 erg·cm-3) and first order
magnetocrystalline anisotropy term ( -8.20×103 erg·cm-3), leading to a large negative effective
magnetic anisotropy energy density ( -2.51×105 erg·cm-3). Consequently, we predict that growing
TbIG on NGG substrate may yield PMA .
Figure 3 show s the substrates on which one may expect PMA rare earth iron garnet films (or
negative Keff) due to strain only. Figures 3(a)-(d) compare the calculated effective energy densities
as a function of strain type and sign for YIG, TmIG, YbIG, TbIG. For comparing the calculation
results presented here with the experimentally reported values for the anisotropy energy density of
TmIG, we added the K eff directly from the experimental data in20 to Fig. 3(b). As shown on Fig.
3(b), the experimental TmIG thin film shows positive K eff as the result of tensile in -plane strain
and large negative magnetostriction constant.
Figure 4(a) -(f) shows the calculated effective energy densities as a function of strain type and sign
for GdIG, SmIG, EuIG, HoIG, DyIG, ErIG, respectively. K eff may get a positive or a negative value
in both compressive and tensile strain cases due to vary ing signs of λ 111 constants of rare earth iron
garnets. In almost all cases that yield PMA on the given substrates, PMA iron garnets form under
compressive lattice strain. The only exception s in which tensile strain can yield PMA in garnet thin
films is Tb IG on SGGG and TbIG on NGG . In both of those cases, a small tensile strain enhances
PMA but the magnetocrystalline anisotropy could already overcome shape and yield PMA without
lattice strain. Therefore, experimental studies should target compressive latti ce strain.
16
Figure 3. Effect of substrate strain on the effective anisotropy energy densities of ( a) YIG, ( b)
TmIG, the data inserted on the graph, with red square symbol , GGG (Exp.) , is the experimental
value of effective anisotropy energy of TmIG on GGG based on ref.20 (c) YbIG, ( d) TbIG. Note
that the axes scales are different in each part.
17
Figure 4. (a) GdIG, ( b) SmIG, ( c) EuIG, ( d) HoIG, ( e) DyIG, ( f) ErIG films on GGG, YAG,
SGGG, TGG, and NGG substrates. Note that the axes scales are different in each part.
18
Based on Fig. 2 -4, the calculations in this paper numerically match with the reported values in the
experimental demonstrations in literature both in sign and order of magnitude . However, since
there are inconsistent sign conventions for predictin g the magnetic anisotropy state of the iron
garnet samples in the literature so far, some of the previous studies draw dif ferent conclusions on
the anisotropy despite the similar K eff.
In case of TmIG, as shown in Fig. 3(b), our model p redicts that there is a tensile strain -induced
anisotropy resulting from the difference in film and substrate lattice parameters and the film’ s
negative magnetostriction constant. The experimental results of magnetic anisotropy in
TmIG/GGG8,20,48 are consistent with our model predictions . Previous studies identify PMA , if the
film Keff is positive. A shortcoming of this approach is that such a definition would also identify
YIG/GGG as PMA although its in-plane easy axis behavior has been experimentally de monstrated
in numerous studies3,32,49. Keff < 0 for PMA definition would be thermodynamically more
appropriate and would also accurately explain almost all cases including YIG/GGG. Further
explanation about TmIG exceptional case is included in the Supplementary Information.
Sensitivity of Anisotropy to Variations in Saturation Magnetic Moment and Film Relaxation
The films with predicted effective anisotropy energy and field may come out differently when
fabricated due to unintentional variability in fabrication process conditions , film stoichiometry
(rare earth ion to iron ratio and oxygen deficiency) as well as process -induced non -equilibrium
phases in the garnet films. These changes may partially or completely relax the films or increase
strain further due to secondary crystallin e phases. Practical minor changes in strain may
dramatically alter both the sign and the magnitude of magnetoelastic ani sotropy energy and may
cause a film predicted as PMA to come out with in -plane easy axis. On the other hand, o ff-
stoichiometry may cause reduction in saturation magnetic moment. Reduction in saturation
magnetic moment decreases shape anisotropy term quadratically (Kshape = 2πM s2), which implies
that a 1 0% reduction in Ms leads to a 19% decrease in K shape and anisotropy field may increase (HA
= 2K eff/Ms). Therefore, sample fabrication issues and the consequent changes in anisotropy terms
may weaken or completely eliminate the PMA of a film/substrate pair and alter anisotropy field .
While these effects may arise unintentionally, one can also use these effects deliberately for
engineering garnet films for devices. Therefore , the sensitivity of anisotropy properties of garnet 19
thin films such as anisotropy field and effective anisotropy energy density need s to be evaluated
with respect to changes in film strain and saturation magnetic moment.
Figure 5 shows the sensitivity of the effective magnetic anisotropy energy density to deviation of
both strain and saturation magnetization, M s for five PMA film/substrate combinations: (a)
HoIG/GGG, (b) YIG/YAG, (c) SmIG/ SGGG, (d) HoIG/TGG , and (e) SmIG/NGG ). The negative
sign of effective magnetic anisotropy energy indicates PMA . The change of anisotropy energy from
negative to positive indicates a transition from PMA to in -plane easy axis . In these plots, calculated
anisotropy energies are presented for saturation moments and strains scanned from 60% to 140%
of tabulated bulk garnet Ms and of the strain s of fully lattice -matched films on the substrate s. The
color scale indicates the anisotropy energy density in erg·cm-3. Although magnetocrystalline
anisotropy energies are negative for all of the thin film rare earth garnets considered here, these
terms are negligible with respect to shape anisotropy (K 1(300 K) ~ -5% of K shape). Therefore,
magnetoelastic anisotropy term must be large enough to overcome shape anisotropy. A derivation
of anisotropy energy as a function of Ms and strain in equations ( 8)-(10) shows that the negative
λ111 values and negative strain states (compressive strain) for garnet films in Fig. 5(a)-(e) (HoIG,
YIG, SmIG) enable these films to have PMA . To retain PMA state; λ 111 must be negative and large
assuming elastic moduli and the Poisson’s ratio are constant . The necessary condition for
maintai ning PMA is shown in equation 11 .
Keff=Kindu +Kshape +K1 (8)
Keff=−3
2λ111Y
1−vε||+2πMs2+K1 (9)
Keff=3
2λ111Y
1−v|ε|||+2πMs2+K1 (10)
Keff<0 if |3
2λ111Y
1−v|ε|||+K1|>2πMs2 (11)
Relaxing each fully strained and lattice -matched thin film towards unstrained state (ε → 0 or
moving from left to right on each plot in Fig. 5 causes the magnetoelastic anisotropy energy term
to decrease in magnitude and gradually vanish . The total anisotropy energy decreases in intensity
for decreasing strain and constant M s. When M s increases, shape anisotropy term also increases
and overcomes magnetoelastic anisotropy term. As a result, higher M s for relaxed films (i.e.
relatively thick and iron -rich garnets) may lose PMA. Therefore, one needs to optimize the film 20
stoichiometry and deposition process conditions, especially growth temp erature, oxygen partial
pressure and film thickness, to ensure that the films are strained and stoichiometric. Since strains
are less than 1% in Fig. 5(a), 5(c)-(e), these samples are predicted to be exp erimentally more
reproducible. For YIG on YAG, as shown in Fig. 5 (b), the strains are around 3%, which may be
challenging to reproduce. The cases presented in Fig. 5 (a)-(e) are the only cases among 50
film/substrate pairs where reasonable changes in M s and strain may lead to complete loss of PMA.
The rest of the cases have not been found as sensitive to strain and M s variability and those
predicted to be PMA are estimated to have stable anisotropy. Effective a nisotropy energy plots
similar to Fig. 5(a) -(e) are presented in the supplementary figures for all 50 film/substrate pairs.
While PMA is a useful metric for garnet films, the effective anisotropy energy of the films should
also not be too high ( < a few 105 erg·cm-3) otherwise the saturation fields for these films would
reach or exceed 0.5 Tesla (5000 Oe). Supplementary figures present the calculated effective
anisotropy energy and anisotropy field values for all 50 film/substrate pairs for changing strain and
Ms values. These figures indicate that one can span anisotropy fields of about 300 Oersteds up to
12.6 Tesla in PMA garnets. For practical integrated magnonic devices, the effective anisotropy
energy should be large enough to have robust PMA although it shoul d not be too high such that
effective anisotropy fields (i.e. saturation fields) would still be small and feasible. Engineered strain
and M s through controlled oxygen stoichiometry may help keep anisotropy field low while
retaining PMA. In addition, according to the recently published paper on magnetic anisotropy of
HoIG50, the lattice matching in case of the thick samples becomes challenging to sustain, and the
strain relaxes inside the film. Thus, the decrease in the anisotropy field is one consequences of the
lower strain state, which is an advantage for magnonics or spin -orbit torque devices. Below a
critical thickness, HoIG gr own on GGG has PMA. However, as the film reaches this critical
thickness, the 40% or more strain relaxation is expected and the easy axis becomes in -plane. So
thinner films are preferred to be grown in integrated device applications. 21
Figure 5. Effect of partial film relaxation and saturation magnetic moment variability on the
effective anisotropy energy density of the films . Variation of effective magnetic anisotropy energy
densities for (a) HoIG on GGG, ( b) YIG on YAG, ( c) SmIG on SGGG, ( d) HoIG on TGG and ( e)
SmIG on NGG are presented when strain relaxation and magnetic saturation moments change
independently. Film strain may vary from a completely lattice -matched state to the substrate to a
relaxed state or a highly strained state due to microparticle nucleation . Strain variability alter s
magnetoelastic anisotropy and cause a PMA film become in -plane easy axis. On the other hand,
22
magnetic saturation moments may deviate from the tabulated values because of process -induced
off-stoich iometry in the films (i.e. rare earth ion to iron ratio or iron deficiency or excess , oxygen
deficiency) . Relaxing the films reduces the magnetoelastic anisotropy term and diminishes PMA.
Increasing M s strengthens shape anisotropy and eliminates PMA for lo w enough strains for all five
cases presented.
Minimizing Gilbert damping coefficient in garnet thin films is also an important goal for spintronic
device applications. First principles predictions of physical origins of Gilbert damping 51 indicate
that magnetic materials with lower M s tend to have lower damping. Based on this prediction, DyIG,
HoIG and GdIG films are predicted to have lower Gilbert damping with respect to the others. Since
the compen sation temperatures of these films could be engineered near room temperature, one may
optimize their damping for wide bandwidths all the way up to terahertz (THz)52 spin waves or
magnons . The first principles predictions also indicate that higher magnetic susceptibility (χm) in
the films helps reduce damping (i.e. lower saturation field). Therefore, the PMA garnet films with
lower anisotropy fields are estimated to have lower Gilbert dampi ng parameters with respect to
PMA garnets with higher anisotropy fields .
Conclusion
Shape, magnetoelastic and magnetocrystalline m agnetic anisotropy energy terms have been
calculated for ten different garnet thin films epitaxially grown on five different garnet substrates.
Negative K eff (effective magnetic anisotropy energy) corresponds to perpendicular magnetic
anisotropy in the convention used here . By choosing a substrate with a lattice parameter smaller
than that of the film, one can induce compressive strain in the films to the extent that one can
always overcome shape anisotropy and achieve PMA for large and negative λ 111. Among the PMA
films predicted, SmIG possesses a high anisotropy energy density and this film is estimated to be
a robus t PMA when grown on all five different substrates.
In order to obtain PMA, magnetoelastic anisotropy term must be large enough to overcome shape
anisotropy. Magnetoelastic anisotropy overcomes shape anisotropy when the strain type
(compressive or tensile) and magnetoelastic anisotropy constants λ 111 of the garnet film have the
correct signs (not necessarily opposite or same) and the magnetoelastic anisotropy term has a
magnitude larger than shape anisotropy. Both compressive and tensile -strained films can , in
principle, become PMA as long as shape anisotropy can be overcome with large magnetoelastic 23
strain effects. Here, in almost all cases that yield PMA on the given substrates, PMA iron garnets
form under compressive lattice strain, except TbIG on SGGG and TbIG on NGG . These two cases
have tensile strain and relatively large magnetocrystalline anisotropy, which could already
overcome shape anisotropy without strain. Experiments are therefore suggested to target mainly
compressive lattice strain.
20 different garnet film/substrate pairs have been predicted to exhibit PMA and their properties are
listed on Table 2 . For 7 of these 20 potential PMA cases, we could find unambiguous experimental
demonstration of PMA. Among the 20 PMA cases, HoIG/GGG, YIG/Y AG, SmIG/SGGG,
HoIG/ TGG and SmIG/ NGG cases have been found to be sensitive to fabrication process or
stoiochiometry -induced variations in M s and strain. In order to control effective anisotropy in rare
earth iron garnets (RIGs), shape anisotropy could be tuned by doping garnet film s with Ce 53, Tb
31 and Bi 54 or by micro/nano -patterning . Saturation magnetization could also be increased
significantly by doping, which results in increasing the shape anisotropy in the ma gnetic thin films.
Among the cases predicted to possess PMA , anisotropy fields ranging from 310 Oe (0.31 T) to
12.6 T have b een calculated . Such a wide anisotropy field range could be spanned and engineered
through strain state, stoichiometry as we ll as substrate choice. For integrated magnonic devices and
circuits, garnets with low M s and lower anisotropy field s (HA < 0.5 T) would require less energy
for switching and would be more appropriate due to their lower estimated Gilbert damping.
Methods
Calculation of anisotropy energy density. We used Keff = K indu + K shape + K 1 equation to calculate
the total anisotropy energy density for each thin film rare earth iron garnet/substrate pair. Each
anisotropy term consist of the following parameters: Keff=−3
2λ111Y
1−vε||+2πMs2+K1. The
energy density is calculated based on the parameters reported in previous references16,34,36,40,41,43.
First order magnetocrystalline anisotropy, K 1, is an intrinsic, temperature -dependent constant
reported for each REIG material. Young’s modulus (Y), poison ratio (ν) and magnetostriction
constant (λ111) parameters evolving in the magnetoelastic anisotropy energy density term (first
term) are considered to be constant according to the values previous ly reported . For shape
anisotropy energy calculations (second term), bulk s aturation magnetization (Ms) for each film was
used. Since each film may exhibit variability in M s with respect to bulk, the model presented here
yields the most accurate predictions when the actual film Ms, λ 111, Y, ν and K 1, and in -plane strain 24
are enter ed for each term. The original Microsoft Excel and MATLAB files used for generating
the data for F igure s 1-5 are presented in the supplementary files .
25
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Acknowledgments
M.C.O. acknowledges BAGEP 2017 Award and TUBITAK Grant No. 117F416.
Competing interests
There is no financial and non -financial competing interest among the authors.
Author contributions
M.C.O. designed the study. S.M.Z. performed the calculations and evaluated and analyzed the
results with M.C.O. Both authors discussed the results and wrote the manuscript together.
|
2211.12889v1.The_fractional_Landau_Lifshitz_Gilbert_equation.pdf | The fractional Landau-Lifshitz-Gilbert equation
R.C. Verstraten1, T. Ludwig1, R.A. Duine1,2, C. Morais Smith1
1Institute for Theoretical Physics, Utrecht University,
Princetonplein 5, 3584CC Utrecht, The Netherlands
2Department of Applied Physics, Eindhoven University of Technology,
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
(Dated: November 24, 2022)
The dynamics of a magnetic moment or spin are of high interest to applications in technology.
Dissipation in these systems is therefore of importance for improvement of efficiency of devices,
such as the ones proposed in spintronics. A large spin in a magnetic field is widely assumed to
be described by the Landau-Lifshitz-Gilbert (LLG) equation, which includes a phenomenological
Gilbert damping. Here, we couple a large spin to a bath and derive a generic (non-)Ohmic damping
term for the low-frequency range using a Caldeira-Leggett model. This leads to a fractional LLG
equation, where the first-order derivative Gilbert damping is replaced by a fractional derivative of
orders≥0. We show that the parameter scan be determined from a ferromagnetic resonance
experiment, where the resonance frequency and linewidth no longer scale linearly with the effective
field strength.
Introduction. — The magnetization dynamics of mate-
rials has attracted much interest because of its techno-
logical applications in spintronics, such as data storage
or signal transfer [1–3]. The right-hand rule of magnetic
forces implies that the basic motion of a magnetic mo-
ment or macrospin Sin a magnetic field Bis periodic
precession. However, coupling to its surrounding (e.g.,
electrons, phonons, magnons, and impurities) will lead
to dissipation, which will align SwithB.
Spintronics-based devices use spin waves to carry sig-
nals between components [4]. Contrary to electronics,
which use the flow of electrons, the electrons (or holes)
in spintronics remain stationary and their spin degrees
of freedom are used for transport. This provides a sig-
nificant advantage in efficiency, since the resistance of
moving particles is potentially much larger than the dis-
sipation of energy through spins. The spin waves con-
sist of spins precessing around a magnetic field and they
are commonly described by the Landau-Lifshitz-Gilbert
(LLG) equation [5]. This phenomenological description
also includes Gilbert damping, which is a term that
slowly realigns the spins with the magnetic field. Much
effort is being done to improve the control of spins for
practical applications [6]. Since efficiency is one of the
main motivations to research spintronics, it is important
to understand exactly what is the dissipation mechanism
of these spins.
Although the LLG equation was first introduced phe-
nomenologically, since then it has also been derived from
microscopic quantum models [7, 8]. Quantum dissipation
is a topic of long debate, since normal Hamiltonians will
always have conservation of energy. It can be described,
for instance, with a Caldeira-Leggett type model [9–13],
where the Hamiltonian of the system is coupled to a bath
of harmonic oscillators. These describe not only bosons,
but any degree of freedom of an environment in equilib-
rium. These oscillators can be integrated out, leading toan effective action of the system that is non-local and ac-
counts for dissipation. The statistics of the bath is cap-
tured by the spectral function J(ω), which determines
the type of dissipation. For a linear spectral function
(Ohmic bath), the first-order derivative Gilbert damping
is retrieved.
The spectral function is usually very difficult to calcu-
late or measure, so it is often assumed for simplicity that
the bath is Ohmic. However, J(ω) can have any contin-
uous shape. Hence, a high frequency cutoff is commonly
put in place, which sometimes justifies a linear expan-
sion. However, a general expansion is that of an sorder
power-law, where scould be any positive real number.
A spectral function with such a power-law is called non-
Ohmic, and we refer to sas the “Ohmicness” of the bath.
It is known that non-Ohmic baths exist [14–23] and that
they can lead to equations of motion that include frac-
tional derivatives [24–28]. Because fractional derivatives
are non-local, these systems show non-Markovian dynam-
ics which can be useful to various applications [29–31].
Here, we show that a macroscopic spin in contact
with a non-Ohmic environment leads to a fractional LLG
equation, where the first derivative Gilbert damping gets
replaced by a fractional Liouville derivative. Then, we ex-
plain how experiments can use ferromagnetic resonance
(FMR) to determine the Ohmicness of their environ-
ment from resonance frequency and/or linewidth. This
will allow experiments to stop using the Ohmic assump-
tion, and use equations based on measured quantities
instead. The same FMR measurements can also be done
with anisotropic systems. Aligning anisotropy with the
magnetic field may even aid the realization of measure-
ments, as this can help reach the required effective field
strengths. In practice, the determination of the type of
environment is challenging, since one needs to measure
the coupling strength with everything around the spins.
However, with the experiment proposed here, one canarXiv:2211.12889v1 [cond-mat.mes-hall] 23 Nov 20222
essentially measure the environment through the spin it-
self. Therefore, the tools that measure spins can now also
be used to determine the environment. This information
about the dissipation may lead to improved efficiency,
stability, and control of applications in technology.
Derivation of a generalized LLG equation. — We con-
sider a small ferromagnet that is exposed to an external
magnetic field. Our goal is to derive an effective equa-
tion of motion for the magnetization. For simplicity, we
model the magnetization as one large spin (macrospin)
ˆS. Its Hamiltonian (note that we set /planckover2pi1andkBto
one) reads ˆHs=B·ˆS−KˆS2
z, where the first term
(Zeeman) describes the coupling to the external mag-
netic fieldB, and the second term accounts for (axial)
anisotropy of the magnet. However, since a magnet con-
sists of more than just a magnetization, the macrospin
will be in contact with some environment. Following
the idea of the Caldeira-Leggett approach [9–13, 32], we
model the environment as a bath of harmonic oscillators,
ˆHb=/summationtext
αˆp2
α/2mα+mαω2
αˆx2
α/2, where ˆxαand ˆpαare
position and momentum operators of the α-th bath oscil-
lator with mass mαand eigenfrequency ωα>0. Further-
more, we assume the coupling between the macrospin and
the bath modes to be linear, ˆHc=/summationtext
αγαˆS·ˆxα, where
γαis the coupling strength between macrospin and the
α-th oscillator. Thus, the full Hamiltonian of macrospin
and environment is given by ˆH=ˆHs+ˆHc+ˆHb.
Next, we use the Keldysh formalism in its path-integral
version [33, 34], which allows us to derive an effective ac-
tion and, by variation, an effective quasi-classical equa-
tion of motion for the macrospin. For the path-integral
representation of the macrospin, we use spin coherent
states [34]|g/angbracketright= exp(−iφˆSz) exp(−iθˆSy) exp(−iψˆSz)|↑/angbracketright,
whereφ,θ, andψare Euler angles and |↑/angbracketrightis the eigen-
state of ˆSzwith the maximal eigenvalue S. Spin co-
herent states provide an intuitive way to think about
the macrospin as a simple vector S=/angbracketleftg|ˆS|g/angbracketright=
S(sinθcosφ,sinθsinφ,cosθ) with constant length Sand
the usual angles for spherical coordinates θandφ. For
spins, the third Euler angle ψpresents a gauge freedom,
which we fix as in Ref. [35] for the same reasons explained
there.
After integrating out the bath degrees of freedom, see
Sup. Mat. [36] for details, we obtain the Keldysh partition
functionZ=/integraltext
Dgexp[iS], with the Keldysh action
S=/contintegraldisplay
dt/bracketleftbig
S˙φ(1−cosθ)−Beff(Sz)·S/bracketrightbig
−/contintegraldisplay
dt/contintegraldisplay
dt/primeS(t)α(t−t/prime)S(t/prime). (1)
The first term, called Berry connection, takes the role
of a kinetic energy for the macrospin; it arises from
the time derivative acting on the spin coherent states
(−i∂t/angbracketleftg|)|g/angbracketright=S˙φ(1−cosθ). The second term is the po-
tential energy of the macrospin, where we introduced aneffective magnetic field, Beff(Sz) =B−KSzez, given by
the external magnetic field and the anisotropy. The third
term arises from integrating out the bath and accounts
for the effect of the environment onto the macrospin;
that is, the kernel function α(t−t/prime) contains informa-
tion about dissipation and fluctuations. Dissipation is
described by the retarded and advanced components
αR/A(ω) =/summationtext
α(γ2
α/2mαω2
α)ω2/[(ω±i0)2−ω2
α], whereas
the effect of fluctuations is included in the Keldysh com-
ponent,αK(ω) = coth(ω/2T) [αR(ω)−αA(ω)]. This is
determined by the fluctuation-dissipation theorem, as we
assume the bath to be in a high-temperature equilibrium
state [33, 34, 37].
From the Keldysh action, Eq. (1), we can now de-
rive an equation of motion for the macrospin by taking
a variation. More precisely, we can derive quasi-classical
equations of motion for the classical components of the
anglesθandφby taking the variation with respect to
their quantum components [38]. The resulting equations
of motion can be recast into a vector form and lead to a
generalized LLG equation
˙S(t) =S(t)×/bracketleftbigg
−Beff[Sz(t)] +/integraldisplayt
−∞dt/primeα(t−t/prime)S(t/prime) +ξ(t)/bracketrightbigg
,
(2)
with the dissipation kernel [39] given by
α(ω) =/integraldisplay∞
−∞dε
πεJ(ε)
(ω+i0)2−ε2, (3)
where we introduced the bath spectral density J(ω) =/summationtext
α(πγ2
α/2mαωα)δ(ω−ωα) [33, 36]. The last term in
Eq. (2) contains a stochastic field ξ(t), which describes
fluctuations (noise) caused by the coupling to the bath;
the noise correlator for the components of ξ(t) is given
by/angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδmnαK(t−t/prime). Next, to get a
better understanding of the generalized LLG equation,
we consider some examples of bath spectral densities.
Fractional Landau-Lifshitz-Gilbert equation. — For the
generalized LLG equation (2), it is natural to ask: In
which case do we recover the standard LLG equation?
We can recover it for a specific choice of the bath spectral
densityJ(ω), which we introduced in Eq. (3). Roughly
speaking,J(ω) describes two things: first, in the delta
functionδ(ω−ωα), it describes at which energies ωα
the macrospin can interact with the bath; second, in
the prefactor πγ2
α/2mαωα, it describes how strongly the
macrospin can exchange energy with the bath at the fre-
quencyωα. In our simple model, the bath spectral den-
sity is a sum over δ-peaks because we assumed excitations
of the bath oscillators to have an infinite life time. How-
ever, also the bath oscillators will have some dissipation
of their own, such that the δ-peaks will be broadened. If,
furthermore, the positions of the bath-oscillator frequen-
ciesωαis dense on the scale of their peak broadening, the
bath spectral density becomes a continuous function in-
stead of a collection of δ-peaks. In the following, we focus3
on cases where the bath spectral density is continuous.
Since the bath only has positive frequencies, we have
J(ω≤0) = 0. Even though J(ω) can have any pos-
itive continuous shape, one might assume that it is an
approximately linear function at low frequencies; that is,
J(ω) =α1ωΘ(ω)Θ(Ωc−ω), (4)
where Θ(ω) = 1 forω > 0 and Θ(ω) = 0 forω < 0 and
Ωcis some large cutoff frequency of the bath such that we
haveωsystem/lessmuchT/lessmuchΩc. Reservoirs with such a linear
spectral density are also known as Ohmic baths. Insert-
ing the Ohmic bath spectral density back into Eq. (3),
while sending Ω c→∞ , we recover the standard LLG
equation,
˙S(t) =S(t)×/bracketleftBig
−Beff[Sz(t)] +α1˙S(t) +ξ(t)/bracketrightBig
,(5)
where the first term describes the macrospin’s precession
around the effective magnetic field, the second term—
known as Gilbert damping—describes the dissipation of
the macrospin’s energy and angular momentum into the
environment, and the third term describes the fluctu-
ations with/angbracketleftξm(t)ξn(t/prime)/angbracketright= 4α1Tδmnδ(t−t/prime), which
are related to the Gilbert damping by the fluctuation-
dissipation theorem. Note that the same results can
be obtained without a cutoff frequency by introducing
a counter term, which effectively only changes the zero-
energy level of the bath, see Sup. Mat. [36] for details.
The assumption of an Ohmic bath can sometimes be
justified, but is often chosen out of convenience, as it is
usually the simplest bath type to consider. To our knowl-
edge, there has been little to no experimental verification
whether the typical baths of magnetizations in ferromag-
nets are Ohmic or not. To distinguish between Ohmic
and non-Ohmic baths, we need to know how the mag-
netization dynamics depends on that difference. Hence,
instead of the previous assumption of a linear bath spec-
tral density (Ohmic bath), we now assume that the bath
spectral density has a power-law behavior at low frequen-
cies,
J(ω) = ˜αsωsΘ(ω)Θ(Ωc−ω), (6)
where we refer to sas Ohmicness parameter [40]. It is
convenient to define αs= ˜αs/sin(πs/2) and we should
note that the dimension of αsdepends on s. Fors= 1
we recover the Ohmic bath. Correspondingly, baths with
s < 1 are called sub-Ohmic and baths with s > 1 are
called super-Ohmic. For 0 <s< 2 and Ωc→∞ , we find
the fractional LLG equation
˙S(t) =S×[−Beff[Sz(t)] +αsDs
tS(t) +ξ(t)],(7)
whereDs
tis a (Liouville) fractional time derivative of or-
dersand the noise correlation is given by /angbracketleftξm(t)ξn(t/prime)/angbracketright=
2αsTδmn(t−t/prime)−s/Γ(1−s); for a detailed calculation,see the Sup. Mat. [36]. Indeed, in the limit of s→1, we
recover the regular LLG equation. The fractional LLG
equation (7) seems quite similar to the standard LLG
equation (5). However, the first-order time derivative in
the Gilbert damping is replaced by a fractional s-order
time derivative in the fractional Gilbert damping; this
has drastic consequences for the dissipative macrospin
dynamics.
Fractional Gilbert damping. — Fractional derivatives
have a long history [29], and many different definitions
exist for varying applications [26, 27, 29]. From our mi-
croscopic model, we found the Liouville derivative [41],
which is defined as
Ds
tS(t) =dn
dtn1
Γ(n−s)/integraldisplayt
−∞dt/prime(t−t/prime)n−1−sS(t/prime),(8)
wherenis the integer such that n≤s<n + 1. This can
be interpreted as doing a fractional integral followed by
an integer derivative. The fractional integral is a direct
generalization from the rewriting of a repeated integral,
by reversing the order of integration into a single one,
which leads to extra powers of ( t−t/prime).
To provide some intuition to the effects of fractional
friction, we propose a thought experiment. Suppose an
object is traveling at constant speed, then x(t) =vt.
Hence, the friction force acting on the object goes like
Ds
tx(t)∝vt1−s. Therefore, we find three regimes. For
s= 1, the friction is constant in time. For s<1, the fric-
tion force increases with time. Hence, longer movements
will be less common. For s > 1, the friction decreases
with time, so longer movements will be more likely once
set in motion.
Within the fractional LLG equation, we thus see two
important new regimes. For s < 1 (sub-Ohmic), the
friction is more likely to relax (localize) the spin (e.g.
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.10.51510
[ωd-(B 0-KS)]/[α sS(B 0-KS)s][αsS(B 0-KS)s]2sin2θ/Ω2s0.20.40.60.81.1.21.41.61.8
FIG. 1. A lin-log plot of the amplitude sin2θas a function of
driving frequency ωdplotted in dimensionless units for several
values ofs. The resonance peaks change, depending on s.
The resonance frequency ωresand linewidth ∆ H/2have been
overlayed with crosses. The red dashed crosses have been
calculated numerically, whereas the black solid crosses are the
derived results from Eqs. (10) to (12).4
sub-diffusion) towards the B-field direction. For small
movements, the friction could be very small, whereas it
would greatly increase for bigger movements. This could
describe a low dissipation stable configuration. For s >
1 (super-Ohmic), the friction could reduce as the spin
moves further, which in other systems is known to cause
L´ evy-flights or super-diffusion [24, 25, 42]. This might
lead the system to be less stable, but can potentially also
greatly reduce the amount of dissipation for strong signal
transfer: In a similar way to the design of fighter-jets,
unstable systems can be easily changed by small inputs,
which leads to more efficient signal transfer.
Ferromagnetic Resonance. — FMR is the phenomenon
where the spin will follow a constant precession in a ro-
tating external magnetic field. The angle θfrom thez-
axis at which it will do so in the steady state will vary
according to the driving frequency ωdof the magnetic
field. Close to the natural frequency of the precession,
one generally finds a resonance peak [43]. We assume a
magnetic field of the form
Beff(t) =
Ω cos(ωdt)
Ω sin(ωdt)
B0−KSz
, (9)
where Ω is the strength of the rotating component, and
we will neglect thermal noise. We search for a steady
state solution of S(t) in the rotating frame where Beff(t)
is constant. We will assume a small θapproximation
where the ground state is in the positive zdirection, i.e.
0<Ω/lessmuchB0−KSandαsS/lessmuch(B0−KS)1−s. Then (see
Sup. Mat. [36] for details of the calculations), we find
that the resonance occurs at a driving frequency
ωres≈(B0−KS) + (B0−KS)sαsScos/parenleftBigπs
2/parenrightBig
.(10)
It should be noted that this is different from what was to
be expected from any scaling arguments, since the cosine
term is completely new compared to previous results [43],
and it vanishes precisely when s= 1. However, this new
non-linear term scales as ( B0−KS)s, which is an easily
controllable parameter. In the limit where B0−KSis
small (resp. large), the linear term will vanish and the
s-power scaling can be measured for the sub(resp. super)-
Ohmic case. The amplitude at resonance is found to be
sin2θres≈Ω2
/bracketleftbig
αsS(B0−KS)ssin/parenleftbigπs
2/parenrightbig/bracketrightbig2, (11)
and the Full Width at Half Maximum (FWHM) linewidth
is given by
∆H/2≈2αsS(B0−KS)ssin/parenleftBigπs
2/parenrightBig
. (12)
Depending on the experimental setup, it might be eas-
ier to measure either the resonance location or the width
0.0 0.5 1.0 1.5 2.001234
ωres/(B0-KS)ΔH/2/[αsS(B0-KS)s]
0.01 0.10 1 10 1000.0110s0.20.40.60.81.1.21.41.61.8FIG. 2. A plot of the linewidth in Eq. (13) as a function of res-
onance frequency for several values of s. The inset shows the
same plot in a log-log scale, where the slope of the linewidth
is precisely the Ohmicness sof the bath.
of the peak. Nevertheless, both will give the opportu-
nity to see the sscaling inB0−KS. The presence
of the anisotropy provides a good opportunity to reach
weak or strong field limits. In fact, the orientation of the
anisotropy can help to add or subtract from the magnetic
field, which should make the required field strengths more
reachable for experiments. Some setups are more suit-
able for measuring the width as a function of resonance
frequency. When s= 1, this relation can be directly
derived from Eqs. (10) and (12). However, when s/negationslash= 1,
the relation can only be approximated for strong or weak
damping. For small αsS, we see that
∆H/2≈2αsS(ωres)ssin/parenleftBigπs
2/parenrightBig
. (13)
The resonance peaks have been calculated numerically
in FIG. 1 in dimensionless values. The red dashed lines
show the location of the numerically calculated peak and
the FWHM line width. The black solid lines show the
location of the analytically approximated result for the
peak location and FWHM line width [Eqs. (10) and (12)].
For smallαsSand Ω, we see a good agreement be-
tween the analytical results and the numerical ones, al-
though sub-Ohmic seems to match more closely than
super-Ohmic. This could be due to the greater sta-
bility of sub-Ohmic systems, since the approximations
might affect less a stable system. As one might expect
from the thought experiment presented earlier, we can
see in FIG. 1 that sub-Ohmic systems require higher,
more energetic, driving frequencies to resonate, whereas
super-Ohmic systems already resonate at lower, less en-
ergetic, driving frequencies. In FIG. 2, we provide a plot
of Eq. (13) to facilitate further comparison with experi-
ments. If the assumption of Gilbert damping was correct,
all that one would see is a slope of one in the log-log inset.
Conclusion. — By relaxing the Ohmic Gilbert damp-
ing assumption, we have shown that the low-frequency
regime of magnetization dynamics can be modeled by a5
fractional LLG equation. This was done by coupling the
macrospin to a bath of harmonic oscillators in the frame-
work of a Caldeira-Leggett model. The Keldysh formal-
ism was used to compute the out-of-equilibrium dynam-
ics of the spin system. By analyzing an FMR setup, we
found ans-power scaling law in the resonance frequency
and linewidth of the spin, which allows for a new way to
measure the value of s. This means that experiments in
magnetization dynamics and spintronics can now avoid
the assumption of Gilbert damping and instead measure
the Ohmicness of the environment. This could aid in a
better understanding of how to improve efficiency, stabil-
ity, and control of such systems for practical applications.
Acknowledgments. — This work was supported by the
Netherlands Organization for Scientific Research (NWO,
Grant No. 680.92.18.05, C.M.S. and R.C.V.) and (partly)
(NWO, Grant No. 182.069, T.L. and R.A.D.).
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components, we would only obtain a noiseless quasi-
classical equation of motion because the information
about noise (fluctuations) is included in the Keldysh
part ofα(t−t/prime), which appears in the action only with
even powers of quantum components. However, there is a
way [44] that allows us to retain information about noise
in the quasi-classical equation of motion; see also [33, 34].
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[39] The dissipation kernel αis closely related to the retarded
αRand advanced αAcomponents. Namely, it is given by
α(ω) =−αR(ω)−αA(−ω).
[40] From a mathematical perspective, any continuous but
not smooth function can still be expanded in a power-law
for small enough parameters. Hence, the only assumption
that we make in this model is that the frequencies in the
system are very small. Then, there will always exist an
0< s∈Rsuch that this expansion holds. In contrast,6
the Ohmic expansion can only be made for smooth con-
tinuous functions.
[41] This definition does not have any boundary conditions,
as they would have to be at −∞ and would dissipate
before reaching a finite time. One can, however, enforce
boundary conditions by applying a very strong magnetic
field for some time such that the spin aligns itself, and
then quickly change to the desired field at t= 0.
[42] A. A. Dubkov, B. Spagnolo, and V. V. Uchaikin, Interna-
tional Journal of Bifurcation and Chaos 18, 2649 (2008).
[43] T. Ludwig, I. S. Burmistrov, Y. Gefen, and A. Shnirman,
Physical Review Research 2, 023221 (2020).
[44] A. Schmid, Journal of Low Temperature Physics 49, 609
(1982).The fractional Landau-Lifshitz-Gilbert equation
Supplementary Material
R.C. Verstraten1, T. Ludwig1, R.A. Duine1,2, C. Morais Smith1
1Institute for Theoretical Physics, Utrecht University,
Princetonplein 5, 3584CC Utrecht, The Netherlands
2Department of Applied Physics, Eindhoven University of Technology,
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
(Dated: November 24, 2022)
CONTENTS
I. Keldysh microscopic model 1
A. Hamiltonian 1
B. Keldysh partition function 2
C. Quasi-classical equation of motion 4
D. Generalized Landau-Lifshitz-Gilbert equation 8
II. Fractional derivative from non-Ohmic spectral function 9
A. Calculating the effective Greens functions 9
B. Ohmic spectral function 11
C. Sub-Ohmic spectral function 13
D. Super-Ohmic spectral function 15
E. Comparison Ohmic versus non-Ohmic 17
III. FMR powerlaw derivation 18
A. Ferromagnetic Resonance 18
B. Resonance frequency and amplitude 20
C. Calculating the FWHM linewidth 21
IV. Dimensional analysis 22
References 22
I. KELDYSH MICROSCOPIC MODEL
For pedagogical reasons we start with a microscopic derivation of the usual LLG equation before going into the
fractional one. In this section, we combine spin coherent states with the Keldysh formalism [1, 2] to derive a stochastic
Langevin-like equation of motion of a (macro) spin [3].
A. Hamiltonian
In the main text, we introduced a spectral function J(ω) with a cutoff frequency Ω c. This was originally done from
the perspective that any spectral function could be expanded to linear order; hence, the model would only be valid
up to some highest frequency. However, the cutoff is also important for the model to be realistic, since any physical
spectral function should vanish as ω→∞ . In the main text, we stated that the same results can be obtained by
introducing a constant counter term in the Hamiltonian. This is a term which exactly completes the square of the
coupling term and the harmonic potential of the bath and can be seen as a normalization of the zero-energy level. If
we instead start the model with this counter term and drop the cutoff, we will get a Greens function αct(ω), which is
precisely such that the original Greens function can be written as α(ω) =α(0) +αct(ω), i.e., the counter term in the
Hamiltonian removes the zero frequency contribution of the Greens function. This α(ω= 0) generates a term in the
equation of motion that goes as/integraltext∞
0d/epsilon1J(/epsilon1)
π/epsilon1[S(t)×S(t)]. Since the integral is finite, with a frequency cutoff in J(ω),
the entire term is zero due to the cross product. This means that the equation of motion will be identical if we startarXiv:2211.12889v1 [cond-mat.mes-hall] 23 Nov 20222
either from the regular Hamiltonian with a frequency cutoff, or with a counter term and no cutoff. Here, we choose
to show the method that includes a counter term, because then we do not need to calculate terms which would have
canceled either way.
The microscopic system that we describe is a large spin in an external magnetic field, where the spin is linearly
coupled to a bath of harmonic oscillators in the same way as in Refs. [4–9]. Therefore, our Hamiltonian has the form
of a system, coupling, bath, and counter term; H(t) =Hs+Hc+Hb+Hct, where
Hs=B·ˆS−KS2
z,
Hc=/summationdisplay
αγαˆS·ˆxα,
Hb=/summationdisplay
αˆp2
α
2mα+mαω2
α
2ˆx2
α,
Hct=/summationdisplay
αγ2
α
2mαω2αˆS2. (1)
Here,Bis the (effective) magnetic field, ˆSis the spin, Kis thez-axis anisotropy, γαis the coupling strength, and
αis the index over all harmonic oscillators which have position ˆxα, momentum ˆpα, massmαand natural frequency
ωα. Notice that the counter term is constant, since S2is a conserved quantity, and that we have indeed completed
the square, such that
H(t) =B·ˆS−KS2
z+/summationdisplay
αˆp2
α
2mα+/bracketleftBigg/radicalbigg
mαω2α
2ˆxα+/radicalBigg
γ2α
2mαω2αˆS/bracketrightBigg2
. (2)
B. Keldysh partition function
We will use the Keldysh formalism to derive a quasi-classical equation of motion. Since this is an out-of-equilibrium
system, a common choice would be to use the Lindblad formalism with a master equation [10]. However, Lindblad can
only describe Markovian systems, which will not be the case when we introduce a non-Ohmic bath. In the Keldysh
formalism, one starts with an equilibrium density matrix in the far past (effectively infinite on the relevant time scale).
This then gets evolved with the time evolution operator as usual. However, in contrast to ordinary path integrals,
once the present has been reached, one evolves back to the infinite past. Since there is infinite time for evolution,
we can reach out-of-equilibrium states adiabatically. The benefit of integrating back to the infinite past is that we
begin and end with the same in-equilibrium system, which means equilibrium techniques can be used, at the cost of
having both the forward ( O+) and backward ( O−) quantities to take care of. To reach useful results, one can apply
a Keldysh rotation to the classical ( Oc= (O++O−)/2) and quantum ( Oq=O+−O−) components with the added
notation/vectorO=/parenleftbigg
Oc
Oq/parenrightbigg
. To derive a quasi-classical equation of motion, the action can be expanded in all the quantum
components, after which the Euler-Lagrange equation for the quantum components provides the equation of motion
in terms of the classical components.
FIG. 1. Figure extracted from Ref. [2]. The Keldysh contour starts at t=−∞, evolves forward to some time t, and then
evolves backwards in time to t=−∞.
To begin, we write down the Keldysh partition function
Z= Tr/braceleftbigg
TKexp/bracketleftbigg
−i/contintegraldisplay
KdtH(t)/bracketrightbigg
ρ0/bracerightbigg
, (3)3
whereTKis the Keldysh time ordering, ρ0is the density matrix at t=−∞, and the integral runs over the Keldysh
contour, as shown in FIG. 1. After discretizing the Keldysh time integral in the way of FIG. 1, we can rewrite the
trace as path-integrals over the spin coherent state |g/angbracketrightand the oscillators |ˆxα/angbracketrightand|ˆpα/angbracketright. This yields
Z=/integraldisplay
Dg/productdisplay
α/integraldisplay
Dˆxα/integraldisplay
DˆpαeiS[g,{ˆxα},{ˆpα}], (4)
with the Keldysh action
S[g,{ˆxα},{ˆpα}] =/contintegraldisplay
Kdt/bracketleftBigg
(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2
z,g
+/summationdisplay
α/parenleftBigg
−γαSg·ˆxα+ˆpα·˙ˆxα−γ2
αS2
g
2mαω2α−ˆp2
α
2mα−mαω2
α
2ˆx2
α/parenrightBigg/bracketrightBigg
, (5)
where we defined Sg=/angbracketleftg|ˆS|g/angbracketright.
The continuous path-integral seems to miss the boundary term /angbracketleftˆx1,α,g1|ρ0|ˆx2N,α,g2N/angbracketright/angbracketleftˆp2N,α|ˆx2N,α/angbracketright, but it is
included in the Keldysh contour, as it connects the beginning and final contour time at t=−∞; see Ref. [2].
Now, we will integrate out the bath degrees of freedom, beginning by completing the square and performing the
Gaussian integral over ˆpα. The Gaussian contribution in ˆpαwill act as a constant prefactor, so it will drop out of
any calculation of an observable due to the normalization. Hence, we can effectively set it to one to find
/integraldisplay
Dˆpαexp/bracketleftbigg
−i/contintegraldisplay
Kdt/parenleftbiggˆp2
α
2mα−ˆpα·˙ˆxα/parenrightbigg/bracketrightbigg
= exp/bracketleftbigg
i/contintegraldisplay
Kdt/parenleftBig
−mα
2ˆxα∂2
tˆxα/parenrightBig/bracketrightbigg
, (6)
where we also did a partial integration in ˆxα. Next we will perform a similar approach for the positions, but it is
useful to apply the Keldysh rotation first. Note that we can directly rewrite the integral over the Keldysh contour as
a regular time integral over the quantum components. However, one must still rewrite the contents of the integral in
terms of the quantum and classical parts of the variables, since the Keldysh rotation does not immediately work for
products. The action can first be written as
iS[g,{ˆxα}] =i/integraldisplay
dt/parenleftBig
[(−i∂t/angbracketleftg|)|g/angbracketright]q−[B·Sg]q+K/bracketleftbig
S2
z,g/bracketrightbigq
−/summationdisplay
α/braceleftBigg
[γαSg·ˆxα]q+γ2
α[S2
g]q
2mαω2α+/bracketleftBigmα
2ˆxα/parenleftbig
∂2
t+ω2
α/parenrightbigˆxα/bracketrightBigq/bracerightBigg/parenrightBigg
. (7)
We can then derive that
−[γαSg·ˆxα]q=−γα/bracketleftbig
S+
g·ˆx+
α−S−
g·ˆx−
α/bracketrightbig
=−γα/bracketleftbigg/parenleftbigg
Sc
g+1
2Sq
g/parenrightbigg
·/parenleftbigg
ˆxc
α+1
2ˆxq
α/parenrightbigg
−/parenleftbigg
Sc
g−1
2Sq
g/parenrightbigg
·/parenleftbigg
ˆxc
α−1
2ˆxq
α/parenrightbigg/bracketrightbigg
=−γα/bracketleftbig
Sc
gˆxq
α+Sq
gˆxc
α/bracketrightbig
=−γα/bracketleftbigg/parenleftbigSc
gSq
g/parenrightbig
τx/parenleftbigg
ˆxc
α
ˆxq
α/parenrightbigg/bracketrightbigg
, (8)
where we introduced τx=/parenleftbigg
0 1
1 0/parenrightbigg
in the Keldysh (classical, quantum) space represented by an upper index candq
respectively. Next, we want to derive a similar form for the part of the action that is quadratic in ˆxα. Since these are
harmonic oscillators in equilibrium, we can refer the reader to Ref. [2], noting that a unit mass was used there, and
conclude that
/bracketleftBig
−mα
2ˆxα/parenleftbig
∂2
t+ω2
α/parenrightbigˆxα/bracketrightBigq
=/parenleftbigˆxc
αˆxq
α/parenrightbig/parenleftBigg
0/bracketleftbig
G−1
α/bracketrightbigA
/bracketleftbig
G−1
α/bracketrightbigR/bracketleftbig
G−1
α/bracketrightbigK/parenrightBigg/parenleftbigg
ˆxc
α
ˆxq
α/parenrightbigg
, (9)
where the retarded and advanced Greens functions read
[G−1
α]R/A(t−t/prime) =δ(t−t/prime)mα
2[(i∂t±i0)2−ω2
α]. (10)
The±i0 is introduced because we need an infinitesimal amount of dissipation on the bath for it to remain in equilibrium
and the sign is tied to causality. This is because there is also an infinitesimal amount of energy transfer from the4
macroscopic spin to each of the oscillators. This results in an extra first-order derivative term, which is found by
multiplying out the square with i0. One might want to set these terms to zero immediately, but as it turns out,
these are very important limits, which shift away poles from integrals that we need to compute later. Once that is
done, the limits are no longer important for the final result, and they may finally be put to zero. Since the bath is in
equilibrium, we can use the fluctuation dissipation theorem to compute the Keldysh component using
GK
α(ω) =/bracketleftbig
GR
α(ω)−GA
α(ω)/bracketrightbig
coth/parenleftBigω
2T/parenrightBig
.
The ˆxdependent part of the action is now given by
iSX=i/integraldisplay
dt/bracketleftbigg
−γα/parenleftbigSc
gSq
g/parenrightbig
τx/parenleftbigg
ˆxc
α
ˆxq
α/parenrightbigg
+/parenleftbigˆxc
αˆxq
α/parenrightbig
G−1
α/parenleftbigg
ˆxc
α
ˆxq
α/parenrightbigg/bracketrightbigg
, (11)
which we can compute by completing the square to find
iSX=i/integraldisplay
dt/bracketleftbigg
−γ2
α
4/vectorST
g/parenleftbigg
0GA
α
GR
αGK
α/parenrightbigg
/vectorSg/bracketrightbigg
. (12)
Before we write down the final effective action, we also have to rewrite the quadratic part in Sin a similar vector
form, which is
−γ2
α[S2
g]q
2mαω2α=−γ2
α
2mαω2α/parenleftbigSc
gSq
g/parenrightbig
τx/parenleftbiggSc
g
Sq
g/parenrightbigg
. (13)
Combining everything together, we find that the partition function of the system is given by Z=/integraltext
Dg eiS[g], with
the effective action
iS[g] =i/integraldisplay
dt/braceleftBigg
/bracketleftbig
(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2
z,g/bracketrightbigq−/integraldisplay
dt/prime/vectorST
g(t)/parenleftbigg
0αA
αRαK/parenrightbigg
(t−t/prime)/vectorSg(t/prime)/bracerightBigg
, (14)
whereαA/R(t−t/prime) =/summationtext
α/parenleftBig
γ2
α
4GA/R
α(t−t/prime) +γ2
α
2mαω2αδ(t−t/prime)/parenrightBig
andαK(t−t/prime) =/summationtext
αγ2
α
4GK
α(t−t/prime).
C. Quasi-classical equation of motion
In the quasi-classical regime, we are interested in solutions where the quantum components ( q) are small compared
to the classical components ( c). We can thus neglect terms of O[(q)3], but we must be careful with ( q)2. We can
use a Hubbard-Stratonovich transformation to convert ( q)2terms into an expression with just ( q), but with a new
fieldξadded to the path integral [3]. The action will then contain only terms of linear order in ( q), which means the
partition function has the form Z∼/integraltext
DcDq exp[if(c)q] =/integraltext
Dc1
2πδ[f(c)]. Hence, only solutions that satisfy f(c) = 0
contribute to the path integral. Within that subset, we want to minimize the action.
In order to derive the equation of motion of the system, we must understand the relation between |g/angbracketrightandSg=
/angbracketleftg|S|g/angbracketright. Using the Euler angle representation [1], we can describe |g/angbracketrightas
|g/angbracketright=g|↑/angbracketright=e−iφSze−iθSye−iψSz|↑/angbracketright=e−iφSze−iθSy|↑/angbracketrighte−iψS(15)
and similarly
/angbracketleftg|=eiψS/angbracketleft↑|eiθSyeiφSz. (16)
Note that the ψangle is now independent of the quantum state |↑/angbracketright, since this angle is describing the rotation of the
vector pointing in the spin direction, which is symmetric. Hence, this will yield a gauge symmetry.
Using the Euler angle representation in the first terms of Eq. (14), we see that
(−i∂t/angbracketleftg|)|g/angbracketright=/parenleftBig
˙ψSeiψS/angbracketleft↑|eiθSyeiφSz+eiψS/angbracketleft↑|˙θSyeiθSyeiφSz+eiψS/angbracketleft↑|eiθSy˙φSzeiφSz/parenrightBig
e−iφSze−iθSy|↑/angbracketrighte−iψS
=˙ψS+˙θ/angbracketleft↑|Sy|↑/angbracketright+˙φ/angbracketleft↑|eiθSySze−iθSy|↑/angbracketright. (17)5
We note that/angbracketleft↑|Sy|↑/angbracketright= 0, while the last term includes a rotation of the spin up state by θdegrees in the ydirection
and then measures the Szcomponent of that state, which is Scosθ. Hence,
(−i∂t/angbracketleftg|)|g/angbracketright=˙ψS+˙φScosθ. (18)
We now define a new variable χsuch thatψ=χ−φ, which results in
(−i∂t/angbracketleftg|)|g/angbracketright= ˙χS−˙φ(1−cosθ)S. (19)
Making use of the Euler angle representation, we also see that
Sg=S
sinθcosφ
sinθsinφ
cosθ
. (20)
We see thatB·Sg=S[Bxsinθcosφ+Bysinθsinφ+Bzcosθ]. Similarly, KS2
z,g=KS2cos2θ. Now, we still have to
compute the quantum parts of these quantities. We first note that
Sq
g,x/S= [sinθcosφ]q= 2 cosθcsinθq
2cosφccosφq
2−2 sinθccosθq
2sinφcsinφq
2;
Sq
g,y/S= [sinθsinφ]q= 2 sinθccosθq
2cosφcsinφq
2+ 2 cosθcsinθq
2sinφccosφq
2;
Sq
g,z/S= [cosθ]q=−2 sinθcsinθq
2;
[cos2θ]q=−2 sinθccosθcsinθq. (21)
Next, we will choose a gauge for χas in Ref. [11], which is
˙χc=˙φc(1−cosθc)
χq=φq(1−cosθc). (22)
Definingp= 1−cosθ, we see that [(−i∂t/angbracketleftg|)|g/angbracketright]q=/bracketleftBig
˙χS−˙φpS/bracketrightBig
q=S/bracketleftBig
φq˙pc−˙φcpq/bracketrightBig
. Now,pq= 2 sinθcsinθq
2and
˙pc=˙θcsinθccosθq
2+˙θq
2cosθcsinθq
2,which leads to
[(−i∂t/angbracketleftg|)|g/angbracketright]q=S/bracketleftBig
φq˙pc−˙φcpq/bracketrightBig
=S/bracketleftBigg
φq˙θcsinθccosθq
2+φq˙θq
2cosθcsinθq
2−2˙φcsinθcsinθq
2/bracketrightBigg
. (23)
Next, we want to express B·Sq
gin terms of Euler angles. We see that
B·Sq
g=S[Bxsinθcosφ+Bysinθsinφ+Bzcosθ]q
= 2S/bracketleftBig
Bx/parenleftbigg
cosθcsinθq
2cosφccosφq
2−sinθccosθq
2sinφcsinφq
2/parenrightbigg
+By/parenleftbigg
sinθccosθq
2cosφcsinφq
2+ cosθcsinθq
2sinφccosφq
2/parenrightbigg
−Bzsinθcsinθq
2/bracketrightBig
, (24)
where we used the results from Eq. (21). Similarly, we have
K/bracketleftbig
S2
z,g/bracketrightbigq=KS2[cos2θ]q=−2KS2sinθccosθcsinθq. (25)
Combining these results, we conclude that
/bracketleftbig
(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2
z,g/bracketrightbigq=S/bracketleftBigg
φq˙θcsinθccosθq
2+φq˙θq
2cosθcsinθq
2−2(−Bz+KScosθc+˙φc) sinθcsinθq
2
−2Bx/parenleftbigg
cosθcsinθq
2cosφccosφq
2−sinθccosθq
2sinφcsinφq
2/parenrightbigg
−By/parenleftbigg
sinθccosθq
2cosφcsinφq
2+ cosθcsinθq
2sinφccosφq
2/parenrightbigg/bracketrightBigg
. (26)6
Remark that this expression only contains odd powers of ( q), so that we can neglect all higher-order terms to get
/bracketleftBig
(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2
z,g/bracketrightBigq
=S/bracketleftBig
−θqsinθc(−Bz+KScosθc+˙φc)
−θqcosθc(Bxcosφc+Bysinφc) +φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig
. (27)
Now, we focus on the part of the action in Eq. (14) that comes from the bath, given by
iSb[g] =−i/integraldisplay
dt/integraldisplay
dt/prime/vectorST
g(t)/parenleftbigg
0αA
αRαK/parenrightbigg
(t−t/prime)/vectorSg(t/prime). (28)
Let us first consider what Sq
gandSc
gare in terms of φandθ. By performing some trigonometric operations on each
of the components, we find that
Sc
g=S
sinθccosθq
2cosφccosφq
2−cosθcsinθq
2sinφcsinφq
2
sinθccosθq
2sinφccosφq
2+ cosθcsinθq
2cosφcsinφq
2
cosθccosθq
2
(29)
and
Sq
g= 2S
cosθcsinθq
2cosφccosφq
2−sinθccosθq
2sinφcsinφq
2
sinθccosθq
2cosφcsinφq
2+ cosθcsinθq
2sinφccosφq
2
−sinθcsinθq
2
. (30)
By expanding in the quantum components of Sc
gandSq
g, we see that
Sc
g= (q)0+O/parenleftbig
(q)2/parenrightbig
,
Sq
g= (q)1+O/parenleftbig
(q)3/parenrightbig
.
Since the action only contains terms with at least one Sq
g, we know that the only way to obtain a term of order ( q)2
is from (Sq
g)2. Hence, we may neglect all terms beyond linear ( q) inS(c/q)
g in the quasi-classical regime. This results
in
Sc
g=S
sinθccosφc
sinθcsinφc
cosθc
, (31)
Sq
g=S
θqcosθccosφc−φqsinθcsinφc
φqsinθccosφc+θqcosθcsinφc
−θqsinθc
. (32)
A useful remark for later is that this shows that
Sq
g=θq∂
∂θcSc
g+φq∂
∂φcSc
g. (33)
Going back to iSb[g], we can rewrite this as a convolution, in the sense that
iSb[g] =−i/integraldisplay
dt/bracketleftbig
Sc
g(t)·/parenleftbig
αA∗Sq
g/parenrightbig
(t) +Sq
g(t)·/parenleftbig
αR∗Sc
g/parenrightbig
(t) +Sq
g(t)·/parenleftbig
αK∗Sq
g/parenrightbig
(t)/bracketrightbig
, (34)
where (f∗g)(t) =/integraltext∞
−∞dt/primef(t−t/prime)g(t/prime). We see that the first two terms contain precisely one quantum component,
but the last term has two quantum components. When writing down the Euler-Lagrange equation of motion, it is
important to realize that the convolution operation will act as if it is a simple multiplication, since the convolution
obeys
d
dx(f(x)∗g)(t) =/parenleftbiggdf
dx∗g/parenrightbigg
(t). (35)7
We now concentrate on the ( q)2part of this action, for which we would like to use a Hubbard-Stratonovich transfor-
mation in order to reduce this to linear in ( q). Recall that a Hubbard-Stratonovich transformation is given by
exp/bracketleftBig
−a
2x2/bracketrightBig
=/radicalbigg
1
2πa/integraldisplay
Dξexp/bracketleftbigg
−ξ2
2a−ixξ/bracketrightbigg
. (36)
However, we see that our action does not contain any purely quadratic terms, but rather a Greens functional shape
asSq
g(t)αK(t−t/prime)Sq
g(t/prime). Hence, to use a Hubbard-Stratonovich like transformation, we must derive it from a Greens
function exponential, similarly to Ref. [3]. Assuming that this is renormalizable and that αKcan be rewritten into a
distribution, we have
1 =/integraldisplay
Dξexp/bracketleftbigg
−1
2/integraldisplay
dt/integraldisplay
dt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)/bracketrightbigg
=/integraldisplay
Dξexp/bracketleftbigg
−1
2/integraldisplay
dt/integraldisplay
dt/prime/parenleftbigg
ξ(t)−2/integraldisplay
dt/prime/primeSq
g(t/prime/prime)αK(t/prime/prime−t)/parenrightbigg
[−2iαK]−1(t−t/prime)/parenleftbigg
ξ(t/prime)−2/integraldisplay
dt/prime/prime/primeαK(t/prime−t/prime/prime/prime)Sq
g(t/prime/prime/prime)/parenrightbigg/bracketrightbigg
=/integraldisplay
Dξexp/bracketleftbigg
−1
2/integraldisplay
dt/integraldisplay
dt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)
−iSq
g(t)δ(t−t/prime)ξ(t/prime)−iξ(t)δ(t−t/prime)Sq
g(t/prime)−2iSq
g(t)αK(t−t/prime)Sq
g(t/prime)/bracketrightbig
=/integraldisplay
Dξexp/bracketleftbigg
−1
2/integraldisplay
dt/integraldisplay
dt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)−2iSq
g(t)δ(t−t/prime)ξ(t/prime)−2iSq
g(t)αK(t−t/prime)Sq
g(t/prime)/bracketrightbigg
,
where we used that/integraltext
dt/primeαK(t−t/prime)[αK]−1(t/prime−t/prime/prime) =δ(t−t/prime/prime) and that 2 iαKis positive real. Therefore, we find that
exp/bracketleftbigg
−i/integraldisplay
dt/integraldisplay
dt/primeSq
g(t)αK(t−t/prime)Sq
g(t/prime)/bracketrightbigg
=/integraldisplay
Dξexp/bracketleftbigg
−1
2/integraldisplay
dt/integraldisplay
dt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)/bracketrightbigg
·exp/bracketleftbigg
i/integraldisplay
dtSq
g(t)ξ(t)/bracketrightbigg
. (37)
The double integral in the first exponential signifies the statistical properties of ξ. For instance, if αKis delta-like,
thenξwould have Gaussian statistics (e.g. white noise), but in general we will have time correlated noise defined by
αK[3], such that
/angbracketleftξ(t)ξ(t/prime)/angbracketright=−2iαK(t−t/prime). (38)
Since there is no gdependence in the double ξexponential, we will leave it out of S[g] and only remember these
statistics. Our partition function is then given by
Z=/integraldisplay
Dξexp (iSn[ξ])/integraldisplay
Dgexp (iSsc[g,ξ]), (39)
where the noise action is given by
iSn[ξ] =−1
2/integraldisplay
dt/integraldisplay
dt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime) (40)
and the semi-classical action is given by
iSsc[g,ξ] =i/integraldisplay
dtS/bracketleftBig
−θqsinθc(−Bz+KScosθc+˙φc)−θqcosθc(Bxcosφc+Bysinφc)
+φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig
+i/integraldisplay
dt/bracketleftbig
ξ(t)Sq
g(t)/bracketrightbig
−i/integraldisplay
dt/bracketleftbig
Sc
g(t)·/parenleftbig
αA∗Sq
g/parenrightbig
(t) +Sq
g(t)·/parenleftbig
αR∗Sc
g/parenrightbig
(t)/bracketrightbig
, (41)
whereSc
g(t) andSq
g(t) include only up to first-order corrections in quantum components. Assuming that αA/Rcan
be written in terms of distributions, we can define the distribution αdiss(t) =−αR(t)−αA(−t) and rewrite the
semi-classical action as
iSsc[g,ξ] =i/integraldisplay
dtS/bracketleftBig
−θqsinθc(−Bz+KScosθc+˙φc)−θqcosθc(Bxcosφc+Bysinφc)
+φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig
+i/integraldisplay
dt/bracketleftbig/parenleftbig
αdiss∗Sc
g/parenrightbig
(t) +ξ(t)/bracketrightbig
Sq
g(t). (42)8
Recall that, using the Euler angles, we have/integraltext
Dg=/integraltext
DθDφ sin(θ). Technically, the factor of sin( θ) would end up
in the action. However, since one could define ρ= cos(θ) as a new variable in order to avoid this, we know that this
term is not relevant to the physics. Hence, we can disregard it.
Since all terms in iSsc[g,ξ] are either linear in θqorφq, we find two Euler-Lagrange equations of the form
δLsc
δθq= 0 andδLsc
δφq= 0. (43)
Remembering Eq. (33), we see thatδSq
g(t)
δθq=δSc
g(t)
δθcandδSq
g(t)
δφq=δSc
g(t)
δφc. Hence, the e.o.m. can be rearranged to yield
˙φc=1
Ssinθc/bracketleftbig
−B(Sc
z) +/parenleftbig
αdiss∗Sc
g/parenrightbig
(t) +ξ(t)/bracketrightbig
·δSc
g(t)
δθc(44)
and
˙θc=−1
Ssinθc/bracketleftbig
−B(Sc
z) +/parenleftbig
αdiss∗Sc
g/parenrightbig
(t) +ξ(t)/bracketrightbig
·δSc
g(t)
δφc, (45)
whereB(Sc
z) =
Bx
By
Bz−KSc
z
.
D. Generalized Landau-Lifshitz-Gilbert equation
We want to show that the equations found by the microscopic model are in fact precisely of the LLG form. For
this, we will have to start from the LLG equation, and introduce the same two Euler angles θandφfor the spin, and
show that this gives rise to the same set of equations as previously deduced.
We begin with the generalized LLG equation
˙S(t) =S(t)×[−B(Sz) + (αdiss∗S) (t) +ξ(t)], (46)
whereαdiss(t) =−αR(t)−αA(−t),/angbracketleftξ(t)ξ(t/prime)/angbracketright=−2iαK(t−t/prime) andB(Sz) = (Bx,By,Bz−KSz)T. Since the velocity
ofSis always perpendicular to S, we know that the magnitude of Sis constant. Hence, we can go to spherical
coordinates, such that
S=S
sinθcosφ
sinθsinφ
cosθ
. (47)
Inserting this into the LLG equation, we firstly see that
˙S=˙θ∂S
∂θ+˙φ∂S
∂φ=˙θS
cosθcosφ
cosθsinφ
−sinθ
+˙φS
−sinθsinφ
sinθcosφ
0
.
Now, we notice that the RHS of the LLG equation can, without loss of generality, be written as S(t)×rwith
r= (x,y,z )T. Working this out explicitly, we find that the LLG equation ˙S=S×rbecomes
S
˙θcosθcosφ−˙φsinθsinφ
˙θcosθsinφ+˙φsinθcosφ
−˙θsinθ
=S
zsinθsinφ−ycosθ
xcosθ−zsinθcosφ
ysinθcosφ−xsinθsinφ
. (48)
We note that the equation corresponding to the zcomponent can be written as
˙θ=−1
sinθr·
−sinθsinφ
sinθcosφ
0
=−1
Ssinθr·∂S
∂φ. (49)9
Now, we add up the ˆ xand ˆyequations, such that the ˙θcancels (i.e.−ˆxsinφ+ ˆycosφ). This yields
˙φsinθ(sin2φ+ cos2φ) =−zsinθ(sin2φ+ cos2φ) +ycosθsinφ+xcosθcosφ,
which simplifies to
˙φ=1
sinθr·
cosθcosφ
cosθsinφ
−sinθ
=1
Ssinθr·∂S
∂θ. (50)
By inserting r=−B(Sz) + (αdiss∗S) (t) +ξ(t), we see that this is identical to the equations derived from the
microscopic model
˙φc=1
Ssinθc/bracketleftbig
−B(Sz) +/parenleftbig
αdiss∗Sc
g/parenrightbig
(t) +ξ(t)/bracketrightbig
·δSc
g(t)
δθc; (51)
˙θc=−1
Ssinθc/bracketleftbig
−B(Sz) +/parenleftbig
αdiss∗Sc
g/parenrightbig
(t) +ξ(t)/bracketrightbig
·δSc
g(t)
δφc. (52)
Therefore, we may conclude that our microscopic model is described by the generalized LLG equation.
For the fractional LLG equation, we are in particular interested in the case where αdiss∗S=αsDs
tS, whereDs
tis
a fractional derivative. For instance, assuming 0 <s< 1, the Liouville fractional derivative is given by
Ds
tf(t) =1
Γ(1−s)/integraldisplayt
−∞(t−t/prime)−sf/prime(t/prime)dt/prime. (53)
So, ifαdiss=αsΘ(t)
Γ(1−s)t−s∂t, then, because of the convolution with S, we would find a fractional LLG equation, whereas
αdiss=α1δ(t)∂twould give the regular LLG equation.
II. FRACTIONAL DERIVATIVE FROM NON-OHMIC SPECTRAL FUNCTION
Here, we will compute the type of dissipation which comes from the spectral function. We will first derive the
spectral function from microscopic quantities to see how it ends up in the Greens function. Then, we will calculate
the dissipation for three different cases.
A. Calculating the effective Greens functions
We recall that
αA/R(t−t/prime) =/summationdisplay
α/parenleftbiggγ2
α
4GA/R
α(t−t/prime) +δ(t−t/prime)γ2
α
2mαω2α/parenrightbigg
,
where
[G−1
α]R/A(t−t/prime) =δ(t−t/prime)mα
2[(i∂t±i0)2−ω2
α].
By the fluctuation dissipation theorem, we also have
αK(ω) =/bracketleftbig
αR(ω)−αA(ω)/bracketrightbig
coth/parenleftBigω
2T/parenrightBig
.
We are interested in finding closed forms for αR/A/K(t−t/prime). Using the relation
/integraldisplay
dt/primeG−1(t−t/prime)G(t/prime−t/prime/prime) =δ(t−t/prime/prime), (54)
we note that
mα
2[(i∂t±i0)2−ω2
α]GR/A
α(t−t/prime/prime) =δ(t−t/prime/prime). (55)10
The Fourier transform1yields
GR/A
α(ω) =1
mα
2[(ω±i0)2−ω2α]. (56)
We therefore find that
αR/A(ω) =/summationdisplay
α/parenleftbiggγ2
α
4GA/R
α(ω) +γ2
α
2mαω2α/parenrightbigg
=/summationdisplay
αγ2
α
2mα/bracketleftbigg1
(ω±i0)2−ω2α+1
ω2α/bracketrightbigg
=/summationdisplay
αγ2
α
2mαω2αω2
(ω±i0)2−ω2α.(57)
The spectral function is given by the imaginary part of the Fourier transform of the dynamical susceptibility
χ(ω) =δ
δS(ω)/summationdisplay
αγαˆxα(ω). (58)
The Hamiltonian e.o.m. for the bath can be found by going back to our starting Hamiltonian
H=B·S+/summationdisplay
αγαS·ˆxα+/summationdisplay
αˆp2
α
2mα+mαω2
α
2ˆx2
α+/summationdisplay
αγ2
α
2mαω2αS2.
The e.o.m. reads
˙ˆxα=ˆpα
mαand ˙ˆpα=−γαS−mαω2
αˆxα. (59)
Combining both equations and taking the Fourier transform, we find
ˆxα(ω) =γα
mα[(ω+i0)2−ω2α]S(ω), (60)
where we have taken an infinitesimal amount of dissipation + i0 on the oscillators into account. This leads to
χ(ω) =/summationdisplay
αγ2
α
mα[(ω+i0)2−ω2α]=/summationdisplay
αγ2
α
2mαωα/parenleftbigg1
ω+i0−ωα−1
ω+i0 +ωα/parenrightbigg
. (61)
We remark that
Im1
x+i0=−πδ(x), (62)
which leads to
J(ω) = Imχ(ω) =−/summationdisplay
απγ2
α
2mαωα[δ(ω−ωα)−δ(ω+ωα)] =−π
2/summationdisplay
αγ2
α
mαωαδ(ω−ωα), (63)
where we used that all oscillator frequencies are positive. We can identify the spectral function in αR/Aas
αR/A(ω) =/summationdisplay
αγ2
α
2mαω2αω2
(ω±i0)2−ω2α=−/integraldisplay∞
0dε
πω2ε−1J(ε)
(ω±i0)2−ε2. (64)
Now, we will assume a particular shape for J(ε). This can be either Ohmic or non-Ohmic, but in general we may
assume a power-law behavior as some J(ε) =αsεs.
1We use the convention f(ω) =/integraltext∞
−∞dteiωtf(t) andf(t) =1
2π/integraltext∞
−∞dωe−iωtf(ω), withδ(t) =1
2π/integraltext∞
−∞dωeiωt.11
B. Ohmic spectral function
Beginning with the Ohmic case J(ε) =α1ε, we see that
2iImαR/A(ω) =αR/A(ω)−/bracketleftBig
αR/A/bracketrightBig∗
(ω)
=−α1/integraldisplay∞
0dε
π/bracketleftbiggω2
(ω±i0)2−ε2−ω2
(ω∓i0)2−ε2/bracketrightbigg
=−α1/integraldisplay∞
−∞dε
2π/bracketleftbiggω2
(ω±i0)2−ε2−ω2
(ω∓i0)2−ε2/bracketrightbigg
=−α1ω2/integraldisplay∞
−∞dε
2π/bracketleftbigg1
(ω±i0 +ε)(ω±i0−ε)−1
(ω∓i0 +ε)(ω∓i0−ε)/bracketrightbigg
=±4i0ω3α1/integraldisplay∞
−∞dε
2π1
(ω±i0 +ε)(ω±i0−ε)(ω∓i0 +ε)(ω∓i0−ε), (65)
which has four poles at ε=±1(ω±2i0). Since the integral scales as <1/|ε|, we can add an infinite radius half circle
to complete a complex contour integral. Notice from symmetry that we will always have one of each of the four poles.
We can thus drop the ±signs inside since this only changes the notation order in the fraction. We thus find that
2iImαR/A(ω) =±4i0ω3α1/integraldisplay∞
−∞dε
2π1
(ε+ω+i0)(ε−ω−i0)(ε+ω−i0)(ε−ω+i0). (66)
Completing the contour along the top, we find poles at ε=±ω+i0, which yields
2iImαR/A(ω) =∓0·4ω3α1/bracketleftbigg1
(ω+i0 +ω+i0)(ω+i0 +ω−i0)(ω+i0−ω+i0)
+1
(−ω+i0 +ω+i0)(−ω+i0−ω−i0)(−ω+i0−ω+i0)/bracketrightbigg
=∓0·4ω3α1/bracketleftbigg1
2(ω+i0)(2ω)(2i0)+1
(2i0)(−2ω)2(−ω+i0)/bracketrightbigg
=±i
2ω3α1/bracketleftbigg1
ω2+i0ω+1
ω2−i0ω/bracketrightbigg
=±i
2ω3α1/bracketleftbiggω2−i0ω+ω2+i0ω
ω4/bracketrightbigg
=±iωα1. (67)
Hence, Im αR/A(ω) =±α1ω/2. Similarly,
2 ReαR/A(ω) =αR/A(ω) +/bracketleftBig
αR/A/bracketrightBig∗
(ω)
=−α1/integraldisplay∞
0dε
π/bracketleftbiggω2
(ω±i0)2−ε2+ω2
(ω∓i0)2−ε2/bracketrightbigg
=−α1/integraldisplay∞
−∞dε
2π/bracketleftbiggω2
(ω±i0)2−ε2+ω2
(ω∓i0)2−ε2/bracketrightbigg
=−α1/integraldisplay∞
−∞dε
2π/bracketleftbiggω2
(ω±i0 +ε)(ω±i0−ε)+ω2
(ω∓i0 +ε)(ω∓i0−ε)/bracketrightbigg
=α1/integraldisplay∞
−∞dε
πω2(ε2−ω2)
(ε+ω+i0)(ε−ω−i0)(ε+ω−i0)(ε−ω+i0). (68)12
Since the integral scales as 1 /|ε|, we can freely add the infinite circular contour along the top. Applying the residue
theorem, we find
2 ReαR/A(ω) = 2iα1/bracketleftbiggω2((ω+i0)2−ω2)
(ω+i0 +ω+i0)(ω+i0 +ω−i0)(ω+i0−ω+i0)
+ω2((−ω+i0)2−ω2)
(−ω+i0 +ω+i0)(−ω+i0−ω−i0)(−ω+i0−ω+i0)/bracketrightbigg
= 2α1i/bracketleftbiggω2(2ωi0)
2(ω+i0)(2ω)(2i0)+ω2(−2ωi0)
(2i0)(−2ω)2(−ω+i0)/bracketrightbigg
=α1
2i/bracketleftbiggω2
ω+i0−ω2
ω−i0/bracketrightbigg
=α1
2i/bracketleftbiggω2(−2i0)
ω2/bracketrightbigg
=α10, (69)
which means that Re αR/A(ω) =α1
2·0 = 0. Hence,
αR/A(ω) =±α1iω
2(70)
and sinceαdiss(t) =−αR(t)−αA(−t) we have
αdiss(ω) =−αR(ω)−αA(−ω) =−α1iω. (71)
In the LLG equation, we thus find
(αdiss∗S) (t) =1
2π/integraldisplay
dωe−iωt(αdiss∗S) (ω)
=1
2π/integraldisplay
dωe−iωtαdiss(ω)S(ω)
=1
2π/integraldisplay
dωe−iωt(−α1iω)S(ω)
=α1∂
∂t1
2π/integraldisplay
dωe−iωtS(ω)
=α1˙S(t). (72)
Furthermore, we have that
αK(ω) = [αR(ω)−αA(ω)] coth/parenleftBigω
2T/parenrightBig
=iα1ωcoth/parenleftBigω
2T/parenrightBig
. (73)
We have two regimes from the cotangent which crossover around ω≈2T. If the temperature is large enough that we
can approximate coth/parenleftbigω
2T/parenrightbig
≈2T
ω, then we have αK(ω)≈2iα1T. Therefore, we find
αK(t) = 2iα1T1
2π/integraldisplay∞
−∞dωe−iωt= 2iα1Tδ(t), (74)
and thus
/angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδm,nαK(t−t/prime) = 4α1Tδm,nδ(t−t/prime). (75)
We see that Eq. (72) and Eq. (75) combine to give the regular LLG equation with first-order dissipation and white
noise fluctuation:
˙S(t) =S(t)×/bracketleftBig
−B+α1˙S(t) +ξ(t)/bracketrightBig
. (76)13
C. Sub-Ohmic spectral function
We now consider the case where
J(ε) =αssin/parenleftBigπs
2/parenrightBig
εs, (77)
with 0<s< 1. In this case,
αR/A(ω) =−αssin/parenleftBigπs
2/parenrightBig/integraldisplay∞
0dε
πω2εs−1
(ω±i0)2−ε2. (78)
Considering the LLG equation, the relevant dissipation term is ( αdiss∗S)(t). In terms of the Fourier transform of
αdiss(t), we find that
(αdiss∗S)(t) =/integraldisplay∞
−∞dt/primeαdiss(t−t/prime)S(t/prime)
=−/integraldisplay∞
−∞dt/prime[αR(t−t/prime) +αA(t/prime−t)]S(t/prime)
=−1
2π/integraldisplay∞
−∞dt/prime/integraldisplay∞
−∞dω/bracketleftBig
e−iω(t−t/prime)αR(ω) +eiω(t−t/prime)αA(ω)/bracketrightBig
S(t/prime)
=−1
2π/integraldisplay∞
−∞dt/prime/integraldisplay∞
−∞dωe−iω(t−t/prime)/bracketleftbig
αR(ω) +αA(−ω)/bracketrightbig
S(t/prime)
=αssin/parenleftbigπs
2/parenrightbig
2π2/integraldisplay∞
−∞dt/prime/integraldisplay∞
−∞dω/integraldisplay∞
0dεe−iω(t−t/prime)/bracketleftbiggω2εs−1
(ω+i0)2−ε2+(−ω)2εs−1
(−ω−i0)2−ε2/bracketrightbigg
S(t/prime)
=−αssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
−∞dt/prime/integraldisplay∞
0dε/integraldisplay∞
−∞dω/bracketleftbigg
e−iω(t−t/prime)ω2εs−1
ε2−(ω+i0)2S(t/prime)/bracketrightbigg
. (79)
Notice that we have two poles at ω=±ε−i0, below the real axis. If t−t/prime<0, then the exponential will go to zero as
ω→+i∞. Hence we could close the ωintegration with a complex contour as an infinite half-circle along the top, and
get zero from the Cauchy theorem. If t−t/prime>0, however, we see that the exponential goes to zero when ω→−i∞.
Hence, we can close the ωintegration along the bottom. Thus, using the residue theorem (reversing the integration
direction), we find
(αdiss∗S)(t) =−αssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
−∞dt/prime/integraldisplay∞
0dε2πiΘ(t−t/prime)
/bracketleftBig
e−i(ε−i0)(t−t/prime)(ε−i0)2εs−1
(ε−i0 +i0 +ε)+e−i(−ε−i0)(t−t/prime)(−ε−i0)2εs−1
(−ε−i0 +i0−ε)/bracketrightBig
S(t/prime)
=−iαssin/parenleftbigπs
2/parenrightbig
π/integraldisplayt
−∞dt/prime/integraldisplay∞
0dε/bracketleftbigg
e−iε(t−t/prime)εs+1
ε+eiε(t−t/prime)εs+1
−ε/bracketrightbigg
S(t/prime)
=−iαssin/parenleftbigπs
2/parenrightbig
π/integraldisplayt
−∞dt/prime/integraldisplay∞
0dε/bracketleftBig
e−iε(t−t/prime)−eiε(t−t/prime)/bracketrightBig
εsS(t/prime)
=−2αssin/parenleftbigπs
2/parenrightbig
π/integraldisplayt
−∞dt/prime/integraldisplay∞
0dεsin[ε(t−t/prime)]εsS(t/prime)
=−2αssin/parenleftbigπs
2/parenrightbig
π/integraldisplay∞
0dε/braceleftbigg/bracketleftbig
εs−1cos[ε(t−t/prime)]S(t/prime)/bracketrightbigt/prime=t
t/prime=t0−/integraldisplayt
t0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg
=−2αssin/parenleftbigπs
2/parenrightbig
π/integraldisplay∞
0dε/braceleftbigg
εs−1S(t)−εs−1cos[ε(t−t0)]S(t0)−/integraldisplayt
t0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg
.(80)
The first term vanishes because of the cross product with S(t) in the LLG equation. The second term is where we
had to be careful. Here, we should realize that the −∞ is physically only indicating that it is a time very far in the
past. So, to avoid unphysical infinities, we introduced a finite initial time t0and we will take t0→−∞ later. For
this, we need to introduce some fractional derivative notation. We define the Riemann-Liouville (RL) and Caputo (C)14
derivatives of order s, with an integer nsuch thatn≤s<n + 1, as
RL
t0Ds
tf(t) =dn
dtn1
Γ(n−s)/integraldisplayt
t0dt/prime(t−t/prime)n−1−sf(t/prime), (81)
C
t0Ds
tf(t) =1
Γ(n−s)/integraldisplayt
t0dt/prime(t−t/prime)n−1−sf(n)(t/prime), (82)
where we reserve the simpler Ds
tnotation for the Liouville derivative that was used in the main text.
Now, rescaling ε→ε/(t−t0) andε→ε/(t−t/prime) in the second and third terms of Eq. (80) respectively, we have
(αdiss∗S)(t) = 2αssin/parenleftbigπs
2/parenrightbig
π/integraldisplay∞
0dεcos(ε)εs−1/bracketleftbigg
(t−t0)−sS(t0) +/integraldisplayt
t0dt/prime(t−t/prime)−s˙S(t/prime)/bracketrightbigg
= 2αssin/parenleftbigπs
2/parenrightbig
π/bracketleftBig
cos/parenleftBigπs
2/parenrightBig
Γ(s)/bracketrightBig/bracketleftbigg
(t−t0)−sS(t0) +/integraldisplayt
t0dt/prime(t−t/prime)−s˙S(t/prime)/bracketrightbigg
= 2αssin/parenleftbigπs
2/parenrightbig
πcos/parenleftBigπs
2/parenrightBig
Γ(s)/bracketleftbig
(t−t0)−sS(t0) + Γ(1−s)C
t0Ds
tS(t)/bracketrightbig
= 2αssin/parenleftbigπs
2/parenrightbig
πcos/parenleftBigπs
2/parenrightBig
Γ(s)Γ(1−s)/bracketleftbigg(t−t0)−s
Γ(1−s)S(t0) +C
t0Ds
tS(t)/bracketrightbigg
= 2αssin/parenleftbigπs
2/parenrightbig
πcos/parenleftBigπs
2/parenrightBigπ
sin(πs)RL
t0Ds
tS(t)
=αsRL
t0Ds
tS(t) =αsDs
tS(t), (83)
where we used several identities from Sec. 6 in the Sup. Mat. of Ref. [12] and in the last line we sent t0→−∞ .
For the noise correlation, we have to compute the Keldysh component. This is
αK(t) =1
2π/integraldisplay∞
−∞dωe−iωtαK(ω)
=1
2π/integraldisplay∞
−∞dωe−iωt[αR(ω)−αA(ω)] coth/parenleftBigω
2T/parenrightBig
=αssin/parenleftbigπs
2/parenrightbig
2π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωe−iωt/bracketleftbiggω2εs−1
ε2−(ω+i0)2−ω2εs−1
ε2−(ω−i0)2/bracketrightbigg
coth/parenleftBigω
2T/parenrightBig
(now sendω→−ωin the advanced part)
=αssin/parenleftbigπs
2/parenrightbig
2π2/integraldisplay∞
0dε/integraldisplay∞
−∞dω/bracketleftbigge−iωtω2εs−1
ε2−(ω+i0)2+eiωtω2εs−1
ε2−(−ω−i0)2/bracketrightbigg
coth/parenleftBigω
2T/parenrightBig
=αssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωcos(ωt)ω2εs−1
ε2−(ω+i0)2coth/parenleftBigω
2T/parenrightBig
. (84)
Now, we send ω→ω/tandε→ε/t, which yields
αK(t) =αst−1−ssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωcos(ω)ω2εs−1
ε2−(ω+i0)2coth/parenleftBigω
2tT/parenrightBig
. (85)
Taking the high temperature limit, we get
αK(t) =αst−1−ssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωcos(ω)ω2εs−1
ε2−(ω+i0)22tT
ω
= 2Tαst−ssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωcos(ω)ωεs−1
ε2−(ω+i0)2
=−2Tαst−ssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωcos(ω)ωεs−1
(ω+i0−ε)(ω+i0 +ε). (86)15
Now, we want to close the integral over ωwith an infinite half-circle. For this, we need fast enough convergence of
the integrand to zero. Splitting the cosine into two exponential parts cos( ω) = (eiω+e−iω)/2, we see that the first
term goes to zero when ω→i∞and that the second term goes to zero when ω→−i∞. Since we have poles at
ω=±ε−i0, the integral along the top half-plane vanishes. The integral along the bottom is then computed as
αK(t) =−Tαst−ssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωe−iωωεs−1
(ω+i0−ε)(ω+i0 +ε)
=−Tαst−ssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε−2πi/bracketleftbigge−i(ε−i0)(ε−i0)εs−1
ε−i0 +i0 +ε+e−i(−ε−i0)(−ε−i0)εs−1
−ε−i0 +i0−ε/bracketrightbigg
= 2iTαst−ssin/parenleftbigπs
2/parenrightbig
π/integraldisplay∞
0dε/bracketleftbigge−iεεs
2ε+eiεεs
2ε/bracketrightbigg
= 2iTαst−ssin/parenleftbigπs
2/parenrightbig
π/integraldisplay∞
0dεcos(ε)εs−1
= 2iTαst−ssin/parenleftbigπs
2/parenrightbig
πcos/parenleftBigπs
2/parenrightBig
Γ(s)
=iTαst−ssin (πs)
πΓ(s)
=iTαst−s
Γ(1−s). (87)
We therefore find that
/angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδm,nαK(t−t/prime) = 2αsTδm,n(t−t/prime)−s
Γ(1−s), (88)
where we assumed that t≥t/prime. Therefore, we have now found the fractional LLG equation
˙S(t) =S(t)×[−B+αsDs
tS(t) +ξ(t)]. (89)
D. Super-Ohmic spectral function
We now consider the case where
J(ε) =αssin/parenleftBigπs
2/parenrightBig
εs, (90)
with 1< s < 2. In this case, everything is equivalent to the sub-Ohmic case, up to Eq. (80), where we wanted to
rewrite the dissipation into a fractional derivative. We had to introduce a finite initial time t0, which lead to
(αdiss∗S)(t) =−2αssin/parenleftbigπs
2/parenrightbig
π/integraldisplayt
−∞dt/prime/integraldisplay∞
0dεsin[ε(t−t/prime)]εsS(t/prime)
=−2αssin/parenleftbigπs
2/parenrightbig
π/integraldisplay∞
0dε/braceleftbigg/bracketleftbig
εs−1cos[ε(t−t/prime)]S(t/prime)/bracketrightbigt/prime=t
t/prime=t0−/integraldisplayt
t0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg
=−2αssin/parenleftbigπs
2/parenrightbig
π/integraldisplay∞
0dε/braceleftbigg
εs−1S(t)−εs−1cos[ε(t−t0)]S(t0)−/integraldisplayt
t0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg
.(91)
The first term vanishes because of the cross product with S(t) in the LLG equation. However, the second term is
more problematic compared to the sub-Ohmic case, since the identity used to rewrite it only holds for s <1. We
solve this by writing it as a time derivative
εs−1cos[ε(t−t0)]S(t0) =d
dtεs−2sin[ε(t−t0)]S(t0), (92)16
and then switching the ordering of the derivative and integral. Performing also one more partial integration in t/prime, we
get
(αdiss∗S)(t)
= 2αssin/parenleftbigπs
2/parenrightbig
π/integraldisplay∞
0dε/braceleftbiggd
dtεs−2sin[ε(t−t0)]S(t0) +/integraldisplayt
t0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg
= 2αssin/parenleftbigπs
2/parenrightbig
π/integraldisplay∞
0dε/parenleftbiggd
dtεs−2sin[ε(t−t0)]S(t0) +/bracketleftBig
εs−2sin[ε(t−t/prime)]˙S(t/prime)/bracketrightBigt/prime=t
t/prime=t0−/integraldisplayt
t0dt/prime/braceleftBig
−sin[ε(t−t/prime)]εs−2¨S(t/prime)/bracerightBig/parenrightbigg
= 2αssin/parenleftbigπs
2/parenrightbig
π/integraldisplay∞
0dε/braceleftbiggd
dtεs−2sin[ε(t−t0)]S(t0) +εs−2sin[ε(t−t0)]˙S(t0) +/integraldisplayt
t0dt/primesin[ε(t−t/prime)]εs−2¨S(t/prime)/bracerightbigg
. (93)
Now, rescaling ε→ε/(t−t0) andε→ε/(t−t/prime) respectively, we have
(αdiss∗S)(t)
= 2αssin/parenleftbigπs
2/parenrightbig
π/integraldisplay∞
0dεsin(ε)εs−2/bracketleftbiggd
dt(t−t0)1−sS(t0) + (t−t0)1−s˙S(t0) +/integraldisplayt
t0dt/prime(t−t/prime)1−s¨S(t/prime)/bracketrightbigg
= 2αssin/parenleftbigπs
2/parenrightbig
π/bracketleftbigg
sin/parenleftbiggπ(s−1)
2/parenrightbigg
Γ(s−1)/bracketrightbigg/bracketleftbigg
(1−s)(t−t0)−sS(t0) + (t−t0)1−s˙S(t0) +/integraldisplayt
t0dt/prime(t−t/prime)1−s¨S(t/prime)/bracketrightbigg
=−2αssin/parenleftbigπs
2/parenrightbig
πcos/parenleftBigπs
2/parenrightBig
Γ(s−1)/bracketleftBig
(1−s)(t−t0)−sS(t0) + (t−t0)1−s˙S(t0) + Γ(2−s)C
t0Ds
tS(t)/bracketrightBig
=−2αssin/parenleftbigπs
2/parenrightbig
πcos/parenleftBigπs
2/parenrightBig
Γ(s−1)Γ(1−(s−1))/bracketleftbigg(t−t0)−s
Γ(1−s)S(t0) +(t−t0)1−s
Γ(2−s)˙S(t0) +C
t0Ds
tS(t)/bracketrightbigg
=−αssin (πs)
ππ
sin[π(s−1)]RL
t0Ds
tS(t)
=αsDs
tS(t), (94)
where we used several identities from Sec. 6 in the Sup. Mat. of Ref. [12], Ref. [13, p.893], and in the last line we
sentt0→−∞ .
For the noise correlation, we have to compute the Keldysh component. This is
αK(t) =1
2π/integraldisplay∞
−∞dωe−iωtαK(ω)
=1
2π/integraldisplay∞
−∞dωe−iωt[αR(ω)−αA(ω)] coth/parenleftBigω
2T/parenrightBig
=αssin/parenleftbigπs
2/parenrightbig
2π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωe−iωt/bracketleftbiggω2εs−1
ε2−(ω+i0)2−ω2εs−1
ε2−(ω−i0)2/bracketrightbigg
coth/parenleftBigω
2T/parenrightBig
. (95)
Now, we send ω→−ωin the advanced part
αK(t) =αssin/parenleftbigπs
2/parenrightbig
2π2/integraldisplay∞
0dε/integraldisplay∞
−∞dω/bracketleftbigge−iωtω2εs−1
ε2−(ω+i0)2+eiωtω2εs−1
ε2−(−ω−i0)2/bracketrightbigg
coth/parenleftBigω
2T/parenrightBig
=αssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωcos(ωt)ω2εs−1
ε2−(ω+i0)2coth/parenleftBigω
2T/parenrightBig
. (96)
Then, we insert cos( ωt) =d
dtsin(ωt)
ωand we send ω→ω/tandε→ε/t, which yields
αK(t) =d
dtαssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωsin(ωt)ωεs−1
ε2−(ω+i0)2coth/parenleftBigω
2T/parenrightBig
=d
dtαst−ssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωsin(ω)ωεs−1
ε2−(ω+i0)2coth/parenleftBigω
2tT/parenrightBig
. (97)17
Taking the high-temperature limit, we get
αK(t) =d
dtαst−ssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωsin(ω)ωεs−1
ε2−(ω+i0)22tT
ω
=d
dt2Tαst1−ssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωsin(ω)εs−1
ε2−(ω+i0)2
=−2Tαs(1−s)t−ssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωsin(ω)εs−1
(ω+i0−ε)(ω+i0 +ε). (98)
Now, we want to close the integral over ωwith an infinite half-circle. For this, we need fast enough convergence of
the integrand to zero. Splitting the cosine into two exponential parts sin( ω) = (eiω−e−iω)/2i, we see that the first
term goes to zero when ω→i∞and that the second term goes to zero when ω→−i∞. Since we have poles at
ω=±ε−i0, the integral along the top half-plane vanishes. The integral along the bottom is then computed as
αK(t) =−iTαs(1−s)t−ssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε/integraldisplay∞
−∞dωe−iωεs−1
(ω+i0−ε)(ω+i0 +ε)
=−iTαs(1−s)t−ssin/parenleftbigπs
2/parenrightbig
π2/integraldisplay∞
0dε−2πi/bracketleftbigge−i(ε−i0)εs−1
ε−i0 +i0 +ε+e−i(−ε−i0)εs−1
−ε−i0 +i0−ε/bracketrightbigg
=−2Tαs(1−s)t−ssin/parenleftbigπs
2/parenrightbig
π/integraldisplay∞
0dε/bracketleftbigge−iεεs−2
2−eiεεs−2
2/bracketrightbigg
=i2Tαs(1−s)t−ssin/parenleftbigπs
2/parenrightbig
π/integraldisplay∞
0dεsin(ε)εs−2
=−i2Tαs(1−s)t−ssin/parenleftbigπs
2/parenrightbig
πcos/parenleftBigπs
2/parenrightBig
Γ(s−1)
=iTαst−ssin (πs)
πΓ(s)
=iTαst−s
Γ(1−s). (99)
We therefore find the same expression as in the sub-Ohmic case, which leads to
/angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδm,nαK(t−t/prime) = 2αsTδm,n(t−t/prime)−s
Γ(1−s). (100)
Therefore, we have now also found the fractional LLG equation in the super-Ohmic case.
E. Comparison Ohmic versus non-Ohmic
In the Ohmic case, we started with J(ω) =α1ωand ended up with a α1˙S(t) friction term, and noise correlation
4α1kBTδ(t−t/prime). On the other hand, in the non-Ohmic case, we started with J(ε) =αssin/parenleftbigπs
2/parenrightbig
εsand ended up with
a frictionαsDs
tS(t) and noise correlation 2 αskBT(t−t/prime)−s
Γ(1−s). Although it is clear that both the non-Ohmic J(ω) and
the friction term will go to the Ohmic case when s→1, the noise is less straightforward. We can see that, as s→1,
the Gamma function will blow up, hence sending the correlation to zero, with the exception of t=t/prime. In this case,
the numerator blows up even before taking the limit of s→1. Hence, we can expect this function to behave as a
delta function. To find the correct prefactor, we integrate the distribution with a test-function f(t/prime) = 1 to find
lim
s→1/integraldisplay∞
−∞d(t−t/prime)(t−t/prime)−s
Γ(1−s)·1 = lim
s→1/bracketleftbigg(t−t/prime)1−s
(1−s)Γ(1−s)/bracketrightbigg∞
(t−t/prime)=−∞
= lim
s→1/bracketleftbigg(t−t/prime)1−s
Γ(2−s)/bracketrightbigg∞
(t−t/prime)=−∞
=/bracketleftbigg(t−t/prime)
|t−t/prime|Γ(1)/bracketrightbigg∞
(t−t/prime)=−∞
= 2.
Hence, we see that the limit of the noise correlation becomes 4 α1kBTδ(t−t/prime), which is precisely as in the Ohmic case.18
III. FMR POWERLAW DERIVATION
Here, we calculate the response of this spin-bath system to a rotating magnetic field. We will find the steady state
solutions and calculate several quantities that experiments could measure.
A. Ferromagnetic Resonance
We will study the effects of a rotating magnetic field on the fractional LLG (FLLG) equation
˙S(t) =S(t)×/braceleftbig
−Beff[t,S(t)] +αsDs
tS(t) +ξ(t)/bracerightbig
. (101)
For simplicity, we assume that the temperature of the bath is low compared to the energy of the external fields, such
that the thermal noise ξ(t) may be neglected. We apply a rotating magnetic field
Beff[t,S(t)] =
Ω cos(ωdt)
Ω sin(ωdt)
B0−KSz(t)
(102)
and use spherical coordinates
S=S
sinθcosφ
sinθsinφ
cosθ
. (103)
We will assume a small θapproximation, where the ground state is in the positive zdirection, i.e. 0 <Ω/lessmuchB0−KS
andαsS/lessmuch(B0−KS)1−s. As shown in Section I D, we may rewrite the FLLG equation of this form in spherical
coordinates as
˙φ=1
sinθ[−Beff[t,S(t)] +αsDs
tS(t)]·
cosθcosφ
cosθsinφ
−sinθ
; (104)
˙θ= [−Beff[t,S(t)] +αsDs
tS(t)]·
sinφ
−cosφ
0
. (105)
In the rotating frame, where B(t) is constant, we could expect the system to go to a steady state after some time.
Hence, we introduce a new coordinate such, that φ=ωdt−ϕ. We may then set ˙ ϕ=˙θ= 0 to find the steady state in
the rotating frame, where
ωdsinθ= [−Beff[t,S(t)] +αsDs
tS(t)]·
cosθcos(ωdt−ϕ)
cosθsin(ωdt−ϕ)
−sinθ
; (106)
0 = [−Beff[t,S(t)] +αsDs
tS(t)]·
sin(ωdt−ϕ)
−cos(ωdt−ϕ)
0
. (107)
We note that S(t) is now only time dependent in the rotating-frame term, which means that we can explicitly calculate
the fractional derivative. The Liouville derivative works well with Fourier transforms, hence the fractional derivative
of a trigonometric function is given by
Ds
tsin(ωt) =|ω|ssin/parenleftBig
ωt+ sign(ω)πs
2/parenrightBig
, (108)
and similarly for a cosine. We remark that the Liouville derivative of a constant can only be described by setting the
initial time to some finite t0, in which case it becomes zero2. Combining this with the steady state expression for S,
2Since physically the infinite time only models a long time in the past, we will apply it as such. Formally, there are some restrictions on
the functions that the Liouville derivative can be applied to. These include restrictions such as its integral over the whole domain being
finite. Since this is not the case for a non-zero constant on an infinite interval, we have to regularize the lower integral boundary −∞
as a finitet0.19
we find
αsDs
tS(t) =αsDs
tS
sinθcos(ωdt−ϕ)
sinθsin(ωdt−ϕ)
cosθ
=αsSsinθ|ωd|s
cos/parenleftbig
ωdt−ϕ+ sign(ωd)πs
2/parenrightbig
sin/parenleftbig
ωdt−ϕ+ sign(ωd)πs
2/parenrightbig
0
. (109)
Hence, we find that
ωdtanθ= [−Beff[t,S(t)] +αsDs
tS(t)]·
cos(ωdt−ϕ)
sin(ωdt−ϕ)
−tanθ
=
−
Ω cos(ωdt)
Ω sin(ωdt)
B0−KScosθ
+αsSsinθ|ωd|s
cos/parenleftbig
ωdt−ϕ+ sign(ωd)πs
2/parenrightbig
sin/parenleftbig
ωdt−ϕ+ sign(ωd)πs
2/parenrightbig
0
·
cos(ωdt−ϕ)
sin(ωdt−ϕ)
−tanθ
=−Ω cosϕ+B0tanθ−KSsinθ+αsSsinθ|ωd|scos/parenleftBig
sign(ωd)πs
2/parenrightBig
(110)
and
0 = [−Beff[t,S(t)] +αsDs
tS(t)]·
sin(ωdt−ϕ)
−cos(ωdt−ϕ)
0
=
−
Ω cos(ωdt)
Ω sin(ωdt)
B0−KScosθ
+αsSsinθ|ωd|s
cos/parenleftbig
ωdt−ϕ+ sign(ωd)πs
2/parenrightbig
sin/parenleftbig
ωdt−ϕ+ sign(ωd)πs
2/parenrightbig
0
·
sin(ωdt−ϕ)
−cos(ωdt−ϕ)
0
= Ω sinϕ−αsSsinθ|ωd|ssin/parenleftBig
sign(ωd)πs
2/parenrightBig
. (111)
With some rearranging, we have
ϕ= arcsin/bracketleftbiggαsS
Ω|ωd|ssin/parenleftBig
sign(ωd)πs
2/parenrightBig
sinθ/bracketrightbigg
(112)
and
Ω cosϕ= Ω cos arcsin/bracketleftbiggαsS
Ω|ωd|ssin/parenleftBig
sign(ωd)πs
2/parenrightBig
sinθ/bracketrightbigg
=/radicalbigg
Ω2−/bracketleftBig
αsS|ωd|ssin/parenleftBig
sign(ωd)πs
2/parenrightBig
sinθ/bracketrightBig2
= (B0−KScosθ−ωd) tanθ+αsSsinθ|ωd|scos/parenleftBig
sign(ωd)πs
2/parenrightBig
. (113)
Squaring this, we get
Ω2= (B0−KScosθ−ωd)2tan2θ+ (αsS|ωd|s)2sin2θ+ 2αsS|ωd|s(B0−KScosθ−ωd) cos/parenleftBigπs
2/parenrightBigsin2θ
cosθ,
and multiplying by cos2θ= 1−sin2θ, we have
(1−sin2θ)Ω2= (B0−KS/radicalbig
1−sin2θ−ωd)2sin2θ+/bracketleftBig
(αsS|ωd|s)2−2αsS|ωd|sKScos/parenleftBigπs
2/parenrightBig/bracketrightBig
sin2θ(1−sin2θ)
+ 2αsS|ωd|s(B0−ωd) cos/parenleftBigπs
2/parenrightBig
sin2θ/radicalbig
1−sin2θ. (114)
We could go further and make this into (effectively) a 4th order equation for sin2θ. However, since we are in a small
θlimit, we will solve this equation up to first order in sin2θ, which yields
Ω2= sin2θ/braceleftBig
Ω2(B0−KS−ωd)2+/bracketleftBig
(αsS|ωd|s)2−2αsS|ωd|sKScos/parenleftBigπs
2/parenrightBig/bracketrightBig
+ 2αsS|ωd|s(B0−ωd) cos/parenleftBigπs
2/parenrightBig/bracerightBig
. (115)20
Hence, we find that
sin2θ=Ω2
Ω2+ (B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftbigπs
2/parenrightbig
=Ω2
(B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftbigπs
2/parenrightbig+O(Ω4). (116)
Note that theO(Ω4) should formally be dimensionless, but we explain what we mean with terms being small in
Section IV, as this is more subtle with fractional dimensions.
B. Resonance frequency and amplitude
Since we are studying ferromagnetic resonance, we want to find the driving frequency for which we get the largest
response from the magnetic system. Since we know that the resonance for an Ohmic system is at ωd=B0−KS, we
will expand the formula around this point to find the new maximum. To compute the resonance frequency ωres, we
thus first assume that ωres≈(B0−KS)(1+y) withysmall, such that|ωres|s≈(B0−KS)s(1+sy) (forB0−KS > 0).
This results in
sin2θ=Ω2
(B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftbigπs
2/parenrightbig
≈Ω2
(B0−KS)2y2+ (αsS)2(B0−KS)2s(1 + 2sy)−2αsS(B0−KS)s+1(1 +sy)ycos/parenleftbigπs
2/parenrightbig. (117)
Now, we put the derivative with respect to yequal to zero, to get
(B0−KS)2y+s(αsS)2(B0−KS)2s−αsS(B0−KS)s+1(1 + 2sy) cos/parenleftBigπs
2/parenrightBig
= 0. (118)
Hence, we find that
y=−s(αsS)2(B0−KS)2s+αsS(B0−KS)s+1cos/parenleftbigπs
2/parenrightbig
(B0−KS)2−2sαsS(B0−KS)s+1cos/parenleftbigπs
2/parenrightbig
=αsS(B0−KS)s−1cos/parenleftBigπs
2/parenrightBig
+O(αsS)2, (119)
which results in
ωres≈(B0−KS)/bracketleftBig
1 +αsS(B0−KS)s−1cos/parenleftBigπs
2/parenrightBig/bracketrightBig
= (B0−KS) +αsS(B0−KS)scos/parenleftBigπs
2/parenrightBig
. (120)
We see that the resonance frequency gets shifted by a small amount, depending on s, which scales non-linearly.
Inserting this result into Eq. (116), we can now also find an approximation for the amplitude at resonance:
sin2θres=Ω2
(B0−KS−ωres)2+ (αsS|ωres|s)2+ 2αsS|ωres|s(B0−KS−ωres) cos/parenleftbigπs
2/parenrightbig
≈Ω2/braceleftBig/bracketleftBig
αsS(B0−KS)scos/parenleftBigπs
2/parenrightBig/bracketrightBig2
+/parenleftBig
αsS/vextendsingle/vextendsingle/vextendsingle(B0−KS) +αsS(B0−KS)scos/parenleftBigπs
2/parenrightBig/vextendsingle/vextendsingle/vextendsingles/parenrightBig2
−2αsS/vextendsingle/vextendsingle/vextendsingle(B0−KS) +αsS(B0−KS)scos/parenleftBigπs
2/parenrightBig/vextendsingle/vextendsingle/vextendsingles
αsS(B0−KS)scos2/parenleftBigπs
2/parenrightBig/bracerightBig−1
=Ω2
[αsS(B0−KS)s]2/bracketleftBig
cos2/parenleftBigπs
2/parenrightBig
+/vextendsingle/vextendsingle/vextendsingle1 +αsS(B0−KS)s−1cos/parenleftBigπs
2/parenrightBig/vextendsingle/vextendsingle/vextendsingle2s
−2/vextendsingle/vextendsingle/vextendsingle1 +αsS(B0−KS)s−1cos/parenleftBigπs
2/parenrightBig/vextendsingle/vextendsingle/vextendsingles
cos2/parenleftBigπs
2/parenrightBig/bracketrightBig−1
=Ω2
[αsS(B0−KS)s]2/bracketleftBig
1−cos2/parenleftBigπs
2/parenrightBig
+O(αsS)/bracketrightBig−1
≈Ω2
/bracketleftbig
αsS(B0−KS)ssin/parenleftbigπs
2/parenrightbig/bracketrightbig2. (121)
Since the sine function decreases as smoves away from one, we see that the amplitude actually increases for non-Ohmic
environments.21
C. Calculating the FWHM linewidth
Next, we are interested not only in the location of the resonance, but also how sensitive the resonance is to the
driving frequency. One way to describe this is by using the Full Width at Half Maximum measure. This provides a
well-defined line width independently of the shape of the peak. It is found by measuring the width of the peak at
half the height of its maximum. This can be measured in the laboratories, but it can also be computed. Since our
function of interest is of the form sin2θ(ωd) = Ω2/g(ωd), it makes sense to approximate the inverse function instead
of the regular one. To this end, we will translate the FWHM measurement to the inverse function, and then Taylor
expandg(ωd) near resonance as a parabola to solve for the new condition of this inverse function. Notice that from
Eq. (116), we have
g(ωd) = (B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftBigπs
2/parenrightBig
. (122)
The FWHM condition is
Ω2
g(ωd)= sin2θ(ωd) =sin2θ(ωres)
2=Ω2
2g(ωres), (123)
hence we must solve for 2 g(ωres) =g(ωd). To this end, let us assume that ωd=ωres+yand expand g(ωd) iny. We
will use that
|a+y|n≈an+nan−1y+1
2n(n−1)an−2y2
for smallyanda>0. Then,
g(ωres+y)
= (B0−KS−ωres−y)2+ (αsS|ωres+y|s)2+ 2αsS|ωres+y|s(B0−KS−ωres−y) cos/parenleftBigπs
2/parenrightBig
≈(B0−KS−ωres)2+ (αsSωs
res)2+ 2αsSωs
res(B0−KS−ωres) cos/parenleftBigπs
2/parenrightBig
+y/parenleftBig
−2(B0−KS−ωres) + 2s(αsS)2ω2s−1
res−2αsScos/parenleftBigπs
2/parenrightBig/braceleftbig
ωs
res+sωs−1
res[ωres−(B0−KS)]/bracerightbig/parenrightBig
+y2/bracketleftBig
1 +s(2s−1)(αsS)2ω2s−2
res−2sαsSωs−1
rescos/parenleftBigπs
2/parenrightBig
+s(s−1)αsSωs−2
res(B0−KS−ωres) cos/parenleftBigπs
2/parenrightBig/bracketrightBig
=g(ωres) +y/parenleftBigg
2αsS(B0−KS)scos/parenleftBigπs
2/parenrightBig
+ 2s(αsS)2(B0−KS)2s−1/bracketleftBig
1 +αsS(B0−KS)s−1cos/parenleftBigπs
2/parenrightBig/bracketrightBig2s−1
−2αsS(B0−KS)scos/parenleftBigπs
2/parenrightBig/braceleftBigg/bracketleftBig
1 +αsS(B0−KS)s−1cos/parenleftBigπs
2/parenrightBig/bracketrightBigs
+sαsScos/parenleftBigπs
2/parenrightBig
(B0−KS)s−1/bracketleftBig
1 +αsS(B0−KS)s−1cos/parenleftBigπs
2/parenrightBig/bracketrightBigs−1/bracerightBigg/parenrightBigg
+y2/braceleftBigg
1 +s(2s−1)(αsS)2(B0−KS)2s−2/bracketleftBig
1 +αsS(B0−KS)s−1cos/parenleftBigπs
2/parenrightBig/bracketrightBig2s−2
−2sαsS(B0−KS)s−1cos/parenleftBigπs
2/parenrightBig/bracketleftBig
1 +αsS(B0−KS)s−1cos/parenleftBigπs
2/parenrightBig/bracketrightBigs−1
−s(s−1)(αsS)2(B0−KS)2s−2cos2/parenleftBigπs
2/parenrightBig/bracketleftBig
1 +αsS(B0−KS)s−1cos/parenleftBigπs
2/parenrightBig/bracketrightBigs−2/bracerightBigg
≈g(ωres) +y/braceleftBig
2s(αsS)2(B0−KS)2s−1/bracketleftBig
1−2 cos2/parenleftBigπs
2/parenrightBig/bracketrightBig/bracerightBig
+y2/braceleftBig
1−2s(αsS)(B0−KS)s−1cos/parenleftBigπs
2/parenrightBig
+ (αsS)2(B0−KS)2s−2/bracketleftBig
s(2s−1)−3s(s−1) cos2/parenleftBigπs
2/parenrightBig/bracketrightBig/bracerightBig
+O(αsS)3.
(124)
Now, we set 2 g(ωres) =g(ωres+y) =g(ωres) +by+ay2, and remark that g(ωres) = (αsS)2(B0−KS)2ssin2/parenleftbigπs
2/parenrightbig
, in
order to find that
y=−b±/radicalbig
b2+ 4ag(ωres)
2a=⇒∆FWHM =/radicalbig
b2+ 4ag(ωres)
a. (125)22
Hence, we find that the lowest-order contribution to the linewidth is given by
∆FWHM≈/radicalBig
4(αsS)2(B0−KS)2ssin2/parenleftbigπs
2/parenrightbig
+O(αsS)3
1 +O(αsS)
= 2(αsS)(B0−KS)ssin/parenleftBigπs
2/parenrightBig
+O(αsS)2. (126)
IV. DIMENSIONAL ANALYSIS
The fractional derivative in the LLG equation has an impact on the dimensions of quantities. We can firstly see
that in the chosen units, we have [ B0−KS] = [ωd] = time−1. Assuming Sto be dimensionless, then [ ωd] = [αsDs
tS] =
[αs][ωd]s, hence [αs] = [ωd]1−s. We can now start to understand what we mean when we say that certain quantities
are small, since this has to be relative to something else. For instance, when we say αsSis small, we understand this
asαsS/lessmuch(B0−KS)1−s. For Ω it is simpler, since there is no fractional derivative acting with it. Hence, for Ω small
we simply mean Ω /lessmuchB0−KS. We can now also define some dimensionless variables, such as α/prime
s=αsS(B0−KS)s−1
and Ω/prime= Ω/(B0−KS). We have used these variables in the figures to show the general behavior of the quantities.
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1712.03550v1.Magnetic_field_gradient_driven_dynamics_of_isolated_skyrmions_and_antiskyrmions_in_frustrated_magnets.pdf |
Magnetic field gradient driven dynamics of isolated skyrmions and antiskyrmions in
frustrated magnets
J. J. Liang1, J. H. Yu1, J. C hen1, M. H. Qin1,*, M. Zeng1, X. B. Lu1, X. S. Gao1,
and J. –M. Liu2,†
1Institute for Advanced Materials , South China Academy of Advanced Optoelectronics and
Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,
South China Normal University, Guangzhou 510006, China
2Laboratory of Solid State Microstr uctures and Innovative Center for Advanced
Microstructures , Nanjing University, Nanjing 210093, China
[Abstract] The study of skyrmion/antiskyrmion motion in magnetic materials is very
important in particular for the spintronics applications. In this work , we stud y the dynamics of
isolated skyrmions and antiskyrmions in frustrated magnets driven by magnetic field gradient ,
using the Landau -Lifshitz -Gilbert simulations on the frustrated classical Heisenberg model on
the triangular lattice . A Hall-like motio n induced by the gradient is revealed in bulk system,
similar to that in the well -studied chiral magnets. More interestingly, our work suggest s that
the lateral confinement in nano -stripes of the frustrated system can completely suppress the
Hall motion an d significantly speed up the motion along the gradient direction. The simulated
results are well explained by the Thiele theory . It is demonstrated that t he acceleration of the
motion is mainly determined by the Gilbert damping constant , which provides use ful
information for finding potential materials for skyrmion -based spintronics .
Keywords: skyrmion dynamics, field gradient, frustrated magnets
PACS numbers: 12.39.Dc, 66.30.Lw .Kw, 75.10.Jm
Email: *qinmh@scnu.edu.cn , † liujm@nju.edu.cn I. INTRODUCTION
Magnetic skyrmions which are topological defects with vortex -like spin structures have
attracted extensive attention since their discovery in chiral magnets due to their interesting
physics and potential applications in spintronic devices.1-4 Specifically, the interesting
characters of skyrmions such as the topological protection5, the ultralow critical currents
required to drive skyrmions (~105 Am-2, several orders of smaller than that for domain -wall
manipulation )3,6, and th eir nanoscale size make s them proposed to be promising candidates
for low po wer consumption magnetic memories and high -density data processing devices.
Theoretically, the cooperation of the energy competition among the ferromagnetic,
Dzyaloshinskii -Moriya ( DM), and the Zeeman couplings and the thermal fluctuations is
suggested to stabilize the skyrmions .7,8 Moreover, the significant effects of the uniaxial stress
on the stabilization of the skyrmion lattice have been revealed in earlier works.9-12 On the
skyrmion dynamics , it has been suggested that the skyrmions in chiral magnets can be
effectively modulated by spin-polarized current,13-16 microwave fields,17 magnetic field
gradients,18,19 electric field gradients,20,21 temperature gradient s22 etc. So far, some of these
manipulations have been realized in experiments.23
Definitely , finding new magnetic systems with skyrmions is essential both in application
potential and in basic physical research.24 More recently, frustrated magnets have been
suggested theoretically to host skyrmion lattice phase . For example, skyrmion crystals an d
isolated skyrmions have been reported in the frustrated Heisenberg model on the triangular
lattice.25,26 In this system, i t is suggested that the skyrmion crystals are stabilized by the
competing ferromagnetic nearest -neighbor (NN) and antifer romagnetic next-nearest -neighbo r
(NNN) interaction s and thermal fluctuations at finite temperatures (T) under applied magnetic
field h. Furthermore, the uniaxial anisotropy strongly affect s the spin orders in triangular
antiferromagnets and stabilize s the isolated sk yrmions even at zero T.26,27
Compared with the skyrmions in chiral magnets, those in frustrated magnets hold two
additional merits. On the one hand, the skyrmion lattice constant is typically an order of
magnitude smaller than that of chiral magnets, and higher -density data processing devices are
expected. On the other hand, the skyrmions are with two additional degrees -of-freedomvorticity and helicity ) due to the fact that the exchange interactions are
insensitive to the direction of spin rotation . As a result, both skyrmion and anti skyrmion
lattices are possible in frustrated magnets which keep the Z2 mirror symmetry in the xy spin
component. Furthermore , the dynamics of skyrmions /antiskyrmions is probably different from
that of chiral magnets , as revea led in earlier work which studied the current -induced dynamics
in nanostripes of frustrated magnets.28 It has been demonstrated that the spin states formed at
the edges create multiple edge channels and guide the skyrmion /antiskyrmion motion.
It is noted that spin-polarized current may not drive the skyrmion well for insulating
materials , and other control parameters such as field gradient are preferred . In chiral magnets,
for example, the gradient can induce a Hall -like motion of skyrmions, i. e., the mai n velocity
v (perpendicular to the gradient direction ) is induced by the gradient, and a low velocity v||
(parallel to the gradient direction ) is induced by the damping effect . Thus, the gradient -driven
motion of skyrmions and antiskyrmions in frustrated systems is also expected . Furthermore , it
has been suggested that the confined geometry suppress es the current -induced Hall motion of
skyrmions and speed s up the motion along the current direction , which is instructive for
future application s.29 In some ex tent, the gradient -driven motion could also be strongly
affected by confining potential in narrow constricted geometries. Thus, as a first step, the
field-gradient -induced dynamics of skyrmions and antiskyrmions in bulk frustrated magnets
as well as in constricted geometries urgently deserves to be revealed theoretically . However,
few works on this subject have been reported, as far as we know.
In this work , we stud y the skyrmion /antiskyrmion dynamics in frustrated magnets
induced by magnetic field gradien ts using Landau -Lifshitz -Gilbert (LLG) simulations and
Thiele approach based on the frustrated classical Heisenberg model on two -dimensional
triangular lattice . A Hall-like motion is revealed in bulk system, similar to that in chiral
magnets. More interest ingly, our work demonstrates that the edge confinement in nanostripes
of frustrated magnets c an completely suppress the Hall motion and significantly accelerate the
motion along the gradient direction.
The remainder of this manuscript is organized as foll ows: in Sec. II the model and the
calculation method will be described. Sec. III is attributed to the results and discussion, and
the conclusion is presented in Sec. IV .
II. MODEL AND METHODS
Following the earlier work,28 we consider the Hamiltonian
22'
12
, ,z z z
i j i j i i i i
i j i i i ijH J J h S D S D S S S S S
, (1)
where Si is the classical Heisenberg spin with unit length on site i. The first term is the
ferromagnetic NN interaction with J1 = 1 (we use J1 as the energy unit, for simplicity) , and t he
second term is the antiferromagnetic NNN interacti on with J2 = 0.5 , and the third term is the
Zeeman coupling with a linear gradient field h = h0 + g·r (h0 = 0.4, r is the coordinate , and g
is the gradient vector with a strength g) applied along the [001] direction ,28 and the fourth
term is the bulk uniaxial anisotropy energy with D = 0.2 , and the last term is the easy plane
anisotropy energy of the edges with D' = 2. D' is only consider ed at the edges for the
nanostripes system , which may give rise to several types of edge states, as uncovered in
earlier work .28 However, it has been confirmed that the skyrmion s/antiskyrmions in
nanostripes move with the same speed when they are captured by one of these edge state s. In
this work, we mainly concern the gradient -driven moti on of isolated skyrmions /antiskyrmion.
We study the spin dynamics at zero T by numerically solving the LLG equation:
ii
i i idd
dt dt SSS f S
, (2)
with the local effective field fi = (∂H/∂ Si). Here, γ = 6 is the gyromagnetic ratio , α is the
Gilbert damping coefficient. We use the fourth -order Runge -Kutta method to solve the LLG
equation. The initial spin configurations are obtained by solving the LLG equation at g = 0.
Subsequently, t he spin dynamics are investigated under gradient fields. Furthermore, the
simulated results are further explained using the approach proposed by Thiele.29 The
displacement of the skyrmion/antiskyrmion is characteri zed by the position of its center (X,
Y): (1 )d d (1 )d d
,.
(1 )d d (1 )d dzz
zzx S x y y S x y
XY
S x y S x y
(3)
Then, the velocity v = (vx, vy) is numerically calculated by
d d , d d .xyv X t v Y t
(4)
At last, v and v|| are obtaine d through a s imple coordinate transformation .
III. RESULTS AND DISCUSSION
First, we investigate the spin configurations of possible isolated skyrmions and
antiskyrmions with various vorticities and helicities obtained by LLG simulations of bulk
system ( D' = 0) at zero g. Specifically, four typical isolated skyrmions with the topological
charge Q = 1 have been observed in our simulations, as depicted in Fig 1(a). The first two
skyrmions are N éel-type ones with different helicities, and the remaining two sk yrmions are
Bloch -type ones. Furthermore, isolated antiskyrmoins are also possible in this system, and
their spin configurations with Q = 1 are shown in Fig. 1(b).
After the relaxation of the spin configurations at g = 0, the magnetic field gradient is
applied along the direction of θ = /6 (θ is the angle between the gradient vector and the
positive x axis, as shown in Fig. 2(a)) to study the dynamics of isolated skyrmions and
antiskyrmions in bulk system . The LLG simulation is performed on a 28 × 28 triangular
lattice with the periodic boundary condition applied along the y' direction perpendicular to the
gradient . Furthermore , we constrain the spin directions at the edge s along the x direction by Sz
= 1 (red circles in Fig. 2(a) ) to reduce the finite lat tice size effect . Similar to that in chiral
magnets, the skyrmion /antiskyrmion motion can be also driven by the magnetic field
gradients in frustrated magnets. Fig. 2 (b) and Fig. 2( c) give respectively the calculated v|| and
v as functions of g at α = 0.04 . v|| of the skyrmion equals to that of the antiskyrmion , and
both v|| and v increase linearl y with g. For a fixed g, the value of v is nearly an order of
magnitude larger than that of v||, clearly exhibiting a Hall -like motion of the skyrmions/a ntiskyrmions. It is noted that v is caused by the gyromagnetic force which
depends on the sign of the topological charge. Thus, along the y' direction, the skyrmion and
antiskyrmion move oppositely under the field gradient , the same as earlier report.18 Moreover,
v|| is resulted from the dissipative force which is associated with the Gilbert damping. For
example , the linear dependence of v|| on the Gilbert damping constant α has been revealed in
chiral magnets,18 which still hold s true for the frustrated magnets. The dependence of velocity
on α at g = 103 is depicted in Fig. 3 , which clearly demonstrates that v|| increases linearly and
v is almost invariant with the increa se of α.
Subsequently , the simulated results are qualitatively explained by Thiele equations:
|| || '', and ,HHv Gv Gv vXY
(5)
with the skyrmion/antiskyrmoin center ( X', Y') in the x'y' coordinate system. Here,
2 2 2 S S S S S Sd S 4 , and d d . G r Q r rx y x x y y
(6)
For the frustrated bulk magnets with the magnetic field gradient applied along the x' direction,
there are
''1 , and 0.z
i
iHHg S gqXY
(7)
For α << 1, q is almost invariant and the velocities can be estimated from
|| 2,. v gq v gqGG
(8)
Thus, a proportional relation between the velocity and field gradient is clearly demonstrated .
Furthermore, v is inversely propor tional to G and/or the topological charge Q, resulting in
the fact that the skyrmion and antiskyrmion move along the y' direction oppositely, as
revealed in our simulations. For current -induced motion of skyrmions, the lateral confine ment can suppress the Hall
motion and accelerate the motion along the current direction.29-31 The confinement effects on
the h-gradient driven skyrmion/antiskyrmion motion are also investigated in the nanostripes
of frustrated magnets. For this case, the L LG simulation is performed on an 84 × 30
triangular -lattice with an open boundary condition along the y direction. For convenience, the
field gradient is applied along the x direction. The easy plane anisotropy with D' = 2 is
considered at the lateral edges , which gives rise to the edge state and in turn confines the
skyrmions/antiskyrmions . Fig. 4(a) gives the time dependence of the y coordinate of the
skyrmion center for α = 0.04 and g = 103. It is clearly shown that t he isolated skyrmion
jumps into the channel at Y = 17 and then moves with a constant speed along the gradient
(negative x, for this case) direction . Furthermore, the position of the channel changes only a
little due to the small range of the gradient consi dered in this work , which never affects our
main conclusions.
More interestingly, the skyrmion/antiskyrmion motion along the gradient direction can be
significant accelerated by the lateral confinement, as shown in Fig. 4(b) which gives v|| (vx) as
a fun ction of g at α = 0.04. For a fixed g, v|| of the nanostripes is almost two orders of
magnitude larger than that of bulk system. When the skyrmion/antiskyrmion is captured by
the edge state ( under which v = 0), the equation (5) gives v|| = gqγ/αΓ. It is s hown that v|| is
inversely proportional to α in this confined geometry, and small α result s in a high speed of
motion of the skyrmion/antiskyrmion . The inversely proportional relation between v|| and α
has also been confirmed in our LLG simulations, as cle arly shown in Fig. 4(c) which gives the
simulated v|| as a function of 1/α at g = 103.
At last, we study the effect of the reversed gradient g on the skyrmion/antiskyrmion
motion, and its trail is recorded in Fig. 4(d). It is clearly shown that the rever sed gradient
moves the skyrmion/antiskyrmion out of the former channel near one lateral edge and drives
it to the new channel near the other lateral edge. Subsequently, the skyrmion/antiskyrmion is
captured by the n ew channel and moves reversely, resulting in the loop -like trail. As a result,
our work suggests that one may modulate the moving channel by reversing the field gradient,
which is meaningful for future applications such as in data eras ing/restoring .
Anyway, it is suggested theoretically that the confined geometry in nanostripes of frustrated magnets could significantly speed up the field-driven motion of the isolated
skyrmions/antiskyrmions, especially for system with small Gilbert damping constant.
Furthermore , we would also like to point out th at th is acceleration is probably available in
other confined materials such as chiral magnets, which deserves to be checked in future
experiments.
IV. CONCLUSION
In conclusion, we have studied the magnetic -field-gradient -drive n motion of the isolated
skyrmions and antiskyrmions in the frustrated triangular -lattice spin model using
Landau -Lifshitz -Gilbert simulations and Thiele theory . The Hall -like motion is revealed in
bulk system, similar to that in chiral magnets. More interestingly, it is suggested that the
lateral confinement in the nanostripes of the frustrated system can suppress the Hall motion
and significantly speed up the motion along the gradient direction. The acceleration of the
motion is mainly determined by the Gilbert damping constant , whic h is helpful for finding
potential materials for skyrmion -based spintronics .
Acknowledgement s:
The work is supported by the National Key Projects for Basic Research of China (Grant
No. 2015CB921202 ), and the National Key Research Programme of China (Gra nt No.
2016YFA0300101), and the Natural Science Foundation of China ( Grant No. 51332007 ), and
the Science and Technology Planning Project of Guangdong Province (Grant No.
2015B090927006) . X. Lu also thanks for the support from the project for Guangdong
Province Universities and Colleges Pearl River Scholar Funded Scheme (2016).
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Rep. 5 7643
FIGURE CAPTIONS
Fig.1. Typical LLG snapshot of the spin configuration s of skyrmion s and antiskyrmion s. (a)
skyrmion structure s and (b) antiskyrmion structure s with different helicit ies.
Fig.2. (a) Effective model on the triangular lattice. (b) v|| and ( c) v as functions of g at α =
0.04 in bulk system .
Fig.3. (a) v|| and ( b) v as functions of α at g = 103.
Fig.4. (a) Time dependence of Y coordinate of the skyrmion center at α = 0.01 and g = 103. v||
as a function of (b) g at α = 0.04 , and (c) α at g = 103 in the nanostripes of frustrated magnets .
(d) The trail of skyrmion /antiskyrmion . The red line records the motion with gradient g along
the positive x direction , while the blu e line records the motion with the revers ed g.
Fig.1. Typical LLG snapshot of the spin configuration s of skyrmion s and antiskyrmion s. (a)
skyrmion structure s and (b) antiskyrmion structures with different helicit ies.
Fig.2. (a) Effective model on the triangular lattice. (b) v|| and (c) v as functions of g at α =
0.04 in bulk system .
Fig.3. (a) v|| and ( b) v as functions of α at g = 103.
Fig.4. (a) Time dependence of Y coordinate of the skyrmion center at α = 0.01 and g = 103. v||
as a function of (b) g at α = 0.04, and (c) α at g = 103 in the nanostr ipes of frustrated magnets .
(d) The trail of skyrmion /antiskyrmion . The red line records the motion with gradient g along
the positive x direction , while the blu e line records the motion with the reversed g.
|
1711.07455v1.Spin_Pumping_in_Ion_beam_Sputtered_Co__2_FeAl_Mo_Bilayers_Interfacial_Gilbert_Damping.pdf | Spin Pumping in Ion-beam Sputtered Co2FeAl/Mo Bilayer s:
Interfacial Gilbert Damping
Sajid Husain, Vineet Barwal, and Sujeet Chaudhary*
Thin Film Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi 110016 (INDIA)
Ankit Kumar, Nilamani Behera, Serkan Akansel, and Peter Svedlindh
Ångström Laboratory, Department of Engineering Sciences, Box 534, SE -751 21 Uppsala, Sweden
Abstract
The spin pumping mechanism and associated interfacial Gilbert damping are demonstrated in
ion-beam sputtered Co2FeAl (CFA)/Mo bilayer thin films employing ferromagnetic resonance
spectroscopy . The d ependence of the net spin current transportation on Mo layer thickness, 0 to
10 nm, and the enhancement of the net effective Gilbert damping are reported . The experimental
data has been analyzed using spin pumping theory in terms of spin current pumped through the
ferromagnet/nonmagnetic metal interface to deduce the effective spin mixing conductance and
the spin -diffusion length , which are estimated to be 1.16(±0.19 )×1019 m−2 and 3.50±0.35nm,
respectively. The damping constant is found to be 8 .4(±0.3)×10-3 in the Mo(3.5nm) capped
CFA(8nm) sample corresponding to a ~42% enhancement of the original Gilbert damping
(6.0(± 0.3)×10-3) in the uncapped CFA layer. This is further confirm ed by insertin g a Cu dusting
layer which reduce s the spin transport across the CFA/Mo interface. The Mo layer thickness
dependent net spin current density is found to lie in the ra nge of 1-3 MAm-2, which also provides
additional quantitative evidence of spin pumping in this bilayer thin film system .
*Author for correspondence: sujeetc@physics.iitd.ac.in
I. INTRODUCTION
Magnetic damping is an exceedingly importan t property for spintronic devices due to its
influence on power consumption and information writing in the spin-transfer torque random
access memor ies ( STT-MRAMs) [1][2]. It is therefore of high importance to study the
generation, manipulation , and detection of the flow of spin angular momentum to enable the
design of efficient spin-based magneti c memories and logic devices [3]. The transfer of spin
angular momentum known as spin pumping in ferromag netic (FM)/ nonmagnetic (NM) bilayer s
provide s information of how the precession of the magnetization transfer s spin angular
momentum into the adjacent nonmagnetic metallic layer [4]. This transfer ( pumping ) of spin
angular momentum slows down the precession and leads to an enhance ment of the effective
Gilbert damping constant in FM/NM bilayers . This enhancement has been an area of intensive
research since the novel mechanism (theory) of spin pumping was proposed by Arne Brataas et
al. [5] [6]. The amount of spin pumping is quantified by the magnitude of the spin current
density at the FM/NM interface and theoretically [7] described as
4eff
S effdgdt mJm
where
m is the magnetization unit vector,
eff
SJ is the effective spin current density pumped into the NM
layer from the FM layer (portrayed in Fig. 1), and
effg is the spin mixing conductance which is
determined by the reflection coefficient s of conductance channels at FM/NM interface [5].
To date, a number of NM metals , such as Pt, Au, [5], Pd [8][9],-Ta [10] and Ru [11],
etc. have been e xtensively investigated with regards to their performance as spin sink material
when in contact with a FM . It is to be noted here that none of the Pt, Pd, Ru, and Au is an earth
abundant material [12]. Thus , there is a natural need to search for new non-magnetic material s
which could generate large spin current at the FM/NM in terface . In this study , we have explored the potential of the transition metal molybdenum (Mo) as a new candidate material for spin
pumping owing to the fact that Mo possesses a large spin-orbit coupling [13]. To the best of our
knowledge, Mo has not been used till date for the study of spin pumping effect in a FM/NM
bilayer system .
In a FM/NM bilayer and/or multilayer system s, there are several mechanisms for
dissipation of the spin angular momentum which are categorized as intrinsic and extrinsic . In the
intrinsic category , the magnon -electron coupling , i.e., spin -orbit coupling (SOC) contributes
significant ly [14]. Among the extrinsic category , the two-magnon scattering (TMS ) mechanism
is linked to the inhomogeneity and interface/surface roughness of the heterostructure ,
etc. [15] [16] [17]. For large SOC , interfacial d-d hybridization between the NM and FM layers
is highly desirable [16]. Thus, the FM -NM interfacial hybridization is expected to result in
enhancement of the transfer of spin angular momentum from the FM to the NM layer , and hence
the NM layer can act as a spin reservoir (sink) [18]. But, the NM metallic layer does not always
act as a perfect spin reservoir due to the spin accumulation effect which prevents transfer of
angular momentum to some extent and a s a result , a backflow of spin-current towards the
FM [6] is estabished . While the flow of spin angular momentum through the FM/NM interface
is determine d by the effective spin-mixing conductance
()effg at the interface , the spin backflow
is governed by the spin diffusion length
()d . It is emphasized here that t hese parameters (
effg
and
d ) are primarily tuned by appropriate selection of a suitable NM layer.
In this work, we have performed ferromagnetic resonance (FMR) measurements to
explore the spin pumping phenomenon and associated interfacial Gilbert damping enhancement
in the Co2FeAl(8nm) /Mo(
Mot) bilayer system ,
Mot is the thickness of Mo , which is varie d from 0 to 10 nm. The
Mot dependent net spin current transfer across the interface and spin diffusion
length of Mo are estimated . The choice of employing the Heusler alloy CoFe 2Al (CFA) as a thin
FM layer lies in its half metallic character anticipated at room temperature [19] [20], a trait
which is highly desirable in any spintronic device operating at room temperature.
II. EXPERIMENTAL DETAILS
The CFA thin films with fixed thickness of 8 nm were grown on naturally oxidized Si(100)
substrate at 573K temperature using an ion-beam sputtering deposition system ( NORDIKO -
3450). The substrate temperature (573K) has been selected following the growth optimization
reported in our previous reports [21] [20] [22]. On the top of the CFA layer , a Mo film with
thickness
Mot (
Mot=0, 0.5, 1.0, 1.5, 2.0, 3.0, 4, 5, 7, 8 and 10 nm) was deposited in situ at room
temperature . In addition , a trilayer structure of CFA(8)/Cu(1)/Mo(5) was also prepared to
understand and confirm the effect of an additional interface on the Gilbert damping (spin
pumping ). Numbers in parenthesis are film thicknesses in nm. All the samples were prepared at a
constant working pressure of ~8.5×10-5 Torr (base vacuum ~ 1.010-7 Torr); Ar gas was directly
fed at 4 sccm into the rf-ion source operated at 75W with the deposition rate s of 0.03nm/s and
0.02nm/s for CFA and Mo, respectively . The deposition rate for Cu was 0.07nm/s at 80 W. The
samples were then cut to 1×4 mm2 to record the FMR spectra employing a homebuilt FMR set-
up [21] [23]. The data was collected in DC-magnetic field sweep mode by keeping the
microwave frequency fixed . The saturation magnetization was measured using the Quantum
Design make Physical Property Measurement System (Model PPMS Evercool -II) with the
vibrating sample magnetometer option (QD PPMS -VSM). The film density, thickness and
interface /surface roughness were estimated by simulating the specular X -ray reflectivity (XRR)
spectra using the PANalytical X’Pert reflectivity software (Ver. 1.2 with segmented fit). To determine surface morphology /microstru cture (e.g., roughness) , topographical imaging was
performed using the ‘Bruker dimension ICON scan assist’ atomic fo rce microscope (AFM). All
measurements were performed at room temperature.
III. RESULTS AND DISCUSSIONS
A. X-ray Reflectivity and A tomic Force Microscopy : Interface/surface analysis
Figure 2 shows the specular XRR spectra recorded on all the CFA(8)/Mo(
Mot ) bilayer thin films .
The fitting parameters were accurately determined by simu lating (red lines) the experimental
curves (filled circles) and are presented in Table -I. It is evident that for the smallest NM layer
thickness , Mo(0.5nm) , the estimated value s of the roughness from XRR and AFM are slightly
larger in comparison to the thickness of the Mo layer which indicates that the surface coverage of
Mo layer is not enough to cover all of the CFA surface in the CFA(8)/Mo(0.5) bilayer sample
(modeled in Fig. 3(a)). For
Mot ≥ 1nm, the film roughness is smaller than the thickness
(indicating that t he Mo layer coverage is uniform as modeled in Fig s. 3(b)-(c)). For the thicker
layer s of Mo (
Mot≥ 5nm) the estimated values of the surface roughness as estimated from both
XRR and AFM are found to be similar ~0.6nm (c.f. the lowest right panel in Fig. 2).
B. Ferromagnetic Resonance Study
The FMR spectra were recorded on al l sample s in 5 to 11 GHz range of microwave frequenc ies.
Figure. 4(a) shows the FMR spectra recorded on the CFA(8)/Mo( 5) bilayer thin film . The FMR
spectra
()FMRI were fitted with the derivative of symmetric and anti -symmetric Lorentzian
function s to extract the line -shape parameters, i.e., resonant field
rH and linewidth
H , given
by [24] [21]:
22
2
2222
22()
() ()22 2
( ) ( )22FMR
dc rdc r
dcS ext A ext
ext ext
r dc rUIH
HH HHH HH
SA
HF H F H
HHSA
HHHH
H
, (1)
where
S extFH and
A extFH are the symmetric and anti -symmetric Lorentzian functions,
respectively, with S and A being the corresponding coefficients. Symbol ‘U’ refers to the raw
signal voltage from the VNA . The linewidth
H is the full width at half maxim um (FWHM) ,
and
dcH is the applied DC-magnetic field.
The f vs.
0 rH plots are shown in Fig. 4(b). These are fitted using t he Kittel’s formul a [25]:
0()2r K r K eff f H H H H M
, (2)
where 𝛾 is the gyromagnetic ratio ;
/Bg
(1.76×1011s-1T-1) with
g being the Lande’s
splitting factor ; taken as 2,
0 effM is the effective saturation magnetization, and
0 KH is the
uniaxial anisotropy field. The value s of
0 effM are comparable to the values of
0 SM (obtained
from VSM measurements) as is shown in Fig . 4(c). Figure 4(e) shows the variation of
0 KH
with
Mot from which the decrease in
0 KH with increas e in
Mot is clearly evident . This observed
reduction in
0 KH could possibly stem from the spin a ccumulation increasing with increasing
Mot
[26]. The FMR spectra was also recorded on CFA (8)/Cu(1)/Mo(5) trilayer thin film for the
comparison with the results of CFA (8)/Mo(5) bilayer. The magnitudes of
0 effM (
0 kH ) for
CFA (8)/Mo(5) and CFA (8)/Cu(1)/Mo(5) are found to be 1.33±0.08 T (0.55±0.15mT) and 1.30±0.04 T (3.21±0.13 mT) , respectivel y. Further, Fig. 4(d) shows the
0 rH vs.
Mot behavior at
different constant frequenc ies ranging between 5 to 11 GHz. The observed values of
0 rH are
constant for all the Mo capped layers which clearly indicate s that the dominant contribution to
the observed resonance spectra arises from the intrinsic effect, i.e., magnon -electron
scattering [27].
C. Mo t hickness -dependen t spin pumping
Figure 5(a) shows t he linewidth
0µH vs. f (for clarity, the results are shown only for a
few sel ected film samples ). The frequency dependent linewidth can mainly have two
contributions ; the intrinsic magnon -electron scattering contribution, and the extrinsic two-
magnon scattering (TMS ) contribution. The extrinsic TMS contribution in linewidth has been
analysed (not presented here) using the methods given by Arias and Mills [28]. A similar
analysis was reported in one of our previous studies on the CFA/Ta system [21]. For the present
case, t he linewidth analysis shows that inclusion of the TMS part does not affect the Gilbert
damping , which means the TMS contribution is negligible in our case. Now , the effective Gilbert
damping constant
eff can be estimated using,
0
04efffHH
. (2)
Here,
0H is the frequency independent contribution from sample inhomogeneity , while the
second term corresponds to the frequency dependent contribution associated with the intrinsic
Gilbert relaxation . Here ,
eff , defined as
eff SP CFA , is the effective Gilbert damping which includes the intrinsic value of CFA
()eff and a spin pumping contribution (
SP ) from the
CFA/Mo bilayer .
The extracted effective Gilbert damping constant values for different
Mot are shown in
Fig 5(b). An enhancement of the Gilbert damping constant with the increase of the Mo layer
thickness is clearly observed , which is anticipated owing to the transfer of spin angular momenta
by spin pumping from CFA to the Mo layer at the CFA /Mo interface . The value of
eff is found
to increase up to 8.4(± 0.3)×10-3 with the increase in
Mot (≥ 3.5nm) , which corresponds to ~42%
enhancement of the damping constant due to spin pumpin g. It is remarkable that such a large
change in Gilbert damping is observed for the CFA/Mo bilayer ; the change is comparabl e to
those reported when a high SOC NM such as Pt [8], Pd [29] [9], Ru [11], and Ta [30] is
employed in FM/NM bilayer s. Here , we would like to mention that the enhancement of the
Gilbert dampin g can , in principle , also be explained by extrinsic two-magnon scattering (TMS )
contribution s in CFA/Mo(
Mot ) bilaye rs by considering the variation of
rH with NM
thickness [27]. In our case, the
0 rH is constant for all
Mot (c.f. Fig . 4(d)). Thus the extrinsic
contribution induced increase in
eff is negligibly small and hence the enhancement of the
damping is dominated by the spin pumping mechanism . The estimated values of
0 0µH are
found to vary from 0.6 to 2.5 mT in the CFA/Mo(
Mot ) thin films . The variation in
0 0µH is
assigned to the finite, but small, statistical variations in sputtering conditions between samples
with different
Mot.
Further, to affirm the spin pumping in the CFA/Mo bilayer system, a copper (Cu) dusting
layer was inserted at the CFA/Mo interface. Fig ure 5(c) compares the linewidth vs. f plot of the CFA (8)/Mo(5) and CFA (8)/Cu(1)/Mo(5) heterostructures. The Gilbert damping was found to
decrease from 8.4(± 0.3)×10-3 to 6.4(± 0.3)×10-3 after inserting the Cu (1) thin layer, which is
comparable to the value of the uncapped CFA (3.5) sample. It may be noted that Cu has a very
large spin diffusion length (
d~300nm) but weak SOC strength [32]. Due to the weak SOC, the
asymmetry in the band structure at the FM/Cu interface would thus lead to a non -equilibrium
spin accumulation at the CFA/ Cu interface [33]. This spin accumulation opposes the transfer of
angular momentum into the Mo layer and hence the Gilbert damping value , after insertion of the
dusting layer , is found very similar to that of the single layer CFA film. It is also known that
enhancement of damping in the FM layer (when coupled to the NM layer ) can occur due to the
magnetic proximity effect [34]. However, we did not find any evidence in favor of the proximity
effect as the effective saturation magnetization did not show any increase on the inserti on of the
ultrathin Cu dusting layer at CFA/Mo bilayer interface , which support s our claim of absence of
spin pumping in the CFA/Cu/Mo trilayer sample .
The flow of angular momentum across the FM/NM bilayer interface is determined by the
effective complex spin-mixing conductance
g Re(g ) Im(g )eff eff eff i , defined as the flow of
angular momentum per unit area through the FM/NM metal interface created by the precessing
moment s in the FM layer . The term effective spin-mixing conductance is being used because it
contain s the forward and backflow of spin momentum at the FM/NM interface. The imaginary
part of the spin-mixing conductance is usually assumed to be negligibly small
Re(g ) Im(g )eff eff
as compared to the real part [35] [36], and therefore, to determine the real
part of the spin-mixing conductance , the obtained
Mot dependent G ilbert damping is fit ted with
the relation [29], 21Re(g ) 14Mo
dt
B
eff CFA eff
S CFAgeMt
, (3)
where
CFAis the damping for a single layer CFA without Mo capping layer,
Re(g )eff is given
in unit s of m-2,
B is the Bohr magneton, and
CFAt is a CFA layer thickness . The exponential
term describes the reflection of spin -current from Mo/air interface . Figure 5(b) shows the
variation of the effective Gilbert damping constant with
Mot and the fit using Eqn. (3) (red line) .
The values of
Re(g )eff and
d are found to be 1. 16(±0.19 )×1019 m-2 and 3.5±0.35 nm,
respectively. The value of the spin-mixing conductance is comparable to those recent ly reported
in FM/Pt (Pd) thin films such as Co/Pt ( 1-4 ×1019 m-2) [8] [33], YIG/Pt (9.7 ×1018 m-2) [37],
Fe/Pd (1×1020 m-2) [9], and Py/Pd(Pt) (1.4(3.2) ×1018 m-2) [34].
We now calculate the net intrinsic interfacial spin mixing conductance
G which
depends on the thickness and the nature of the NM layer as per the relation [9] [38],
11
4( ) Re(g ) 1 tanh3Mo
Mo eff
dtGt
, (4)
where
24( / )Z e c
is a material dependent param eter (Z is the atomic number of Mo i.e., 42
and c is the speed of light) whose value for Mo is 0.0088 . Using Eq. (4),
()Mo Gt values have
been compu ted for variou s
Mot ; the results are shown in Fig. 6(a). The
Mot dependence of
G
clearly suggest s that the spin mixing conductance critically depend s on the NM layer properties .
For bilayers with
Mot 6 nm,
G attains its saturation value, which is quite comparable with
those reported for Pd and Pt [34] [37]. Understandably, such a large value of the spin mixing conductance will yield a large spin current into the adjacent NM layer [6] [7] [37] [33]. In the
next section, we have estimated the spin current from the experimental FMR data and discuss the
same with regards to spin pumping in further detail .
D. Spin current generation in Mo due to spin pumping
The enhancement of the Gilbert damping observed in the CFA (8)/Mo(
Mot) bilayers (Fig. 5(b)) is
generally interpreted in terms of the spin -current generated in Mo layer by the spin pumping
mechanism at the bilayer interface (Fig. 1). The associated net effective spin current density in
Mo is described by the relation [38] [39]:
00 02 22 2
2
0224 2( ) G ( )8 4eff eff rf eff
S Mo Mo
effeffMM h eJ t t
M
, (5)
where
2f and
rfhis the rf-field (26 A/m) in the strip -line of our co-planar waveguide.
G ( )Mot
is the net intrinsic inte rfacial spin mixing conductance discussed in the previous
section (Fig. 6). The estimated values of
()eff
S MoJt for differen t microwave frequencies are shown
in Fig. 7. It is clearly observed that the spin current density increase s with the increase in
Mot, the
increase becomes relatively less at higher
Mot , which indicate s the progressive spin current
generation in Mo . Such an appreciable change in current density directly provide s evidence of
the interfacial enhancement of the Gilbert damping in these CFA/Mo bilayers .
Further, it would be interesting to investigate the effect on the spin current generation in
Mo layer if an ultra thin dusting layer of Cu is inserted at the CFA/Mo interface . In princip le, on
insertion of a thin Cu layer , the spin pumping should cease because of the unmatched band
structure between the CFA /Cu and Cu/M o interfaces owing to the insignificant SOC in Cu. This is in consonance with the observed decrease in Gilbert damping back to the value for the
uncapped CFA layer (c.f. Fig. 5(c) and associated discussion ). The spi n-mixing conductance of
the trilayer heterostructure can be evaluated by
0 g/eff B eff S CFAg M t [29], where
sp eff CFA
is the spin-pump ing induced Gilbert damping contribution which for the
CFA/Cu/Mo trilayer is quite small , i.e., 4.0(±0.3) ×10-4 after Cu insertion. For the trilayer,
geff is
found to be 1.49 (±0.12) ×1017 m-2 which is two order s of magnitude small er compared to that of
the CFA/Mo bilayers. Furthermore, u sing the values of
geff ,
0 effM and
eff for the
CFA/Cu/Mo trilayer hetero structure in Eqn. (5) and for f = 9GHz, the spin current density is
found be 0.0278 (±0.001 3) MA/m2, which is two order of magnitude smaller than that in the
CFA/Mo bilayers. Thus, t he reduction in
eff and
eff
SJ subsequent to Cu dusting is quite
comparable to previously reported results [33] [40].
IV. CONCLUSIONS
We have systematically investigated the changes in the spin dynamics in the ion-beam sputtered
Co2FeAl ( CFA )/Mo(
Mot) bilayer s for various
Mot at constant CFA thickness of 8nm . Increasing
the Mo layer thickness to its spin diffusion length; CFA (8)/Mo(
Mot =
d), the effective Gilbert
damping constant increases to 8.4(± 0.3)×10-3 which corresponds to about ~42% enhancement
with respect to the
eff value of 6.0(± 0.3)×10-3 for the uncapped CFA layer (i.e., without the top
Mo layer ). We interpret our results based on the spin -pumping effect s where in the effective spin-
mixing conductance , and spin -diffusion length are found to be 1.16(±0.19 )×1019 m−2 and
3.50±0.35nm, respectively. The spin pumping is further confirmed by inserting an ultrathin Cu
layer at the CFA/Mo interface. The overall effect of the damping constant enhancement observed when Mo is deposited over CFA is remarkably comparable to the far less -abundant non-
magnetic metals that are currently being used for spin pumping applications . From this view
point , the demonstration of the new material , i.e., Mo, as a suitable spin pumping medium is
indispensable for the development of novel STT spintronic devices .
ACKNOWLEDGMENT S
One of the authors SH acknowledge s the Department of Science and Technology , Govt. of India
for providing the INSPIRE Fellowship. Authors thank the NRF facilit ies of IIT Delhi for AFM
imaging . This work was in part supported by Knut and Alice Wallenberg (KAW) Foundation
Grant No. KAW 2012.0031 . We also acknowledge the Ministry of Information Technology,
Government of India for providing the financial grant .
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Brown, N. X. Sun, and A. Samples, Phys. Rev. B 91, 214416 (2015). Table: 1 Summary of XRR simulated parameters , i.e., ,
FMt ,
Mot , and σ for the bilayer thin
films [Si/CFA( 8)/Mo(
Mot)]. Here ,
FMt ,
Mot, and σ refer to the density, thickness, and
interface width of the individual layers , respectively.
CFA (Nominal thickness = 8 nm) Mo MoOx
S.No.
1
2
3
4
5
6
7
8
9
10 (gm/cc)±0.06
7.35
7.31
7.50
7.50
7.00
7.00
7.29
7.22
7.00
7.64 tFM(nm)±0.01
7.00
8.17
7.22
8.18
7.00
8.28
8.00
7.79
8.12
8.00 σ(nm) ±0.03
0.20
0.35
0.80
0.37
1.00
0.56
0.44
0.98
0.15
0.17 (gm/cc)±0.0 5
6.05
8.58
10.50
9.94
9.50
10.45
9.43
10.50
9.29
9.23 tMo(nm)±0.01
0.58
1.00
1.50
2.00
3.00
3.46
4.86
6.47
8.21
10.26 σ(nm) ±0.03
0.94
0.54
0.52
0.64
0.60
0.78
0.26
0.67
0.64
0.67 (gm/cc)±0.06
4.07
4.04
5.00
4.38
5.17
4.38
6.50
4.81
4.00
5.00 t(nm)±0.01
0.97
0.82
1.08
0.85
1.00
1.01
0.98
1.03
0.96
1.17 σ(nm) ±0.03
0.59
0.35
0.5
0.45
0.37
0.4
0.56
0.62
0.8
0.73
Figure captions
FIG. 1. (color online) Schematic of the CFA/Mo bilayer structure used in our work portrayed
for an example of spin current density
eff
SJ generated at the CFA/Mo interface by spin pumping .
FIG. 2 XRR spectra and the AFM topographical images of Si/CFA( 8)/Mo(
Mot ). In the
respective XRR spectra, circles represent the recorded experimental data points, and lines
represent the simulated profiles. The estimated values of the surface roughness in the entire
sample series as obtained from XRR and AFM topographical measurements are compared in the
lowest right panel. The simulated parameters are presented in the Table -I. All AFM images were
recorded on a scan area of 10×10 m2.
FIG.3 : The atomic representation (model) of the growth of th e Mo layer (yellow sphere) on
top of the CFA (blue spheres) layer. The film changes from discontinuous to continuous as the
thickness of the Mo layer is increased. Shown are the 3 different growth stages of the films: (a)
least coverage (b) partial coverage and (c) full coverage .
FIG. 4: (a) Typical FMR spectra recorded at various frequencies (numbers in graph are the
microwave frequencies in GHz) for the Si/SiO 2/CFA(8)/Mo(5) bilayer sample (symbols
correspond to experimental data and red lines are fits to the Eqn. (1)) Inset: FMR spectra of CFA
single layer (filled circles) and CFA(8)/Mo(2) bilayer (open circles) samples measured at 5GHz
showing the increase in linewidth due to spin pumping. (b) The resonance field
0 rH vs. f for all
the samples ( red lines are the fits to the Eqn. (2). (c) Effective magnetization (scale on left) and
saturation magnetization (scale on right) vs.
Mot . The solid line represents the bulk value of the
saturation magnetization of Co 2FeAl . (d) The resonance field
0 rH vs.
Mot at different constant frequencies for CFA(8)/Mo(
Mot ) bilayer thin films. (e) Anisotropy field
0 KH vs.
Mot . (f)
Comparison of
0 rH vs. f for the CFA(8)/Mo(5) and CFA(8)/Cu(1)/Mo(5) samples.
FIG. 5: (a) Linew idth vs. frequency for Si/SiO 2/CFA(8)/Mo(
Mot ) bilayer thin films. (b)
Effective Gilbert damping constant vs. Mo layer thicknesses. (c)
0H vs. f for CFA(8)/ Mo(5)
and CFA(8)/Cu(1)/Mo(5) films.
FIG. 6 : Intrinsic s pin-mixing conductance vs.
Mot of the CFA (8)/Mo(
Mot) bilayers .
FIG. 7. The effective spin current density (generated in Mo) vs.
Mot at different microwave
frequencies calculated using Eqn. (5)
FIG. 1
FIG. 2
FIG. 3
FIG. 4
FIG. 5
FIG. 6
FIG. 7
|
1703.03485v2.Long_time_dynamics_of_the_strongly_damped_semilinear_plate_equation_in___mathbb_R___n__.pdf | arXiv:1703.03485v2 [math.AP] 10 Apr 2017LONG-TIME DYNAMICS OF THE STRONGLY DAMPED SEMILINEAR PLATE
EQUATION IN Rn
AZER KHANMAMEDOV AND SEMA YAYLA
Abstract. We investigate the initial-value problem for the semilinea r plate equation containing local-
ized strong damping, localized weak damping and nonlocal no nlinearity. We prove that if nonnegative
damping coefficients are strictly positive almost everywher e in the exterior of some ball and the sum
of these coefficients is positive a.e. in Rn, then the semigroup generated by the considered problem
possesses a global attractor in H2(Rn)×L2(Rn). We also establish boundedness of this attractor in
H3(Rn)×H2(Rn).
1.Introduction
In this paper, our main purpose is to study the long-time dynamics (in terms of attractors) of the
plate equation
utt+γ∆2u−div(β(x)∇ut)+α(x)ut+λu−f(/ba∇dbl∇u(t)/ba∇dblL2(Rn))∆u+g(u) =h(x), (t,x)∈R+×Rn, (1.1)
with initial data
u(0,x) =u0(x),ut(0,x) =u1(x),x∈Rn, (1.2)
whereγ >0,λ >0,h∈L2(Rn) and the functions α(·), β(·), f(·) andg(·) satisfy the following
conditions:
α, β∈L∞(Rn),α(·)≥0, β(·)≥0 a.e. in Rn, (1.3)
α(·)≥α0>0 andβ(·)≥β0>0 a.e. in {x∈Rn:|x| ≥r0}, for somer0>0, (1.4)
α(·)+β(·)>0 a.e. in Rn, (1.5)
f∈C1(R+), f(z)≥0, for allz∈R+, (1.6)
g∈C1(R),|g′(s)| ≤C/parenleftig
1+|s|p−1/parenrightig
,p≥1, (n−4)p≤n, (1.7)
g(s)s≥0, for every s∈R. (1.8)
The problem (1.1)-(1.2) can be reduced to the following Cauchy prob lem for the first order abstract
differential equation in the space H2(Rn)×L2(Rn):
/braceleftbiggd
dtθ(t) =Aθ(t)+F(θ(t)),
θ(0) =θ0,
whereθ(t) = (u(t),ut(t)),θ0= (u0,u1),A(u, v) = (v,−γ∆2u+div(β(·)∇v)−α(·)v−λu),D(A) =/braceleftbig
(u,v)∈H3(Rn)×H2(Rn) :γ∆2u−div(β(·)∇v)∈L2(Rn)/bracerightbig
andF(u,v) = (0,f(/ba∇dbl∇u/ba∇dblL2(Rn))∆u
−g(u)+h). Defining suitable equivalent norm in H2(Rn)×L2(Rn), it is easy to see that the operator
A, thanks to (1.3), is maximal dissipative in H2(Rn)×L2(Rn) and consequently, due to Lumer-Phillips
Theorem(see [1, Theorem4.3]), it generatesa linearcontinuoussem igroup/braceleftbig
etA/bracerightbig
t≥0. Also, by (1.6)-(1.7),
we find that the nonlinear operator F:H2(Rn)×L2(Rn)→H2(Rn)×L2(Rn) is Lipschitz continuous
on bounded subsets of H2(Rn)×L2(Rn). So, applying semigroup theory (see, for example [2, p. 56-58]),
and taking advantage of energy estimates, we have the following we ll-posedness result.
2000Mathematics Subject Classification. 35B41, 35G20, 74K20.
Key words and phrases. wave equation, plate equation, global attractor.
12 AZER KHANMAMEDOV AND SEMA YAYLA
Theorem 1.1. Assume that the conditions (1.3), (1.6), (1.7) and (1.8) hol d. Then, for every (u0,u1)∈
H2(Rn)×L2(Rn), the problem (1.1)-(1.2) has a unique weak solution u∈C/parenleftbig
[0,∞);H2(Rn)/parenrightbig
∩C1/parenleftbig
[0,∞);L2(Rn)/parenrightbig
, which depends continuously on the initial data and satisfie s the energy equality
E(u(t))+/integraldisplay
RnG(u(t,x))dx+1
2F/parenleftig
/ba∇dbl∇u(t)/ba∇dbl2
L2(Rn)/parenrightig
−/integraldisplay
Rnh(x)u(t,x)dx
+t/integraldisplay
s/integraldisplay
Rnα(x)|ut(τ,x)|2dxdτ+t/integraldisplay
s/integraldisplay
Rnβ(x)|∇ut(τ,x)|2dxdτ
=E(u(s))+/integraldisplay
RnG(u(s,x))dx+1
2F/parenleftig
/ba∇dbl∇u(s)/ba∇dbl2
L2(Rn)/parenrightig
−/integraldisplay
Rnh(x)u(s,x)dx,∀t≥s≥0,(1.9)
whereF(z) =z/integraltext
0f(√s)dsfor allz∈R+,G(z) =z/integraltext
0g(s)dsfor allz∈RandE(u(t)) =1
2/integraltext
Rn(|ut(t,x)|2+
γ|∆u(t,x)|2+λ|u(t,x)|2)dx. Moreover, if (u0,u1)∈D(A), thenu(t,x)is a strong solution satisfying
(u,ut)∈C([0,∞);D(A))∩C1[0,∞);H2(Rn)×L2(Rn).
Thus, duetoTheorem1.1,theproblem(1.1)-(1.2)generatesastr onglycontinuoussemigroup {S(t)}t≥0
inH2(Rn)×L2(Rn) by the formula ( u(t),ut(t)) =S(t)(u0,u1), whereu(t,x) is a weak solution of
(1.1)-(1.2) with the initial data ( u0,u1).
Attractors for hyperbolic and hyperbolic like equations in unbounde d domains have been extensively
studied by many authors over the last few decades. To the best of our knowledge, the first works in
this area were done by Feireisl in [3] and [4], for the wave equations wit h the weak damping (the case
γ= 0,β≡0 andf≡1 in (1.1)) . In those articles the author, by using the finite speed pr opagation
property of the wave equations, established the existence of the global attractors in H1(Rn)×L2(Rn).
The global attractors for the wave equations involving strong dam ping in the form −∆ut, besides weak
damping, were investigated in [5] and [6], where the authors, by using splitting method, proved the
existence of the global attractors in H1(Rn)×L2(Rn), under different conditions on the nonlinearities.
Recently, in [7], the results of [5] and [6] have been improved for the w ave equation involving additional
nonlocal nonlinear term in the form −(a+b/ba∇dbl∇u(t)/ba∇dbl2
L2(Rn))∆u(a≥0, b >0). For the plate equation
with only weak damping and local nonlinearity (the case γ= 1,β≡0 andf≡0 in (1.1)), attractors
were investigated in [8] and [9], where the author, inspired by the met hods of [10] and [11], proved the
existence, regularityand finite dimensionality ofthe global attract orsinH2(Rn)×L2(Rn). The situation
becomes more difficult when the equation contains localized damping te rms and nonlocal nonlinearities.
Recently, in [12] and [13], the plate equation with localized weak damping (the caseβ≡0 in (1.1)) and
involving nonlocal nonlinearities as −f(/ba∇dbl∇u/ba∇dblL2(Rn))∆uandf(/ba∇dblu/ba∇dblLp(Rn))|u|p−2uhave been considered.
In these articles, the existence of global attractors has been pr oved when the coefficient α(·) of the weak
damping term is strictly positive (see [12]) or, in addition to (1.3), is pos itive (see [13]) almost everywhere
inRn. However, in the case when α(·) vanishes in a set of positive measure, the existence of the global
attractor for (1.1) with β≡0 remained as an open question (see [12, Remark 1.2]). On the other hand, in
the case when α≡0 and even β≡1, the semigroup {S(t)}t≥0generated by (1.1)-(1.2) does not possess
a global attractor in H2(Rn)×L2(Rn). Indeed, if {S(t)}t≥0possesses a global attractor, then the linear
semigroup/braceleftbig
etA/bracerightbig
t≥0decay exponentially in the real and consequently, complex space H2(Rn)×L2(Rn),
which, due to Hille-Yosida Theorem (see [1, Remark 5.4]), implies neces sary condition iR⊂ρ(A). This
condition is equivalent to the solvability of the equation ( iµI−A)(u,v) = (y,z) inH2(Rn)×L2(Rn),
for every (y,z) inH2(Rn)×L2(Rn) andµ∈R. Choosing µ=√
λandy= 0, we have v=i√
λuand
∆(∆u−iu) =z. If the last equation for every z∈L2(Rn) has a solution u∈H3(Rn), then denoting
ϕ= ∆u−iu, we can say that the equation ∆ ϕ=zhas a solution in H1(Rn), for every z∈L2(Rn).
However, the last equation, as shown in [6], is not solvable in H1(Rn) for somez∈L2(Rn). Hence, the
necessary condition iR⊂ρ(A) does not hold. Thus, in the case when α≡0 andβ≡1, the problem
(1.1)-(1.2) does not have a global attractor, and in the case when β≡0 andα(·) vanishes in a set of
positive measure, the existence of the global for (1.1)-(1.2) is an o pen question.LONG-TIME DYNAMICS 3
In this paper, we impose conditions (1.3)-(1.5) on damping coefficient sα(·) andβ(·), which, unlike the
conditions imposed in the previous articles dealing with the wave and pla te equations involving strong
damping and/or nonlocal nonlinearities, allow both of them to be vanis hed in the sets of positive measure
such that in these sets the strong damping and weak damping comple te each other. Thus, our main result
is as follows:
Theorem 1.2. Under the conditions (1.3)-(1.8) the semigroup {S(t)}t≥0generated by the problem (1.1)-
(1.2) possesses a global attractor AinH2(Rn)×L2(Rn)andA=Mu(N). HereMu(N)is unstable
manifold emanating from the set of stationary points N(for definition, see [14, 359] ). Moreover, the
global attractor Ais bounded in H3(Rn)×H2(Rn).
The plan of the paper is as follows: In the next section, after the pr oof of two auxiliary lemmas, we
establish asymptotic compactness of {S(t)}t≥0in the interior domain. Then, we prove Lemma 2.3, which
plays a key role for the tail estimate, and thereby we show that the solutions of (1.1)-(1.2) are uniformly
(with respect to the initial data) small at infinity for large time. This f act, together with asymptotic
compactness in the interior domain, yields asymptotic compactness of{S(t)}t≥0in the whole space, and
by applying the abstract result on the gradient systems, we estab lish the existence of the global attractor
(see Theorem 2.3). In Section 3, by using the invariance of the globa l attractor, we show that it has an
additional regularity.
2.Existence of the global attractor
We begin with the following lemmas:
Lemma 2.1. Assume that the condition (1.6) holds. Also, assume that the sequence {vm}∞
m=1is weakly
star convergent in L∞/parenleftbig
0,∞;H2(Rn)/parenrightbig
, the sequence {vmt}∞
m=1is bounded in L∞/parenleftbig
0,∞;L2(Rn)/parenrightbig
and the
sequence/braceleftig
/ba∇dbl∇vm(t)/ba∇dblL2(Rn)/bracerightig∞
m=1is convergent, for all t≥0. Then, for every r>0andφ∈C1
0(B(0,r))
lim
m→∞limsup
l→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay
0/integraldisplay
B(0,r)τ/parenleftig
f(/ba∇dbl∇vm(τ)/ba∇dblL2(Rn))∆vm(τ,x)−f(/ba∇dbl∇vl(τ)/ba∇dblL2(Rn))∆vl(τ,x)/parenrightig
×φ(x)(vmt(τ,x)−vlt(τ,x))dxdτ|= 0,∀t≥0,
whereB(0,r) ={x∈Rn:|x|<r}.
Proof.Firstly, we have
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay
0/integraldisplay
B(0,r)τ/parenleftig
f(/ba∇dbl∇vm(τ)/ba∇dblL2(Rn))∆vm(τ,x)−f(/ba∇dbl∇vl(τ)/ba∇dblL2(Rn))∆vl(τ,x)/parenrightig
×φ(x)(vmt(τ,x)−vlt(τ,x))dxdτ|
≤1
2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay
0τf/parenleftig
/ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightigd
dτ/integraldisplay
B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dxdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingleKm,l
r(t)/vextendsingle/vextendsingle, (2.1)
whereKm,l
r(t) =t/integraltext
0τ/parenleftig
f(/ba∇dbl∇vm(τ)/ba∇dblL2(Rn))−f(/ba∇dbl∇vl(τ)/ba∇dblL2(Rn))/parenrightig/integraltext
B(0,r)φ(x)∆vm(τ,x)
×(vmt(τ,x)−vlt(t,x))dxdτ−t/integraltext
0/integraltext
B(0,r)τf/parenleftig
/ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightig
∇φ(x)·∇(vm(τ,x)−vl(τ,x))
×(vmt(τ,x)−vlt(τ,x))dxdτ. Applying [15, Corollary 4], we have that the sequence {vm}∞
m=1is rela-
tively compact in C/parenleftbig
[0,T];H2−ε(B(0,r))/parenrightbig
, for everyε>0,T >0 andr>0. So,
vm→vstrongly in C/parenleftbig
[0,T];H2−ε(B(0,r))/parenrightbig
, (2.2)4 AZER KHANMAMEDOV AND SEMA YAYLA
for somev∈C/parenleftbig
[0,T];H2−ε(B(0,r))/parenrightbig
. Hence, we find
lim
m→∞limsup
l→∞/vextendsingle/vextendsingleKm,l
r(t)/vextendsingle/vextendsingle= 0,∀t≥0. (2.3)
Now, denoting fε(u) =/braceleftbigg
f(u), u≥ε
f(ε),0≤u<εforε>0, we get
/vextendsingle/vextendsingle/vextendsinglef/parenleftig
/ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightig
−fε/parenleftig
/ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightig/vextendsingle/vextendsingle/vextendsingle≤max
0≤s1,s2≤ε|f(s1)−f(s2)|,
and then, for the first term on the right hand side of (2.1), we obta in
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay
0τf/parenleftig
/ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightigd
dτ/integraldisplay
B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dxdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay
0τfε/parenleftig
/ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightigd
dτ/integraldisplay
B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dxdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+c1t2max
0≤s1,s2≤ε|f(s1)−f(s2)|,∀t≥0. (2.4)
Let us estimate the first term on the right hand side of (2.4). By usin g integration by parts, we have
t/integraldisplay
0τfε/parenleftig
/ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightigd
dτ/integraldisplay
B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dxdτ
=tfε/parenleftig
/ba∇dbl∇vl(t)/ba∇dblL2(Rn)/parenrightig/integraldisplay
B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dx
−t/integraldisplay
0fε/parenleftig
/ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightig/integraldisplay
B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dxdτ
−t/integraldisplay
0τd
dt/parenleftig
fε/parenleftig
/ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightig/parenrightig/integraldisplay
B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dxdτ. (2.5)
By the conditions of the lemma and the definition of fε, it follows that/braceleftig
fε/parenleftig
/ba∇dbl∇vm(·)/ba∇dblL2(Rn)/parenrightig/bracerightig∞
m=1is
bounded in W1,∞(0,∞). Then, considering (2.2) in (2.5), we get
lim
m→∞limsup
l→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay
0τfε/parenleftig
/ba∇dbl∇vl(τ)/ba∇dblL2(Rn)/parenrightigd
dτ/integraldisplay
B(0,r)φ(x)|∇vm(τ,x)−∇vl(τ,x)|2dxdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0.(2.6)
Taking into account (2.3), (2.4) and (2.6) in (2.1), we obtain
limsup
m→∞limsup
l→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet/integraldisplay
0/integraldisplay
B(0,r)τ/parenleftig
f(/ba∇dbl∇vm(t)/ba∇dblL2(Rn)))∆vm(t,x)−f(/ba∇dbl∇vl(t)/ba∇dblL2(Rn)))∆vl(t,x)/parenrightig
×φ(x)(vmt(τ,x)−vlt(τ,x))dxdτ| ≤c1t2max
0≤s1,s2≤ε|f(s1)−f(s2)|,∀t≥0,
which yields the claim of the lemma, since ε>0 is arbitrary. /squareLONG-TIME DYNAMICS 5
Lemma 2.2. Assume that the condition (1.7) holds. Also, let the sequenc e{vm}∞
m=1be weakly star
convergent in L∞/parenleftbig
0,∞;H2(Rn)/parenrightbig
and the sequence {vmt}∞
m=1be bounded in L∞/parenleftbig
0,∞;L2(Rn)/parenrightbig
. Then,
for everyr>0andφ∈L∞(B(0,r))
lim
m→∞lim
l→∞t/integraldisplay
0/integraldisplay
B(0,r)τ(g(vm(τ,x))−g(vl(τ,x)))φ(x)(vmt(τ,x)−vlt(τ,x))dxdτ= 0,∀t≥0.
Proof.We have
t/integraldisplay
0/integraldisplay
B(0,r)τ(g(vm(τ,x))−g(vl(τ,x)))φ(x)(vmt(τ,x)−vlt(τ,x))dxdτ
=t/integraldisplay
0/integraldisplay
B(0,r)τφ(x)g(vm(τ,x))vmt(τ,x)dxdτ+t/integraldisplay
0/integraldisplay
B(0,r)τφ(x)g(vl(τ,x))vlt(τ,x)dxdτ
−t/integraldisplay
0/integraldisplay
B(0,r)τφ(x)g(vm(τ,x))vlt(τ,x)dxdτ−t/integraldisplay
0/integraldisplay
B(0,r)τφ(x)g(vl(τ,x))vmt(τ,x)dxdτ. (2.7)
Let us estimate the first two terms on the right hand side of (2.7). A pplying integration by parts, we get
t/integraldisplay
0/integraldisplay
B(0,r)τφ(x)g(vm(τ,x))vmt(τ,x)dxdτ+t/integraldisplay
0/integraldisplay
B(0,r)τφ(x)g(vl(τ,x))vlt(τ,x)dxdτ
=t/integraldisplay
0τd
dτ
/integraldisplay
B(0,r)φ(x)G(vm(τ,x))dx
dτ+t/integraldisplay
0τd
dτ
/integraldisplay
B(0,r)φ(x)G(vl(τ,x))dx
dτ
=t/integraldisplay
B(0,r)φ(x)G(vm(t,x))dx+t/integraldisplay
B(0,r)φ(x)G(vl(τ,x))dx
−t/integraldisplay
0/integraldisplay
B(0,r)φ(x)G(vm(τ,x))dxdτ−t/integraldisplay
0/integraldisplay
B(0,r)φ(x)G(vl(τ,x))dxdτ. (2.8)
By the conditions of the lemma, we obtain
/braceleftbigg
vm→vweakly star in L∞/parenleftbig
0,∞;H2(Rn)/parenrightbig
,
vmt→vtweakly star in L∞/parenleftbig
0,∞;L2(Rn)/parenrightbig
,(2.9)
for somev∈L∞/parenleftbig
0,∞;H2(Rn)/parenrightbig
∩W1,∞/parenleftbig
0,∞;L2(Rn)/parenrightbig
.Applying [15, Corollary 4], by (2.9), we have
vm→vstrongly in C/parenleftbig
[0,T];H2−ε(B(0,r))/parenrightbig
,
for everyε>0 andT >0. Hence, taking into account (1.7), we get
G(vm)→G(v) strongly in C/parenleftbig
[0,T];L1(B(0,r))/parenrightbig
. (2.10)
Then, passing to the limit in (2.8) and using (2.10), we obtain
lim
m→∞lim
l→∞
t/integraldisplay
0/integraldisplay
B(0,r)τφ(x)g(vm(τ,x))vmt(τ,x)dxdτ+t/integraldisplay
0/integraldisplay
B(0,r)τφ(x)g(vl(τ,x))vlt(τ,x)dxdτ
= 2t/integraldisplay
B(0,r)φ(x)G(v(t,x))dx−2t/integraldisplay
0/integraldisplay
B(0,r)φ(x)G(v(τ,x))dxdτ. (2.11)6 AZER KHANMAMEDOV AND SEMA YAYLA
Now, for the last two terms on the right hand side of (2.7), consider ing (2.9), we get
lim
m→∞lim
l→∞
−t/integraldisplay
0/integraldisplay
B(0,r)τφ(x)g(vm(τ,x))vlt(τ,x)dxdτ−t/integraldisplay
0/integraldisplay
B(0,r)τφ(x)g(vl(τ,x))vmt(τ,x)dxdτ
=−2t/integraldisplay
0/integraldisplay
B(0,r)τφ(x)g(v(τ,x))vt(τ,x)dxdτ
=−2t/integraldisplay
B(0,r)φ(x)G(v(t,x))dx+2t/integraldisplay
0/integraldisplay
B(0,r)φ(x)G(v(τ,x))dxdτ. (2.12)
Hence, considering (2.11)-(2.12) and passing to the limit in (2.7), we o btain the claim of the lemma. /square
Now, we can prove the asymptotic compactness of {S(t)}t≥0in the interior domain.
Theorem 2.1. Assume that the conditions (1.3)-(1.8) hold and Bis a bounded subset of H2(Rn)×
L2(Rn). Then every sequence of the form {S(tk)ϕk}∞
k=1,where{ϕk}∞
k=1⊂ B,tk→ ∞,has a convergent
subsequence in H2(B(0,r))×L2(B(0,r)), for every r>0.
Proof.We will use the asymptotic compactness method introduced in [16]. Co nsidering (1.3), (1.6), (1.7)
and (1.8) in (1.9), we have
sup
t≥0sup
ϕ∈B/ba∇dblS(t)ϕ/ba∇dblH2(Rn)×L2(Rn)<∞. (2.13)
Due to the boundedness of the sequence {ϕk}∞
k=1inH2(Rn)×L2(Rn), by (2.13), it follows that the
sequence {S(·)ϕk}∞
k=1is bounded in L∞/parenleftbig
0,∞;H2(Rn)×L2(Rn)/parenrightbig
. Then for any T≥1 there exists a
subsequence {km}∞
m=1such thattkm≥T, and
vm→vweakly star in L∞/parenleftbig
0,∞;H2(Rn)/parenrightbig
,
vmt→vtweakly star in L∞/parenleftbig
0,∞;L2(Rn)/parenrightbig
,
/ba∇dbl∇vm/ba∇dbl2
L2(Rn)→qweakly star in W1,∞(0,∞),
vm→vstrongly in C/parenleftbig
[0,T];H2−ε(B(0,r))/parenrightbig
,ε>0,(2.14)
for somev∈L∞/parenleftbig
0,∞;H2(Rn)/parenrightbig
∩W1,∞/parenleftbig
0,∞;L2(Rn)/parenrightbig
andq∈W1,∞(0,∞), where (vm(t),vmt(t)) =
S(t+tkm−T)ϕkm.
Now, taking into account (1.4) in (1.9), we find
∞/integraldisplay
0/ba∇dblvmt(t)/ba∇dbl2
L2(Rn\B(0,r0))dt+∞/integraldisplay
0/ba∇dbl∇vmt(t)/ba∇dbl2
L2(Rn\B(0,r0))dt≤c1. (2.15)
By (1.1), we have
vmtt(t,x)−div(β(x)∇vmt(t,x))+γ∆2vm(t,x)+α(x)vmt(t,x)+λvm(t,x)
=f(/ba∇dbl∇vm(t)/ba∇dblL2(Rn)))∆vm(t,x)−g(vm(t,x))+h(x). (2.16)
Letη∈C∞(Rn), 0≤η(x)≤1,η(x) =/braceleftbigg
0,|x| ≤1
1,|x| ≥2andηr(x) =η/parenleftbigx
r/parenrightbig
. Multiplying (2.16) with
η2
rvmand integrating the obtained equality over (0 ,T)×Rn, we get
T/integraldisplay
0/parenleftig
γ/ba∇dblηr∆vm(t)/ba∇dbl2
L2(Rn)+λ/ba∇dblηrvm(t)/ba∇dbl2
L2(Rn)/parenrightig
dt
=−1
2/integraldisplay
Rnη2
r(x)β(x)|∇vm(T,x)|2dx+1
2/integraldisplay
Rnη2
r(x)β(x)|∇vm(0,x)|2dxLONG-TIME DYNAMICS 7
−2
rn/summationdisplay
i=1T/integraldisplay
0/integraldisplay
Rnβ(x)vmtxi(t,x)ηrηxi/parenleftigx
r/parenrightig
vm(t,x)dxdt
+T/integraldisplay
0/ba∇dblηrvmt(t)/ba∇dbl2
L2(Rn)dt−/integraldisplay
Rnη2
r(x)vmt(T,x)vm(T,x)dx+/integraldisplay
Rnη2
r(x)vmt(0,x)vm(0,x)dx
−4γ
rn/summationdisplay
i=1T/integraldisplay
0/integraldisplay
Rnηr(x)ηxi/parenleftigx
r/parenrightig
∆vm(t,x)vmxi(t,x)dxdt−γT/integraldisplay
0/integraldisplay
Rn∆/parenleftbig
η2
r(x)/parenrightbig
∆vm(t,x)vm(t,x)dxdt
−1
2/integraldisplay
Rnη2
r(x)α(x)|vm(T,x)|2dx+1
2/integraldisplay
Rnη2
r(x)α(x)|vm(0,x)|2dx
−T/integraldisplay
0f(/ba∇dbl∇vm(t)/ba∇dblL2(Rn)))/integraldisplay
Rnη2
r(x)|∇vm(t,x)|2dxdt
−2
rn/summationdisplay
i=1T/integraldisplay
0f(/ba∇dbl∇vm(t)/ba∇dblL2(Rn)))/integraldisplay
Rnηrηxi/parenleftigx
r/parenrightig
vmxi(t,x)vmdxdt
−T/integraldisplay
0/integraldisplay
Rng(vm(t,x))η2
r(x)vm(t,x)dxdt
+T/integraldisplay
0/integraldisplay
Rnh(x)η2
r(x)vm(t,x)dxdt. (2.17)
Taking into account (1.3), (1.6), (1.8), (1.9), (2.13) and (2.15) in (2 .17), we obtain
lim sup
m→∞T/integraldisplay
0/parenleftig
γ/ba∇dbl∆vm(t)/ba∇dbl2
L2(Rn\B(0,2r))+λ/ba∇dblvm(t)/ba∇dbl2
L2(Rn\B(0,2r))/parenrightig
dt
≤c2/parenleftigg
1+√
T
r+T
r+T/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightigg
,∀r≥r0. (2.18)
Now, by (1.1), we have
vmtt(t,x)−vltt(t,x)−div(β(x)·∇(vmt(t,x)−vlt(t,x)))+γ∆2(vm(t,x)−vl(t,x))
+α(x)(vmt(t,x)−vlt(t,x))+λ(vm(t,x)−vl(t,x))
=f(/ba∇dbl∇vm(t)/ba∇dblL2(Rn)))∆vm(t,x)−f(/ba∇dbl∇vl(t)/ba∇dblL2(Rn)))∆vl(t,x)−g(vm)+g(vl). (2.19)
Multiplying (2.19) by/summationtextn
i=1xi(1−η4r)(vm−vl)xi+1
2(n−1)(1−η4r)(vm−vl), integrating the ob-
tained equality over (0 ,T)×Rnand taking into account (2.13), we obtain
3γ
2T/integraldisplay
0/ba∇dbl∆(vm(t)−vl(t))/ba∇dbl2
L2(B(0,4r))dt+1
2T/integraldisplay
0/ba∇dblvmt(t)−vlt(t)/ba∇dbl2
L2(B(0,4r))dt
≤c3(1+T+rT)/ba∇dblvm−vl/ba∇dblC[0,T];H1(B(0,8r))
+c3/parenleftig√
T+r√
T/parenrightig/vextenddouble/vextenddouble/vextenddouble/radicalbig
β(∇vmt−∇vlt)/vextenddouble/vextenddouble/vextenddouble
L2((0,T)×B(0,8r))
+c3/parenleftig
/ba∇dblvmt−vlt/ba∇dbl2
L2(0,T;L2(B(0,8r)\B(0,4r)))+/ba∇dblvm−vl/ba∇dbl2
L2(0,T;H2(B(0,8r)\B(0,4r)))/parenrightig
.(2.20)8 AZER KHANMAMEDOV AND SEMA YAYLA
Thus, considering (2.14), (2.15), (2.18) and passing to the limit in (2.2 0) , we get
limsup
m→∞limsup
l→∞T/integraldisplay
0/bracketleftig
/ba∇dbl∆(vm(t)−vl(t))/ba∇dbl2
L2(B(0,4r))+/ba∇dblvmt(t)−vlt(t)/ba∇dbl2
L2(B(0,4r))/bracketrightig
dt
≤c4/parenleftigg
1+√
T
r+T
r+r√
T+T/ba∇dblh/ba∇dblL2(Rn\B(0,2r))/parenrightigg
,∀r≥r0. (2.21)
Now,multiplying(2.19)by(1 −η2r)4t/bracketleftbig
2(vmt−vlt)+α0η4
r(vm−vl)/bracketrightbig
andintegratingtheobtainedequal-
ity over (0,T)×Rn,we obtain
γT/ba∇dbl∆(vm(T)−vl(T))/ba∇dbl2
L2(B(0,2r))+T/ba∇dblvmt(T)−vlt(T)/ba∇dbl2
L2(B(0,2r))+
+Tλ/ba∇dblvm(T)−vl(T)/ba∇dbl2
L2(B(0,2r))≤T/integraldisplay
0/ba∇dblvmt(t)−vlt(t)/ba∇dbl2
L2(B(0,4r))dt
+γT/integraldisplay
0/ba∇dbl∆(vm(t)−vl(t))/ba∇dbl2
L2(B(0,4r))dt+λT/integraldisplay
0/ba∇dblvm(t)−vl(t)/ba∇dbl2
L2(B(0,4r))dt
+2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleT/integraldisplay
0/integraldisplay
B(0,4r)t/parenleftig
f(/ba∇dbl∇vm(t)/ba∇dblL2(Rn))∆vm(t,x)−f(/ba∇dbl∇vl(t)/ba∇dblL2(Rn))∆vl(t,x))/parenrightig
×(1−η2r)4(vmt(t,x)−vlt(t,x))dxdt/vextendsingle/vextendsingle/vextendsingle
+2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleT/integraldisplay
0/integraldisplay
B(0,4r)t(g(vm(t,x))−g(vl(t,x)))(1−η2r)4(vmt(t,x)−vlt(t,x))dxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+α0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleT/integraldisplay
0/integraldisplay
B(0,4r)t/parenleftig
f(/ba∇dbl∇vm(t)/ba∇dblL2(Rn))∆vm(t,x)−f(/ba∇dbl∇vl(t)/ba∇dblL2(Rn))∆vl(t,x))/parenrightig
×(1−η2r)4η4
r(x)(vm(t,x)−vl(t,x))dxdt/vextendsingle/vextendsingle/vextendsingle
+α0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleT/integraldisplay
0/integraldisplay
B(0,4r)t(g(vm(t,x))−g(vl(t,x)))(1−η2r)4η4
r(x)(vm(t,x)−vl(t,x))dxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+c5T
rT/integraldisplay
0/integraldisplay
B(0,4r)\B(0,r)|∇(vmt(t,x)−vlt(t,x))|2dxdt
+c5T
rT/integraldisplay
0/integraldisplay
B(0,4r)\B(0,r)|vmt(t,x)−vlt(t,x)|2dxdt
+c5T/ba∇dblvm−vl/ba∇dbl2
C([0,T];H1(B(0,4r))),∀r≥r0,∀T≥1. (2.22)
Then, taking into account (2.14), (2.15), (2.21), Lemma 2.1 and Lem ma 2.2, and passing to the limit in
(2.22), we find
limsup
m→∞limsup
l→∞/parenleftig
/ba∇dblvm(T)−vl(T)/ba∇dbl2
H2(B(0,2r))+/ba∇dblvmt(T)−vlt(T)/ba∇dbl2
L2(B(0,2r))/parenrightig
≤c6/parenleftbigg1
T+1√
Tr+1
r+r√
T+/ba∇dblh/ba∇dblL2(Rn\B(0,2r))/parenrightbigg
,∀r≥r0,∀T≥1. (2.23)LONG-TIME DYNAMICS 9
Thus, by the definition of vm, the inequality (2.23) yields
limsup
m→∞limsup
l→∞/ba∇dblS(tkm)ϕkm−S(tkl)ϕkl/ba∇dbl2
H2(B(0,r))×L2(B(0,r))
≤c7/parenleftbigg1
T+1√
Tr+1
r+r√
T+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg
,∀r≥2r0,∀T≥1. (2.24)
Passing to the limit as T→ ∞in (2.24), we obtain
liminf
l→∞liminf
m→∞/ba∇dblS(tk)ϕk−S(tm)ϕm/ba∇dbl2
H2(B(0,r))×L2(B(0,r))
≤c7/parenleftbigg1
r+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg
,∀r≥2r0,
which gives
liminf
l→∞liminf
m→∞/ba∇dblS(tk)ϕk−S(tm)ϕm/ba∇dbl2
H2(B(0,r))×L2(B(0,r))
≤c7/parenleftbigg1
/tildewider+/ba∇dblh/ba∇dblL2(Rn\B(0,/tildewider))/parenrightbigg
,∀/tildewider≥r≥2r0. (2.25)
Consequently, by passing to the limit as /tildewider→ ∞in (2.25), we deduce
liminf
l→∞liminf
m→∞/ba∇dblS(tk)ϕk−S(tm)ϕm/ba∇dblH2(B(0,r))×L2(B(0,r))= 0,∀r>0. (2.26)
Letriր ∞asi→ ∞. Takingr=riin (2.26) and using the arguments at the end of the proof of [17,
Lemma 3.4], we can say that there exist subsequences/braceleftig
k(i)
m/bracerightig
such that
/braceleftig
k(1)
m/bracerightig
⊃/braceleftig
k(2)
m/bracerightig
⊃...⊃/braceleftig
k(i)
m/bracerightig
⊃..
and /braceleftig
S(tk(i)
m)ϕk(i)
m/bracerightig
converges in H2(B(0,ri))×L2(B(0,ri)).
Thus, the diagonal subsequence/braceleftig
S(tk(m)
m)ϕk(m)
m/bracerightig
converges in H2(B(0,r))×L2(B(0,r)), for every
r>0. /square
To establish the tail estimate, we need the following lemma.
Lemma 2.3. Let the conditions (1.3)-(1.6) hold and Bbe a bounded subset of H2(Rn).Then for every
ε >0there exist a constant δ≡δ(ε)>0and functions ψε∈L∞(Rn),ϕε∈C∞(Rn), such that
0≤ψε≤min/braceleftbig
1,δ−1β/bracerightbig
a.e. inRn,0≤ϕε≤1inRn, supp(ϕε)⊂ {x∈Rn:α(x)≥δa.e. inRn}and
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglef/parenleftig
/ba∇dbl∇u/ba∇dblL2(Rn)/parenrightig
−fδ/parenleftigg/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig
ψε∇u/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/ba∇dbl√ϕε∇u/ba∇dbl2
L2(Rn)/parenrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<ε, (2.27)
for everyu∈B,wherefδis the function defined in the proof of Lemma 2.1.
Proof.LetA0={x∈B(0,r0) :α(x) = 0}andAk=/braceleftbig
x∈B(0,r0) : 0≤α(x)<1
k/bracerightbig
. It is easy to see
thatAk+1⊂Ak, andA0=∩
k>0Ak.Hence, lim
k→∞mes(Ak) =mes(A0). So, forδ >0, there exists kδsuch
that
mes(Akδ\A0)<δ
3. (2.28)
SinceAkδisameasurablesubsetof B(0,r0),thereexistsanopenset O(1)
δ⊂B(0,r0)suchthatAkδ⊂O(1)
δ
and
mes/parenleftig
O(1)
δ\Akδ/parenrightig
<δ
3. (2.29)
Now, letηδ∈C0(Rn) such that 0 ≤ηδ≤1, ηδ|O(1)
δ= 1 and supp( ηδ)⊂O(2)
δ,whereO(1)
δ⋐O(2)
δand
mes/parenleftig
O(2)
δ\O(1)
δ/parenrightig
<δ
3. (2.30)
Then setting ϕδ:= 1−ηδ, we haveϕδ∈C(Rn), 0≤ϕδ≤1,ϕδ|Rn\O(2)
δ= 1 and supp( ϕδ)⊂Rn\O(1)
δ.10 AZER KHANMAMEDOV AND SEMA YAYLA
By (2.28)-(2.30), we obtain
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
O(2)
δϕδ|∇u(x)|2dx−/integraldisplay
O(2)
δ\A0|∇u(x)|2dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
O(2)
δ\O(1)
δϕδ|∇u(x)|2dx−/integraldisplay
O(2)
δ\A0|∇u(x)|2dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
≤2/integraldisplay
O(2)
δ\A0|∇u(x)|2dx≤2c/ba∇dblu/ba∇dbl2
H2(Rn)/parenleftig
mes/parenleftig
O(2)
δ\A0/parenrightig/parenrightign∗
<2cδn∗/ba∇dblu/ba∇dbl2
H2(Rn), (2.31)
for everyu∈H2(Rn), wheren∗=
1, n = 1,
q,0<q<1, n= 2,
2
n, n ≥3andc>0.
Now, by (1.5), it follows that
β >0 a.e. inA0.
Hence, by Lebesgue dominated convergence theorem, there exis tsλδ>0 such that
/integraldisplay
A0λδ
λδ+β(x)dx<δ,
which yields/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
A0|∇u(x)|2dx−/integraldisplay
A0β(x)
λδ+β(x)|∇u(x)|2dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<cδn∗/ba∇dblu/ba∇dbl2
H2(Rn). (2.32)
Thus, denoting ψδ=/braceleftigg
β(x)
λδ+β(x),x∈A0,
0,x∈Rn\A0,by (2.31) and (2.32), we get
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/ba∇dbl∇u/ba∇dbl2
L2(Rn)−/vextenddouble/vextenddouble/vextenddouble/radicalbig
ψδ∇u/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)−/ba∇dbl√ϕδ∇u/ba∇dbl2
L2(Rn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
<3cδn∗/ba∇dblu/ba∇dbl2
H2(Rn),
and consequently/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/ba∇dbl∇u/ba∇dblL2(Rn)−/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig
ψδ∇u/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/ba∇dbl√ϕδ∇u/ba∇dbl2
L2(Rn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
≤/radicaligg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/ba∇dbl∇u/ba∇dbl2
L2(Rn)−/vextenddouble/vextenddouble/vextenddouble/radicalbig
ψδ∇u/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)−/ba∇dbl√ϕδ∇u/ba∇dbl2
L2(Rn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
<√
3cδ1
2n∗/ba∇dblu/ba∇dblH2(Rn).
The last inequality, together with the differentiability of the function f, yields (2.27). /square
Now, let us proof the following tail estimate.
Theorem 2.2. Assume that the conditions (1.3)-(1.8) hold and Bis a bounded subset of H2(Rn)×
L2(Rn). Then for any ε>0there existT≡T(B,ε)andR≡R(B,ε)such that
/ba∇dblS(t)ϕ/ba∇dblH2(Rn\B(0,r))×L2(Rn\B(0,r))<ε,
for everyt≥T,r≥Randϕ∈ B.LONG-TIME DYNAMICS 11
Proof.Let (u0,u1)∈ Band (u(t),ut(t)) =S(t)(u0,u1). Multiplying (1.1) with η2
rut, integrating the
obtained equality over Rnand taking into account (2.13) ,we get
1
2d
dt/parenleftig
/ba∇dblηrut(t)/ba∇dbl2
L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2
L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2
L2(Rn)/parenrightig
+d
dt
/integraldisplay
Rnη2
r(x)G(u(t,x))dx
+/vextenddouble/vextenddouble/vextenddouble/radicalbig
βηr∇ut(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/vextenddouble/vextenddouble√αηrut(t)/vextenddouble/vextenddouble2
L2(Rn)
−/integraldisplay
Rnf(/ba∇dbl∇u(t)/ba∇dblL2(Rn))∆uη2
rutdx
≤c2/parenleftbigg1
r+1
r/vextenddouble/vextenddouble/vextenddouble/radicalbig
βηr∇ut(t)/vextenddouble/vextenddouble/vextenddouble
L2(B(0,2r))+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg
,∀r≥r0. (2.33)
Now, let us estimate the last term on the left hand side of (2.33). By L emma 2.3, we have
−f(/ba∇dbl∇u(t)/ba∇dblL2(Rn))/integraldisplay
Rn∆uη2
rutdx
≥ −ε/ba∇dblηr∆u(t)/ba∇dblL2(Rn)/ba∇dblηrut(t)/ba∇dblL2(Rn)−c3
r
+1
2fδ/parenleftigg/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig
ψε∇u(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/vextenddouble/vextenddouble/vextenddouble/radicalbig
φε∇u(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)/parenrightigg
d
dt/parenleftig
/ba∇dblηr∇(u(t))/ba∇dbl2
L2(Rn)/parenrightig
.(2.34)
Moreover,forthe last termon the righthandside of(2.34), byusin gthe definition of fδand the properties
ofψεandϕε, we obtain
1
2fδ/parenleftigg/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig
ψε∇u(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/ba∇dbl√ϕε∇u(t)/ba∇dbl2
L2(Rn)/parenrightigg
d
dt/parenleftig
/ba∇dblηr∇(u(t))/ba∇dbl2
L2(Rn)/parenrightig
≥1
2d
dt/parenleftigg
fδ/parenleftigg/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig
ψε∇u(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/ba∇dbl√ϕε∇u(t)/ba∇dbl2
L2(Rn)/parenrightigg
/ba∇dblηr∇(u(t))/ba∇dbl2
L2(Rn)/parenrightigg
−c4/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/radicalbig
β∇ut(t)/vextenddouble/vextenddouble/vextenddouble
L2(Rn)+/vextenddouble/vextenddouble√αut(t)/vextenddouble/vextenddouble
L2(Rn)/parenrightbigg
/ba∇dblηr∇(u(t))/ba∇dbl2
L2(Rn). (2.35)
Considering (2.34) and (2.35) in (2.33), we obtain
1
2d
dt/parenleftig
/ba∇dblηrut(t)/ba∇dbl2
L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2
L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2
L2(Rn)/parenrightig
+d
dt
/integraldisplay
Rnη2
r(x)G(u(t,x))dx
+/vextenddouble/vextenddouble/vextenddouble/radicalbig
βηr∇ut(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/vextenddouble/vextenddouble√αηrut(t)/vextenddouble/vextenddouble2
L2(Rn)
+1
2d
dt/parenleftigg
fδ/parenleftigg/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig
ψε∇u(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/vextenddouble/vextenddouble/vextenddouble/radicalbig
φε∇u(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)/parenrightigg
/ba∇dblηr∇(u(t))/ba∇dbl2
L2(Rn)/parenrightigg
−c4/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/radicalbig
β∇ut(t)/vextenddouble/vextenddouble/vextenddouble
L2(Rn)+/vextenddouble/vextenddouble√αut(t)/vextenddouble/vextenddouble
L2(Rn)/parenrightbigg
/ba∇dblηr∇(u(t))/ba∇dbl2
L2(Rn).
−ε/ba∇dblηr∆u(t)/ba∇dblL2(Rn)/ba∇dblηrut(t)/ba∇dblL2(Rn)
≤c2/parenleftbigg1
r+1
r/vextenddouble/vextenddouble/vextenddouble/radicalbig
βηr∇ut(t)/vextenddouble/vextenddouble/vextenddouble
L2(B(0,2r))+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg
. (2.36)
Multiplying (1.1) with µη2
ru, integrating the obtained equality over Rnand taking into account (1.6),
(1.8) and (2.13), we get
µγ/ba∇dblηr∆u(t)/ba∇dbl2
L2(Rn)+µλ/ba∇dblηru(t)/ba∇dbl2
L2(Rn)−µ/ba∇dblηrut(t)/ba∇dbl2
L2(Rn)
+µ
2d
dt/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/radicalbig
βηr∇u(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/vextenddouble/vextenddouble√αηru(t)/vextenddouble/vextenddouble2
L2(Rn)/parenrightbigg12 AZER KHANMAMEDOV AND SEMA YAYLA
+µd
dt
/integraldisplay
Rnη2
r(x)ut(t,x)u(t,x)dx
≤c5/parenleftbigg1
r+1
r/vextenddouble/vextenddouble/vextenddouble/radicalbig
βηr∇ut(t)/vextenddouble/vextenddouble/vextenddouble
L2(B(0,2r))+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg
. (2.37)
Summing (2.36) and (2.37), applying Young inequality and choosing εandµsmall enough, we obtain
d
dt
/ba∇dblηrut(t)/ba∇dbl2
L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2
L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2
L2(Rn)+/integraldisplay
Rn\B(0,r)η2
r(x)G(u(t,x))dx
+1
2d
dt/parenleftigg
fδ/parenleftigg/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig
ψε∇u(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/vextenddouble/vextenddouble/vextenddouble/radicalbig
φε∇u(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)/parenrightigg
/ba∇dblηr∇(u(t))/ba∇dbl2
L2(Rn)/parenrightigg
+µ
2d
dt/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/radicalbig
βηr∇u(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/vextenddouble/vextenddouble√αηru(t)/vextenddouble/vextenddouble2
L2(Rn)/parenrightbigg
+c6/parenleftig
/ba∇dblηrut(t)/ba∇dbl2
L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2
L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2
L2(Rn)/parenrightig
≤c7/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/radicalbig
β∇ut(t)/vextenddouble/vextenddouble/vextenddouble
L2(Rn)+/vextenddouble/vextenddouble√αut(t)/vextenddouble/vextenddouble
L2(Rn)/parenrightbigg
/ba∇dblηr∇(u(t))/ba∇dbl2
L2(Rn)
+c7/parenleftbigg1
r+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg
,∀r≥r0,
whereci(i= 6,7) are positive constants. By denoting
Φ(t) :=/ba∇dblηrut(t)/ba∇dbl2
L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2
L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2
L2(Rn)+/integraldisplay
Rnη2
r(x)G(u(t,x))dx
+1
2fδ/parenleftigg/radicaligg/vextenddouble/vextenddouble/vextenddouble/radicalbig
ψε∇u(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/vextenddouble/vextenddouble/vextenddouble/radicalbig
φε∇u(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)/parenrightigg
/ba∇dblηr∇(u(t))/ba∇dbl2
L2(Rn)
+µ
2/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/radicalbig
βηr∇u(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn\B(0,r))+/vextenddouble/vextenddouble√αηru(t)/vextenddouble/vextenddouble2
L2(Rn\B(0,r))/parenrightbigg
,
we get
d
dtΦ(t)+c6/parenleftig
/ba∇dblηrut(t)/ba∇dbl2
L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2
L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2
L2(Rn)/parenrightig
≤c7/parenleftbigg/vextenddouble/vextenddouble/vextenddouble/radicalbig
β∇ut(t)/vextenddouble/vextenddouble/vextenddouble
L2(Rn)+/vextenddouble/vextenddouble√αut(t)/vextenddouble/vextenddouble
L2(Rn)/parenrightbigg
/ba∇dblηr∇(u(t))/ba∇dbl2
L2(Rn)
+c7/parenleftbigg1
r+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg
,∀r≥r0. (2.38)
Moreover, there exist /hatwidec≡/hatwidec(B)>0 such that
/ba∇dblηrut(t)/ba∇dbl2
L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2
L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2
L2(Rn)
≤Φ(t)≤/hatwidec/parenleftig
/ba∇dblηrut(t)/ba∇dbl2
L2(Rn)+γ/ba∇dblηr∆u(t)/ba∇dbl2
L2(Rn)+λ/ba∇dblηru(t)/ba∇dbl2
L2(Rn)/parenrightig
. (2.39)
So, considering (2.39) in (2.38), we have
d
dtΦ(t)+H(t)Φ(t)
≤c7/parenleftbigg1
r+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg
,
whereH(t) =c8−c7/parenleftig/vextenddouble/vextenddouble√β∇ut(t)/vextenddouble/vextenddouble
L2(Rn)+/ba∇dbl√αut(t)/ba∇dblL2(Rn)/parenrightig
andc8>0. Then, by Gronwallinequality,
we obtain
Φ(t)≤e−/integraltextt
0H(τ)dτΦ(0)+c7/parenleftbigg1
r+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg/integraldisplayt
0e−/integraltextt
τH(σ)dσdτ. (2.40)LONG-TIME DYNAMICS 13
Furthermore, applying Young inequality and taking into account (1.9 ), we have
e−/integraltextt
τH(σ)dσ≤e−1
2c8(t−τ)+c9/integraltextt
τ/parenleftBig
/ba∇dbl√β∇ut(t)/ba∇dbl2
L2(Rn)+/ba∇dbl√αut(t)/ba∇dbl2
L2(Rn)/parenrightBig
dσ
≤c10e−1
2c8(t−τ),∀t≥τ≥0. (2.41)
Therefore, considering (2.41) in (2.40), we get
Φ(t)≤c10e−1
2c8tΦ(0)+c11/parenleftbigg1
r+/ba∇dblh/ba∇dblL2(Rn\B(0,r))/parenrightbigg
,∀t≥0,
which completes the proof of the theorem. /square
Now, we are in a position to prove the existence of the global attrac tor.
Theorem 2.3. Let the conditions (1.3)-(1.8) hold. Then the semigroup {S(t)}t≥0generated by the
problem (1.1)-(1.2) possesses a global attractor AinH2(Rn)×L2(Rn)andA=Mu(N).
Proof.By Theorem 2.1 and Theorem 2.2, it follows that every sequence of th e form{S(tk)ϕk}∞
k=1, where
{ϕk}∞
k=1⊂ B,tk→ ∞,andBis bounded subset of H2(Rn)×L2(Rn), has a convergent subsequence
inH2(Rn)×L2(Rn). Since, by (1.6) and (1.8), the set N, which is the set of stationary points of
{S(t)}t≥0is bounded in H2(Rn)×L2(Rn), to complete the proof, it is enough to show that the pair/parenleftbig
S(t),H2(Rn)×L2(Rn)/parenrightbig
is a gradient system (see [14]).
Now, for (u(t),ut(t)) =S(t)(u0,u1), let the equality
L(u(t),ut(t)) =L(u0,u1),∀t≥0,
hold, where L(u,v) =1
2/integraltext
Rn(|v(x)|2+γ|∆u(x)|2+λ|u(x)|2)dx+/integraltext
RnG(u(x))dx+1
2F/parenleftig
/ba∇dbl∇u/ba∇dbl2
L2(Rn)/parenrightig
−
/integraltext
Rnh(x)u(x)dx.Then considering (1.3) and (1.9), we have
αut(t,·) = 0 andβ∇ut(t,·) = 0 a.e. in Rn,
fort≥0.Taking into account (1.5), from the above equalities, it follows that
ut(t,·)utxi(t,·) = 0 a.e. in Rn,
and consequently
∂
∂xi/parenleftbig
u2
t(t,·)/parenrightbig
= 0 a.e. in Rn,
fori=1,nandt≥0.The last equality means that u2
t(t,·) is independent of variable x, for everyt≥0.
Hence, byut(t,·)∈L2(Rn), we have
ut(t,·) = 0 a.e. in Rn,
fort≥0. So,
(u(t),ut(t)) = (ϕ,0),∀t≥0,
where (ϕ,0)∈ N. Thus, the pair/parenleftbig
S(t),H2(Rn)×L2(Rn)/parenrightbig
is a gradient system. /square
3.Regularity of the global attractor
We start with the following lemma.
Lemma 3.1. Let the condition (1.7) hold and Kbe a compact subset of H2(Rn). Then for every ε>0
there exists a constant Cǫ>0such that
/ba∇dblg(u1)−g(u2)/ba∇dblL2(Rn)≤ε/ba∇dblu1−u2/ba∇dblH2(Rn)+Cǫ/ba∇dblu1−u2/ba∇dblL2(Rn), (3.1)
for everyu1,u2∈K.14 AZER KHANMAMEDOV AND SEMA YAYLA
Proof.ByMeanValueTheorem,H¨ olderinequalityandtheembedding H2(Rn)֒→L2n
(n−4)+(Rn)∩L2(Rn),
we have
/ba∇dblg(u)−g(v)/ba∇dbl2
L2(Rn)=/integraldisplay
Rn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/integraldisplay
0g′(τu(x)+(1−τ)v(x))dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
|u(x)−v(x)|2dx
≤1/integraldisplay
0/integraldisplay
{x∈Rn:|τu(x)+(1−τ)v(x)|>M}|g′(τu(x)+(1−τ)v(x))|2|u(x)−v(x)|2dxdτ
+/ba∇dblg′/ba∇dblC[−M,M]/ba∇dblu−v/ba∇dbl2
L2(Rn)
≤c1/integraldisplay
{x∈Rn:|u(x)|+|v(x)|>M}/parenleftig
1+|u(x)|2(p−1)+|v(x)|2(p−1)/parenrightig
|u(x)−v(x)|2dx
+/ba∇dblg′/ba∇dblC[−M,M]/ba∇dblu−v/ba∇dbl2
L2(Rn)
≤c2
/integraldisplay
{x∈Rn:|u(x)|+|v(x)|>M}/parenleftig
1+|u(x)|2(p−1)q+|v(x)|2(p−1)q/parenrightig
dx
1
q
×/ba∇dblu−v/ba∇dblH2(Rn)+/ba∇dblg′/ba∇dblC[−M,M]/ba∇dblu−v/ba∇dbl2
L2(Rn), (3.2)
whereq= max/braceleftbig
1,n
4/bracerightbig
.
Since, by (1.7), H2(Rn)֒→L2(p−1)q(Rn), we have that Kis compact subset of L2(p−1)q(Rn).Hence,
lim
M→∞sup
u,v∈K/integraldisplay
{x∈Rn:|u(x)|+|v(x)|>M}/parenleftig
1+|u(x)|2(p−1)q+|v(x)|2(p−1)q/parenrightig
dx= 0. (3.3)
Thus, (3.2) and (3.3) give us (3.1). /square
Theorem 3.1. The global attractor Ais bounded in H3(Rn)×H2(Rn).
Proof.Letϕ∈ A. SinceAis invariant, there exists an invariant trajectory Γ = {(u(t),ut(t)) :t∈R}
⊂ Asuch that (u(0),ut(0)) =ϕ(see [18, p. 159]). Now, let us define
v(t,x) :=u(t+σ,x)−u(t,x)
σ,σ>0.
Then, by (1.1), we get
vtt(t,x)+γ∆2v(t,x)−div(β(x)∇vt)+α(x)vt(t,x)+λv(t,x)
−f/parenleftig
/ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightig
∆v(t,x)−f(/ba∇dbl∇u(t+σ)/ba∇dblL2(Rn))−f/parenleftig
/ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightig
σ∆u(t+σ,x)
+g(u(t+σ,x))−g(u(t,x))
σ= 0, (t,x)∈R×Rn. (3.4)
Multiplying (3.4) by vtand integrating the obtained equality over Rn, we find
d
dtE(v(t))+/vextenddouble/vextenddouble/vextenddouble/radicalbig
β∇vt(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/vextenddouble/vextenddouble√αvt(t)/vextenddouble/vextenddouble2
L2(Rn)
≤ −1
2f/parenleftig
/ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightigd
dt/parenleftig
/ba∇dbl∇v(t)/ba∇dbl2
L2(Rn)/parenrightig
+f(/ba∇dbl∇u(t+σ)/ba∇dblL2(Rn))−f/parenleftig
/ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightig
σ
×/integraldisplay
Rn∆u(t+σ,x)vt(t,x)dx−1
σ/integraldisplay
Rn(g(u(t+σ,x))−g(u(t,x)))vt(t,x)dx
≤ −1
2f/parenleftig
/ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightigd
dt/parenleftig
/ba∇dbl∇v(t)/ba∇dbl2
L2(Rn)/parenrightig
+c1/ba∇dbl∇v(t)/ba∇dblL2(Rn)/ba∇dblvt(t)/ba∇dblL2(Rn)+1
σ/ba∇dblg(u(t+σ))−g(u(t))/ba∇dblL2(Rn)/ba∇dblvt(t)/ba∇dblL2(Rn).LONG-TIME DYNAMICS 15
Taking into account Lemma 3.1 in the last inequality, we obtain
d
dtE(v(t))+/vextenddouble/vextenddouble/vextenddouble/radicalbig
β∇vt(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/vextenddouble/vextenddouble√αvt(t)/vextenddouble/vextenddouble2
L2(Rn)
≤ −1
2f/parenleftig
/ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightigd
dt/parenleftig
/ba∇dbl∇v(t)/ba∇dbl2
L2(Rn)/parenrightig
+c1/ba∇dbl∇v(t)/ba∇dblL2(Rn)/ba∇dblvt(t)/ba∇dblL2(Rn)+/parenleftig
ε/ba∇dblv(t)/ba∇dblH2(Rn)+Cε/ba∇dblv(t)/ba∇dblL2(Rn)/parenrightig
/ba∇dblvt(t)/ba∇dblL2(Rn),(3.5)
for anyε>0. Moreover, by (2.13), we have
/ba∇dblv(t)/ba∇dblL2(Rn)=/vextenddouble/vextenddouble/vextenddouble/vextenddoubleu(t+σ,x)−u(t,x)
σ/vextenddouble/vextenddouble/vextenddouble/vextenddouble
L2(Rn)≤sup
0≤t<∞/ba∇dblut(t)/ba∇dblL2(Rn)</hatwideC,∀t∈R. (3.6)
Then, considering (3.6) in (3.5), we get
d
dtE(v(t))+/vextenddouble/vextenddouble/vextenddouble/radicalbig
β∇vt(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/vextenddouble/vextenddouble√αvt(t)/vextenddouble/vextenddouble2
L2(Rn)
≤ −1
2f/parenleftig
/ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightigd
dt/parenleftig
/ba∇dbl∇v(t)/ba∇dbl2
L2(Rn)/parenrightig
+/parenleftig
c2/ba∇dblv(t)/ba∇dbl1
2
H2(Rn)+ε/ba∇dblv(t)/ba∇dblH2(Rn)+/tildewiderCε/parenrightig
/ba∇dblvt(t)/ba∇dblL2(Rn). (3.7)
Now, let us estimate the first term on the right hand side of (3.7). By (2.13) and (3.6), we have
−1
2f/parenleftig
/ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightigd
dt/parenleftig
/ba∇dbl∇v(t)/ba∇dbl2
L2(Rn)/parenrightig
≤c3max
0≤s1,s2≤ε|f(s1)−f(s2)|/ba∇dblvt(t)/ba∇dblL2(Rn)/ba∇dbl∆v(t)/ba∇dblL2(Rn)
−1
2fε/parenleftig
/ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightigd
dt/parenleftig
/ba∇dbl∇v(t)/ba∇dbl2
L2(Rn)/parenrightig
≤c3max
0≤s1,s2≤ε|f(s1)−f(s2)|/ba∇dblvt(t)/ba∇dblL2(Rn)/ba∇dbl∆v(t)/ba∇dblL2(Rn)
−1
2d
dt/parenleftig
fε/parenleftig
/ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightig
/ba∇dbl∇v(t)/ba∇dbl2
L2(Rn)/parenrightig
+c4/ba∇dblv(t)/ba∇dblH2(Rn), (3.8)
for anyε>0, wherefεis the function defined in the proof of Lemma 2.1. Considering (3.8) in ( 3.7), we
obtain
d
dt/parenleftbigg
E(v(t))+1
2fε/parenleftig
/ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightig
/ba∇dbl∇v(t)/ba∇dbl2
L2(Rn)/parenrightbigg
+/vextenddouble/vextenddouble/vextenddouble/radicalbig
β∇vt(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)+/vextenddouble/vextenddouble√αvt(t)/vextenddouble/vextenddouble2
L2(Rn)
≤c3max
0≤s1,s2≤ε|f(s1)−f(s2)|/ba∇dblvt(t)/ba∇dblL2(Rn)/ba∇dbl∆u(t)/ba∇dblL2(Rn)+c4/ba∇dblv(t)/ba∇dblH2(Rn)
+/parenleftig
c2/ba∇dblv(t)/ba∇dbl1
2
H2(Rn)+ε/ba∇dblv(t)/ba∇dblH2(Rn)+/tildewiderCε/parenrightig
/ba∇dblvt(t)/ba∇dblL2(Rn). (3.9)
Letηr(x) be the cut-off function defined in the proof of Theorem 2.1. Multiply ing (3.4) by/summationtextn
i=1xi(1−η2r0)vxi+1
2(n−1)(1−η2r0)v, and integrating over Rn, by (2.13) and (3.6), we get
3
2γ/ba∇dbl∆(v(t))/ba∇dbl2
L2(B(0,2r0))+1
2/ba∇dblvt(t)/ba∇dbl2
L2(B(0,2r0))
+d
dt
/summationdisplayn
i=1/integraldisplay
Rnxi(1−η2r0(x))vxi(t,x)vt(t,x)dx+1
2(n−1)/integraldisplay
Rn(1−η2r0(x))vt(t,x)v(t,x)dx
≤c5/ba∇dblvt(t)/ba∇dbl2
L2(B(0,4r0)\B(0,2r0))+c5/ba∇dbl∆v(t)/ba∇dbl2
L2(B(0,4r0)\B(0,2r0))+c5/ba∇dblv(t)/ba∇dbl1
2
H2(B(0,4r0))
+c5/vextenddouble/vextenddouble/vextenddouble/radicalbig
β∇vt(t)/vextenddouble/vextenddouble/vextenddouble
L2(B(0,4r0))/parenleftig
/ba∇dblv(t)/ba∇dbl1
2
H2(B(0,4r0))+/ba∇dblv(t)/ba∇dblH2(B(0,4r0))/parenrightig
+c5/vextenddouble/vextenddouble√αvt(t)/vextenddouble/vextenddouble
L2(B(0,4r0))/parenleftig
/ba∇dblv(t)/ba∇dbl1
2
H2(B(0,4r0))+1/parenrightig
+c5/ba∇dblv(t)/ba∇dbl3
2
H2(B(0,4r0))+c5. (3.10)16 AZER KHANMAMEDOV AND SEMA YAYLA
Multiplying (3.4) by η2
r0vand integrating over Rn, we find
d
dt
/integraldisplay
Rnη2
r0v(t,x)vt(t,x)dx+1
2/vextenddouble/vextenddouble√αηr0v(t)/vextenddouble/vextenddouble2
L2(Rn)+1
2/vextenddouble/vextenddouble/vextenddouble/radicalbig
βηr0∇v(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)
+/ba∇dbl∆v(t)/ba∇dbl2
L2(Rn\B(0,r0))+λ/ba∇dblv(t)/ba∇dbl2
L2(Rn\B(0,r0))−/ba∇dblvt(t)/ba∇dbl2
L2(Rn\B(0,r0))
≤c6/ba∇dblv(t)/ba∇dbl3
2
H2(Rn)+c6/vextenddouble/vextenddouble/vextenddouble/radicalbig
β∇vt(t)/vextenddouble/vextenddouble/vextenddouble
L2(Rn)/parenleftig
1+/ba∇dblv(t)/ba∇dbl1
2
H2(B(0,4r0))/parenrightig
+c6. (3.11)
Multiplying (3.10) and (3.11) by δ2andδ, respectively, then summing the obtained inequalities with
(3.9), choosing ε>0 andδ>0 sufficiently small and applying Young inequality, we get
d
dtΨ(t)+c7E(v(t))≤c8,∀t∈R, (3.12)
where
Ψ(t) :=E(v(t))+1
2fε/parenleftig
/ba∇dbl∇u(t)/ba∇dblL2(Rn)/parenrightig
/ba∇dbl∇v(t)/ba∇dbl2
L2(Rn)
+δ
/integraldisplay
Rnη2
r0v(t,x)vt(t,x)dx+1
2/vextenddouble/vextenddouble√αηr0v(t)/vextenddouble/vextenddouble2
L2(Rn)+1
2/vextenddouble/vextenddouble/vextenddouble/radicalbig
βηr0∇v(t)/vextenddouble/vextenddouble/vextenddouble2
L2(Rn)
+δ2
/summationdisplayn
i=1/integraldisplay
Rnxi(1−η2r0(x))vxi(t,x)vt(t,x)dx+1
2(n−1)/integraldisplay
Rn(1−η2r0(x))vt(t,x)v(t,x)dx
,
and the positive constant c8, as the previous ci/parenleftbig
i=1,7/parenrightbig
, is independent of the trajectory Γ .
Sinceδ>0 is sufficiently small, there exist constants c>0,/tildewidec>0 such that
cE(v(t))≤Ψ(t)≤/tildewidecE(v(t)),∀t∈R. (3.13)
Taking into account (3.13) in (3.12), we obtain
d
dtΨ(t)+c9Ψ(t)≤c8,∀t∈R,
which yields
Ψ(t)≤e−c9(t−s)Ψ(s)+c8
c9,∀t≥s.
Passing to the limit as s→ −∞and considering (3.13), we get
E(v(t))≤c10,∀t∈R.
By using the definition of v, after passing to the limit as σ→0 in the last inequality, we find
E(ut(t))≤c10,∀t∈R. (3.14)
Considering (3.14) in (1.1), we obtain
/ba∇dblu(t)/ba∇dblH3(Rn)≤c11,∀t∈R.
Thus, the last inequality, together with (3.14), yields
/ba∇dblϕ/ba∇dblH3(Rn)×H2(Rn)≤c12,∀ϕ∈ A,
which completes the proof of the theorem. /squareLONG-TIME DYNAMICS 17
References
[1] A. Pazy, Semigroups of linear operators and application s to partial di erential equations, Springer, New York, 1983 .
[2] T. Cazenave and A. Haraux, An introduction to semilinear evolution equations, Oxford University Press, New York,
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[3] E. Feireisl, Attractors for semilinear damped wave equa tions on R3, Nonlinear Anal., 23 (1994) 187–195.
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unbounded domain, J.Differential Equations, 225 (2006) 528 -548.
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Department of Mathematics, Faculty of Science, Hacettepe U niversity, Beytepe 06800 ,Ankara,
Turkey
E-mail address :azer@hacettepe.edu.tr
Department of Mathematics, Faculty of Science, Hacettepe U niversity, Beytepe 06800 ,Ankara,
Turkey
E-mail address :semasimsek@hacettepe.edu.tr |
1607.04983v3.Magnetic_Skyrmion_Transport_in_a_Nanotrack_With_Spatially_Varying_Damping_and_Non_adiabatic_Torque.pdf | 1
Magnetic Skyrmion Transport in a Nanotrack With Spatially
Varying Damping and Non-adiabatic Torque
Xichao Zhang1,2, Jing Xia1, G. P. Zhao3, Xiaoxi Liu4, and Yan Zhou1
1School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518172, China
2School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China
3College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China
4Department of Information Engineering, Shinshu University, Wakasato 4-17-1, Nagano 380-8553, Japan
Reliable transport of magnetic skyrmions is required for any future skyrmion-based information processing devices. Here we
present a micromagnetic study of the in-plane current-driven motion of a skyrmion in a ferromagnetic nanotrack with spatially
sinusoidally varying Gilbert damping and/or non-adiabatic spin-transfer torque coefficients. It is found that the skyrmion moves in
a sinusoidal pattern as a result of the spatially varying Gilbert damping and/or non-adiabatic spin-transfer torque in the nanotrack,
which could prevent the destruction of the skyrmion caused by the skyrmion Hall effect. The results provide a guide for designing
and developing the skyrmion transport channel in skyrmion-based spintronic applications.
Index Terms —magnetic skyrmions, racetrack memories, micromagnetics, spintronics.
I. I NTRODUCTION
Magnetic skyrmions are quasiparticle-like domain-wall
structures with typical sizes in the sub-micrometer regime [1]–
[7]. They are theoretically predicted to exist in magnetic metals
having antisymmetric exchange interactions [8], and confirmed
by experiments [9], [10] just after the turn of the twenty-
first century. Isolated skyrmions are expected to be used to
encode information into bits [11], which might lead to the
development of novel spintronic applications, such as the
racetrack memories [12]–[19], storage devices [20]–[22], and
logic computing devices [23].
The write-in and read-out processes of skyrmions in thin
films are realizable and controllable at low temperatures [24]–
[26]. A recent experiment has realized the current-induced
creation and motion of skyrmions in Ta/CoFeB/TaO trilayers
at room temperature [27]. Experimental investigations have
also demonstrated the increased stability of skyrmions in mul-
tilayers [28]–[30], which makes skyrmions more applicable to
practical room-temperature applications.
However, the skyrmion experiences the skyrmion Hall effect
(SkHE) [31], [32], which drives it away from the longitudinal
direction when it moves in a narrow nanotrack. As a con-
sequence, in the high-speed operation, the transverse motion
of a skyrmion may result in its destruction at the nanotrack
edges [18], [33]–[36]. Theoretical and numerical works have
proposed several intriguing methods to reduce or eliminate the
detrimental transverse motion caused by the SkHE. For ex-
ample, one could straightforwardly enhance the perpendicular
magnetic anisotropy near the nanotrack edges to better confine
the skyrmion motion [33]. An alternative solution is to trans-
port skyrmions on periodic substrates [37]–[40], where the
skyrmion trajectory can be effectively controlled. Moreover, by
constructing antiferromagnetic skyrmions [34], [35] and anti-
The first two authors contributed equally to this work. Corre-
sponding authors: X. Liu (email: liu@cs.shinshu-u.ac.jp) and Y . Zhou
(email: zhouyan@cuhk.edu.cn).ferromagnetically exchange-coupled bilayer skyrmions [18],
[36], the SkHE can be completely suppressed. Recently, it is
also found that the skyrmionium can perfectly move along
the driving force direction due to its spin texture with a zero
skyrmion number [41], [42].
In this paper, we propose and demonstrate that a skyrmion
guide with spatially sinusoidally varying Gilbert damping
and/or non-adiabatic spin-transfer torque (STT) coefficients
can be designed for transporting skyrmions in a sinusoidal
manner, which is inspired by a recent study on the magnetic
vortex guide [43], where the vortex core motion is controlled
via spatially varying Gilbert damping coefficient. The results
provide a guide for designing and developing the skyrmion
transport channel in future spintronic devices based on the
manipulation of skyrmions.
II. M ETHODS
Our simulation model is an ultra-thin ferromagnetic nan-
otrack with the length land the width w, where the thick-
ness is fixed at 1nm. We perform the simulation using
the standard micromagnetic simulator, i.e., the 1.2 alpha 5
release of the Object Oriented MicroMagnetic Framework
(OOMMF) [44]. The simulation is accomplished by a set of
built-in OOMMF extensible solver (OXS) objects. We employ
the OXS extension module for modeling the interface-induced
antisymmetric exchange interaction, i.e., the Dzyaloshinskii-
Moriya interaction (DMI) [45]. In addition, we use the updated
OXS extension module for simulating the in-plane current-
induced STTs [46]. The in-plane current-driven magnetization
dynamics is governed by the Landau-Lifshitz-Gilbert (LLG)
equation augmented with the adiabatic and non-adiabatic
STTs [44], [47]
dM
dt=
0MHeff+
MS(MdM
dt) (1)
+u
M2
S(M@M
@xM) u
MS(M@M
@x);arXiv:1607.04983v3 [cond-mat.mes-hall] 15 Dec 20162
Fig. 1. (a) The magnetic damping coefficient (x)and non-adiabatic STT
coefficient (x)as functions of xin the nanotrack. (b) Trajectories of current-
driven skyrmions with = =2 = 0:15, = = 0:3, and = 2 = 0:6.
Dot denotes the skyrmion center. Red cross indicates the skyrmion destruction.
(c) Skyrmion Hall angle as a function of xfor skyrmion motion with =
=2 = 0:15, = = 0:3, and = 2 = 0:6. The dashed lines indicate
=14. (e) Real-space top-views of skyrmion motion with = =2 =
0:15, = = 0:3, and = 2 = 0:6.wandvdenote the nanotrack
width and velocity direction, respectively. The dashed line indicates the central
line of the nanotrack. The skyrmion is destroyed at t= 870 ps when =
2 = 0:6. The out-of-plane magnetization component is represented by the
red ( z)-white ( 0)-green ( +z) color scale.
where Mis the magnetization, MSis the saturation magne-
tization,tis the time,
0is the Gilbert gyromagnetic ratio,
is the Gilbert damping coefficient, and is the strength of
the non-adiabatic STT. The adiabatic STT coefficient is given
byu, i.e., the conduction electron velocity. The effective field
Heffis expressed as
Heff= 1
0@E
@M; (2)
where0is the vacuum permeability constant. The average
energy density Econtains the exchange, anisotropy, demag-
netization, and DMI energies, which is given as
E=A[r(M
MS)]2 K(nM)2
M2
S 0
2MHd(M) (3)
+D
M2
S(Mz@Mx
@x+Mz@My
@y Mx@Mz
@x My@Mz
@y);
whereA,K, andDare the exchange, anisotropy, and DMI
energy constants, respectively. nis the unit surface normal
vector, and Hd(M)is the demagnetization field. Mx,My
andMzare the three Cartesian components of M.
The model is discretized into tetragonal volume elements
with the size of 2nm2nm1nm, which ensures a
good compromise between the computational accuracy and ef-
ficiency. The magnetic parameters are adopted from Refs. [14],
Fig. 2. (a)vx, (b)vy, and (c) as functions of and given by Eq. (11)
and Eq. (12), respectively. vxandvyare reduced by u.
[23]:
0= 2:211105m/(As),A= 15 pJ/m,D= 3mJ/m2,
K= 0:8MJ/m3,MS= 580 kA/m. In all simulations, we
assumeu= 100 m/s andw= 50 nm. The skyrmion is initially
located at the position of x= 100 nm,y= 25 nm.
The Gilbert damping coefficient is defined as a function
of the longitudinal coordinate xas follows [Fig. 1(a)]
(x) = ampf1 + sin [2(x= )]g+ min; (4)
where amp= ( max min)=2is the amplitude of the
function. maxand minstand for the maximum and mini-
mum values of the function, respectively. denotes the
wavelength of the function. It is worth mentioning that the
spatially varying can be achieved by gradient doping of
lanthanides impurities in ferromagnets [43], [48], [49]. Exper-
iments have found that is dependent on the interface [50].
Thus it is also realistic to construct the varying by techniques
such as interface engineering. Indeed, as shown in Ref. [51],
local control of in a ferromagnetic/non-magnetic thin-film
bilayer has been experimentally demonstrated by interfacial
intermixing induced by focused ion-beam irradiation.
In a similar way, the non-adiabatic STT coefficient is
also defined as a function of the longitudinal coordinate xas
follows [Fig. 1(a)]
(x) = ampf1 + sin [2(x= ) ']g+ min;(5)
where amp= ( max min)=2is the amplitude of the
function. maxand minstand for the maximum and minimum
values of the function, respectively. and'denote the
wavelength and phase of the function, respectively. Since
the value of depends on the material properties [52], it is
expected to realize the spatial varying by constructing a
superlattice nanotrack using different materials, similar to the
model given in Ref. [43]. Note that the effect of varying
has also been studied in spin torque oscillators [53].
III. R ESULTS
A. Nanotrack with spatially uniform and
We first recapitulate the in-plane current-driven skyrmion
motion in a nanotrack with spatially uniform and . As
shown in Fig. 1(b), the skyrmion moves along the central line
of the nanotrack when = = 0:3. However, due to the
SkHE, it shows a transverse shift toward the upper and lower
edges when = 2 = 0:6and = =2 = 0:15, respectively.3
The skyrmion is destroyed by touching the upper edge when
= 2 = 0:6att= 870 ps.
The skyrmion Hall angle , which characterizes the trans-
verse motion of the skyrmion caused by the SkHE, is defined
as
= tan 1(vy=vx): (6)
Figure 1(c) shows as a function of xfor the skyrmion motion
with = =2 = 0:15, = = 0:3, and = 2 = 0:6. It
can be seen that = 0when = = 0:3, indicating
the moving skyrmion has no transverse motion [Fig. 1(d)].
When = =2 = 0:15,increases from 15to0,
indicating the moving skyrmion has a transverse shift toward
the lower edge which is balanced by the transverse force due to
the SkHE and the edge-skyrmion repulsive force [Fig. 1(d)].
When = 2 = 0:6,decreases from 15to3within
870 ps, indicating the moving skyrmion shows a transverse
motion toward the upper edge. At t= 870 ps, the skyrmion
is destroyed as it touches the upper edge of the nanotrack
[Fig. 1(d)]. It should be noted that the skyrmion profile is
rigid before it touches the nanotrack edge. In order to better
understand the transverse motion caused by the SkHE, we also
analyze the in-plane current-driven skyrmion motion using the
Thiele equation [54]–[57] by assuming the skyrmion moves in
an infinite film, which is expressed as
G(v u) +D( u v) =0; (7)
where G= (0;0; 4Q)is the gyromagnetic coupling vector
with the skyrmion number
Q=1
4Z
m@m
@x@m
@y
dxdy: (8)
m=M=M Sis the reduced magnetization and Dis the
dissipative tensor
D= 4DxxDxy
DyxDyy
: (9)
u= (u;0)is the conduction electron velocity, and vis the
skyrmion velocity. For the nanoscale skyrmion studied here,
we have
Q= 1;Dxx=Dyy= 1;Dxy=Dyx= 0: (10)
Hence, the skyrmion velocity is given as
vx=u( + 1)
2+ 1; vy=u( )
2+ 1: (11)
The skyrmion Hall angle is thus given as
= tan 1(vy=vx) = tan 1
+ 1
: (12)
By calculating Eq. (11), we show vxas functions of and
in Fig. 2(a). vxranges between 0:5uand1:21u, indicating the
skyrmion always moves in the +xdirection. When = 0:42
and = 1,vxcan reach the maximum value of vx= 1:21u.
Similarly, we show vyas functions of and in Fig. 2(b).
vyranges between 0:5uandu, indicating the skyrmion can
move in both the ydirections. When < ,vy>0, the
skyrmion shows a positive transverse motion, while when >
,vy<0, the skyrmion shows a negative transverse motion.
Fig. 3. (a) Trajectories of current-driven skyrmions with amp =
0:315;0:225;0:215. = 2wand = 0:3. (b) as a function of x
for skyrmion motion with amp= 0:315;0:225;0:215. = 2wand
= 0:3. (c) Trajectories of current-driven skyrmions with =w;2w;4w.
amp= 0:225 and = 0:3. (d)as a function of xfor skyrmion motion
with =w;2w;4w. amp= 0:225 and = 0:3.
Fig. 4. (a) Trajectories of current-driven skyrmions with amp =
0:315;0:225;0:215. = 2w,'= 0, and = 0:3. (b) as a function
ofxfor skyrmion motion with amp= 0:315;0:225;0:215. = 2w,
'= 0 , and = 0:3. (c) Trajectories of current-driven skyrmions with
=w;2w;4w. amp= 0:225,'= 0, and = 0:3. (d)as a function
ofxfor skyrmion motion with =w;2w;4w. amp= 0:225,'= 0, and
= 0:3.
By calculating Eq. (12), we also show as functions of
and in Fig. 2(c), where varies between = 45and
= 45. Obviously, one has = 0,<0, and >0
for = , > , and < , respectively, which agree with
the simulation results for the nanotrack when the edge effect
is not significant, i.e., when the skyrmion moves in the interior
of the nanotrack. For example, using Eq. (12), the skyrmion
has = 14and = 14for = 2 = 0:6and =
=2 = 0:15, respectively, which match the simulation results
att0ps where the edge effect is negligible [Fig. 1(c)].
B. Nanotrack with spatially varying or
We first demonstrate the in-plane current-driven skyrmion
motion in a nanotrack with spatially varying and spatially
uniform , i.e., is a function of x, as in Eq. (4), and =
0:3. Figure 3(a) shows the trajectories of the current-driven
skyrmions with different (x)functions where = 2wand
= 0:3. For max= 0:75, min= 0:12, i.e., amp= 0:315,
the skyrmion moves in the rightward direction in a sinusoidal
pattern. For max= 0:6, min= 0:15, i.e., amp= 0:225, the
maximum transverse shift of skyrmion is reduced in compared
to that of amp= 0:315. For max= 0:45, min= 0:2, i.e.,
amp= 0:125, the amplitude of the skyrmion trajectory further4
Fig. 5. Trajectories of current-driven skyrmions with '= 02. amp=
amp= 0:225 and = = 2w.
decreases. as a function of xcorresponding to Fig. 3(a) for
different (x)functions are given in Fig. 3(b). Figure 3(c)
shows the trajectories of the current-driven skyrmions with
different where amp= 0:225and = 0:3.as a function
ofxcorresponding to Fig. 3(c) for different are given in
Fig. 3(d).
We then investigate the in-plane current-driven skyrmion
motion in a nanotrack with spatially uniform and spatially
varying , i.e., is a function of x, as in Eq. (5), and =
0:3. Figure 4(a) shows the trajectories of the current-driven
skyrmions with different (x)functions where = 2w,'=
0and = 0:3. The results are similar to the case with spatially
varying . For max= 0:75, min= 0:12, i.e., amp= 0:315,
the skyrmion moves in the rightward direction in a sinusoidal
pattern. For max= 0:6, min= 0:15, i.e., amp= 0:225, the
maximum transverse shift of skyrmion is reduced in compared
to that of amp= 0:315. For max= 0:45, min= 0:2, i.e.,
amp= 0:125, the amplitude of the skyrmion trajectory further
decreases. as a function of xcorresponding to Fig. 4(a) for
different (x)functions are given in Fig. 4(b). Figure 4(c)
shows the trajectories of the current-driven skyrmions with
different where amp= 0:225and = 0:3.as a function
ofxcorresponding to Fig. 4(c) for different are given in
Fig. 4(d).
From the skyrmion motion with spatially varying or
spatially varying , it can be seen that the amplitude of
trajectory is proportional to ampor amp. The wavelength of
trajectory is equal to ; , while the amplitude of trajectory is
proportional to ; .also varies with xin a quasi-sinusoidal
manner, where the peak value of (x)is proportional to amp,
amp, and ; . As shown in Fig. 2(c), when is fixed at a
value between maxand min, larger ampwill lead to larger
peak value of (x). On the other hand, a larger ; allows
a longer time for the skyrmion transverse motion toward a
certain direction, which will result in a larger amplitude of
trajectory as well as a larger peak value of (x).
Fig. 6. as a function of xfor skyrmion motion with '= 02.
amp= amp= 0:225 and = = 2w.
C. Nanotrack with spatially varying and
We also demonstrate the in-plane current-driven skyrmion
motion in a nanotrack with both spatially varying and ,
i.e., both and are functions of x, as given in Eq. (4) and
Eq. (5), respectively.
Figure 5 shows the trajectories of the current-driven
skyrmions with spatially varying and where amp=
amp= 0:225and = = 2w. Here, we focus on the effect
of the phase difference between the (x)and (x)functions.
For'= 0 and'= 2, as the (x)function is identical to
the (x)function, the skyrmion moves along the central line
of the nanotrack. For 0<'< 2, as (x)could be different
from (x)at a certainx, it is shown that the skyrmion moves
toward the right direction in a sinusoidal pattern, where the
phase of trajectory is subject to '. Figure 6 shows as a
function of xcorresponding to Fig. 5 for '= 02where
amp= amp= 0:225 and = = 2w. It shows that
= 0when'= 0 and'= 2, while it varies with xin
a quasi-sinusoidal manner when 0<'< 2. The amplitude
of trajectory as well as the peak value of (x)reach their
maximum values when '=.
IV. C ONCLUSION
In conclusion, we have shown the in-plane current-driven
motion of a skyrmion in a nanotrack with spatially uniform
and , where is determined by and , which can vary
between = 45and = 45in principle. Then, we
have investigated the in-plane current-driven skyrmion motion
in a nanotrack with spatially sinusoidally varying or .
The skyrmion moves on a sinusoidal trajectory, where the
amplitude and wavelength of trajectory can be controlled by
the spatial profiles of and . The peak value of (x)is
proportional to the amplitudes and wavelengths of (x)and
(x). In addition, we have demonstrated the in-plane current-
driven skyrmion motion in a nanotrack having both spatially
sinusoidally varying and with the same amplitude and
wavelength. The skyrmion moves straight along the central5
line of the nanotrack when (x)and (x)have no phase
difference, i.e., '= 0. When'6= 0, the skyrmion moves
in a sinusoidal pattern, where the peak value of (x)reaches
its maximum value when '=. This work points out the
possibility to guide and control skyrmion motion in a nan-
otrack by constructing spatially varying parameters, where the
destruction of skyrmion caused by the SkHE can be prevented,
which enables reliable skyrmion transport in skyrmion-based
information processing devices.
ACKNOWLEDGMENT
X.Z. was supported by JSPS RONPAKU (Dissertation
Ph.D.) Program. G.P.Z. was supported by the National Natural
Science Foundation of China (Grants No. 11074179 and No.
10747007), and the Construction Plan for Scientific Research
Innovation Teams of Universities in Sichuan (No. 12TD008).
Y .Z. was supported by the Shenzhen Fundamental Research
Fund under Grant No. JCYJ20160331164412545.
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1701.08076v2.Structural_scale__q__derivative_and_the_LLG_Equation_in_a_scenario_with_fractionality.pdf | Structural scale q derivative and the LLG-Equation in a scenario with
fractionality
J.Weberszpil
Universidade Federal Rural do Rio de Janeiro, UFRRJ-IM/DTL and
Av. Governador Roberto Silveira s/n- Nova Iguaçú, Rio de Janeiro, Brasil, 695014.
J. A. Helayël-Netoy
Centro Brasileiro de Pesquisas Físicas-CBPF-Rua Dr Xavier Sigaud 150, and
290-180, Rio de Janeiro RJ Brasil.
(Dated: November 5, 2018)
In the present contribution, we study the Landau-Lifshitz-Gilbert equation
with two versions of structural derivatives recently proposed: the scale
q derivative in the non-extensive statistical mechanics and the axiomatic met-
ric derivative, which presents Mittag-Leffler functions as eigenfunctions. The
use of structural derivatives aims to take into account long-range forces, possi-
ble non-manifest or hidden interactions and the dimensionality of space. Hav-
ing this purpose in mind, we build up an evolution operator and a deformed
version of the LLG equation. Damping in the oscillations naturally show up
without an explicit Gilbert damping term.
Keywords: Structural Derivatives, Deformed Heisenberg Equation, LLG Equation, Non-extensive
Statistics, Axiomatic Deformed Derivative
I. INTRODUCTION
In recent works, we have developed connections and a variational formalism to treat deformed or metric
derivatives, considering the relevant space-time/ phase space as fractal or multifractal [1] and presented a
Electronic address: josewebe@gmail.com
yElectronic address: helayel@cbpf.br arXiv:1701.08076v2 [math-ph] 28 Feb 20172
variational approach to dissipative systems, contemplating also cases of a time-dependent mass [2].
The use of deformed-operators was justified based on our proposition that there exists an intimate
relationship between dissipation, coarse-grained media and a limit energy scale for the interactions. Con-
cepts and connections like open systems, quasi-particles, energy scale and the change in the geometry
of space–time at its topological level, nonconservative systems, noninteger dimensions of space–time con-
nected to a coarse-grained medium, have been discussed. With this perspective, we argued that deformed
or, we should say, Metric or Structural Derivatives, similarly to the Fractional Calculus (FC), could allows
us to describe and emulate certain dynamics without explicit many-body, dissipation or geometrical terms
in the dynamical governing equations. Also, we emphasized that the paradigm we adopt was different
from the standard approach in the generalized statistical mechanics context [3–5], where the modification
of entropy definition leads to the modification of the algebra and, consequently, the concept of a derivative
[1, 2]. This was set up by mapping into a continuous fractal space [6–8] which naturally yields the need
of modifications in the derivatives, that we named deformed or, better, metric derivatives [1, 2]. The
modifications of the derivatives, accordingly with the metric, brings to a change in the algebra involved,
which, in turn, may lead to a generalized statistical mechanics with some adequate definition of entropy.
The Landau-Lifshitz-Gilbert (LLG) equation sets out as a fundamental approach to describe physics
in the field of Applied Magnetism. It exhibits a wide spectrum of effects stemming from its non-linear
structure, and its mathematical and physical consequences open up a rich field of study. We pursue the
investigation of the LLG equation in a scenario where complexity may play a role. The connection between
LLG and fractionality, represented by an deformation parameter in the deformed differential equations,
has not been exploited with due attention. Here, the use of metric derivatives aims to take into account
long-range forces, possible non-manifest or hidden interactions and/or the dimensionality of space.
In this contribution, considering intrinsically the presence of complexity and possible dissipative effects,
and aiming to tackle these issues, we apply our approach to study the LLG equation with two metric
or structural derivatives, the recently proposed scale q derivative [2] in the nonextensive statistical
mechanics and, as an alternative, the axiomatic metric derivative (AMD) that has the Mittag-Leffler
function as eigenfunction and where deformed Leibniz and chain rule hold - similarly to the standard
calculus - but in the regime of low-level of fractionality. The deformed operators here are local. We3
actually focus our attention to understand whether the damping in the LLG equation can be connected
to some entropic index, the fractionality or even dimensionality of space; in a further step, we go over
into anisotropic Heisenberg spin systems in (1+1) dimensions with the purpose of modeling the weak
anisotropy effects by means of some representative parameter, that depends on the dimension of space or
the strength of the interactions with the medium. Some considerations about an apparent paradox in the
magnetization or angular damping is given.
Our paper is outlined as follows: In Section 2, we briefly present the scale q derivative in a nonex-
tensive context, building up the q deformed Heisenberg equation and applying to tackle the problem of
the LLG equation; in Section 3, we apply the axiomatic derivative to build up the deformed Heisen-
berg equation and to tackle again the problem of LLG equation. We finally present our Conclusions and
Outlook in Section 4.
II. APPLYING SCALE q DERIVATIVE IN A NONEXTENSIVE CONTEXT
Here, in this Section, we provide some brief information to recall the main forms of scale q derivative.
The readers may see ref. [1, 2, 6] for more details.
Some initial claims here coincide with our work of Refs. [1, 2] and the approaches here are in fact based
on local operators [1].
The local differential equation,
dy
dx=yq; (1)
with convenient initial condition, yields the solution given by the q-exponential, y=eq(x)[3–5].
The key of our work here is the Scale q derivative (Sq-D) that we have recently defined as
D
(q)f(x)[1 + (1 q)x]df(x)
dx: (2)
The eigenvalue equation holds for this derivative operator, as the reader can verify:
D
(q)f(x) =f(x): (3)4
A.q deformed Heisenberg Equation in the Nonextensive Statistics Context
With the aim to obtain a scale q deformed Heisenberg equation, we now consider the scale q
derivative [2]
dq
dtq= (1 + (1 q)xd
dx(4)
and the Scale - q Deformed Schrödinger Equation [2],
i~D
q;t = ~2
2mr2 V =H ; (5)
that, as we have shown in [2], is related to the nonlinear Schrödinger equation referred to in Refs. [10]
as NRT-like Schrödinger equation (with q=q0 2compared to the q index of the reference) and can be
thought as resulting from a time scale q deformed-derivative applied to the wave function .
Considering in eq.(5), (~r;t) =Uq(t;t0) (~r;t0), theq evolution operator naturally emerges if we take
into account a time scale q deformed-derivative (do not confuse with formalism of discrete scale time
derivative):
Uq(t;t0) =e( i
~MqHqt)
q: (6)
Here,Mqis a constant for dimensional regularization reasons. Note that the q-deformed evolution
operator is neither Hermitian nor unitary, the possibility of a q unitary asUy
q(t;t0)
qUq(t;t0) =1could
be thought to come over these facts. In this work, we assume the case where the commutativity of Uqand
Hholds, but the q unitarity is also a possibility.
Now, we follow similar reasonings that can be found in Ref.[12] and considering the Sq-D.
So, with these considerations, we can now write a nonlinear Scale q deformed Heisenberg Equation
as
D
t;q^A(t) = i
~Mq[^A;H]; (7)
where we supposed that UqandHcommute and Mqis some factor only for dimensional equilibrium.5
B.q deformed LLG Equation
To build up the scale q deformed Landau-Lifshitz-Gilbert Equation, we consider eq.(7), with ^A(t) =
^Sq
D
t;q^Sq(t) = i
~Mq[^Sq;H]; (8)
where we supposed that UqandHcommute.
H= gqB
~Mq^Sq~Heff: (9)
Here,~Heffis some effective Hamiltonian whose form that we shall clearly write down in the sequel.
The scale q deformed momentum operator is here defined as bp
q0= i~Mq0[1 +(1 q0)x]@q
@xq:
Considering this operator, we obtain a deformed algebra, here in terms of commutation relation between
coordinate and momentum
^xq
i;^pq
j
={[1 +(1 q0)x]~Mq0{jI (10)
and, for angular momentum components, as
h
^Lq
i;^Lq
ji
={[1 +(1 q0)x]~Mq^Lq
k: (11)
Theq0factor in ^xq0
{;^pq0
j;^Lq0
i;^Lq0
j;Mq0is only an index and qis not necessarily equal to q0.
The resulting scale q deformed LLG equation can now be written as
D
t;q^Sq(t) = [1 +(1 q0)x]gqB
~Mq^Sq~Heff: (12)
Take ^mq
q^Sq;
q0[1+(1 q0)x]gqB
~Mq.
If we consider that the spin algebra is nor affected by any emergent effects, we can take q0= 1.
Considering the eq.(7) with ^A(t) = ^Sqand ^mq=j
qj^Sqandq0= 1; we obtain the q time deformed
LLG dynamical equation for magnetization as
D
t;q^mq(t) = j
j^mq~Heff: (13)6
Considering ~Heff=H0^k;we have the solution:
mx;q=cosq(0) cosq(
H0t) +sinq(0) sinq(
H0t): (14)
In the figure, 0= 0:
Figure 1: Increase/Damping- cosq(x)
.
III. APPLYING AXIOMATIC DERIVATIVE AND THE DEFORMED HEISENBERG
EQUATION
Now, to compare results with two different local operators, we apply the axiomatic metric derivative.
Following the steps on [12] and considering the axiomatic MD [13], there holds the eigenvalue equation
D
xE (x ) =E (x );whereE (x )is the Mittag-Leffler function that is of crucial importance to
describe the dynamics of complex systems. It involves a generalization of the exponential function and
several trigonometric and hyperbolic functions. The eigenvalue equation above is only valid if we consider
very close to 1:This is what we call low-level fractionality [13]. Our proposal is to allow the use o Leibniz
rule, even if it would result in an approximation. So, we can build up an evolution operator:
U (t;t0) =E ( i
~ Ht ); (15)
and for the deformed Heisenberg Equation
D
tAH
(t) = i
~ [AH
;H]; (16)7
where we supposed that U andHcommute.
To build up the deformed Landau-Lifshitz-Gilbert Equation, we use the eq. (16), and considering and
spin operator ^S (t), in such a way that we can write the a deformed Heisenberg equation as
D
t^S (t) = i
~ [^S ;H]; (17)
whith
H= g B
~ ^S ~Heff: (18)
Here,~Heffis some effective Hamiltonian whose form that we will turn out clear forward.
Now, consider the deformed momentum operator as [9, 11, 12]
bp = i(~) Mx; @
@x : (19)
Takingthisoperator, weobtainadeformedalgebra, hereintermsofcommutationrelationforcoordinate
and momentum
^x
i;^p
j
={ ( + 1)~ M {jI (20)
and for angular momentum components as
h
^L
i;^L
ji
={ ( + 1)~ M ^L
k: (21)
The resulting the deformed LLG equation can now be written as
J
0D
t^S (t) = M ( + 1)g B
~ ^S ~Heff: (22)
If we take ^m
^S ,
M ( +1)g B
~ , we can re-write the equation as the deformed LLG
J
0D
t^m (t) = j
j^m ~Heff; (23)
with~Heff=H0^k. We have the Solution of eq.(23):
m x=Acos0E2 ( !2
0t2 ) +Asin0:x:E 2 ;1+ ( !2
0t2 ): (24)8
In the figure below, the reader may notice the behavior of the magnetization, considering 0= 0.
Figure 2: a) Damping of oscillations. In the figure = 1. b) Increase of oscillations
.
For = 1;the solution reduces to mx=Acos(!0t+0), the standard Simple Harmonic Oscillator
solution for the precession of magnetization.
The presence of complex interactions and dissipative effects that are not explicitly included into the
Hamiltonian can be seen with the use of deformed metric derivatives. Without explicitly adding up
the Gilbert damping term, the damping in the oscillations could reproduce the damping described by
the Gilbert term or could it disclose some new extra damping effect. Also, depending on the relevant
parameter, the q entropic parameter or for , the increasing oscillations can signally that it is sensible
to expect fractionality to interfere on the effects of polarized currents as the Slonczewski term describes.
We point out that there are qualitative similarities in both cases, as the damping or the increasing of the
oscillations, depending on the relevant control parameters. Despite that, there are also some interesting
differences, as the change in phase for axiomatic derivative application case.
Here, we cast some comments about an apparent paradox: If we make, as usually done in the literature
for LLG, the scalar product in eq. (13) with, ^mq;we obtain an apparent paradox that the modulus of
^mqdoes not change. On the other hand, if instead of ^m ;we proceed now with a scalar product with ~Heff
and we obtain thereby the indications that the angle between ^m and~Heffdoes not change. So, how to
explain the damping in osculations for ^mq?This question can be explained by the the following arguments.
Even the usual LLG equation, with the term of Gilbert, can be rewritten in a form similar to eq. LLG9
without term of Gilbert. See eq. (2.7) in the Ref. [14]. The effective ~Hefffield now stores information
about the interactions that cause damping. In our case, when carrying out the simulations, we have taken
~Heffas a constant effective field. Here, we can argue that the damping term, eq. (2.8) in Ref. [14] being
small, this would cause the effective field ~Heff= !H(t) + !k( !S !H)to be approximately !H(t). In this
way, the scalar product would make dominate over the term of explicit dissipation. This could, therefore,
explain the possible inconsistency.
IV. CONCLUSIONS AND OUTLOOK
In short:
Here, we tackle the problem of LLG equations considering the presence of complexity and dissipation
or other interactions that give rise to the term proposed by Gilbert or the one by Slonczewski.
With this aim, we have applied scale - q derivative and the axiomatic metric derivative to build up
deformed Heisenberg equations. The evolution operator naturally emerges with the use of each case of the
structural derivatives. The deformed LLG equations are solved for a simple case, with both structural or
metric derivatives.
Also, in connection with the LLG equation, we can cast some final considerations for future investiga-
tions:
Does fractionality simply reproduce the damping described by the Gilbert term or could it disclose
some new effect extra damping?
Is it sensible to expect fractionality to interfere on the effects of polarized currents as the Slonczewski
term describes?
These two points are relevant in connection with fractionality and the recent high precision measure-
ments in magnetic systems may open up a new venue to strengthen the relationship between the fractional
properties of space-time and Condensed Matter systems.
[1] J. Weberszpil, Matheus Jatkoske Lazo and J.A. Helayël-Neto, Physica A 436, (2015) 399–404.10
[2] Weberszpil, J.; Helayël-Neto, J.A., Physica. A (Print), v. 450, (2016) 217-227; arXiv:1511.02835 [math-ph].
[3] C. Tsallis, J. Stat. Phys. 52, (1988) 479-487.
[4] C. Tsallis, Brazilian Journal of Physics, 39, 2A, (2009) 337-356.
[5] C. Tsallis, Introduction to Nonextensive Statistical Mechanics - Approaching a Complex World (Springer,
New York, 2009).
[6] Alexander S. Balankin and Benjamin Espinoza Elizarraraz, Phys. Rev. E 85, (2012) 056314.
[7] A. S. Balankin and B. Espinoza, Phys. Rev. E 85, (2012) 025302(R).
[8] Alexander Balankin, Juan Bory-Reyes and Michael Shapiro, Phys A, in press, (2015)
doi:10.1016/j.physa.2015.10.035.
[9] Weberszpil, J. ; Helayël-Neto, J. A., Advances in High Energy Physics, (2014), p. 1-12.
[10] F. D. Nobre, M. A. Rego-Monteiro, and C. Tsallis, Phys. Rev. Lett. 106, (2011) 140601.
[11] J.Weberszpil, C.F.L.Godinho, A.ChermanandJ.A.Helayël-Neto, In: 7thConferenceMathematicalMethods
in Physics - ICMP 2012, 2012, Rio de Janeiro. Proceedings of Science (PoS). Trieste, Italia: SISSA. Trieste,
Italia: Published by Proceedings of Science (PoS), 2012. p. 1-19.
[12] J. Weberszpil and J. A. Helayël-Neto, J. Adv. Phys. 7, 2 (2015) 1440-1447, ISSN 2347-3487.
[13] J. Weberszpil, J. A. Helayël-Neto, arXiv:1605.08097 [math-ph]
[14] M. Lakshmanan, Phil. Trans. R. Soc. A (2011) 369, 1280–1300 doi:10.1098/rsta.2010.0319 |
2101.09400v2.Oscillation_time_and_damping_coefficients_in_a_nonlinear_pendulum.pdf | Oscillation time and damping coecients in a
nonlinear pendulum
Jaime Arango
March 24, 2022
Abstract
We establish a relationship between the normalized damping co-
ecients and the time that takes a nonlinear pendulum to complete
one oscillation starting from an initial position with vanishing velocity.
We establish some conditions on the nonlinear restitution force so that
this oscillation time does not depend monotonically on the viscosity
damping coecient.
ASC2020: 34C15, 34C25
Keywords. oscillation time, damping, damped oscillations
This paper is dedicated to the memory of Prof. Alan
Lazer (1938-2020), University of Miami. It was my pleasure
to discuss with him some of the results presented here
1 Introduction
The pendulum is perhaps the oldest and fruitful paradigm for the study of
an oscillating system. The apparent regularity of an oscillating mass going to
and fro through the equilibrium position has fascinated the scientists well be-
fore Galileo. There are plenty of mathematical models accounting for almost
any observed behavior of the pendulum's oscillation. From the sheer amount
of the literature on the subject, one would expect that there is no reasonable
question regarding a pendulum that has no been already answered. And
that might be true. Yet, for whatever reason, it is not impossible to take
on a question whose answer does not seem to follow immediately from the
classical sources.
In a typical experimental setup with no noticeable damping, the oscilla-
tions of a pendulum are periodic. Now, if the damping cannot be neglected,
1arXiv:2101.09400v2 [math.CA] 24 Jun 2021we still observe oscillations, even though they are non periodic. However,
we can measure the time spent by a complete oscillation, and this time is a
natural generalization of the period. But, how does depend this oscillation
time on the characteristic of the medium, say on the viscosity of the sur-
rounding atmosphere? It seems that there is no much information on how
the damping a ects the oscillation time. There are plenty of new publica-
tions regarding damping and oscillations, ranging from analytical solutions
([5], [3],[6]), to very clever experimental setups (see for example [4]). The
nature of the damping has been also extensively considered ([8], [2]), but the
dependence of the oscillation time on the damping or on the non-linearity
seems to be less investigated.
For the sake of simplicity we analyze the oscillation time in the frame of a
model that appear in almost any text book of ordinary di erential equations
(see for example [1]):
x+ 2 _x+x(1 +f(x)) = 0; (1)
wherex=x(t) measures the pendulum's deviation with respect to a vertical
axis of equilibrium and 0 denote the viscous damping coecient. The
termxf(x) models the nonlinear part of the restoring force. We've rescaled
the time so that the period of the linear undamped oscillation is exactly 2 .
The math of the solutions x=x(t) is classical. If fis smooth and x0
andv0are given real values, then there exists a unique solution satisfying the
given conditions x(0) =x0and _x(0) =v0:Moreover, if f(0) = 0, then x= 0
is a stable equilibrium solution of (1). As a consequence, x(t) is de ned
for allt0 providedjx0j 1 andjv0j 1. Notice that the points of
vanishing derivative of a solution x=x(t) to (1) are isolated and those points
correspond, either to local maxima or to local minima. Denote by (x0; ) the
amount of time spent (by the mass) completing one oscillation starting from
x0with vanishing velocity ( v0= 0). To be precise, if x=x(t) starts from x0
with vanishing velocity, then xreaches a local maximum at t= 0, and the
oscillation is completed when xreaches the next local maximum. Certainly,
the oscillation time generalizes the period of solutions for the undamped
model ( = 0). In this investigation we analyze the dependence of onx0
and on under the following working hypothesis:
Assumption 1.1. On small neighborhood of 0the function fis even and
for some constant a>0we have
f(x) = ax2+O
jxj4
;
We shall show that for x0 xed, reaches a positive minimum at some
0< 0<1:It does not seem obvious that an increase in the damping
20.0 0.1 0.2 0.3 0.4 0.5
Damping coefficient α2π6.46.97.5τ(x0,α,0)
x0= 0.1
x0= 0.6
x0= 1.2
x0= 1.6Figure 1: Numerical simulation (x0; ) depending on for several values of
x0. The nonlinear term fwas chosen so that x(1 +f(x)) = sinx:
coecient might cause a decrease in . It is also worth noticing that the
existence of a minimum of is a consequence the sign of the constant ain
the above assumption. Indeed, according to numerical experiments carried
out by the author, does not reach a positive minimum if a < 0. The
author is not aware of a similar result in the current literature nor whether
this phenomena has been experimentally addressed. The whole paper was
written with the aim at the mathematical pendulum x(1 +f(x)) = sinx:
In that case, Figure 1 summarize our ndings by picturing the numerically
simulated value for (x0; ). Interestingly, our qualitative analysis accurately
re
ects variations of that are not easy to spot numerically. For instance,
the minimum of (x0; ) forx0= 0:1 is not evident in Figure 1.
The arguments and proofs in this paper are entirely based on well estab-
lished techniques of ODE theory. However, the main result (Theorem 3.1)
rests on delicate estimates involving a di erential equation describing the
dependence of the solution x=x(t) with respect to .
2 Underdamped oscillations
De nitions of underdamped oscillations in linear systems naturally carry over
to solutions of (1). From now on, x(;x0; ) stands for the unique solution
to (1) satisfying the initial condition x(0) =x0and _x(0) = 0:We also write
(x0; ) to highlight the dependence of the oscillation time on x0and . We
3will write simply orxwhen no confusion can arise. It is convenient to
represent (1) in the phase space ( x;v) with _x=v:
_x=v
_v= 2 v x xf(x):(2)
Equation (2) is explicitly solvable whenever f0, and in that case, its
solution is given by
xl(t) =e t
!(!cos!t+ sin!t)x0
vl(t) = e t
!sin!tx 0(3)
where!=p
1 2. Moreover, the oscillation time lis given by
l=2
!=2p
1 2:
Notice that lis an increasing function that solely depends on .
Though a closed-form solution of (1) is either not known or impractical,
we could express the relevant solutions implicitly. To that end, we rewrite (2)
so that the nonlinear term xf(x) assumes the role of a non homogeneous
forcing term. The expression for the solution ( x;v) is implicitly given by
x(t) =xl(t) 1
!Zt
0e (t s)sin!(t s)x(s)f(x(s))ds
v(t) =vl(t) 1
!Zt
0e (t s)(!cos!(t s) sin!(t s))x(s)f(x(s))ds
(4)
Next, we estimate the solutions of (2) in the conservative case ( = 0) in
which all solutions are periodic and the period is given by (x0;0):
Lemma 2.1. If(x;v)stands for the solution to (2)with = 0that satis es
(x(0);v(0)) = (x0;0), then there exists >0so that for alljx0jand all
0t we have
x(t) =x0cost+R1(t;x0); v(t) = x0sint+R2(t;x0); (5)
where
jRi(t;x0)jconstjx3
0j; i = 1;2:
4Proof. Letting = 0 in (4) we obtain
R1(t;x0) = Zt
0cos(t s)x(s)f(x(s))ds: (6)
Since (0;0) is a stable equilibrium solution to (2), there exists >0 and>0
so that any solution ( x;v) to (2) starting at ( x0;0), withjx0jsatis es
jx(t)j. Now write F(z) = zf(z) and notice that for some 2( ;)
we have
F(x(s)) =F(x0coss+R1(s;x0)) =F(x0coss) +R1(s;x0)F0():
Next, identity (6), Assumption (1.1) and some standard estimations yield
jR1(t;x0)j2ajx3
0j+c2Zt
0jR1(s;x0)jds
wherec2= maxz2[ ;]jF0(z)j. The rst claim follows now from Gronwall's
inequality. The proof of the estimation for R2is analogous.
At this point it is appropriated to de ne the half oscillation time ^ =
^ (x0; ) to be the time spent by the solution x(t;x0; ); t0;reaching the
next local minimum. If = 0 andfis even, the symmetry of the solution
(1) yields. 2^ = :
Lemma 2.2. If^ = ^ (x0; )denote the half oscillation time and ais the
constant of Assumption 1.1, then
^ (x0; )>p
1 2and lim
x0!0+^ (x0;0) =+a
8x2
0+o(x3
0):
Proof. We introduce introduce the polar coordinates
r=p
x2+v2;tan=x
v;
to obtain
_=
1 + sin 2+ sin2f(x)
_r= v
r(2 v+xf(x))(7)
As a consequence of equation (7) we obtain the following expression for
the half oscillation time ^ = ^ (x0; )
^ =Z
0d
1 + sin 2+ sin2f(x()): (8)
5Now, the e ect of the nonlinearity on the oscillation time is clear. By As-
sumption 1.1 we obtain
^ (x0; )>Z
0d
1 + sin 2=p
1 2:
For = 0 we use estimation (5) to obtain
^ (x0;0) =Z
0d
1 ax2
0sin2cos2t()+o(x3
0):
Now a straightforwards computation yields
lim
x0!0+^ (x0;0) =;lim
x0!0+@^
@x0(x0;0) = 0:
Now, the expression for@2^
@x2
0(x0;0) is somewhat cumbersome. However, taking
into account that lim x0!0t() =, we readily obtain
lim
x0!0+@2^
@x2
0(x0;0) =Z
02asin2cos2d=2a
8;
and the second claim of the lemma follows by the second order Taylor ex-
pansion of ^ (x0;0) aroundx0
A reasoning analogous to that in the proof of the preceding lemma shows
that
(x0; )>2p
1 2 l:
This inequality is illustrated in Figure 2 when a= 1. Had we considered in
Assumption 1.1 negative values for a, then the inequality would reverse to
(x0; )< las it is depicted in Figure 2.
3 The role of the viscous damping
It is not dicult at all to obtain a di erential equation describing the move-
ment of the pendulum depending on the viscous damping coecient. Indeed,
writing
X(t;x0; ) =@x
@ (t;x0; ); V (t;x0; ) =@v
@ (t;x0; ):
Derivation of equation (2) with respect to yields:
_X=V
_V= 2 V X 2v (xf0(x) +f(x))X:(9)
6As for the initial conditions we have
X(0;x0; ) = 0; V (0;x0; ) = 0:
Let us write G(x) = d
dx(xf(x)). Again, as we did with equation (2),
equation (9) can be seen as a linear homogeneous part plus the forcing term
2v+G(x)X:The solution X;V is implicitly given by
X(t) =1
!Zt
0e (t s)sin!(t s)
2v(s) +G(x(s))X(s)gds
V(t) =1
!Zt
0e (t s)(!cos!(t s) sin!(t s))
2v(s) +G(x(s))X(s)gds
In particular, for = 0 the above expressions reduce to
X(t) =Zt
0sin (t s)
2v(s) +G(x(s))X(s)
ds
V(t) =Zt
0cos (t s)
2v(s) +G(x(s))X(s)
ds(10)
The following lemma does the heavy lifting to deliver the main result of
the paper.
Lemma 3.1. Under Assumption 1.1, if ^ = ^ (x0;0)denotes the half oscil-
lation time when = 0, then for 0<x 0 1we haveV(^ ;x0;0)>0:
Proof. We start with an auxiliary estimate for X(t) in equation (10). By
Lemma (2.1) and by Assumption 1.1, for 0 <twe have
X(t) =x0( tcost+ sint) + 3ax2
0Zt
0sin (t s) cos2sX(s)ds+O(jx0j4)
(11)
Notice that X1(t)x0( tcost+ sint) does not vanish on (0 ;) and that
G(x(s))>0 provided 0 <x 0 1. Further, the initial conditions for X(t) at
t= 0 and equation (9) yield that
X(0) = 0 = _X(0) = X(0) and...
X(0) = 2x0(1 +f(x0))>0;
meaning that X(t) is positive on an interval (0 ;) with>0. We claim that
X(t)>0 for 0< t:On the contrary, there exists < t 0< such that
X(t0) = 0 andX(t)>0 fort2(0;t0). Now, by Lemma 2.2 we know that
7^ > . Therefore, the polar angle (t) in (7) satis es < (t)<0 for all
0<t< and a fortiori v(t)<0 on (0;]. But this is a contradiction to the
rst equation of (10) evaluated at t=t0since fors2(0;t0) we have
sin (t0 s)
2v(s) +G(x(s))X(s)
>0:
Next, by the equation (11) it follows immediately that X(t) =X1(t) +
O(jx0j3). Analogously, for V(t) we obtain
V(t) =x0tsint+ 3ax2
0Zt
0cos (t s) cos2sX1(s)ds+O(jx0j4)
V1(t) +V2(t) +O(jx0j4)
whereV1(t)x0tsint. Now,V2(t) can be explicitly evaluated. For the
reader's convenience, we write the complete expression for V2:
V2(t) =3ax3
0
1
32(6t2+ 5) cost 3
32tsin 3t
1
16tsint 17
128cos 3t+37
128cost
:
Moreover, it is somewhat tedious but straightforward to show that V2is
positive and increasing on a small neighborhood of . By Lemma 2.2 ^ > ,
therefore
V2(^ )>V 2() =9ax3
02
16:
Again, by Lemma 2.2 we obtain
V1(^ ) =V1() + (^ )V0
1() +O(jx0j4)
= ax3
02
8+O(jx0j4);
so thatV(^ ) =V1(^ ) +V2(^ )>0:
Now we are in a position to show the main result of the paper
Theorem 3.1. Under Assumption 1.1, there exists a > 0such that for
0< x 0< xed, the oscillation time (x0; ), for 0< < 1, reaches a
positive minimum at some 0< < 1. Moreover,
lim
!1 (x0; ) =1:
8Proof. We let 0<x 0 1 xed by now and denote by ( x;v) be the solution
of equation (2). By de nition of ^ we havev(^ ; ) = 0, so that the Implicit
Function Theorem yields
@^
@ _v(^ ; ) +V(^ ; ) = 0;
therefore
@^
@ =V(^ ; )
x(^ ; ) (1 +f(x(^ ; ))):
Sincex(^ ; ) is negative, it follows from Lemma 3.1 that and@^
@ j =0<0.
Now we shall show that the last inequality holds for the oscillation time .
To do that, write ^ x0= x(^ ( ;x 0);x0) and see that
( ;x 0) = ^ ( ;x 0) + ^ ( ;^x0):
That is to say, the half oscillation time depends on jx0jonly. Notice that
^x0x0and the equality holds in the conservative case = 0 only. Therefore
@
@ ( ;x 0) =@^
@ ( ;x 0) +@^
@ ( ;^x0) @^x0
@ ( ;x 0)@^
@ ( ;x 0) = 0:
Moreover, since
@^x0
@ ( ;x 0) =v(^ ( ;x 0);x0) = 0;
we have that
lim
x0!0+@
@ ( ;x 0) = 2 lim
x0!0+@^x0
@ ( ;x 0)
Finally, by the rst claim of Lemma 2.2, ( ;x 0) must attain a minimum at
some 0< < 1.
4 Conclusions and nal remarks
An oscillating mass exhibits gradually diminishing amplitude in the presence
of damping. The time spent by the mass completing one oscillation depends
on several factors, as the model for the restoring force, how the oscillation
starts, and the nature of the damping. For the sake of our discussion we
consider a vertical pendulum with a nonlinear restoring force resembling the
mathematical pendulum, letting the oscillation start at a small amplitude
with vanishing velocity and a viscous damping model with a (normalized)
viscosity coecient . We have proved that the oscillation time ( )
does not depend monotonically on , meaning that there exists a threshold
9Figure 2: Numerical simulation of the oscillation time depending on
the damping coecient with starting amplitude x0= 0:2 and non linear
restoring term given by f(x) = ax2,a=1. The curve with the round
marker (blue in the online version) corresponds to the oscillation time lof
the linear case f0
0(which depends on the starting amplitude of the oscillation) such that
reaches a local minimum at 0(see Figure 1). It is worth noticing that this
behavior cannot be observed if the restitution force is linear, i. e., what we
report in this paper is essentially a nonlinear phenomenon.
The proof of existence of a positive minimum for the oscillation time rests
heavily on the fact that the constant ain Assumption 1.1 is positive. Just to
experiment the e ect of changing the sign of the constant a, we carried out
some numerical simulations of with the nonlinear term f(x) = ax2for
a= 1; 1. The corresponding equations are particular cases of an unforced
Dung oscillator [7]. The numerical results are shown in Figure 2. Just for
the sake of the numerical experimentation we also considered negative values
for . Ifa= 1 we see that reaches its minimum at a positive value for
. By contrast, if a= 1 no minimum seems to exist. The curve with the
round marker (blue in the online version) corresponds to the oscillation time
of the linear case l=2p
1 2.
The numerical experimentation of the oscillation time (not shown in
this paper) assuming a quadratic damping exhibits the same behavior as
the graphics of Figure 2. If the readers are curious about the numerical
experiments, they could take a look at the author's GitHub page
https://github.com/arangogithub/Oscillation-time
and download a Jupyter notebook with the python code featuring the results
shown in Figures 1 and 2
10Acknowledgment
The author would like to give the reviewer his very heartfelt thanks for
carefully reading the manuscript and for pointing out several inaccuracies of
the document.
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11 |
1307.2648v2.Scaling_of_spin_Hall_angle_in_3d__4d_and_5d_metals_from_Y3Fe5O12_metal_spin_pumping.pdf | 1
Scaling of spin Hall angle in 3 d, 4d and 5d metals from Y3Fe5O12/metal spin
pumping
H. L. Wang†, C. H. Du†, Y . Pu, R. Adur, P. C. Hammel*, F. Y . Yang*
Department of Physics, The Ohio State University, Columbus, OH 43210 ,USA
†These authors made equal contributions to this work
*E-mails:fyyang@physics.osu.edu; hammel@physics.osu.edu
Abstract
We have investigated spin pumping from Y3Fe5O12 thin films into Cu, Ag, Ta, W, Pt and Au
with varying spin-orbit coupling strengths . From measurements of Gilbert damping
enhance ment and inverse spin Hall signals spanning three orders of magnitude , we determine
the spin Hall angles and interfacial spin mixing conductance s for the six metals . For noble
metals Cu, Ag and Au (same d-electron counts) , the spin Hall angles vary as Z4 (Z: atomic
number ), corroborating the role of spin -orbit coupling . In contrast, amongst the four 5 d
metals, the variation of the spin Hall angle is dominated by the sensitivity of the d-orbital
moment to the d-electron count, confirming theoretical predictions.
PACS: 75.47.Lx, 76.50.+g, 75.70.Ak, 61.05.cp
2
Spin pumping of p ure spin c urrent s from a ferromagnet ( FM) into a nonmagnetic
material ( NM) provide s a promising route toward energy -efficient spintronic devices . The
inverse spin Hall effect (ISHE) in FM/Pt bilayer systems [1-13] is the most widely used tool
for detecting s pin currents generated by either ferromagnetic resonance ( FMR ) or a thermal
gradient. The intense interest in spin pumping emphasizes the pressing need for quantitative
understanding of ISHE in normal metals other than Pt [10]. To date, spin Hall angles (SH)
have been measured for several metals and alloys by spin Hall or ISHE measurements,
mostly using metallic FMs [ 14]. Due to current shunting of the metallic FMs and potential
confounding effects of anisotropic magnetoresistance ( AMR ) or anomalous Hall effect
(AHE ), the reported values of SH vary significantly , sometimes by more than one order of
magnitude for the same materials [14]. Here we report a systematic study of FMR spin
pumping from insulating Y3Fe5O12 (YIG) epitaxial thin films grown by sputtering [15-22]
into six normal metal s, Cu, Ag, Ta, W, Pt and Au , that span a wide range in two key
parameters : a factor of ~50 in spin-orbit coupling strength [ 23] and over two orders of
magnitude variation in spin diffusion length (SD) [4, 24-26]. Due to their weak spin-orbit
coupling and relatively long spin diffusion length s, Cu and Ag present a significant challenge
for ISHE detection of spin pumping . ISHE voltages (VISHE) exceeding 5 mV are generated in
our YIG/Ta and YIG /W bilayer s and here we report ~1 V spin pumping signals in YIG/Cu
and YIG /Ag bilayer s. The recently rep orted proximity effect in Pt [ 9, 13 ] should lead to at most
V-level contribution to the measured VISHE, which is negligible compared with the observed
mV-level VISHE in our YIG/Pt bilayer and should not be a factor in other five metals. The large 3
dynamic range that this sensitivity provides and the insulating nature of YIG films enable
quantitative determination of spin mixing conductance s across the YIG/metal interfa ces [5,
12] and spin Hall angle s of these 3d, 4d and 5 d metals .
We characterize the structural quality of our epitaxial YIG films deposited on
(111) -oriented Gd3Ga5O12 (GGG) substrates using off-axis ultrahigh vacuum (UHV)
sputtering [15-22] by high-resolution x -ray diffraction (XRD). A representative θ-2θ scan of a
20-nm YIG film shown in Fig. 1 a demonstrates phase purity and clear Laue oscillations ,
indicat ing high uniformity of the film . We find an out-of-plane lattice constant of the YIG
film, c = 12.393 Å , very close to the bulk value of 12.376 Å . The XRD rocking curve in the
left inset to Fig. 1a give s a full width at half maximum (FWHM) of only 0.0092, near the
resolution limit of our high-resolution XRD, demonstrating the excellent crystalline quality of
the YIG film . The atomic force microscopy (AFM) image in the right inset to Fig. 1a shows a
smooth surface with a roughness of 0.15 nm. Figure 1b shows a representative FMR
derivative absorption spectrum f or a 20-nm YIG film used in this study taken by a Bruker
EPR spectrometer in a cavity at a radio -frequency (rf) f = 9.65 GHz and an input microwave
power Prf = 0.2 mW with a magnetic field H applied in the film plane. The peak -to-peak
linewidth ( H) obtained from the spectrum is 1 1.7 Oe and an effective saturation
magnetization 4π𝑀eff = 1786 ± 36 Oe is extracted from fitt ing the angular dependence of
resona nce field [27]. Due to the small magnetic anisotropy of YIG, the satura tion
magnetization 4𝑀s can be approximated at 1786 Gauss which agrees well with the value
reported for single crystal YIG , indicating the high magnetic quality of our YIG films [28]. In 4
this letter , all six YIG/metal bilayers are made from the same 20 -nm YIG film characterized
in Figs. 1b and 1c.
Our s pin pumping measurements are carried out in the center of the EPR cavity on the
six YIG/ metal bilayer s at room temperature (approximate dimension s of 1.0 mm 5 mm).
The thickness of the metal layers is 5 nm for Ag, Ta, W, Pt, Au and 10 nm for Cu, and all are
made by UHV off-axis sputtering . Resistivity () measurements confirm that the Ta and W
films are -phase [29, 30] (high resistivity , see Table I) . During the spin pumping
measurements, a DC magnetic field H is applied in the xz-plane and the ISHE voltage is
measured across the ~5-mm long metal layer along the y-axis, as illustrated in Fig. 1c. At
resonan ce, the precessing YIG magnetization (M) transfers angular momentum to the
conduction electrons in the normal metal. The resulting pure spin current Js is injected into
the metal layer along the z-axis with spin polarization 𝜎 parallel to M and then converted to a
charge current Jc SHJs𝜎 by the ISHE via the spin-orbit interaction .
Figure 2 show s VISHE vs. H spectra of the six YIG/metal bilayers at θH = 90 and 270
(both with in -plane field) at Prf = 200 mW. For YIG/Ta and YIG /W bilayer s, |VISHE| exceeds 5
mV (1 mV/mm) . For YIG /Pt, YIG/Au and YIG /Ag, VISHE = 2.10 mV, 72.6 V and 1. 49 V,
respectively . Due to the opposi te sign s of SH, Pt, Au and Ag give positive VISHE while Ta and
W give negative VISHE at θH = 90. This agrees with the predicted signs of SH [31, 32] of the
metals . When H is reversed from θH = 90 to 270, all the VISHE signal s change sign as
expected from the ISHE . The rf-power dependenc ies of VISHE are shown in the upper insets to
Figs. 2a -2f at θH = 90, each of which shows a linear dependence, indicat ing that the 5
observed spin pumping signals are in the linear regime. Furthermore, the large spin pumping
signals provided by the YIG films e nable the observation of VISHE = 0.99 V in the YIG/Cu
bilayer (Fig. 2f) . Due to the much weak er spin-orbit coupling [24] in Cu compared to th ose
5d metals , there is no previous report of ISHE detection of spin pumping in FM/Cu structure s.
This first observation of VISHE in YIG/Cu enables the determination of spin Hall angle in Cu .
When H is rotated from in -plane to out -of-plane , M remains essentially parallel to H at
all angles since the FMR resonance field Hres (between 2500 and 5000 Oe ) always exceeds
4Meff (1786 Oe). The lower insets to Figs. 2a-2f show the angular dependenc ies of the
normalized VISHE for the six bilayer s; all clearly exhibit the expected sinusoidal shape (VISHE
Js𝜎 JsM JsH sinθH), confirming that the observed ISHE voltage s arise from FMR
spin pumping. Since YIG is insulating we can rule out artifacts due to thermoelectric or
magnetoelectric effects, such as AMR or AHE , enabling more straightforward measurement
of the inverse spin H all effect than using metallic FMs.
Figure s 3a-3f show the FMR derivative absorption spectr a (f = 9.65 GHz) of the
20-nm YIG film s before and after the deposition of the metal s. The FMR linewi dths are
clearly enhanced in YIG/metal bilayers relative to the bare YIG films . The linewidth
enhancement [10, 11] is a consequence of FMR spin pumping: the coupling that transfer s
angular momentum from YIG to the metal adds to the damping of the precessing YIG
magnetization , thus increas ing the linewidth. In order to accurately determine the
enhance ment of Gilbert damping, we measured the frequency dependenc ies of the linewidth
of a bare YIG film and the six YIG/ metal bilayers using a microstrip transmission line . In all 6
cases the linewidth increases linear ly with frequency (Fig. 3g). The Gilbert damping constant
can be obtained using [ 33],
Δ𝐻=Δ𝐻inh+4𝜋𝛼𝑓
√3𝛾, (1)
where Hinh is the inhomog eneous broadening and is the gyromagnetic ratio . Table I shows
the damping enhancement sp due to spin pumping : sp=YIG/NM−YIG, where YIG/NM
and YIG = (9.1 ± 0.6) 10-4 are obtained from the least-squares fits in Fig. 3g .
The observed ISHE voltages depend on several materials parameters [4, 11],
𝑉ISHE=−𝑒𝜃SH
𝜎N𝑡N+𝜎F𝑡F𝜆SDtanh(𝑡N
2𝜆SD)𝑔↑↓𝑓𝐿𝑃(𝛾ℎrf
2𝛼𝜔)2
, (2)
where e is the electron charge , N (F) is the conductivity of the NM (FM), tN (tF) is the
thickness of the NM (FM) layer, 𝑔↑↓ is the interfacial spin mixing conductance , = 2f is
the FMR frequency , L is the sample length, and hrf = 0.25 Oe [34] in our FMR cavity at Prf =
200 mW. The factor P arises from the ellipticity of the magnetization precession [10],
𝑃=2𝜔[𝛾4𝜋𝑀s+√(𝛾4𝜋𝑀s)2+4𝜔2]
(𝛾4𝜋𝑀s)2+4𝜔2= 1.21 (3)
for all the FMR and spin pumping measurements. The spin mixing conductance can be
determined from the damp ing enhancement [10-12],
𝑔↑↓=4𝜋𝑀s𝑡F
𝑔𝜇B(YIG/NM−YIG) (4)
where 𝑔 and 𝜇B are the Landé 𝑔 factor and Bohr magneton , respectively.
Although the reported spin diffusion length var ies from a few nm to a few hundred
nm across the range of metals we have measured , the term 𝜆SDtanh(𝑡N
2𝜆SD) in Eq. (2) is rather
insensitive to the value of SD for a given tN (e.g., 5 nm ) due to the limitation of film
thickness ; for example, 𝜆SDtanh(𝑡N
2𝜆SD) = 1.70 nm for SD = 2 nm and 2.50 nm for SD = 7
[10]. In this calculation, w e assume 𝜆SD = 10 nm for Pt [4], 2 nm for W and Ta [25], 60 nm
for Au [24], 700 nm for Ag [26] and 500 nm for Cu [24]. Electrical conduction in YIG can be
neglected. From Eqs. (2)-(4), 𝑔↑↓ can be obtained from the Gilbert damping enhancement
and SH can be calculated for the six metals (Table I) [ 35]. Consequently , the spin current
density Js can be estimated using [11],
𝐽s=𝑡N𝜎N+𝑡F𝜎F
𝜃SH𝜆SDtanh(𝑡N
2𝜆SD)𝑉ISHE
𝐿. (5)
The power of inverse spin Hall effect as a probe of spin pumping calls for a
quantitative understanding to enable more precise and detailed experiments. Spin Hall angles
have been measured in several normal metals by spin Hall or ISHE measurements, mostly
using metallic FMs [10, 14, 32, 35]. Due to the impact of AMR or AHE in electrical ly
conducting metallic FMs in the heterostructures , and the variation of sample quality among
different groups, the reported values of SH vary significantly for the same materials, in some
cases, by more than one order of magnitude [ 14]. Here, we report measurements of the spin
Hall angle for various 3d, 4d and 5 d metals using Eq. (2) from the large ISHE signals and,
independently, spin -pumping enhancement of Gilbert damping (to obtain 𝑔↑↓ and ) of the
insulating YIG thin film . This set of experimental data can be compared to discern trends and
uncover the roles of various materials parameters in spin -orbit coupling , including the atomic
number as well as d-electron co unt in transition metals [ 23, 31]. We first show in Fig. 4a the
linear dependence of SH on Z4 for Cu, Ag and Au , reflecting the key role of atomic number
in spin-orbit coupling [23] and spin Hall physics in metals having a particular d-electron
configuration . We note that the spin Hall angles of the four 5 d metals do not vary as Z4 at all , 8
indicating that the d-orbital filling plays the dominant role [ 31]. Figure 4b shows SH vs. Z for
Ta, W, Pt and Au , which matches well with the theoretical calculations by Tanaka et al. [31],
including the sign change and relative magnitude of spin Hall effect in 5d metals . These two
results highlight and distinguish the roles of atomic number and d-orbital filling in spin Hall
physics in transition metals , and clarify their relative importance .
In conclusion , FMR spin pumping measurements on YIG/NM bilayers give mV-level
ISHE voltages in YIG /Pt, YIG/Ta and YIG/W bilayer s and robust spin pumping signals in
YIG/Cu and YIG/Ag . YIG/NM interfacial spin mixing conductance s are determined by the
enhanced Gilbert damping which are measured by frequency dependence of FMR linewi dth
before and after the deposition of metals. The inferred spin Hall angles of the six metals
imply the i mportant roles of atomic number and d-electron configuration in spin Hall physics .
This work is supported by the Center for Emergent Materials at the Ohio State
University, a NSF Materials Research Science and Engineering Center (DMR -0820414)
(HLW, YP, and FYY) and by the Department of Energy through grant DE -FG02 -03ER46054
(RA, PCH). Partial support is provided by Lake Shore Cryogenics Inc. (CHD) and the
NanoSystems Laboratory at the Ohio State University.
9
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13
Table I . ISHE voltages at f = 9.65 GHz and Prf = 200 mW , FMR linewidth changes at f =
9.65 GHz , Gilbert damping enhancement due to spin pumping sp=YIG/NM−YIG
(YIG= 9.1 ± 0.6 10-4) and resistivity of the six YIG/metal bilayers , and the calculated
interfacial spin mixing conductance, spin Hall angle, and spin current density for each metal .
Bilayer VISHE H
change sp ( m) 𝑔↑↓(m-2) SH 𝐽𝑠 (A/m2)
YIG/Pt 2.10
mV 24.3 Oe (3.6 ± 0.3)
10-3 4.8 10-7 (6.9 ± 0.6)
1018 0.10 ± 0.01 (2.0 ± 0.2)
107
YIG/Ta -5.10
mV 16.5 Oe (2.8 ± 0.2)
10-3 2.9 10-6 (5.4 ± 0.5)
1018 -0.069 ± 0.006 (1.6 ± 0.2)
107
YIG/W -5.26
mV 12.3 Oe (2.4 ± 0.2)
10-3 1.810-6 (4.5 ± 0.4)
1018 -0.14 ± 0.01 (1.4 ± 0.1)
107
YIG/Au 72.6
V 5.50 Oe (1.4 ± 0.1)
10-3 4.9 10-8 (2.7 ± 0.2)
1018 0.084 ± 0.007 (7.6 ± 0.7)
106
YIG/Ag 1.49
V 1.30 Oe (2.7 ± 0.2)
10-4 6.6 10-8 (5.2 ± 0.5)
1017 0.0068 ± 0.0007 (1.5 ± 0.1)
106
YIG/Cu 0.99
V 3.70 Oe (8.1 ± 0.6)
10-4 6.3 10-8 (1.6 ± 0.1)
1018 0.0032 ± 0.0003 (4.6 ± 0.4)
106
14
Figure Captions:
Figure 1. (a) Semi -log θ-2θ XRD scan of a 20-nm thick YIG film near the YIG (444) peak
(blue line) , which exhibits clear Laue oscillations corresponding to the film thickness. Left
inset: rocking curve of the YIG film measured at 2θ = 50.639 for the first satellite peak
(green arrow) to the left of the main YIG (444) peak gives a FWHM of 0. 0092. Righ t inset:
AFM image of the YIG film with a roughness of 0.15 nm. (b) A representative
room -temperature FMR derivative spectrum of a 20 -nm YIG film with an in -plane field at Prf
= 0.2 mW, which gives a peak -to-peak linewidth of 11.7 Oe. (c) Schematic of experimental
setup for ISHE voltage measurements.
Figure 2 . VISHE vs. H spectra of (a) YIG/ Pt, (b) YIG/Ta, (c) YIG/W, (d) YIG/Au, (e)
YIG/Ag, and (f) YIG/Cu bilayers at θH = 90(red) and 270 (blue) using Prf = 200mW . Top
insets: rf-power dependencies of the corresponding VISHE at θH = 90. Bottom insets: a ngular
dependenc ies (θH) of VISHE normalized by the magnitude of VISHE at θH = 90, where t he
green curves are sin θH for Pt, Au, Ag, Cu, and -sinθH for Ta and W.
Figure 3 . FMR derivative absorption spectr a of the 20-nm YIG films before (blue) and
after (red) the deposition at f = 9.65 GHz of (a) Pt, (b) Ta, (c) W, (d) Au, (e) Ag, and (f) Cu. (g)
Frequency dependence of peak -to-peak FMR linewi dth of a bare YIG film and the six
YIG/ metal bilayers .
Figure 4. (a) Spin Hall angles as a function of Z4 for Cu, Ag and Au , reflecting the Z4
dependence of SH for noble metals with the same d-orbital filling . (b) Spin Hall angles for 5d
transition metals Ta, W, Pt and Au, of which b oth the signs and relative magnitudes agree
well with the theoretical predictions in Ref. 3 1. 15
Figure 1.
100102104
49 50 51 52 53Intensity (c/s)
2 (deg)(a)
GGG(444)YIG(444)
25.6 25.65
(deg)FWHM=
0.0092o
2400 2600 2800-2-1012dIFMR/dH (a.u.)
H (Oe)H = 11.7 Oe
Prf = 0.2 mW(b)
(c)
roughness:
0.15 nm16
Figure 2.
-2000-1000010002000VISHE (V)
(a)YIG/Pt
H = 90o
H = 270o010002000
0100 200Prf (mW)
-6000-4000-20000200040006000VISHE (V)
(b)YIG/Ta
H = 90oH = 270o
-6000-4000-20000
0100 200Prf (mW)
-6000-4000-20000200040006000
-150 -100 -50 0 50 100
H - Hres (Oe)VISHE (V)
(c)YIG/W
H = 90oH = 270o
-6000-4000-20000
0100 200Prf (mW)-80-4004080VISHE (V)
(d)YIG/Au
H = 90o
H = 270o0204060
0100 200Prf (mW)
-1.5-1-0.500.511.5
(e)YIG/Ag
H = 90o
H = 270oVISHE (V)00.511.5
0100 200Prf (mW)
-1-0.500.51
-150 -100 -50 0 50 100
H - Hres (Oe)VISHE (V)
(f)YIG/Cu
H = 90o
H = 270o00.51
0100 200Prf (mW)-101
0180 360
H (deg)
-101
0180 360
H (deg)
-101
0180 360H (deg)-101
0180 360
H (deg)
-101
0180 360H (deg)
-101
0180 360H (deg)17
Figure 3 .
-2-1012
(a)YIG
YIG/Pt
-2-1012dIFMR/dH (a.u.)
(b)YIG
YIG/Ta
-2-1012
-60-40-20 02040
H - Hres (Oe)(c)YIG
YIG/W(d)YIG
YIG/Au
(e)YIG
YIG/Ag
-40-20 0204060
H -Hres (Oe)(f)YIG
YIG/Cu
01020304050
0 5 10 15 20H (Oe)
f (GHz)YIG/Pt
YIG/Ta
YIG/W
YIG/Au
YIG/Cu
YIG/Ag
YIG
(g)18
Figure 4.
00.040.08
01x1072x1073x1074x107SH
Z 4(a)
CuAgAu
-0.100.1
73747576777879
ZSH
Ta
WPtAu (b) |
0802.2043v2.Light_induced_magnetization_precession_in_GaMnAs.pdf | Light-induced magnetization precession in GaMnAs
E. Rozkotová, P. N ěmeca), P. Horodyská, D. Sprinzl, F. Trojánek, and P. Malý
Faculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 3,
121 16 Prague 2, Czech Republic
V. Novák, K. Olejník, M. Cukr, and T. Jungwirth
Institute of Physics ASCR v.v.i., Cukrovarnická 10, 162 53 Prague, Czech Republic
We report dynamics of the transient polar Kerr rotation (KR) and of the transient
reflectivity induced by femtosecond laser pulses in ferromagnetic (Ga,Mn)As with no
external magnetic field applied. It is s hown that the measured KR signal consist of
several different contributions, among which only the oscillatory signal is directly
connected with the ferromagnetic order in (Ga,Mn)As. The origin of the light-induced magnetization precession is discussed and the magnetization precession damping (Gilbert damping) is found to be strongly influenced by annealing of the sample.
(Ga,Mn)As is the most intensively studied member of the family of diluted magnetic
semiconductors with carrier-mediated ferromagn etism [1]. The sensitiv ity of ferromagnetism
to concentration of charge carriers opens up th e possibility of magneti zation manipulation on
the picosecond time scale using light pulses from ultrafast lasers [2]. Photoexcitation of a
magnetic system can strongly disturb the equili brium between the mobile carriers (holes),
localized spins (Mn ions), and the lattice. This in turn tr iggers a variety of dynamical
processes whose characteristic time scales and st rengths can be investigated by the methods of
time-resolved laser spectroscopy [2]. In part icular, the magnetization reversal dynamics in
various magnetic materials attracts a significant attention because it is directly related to the
speed of data storage in the magnetic reco rding [3]. The laser-induced precession of
magnetization in ferromagnetic (Ga,Mn)As has been recently re ported by two research groups
[4-6] but the physical processes responsible fo r it are still not well unde rstood. In this paper
we report on simultaneous measurements of the light-induced magnetization precession
dynamics and of the dynamics of photoinjected carriers.
The experiments were performed on a 500 nm thick ferromagnetic Ga 1-xMn xAs film
with x = 0.06 grown by the low temperature molecular beam epitaxy (LT-MBE) on a
GaAs(001) substrate. We studied both the as-grown sample, with the Curie temperature TC ≈
60 K and the conductivity of 120 Ω-1cm-1, and the sample annealed at 200°C for 30 hours,
with TC ≈ 90 K and the conductivity of 190 Ω-1cm-1; using the mobility vs. hole density
dependence typical for GaMnAs [7] we can r oughly estimate their hole densities as 1.5 x 1020
cm-3 and 3.4 x 1020 cm-3, respectively. Magnetic properties of the samples were measured
using a superconducting quantum interference de vice (SQUID) with magnetic field of 20 Oe
applied along different crysta llographic directions. The photo induced magnetization dynamics
was studied by the time-resolved Kerr rotation (KR) technique [2] using a femtosecond
titanium sapphire laser (Tsunami, Spectra Physics) . Laser pulses, with th e time width of 80 fs
and the repetition rate of 82 MHz, were tune d to 1.54 eV. The energy fluence of the pump
pulses was typically 15 μJ.cm-2 and the probe pulses were always at least 10 times weaker.
The polarization of the pump pulses was either circular or linear, while the probe pulses were
a) Electronic mail: nemec@karlov.mff.cuni.cz
1linearly polarized (typically along the [010] crystallographic direction in the sample, but
similar results were obtained also for other orientations). The rotation angle of the
polarization plane of the reflected probe pulses was obtained by taking the difference of
signals measured by detectors in an optical br idge detection system [2]. Simultaneously, we
measured also the sum of signals from the detectors, which corresponded to a probe intensity
change due to the pump induced modification of the sample reflectivity. The experiment was
performed with no external magnetic field applie d. However, the sample was cooled in some
cases with no external magnetic field applied or alternatively with a ma gnetic field of 170 Oe
applied along the [-110] direction.
Fig. 1. Dynamics of photoinduced Kerr rotation angle (KR) measured for the as-grown sample at 10 K. (a) KR
measured for σ + and σ - circularly polarized (CP) pump pulses; (b) KR measured for p and s linearly polarized
(LP) pump pulses. Polarization-independent part (c) (polar ization-dependent part (d)) of KR signal, which was
computed from the measured traces as an average of the signals (a half of the difference between the signals)
detected for pump pulses with the opposite CP (LP). Inset: Fourier transform of the oscillations. No external
magnetic field was applied during the sample cooling.
In Fig. 1 we show typical te mporal traces of the transien t angles of KR measured for
the as-grown sample at 10 K. The KR signal wa s dependent on the light polarization but there
were certain features presen t for both the circular (Fig. 1 (a)) and linear (Fig. 1 (b))
polarizations. In Fig. 1 (c) we show the polari zation-independent part of the measured KR
signal, which was the same for circular and line ar polarization of pump pulses. On the other
hand, the amplitude of the polarization-dependent part of the signal (Fig. 1 (d)) was larger for
the circular polarization. The interpretation of the polarization-dependent part of the signal is
significantly complicated by the fact that the circularly polarized light generates spin-
polarized carriers (electrons in particular), whose contribution to the measured KR signal can
even exceed that of ferromagnetic ally coupled Mn spins [8]. In the following we concentrate
on the polarization-independent part of the KR signal (Fig. 1 (c)). This signal can be fitted
well (see Fig. 2) by an exponentially damped sine harmonic oscillation superimposed on a
pulse-like function:
() ( ) ( ) () [ ]()2 1 / exp / exp 1 sin / exp τ τ ϕωτ t t B t t A t KRD − −−+ + − = . (1)
2The oscillatory part of the KR signa l is characterized by the amplitude (A ), damping time ( τD),
angular frequency ( fπω2= ), and phase ( ϕ). The pulse-like part of the KR signal is
described by the amplitude ( B), rise time ( τ1), and decay time ( τ2). In the inset of Fig. 2 we
show the dynamics of the sample reflectivity change ΔR/R. This signal monitored the change
of the complex index of refraction of the sa mple due to carriers photoinjected by the pump
pulse. From the dynamics of ΔR/R we can conclude that the population of photogenerated free
carriers (electrons in par ticular [9]) decays within ≈ 50 ps after the photoinjection. This rather
short lifetime of free electrons is similar to th at reported for the low temperature grown GaAs
(LT-GaAs), which is generally interpreted as a consequence of a high concentration of
nonradiative recombination centers induced by th e low temperature growth mode of the MBE
[9]. It is also clearly apparent from the inset of Fig. 2 that the KR data can be fitted well by
Eq. (1) only for time delays larger than ≈ 50 ps (i.e., just after th e population of photoinjected
free electrons nonradiatively decayed). We will come back to this point later.
Fig. 2. The fitting procedure applied to the polarization-independent part of KR signal. (a) The measured data
from Fig. 1 (c) (points) are fitted (solid line) by a sum of the exponentially damped sine harmonic oscillation
(solid line in part (b)) and the pulse-like KR signal (dashe d line in part (b)). Inset: Dynamics of the reflectivity
change (thick solid line) and the detail of the fitted KR signal.
In Fig. 3(a) we show the intensity dependence of A and B, and in Fig. 3(b) of ω and τD
measured at 10 K. For the increasing inte nsity of pump pulses the magnitudes of A and B
were increasing, ω was decreasing and the values of τD were not changing significantly. The
application of magnetic field applied along the [-110] directi on during the sample cooling
modified the value of ω. For 10 K (and pump intensity I0) the frequency decreased from 24.5
to 20 GHz (open and solid point in Fig. 3 (d ), respectively). The measured temperature
dependence of A and B (Fig. 3 (c)) revealed that the oscillatory signal vanished above TC,
while a certain fraction of the pulse-like KR signal persisted even above TC. This shows that
only the oscillatory part of the KR si gnal was directly c onnected with the ferromagnetic order
in (Ga,Mn)As. (It is worth noti ng that also the polarization-de pendent part of the KR signal
was non-zero even above TC.) The frequency of oscillations was decreasing with the sample
temperature (Fig. 3 (d)), but the values of τD were not changing sign ificantly (not shown
here).
3
Fig. 3. Intensity dependence of ⎪A⎪and ⎪B⎪ (a), ω and τD (b) measured at 10 K; I0 = 15 μJ.cm-2, no external
magnetic field was applied during the sample cooling. (c), (d) Temperature dependence of ⎪A⎪, ⎪B⎪ and ω
(points) measured at pump intensity I0. The open point in (d) was obtained for the sample cooled with no
external magnetic field applied and the data in (c) and the solid points in (d) were obtained for the sample cooled
with magnetic field applied along the [-110]. The lines in (d) are the temperature dependence of the sample
magnetization projections to different crystallographic directions measured by SQUID.
The photoinduced magnetization precession was reported by A. Oiwa et al. , who
attributed it to the precession of ferromagnetically coupled Mn spins induced by a change in
magnetic anisotropy initiated by an increase in hole concentration [4]. It was also shown that
the photoinduced magnetization precession a nd the ferromagnetic resonance (FMR) can
provide similar information [4]. Magnetic an isotropy in (Ga,Mn)As is influenced by the
intrinsic cubic anisotropy, which is arising from its zinc-blende symmetry, and by the uniaxial anisotropy, which is a result of a strain induced by different lattice constants of GaMnAs and
the substrate. For the standard stressed GaMnAs films with Mn content above 2% grown on
GaAs substrates the magnetic easy axes are in -plain. Consequently, the measured polar Kerr
rotation is not sensitive to the steady state magn etization of the sample, but only to the light-
induced transient out-of-plane magnetization due to the polar Kerr effect [2]. In our
experiment, the pump pulses with a fluence I
0 = 15 μJ.cm-2 photoinjected electron-hole pairs
with an estimated concentration Δp = Δn ≈ 8 x 1017 cm-3. This corresponded to Δp/p ≈ 0.5%
and such a small increase in the hole concentrati on is highly improbable to lead to any sizable
change of the sample anisotropy [1]. Another hypothesis about the origin of the light-induced
magnetization precession was reported recently by J. Qi et al. [6]. The authors suggested that
not only the transient increase in local hole concentration Δp but also the local temperature
increase ΔT contributes to the change of anisotr opy constants. This modification of the
sample anisotropy changes in turn the direction of the in-plane magnetic easy axis and,
consequently, triggers a precessional motion of the magnetization around the altered magnetic
anisotropy field . The magnitude of decreases as T (the sample temperature) or ΔT
increases, primarily due to the decrease in the cubic anisotropy constant KMn
anisHMn
anisH
1c [6]. Our samples
exhibit in-plane easy axis behavior typica l for stressed GaMnAs layers grown on GaAs
substrates. To characterize their in-plane anis otropy we measured the temperature dependent
magnetization projections to [110] , [010], and [-110] crystallographic directions – the results
are shown in Fig. 3 (d) and in inset of Fig. 4 for the as-grown and the annealed sample,
respectively. At low temperatures the cubi c anisotropy dominates (as indicated by the
4maximal projection measured along the [010] di rection) but the uniaxial in-plane component
is not negligible and the sample magnetization is slightly tilted from the [010] direction
towards the [-110] direction. Both samples exhibit rotation of magnetizat ion direction in the
temperature region 10-25K, which is in agreement with the expected fast weakening of the
cubic component with an increasing temperatur e. In our experiment, the excitation fluence I0
led to ΔT ≈ 10 K (as estimated from the GaAs specifi c heat of 1 mJ/g/K [6]) that can be
sufficient for a change of the easy axis positi on. This temperature-ba sed hypothesis about the
origin of magnetization precession is supported also by our observation that the oscillations
were not fully developed immediately after the photoinj ection of carriers but only after ≈ 50
ps when phonons were emitted by the nonradiative decay of the population of free electrons
(see inset in Fig. 2). We also point out that the measured precession frequency ω and the
sample magnetization M (measured by SQUID) had very similar temperature dependence (see
Fig. 3(d)).
Fig. 4. Polarization-independent part of KR signal measured for the annealed sample at 10 K; I0 = 15 μJ.cm-2, the
sample was cooled with magnetic field applied along the [-110]. Inset: Temperature dependence of the sample
magnetization projections to different crystallographic directions measured by SQUID.
An example of the results measured for the annealed sample is shown in Fig. 4. The
analysis of the data revealed that at simila r conditions the precession frequency was slightly
higher in the annealed sample (20 GHz and 24 GHz for the as-grown and the annealed
sample, respectively). However, a major effect of the sample annealing was on the oscillation
damping time τD, which increased from 0.4 ns to 1.1 ns. This prolongation of τD can be
attributed to the improved quality of the ann ealed sample, which is indicated by the higher
value of TC and by the more Brillouin-like temperature dependence of the magnetization (cf.
Fig. 3 (d) and inset in Fig. 4). The damping of oscillations is connected with the precession
damping in the Landau-Lifshitz-Gilbert equation [1]. The exact determination of the intrinsic
Gilbert damping coefficient α from the measured data is not straightforward because it is
difficult to decouple the contribution due to the inhomogeneous broadening [10]. In Ref. 6 the
values of α from 0.12 to 0.21 were deduced for the as -grown sample from the analysis of the
oscillatory KR signal. The time-domain KR should provide similar information as the
frequency-domain based FMR, where the relaxa tion rate of the magnetization is connected
with the peak-to-peak ferromagnetic resonance linewidth ΔHpp [10]. Indeed both methods
showed that the relaxation rate of the magnetization is consider ably slower in the annealed
samples (as indicated by the prolongation of τD in our experiment and by the reduction of
ΔHpp in FMR [10]).
5 In conclusion, we studied the transient Ke rr rotation (KR) and the reflectivity change
induced by laser pulses in (Ga ,Mn)As with no external magnetic field applied. We revealed
that the measured KR signals consisted of seve ral different contributions and we showed that
only the oscillatory KR signal was directly connected with the fe rromagnetic order in
(Ga,Mn)As. Our data indicated that the phonons emitted by photoinjected carriers during their
nonradiative recombination in (Ga,Mn)As can be re sponsible for the magnetic anisotropy
change that was triggering the magnetization precession. We also observed that the precession
damping was strongly suppressed in the ann ealed sample, which reflected its improved
magnetic properties. This work was supported by Ministry of Education of the Czech Republic in the framework of the rese arch centre LC510, the research plans
MSM0021620834 and AV0Z1010052, by the Grant Agen cy of the Charles University in
Prague under Grant No. 252445, and by the Grant Agency of Academy of Sciences of the Czech Republic Grants FON/06/E 001, FON/06/E002, and KAN400100652.
References
[1] T. Jungwirth, J. Sinova, J. Maše k, A. H. MacDonald, Rev. Mod. Phys. 78, 809 (2006).
[2] J. Wang, Ch. Sun, Y. Hashimoto, J. Kono, G.A. Khodaparast, L. Cywinski, L.J. Sham, G.D. Sanders, Ch.J. Stanton, H. Munekata, J. Phys.: Condens. Matter
18, R501 (2006).
[3] A.V. Kimel, A. Kirilyuk, F. Hansteen, R.V. Pisarev, T. Ra sing, J. Phys.: Condens. Matter
19, 043201 (2007).
[4] A. Oiwa, H. Takechi, H. Munekata, J. Supercond. 18, 9 (2005).
[5] H. Takechi, A. Oiwa, K. Nomura, T. Kondo, H. Munekata, phys. stat. sol. (c) 3, 4267
(2006).
[6] J. Qi, Y. Xu, N.H. Tolk, X. Liu, J.K. Furdyna, I.E. Perakis, Appl. Phys. Lett. 91, 112506
(2007).
[7] T. Jungwirth et al., Phys. Rev. B 76, 125206 (2007).
[8] A.V. Kimel, G.V. Astakhov, G.M. Schott, A. Kirilyuk, D. R. Yakovlev, G. Karczewski,
W. Ossau, G. Schmidt, L.W. Molenkamp, Th. Rasing, Phys. Rev. Lett. 92, 237203 (2004).
[9] M. Stellmacher, J. Nagle, J.F. Lampin, P. Santoro, J. Van eecloo, A. Alexandrou, J. Appl.
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[10] X. Liu, J.K. Furdyna, J. Phys.: Condens. Matter 18, R245 (2006).
6 |
2101.02794v2.Mechanisms_behind_large_Gilbert_damping_anisotropies.pdf | Mechanisms behind large Gilbert damping anisotropies
I. P. Miranda1, A. B. Klautau2,∗A. Bergman3, D. Thonig3,4, H. M. Petrilli1, and O. Eriksson3,4
1Universidade de São Paulo, Instituto de Física,
Rua do Matão, 1371, 05508-090, São Paulo, SP, Brazil
2Faculdade de Física, Universidade Federal do Pará, Belém, PA, Brazil
3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden and
4School of Science and Technology, Örebro University, Fakultetsgatan 1, SE-701 82 Örebro, Sweden
(Dated: November 22, 2021)
A method with which to calculate the Gilbert damping parameter from a real-space electronic
structure method is reported here. The anisotropy of the Gilbert damping with respect to the
magnetic moment direction and local chemical environment is calculated for bulk and surfaces
of Fe 50Co50alloys from first principles electronic structure in a real space formulation. The size
of the damping anisotropy for Fe 50Co50alloys is demonstrated to be significant. Depending on
details of the simulations, it reaches a maximum-minimum damping ratio as high as 200%. Several
microscopic origins of the strongly enhanced Gilbert damping anisotropy have been examined, where
in particular interface/surface effects stand out, as do local distortions of the crystal structure.
Although theory does not reproduce the experimentally reported high ratio of 400% [Phys. Rev.
Lett. 122, 117203 (2019)], it nevertheless identifies microscopic mechanisms that can lead to huge
damping anisotropies.
Introduction: Magnetic damping has a critical impor-
tanceindeterminingthelifetime,diffusion,transportand
stability of domain walls, magnetic vortices, skyrmions,
and any nano-scale complex magnetic configurations [1].
Given its high scientific interest, a possibility to obtain
this quantity by means of first-principles theory [2] opens
new perspectives of finding and optimizing materials for
spintronic and magnonic devices [3–8]. Among the more
promising ferromagnets to be used in spintronics devices,
cobalt-iron alloys demonstrate high potentials due to the
combination of ultralow damping with metallic conduc-
tivity [4, 9].
Recently, Li et al.[10] reported an observed, gi-
ant anisotropy of the Gilbert damping ( α) in epitaxial
Fe50Co50thin films (with thickness 10 −20nm) reach-
ing maximum-minimum damping ratio values as high as
400%. TheauthorsofRef. [10]claimedthattheobserved
effect is likely due to changes in the spin-orbit coupling
(SOC) influence for different crystalline directions caused
by short-range orderings that lead to local structural dis-
tortions. This behaviour differs distinctly from, for ex-
ample, pure bcc Fe [11]. In order to quantitatively pre-
dict the Gilbert damping, Kambersky’s breathing Fermi
surface (BFS) [12] and torque-correlation (TC) [13] mod-
els are frequently used. These methods have been ex-
plored for elements and alloys, in bulk form or at sur-
faces, mostly via reciprocal-space ab-initio approaches,
in a collinear or (more recently) in a noncollinear con-
figuration [14]. However, considering heterogeneous ma-
terials, such as alloys with short-range order, and the
possibility to investigate element specific, non-local con-
tributions to the damping parameter, there are, to the
best of our knowledge, no reports in the literature that
rely on a real space method.
In this Letter, we report on an implementation of ab
initiodamping calculations in a real-space linear muffin-tin orbital method, within the atomic sphere approxi-
mation (RS-LMTO-ASA) [15, 16], with the local spin
density approximation (LSDA) [17] for the exchange-
correlation energy. The implementation is based on the
BFSandTCmodels, andthemethod(SupplementalMa-
terial - SM, for details) is applied to investigate the re-
ported, huge damping anisotropy of Fe50Co50(100)/MgO
films [10]. A main result here is the identification of
a microscopic origin of the enhanced Gilbert damping
anisotropy of Fe50Co50(100) films, and the intrinsic rela-
tionships to the local geometry of the alloy. Most signifi-
cantly, wedemonstratethatasurfaceproducesextremely
large damping anisotropies that can be orders of magni-
tude larger than that of the bulk. We call the attention
to the fact that this is the first time, as far as we know,
that damping values are theoretically obtained in such a
local way.
Results: We calculated: i)ordered Fe50Co50in theB2
structure (hereafter refereed to as B2-FeCo) ii)random
Fe50Co50alloysinbccorbctstructures, wherethevirtual
crystal approximation (VCA) was applied; iii)Fe50Co50
alloys simulated as embedded clusters in a VCA matrix
(host). In all cases VCA was simulated with an elec-
tronic concentration corresponding to Fe50Co50. The ii)
andiii)alloys were considered as in bulk as well as in
the (001) surface, with bcc and bct structures (here-
after correspondingly refereed as VCA Fe50Co50bcc,
VCA Fe50Co50bct, VCA Fe50Co50(001) bcc and VCA
Fe50Co50(001) bct). The effect of local tetragonal distor-
tions was considered with a localc
a= 1.09ratio (SM for
details). All data for cluster based results, were obtained
from an average of several different configurations. The
total damping for a given site iin real-space ( αt, Eqs. S6
and S7 from SM) can be decomposed in non-local, αij
(i/negationslash=j), and local (onsite), αonsite(orαii,i=j) contri-
butions, each of them described by the tensor elements2
ανµ
ij=g
miπ/integraldisplay
η(/epsilon1)Tr/parenleftBig
ˆTν
iImˆGijˆTµ
jImˆGji/parenrightBig
d/epsilon1,(1)
wheremiis the total magnetic moment localized in the
reference atomic site i,µ,ν={x,y,z},ˆTis the torque
operator, and η(/epsilon1) =∂f(/epsilon1)
∂/epsilon1the derivative of the Fermi
distribution. The scalar αijparameter is defined in the
collinear regime as αij=1
2(αxx
ij+αyy
ij).
To validate our methodology, the here obtained total
damping for several systems (such as bcc Fe, fcc Ni, hcp
and fcc Co and B2-FeCo) were compared with estab-
lished values available in the literature (Table S1, SM),
where an overall good agreement can be seen.
Fig. 1 shows the non-local contributions to the damp-
ing for bcc Fe and B2-FeCo. Although the onsite contri-
butions are around one order of magnitude larger than
the non-local, there are many αijto be added and total
net values can become comparable. Bcc Fe and B2-FeCo
have very different non-local damping contributions. El-
ement resolved αij, reveal that the summed Fe-Fe in-
teractions dominate over Co-Co, for distances until 2a
inB2-FeCo. We observe that αijis quite extended in
space for both bcc Fe and B2-FeCo. The different con-
tributions to the non-local damping, from atoms at equal
distance arises from the reduced number of operations in
the crystal point group due to the inclusion of SOC in
combination with time-reversal symmetry breaking. The
B2-FeCo arises from replacing every second Fe atom in
the bcc structure by a Co atom. It is interesting that this
replacement (i.e. the presence of Co in the environment)
significantly changes the non-local contributions for Fe-
Fe pairs , what can more clearly be seen from the Insetin
Fig. 1, where the non-local damping summed over atoms
at the same relative distance for Fe-Fe pairs in bcc Fe
andB2-FeCo are shown; the non-local damping of Fe-Fe
pairs are distinctly different for short ranges, while long
ranged (further than ∼2.25 Å) contributions are smaller
and more isotropic.
The damping anisotropy, i.e. the damping change,
when the magnetization is changed from the easy axis
to a new direction is1
∆αt=/parenleftBigg
α[110]
t
α[010]
t−1/parenrightBigg
×100%, (2)
whereα[110]
tandα[010]
tare the total damping obtained
for magnetization directions along [110]and [010], re-
spectively. Analogousdefinitionalsoappliesfor ∆αonsite.
1We note that this definition is different to the maximum-
minimum damping ratio, defined asα[110]
t
α[010]
t×100%, from Ref.
[10].We investigated this anisotropy in surfaces and in bulk
systems with (and without) tetragonal structural distor-
tions. Our calculations for VCA Fe50Co50bcc show a
damping increase of ∼13%, when changing the magne-
tization direction from [010]to[110](Table S2 in the
SM). The smallest damping is found for the easy magne-
tization axis, [010], which holds the largest orbital mo-
ment (morb) [18]. For VCA Fe50Co50bcc we obtained
a small variation of ∼2%for the onsite contribution
(α[010]
onsite = 8.94×10−4andα[110]
onsite = 8.76×10−4),
what implies that the anisotropy comes mostly from
the non-local contributions, particularly from the next-
nearest neighbours. For comparison, ∆αt∼3%(with
∆αonsite∼0.4%) in the case of bcc Fe, what corrobo-
rates the reported [11] small bcc Fe anisotropy at room
temperature, andwiththebulkdampinganisotropyrates
[19].
We also inspected the chemical inhomogeneity influ-
ence on the anisotropy, considering the B2-FeCo alloy,
where the weighted average damping (Eq. S7 of SM)
was used instead. The B2-FeCo bcc (∼7%) and VCA
Fe50Co50bcc (∼13%) anisotropies are of similar magni-
tudes. Both B2structure and VCA calculations lead to
damping anisotropies which are significantly lower than
whatwasobservedintheexperiments, anditseemslikely
thatthepresenceofdisorderincompositionand/orstruc-
tural properties of the Fe/Co alloy would be important
to produce large anisotropy effects on the damping.
α
ij
×
10
-4
−3
−2
−1
0
1
2
3
4
Normalized
distance
1.0
1.5
2.0
2.5
3.0
B2
Fe-Co
B2
Fe-Fe
B2
Co-Co
bcc
Fe-Fe
−10
0
10
20
1.0
1.5
2.0
2.5
3.0
Figure 1. (Color online) Non-local damping contributions,
αij, in (Fe-centered) bulk B2-FeCo and bcc Fe, as a function
of the normalized distance in lattice constant units a.Inset:
Non-local contributions from only Fe-Fe pairs summed, for
each distance, in bcc Fe bulk (empty blue dots) and in the
B2-FeCo (full red dots). The onsite damping for Fe (Co) in
B2-FeCo isαFe
onsite = 1.1×10−3(αCo
onsite = 0.8×10−3) and for
bcc Fe it is αFe
onsite = 1.6×10−3. The magnetization direction
isz([001]). Lines are guides for the eyes.
Weanalyzedtheroleoflocaldistortionsbyconsidering3
a hypothetical case of a large, 15%(c
a= 1.15), distortion
on thez-axis of ordered B2-FeCo. We found the largest
damping anisotropy ( ∼24%) when comparing the results
with magnetization in the [001](α[001]
t= 10.21×10−3)
and in the [010](α[010]
t= 7.76×10−3) directions. This
confirms that, indeed, bct-like distortions act in favour of
the∆αtenhancement (and therefore, of the maximum-
minimum damping ratio), but the theoretical data are
not large enough to explain the giant value reported ex-
perimentally [10].
Nevertheless, in the case of an alloy, the local lattice
distortions suggested in Ref. [10] are most to likely occur
in an heterogeneous way [20], with different distortions
for different local environments. To inspect this type
of influence on the theoretical results, we investigated
(Table S3, SM) clusters containing different atomic con-
figurations embedded in a VCA Fe50Co50matrix (with
Fe bulk lattice parameter); distortions were also consid-
ered such that, locally in the clusters,c
a= 1.15(Ta-
ble S4, in the SM). Moreover, in both cases, two types
of clusters have to be considered: Co-centered and Fe-
centered. The αtwas then computed as the sum of the
local and non-local contributions for clusters with a spe-
cific central (Fe or Co) atom, and the average of Fe-
and Co-centered clusters was taken. Fe-centered clus-
tershaveshownlargeranisotropies, onaverage ∼33%for
the undistorted (∼74%for the distorted) compared with
∼8%fortheundistortedCo-centeredclusters( ∼36%for
the distorted). Although these results demonstrate the
importance of both, local distortions as well as non-local
contributions to the damping anisotropy, they are not
still able to reproduce the huge observed [10] maximum-
minimum damping ratio.
We further proceed our search for ingredients that
could lead to a huge ∆αtby inspecting interface effects,
which are present in thin films, grain boundaries, stack-
ing faults and materials in general. Such interfaces may
influence observed properties, and in order to examine
if they are relevant also for the reported alloys of Ref.
[10], we considered these effects explicitly in the calcu-
lations. As a model interface, we considered a surface,
what is, possibly, the most extreme case. Hence, we per-
formed a set of αtcalculations for the Fe50Co50(001),
first on the VCA level. Analogous to the respective bulk
systems, we found that the onsite contributions to the
damping anisotropy are distinct, but they are not the
main cause ( ∆αonsite∼18%). However, the lack of in-
version symmetry in this case gives a surprisingly large
enhancement of ∆αt, thus having its major contribution
coming from the non-local damping terms, in particular
from the next-nearest neighbours. Interestingly, negative
non-local contributions appear when αtis calculated in
the[010]direction. These diminish the total damping
(the onsite contribution being always positive) and gives
rise to a larger anisotropy, as can be seen by comparisonof the results shown in Table I and Table S5 (in the Sup-
plemental Material). In this case, the total anisotropy
was found to be more than ∼100%(corresponding to a
maximum-minimum damping ratio larger than 200%).
A compilation of the most relevant theoretical results
obtained here is shown in Fig. 2, together with the ex-
perimental data and the local density of states (LDOS)
atEFfor each magnetization direction of a typical atom
in the outermost layer (data shown in yellow). As shown
in Fig. 2, the angular variation of αthas a fourfold ( C4v)
symmetry, with the smallest Gilbert damping occurring
at 90◦from the reference axis ( [100],θH= 0◦), for both
surface and bulk calculations. This pattern, also found
experimentally in [10], matches the in-plane bcc crys-
tallographic symmetry and coincides with other mani-
festations of SOC, such as the anisotropic magnetoresis-
tance [10, 21]. Following the simplified Kambersky’s for-
mula [13, 22], in which (see SM) α∝n(EF)and, there-
fore, ∆α∝∆n(EF), we can ascribe part of the large
anisotropy of the FeCo alloys to the enhanced LDOS dif-
ferences at the Fermi level, evidenced by the close corre-
lation between ∆n(EF)and∆αtdemonstrated in Fig. 2.
Thus, as a manifestation of interfacial SOC (the so-called
proximity effect [23]), the existence of ∆αtcan be under-
stoodintermsofRashba-likeSOC,whichhasbeenshown
to play an important role on damping anisotropy [24, 25].
Analogous to the bulk case, the higher morboccurs where
the system presents the smallest αt, and the orbital
moment anisotropy matches the ∆αtfourfold symme-
try with a 90◦rotation phase (see Fig. S3, SM). Note
that a lower damping anisotropy than Co50Fe50(001) is
found for a pure Fe(001) bcc surface, where it is ∼49%
(Table S2, SM), in accordance with Refs. [7, 26], with
a dominant contribution from the onsite damping val-
ues (conductivity-like character on the reciprocal-space
[19, 27]).
The VCA surface calculations on real-space allows to
investigate the layer-by-layer contributions (intra-layer
damping calculation), as shown in Table I. We find that
themajorcontributiontothedampingsurfaceanisotropy
comes from the outermost layer, mainly from the differ-
ence in the minority 3dstates around EF. The deeper
layers exhibit an almost oscillatory ∆αtbehavior, simi-
lar to the oscillation mentioned in Ref. [28] and to the
Friedel oscillations obtained for magnetic moments. The
damping contributions from deeper layers are much less
influenced by the inversion symmetry breaking (at the
surface), as expected, and eventually approaches the typ-
icalbulklimit. Therefore, changesintheelectronicstruc-
tureconsiderednotonlytheLDOSoftheoutermostlayer
but a summation of the LDOS of all layers (including the
deeper ones), which produces an almost vanishing differ-
ence between θH= 0◦andθH= 45◦(also approaching
the bulk limit). The damping anisotropy arising as a sur-
face effect agrees with what was observed in the case of
Fe [7] and CoFeB [29] on GaAs(001), where the damping4
0o45o90o
135o
180o
225o
270o315o[100][110][010]
[−110]
0.10.20.3
Δn(EF)
(st./Ry−at.)
0.0050.0100.015
αt
θH
Figure 2. (Color online) Total damping and LDOS difference
atEF,∆n(EF), as a function of θH, the angle between the
magnetizationdirectionandthe [100]-axis. Squares: (redfull)
VCAFe 50Co50(001)bcc. Triagles: (greenfull)averageover32
clusters (16 Fe-centered and 16 Co-centered), with bcc struc-
ture at the surface layers (SM) embedded in a VCA medium;
(gray open) similar calculations, but with a local lattice dis-
tortion. Circles: (yellow open) ∆n(EF)betweenθH= 0◦and
the current angle for a typical atom in the outermost layer
of VCA Fe 50Co50(001) bcc; (blue full) experimental data [10]
for a 10-nm Fe 50Co50/Pt thin film; (purple full) average bulk
VCA Fe 50Co50bcc; and (brown full) the B2-FeCo bulk. Lines
are guides for the eyes.
anisotropy diminishes as the film thickness increases.
Table I. Total intra-layer damping ( αt×10−3) and anisotropy,
∆αt(Eq. 2), of a typical (VCA) atom in each Fe 50Co50(001)
bcc surface layer for magnetization along [010]and[110]di-
rections. In each line, the sum of all αijin the same layer is
considered. Outermost (layer 1) and deeper layers (2-5).
Layerαt[010]αt[110] ∆αt
1 7.00 14.17 +102.4%
2 1.28 1.16 −9.4%
3 2.83 3.30 +16.6%
4 2.18 1.99 −8.7%
5 2.54 2.53 −0.4%
We also studied the impact of bct-like distortions in
thesurface, initiallybyconsideringtheVCAmodel. Sim-
ilartothebulkcase,tetragonaldistortionsmaybeimpor-
tant for the damping anisotropy at the surface, e.g. when
local structural defects are present. Therefore, localized
bct-like distortions of the VCA medium in the surface,
particularly involving the most external layer were inves-
tigated. The structural model was similar to what was
used for the Fe50Co50bulk, consideringc
a= 1.09(see
SM). Our calculations show that tetragonal relaxations
around a typical site in the surface induce a ∆αt∼75%,
fromα[010]
t= 8.94×10−3toα[110]
t= 15.68×10−3. Themain effect of these distortions is an enhancement of the
absolute damping values in each direction with respect to
the pristine (bcc) system. This is due to an increase on
αonsite, fromα[010]
onsite = 7.4×10−3toα[010]
onsite = 9.5×10−3,
and fromα[110]
onsite = 8.7×10−3toα[110]
onsite = 11.7×10−3;
the resulting non-local contributions remains similar to
theundistortedcase. Theinfluenceofbct-likedistortions
on the large damping value in the Fe50Co50surface is in
line with results of Mandal et al.[30], and is related to
the transition of minority spin electrons around EF.
We then considered explicit 10-atom Fe50Co50clusters
embedded in a VCA FeCo surface matrix. The results
from these calculations were obtained as an average over
16 Fe-centered and 16 Co-centered clusters. We con-
sidered clusters with undistorted bcc crystal structure
(Fig. 2, yellow open circles) as well as clusters with lo-
cal tetragonal distortions (Fig. 2, black open circles). As
shown in Fig. 2 the explicit local tetragonal distortion
influences the damping values ( α[010]
t= 10.03×10−3and
α[110]
t= 14.86×10−3)andtheanisotropy, butnotenough
toreproducethehugevaluesreportedintheexperiments.
A summary of the results obtained for each undis-
torted FeCo cluster at the surface is shown in Fig. 3:
Co-centered clusters in Fig. 3(a) and Fe-centered clusters
in Fig. 3(b). A large variation of αtvalues is seen from
clustertocluster, dependingonthespatialdistributionof
atomic species. It is clear that, αtis larger when there is
a larger number of Fe atoms in the surface layer that sur-
roundsthecentral, referenceclustersite. Thiscorrelation
can be seen by the numbers in parenthesis on top of the
blue symbols (total damping for each of the 16 clusters
that were considered) in Fig. 3. We also notice from the
figure that the damping in Fe-centered clusters are lower
than in Co-centered, and that the [010]magnetization di-
rection exhibit always lower values. In the Insetof Fig. 3
the onsite contributions to the damping, αonsite, and the
LDOS atEFin the central site of each cluster are shown:
a correlation, where both trends are the same, can be ob-
served. The results in Fig. 3 shows that the neighbour-
hood influences not only the local electronic structure at
the reference site (changing n(EF)andαonsite), but also
modifies the non-local damping αij, leading to the cal-
culatedαt. In other words, the local spatial distribution
affects how the total damping is manifested, something
which is expressed differently among different clusters.
This may open up for materials engineering of local and
non-local contributions to the damping.
Conclusions: We demonstrate here that real-space
electronic structure, based on density functional theory,
yield a large Gilbert damping anisotropy in Fe50Co50al-
loys. Theory leads to a large damping anisotropy, when
the magnetization changes from the [010]to the [110]di-
rection, which can be as high as ∼100%(or200%in the
minimum-maximum damping ratio) when surface calcu-
lations are considered. This is in particular found for5
(0)(1)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(3)(3)(4)(4)
(1)(1)(2)(2)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(4)(5)
Figure 3. Damping for the [010](open circles) and [110](full
circles) magnetization directions for distinct types of 10-atom
Fe50Co50bcc clusters, embedded in VCA Fe 50Co50(001) bcc
and without any distortion around the reference atom (for
whichαtandαonsiteare shown). (a) Co-centered and (b)
Fe-centered clusters. The quantity of Fe atoms in the surface
layers (near vacuum) are indicated by the numbers in paren-
thesis and the results have been ordered such that larger val-
ues are to the left in the plots. Insets:αonsitefor the [010]
(red open circles) and [110](blue filled circles) magnetization
directions, and corresponding local density of states, n(EF),
at the Fermi level (green filled and unfilled triangles) at the
central atom (placed in the outermost layer) for both types
of clusters. Lines are guides for eyes.
contributions from surface atoms in the outermost layer.
Hence the results presented here represents one more ex-
ample, in addition to the well known enhanced surface
orbital moment [31], of the so-called interfacial spin-orbit
coupling. This damping anisotropy, which holds a bcc-
like fourfold ( C4v) symmetry, has a close relation to the
LDOS difference of the most external layer at EF(ma-
jorly contributed by the minority dstates), as well as
to the orbital moment anisotropy with a 90◦phase. As
a distinct example of an interface, we consider explicitly
the Fe50Co50cluster description of the alloy. In this case,
besides an onsite contribution, we find that the damp-
ing anisotropy is mostly influenced by non-local next-
nearest-neighbours interactions.
Several Gilbert damping anisotropy origins are also
demonstrated here, primarily related to the presence of
interfaces, alloy composition and local structural distor-
tions (as summarized in Table S6, in the SM [32]). Pri-
marily we find that: ( i) the presence of Co introduces anenhanced spin-orbit interaction and can locally modify
the non-local damping terms; ( ii) the randomness of Co
in the material, can modestly increase ∆αtas a total ef-
fect by creating Co-concentrated clusters with enhanced
damping; ( iii) at the surface, the spatial distribution of
Fe/Co, increases the damping when more Fe atoms are
present in the outermost layer; and ( iv) the existence
of local, tetragonal distortions, which act in favour (via
SOC) of the absolute damping enhancement, by modify-
ing theαonsiteof the reference atom, and could locally
change the spin relaxation time. Furthermore, in rela-
tionship to the work in Ref. [10], we show here that bulk
like tetragonal distortions, that in Ref. [10] were sug-
gested to be the key reason behind the observed huge
anisotropy of the damping, can in fact not explain the
experimental data. Such distortions were explicitly con-
sidered here, using state-of-the-art theory, and we clearly
demonstrate that this alone can not account for the ob-
servations.
Although having a similar trend as the experimen-
tal results of Ref. [10], we do not reproduce the most
extreme maximum-minimum ratio reported in the ex-
periment,∼400%(or∆αt∼300%). The measured
damping does however include effects beyond the intrin-
sic damping that is calculated from our electronic struc-
turemethodology. Other mechanismsare knownto influ-
ence the damping parameter, such as contributions from
eddy currents, spin-pumping, and magnon scattering, to
name a few. Thus it is possible that a significant part
of the measured anisotropy is caused by other, extrin-
sic, mechanisms. Despite reasons for differences between
observation and experiment on films of Fe50Co50alloys,
the advancements presented here provide new insights on
the intrinsic damping anisotropy mechanisms, something
which is relevant for the design of new magnetic devices.
Acknowledgements: H.M.P. and A.B.K. acknowledge
financial support from CAPES, CNPq and FAPESP,
Brazil. The calculations were performed at the computa-
tional facilities of the HPC-USP/CENAPAD-UNICAMP
(Brazil), at the National Laboratory for Scientific Com-
puting (LNCC/MCTI, Brazil), and at the Swedish Na-
tional Infrastructure for Computing (SNIC). I.M. ac-
knowledge financial support from CAPES, Finance Code
001, process n◦88882.332894/2018-01, and in the Insti-
tutional Program of Overseas Sandwich Doctorate, pro-
cess n◦88881.187258/2018-01. O.E. acknowledges sup-
port from the Knut och Alice Wallenberg (KAW) foun-
dation, the Swedish research council (VR), the Founda-
tion for Strategic Research (SSF), the Swedish energy
agency (Energimyndigheten), eSSENCE, STandUPP,
and the ERC (synergy grant FASTCORR). D.T. ac-
knowledges support from the Swedish Research Council
(VR) through Grant No. 2019-03666.6
∗aklautau@ufpa.br
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I. P. Miranda1, A. B. Klautau2,∗A. Bergman3, D. Thonig3,4, H. M. Petrilli1, and O. Eriksson3,4
1Universidade de São Paulo, Instituto de Física,
Rua do Matão, 1371, 05508-090, São Paulo, SP, Brazil
2Faculdade de Física, Universidade Federal do Pará, Belém, PA, Brazil
3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden and
4School of Science and Technology, Örebro University, Fakultetsgatan 1, SE-701 82 Örebro, Sweden
(Dated: November 22, 2021)
I. Theory
The torque-correlation model, first introduced by
Kamberský [1], and later elaborated by Gilmore et al.
[2], can be considered as both a generalization and an
extended version of the breathing Fermi surface model,
which relates the damping of the electronic spin orienta-
tion, with the variation in the Fermi surface when the
local magnetic moment is changed. In this scenario,
and considering the collinear limit of the magnetic or-
dering, due to the spin-orbit coupling (SOC), the tilting
in magnetization ˆmby a small change δˆmgenerates a
non-equilibrium population state which relaxes within a
timeτtowards the equilibrium. We use an angle θ, to
represent the rotation of the magnetization direction δˆm.
IftheBlochstatesofthesystemsarecharacterizedbythe
genericbandindex natwavevector k(withenergies /epsilon1k,n),
it is possible to define a tensorfor the damping, that has
matrix elements (adopting the isotropic relaxation time
approximation)
ανµ=gπ
m/summationdisplay
n,mdk
(2π)3η(/epsilon1k,n)/parenleftbigg∂/epsilon1k,n
∂θ/parenrightbigg
ν/parenleftbigg∂/epsilon1k,m
∂θ/parenrightbigg
µτ
~
(S1)
which accounts for both intraband ( n=m, conductivity-
like) and interband ( n/negationslash=m, resistivity-like) contribu-
tions [2]. Here µ,νare Cartesian coordinate indices,
that will be described in more detail in the discussion
below, while η(/epsilon1k,n) =∂f(/epsilon1)
∂/epsilon1/vextendsingle/vextendsingle/vextendsingle
/epsilon1k,nis the derivative of
the Fermi distribution, f, with respect to the energy
/epsilon1, andn,mare band indices. Therefore, the torque-
correlation model correlates the spin damping to vari-
ations of the energy of single-particle states with respect
to the variation of the spin direction θ, i.e.∂/epsilon1k,n
∂θ. Us-
ing the Hellmann-Feynmann theorem, which states that
∂/epsilon1k,n
∂θ=/angbracketleftψk,n|∂H
∂θ|ψk,n/angbracketright, and the fact that only the spin-
orbit Hamiltonian Hsochanges with the magnetization
direction, the spin-orbit energy variation is given by
∂/epsilon1k,n(θ)
∂θ=/angbracketleftψk,n|∂
∂θ/parenleftbig
eiσ·ˆnθHsoe−iσ·ˆnθ/parenrightbig
|ψk,n/angbracketright(S2)
in which σrepresents the Pauli matrices vector, and
ˆnis the direction around which the local moment hasbeen rotated. The expression in Eq. S2 can be eas-
ily transformed into∂/epsilon1k,n(θ)
∂θ=i/angbracketleftψk,n|[σ·ˆn,Hso]|ψk,n/angbracketright
and we call ˆT= [σ·ˆn,Hso]thetorqueoperator. In
view of this, it is straightforward that, in the collinear
case in which all spins are aligned to the zdirection,
σ·ˆn=σµ(µ=x,y,z), originating the simplest {x,y,z}-
dependent torque operator ˆTµ. Putting together the in-
formation on Eqs. S1 and S2, and using the fact that
the imaginary part of the Greens’ functions can be ex-
pressed, in Lehmann representation, as Im ˆG(/epsilon1±iΛ) =
−1
π/summationtext
nΛ
(/epsilon1−/epsilon1n)2+Λ2|n/angbracketright/angbracketleftn|, then it is possible to write in
reciprocal-space [3]:
ανµ=g
mπ/integraldisplay /integraldisplay
η(/epsilon1)Tr/parenleftBig
ˆTνImˆGˆTµImˆG/parenrightBig
d/epsilon1dk
(2π)3.(S3)
In a real-space formalism, the Fourier transformation
of the Green’s function is used to find a very similar ex-
pression emerges for the damping element ανµ
ijrelative to
two atomic sites iandj(at positions riandrj, respec-
tively) in the material:
ανµ
ij=g
miπ/integraldisplay
η(/epsilon1)Tr/parenleftBig
ˆTν
iImˆGijˆTµ
jImˆGji/parenrightBig
d/epsilon1,(S4)
where we defined mi= (morb+mspin)as the total mag-
netic moment localized in the reference atomic site iin
the pair{i,j}. The electron temperature that enters into
η(ε)is zero and, consequently, the energy integral is per-
formed only at the Fermi energy. In this formalism, then,
the intraband and interband terms are replaced by onsite
(i=j) and non-local ( i/negationslash=j) terms. After calculation of
all components of Eq. S4 in a collinear magnetic back-
ground, we get a tensor of the form
αij=
αxxαxyαxz
αyxαyyαyz
αzxαzyαzz
, (S5)
which can be used in the generalized atomistic Landau-
Lifshitz-Gilbert (LLG) equation for the spin-dynamics
of magnetic moment on site i[4]:∂mi
∂t=mi×/parenleftBig
−γBeff
i+/summationtext
jαij
mj·∂mj
∂t/parenrightBig
. Supposing that all spins are
parallel to the local zdirection, we can define the scalar2
αvalue as the average between components αxxandαyy,
that is:α=1
2(αxx+αyy).
Once one has calculated the onsite ( αonsite) and the
non-local (αij) damping parameters with respect to the
site of interest i, the total value, αt, can be defined as
the sum of all these α’s:
αt=/summationdisplay
{i,j}αij. (S6)
In order to obtain the total damping in an heteroge-
neous atomic system (more than one element type), such
as Fe50Co50(with explicit Fe/Co atoms), we consider the
weighted average between the different total local damp-
ing values ( αi
t), namely:
αt=1
Meff/summationdisplay
imiαi
t, (S7)
wheremiis the local magnetic moment at site i, and
Meff=/summationtext
imiis the summed total effective magnetiza-
tion. This equation is based on the fact that, in FMR ex-
periments, the magnetic moments are excited in a zone-
centered, collective mode (Kittel mode). In the results
presented here, Eq. S7 was used to calculate αtofB2-
FeCo, both in bcc and bct structures.
II. Details of calculations
The real-space linear muffin-tin orbital on the atomic-
sphere approximation (RS-LMTO-ASA) [5] is a well-
established method in the framework of the DFT to de-
scribetheelectronicstructureofmetallicbulks[6,7], sur-
faces [8, 9] and particularly embedded [10] or absorbed
[11–14] finite cluster systems. The RS-LMTO-ASA is
based on the LMTO-ASA formalism [15], and uses the
recursion method [16] to solve the eigenvalue problem
directly in real-space. This feature makes the method
suitable for the calculation of local properties, since it
does not depend on translational symmetry.
The calculations performed here are fully self-
consistent, and the spin densities were treated within
the local spin-density approximation (LSDA) [17]. In all
cases, we considered the spin-orbit coupling as a l·sterm
included in each variational step [18–20]. The spin-orbit
is strictly necessary for the damping calculations due to
its strong dependence on the torque operators, ˆT. In
the recursion method, the continued fractions have been
truncated with the Beer-Pettifor terminator [21] after 22
recursion levels ( LL= 22). The imaginary part that
comes from the terminator was considered as a natural
choice for the broadening Λto build the Green’s func-
tions ˆG(/epsilon1+iΛ), which led to reliable αparameters in
comparison with previous results (see Table S1).To account for the Co randomness in the experimental
Fe50Co50films [22], some systems were modeled in terms
of the virtual crystal approximation (VCA) medium of
Fe50Co50, considering the bcc (or the bct) matrix to have
the same number of valence electrons as Fe50Co50(8.5
e−). However, we also investigated the role of the Co
presence, as well as the influence of its randomness (or
ordering), by simulating the B2(CsCl) FeCo structure
(a=aFe). The VCA Fe50Co50andB2-FeCo bulks were
simulated by a large matrix containing 8393 atoms in
real-space, the first generated by using the Fe bcc lat-
tice parameter ( aFe= 2.87Å) and the latter using the
optimized lattice parameter ( a= 2.84Å). Thisachoice
in VCA Fe50Co50was based on the fact that it is eas-
ier to compare damping results for Fe50Co50alloy and
pure Fe bcc bulk if the lattice parameters are the same,
and the use of the aFehas shown to produce trustwor-
thyαtvalues. On the other hand, bct bulk structures
withc
a= 1.15(B2-FeCo bct and VCA Fe50Co50bct) are
based on even larger matrices containing 49412 atoms.
The respective surfaces were simulated by semi-matrices
of the same kind (4488 and 19700 atoms, respectively),
considering one layer of empty spheres above the outer-
most Fe50Co50(or pure Fe) layer, in order to provide a
basis for the wave functions in the vacuum and to treat
the charge transfers correctly.
We notice that the investigations presented here are
based on a (001)-oriented Fe50Co50film, in which only a
small lattice relaxation normal to the surface is expected
to occur (∼0.1%[23]).
Damping parameters of Fe-centered and Co-centered
clusters, embedded in an Fe50Co50VCA medium, have
been calculated (explicitly) site by site. In all cases,
these defects are treated self-consistently, and the po-
tential parameters of the remaining sites were fixed at
bulk/pristine VCA surface values, according to its envi-
ronment. When inside the bulk, we placed the central
(reference) atom of the cell in a typical site far away
from the faces of the real-space matrix, avoiding any un-
wanted surface effects. We considered as impurities the
nearest 14 atoms (first and second nearest neighbours,
up to 1a) from the central atom, treating also this sites
self-consistently, in a total of 15 atoms. We calculated
10 cases with Fe and Co atoms randomly positioned: 5
with Fe as the central atom (Fe-centered) and 5 with Co
as the central atom (Co-centered). An example (namely
cluster #1 of Tables S3 and S4), of one of these clusters
embedded in bulk, is represented in Fig. S1(a). As the
self-consistent clusters have always a total of 15 atoms,
the Fe (Co) concentration is about 47% (53%) or vice-
versa. On the other hand, when inside the surface, we
placed the central (reference) atom of the cluster in a
typical site of the most external layer (near vacuum),
since this has shown to be the layer where the damping
anisotropy is larger. Therefore, we considered as impu-
rities the reference atom itself and the nearest 9 atoms3
(up to 1a), in a total of 10 atoms (and giving a perfect
50% (50%) concentration). An example of one of these
clusters embedded in a surface is shown in Fig. S1(b).
(a)
(b)
Figure S1. (Color online) Schematic representation of an ex-
ampleof: (a)Fe-centered15-atomclusterembeddedinaVCA
Fe50Co50bcc bulk medium; (b) Co-centered 10-atom cluster
embedded in a VCA Fe 50Co50(001) bcc surface medium. Yel-
low and blue spheres represent Fe and Co atoms, respectively,
while gray atoms represent the VCA Fe 50Co50sites (8.5 va-
lence e−). The Fe(Co) concentration in the clusters are: (a)
53% (47%) and (b) 50% (50%) . The total number of atoms
including the surrounding VCA sites are: (a) 339 and (b) 293.
They were all accounted in the sum to obtain αtat the central
(reference) Fe (a) and Co (b) site.
To simulate a bct-like bulk distortion, the 8 first neigh-
bours of the central atom were stretched in the cdi-
rection, resulting in ac
a= 1.15ratio. On the other
hand, when embedded on the Fe50Co50(001) bcc surface,
the central (reference) atom is placed in the outermost
layer (near vacuum), and we simulate a bct distortion
by stretching the 4 nearest-neighbours (on the second
layer) to reproduce ac
a= 1.09ratio (the maximum per-
centage that the atoms, in these conditions, could be
moved to form a bct-like defect). In this case, a total
of 10 atoms (the nearest 9 atoms from the central one
– up to 1a– and the reference atom) were treated self-
consistently, analogous to as shown in Fig. S1(b). As in
the case of the pristine bcc Fe50Co50clusters embedded
in the VCA surface, we considered a total of 32 10-atom
clusters with different Fe/Co spatial distributions, being
16 Fe-centered, and 16 Co-centered.III. Comparison with previous results
Theab-initio calculation of the Gilbert damping, in
the collinear limit, is not a new feature in the literature.
Mainly, the reported theoretical damping results are for
bulk systems [2, 4, 24–28], but, some of them even stud-
ied free surfaces [29]. Therefore, in order to demonstrate
the reliability of the on-site and total damping calcula-
tions implemented here in real-space, a comparison of
the presently obtained with previous (experimental and
theoretical) results, are shown in Table S1. As can be
seen, our results show a good agreement with previously
obtainedαvalues, including some important trends al-
readypredictedbefore. Forexample, thereducedGilbert
damping of Co hcp with respect to the Co fcc due to
the reduction of the density of states at the Fermi level
[24, 28], (∼10.92states/Ry-atom in the hcp case and
∼16.14states/Ry-atom in the fcc case).
IV. Details of the calculated damping values
The damping values obtained for the systems studied
here are shown in Tables (S2-S5). These data can be use-
ful for the full understanding of the results presented in
the main text. For easy reference, in Table S2 the αtof a
typical atom in each system (bulk or surface) for different
spin quantization axes are shown. These data are plot-
ted in Fig. 2 of the main text. The obtained values show
that, indeed, for bulk systems the damping anisotropies
are not so pronounced as in the case of Fe50Co50(001)
bcc surface.
As observed in Table S2, the increase in αtwhen
changing from the bcc Fe50Co50(c
a= 1) to the bct
Fe50Co50bulk structure (c
a= 1.15) is qualitatively con-
sistent to what was obtained by Mandal et al.[33] (from
αt= 6.6×10−3in the bcc to αt= 17.8×10−3in the
bct, withc
a= 1.33[33]).
Tables S3 and S4 refer to the damping anisotropies
(∆αt) for all Fe-centered and Co-centered clusters stud-
ied here, with different approaches: ( i) bcc clusters em-
bedded in the VCA medium (Table S3) and ( ii) bct-like
clusters embedded in the VCA medium (Table S4).
In comparison with bct-like clusters, we found larger
absoluteαtvalues but lower damping anisotropies. In
all cases, Fe-centered clusters present higher ∆αtper-
centages.
In Table S5 the onsite damping anisotropies ( ∆αonsite)
for each layer of the Fe50Co50(001) bcc surface ("1" repre-
sents the layer closest to vacuum) are shown. In compar-
ison with the total damping anisotropies (Table I of the
main text), much lower percentages are found, demon-
strating that the damping anisotropy effect comes ma-
jorly from the non-local damping contributions.
The most important results concerning the largest
dampinganisotropiesaresummarizedinTableS6, below.4
Table S1. Total damping values ( ×10−3) calculated for some bulk and surface systems, and the comparison with previous
literature results. The onsite contributions are indicated between parentheses, while the total damping, αt, are indicated
without any symbols. All values were obtained considering the [001]magnetization axis. The VCA was adopted for alloys,
except for the Fe 50Co50bcc in theB2structure (see Eq. S7). Also shown the broadening Λvalue considered in the calculations.
Bulks a(Å) This work Theoretical Experimental Λ(eV)
Fe bcc 2.87 4.2(1.6) 1 .3[2]a/(3.6)[4] 1.9[30]/2.2[31]
Fe70Co30bcc 2.87 2.5(0.7) − 3−5[32]d
Fe50Co50bcc 2.87 3.7(1.0)[VCA]/ 2.3(1.0)[B2]1.0[25]c[VCA]/ 6.6[33] [B2] 2.3[27]
Ni fcc 3.52 27.8(57.7) 23 .7[34]/( 21.6[4])b26.0[31]/24.0[35]
Ni80Fe20(Py) fcc 3.52 9.8(12.1) 3 .9[25]c8.0[30]/5.0[35]
Co fcc 3.61 [3] 3.2(5.3) 5 .7[28]/(3.9[4])b11.0[30]∼5×10−2
Co hcp 2.48/4.04 [28] 2.1(6.2) 3 .0[28] 3.7[31]
Co85Mn15bcc [36] 2.87 [28] 6.2(4.2) 6 .6[28] −
Co90Fe10fcc 3.56 [37] 3.6(4.2) − 3.0[35]/4.8[37]
Surfaces a(Å) This work Theoretical Experimental
Fe(001) bcc [110] 2.87 5.8(5.4)e− 7.2[38]h/6.5[39]i
Fe(001) bcc [100] 2.87 3.9(4.4)f∼4[29]g4.2[40]j
Ni(001) fcc 3.52 80.0(129.6)∼10[29]g/12.7[41]m22.1[42]l
PdFe/Ir(111) [43] fcc 3.84 3.9(2.7)n− −
PdCo/Ir(111) [44] fcc 3.84 3.2(14.7)o− −
aWith Λ∼2×10−2eV.
bWith Λ = 5×10−3eV.
cWith Λ∼1.4×10−4eV.
dFor a 28%Co concentration, but the results do not significantly change for a 30%Co concentration. Range including results before and
after annealing.
eOf a typical atom in the more external surface layer (in contact with vacuum), in the [110] magnetization direction.
fOf a typical atom in the more external surface layer (in contact with vacuum), in the [100] magnetization direction.
gFor a (001) bcc surface with thickness of N= 8ML (the same number of slabs as in our calculations), and Λ = 10−2eV.
hAnisotropic damping obtained for a 0.9 nm Fe/GaAs(001) thin film (sample S2 in Ref. [38]) in the [110] magnetization direction.
iAnisotropic damping obtained for a 1.14 nm Fe/InAs(001) thin film in the in-plane [110] hard magnetization axis.
jFor a 25-nm-thick Fe films grown on MgO(001).
kFor epitaxial Fe(001) films grown on GaAs(001) and covered by Au, Pd, and Cr capping layers.
lIntrinsic Gilbert damping for a free 4×[Co(0.2 nm)/Ni(0.6 nm)](111) multilayer. Not the same system as Ni(001), but the nearest system
found in literature.
mFor a Co | Ni multilayer with Ni thickness of 4 ML (fcc stacking).
nOf a typical atom in the Fe layer.
oOf a typical atom in the Co layer.
The alloys with short-range orders (SRO) are described
as FeCo clusters (with explicit Fe and Co atoms) embed-
ded in the Fe50Co50VCA medium – with and without
the bct-like distortion. In this case, the damping is cal-
culated as a weighted average (Eq. S7). As discussed in
the main text, it can be seen from Table S6 that distor-
tions and disorder can increase the anisotropy but the
major effect comes from the surface. We notice that the
number of clusters considered is limited in the statistical
average.
IV. Kambersky’s simplified formula
InordertoconnecttheanisotropyoftheGilbertdamp-
ing to features in the electronic structure, we consider in
the following Kambersky’s simplified formula for Gilbert
damping [47, 48]α=1
γMs/parenleftBigγ
2/parenrightBig2
n(EF)ξ2(g−2)2
τ.(S8)
Here,γis the gyromagnetic ratio, n(EF)represents the
LDOS at the Fermi level, ξis the SOC strength, τis the
electron scattering time, Msis the spin magnetic mo-
ment, andgis the spectroscopic g-factor [35, 49]. Note
that Eq. S8 demonstrates the direct relation between
αandn(EF), often discussed in the literature, e.g., in
Ref. [27]. Our first principles calculations have shown
no significant change in ξ, upon variation of the mag-
netization axis, for the FeCo systems ( ξCo= 71.02meV
andξFe= 53.47meV). Hence, we can soundly relate the
damping anisotropy ∆αtto∆n(EF).
Figure S2 shows how the LDOS difference (per atom)
∆n(E)between the [010]and [110]magnetization di-
rections is developed in pure Fe(001) bcc and in VCA
Fe50Co50(001) bcc surfaces, respectively. In both cases,5
TableS2. Totaldamping( αt×10−3)ofatypicalatomin
each system for the spin quantization axes [010](θH=
90◦) and [110](θH= 45◦); also shown for the [001]and
[111]. Bulk and surface bct systems are simulated with
c
a= 1.15.
Bulks
Bulk αt[010]αt[110] ∆αt
Fe bcc 4.18 4.31 +3.1%a
B2-FeCo bcc 2.28 2.44 +7.2%
B2-FeCo bct 7.76 8.85 +12.4%
VCA Fe 50Co50bcc 3.70 4.18 +13.0%
VCA Fe 50Co50bct 4.69 5.10 +8.7%
αt[010]αt[001] ∆αt
B2-FeCo bct 7.76 10.21 +24.1%
VCA Fe 50Co50bct 4.69 5.75 +22.6%
αt[010]αt[111] ∆αt
Fe bcc 4.18 4.56 +9.1%b
Surfaces
Surface αt[010]αt[110] ∆αt
Fe(001) bcc 3.85 5.75 +49.4%
Fe/GaAs(001) bcc [38] 4.7(7) 7.2(7) +53(27)%c
Fe/MgO(001) bcc [45] 3.20(25) 6.15(20) +92(14)%d
VCA Fe 50Co50(001) bcc 7.00 14.17 +102.4%
VCA Fe 50Co50(001) bct 15.20 14.80−2.6%
αt[010]αt[001] ∆αt
VCA Fe 50Co50(001) bct 15.20 15.56 +2.4%
VCA Fe 50Co50(001) bcc 7.00 9.85 +40.7%
aMankovsky et al.[24] find a damping anisotropy of ∼12%
for bulk Fe bcc at low temperatures ( ∼50K) between
[010] and [011] magnetization directions. For this result,
the definition α=1
2(αxx+αyy)was used.
bThis result agrees with Gilmore et al.[46], which find
that the total damping of pure Fe bcc presents its higher
value in the [111] crystallographic orientation and the
lower value in the [001] direction, except for high scatter-
ing rates. Also agrees with Mankovsky et al.[24] results.
cAnisotropic damping obtained for a 0.9 nm Fe/GaAs(001)
thin film (sample S2 in Ref. [38]) in the [010] and [110]
magnetization directions.
dFor a Fe(15 nm)/MgO(001) film at T= 4.5K in the high-
est applied magnetic field, in which only intrinsic contri-
butions to the anisotropic damping are left.
the chosen layer,denoted as first, is the most external
one (near vacuum). the VCA Fe50Co50(001) bcc we also
calculated ∆n(E)for all layers summed (total DOS dif-
ference).
As can be seen, although in all cases the quantity
∆n(E)exhibits some oscillations, differently from what
we observe forthe pureFe(001) surface case, at the Fermi
energy, there is a non-negligible difference in the minor-
ity spin channel ( 3dstates) for the VCA Fe50Co50(001).
Considering the results presented in Table I (main text)
the larger contribution to the damping anisotropy comes
from the most external layer. The results by Li et al.
[22] indicate a small difference (for two magnetization
directions) of the total density of states at the FermiTable S3. Total damping anisotropy ( ×10−3) of all stud-
ied Co-centered and Fe-centered bcc clusters for the spin-
quantization axis [010]and [110], considering the 15-atom
FeCo cluster together with the VCA medium in the summa-
tion for total damping.
Co-centered
Cluster # αt[010]αt[110] ∆αt
1 10.11 9.65 4.8%
2 8.09 6.96 16.2%
3 7.81 7.02 11.3%
4 7.11 7.02 1.3%
5 7.48 6.88 8.7%
Average 8.12 7.51 8.1%
Fe-centered
Cluster # αt[010]αt[110] ∆αt
1 2.68 2.03 32.0%
2 2.49 2.05 21.5%
3 2.56 1.86 37.6%
4 2.45 1.79 36.9%
5 2.76 2.01 37.3%
Average 2.59 1.95 32.8%
Table S4. Total damping anisotropy ( ×10−3) of all stud-
ied Co-centered and Fe-centered bcc clusters for the spin
quantization axis [010]and [110], with bct-like distortions/parenleftbigc
a= 1.15/parenrightbig
, considering the 15-atom FeCo cluster together
with the VCA medium in the summation for total damping.
Co-centered
Cluster # αt[010]αt[110] ∆αt
1 5.85 4.37 33.9%
2 5.95 4.21 41.3%
3 5.88 4.35 35.2%
4 5.90 4.41 33.8%
5 5.86 4.34 35.0%
Average 5.89 4.34 35.7%
Fe-centered
Cluster # αt[010]αt[110] ∆αt
1 2.36 1.39 69.8%
2 2.27 1.32 72.0%
3 2.22 1.26 76.2%
4 2.25 1.26 78.6%
5 2.42 1.38 75.4%
Average 2.30 1.32 74.2%
Table S5. Onsite damping ( αonsite×10−3) of a typical atom
in each layer of the VCA Fe 50Co50(001) bcc for the spin quan-
tization axis [010]and[110].
Layerαonsite[010]αonsite[110] ∆αonsite
1 7.36 8.70 +18.2%
2 0.63 0.69 +9.5%
3 1.41 1.44 +2.1%
4 0.87 0.86−1.1%
5 0.99 0.97−2.0%6
TableS6. SummaryofthemainFe 50Co50dampinganisotropy
results for: pure ordered ( B2) alloy; pure random (VCA) bulk
alloy; bcc bulk together with short-range order (SRO) clus-
ters (see Table S3); bulk together with bct-like distorted clus-
ters inside (see Table S4); surface calculations, in the pristine
mode and with explicit bct-like clusters embedded (surface +
distortion). The maximum-minimum ratio according to Ref.
[22] isα[110]
t
α[010]
t×100%.
Structure ∆αtMax-min ratio
Ordered alloy bcc 7.2% 107.2%
Ordered alloy bct 24.1% 124.1%
Random alloy bcc 13% 113%
Random alloy bct 22.6% 122.6%
Random alloy + SRO 14.9% 114.9%
Random alloy + SRO + Distortion 47.2% 147.2%
Surface (external layer) 102.4% 202.4%
Surface (ext. layer) + Distortion 75.4% 175.4%
10-nm Co 50Fe50/Pt [22] (exp.) 281.3% 381.3%
−2−1 0 1 2
−0.02 −0.01 0 0.01 0.02Δn(E)[010]−[110] (states/Ry−atom)
Energy (E−EF) (Ry)Fe(001) bcc (first)
VCA Fe50Co50(001) bcc (first)
VCA Fe50Co50(001) bcc (all)
Figure S2. LDOS difference (per atom), ∆n(E), between the
[010]and[110]magnetization directions, for both spin chan-
nels (full lines for majority spin and dashed lines for minority
spin states), in the outermost layer in pure Fe(001) bcc (in
black); outermost layer in VCA Fe 50Co50(001) bcc (in blue);
and all layers summed in VCA Fe 50Co50(001) bcc (in red).
level,N(EF), what the authors claim that could not ex-
plain the giant maximum-minimum damping ratio ob-
served. So, in order to clarify this effect in the VCA
Fe50Co50(001) bcc, ∆n(E)was also calculated for the all
layers summed, what is shown in Fig. S2 (in red). This
difference is in fact smaller if we consider the DOS of
the whole system, with all layers summed. However, if
we consider only the most external layer, then the LDOS
variation is enhanced. This is consistent with our theo-
retical conclusions. As we mention in the main text, this
do not rule out a role also played by local (tetragonal-
like) distortions and other bulk-like factors in the damp-
ing anisotropy.For the outermost layer of Fe(001) bcc, the calculated
LDOSatEFis∼20.42states/Ry-atominthe[110]direc-
tion and∼20.48states/Ry-atom in the [010] direction,
which represents a difference of ∼0.3%and agrees with
the calculations performed by Chen et al.[38].
V. Correlation with anisotropic orbital moment
Besides the close relation exhibited between ∆αt
and∆n(EF), we also demonstrate the existence of an
anisotropic orbital moment in the outermost layer, in
which the fourfold symmetry ( C4v) matches the damp-
ing anisotropy with a 90◦phase. Fig. S3 shows this
correlation between ∆αtand∆morbfor two situations:
(i) for a typical atom in the outermost layer of VCA
Fe50Co50(001) bcc (blue open dots); and ( ii) for a typi-
cal atom in the VCA Fe50Co50bcc bulk, considering the
same ∆morbscale. For case ( i) we find orbital moments
differencesmorethanoneorderofmagnitudehigherthan
case (ii).
0o45o90o
135o
180o
225o
270o315o[100][110][010]
[−110]
0.51.52.5
Δmorb
(µB/atom × 10−3)
0.0050.0100.015
αt
θH
Figure S3. (Color online) Total damping and orbital moment
difference, ∆morbas a function of θH, the angle between the
magnetizationdirectionandthe [100]-axis. Squares: (redfull)
VCA Fe 50Co50(001) bcc. Circles: (blue open) morbdifference
betweenθH= 90◦and the current angle for a typical atom in
the outermost layer of VCA Fe 50Co50(001) bcc; and (yellow
full) samemorbdifference but for a typical atom in the VCA
Fe50Co50bcc bulk (in the same scale). Lines are guides for
the eyes.
VI. Contribution from next-nearest-neighbours
Finally, we show in Fig. S4 the summation of all non-
local damping contributions, αij, for a given normalized
distance in the outermost layer of VCA Fe50Co50(001)7
bcc. As we can see, the next-nearest-neighbours from a
reference site (normalized distanced
a= 1) have very dis-
tinctαijcontributions to αtfor the two different mag-
netization directions ( [010]and[110]), playing an impor-
tant role on the final damping anisotropy. We must note,
however, that these neighbours in a (001)-oriented bcc
surface are localized in the same layer as the reference
site, most affected by the interfacial SOC. Same trend is
observed ford
a= 2, however less intense. This is con-
sistent with our conclusions, about the relevance of the
outermost layer on ∆αt.
−3−2−1 0 1 2
0.5 1 1.5 2 2.5 3 3.5 4∑αij × 10−3
Normalized distance[110]
[010]
Figure S4. (Color online) Summation of all non-local Gilbert
damping parameters ( αij,i/negationslash=j) in each neighboring normal-
izeddistancebetweensites iandjfortheVCAFe 50Co50(001)
bcc in the two most different directions for the damping
anisotropy: [010](θH= 90◦), in blue open squares, and [110]
(θH= 45◦), in red full circles. Lines are guides for the eyes.
∗aklautau@ufpa.br
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0905.4544v2.Hydrodynamic_theory_of_coupled_current_and_magnetization_dynamics_in_spin_textured_ferromagnets.pdf | Hydrodynamic theory of coupled current and magnetization dynamics in
spin-textured ferromagnets
Clement H. Wong and Yaroslav Tserkovnyak
Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
We develop the hydrodynamic theory of collinear spin currents coupled to magnetization dynamics
in metallic ferromagnets. The collective spin density couples to the spin current through a U(1)
Berry-phase gauge eld determined by the local texture and dynamics of the magnetization. We
determine phenomenologically the dissipative corrections to the equation of motion for the electronic
current, which consist of a dissipative spin-motive force generated by magnetization dynamics and
a magnetic texture-dependent resistivity tensor. The reciprocal dissipative, adiabatic spin torque
on the magnetic texture follows from the Onsager principle. We investigate the e ects of thermal
uctuations and nd that electronic dynamics contribute to a nonlocal Gilbert damping tensor in
the Landau-Lifshitz-Gilbert equation for the magnetization. Several simple examples, including
magnetic vortices, helices, and spirals, are analyzed in detail to demonstrate general principles.
PACS numbers: 72.15.Gd,72.25.-b,75.75.+a
I. INTRODUCTION
The interaction of electrical currents with magnetic
spin texture in conducting ferromagnets is presently a
subject of active research. Topics of interest include
current-driven magnetic dynamics of solitons such as do-
main walls and magnetic vortices,1,2,3,4as well as the
reciprocal process of voltage generation by magnetic
dynamics.5,6,7,8,9,10,11,12This line of research has been
fueled in part by its potential for practical applications
to magnetic memory and data storage devices.13Funda-
mental theoretical interest in the subject dates back at
least two decades.5,6,14It was recognized early on6that
in the adiabatic limit for spin dynamics, the conduction
electrons interact with the magnetic spin texture via an
e ective spin-dependent U(1) gauge eld that is a local
function of the magnetic con guration. This gauge eld,
on the one hand, gives rise to a Lorentz force due to
\ ctitious" electric and magnetic elds and, on the other
hand, mediates the so-called spin-transfer torque exerted
by the conduction electrons on the collective magnetiza-
tion. An alternative and equivalent view is to consider
this force as the result of the Berry phase15accumulated
by an electron as it propagates through the ferromagnet
with its spin aligned with the ferromagnetic exchange
eld.8,10,16In the standard phenomenological formalism
based on the Landau-Lifshitz-Gilbert (LLG) equation,
the low-energy, long-wavelength magnetization dynamics
are described by collective spin precession in the e ective
magnetic eld, which is coupled to electrical currents via
the spin-transfer torques. In the following, we develop
a closed set of nonlinear classical equations governing
current-magnetization dynamics, much like classical elec-
trodynamics, with the LLG equation for the spin-texture
\ eld" in lieu of the Maxwell equations for the electro-
magnetic eld.
This electrodynamic analogy readily explains various
interesting magnetoelectric phenomena observed recently
in ferromagnetic metals. Adiabatic charge pumping bymagnetic dynamics17can be understood as the gener-
ation of electrical currents due to the ctitious electric
eld.18In addition, magnetic textures with nontrivial
topology exhibit the so-called topological Hall e ect,19,20
in which the ctitious magnetic eld causes a classical
Hall e ect. In contrast to the classical magnetoresis-
tance, the
ux of the ctitious magnetic eld is a topo-
logical invariant of the magnetic texture.6
Dissipative processes in current-magnetization dynam-
ics are relatively poorly understood and are of central
interest in our theory. Electrical resistivity due to quasi-
one-dimensional (1D) domain walls and spin spirals have
been calculated microscopically.21,22,23More recently, a
viscous coupling between current and magnetic dynam-
ics which determines the strength of a dissipative spin
torque in the LLG equation as well the reciprocal dis-
sipative spin electromotive force generated by magnetic
dynamics, called the \ coecient,"2was also calcu-
lated in microscopic approaches.3,24,25Generally, such
rst-principles calculations are technically dicult and
restricted to simple models. On the other hand, the num-
ber of di erent forms of the dissipative interactions in the
hydrodynamic limit are in general constrained by sym-
metries and the fundamental principles of thermodynam-
ics, and may readily be determined phenomenologically
in a gradient expansion. Furthermore, classical thermal
uctuations may be easily incorporated in the theoretical
framework of quasistationary nonequilibrium thermody-
namics.
The principal goal of this paper is to develop a (semi-
phenomenological) hydrodynamic description of the dis-
sipative processes in electric
ows coupled to magnetic
spin texture and dynamics. In Ref. 11, we drew the anal-
ogy between the interaction of electric
ows with quasis-
tationary magnetization dynamics with the classical the-
ory of magnetohydrodynamics. In our \spin magnetohy-
drodynamics," the spin of the itinerant electrons, whose
ows are described hydrodynamically, couples to the lo-
cal magnetization direction, which constitutes the col-
lective spin-coherent degree of freedom of the electronicarXiv:0905.4544v2 [cond-mat.mes-hall] 16 Nov 20092
uid. In particular, the dissipative coupling between
the collective spin dynamics and the itinerant electrons
is loosely akin to the Landau damping, capturing cer-
tain kinematic equilibration of the relative motion be-
tween spin-texture dynamics and electronic
ows. In our
previous paper,11we considered a special case of incom-
pressible
ows in a 1D ring to demonstrate the essential
physics. In this paper, we establish a general coarse-
grained hydrodynamic description of the interaction be-
tween the electric
ows and textured magnetization in
three dimensions, treating the itinerant electron's degrees
of freedom in a two-component
uid model (correspond-
ing to the two spin projections of spin-1 =2 electrons along
the local collective magnetic order). Our phenomenology
encompasses all the aforementioned magnetoelectric phe-
nomena.
The paper is organized as follows. In Sec. II, we use a
Lagrangian approach to derive the semiclassical equation
of motion for itinerant electrons in the adiabatic approx-
imation for spin dynamics. In Sec. III, we derive the
basic conservation laws, including the Landau-Lifshitz
equation for the magnetization, by coarse-graining the
single-particle equation of motion and the Hamiltonian.
In Sec. IV, we phenomenologically construct dissipative
couplings, making use of the Onsager reciprocity princi-
ple, and calculate the net dissipation power. In particu-
lar, we develop an analog of the Navier-Stokes equation
for the electronic
uid, focusing on texture-dependent
e ects, by making a systematic expansion in nonequi-
librium current and magnetization consistent with sym-
metry requirements. In Sec. V, we include the e ects of
classical thermal
uctuations by adding Langevin sources
to the hydrodynamic equations, and arrive at the central
result of this paper: A set of coupled stochastic di er-
ential equations for the electronic density, current, and
magnetization, and the associated white-noise correlators
of thermal noise. In Sec. VI, we apply our results to
special examples of rotating and spinning magnetic tex-
tures, calculating magnetic texture resistivity and mag-
netic dynamics-generated currents for a magnetic spiral
and a vortex. The paper is summarized in Sec. VII and
some additional technical details, including a microscopic
foundation for our semiclassical theory, are presented in
the appendices.
II. QUASIPARTICLE ACTION
In a ferromagnet, the magnetization is a symmetry-
breaking collective dynamical variable that couples to the
itinerant electrons through the exchange interaction. Be-
fore developing a general phenomenological framework,
we start with a simple microscopic model with Stoner in-
stability, which will guide us to explicitly construct some
of the key magnetohydrodynamic ingredients. Within a
low-temperature mean- eld description of short-ranged
electron-electron interactions, the electronic action isgiven by (see appendix A for details):
S=Z
dtd3r^ y
i~@t+~2
2mer2
2+
2m^
^ :(1)
Here, ( r;t) is the ferromagnetic exchange splitting,
m(r;t) is the direction of the dynamical order param-
eter de ned by ~h^ y^^ i=2 =sm,sis the local spin
density, and ^ (r;t) is the spinor electron eld operator.
For the short-range repulsion U > 0 discussed in ap-
pendix A, ( r;t) = 2Us(r;t)=~and (r;t) =U(r;t),
where=h^ y^ iis the local particle number density.
For electrons, the magnetization Mis in the opposite di-
rection of the spin density: M=
sm, where
<0 is
the gyromagnetic ratio. Close to a local equilibrium, the
magnetic order parameter describes a ground state con-
sisting of two spin bands lled up to the spin-dependent
Fermi surfaces, with the spin orientation de ned by m.
We will focus on soft magnetic modes well below the
Curie temperature, where only the direction of the mag-
netization and spin density are varied, while the
uctu-
ations of the magnitudes are not signi cant. The spin
density is given by s=~(+ )=2 and particle den-
sity by=++ , whereare the local spin-up/down
particle densities along m.scan be essentially constant
in the limit of low spin susceptibility.
Starting with a nonrelativistic many-body Hamilto-
nian, the action (1) is obtained in a spin-rotationally
invariant form. However, this symmetry is broken by
spin-orbit interactions, whose role we will take into ac-
count phenomenologically in the following. When the
length scale on which m(r;t) varies is much greater than
the ferromagnetic coherence length lc~vF=, where
vFis the Fermi velocity, the relevant physics is captured
by the adiabatic approximation. In this limit, we start
by neglecting transitions between the spin bands, treat-
ing the electron's spin projection on the magnetization
as a good quantum number. (This approximation will
be relaxed later, in the presence of microscopic spin-
orbit or magnetic disorder.) We then have two e ec-
tively distinct species of particles described by a spinor
wave function ^ 0, which is de ned by ^ =^U(R)^ 0. Here,
^U(R) is an SU(2) matrix corresponding to the local spa-
tial rotationR(r;t) that brings the z-axis to point along
the magnetization direction: R(r;t)z=m(r;t), so that
^Uy(^m)^U= ^z. The projected action then becomes:
S=Z
dtZ
d3r^ 0y"
(i~@t+ ^a) ( i~r ^a)2
2me
2+
2^z
^ 0 Z
dtF[m];(2)
where
F[m] =A
2Z
d3r(@im)2(3)
is the spin-texture exchange energy (implicitly summing
over the repeated spatial index i), which comes from the3
terms quadratic in the gauge elds that survive the pro-
jection. In the mean- eld Stoner model, the ferromag-
netic exchange sti ness is A=~2=4me. To broaden our
scope, we will treat it as a phenomenological constant,
which, for simplicity, is determined by the mean electron
density.26The spin-projected \ ctitious" gauge elds are
given by
a(r;t) =i~hj^Uy@t^Uji;
a(r;t) =i~hj^Uyr^Uji: (4)
Choosing the rotation matrices ^U(m) to depend only on
the local magnetic con guration, it follows from their
de nition that spin- gauge potentials have the form:
a= @tmamon
(m); ai= @imamon
(m);(5)
where amon
(m) i~hj^Uy@m^Uji. We show in Ap-
pendix B the well known result (see, e.g., Ref. 27) that
amon
is the vector potential (in an arbitrary gauge) of
a magnetic monopole in the parameter space de ned by
m:
@mamon
(m) =qm; (6)
whereq=~=2 is the monopole charge (which is ap-
propriately quantized).
By noting that the action (2) is formally identical to
charged particles in electromagnetic eld, we can imme-
diately write down the following classical single-particle
Lagrangian for the interaction between the spin- elec-
trons and the collective spin texture:
L(r;_r;t) =me_r2
2+_ra(r;t) +a(r;t); (7)
where _ris the spin-electron (wave-packet) velocity. To
simplify our discussion, we are omitting here the spin-
dependent forces due to the self-consistent elds (r;t)
and ( r;t), which will be easily reinserted at a later
stage. See Eq. (29).
The Euler-Lagrange equation of motion for v=
_rderived from the single-particle Lagrangian (7),
(d=dt)(@L=@_r) =@L=@r, gives
me_v=q(e+vb): (8)
The ctitious electromagnetic elds that determine the
Lorentz force are
qei=@ia @tai=qm(@tm@im);
qbi=ijk@jak=qijk
2m(@km@jm):(9)
They are conveniently expressed in terms of the tensor
eld strength
qf@a @a=qm(@m@m) (10)
byei=fi0andbi=ijkfjk=2.ijkis the antisymmet-
ric Levi-Civita tensor and we used four-vector notation,de ning@= (@t;r) anda= (a;a). Here and
henceforth the convention is to use Latin indices to de-
note spatial coordinates and Greek for space-time coor-
dinates. Repeated Latin indices i;j;k are, furthermore,
always implicitly summed over.
III. SYMMETRIES AND CONSERVATION
LAWS
A. Gauge invariance
The Lagrangian describing coupled electron transport
and collective spin-texture dynamics (disregarding for
simplicity the ordinary electromagnetic elds) is
L(rp;vp;m;@m)
=X
p
mev2
p
2+vpa+a!
A
2Z
d3r(@im)2
=X
p
mev2
p
2+v
pa!
A
2Z
d3r(@im)2:(11)
v
p(1;vp),vp=_r, andhere is the spin of indi-
vidual particles labelled by p. The resulting equations
of motion satisfy certain basic conservation laws, due to
spin-dependent gauge freedom, space-time homogeneity,
and spin isotropicity.
First, let us establish gauge invariance due to an ambi-
guity in the choice of the spinor rotations ^U(r;t)!^U^U0.
Our formulation should be invariant under arbitrary di-
agonal transformations ^U0=e ifand ^U0=e ig^z=2on
the rotated fermionic eld ^ 0, corresponding to gauge
transformations of the spin-projected theory:
a=~@fanda=~@g=2; (12)
respectively. The change in the Lagrangian density is
given by
L=j@fandL=j
s@g; (13)
respectively, where j=j++j andjs=~(j+ j )=2
are the corresponding charge and spin gauge currents.
The action S=R
dtd3rLis gauge invariant, up to sur-
face terms that do not a ect the equations of motion,
provided that the four-divergence of the currents vanish,
which is the conservation of particle number and spin
density:
_+rj= 0;_s+rjs= 0: (14)
(The second of these conservation laws will be relaxed
later.) Here, the number and spin densities along with
the associated
ux densities are
=X
pnp++ ;
j=X
pnpvpv; (15)4
and
s=X
pqnp~
2(+ );
js=X
pqnpvpsvs; (16)
wherenp=(r rp) andp=for spins up and down.
In the hydrodynamic limit, the above equations deter-
mine the average particle velocity vand spin velocity
vs, which allows us to de ne four-vectors j= (;v)
andj
s= (s;svs). Microscopically, the local spin-
dependent currents are de ned, in the presence of electro-
magnetic vector potential aand ctitious vector potential
a, by
mev= Reh y
( i~r a ea) i; (17)
wheree<0 is the electron charge.
B. Angular and linear momenta
Our Lagrangian (11) contains the dynamics of m(r)
that is coupled to the current. In this regard, we note
that the time component of the ctitious gauge poten-
tial (B4),a= ~@t'(1 cos)=2, is a Wess-Zumino
action that governs the spin-texture dynamics.4,6,28The
variational equation mmL= 0 gives:
s(@t+vsr)m+mmF= 0: (18)
To derive this equation, we used the spin-density con-
tinuity equation (14) and a gauge-independent identity
satis ed by the ctitious potentials: their variations with
respect to mare given by
ma(m;@m) =qm@m; (19)
where
m@
@m X
@@
@(@m): (20)
One recognizes that Eq. (18) is the Landau-Lifshitz (LL)
equation, in which the spin density precesses about the
e ective eld given explicitly by
hmF= A@2
im: (21)
Equation (18) also includes the well-known reactive spin
torque: = (jsr)m,3which is evidently the change
in the local spin-density vector due to the spin angular
momentum carried by the itinerant electrons. One can
formally absorb this spin torque by de ning an advective
time derivative Dt@t+vsr, with respect to the
average spin drift velocity vs.
Equation (18) may be written in a form that explicitly
expresses the conservation of angular momentum:27,29
@t(smi) +@jij= 0; (22)where the angular-momentum stress tensor is de ned by
ij=svsjmi A(m@jm)i: (23)
Notice that this includes both quasiparticle and collective
contributions, which stem respectively from the trans-
port and equilibrium spin currents.
The Lorentz force equation for the electrons, Eq. (8),
in turn, leads to a continuity equation for the kinetic
momentum density.6To see this, let us start with the
microscopic perspective:
@t(vi) =@tX
pnpvp=X
p( _npvp+np_vp): (24)
Using the Lorentz force equation for the second term, we
have:
meX
pnp_vp=X
pqnp(ei+ijkbkvpj) =X
pqnpfiv
p
=sm(@tm@im) +svsjm(@jm@im)
= (@im)(mF) = A(@im)(@2
jm); (25)
utilizing Eq. (18) to obtain the last line. Coarse-graining
the rst term of Eq. (24), in turn, we nd:
X
p_npvp= @jX
p(r rp)vpivpj! @jX
vivj:
(26)
Putting Eqs. (25) and (26) together, we can nally write
Eq. (24) in the form:
me@t(vi) +@j
Tij+meX
vivj!
= 0;(27)
where
Tij=A
(@im)(@jm) ij
2(@km)2
(28)
is the magnetization stress tensor.6
A spin-dependent chemical potential ^ =^K 1^gov-
erned by local density and short-ranged interactions can
be trivially incorporated by rede ning the stress tensor
as
Tij!Tij+ij
2^T^K 1^: (29)
In our notation, ^ = (+; )T, ^= (+; )Tand ^Kis
a symmetric 22 compressibility matrix in spin space,
which includes the degeneracy pressure as well as self-
consistent exchange and Hartree interactions. In general,
Eq. (29) is valid only for suciently small deviations from
the equilibrium density.
Using the continuity equations (14), we can combine
the last term of Eq. (27) with the momentum density
rate of change:
@t(vi) +@j(vivj) =(@t+vr)vi;(30)5
which casts the momentum density continuity equation
in the Euler equation form:
meX
(@t+vr)vi+@jTij= 0: (31)
We do not expect such advective corrections to @tto
play an important role in electronic systems, however.
This is in contrast to the advective-like time derivative
in Eq. (18), which is rst order in velocity eld and is
crucial for capturing spin-torque physics.
C. Hydrodynamic free energy
We will now turn to the Hamiltonian formulation and
construct the free energy for our magnetohydrodynamic
variables. This will subsequently allow us to develop a
nonequilibrium thermodynamic description. The canon-
ical momenta following from the Lagrangian (11) are
pp@L
@vp=mevp+ap;
@L
@_m=X
pnp@a
@_m=X
pnpamon
(m): (32)
Notice that for our translationally-invariant system, the
total linear momentum
PX
ppp+Z
d3r(r)m=meX
pvp; (33)
where we have used Eq. (5) to obtain the second equality,
coincides with the kinetic momentum (mass current) of
the electrons. The latter, in turn, is equivalent to the lin-
ear momentum of the original problem of interacting non-
relativistic electrons, in the absence of any real or cti-
tious gauge elds. See appendix A. While Pis conserved
(as discussed in the previous section and also follows now
from the general principles), the canonical momenta of
the electrons and the spin-texture eld, Eqs. (32), are
not conserved separately. As was pointed out by Volovik
in Ref. 6, this explains anomalous properties of the lin-
ear momentum associated with the Wess-Zumino action
of the spin-texture eld: This momentum has neither
spin-rotational nor gauge invariance. The reason is that
the spin-texture dynamics de ne only one piece of the
total momentum, which is associated with the coherent
degrees of freedom. Including also the contribution as-
sociated with the incoherent (quasiparticle) background
restores the proper gauge-invariant momentum, P, which
corresponds to the generator of the global translation in
the microscopic many-body description.
Performing a Legendre transformation to Hamiltonianas a function of momenta, we nd
H[rp;pp;m;] =X
pvppp+Z
d3r_m L
=X
p(pp a)2
2me+A
2Z
d3r(@im)2
E+F; (34)
whereEis the kinetic energy of electrons and Fis the
exchange energy of the magnetic order. As could be
expected,Eis the familiar single-particle Hamiltonian
coupled to an external vector potential. According to
a Hamilton's equation, the velocity is conjugate to the
canonical momentum: vp=@H=@ pp. We note that ex-
plicit dependence on the spin-texture dynamics dropped
out because of the special property of the gauge elds:
_m@_ma=a. Furthermore, according to Eq. (19), we
havemmE= (jsr)m, so the LL Eq. (18) can be
written in terms of the Hamiltonian (34) as11
s_m+mmH= 0: (35)
So far, we have included in the spin-texture equa-
tion only the piece coupled to the itinerant electron de-
grees of freedom. The purely magnetic part is tedious
to derive directly and we will include it in the usual LL
phenomenology.29To this end, we rede ne
F[m(r)]!F+F0; (36)
by adding an additional magnetic free energy F0[m(r)],
which accounts for magnetostatic interactions, crystalline
anisotropies, coupling to external elds, as well as energy
associated with localized dorforbitals.30Then the to-
tal free energy (Hamiltonian) is H=E+F, and we in
general de ne the e ective magnetic eld as the thermo-
dynamic conjugate of m:hmH. The LL equation
then becomes
%s_m+mh= 0; (37)
where%sis the total e ective spin density. To enlarge
the scope of our phenomenology, we allow the possibility
that%s6=s. For example, in the s dmodel, an extra
spin density comes from the localized d-orbital electrons.
Microscopically, %s@tmterm in the equation of motion
stems from the Wess-Zumino action generically associ-
ated with the total spin density.
In the following, it may sometimes be useful to separate
out the current-dependent part of the e ective eld, and
write the purely magnetic part as hmmF, so that
h=hm m(jsr)m (38)
and Eq. (37) becomes:
%s_m+ (jsr)m+mhm= 0: (39)6
For completeness, let is also write the equation of motion
for the spin- acceleration:
me(@t+vr)vi=q[m(@tm@im)
+vjm(@jm@im)] r;(40)
retaining for the moment the advective correction to
the time derivative on the left-hand side and reinserting
the force due to the spin-dependent chemical potential,
^=^K 1^. These equations constitute the coupled re-
active equations for our magneto-electric system. The
Hamiltonian (free energy) in terms of the collective vari-
ables is (including the elastic compression piece)
H[;p;m] =X
Z
d3r(p a)2
2me
+1
2Z
d3r^T^K 1^+F[m]; (41)
where p=mev+ais the spin-dependent momentum
that is locally averaged over individual particles.
D. Conservation of energy
So far, our hydrodynamic equations are reactive, so
that the energy (41) must be conserved: P_H=_E+
_F= 0. The time derivative of the electronic energy Eis
_E=Z
d3rX
mev_v+ _mev2
2+
=Z
d3rX
mevj_vj @j(vj)mev2
2+
=Z
d3rX
vj[me(@t+vr)vj+@j]
=Z
d3rX
qv(e+vb)
=Z
d3rX
qve=Z
d3rjse: (42)
The change in the spin-texture energy is given, according
to Eq. (39), by
_F=Z
d3r_mmF=Z
d3r_mhm
=Z
d3r_m[%sm_m+m(jsr)m)]
= Z
d3rjse: (43)
The total energy is thus evidently conserved, P= 0.
When we calculate dissipation in the rest of the paper,
we will omit these terms which cancel each other. The
total energy
ux density is evidently given by
Q=X
mev2
2+
v: (44)IV. DISSIPATION
Having derived from rst principles the reactive cou-
plings in our magneto-electric system, summed up in
Eqs. (39)-(41), we will proceed to include the dissipa-
tive e ects phenomenologically. Let us focus on the lin-
earized limit of small deviations from equilibrium (which
may be spin textured), so that the advective correction
to the time derivative in the Euler Eq. (40), which is
quadratic in the velocity eld, can be omitted. To elimi-
nate the quasiparticle spin degree of freedom, let us, fur-
thermore, treat halfmetallic ferromagnets, so that =+
ands=q, whereq=~=2 is the electron's spin.31From
Eq. (40), the equation of motion for the local (averaged)
canonical momentum is:32
_p=q
jb r; (45)
in a gauge where a= 0, so that _p=me_v qe.33
==K. The Lorentz force due to the applied (real)
electromagnetic elds can be added in the obvious way
to the right-hand side of Eq. (45). Note that since we
are now interested in linearized equations close to equi-
librium,in Eq. (45) can be approximated by its (ho-
mogeneous) equilibrium value.
Introducing relaxation through a phenomenological
damping constant (Drude resistivity)
=me
; (46)
where is the collision time, expressing the ctitious
magnetic eld in terms of the spin texture, Eq. (45) be-
comes:
_pi= q
(m@im)(jr)m @i
ji: (47)
Adding the phenomenological Gilbert damping34to
the magnetic Eq. (37) gives the Landau-Lifshitz-Gilbert
equation:
%s(_m+ m_m) =hm; (48)
where is the damping constant. Eqs. (47) and
(48), along with the continuity equation, _ = rj,
are the near-equilibrium thermodynamic equations for
(;p;m) and their respective thermodynamic conjugates
(;j;h) = (H;pH;mH). This system of equations of
motion may be written formally as
@t0
@
p
m1
A=b [m(r)]0
@
j
h1
A: (49)
The matrix ^ depends on the equilibrium spin texture
m(r). By the Onsager reciprocity principle, ij[m] =
sisj ji[ m], wheresi=if theith variable is even
(odd) under time reversal.7
In the quasistationary description of a nonequilibrium
thermodynamic system, the entropy S[;p;m] is for-
mally regarded as a functional of the instantaneous ther-
modynamic variables, and the probability of a given con-
guration is proportional to eS=kB. If the heat conduc-
tance is high and the temperature Tis uniform and con-
stant, the instantaneous rate of dissipation P=T_Sis
given by the rate of change in the free energy, P=_H=R
d3rP:
P= _ h_m j_p= %s_m2+
j2; (50)
where we used Eq. (47) and expressed the e ective eld
has a function of _mby taking mof Eq. (48):
h=%sm_m %s_m: (51)
Notice that the ctitious magnetic eld bdoes not con-
tribute to dissipation because it does not do work.
So far, there is no dissipative coupling between the
current and the spin-texture dynamics, and the macro-
scopic equations obey the global time-reversal symme-
try. However, we know that dissipative couplings ex-
ists due to the misalignment of the electron's spin with
the collective spin texture and spin-texture resistivity.3,22
Following Ref. 11, we add these well-known e ects phe-
nomenologically by making an expansion in the equations
of motion to linear order in the nonequilibrium quanti-
ties _mandj. To limit the number of terms one can write
down, we will only add terms that are spin-rotationally
invariant and isotropic in real space (which disregards,
in particular, such e ects as the angular magnetoresis-
tance and the anomalous Hall e ect). To second order in
the spatial gradients of m, there are only three dissipa-
tive phenomenological terms with couplings ,0, and
consistent with the above requirements, which could be
added to the right-hand side of Eq. (47).35The momen-
tum equation becomes:
_pi= q
(m@im)(jr)m @i
ji
(@km)2ji 0@im(jr)m q _m@im:(52)
It is known that the \ term" comes from a misalignment
of the electron spin with the collective spin texture, and
the associated dephasing. It is natural to expect thus
that the dimensionless parameter ~= s, where s
is a characteristic spin-dephasing time.3The \terms"
evidently describe texture-dependent resistivity, which
is anisotropic with respect to the gradients in the spin
texture along the local current density. Such term are
also naturally expected, in view of the well-known giant-
magnetoresistance e ect,36in which noncollinear magne-
tization results in electrical resistance. The microscopic
origin of this term is due to spin-texture misalignment,
which modi es electron scattering.
The total spin-texture-dependent resistivity can be putinto a tensor form:
ij[m] =ij
+(@km)2
+0@im@jm
+q
m(@im@jm): (53)
The last term due to ctitious magnetic eld gives a Hall
resistivity. Note that ^
[m] = ^
T[ m], consistent with
the Onsager theorem. We can nally write Eq. (47) as:
_pi=
ij[m]jj @i q _m@im: (54)
As was shown in Ref. 11, since the Onsager relations
require thatb [m] =b [ m]Twithin the current/spin-
texture elds sector, there must be a counterpart to the
term above in the magnetic equation, which is the well-
known dissipative \ spin torque:"
%s(_m+ m_m) =hm q m(jr)m:(55)
The total dissipation Pis now given by
P= %s_m2+ 2q _m(jr)m+
+(@km)2
j2
+0[(jr)m]2
= %s
_m+q
%s(jr)m2
+
+(@km)2
j2
+
0 (q )2
%s
[(jr)m]2: (56)
The second law of thermodynamics requires the total dis-
sipation to be positive, which puts some constraints on
the allowed values of the phenomenological parameters.
We can easily notice, however, that the dissipation (56)
is guaranteed to be positive-de nite if
+0(q )2
%s; (57)
which may serve as an estimate for the spin-texture re-
sistivity due to spin dephasing. This is consistent with
the microscopic ndings of Ref. 23.
V. THERMAL NOISE
At nite temperature, thermal agitation causes
uc-
tuations of the current and spin texture, which are cor-
related due to their coupling. A complete description
requires that we supplement the stochastic equations of
motion with the correlators for these
uctuations. It
is convenient to regard these
uctuations as being due
to the stochastic Langevin \forces" ( ;j;h) on the
right-hand side of Eq. (49). The complete set of nite-
temperature hydrodynamic equations thus becomes:
_= r~j;
_p+q _mirmi= ^
[m]~j r~;
%s(1 + m)_m=~hm q m(~jr)m:(58)8
where (~;~j;~h) = (+;j+j;h+h). The simplest
(while possibly not most realistic) case corresponds to
a highly compressible
uid, such that K!1 . In this
limit,==K!0 and the last two equations com-
pletely decouple from the rst, continuity equation. In
the remainder of this section, we will focus on this special
case. The correlations of the stochastic elds are given
by the symmetric part of the inverse matrix b = b 1,37
which is found by inverting Eq. (58) (reduced now to a
system of two equations):
~j= ^
1(_p+q _mirmi);
~h=%sm_m %s_m q (~jr)m: (59)
Writing formally these equations as (after substituting ~j
from the rst into the second equation)
~j
~h
= b[m(r)]
_p
_m
; (60)
we immediately read out for the matrix elements
b(r;r0) =b(r)(r r0):
ji;ji0(r) =(^
1)ii0;
ji;hi0(r) =q (^
1)ik@kmi0;
hi0;ji(r) = q (^
1)ki@kmi0;
hi;hi0(r) = %sii0+%sii0kmk
(q )2(@kmi)(^
1)kk0(@k0mi0):(61)
According to the
uctuation-dissipation theorem, we
symmetrize b to obtain the classical Langevin
correlators:37
hji(r;t)ji0(r0;t0)i=T=gii0;
hji(r;t)hi0(r0;t0)i=T=q g0
ik@kmi0;
hhi(r;t)hi0(r0;t0)i=T= %sii0
(q )2gkk0(@kmi)(@k0mi0); (62)
whereT= 2kBT(r r0)(t t0) and
^g= [^
1+ (^
1)T]=2;^g0= [^
1 (^
1)T]=2 (63)
are, respectively, the symmetric and antisymmetric parts
of the conductivity matrix ^
1. The short-ranged, -
function character of the noise correlations in space stems
from the assumption of high electronic compressibility.
Contrast this to the results of Ref. 11 for incompressible
hydrodynamics. A presence of long-ranged Coulombic
interactions and plasma modes would also give rise to
nonlocal correlations. These are absent in our treatment,
which disregards ordinary electromagnetic phenomena.
Focusing on the microwave frequencies !characteris-
tic of ferromagnetic dynamics, it is most interesting to
consider the regime where ! 1. This means that
we can employ the drift approximation for the rst of
Eqs. (59):
_pi=me_vi qei qei=q_m(m@im): (64)Substituting this _pin Eq. (59), we can easily nd a closed
stochastic equation for the spin-texture eld:
%s(1 + m)_m+m $_m= (hm+h)m;(65)
where we have de ned the \spin-torque tensor"
$=q2(^
1)kk0(m@km @km)
(m@k0m+ @k0m): (66)
The antisymmetric piece of this tensor modi es the e ec-
tive gyromagnetic ratio, while the more interesting sym-
metric piece determines the additional nonlocal Gilbert
damping:
$= $+ $T
2%s=q2
%sG$; (67)
where
G$=gkk0
(m@km)
(m@k0m) 2@km
@k0m
+ g0
kk0[(m@km)
@k0m @km
(m@k0m)]:
(68)
In obtaining Eq. (65) from Eqs. (59), we have separated
the reactive spin torque out of the e ective eld: h=
hm qm(jr)m. (The remaining piece hmthus re
ects
the purely magnetic contribution to the e ective eld.)
The total stochastic magnetic eld entering Eq. (65),
h=h+qm(jr)m; (69)
captures both the usual magnetic Brown noise38h
and the Johnson noise spin-torque contribution39hJ=
qm(jr)mthat arises due to the substitution j=~j j
in the reactive spin torque q(jr)m. Using correla-
tors (62), it is easy to show that the total e ective eld
uctuations hare consistent with the nonlocal e ec-
tive Gilbert damping tensor (68), in accordance with the
uctuation-dissipation theorem applied directly to the
purely magnetic Eq. (65).
To the leading, quadratic order in spin texture, we can
replacegkk0!kk0=
andg0
kk0!0 in Eq. (68). This ad-
ditional texture-dependent nonlocal damping (along with
the associated magnetic noise) is a second-order e ect,
physically corresponding to the backaction of the magne-
tization dynamics-driven current on the spin texture.11
It should be noted that in writing the modi ed LLG
equation (55), we did not systematically expand it to
include the most general phenomenological terms up to
the second order in spin texture. We have only included
extra spin-torque terms, which are required by the On-
sager symmetry with Eq. (52). The second-order Gilbert
damping (68) was then obtained by solving Eqs. (52) and
(55) simultaneously. (Cf. Refs. 11,40.) This means in
particular, that this procedure does not capture second-
order Gilbert damping e ects whose physical origin is
unrelated to the longitudinal spin-transfer torque physics
studied here. One example of that is the transverse spin-
pumping induced damping discussed in Refs. 41.9
VI. EXAMPLES
A. Rigidly spinning texture
To illustrate the resistivity terms in the electron's
equation of motion (52), we rst consider 1D textures.
Take, for example, the case of a 1D spin helix m(z)
along thezaxis, whose spatial gradient pro le is given by
@zm=^ zm, whereis the wave vector of the spatial
rotation and m?^ z. See Fig. 1. It gives anisotropic re-
sistivity in the xyplane,r()
?, and along the zdirection,
r()
k:
r()
?=(@zm)2=2; r()
k= (+0)2: (70)
FIG. 1: (Color online) The transverse magnetic helix, @zm=
^ zm, with texture-dependent anisotropic resistivity (70).
We assume here translational invariance in the transverse ( xy)
directions. Spinning this helix about the vertical zaxis gen-
erates the dissipative electromotive forces f( )
z, which is spa-
tially uniform and points everywhere along the zaxis. A
magnetic spiral, @zm=^'m=^, spinning around the z
axis, on the other hand, produces a purely reactive electromo-
tive forceez, as discussed in the text, which is oscillatatory
in space along the zaxis.
The ctitious electric eld and dissipative force re-
quire magnetic dynamics. A general texture globally ro-
tating clockwise in spin space in the xyplane according
to_m= !^ zm(which may be induced by applying a
magnetic eld along the zdirection) generates an electric
eld
ei= (m_m)@im= !(m^ zm)@im
= !@imz= !@icos (71)and a force
f( )
i= _m@im= !^ z(m@im)
= !sin2@i'; (72)
where (;') denote the position-dependent spherical an-
gles parametrizing the spin texture. The reactive force
(71) has a simple interpretation of the gradient of the
Berry-phase15accumulation rate [which is locally deter-
mined by the solid angle subtended by m(t)]. In the
case of the transverse helix discussed above, ==2,
'=z !t, so thatez= 0 whilef( )
z= ! is nite.
As an example of a dynamical texture that does not
generate f( )while producing a nite e, consider a spin
spiral along the zaxis, described by @zm=^'m=^,
and rotating in time in the manner described above. It is
clear geometrically that the change in the spin texture in
time is in a direction orthogonal to its gradients in space.
Speci cally, =z,'= !t, so thatf( )
z= 0 while the
electric eld is oscillatory, ez=!sin.
B. Rotating spin textures
We show here that a vortex rotating about its core in
orbital space generates a current circulating around its
core, as well as a current going radially with respect to
the core. Consider a spin texture with a time depen-
dence corresponding to the real-space rotation clockwise
in thexyplane around the origin, such that m(r;t) =
m(r(t);0) with _r=!^ zr=!r^ , where we use polar co-
ordinates (r; ) on the plane normal to the zaxis in real
space [to be distinguished from the spherical coordinates
(;') that parametrize min spin space], we have
_m= (_rr)m=!@ m: (73)
Form(r; ) in polar coordinates, the components of the
electric eld are,
er=!m(@ m@rm); e = 0; (74)
while the components of the force are
f( )
r= !(@rm)(@ m); f( )
= !(@ m)2
r:(75)
In order to nd the ctitious electromagnetic elds, we
need to calculate the following tensors (which depend on
the instantaneous spin texture):
bijm(@im@jm) = sin(@i@j' @j@i');
dij@im@jm=@i@j+ sin2@i'@j': (76)
As an example, consider a vortex centered at the ori-
gin in thexyplane with winding number 1 and positive
polarity, as shown in Fig. 2. Its angular coordinates are
given by
'= ( +!t) +
2; =(r); (77)10
where = arg( r) andis a rotationally invariant func-
tion such that !0 asr!0 and!=2 asr!1 .
Evaluating the tensors in equation (76) for this vortex in
polar coordinates gives drr= (@r)2,d = (sin=r)2,
dr = 0, andbr = (@rcos)=r. The radial electric
eld is then given by
er= !rbr =!@rcos: (78)
The force is in the azimuthal direction:
f( )
r= 0; f( )
= !rd = !sin2
r: (79)
We can interpret this force as the spin texture \dragging"
the current along its direction of motion. Notice that the
forces in Eqs. (78) and (79) are the negative of those in
Eqs. (71) and (72), as they should be for the present case,
since the combination of orbital and spin rotations of our
vortex around its core leaves it invariant, producing no
forces.
FIG. 2: Positive-polarity magnetic vortex con guration pro-
jected on the xyplane. mhas a positive (out-of-plane) z
component near the vortex core. Rotating this vortex about
the origin in real space generates the current in the xyplane
shown in Fig. 3.
The total resistivity tensor (53) is (in the cylindrical
coordinates)
^
=
+(drr+d ) +0^d+q
^b=
r
?
?
;(80)
where
r=
+ (+0)(@r)2+sin
r2
;
=
+(@r)2+ (+0)sin
r2
;
?= q
@rcos
r: (81)Here, the two diagonal components,
rand
, describe
the (dissipative) anisotropic resistivity, while the o -
diagonal component,
?, captures what is called the
topological Hall e ect.19
In the drift approximation, Eq. (64), the current-
density eld j=jr^ r+j^is given by
j= ^
1q(e+f( ));
jr
j
=q!^
1@rcos
sin2=r
= q!sin
r
+
2
?
?
?
r
@r
sin=r
:(82)
More explicitly, we may consider a pro le =(1
e r=a)=2, whereais the radius of the vortex core. The
corresponding current (82) is sketched in Fig. 3.
FIG. 3: We plot here the current in Eq. (82) (all parameters
set to 1). Near the core, the current spirals inward and charges
build up at the center (which is allowed for our compressible
uid).
We note that the ctitious magnetic eld ijkbjk=2
points everywhere in the zdirection, its total
ux
through the xyplane being given by
F=Z
d dr (rbr ) = Z
d dr (@ '@rcos) = 2:
(83)
Note that the integrand is just the Jacobian of the map
from the plane to the sphere de ned by the spin-texture
eld:
((r);'(r)) :R2!S2: (84)
This re
ects the fact that the ctitious magnetic
ux is
generally a topological invariant, corresponding to the 2
homotopy group of the mapping (84).6,42
C. Anisotropic resistivity of a 3D spiral
Consider the texture described by @im=i^ zm,
where the spatial rotation stays in the xyplane, but the11
wave vectorcan be in any direction. The spin texture
forms a transverse helix in the zdirection and a planar
spiral in the xandydirections. Fig. 4 shows such a
con guration for pointing along ( x+y+z)=p
3. The
ctitious magnetic eld bvanishes, but the anisotropic
resistivity still depends nontrivially on the spin texture:
ij=
+(@km)2
ij+0@im@jm
= (
+2)ij+0ij; (85)
which, according to j= ^
1E, would give a transverse
current signal for an electric eld applied along the Carte-
sian axesx,y, orz.
FIG. 4: (Color online) A set of spin spirals which is topo-
logically trivial because r= 0 (and equivalent to the spin
helix, Fig. 1, up to a global real-space rotation), hence the
ctitious magnetic eld b, Eq. (76), is zero. There is, how-
ever, an anisotropic texture-dependent resistivity with nite
o -diagonal components, Eq. (85).
VII. SUMMARY
We have developed semi-phenomenologically the hy-
drodynamics of spin and charge currents interacting with
collective magnetization in metallic ferromagnets, gener-
alizing the results of Ref. 11 to three dimensions and
compressible
ows. Our theory reproduces known re-
sults such as the spin-motive force generated by mag-
netization dynamics and the dissipative spin torque, al-
beit from a di erent viewpoint than previous microscopic
approaches. Among the several new e ects predicted,
we nd both an isotropic and an anisotropic texture-
dependent resistivity, Eq. (53), whose contribution to theclassical (topological) Hall e ect should be described on
par with that of the ctitious magnetic eld. By calculat-
ing the dissipation power, we give a lower bound on the
spin-texture resistivity as required by the second law of
thermodynamics. We nd a more general form, includ-
ing a term of order , of the texture-dependent correction
to nonlocal Gilbert damping, predicted in Ref. 11. See
Eq. (68).
Our general theory is contained in the stochastic hy-
drodynamic equations, Eqs. (58), which we treated in
the highly compressible limit. The most general situ-
ation is no doubt at least as rich and complicated as
the classical magnetohydrodynamics. A natural exten-
sion of this work is the inclusion of heat
ows and re-
lated thermoelectric e ects, which we plan to investigate
in a future work. Although we mainly focused on the
halfmetallic limit in this paper, our theory is in principle
a two-component
uid model and allows for the inclu-
sion of a fully dynamical treatment of spin densities and
associated
ows.31Finally, our hydrodynamic equations
become amenable to analytic treatments when applied to
the important problem of spin-current driven dynamics
of magnetic solitons, topologically stable objects that can
be described by a small number of collective coordinates,
which we will also investigate in future work.
Acknowledgments
We are grateful to Gerrit E. W. Bauer, Arne Brataas,
Alexey A. Kovalev, and Mathieu Taillefumier for stimu-
lating discussions. This work was supported in part by
the Alfred P. Sloan Foundation and the NSF under Grant
No. DMR-0840965.
APPENDIX A: MANY-BODY ACTION
We can formally start with a many-body action, with
Stoner instability built in due to short-range repulsion
between electrons:25
S[ (r;t); (r;t)] =Z
CdtZ
d3r
^ +
i~@t+~2
2mer2
^ U " # # "
;(A1)
where time truns along the Keldysh contour from 1
to1and back. and are mutually independent
Grassmann variables parametrizing fermionic coherent
states and ^ += ( "; #) and ^ = ( "; #)T. The four-
fermion interaction contribution to the action can be de-
coupled via Hubbard-Stratonovich transformation, after12
introducing auxiliary bosonic elds and:
eiSU=~= exp
i
~Z
CdtZ
d3rU " # # "
=Z
D[ (r;t);(r;t)] expi
~Z
CdtZ
d3r
2
4U 2
4U
2^ +^ +
2^ +^^
:(A2)
In obtaining this result, we decomposed the interaction
into charge- and spin-density pieces:
" # # "=1
4(^ +^ )2 1
4(^ +m^^ )2; (A3)
where mis an arbitrary unit vector. It is easy to
show thath (r;t)i=Uh^ +(r;t)^ (r;t)iandh(r;t)i=
Uh^ +(r;t)^^ (r;t)i, when properly averaging over the
coupled quasiparticle and bosonic elds.
The next step in developing mean- eld theory is to
treat the Hartree potential (r;t) and Stoner exchange
(r;t)(r;t)m(r;t) elds in the saddle-point approx-
imation. Namely, the e ective bosonic action
Se [ (r;t);(r;t)] = i~lnZ
D[^ +;^ ]ei
~S(^ +;^ ; ;)
(A4)
is minimized, Se = 0, in order to nd the equations
of motion for the elds and. In the limit of suf-
ciently low electron compressibility and spin suscepti-
bility, the charge- and spin-density
uctuations are sup-
pressed, de ning mean- eld parameters and . Since
a constant only shifts the overall electrochemical po-
tential, it is physically inconsequential. Our theory is de-
signed to focus on the remaining soft (Goldstone) modes
associated with the spin-density director m(r;t), while
(r;t) and ( r;t) are in general allowed to
uctuate
close to their mean- eld values and , respectively.
The saddle-point equation of motion for the collective
spin direction m(r;t) follows from mSe [m] = 0, after
integrating out electronic degrees of freedom. Because
of the noncommutative matrix structure of the action
(A2), it is still a nontrivial problem. The problem sim-
pli es considerably in the limit of large exchange split-
ting , where we can project spins on the local magnetic
direction m. This lays the ground to the formulation dis-
cussed in Sec. II, where the collective spin-density eld
parametrized by the director m(r;t) interacts with the
spin-up/down free-electron eld. The resulting equations
of motion constitute the self-consistent dynamic Stoner
theory of itinerant ferromagnetism.
In the remainder of this appendix, we explicitly show
that the semiclassical formalism developed in Secs. II-
III B is equivalent to a proper eld-theoretical treatment.
The equation of motion for the spin texture follows from
extremizing the e ective action with respect to variations
inm. Because of the constraint on the magnitude of m,
its variation can be expressed as m=m, withbeing an arbitrary in nitesimal vector, so that the
equation of motion is given by mmSe = 0:
0 =mmSe
=1
ZZ
D[^ +;^ ] (mmS)ei
~S[^ +;^ ; ;]
=X
(mma) @S
@a
mmF; (A5)
whereZ=R
D[^ +;^ ]ei
~S[^ +;^ ; ;]and we have used
the path-integral representation of the vacuum expecta-
tion value. aare the spin-dependent gauge potentials
(4) andFthe spin exchange energy, appearing after we
project spin dynamics on the collective eld . Equa-
tion (A5) may be expressed in terms of the hydrody-
namic variables of the electrons. De ning spin-dependent
charge and current densities, j
= (;j), by
= @S
@a
=h i;
j= @S
@a
=1
meRe
( i~r a)
=v;
(A6)
Eq. (A5) reduces to the Landau-Lifshitz Eq. (18). Min-
imizing action (A4) with respect to the and elds
gives the anticipated self-consistency relations:
(r;t) =Uh^ +(r;t)^ (r;t)i=U(++ );
(r;t) =Uh^ +(r;t)^z^ (r;t)i=U(+ ):(A7)
APPENDIX B: THE MONOPOLE GAUGE FIELD
Let (;') be the spherical angles of m, the direction
of the local spin density, and ^ be the spin up/down
(=) spinors given by, up to a phase,
^ +(;') =
cos
2
ei'sin
2
;
^ (;') = ^ +( ;'+) =
sin
2
ei'cos
2
:(B1)
The spinors are related to the spin-rotation matrix ^U(m)
by ^ =^Uji. The gauge eld in mspace, which enters
Eq. (5), is thus given by
amon
(;') = i~^ y
@m^ =~
21 cos
sin
^';(B2)
where we used the gradient on a unit sphere: @m=
^@+^'@'=sin. The magnetic eld corresponding to
this vector potential [extended to three dimensions by
a(m)!a(;')=m] is given on the unit sphere by
@mamon
=@m(a'^') =m
sin@(sina') =~
2m:
(B3)13
It follows from Eqs. (5) and (B2) that the spin-dependent
real-space gauge elds are given by
a= ~
2@'(1 cos): (B4)
Notice that the =monopole eld (B2), as well as the
above gauge elds, are singular on the south/north pole(corresponding to the Dirac string). This is what allows a
magnetic eld with nite divergence. Any other choice of
the monopole gauge eld (B2) would correspond to a dif-
ferent choice of the spinors (B1), translating into a gauge
transformation of the elds (B4). This is immediately
seeing by noticing that amon
(m)!amon
(m) +@mf(m)
corresponds to a(r;t)!a(r;t) +@f(m(r;t)).
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2209.02914v2.Convergence_analysis_of_an_implicit_finite_difference_method_for_the_inertial_Landau_Lifshitz_Gilbert_equation.pdf | CONVERGENCE ANALYSIS OF AN IMPLICIT FINITE
DIFFERENCE METHOD FOR THE INERTIAL
LANDAU-LIFSHITZ-GILBERT EQUATION
JINGRUN CHEN, PANCHI LI, AND CHENG WANG
Abstract. The Landau-Lifshitz-Gilbert (LLG) equation is a widely used model
for fast magnetization dynamics in ferromagnetic materials. Recently, the iner-
tial LLG equation, which contains an inertial term, has been proposed to cap-
ture the ultra-fast magnetization dynamics at the sub-picosecond timescale.
Mathematically, this generalized model contains the rst temporal derivative
and a newly introduced second temporal derivative of magnetization. Conse-
quently, it produces extra diculties in numerical analysis due to the mixed
hyperbolic-parabolic type of this equation with degeneracy. In this work, we
propose an implicit nite di erence scheme based on the central di erence in
both time and space. A xed point iteration method is applied to solve the im-
plicit nonlinear system. With the help of a second order accurate constructed
solution, we provide a convergence analysis in H1for this numerical scheme, in
the`1(0;T;H1
h) norm. It is shown that the proposed method is second order
accurate in both time and space, with unconditional stability and a natural
preservation of the magnetization length. In the hyperbolic regime, signi cant
damping wave behaviors of magnetization at a shorter timescale are observed
through numerical simulations.
1.Introduction
The Landau-Lifshitz-Gilbert (LLG) equation [15, 19] describes the dissipative
magnetization dynamics in ferromagnetic materials, which is highly nonlinear and
has a non-convex constraint. Physically, it is widely used to interpret the experi-
mental observations. However, recent experiments [5, 16, 17] con rm that its valid-
ity is limited to timescales from picosecond to larger timescales for which the angular
momentum reaches equilibrium in a force eld. At shorter timescales, e.g. 100 fs,
the ultra-fast magnetization dynamics has been observed [17]. To account for this,
the inertial Landau-Lifshitz-Gilbert (iLLG) equation is proposed [6, 10, 12]. As a
result, the magnetization converges to its equilibrium along a locus with damping
nutation simulated in [21], when the inertial e ect is activated by a non-equilibrium
initialization or an external magnetic eld.
For a ferromagnet over
2Rd;d= 1;2;3, the observable states are depicted by
the distribution of the magnetization in
. The magnetization denoted by m(x;t) is
a vector eld taking values in the unit sphere S2ofR3, which indicates that jmj= 1
in a point-wise sense. In micromagnetics, the evolution of mis governed by the
LLG equation. In addition to experiment and theory, micromagnetics simulations
Date : September 13, 2022.
2010 Mathematics Subject Classi cation. Primary 35K61, 65M06, 65M12.
Key words and phrases. Convergence analysis, inertial Landau-Lifshitz-Gilbert equation, im-
plicit central di erence scheme, second order accuracy.
1arXiv:2209.02914v2 [math.NA] 12 Sep 20222 JINGRUN CHEN, PANCHI LI, AND CHENG WANG
have become increasingly important over the past several decades. Therefore, nu-
merous numerical approaches have been proposed for the LLG equation and its
equivalent form, the Landau-Lifshitz (LL) equation; see [9, 18] for reviews and ref-
erences therein. In terms of time marching, the simplest explicit methods, such
as the forward Euler method and Runge-Kutta methods, were favored in the early
days, while small time step size must be used due to the stability restriction [22].
Of course, implicit methods avoid the stability constraint and these methods pro-
duce the approximate solution in H1(
) [1, 2]. However, in order to guarantee the
convergence of the schemes, a step-size condition k=O(h2) must be satis ed in
both the theoretical analysis and numerical simulations. To obtain the weak solu-
tion in the nite element framework, an intermediate variable vwith the de nition
v=@tmrepresenting the increment rate at current time is introduced, and to solve
vin the tangent space of mwhere it satis es vm= 0 in a point-wise sense, then
the con guration at the next time step can be obtained. Directly, the strong solu-
tion can be obtained through solving the implicit mid-point scheme [4] and the im-
plicit backward Euler scheme [13] using xed-point iteration methods. By contrast,
the semi-implicit methods have achieved a desired balance between stability and
eciency for the micromagnetics simulations. The Gauss-Seidel projection meth-
ods [11, 20, 27], the linearized backward Euler scheme [8, 14], the Crank-Nicolson
projection scheme [3], and the second order semi-implicit backward di erentiation
formula projection scheme [7, 28] have been developed in recent years. In prac-
tice, all these semi-implicit methods inherit the unconditional stability of implicit
schemes, and achieve the considerable improvement in eciency.
The LLG equation is a nonlinear parabolic system which consists of the gyro-
magnetic term and the damping term. It is a classical kinetic equation that only
contains the velocity; no acceleration is included in the equation. When relaxing
the system from a non-equilibrium state or applying a perturbation, it is natural
that an acceleration term will be present, resulting in the inertial term in the iLLG
equation. More speci cally, the time evolution of m(x;t) is described by @tmand
m@tmwith the addition of an inertial term m@ttm. Thus, the iLLG equa-
tion is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy. To
numerically study the hyperbolic behaviors of the magnetization, the rst-order
accuracy tangent plane scheme (TPS) and the second-order accuracy angular mo-
mentum method (AMM) are proposed in [23]. The xed-point iteration method is
used for the implicit marching. These two methods aim to nd the weak solution.
Furthermore, a second-order accurate semi-implicit method is presented in [21], and
@ttmand@tmare approximated by the central di erence.
In this work, we provide the convergence analysis of the implicit mid-point
scheme on three time layers for the iLLG equation. Subject to the condition
kCh2, it produces a unique second-order approximation in H1(
T). Owing to
the application of the mid-point scheme, it naturally preserves the magnetization
length. Moreover, we propose a xed-point iteration method to solve the nonlinear
scheme, which converges to a unique solution under the condition of kCh2.
Numerical simulations are reported to con rm the theoretic analysis and study the
inertial dynamics at shorter timescales.
The rest of this paper is organized as follows. The iLLG equation and the
numerical method are introduced in Section 2. The detailed convergence analysis
is provided in Section 3. In addition, a xed-point iteration method for solvingCONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 3
the implicit scheme is proposed in Section 4, and the convergence is established
upon the condition kCh2. Numerical tests, including the accuracy test and
observation of the inertial e ect, are presented in Section 5. Concluding remarks
are made in Section 6.
2.The physical model and the numerical method
The intrinsic magnetization of a ferromagnetic body m=m(x;t) :
T:=
(0;T)!S2is modeled by the conventional LLG equation:
@tm= mm+ m@tm; (x;t)2
T; (2.1a)
m(x;0) =m(0); x2
; (2.1b)
@m(x;t) = 0; (x;t)2@
[0;T]; (2.1c)
whererepresents the unit outward normal vector on @
, and 1 is the
damping parameter. If the relaxation starts from a non-equilibrium state or a
sudden perturbation is applied, the acceleration should be considered in the kinetic
equation, which is the inertial e ect observed in various experiments at the sub-
picosecond timescale. In turn, its dynamics is described by the iLLG equation
@tm= m(m+He) + m(@tm+ @ttm); (x;t)2
T; (2.2a)
m(x;0) =m(0); x2
; (2.2b)
@tm(x;0) = 0; x2
; (2.2c)
@m(x;t) = 0; (x;t)2@
[0;T]; (2.2d)
where is the phenomenological inertia parameter, and Heis a perturbation of
an applied magnetic eld. To ease the discussion, the external eld is neglected in
the subsequent analysis and is only considered in micromagnetics simulations. An
additional initial condition @tm(x;0) = 0 is added, which implies that the velocity
is 0 att= 0 and it is a necessary condition for the well-posedness. Then the energy
is de ned as
(2.3)E[m] =1
2Z
jrmj2 2mHe+ j@tmj2
dx:
For constant external magnetic elds, it satis es the energy dissipation law
(2.4)d
dtE[m] = Z
j@tmj2dx0:
Therefore, under the condition of (2.2c), for almost all T02(0;T), we have
(2.5)1
2Z
jrm(x;T0)j2+ j@tm(x;T0)j2
dx1
2Z
jrm(x;0)j2
dx:
Before the formal algorithm is presented, here the spatial di erence mesh and
the temporal discretization have to be stated. The uniform mesh for
is con-
structed with mesh-size hand a time step-size k > 0 is set. Let Lbe the set of
nodesfxl= (xi;yj;zk)gin 3-D space with the indices i= 0;1;;nx;nx + 1;j=
0;1;;ny;ny +1 andk= 0;1;;nz;nz +1, and the ghost points on the bound-
ary of
are denoted by ix= 0;nx+ 1,jy= 0;ny+ 1 andkz= 0;nz+ 1. We
use the half grid points with mi;j;k=m((i 1
2)hx;(j 1
2)hy;(k 1
2)hz). Here
hx= 1=nx,hy= 1=ny,hz= 1=nzandh=hx=hy=hzholds for uniform4 JINGRUN CHEN, PANCHI LI, AND CHENG WANG
spatial meshes. Due to the homogeneous Neumann boundary condition (2.2d), the
following extrapolation formula is derived:
(2.6)mix+1;j;k=mix;j;k;mi;jy+1;k=mi;jy;k;mi;j;k z+1=mi;j;k z;
for any 1inx;1jny;1knz. Meanwhile, the temporal derivatives
are discretized by the central di erence, with the details stated in the following
de nition.
De nition 2.1. For n+1= (x;tn+1)and n+1= (tn+1), de ne
d+
t n= n+1 n
k; d
t n= n n 1
k;
and
D+
t n= n+1 n
k; D
t n= n n 1
k:
Consequently, we denote
dt n+1=1
2(d+
t n+d
t n); Dt n+1=1
2(D+
t n+D
t n):
In particular, the second time derivative is approximated by the central di erence
form
(2.7) dtt = n+1 2 n+ n 1
k2:
Then for the initial condition (2.2c), there holds
(2.8) m(xl;0) =m(xl;k);8l2L;
whereL=f(i;j;k )ji= 1;;nx;j= 1;;ny;k= 1;;nz:g. Denotemn
h(n
0) as the numerical solution. Given grid functions fh;gh2`2(
h;R3), we list
de nitions of the discrete inner product and norms used in this paper.
De nition 2.2. The discrete inner product h;iin`2(
h;R3)is de ned by
(2.9) hfh;ghi=hdX
l2Lfh(xl)gh(xl):
The discrete `2norm andH1
hnorm ofmhare
(2.10) kfhk2
2=hdX
l2Lfh(xl)fh(xl);
and
(2.11) kfhk2
H1
h=kfhk2
2+krhfhk2
2
withrhrepresenting the central di erence stencil of the gradient operator.
Besides, the norm kk1in`1(
h;R3) is de ned by
(2.12) kfhk1= max
l2Lkfh(xl)k1:
Therefore, the approximation scheme of the iLLG equation is presented below.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 5
Algorithm 2.1. Givenm0
h;m1
h2W1;2(
h;S2). Letmn 1
h;mn
h2W1;2(
h;S2),
we computemn+1
hby
(2.13) dtmn+1
h mn
h
dtmn+1
h+ dttmn
h
= mn
hhmn
h;
where mn
h=1
2(mn+1
h+mn 1
h), and hrepresents the standard seven-point stencil
of the Laplacian operator.
The corresponding fully discrete version of the above (2.13) reads as
mn+1
h mn 1
h
2k mn+1
h+mn 1
h
2mn+1
h mn 1
h
2k+ mn+1
h 2mn
h+mn 1
h
k2
= mn+1
h+mn 1
h
2hmn+1
h+mn 1
h
2
: (2.14)
Within three time steps, there have not been many direct discretization methods
to get the second-order temporal accuracy. Due to the mid-point approximation
feature, this implicit scheme is excellent in maintaining certain properties of the
original system.
Lemma 2.1. Given m0
h(xl) = 1, then the sequence fmn
h(xl)gn0produced by
(2.13) satis es
(i)jmn
h(xl)j= 1;8l2L;
(ii)1
2Dtkrhmn+1
hk2
2+ kdtmn+1
hk2
2+1
2 D
tkd+
tmn
hk2
2= 0.
Proof. On account of the initial condition (2.2c), we see that m0(xl) =m1(xl)
holding for all l2L. Taking the vector inner product with (2.13) by ( mn+1
h(xl) +
mn 1
h(xl)), it obvious that we can get
jmn+1
hj=jmn
hj==jm1
hj=jm0
hj= 1;
in the point-wise sense. This con rms (i). In order to verify (ii), we take inner
product with (2.13) by hmn
hand get
1
2Dtkrhmn+1
hk2
2 hmn
hdtmn+1
h; hmn
hi hmn
hdttmn
h; hmn
hi= 0:
Subsequently, taking inner products with dtmn+1
handdttmn+1
hseparately leads to
the following equalities:
kdtmn+1
hk2
2 hmn
hdttmn
h;dtmn+1
hi= hmn
hdtmn+1
h; hmn
hi;
and
1
2D
tkd+
tmn
hk2
2+ hmn
hdttmn
h;dtmn+1
hi= hmn
hdttmn
h; hmn
hi:
A combination of the above three identities yields (ii).
In lemma 2.1, taking k!0 gives
(2.15)d
dt1
2krhmn+1
hk2
2+
2k@tmn
hk2
2
= k@tmn+1
hk2
2;
which is consistent with the continuous energy law (2.4). Accordingly, in the ab-
sence of the external magnetic eld, the discretized version energy dissipation law
would be maintained with a modi cation
(2.16)E(mn+1
h;mn
h) =
2
mn+1
h mn
h
k
2
2+1
4(krhmn+1
hk2
2+krhmn
hk2
2):6 JINGRUN CHEN, PANCHI LI, AND CHENG WANG
Theorem 2.1. Givenmn 1
h;mn
h;mn+1
h2W1;2(
h;S2), we have a discrete energy
dissipation law, for the modi ed energy (2.16) :
(2.17) E(mn+1
h;mn
h)E(mn
h;mn 1
h):
Proof. Denote a discrete function
n:= mn+1
h mn 1
h
2k+ mn+1
h 2mn
h+mn 1
h
k2
1
2h(mn+1
h+mn 1
h):
Taking a discrete inner product with (2.13) by ngives
4k2hmn+1
h mn 1
h;mn+1
h mn 1
hi+
2k3hmn+1
h mn 1
h;mn+1
h 2mn
h+mn 1
hi(2.18)
+
4kD
mn+1
h mn 1
h; h(mn+1
h+mn 1
h)E
=D
mn+1
h+mn 1
h
2n;nE
= 0:
Meanwhile, the following estimates are available:
hmn+1
h mn 1
h;mn+1
h mn 1
hi=kmn+1
h mn 1
hk2
20; (2.19)
hmn+1
h mn 1
h;mn+1
h 2mn
h+mn 1
hi
=D
(mn+1
h mn
h) + (mn
h mn 1
h);(mn+1
h mn
h) (mn
h mn 1
h)E
;
=kmn+1
h mn
hk2
2 kmn
h mn 1
hk2
2;(2.20)
D
mn+1
h mn 1
h; h(mn+1
h+mn 1
h)E
=D
rh(mn+1
h mn 1
h);rh(mn+1
h+mn 1
h)E
=krhmn+1
hk2
2 krhmn 1
hk2
2
=(krhmn+1
hk2
2+krhmn
hk2
2) (krhmn
hk2
2+krhmn 1
hk2
2):(2.21)
Going back to (2.18), we arrive at
2k
mn+1
h mn
h
k
2
2
mn
h mn 1
h
k
2
2
(2.22)
+1
4k
(krhmn+1
hk2
2+krhmn
hk2
2) (krhmn
hk2
2+krhmn 1
hk2
2)
=
mn+1
h mn 1
h
2k
2
20;
which is exactly the energy dissipation estimate (2.17). This nishes the proof of
Theorem 2.1.
Meanwhile, it is noticed that, given the initial pro le of matt= 0, namelym0,
an accurate approximation to m1andm2has to be made. In more details, an
O(k2+h2) accuracy is required for both m1,m2andm1 m0
k,m2 m1
k, which is
needed in the convergence analysis.
The initial pro le m0could be taken as m0=m(;0). This in turn gives a
trivial zero initial error for m0. Form1andm2, a careful Taylor expansion revealsCONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 7
that
m1=m0+k@tm0+k2
2@ttm0+O(k3)
=m0+k2
2@ttm0+O(k3); (2.23)
m2=m0+ 2k@tm0+ 2k2@ttm0+O(k3)
=m0+ 2k2@ttm0+O(k3); (2.24)
in which the initial data (2.2c), @tm(;0)0, has been applied in the derivation.
Therefore, an accurate approximation to m1andm2relies on a precise value of
@ttmatt= 0. An evaluation of the original PDE (2.2a) implies that
m0(@ttm0) =1
m0(m0+H0
e); (2.25)
in which the trivial initial data (2.2c) has been applied again. Meanwhile, motivated
by the point-wise temporal di erentiation identity
(2.26) m@ttm= (@tm)2+1
2@tt(jmj2) = (@tm)2;
and the fact that jmj1, we see that its evaluation at t= 0 yields
(2.27) m0@ttm0= (@tm0)2= 0:
Subsequently, a combination of (2.26) and (2.27) uniquely determines @ttm0:
(2.28) @ttm0= 1
m0(m0(m0+H0
e));
and a substitution of this value into (2.23), (2.24) leads to an O(k3) approximation
tom1andm2.
Moreover, with spatial approximation introduced, an O(k2+h2) accuracy is
obtained for both m1,m2andm1 m0
k,m2 m1
k. This nishes the initialization
process.
3.Convergence analysis
The theoretical result concerning the convergence analysis is stated below.
Theorem 3.1. Assume that the exact solution of (2.2) has the regularity me2
C3([0;T]; [C0(
)]3)\C2([0;T]; [C2(
)]3)\L1([0;T]; [C4(
)]3). Denote a nodal
interpolation operator Phsuch thatPhmh2C1(
), and the numerical solution mn
h
(n0) obtained from (2.13) with the initial error satisfying kep
hk2+krhep
hk2=
O(k2+h2), whereep
h=Phme(;tp) mp
h,p= 0;1;2, andkeq+1
h eq
h
kk2=O(k2+h2),
q= 0;1. Then the following convergence result holds for 2nT
k
ash;k!0+:
kPhme(;tn) mn
hk2+krh(Phme(;tn) mn
h)k2C(k2+h2); (3.1)
in which the constant C>0is independent of kandh.
Before the rigorous proof is given, the following estimates are declared, which
will be utilized in the convergence analysis. In the sequel, for simplicity of notation,
we will use a uniform constant Cto denote all the controllable constants throughout
this part.8 JINGRUN CHEN, PANCHI LI, AND CHENG WANG
Lemma 3.1 (Discrete gradient acting on cross product) .[7]For grid functions fh
andghover the uniform numerical grid, we have
krh(fhgh)k2C
kfhk2krhghk1+kghk1krhfhk2
: (3.2)
Lemma 3.2 (Point-wise product involved with second order temporal stencil) .For
grid functions fhandghover the time domain, we have
fn+1
h 2fn
h+fn 1
h
k2gn
h= fn
h fn 1
h
kgn
h gn 1
h
k
+1
kfn+1
h fn
h
kgn
h fn
h fn 1
h
kgn 1
h
: (3.3)
Now we proceed into the convergence estimate. First, we construct an approxi-
mate solution m:
(3.4) m=me+h2m(1);
in which the auxiliary eld m(1)satis es the following Poisson equation
m(1)=^Cwith ^C=1
j
jZ
@
@3
meds; (3.5)
@zm(1)jz=0= 1
24@3
zmejz=0; @zm(1)jz=1=1
24@3
zmejz=1;
with boundary conditions along xandydirections de ned in a similar way.
The purpose of such a construction will be illustrated later. Then we extend the
approximate pro le mto the numerical \ghost" points, according to the extrapo-
lation formula:
(3.6) mi;j;0=mi;j;1;mi;j;nz +1=mi;j;nz;
and the extrapolation for other boundaries can be formulated in the same man-
ner. Subsequently, we prove that such an extrapolation yields a higher order
O(h5) approximation, instead of the standard O(h3) accuracy. Also see the re-
lated works [24, 25, 26] in the existing literature.
Performing a careful Taylor expansion for the exact solution around the boundary
sectionz= 0, combined with the mesh point values: z0= 1
2h,z1=1
2h, we get
me(xi;yj;z0) =me(xi;yj;z1) h@zme(xi;yj;0) h3
24@3
zme(xi;yj;0) +O(h5)
=me(xi;yj;z1) h3
24@3
zme(xi;yj;0) +O(h5); (3.7)
in which the homogenous boundary condition has been applied in the second step.
A similar Taylor expansion for the constructed pro le m(1)reveals that
m(1)(xi;yj;z0) =m(1)(xi;yj;z1) h@zm(1)(xi;yj;0) +O(h3)
=m(1)(xi;yj;z1) +h
24@3
zme(xi;yj;0) +O(h3); (3.8)
with the boundary condition in (3.5) applied. In turn, a substitution of (3.7)-(3.8)
into (3.4) indicates that
(3.9) m(xi;yj;z0) =m(xi;yj;z1) +O(h5):
In other words, the extrapolation formula (3.6) is indeed O(h5) accurate.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 9
As a result of the boundary extrapolation estimate (3.9), we see that the discrete
Laplacian of myields the second-order accuracy at all the mesh points (including
boundary points):
(3.10)
hmi;j;k= me(xi;yj;zk)+O(h2);80inx+1;0jny+1;0knz+1:
Moreover, a detailed calculation of Taylor expansion, in both time and space, leads
to the following truncation error estimate:
mn+1
h mn 1
h
2k=mn+1
h+mn 1
h
2
mn+1
h mn 1
h
2k+ mn+1
h 2mn
h+mn 1
h
k2
hmn+1
h+mn 1
h
2
+ n; (3.11)
wherek nk2C(k2+h2). In addition, a higher order Taylor expansion in space
and time reveals the following estimate for the discrete gradient of the truncation
error, in both time and space:
(3.12) krh nk2;k n n 1
kk2C(k2+h2):
In fact, such a discrete kkH1
hbound for the truncation comes from the regularity as-
sumption for the exact solution, me2C3([0;T]; [C0(
)]3)\C2([0;T]; [C2(
)]3)\
L1([0;T]; [C4(
)]3), as stated in Theorem 3.1, as well as the fact that m(1)2
C1([0;T]; [C1(
)]3)\L1([0;T]; [C2(
)]3), as indicated by the Poisson equation (3.5).
We introduce the numerical error function en
h=mn
h mn
h, instead of a di-
rect comparison between the numerical solution and the exact solution. The error
function between the numerical solution and the constructed solution mhwill be
analyzed, due to its higher order consistency estimate (3.9) around the boundary.
Therefore, a subtraction of (2.14) from the consistency estimate (3.11) leads to the
error function evolution system:
en+1
h en 1
h
2k=mn+1
h+mn 1
h
2~n
h+en+1
h+en 1
h
2n
h+ n;(3.13)
n
h:= mn+1
h mn 1
h
2k+ mn+1
h 2mn
h+mn 1
h
k2
hmn+1
h+mn 1
h
2
;(3.14)
~n
h:= en+1
h en 1
h
2k+ en+1
h 2en
h+en 1
h
k2
hen+1
h+en 1
h
2
:(3.15)
Before proceeding into the formal estimate, we establish a W1
hbound forn
h,
which is based on the constructed approximate solution m(by (3.14)). Because of
the regularity for me, the following bound is available:
k`
hk1;krh`
hk1;kn
h n 1
h
kk1C; ` =n;n 1: (3.16)
In addition, the following preliminary estimate will be useful in the convergence
analysis.10 JINGRUN CHEN, PANCHI LI, AND CHENG WANG
Lemma 3.3 (A preliminary error estimate) .We have
ke`
hk2
22ke0
hk2
2+ 2Tk` 1X
j=0kej+1
h ej
h
kk2
2;8`kT: (3.17)
Proof. We begin with the expansion:
e`
h=e0
h+k` 1X
j=0ej+1
h ej
h
k;8`kT: (3.18)
In turn, a careful application of the Cauchy inequality reveals that
ke`
hk2
22
ke0
hk2
2+k2k` 1X
j=0ej+1
h ej
h
kk2
2
; (3.19)
k2k` 1X
j=0ej+1
h ej
h
kk2
2k2`` 1X
j=0kej+1
h ej
h
kk2
2Tk` 1X
j=0kej+1
h ej
h
kk2
2; (3.20)
in which the fact that `kThas been applied. Therefore, a combination of
(3.19) and (3.20) yields the desired estimate (3.17). This completes the proof of
Lemma 3.3.
Taking a discrete inner product with the numerical error equation (3.13) by ~n
h
gives
1
2khen+1
h en 1
h;~n
hi=hmn+1
h+mn 1
h
2~n
h;~n
hi
+hen+1
h+en 1
h
2n
h;~n
hi+h n;~n
hi: (3.21)
The analysis on the left hand side of (3.21) is similar to the ones in (2.19)-(2.21):
1
2khen+1
h en 1
h;~n
hi=
2k3hen+1
h en 1
h;en+1
h 2en
h+en 1
hi
+
4k2hen+1
h en 1
h;en+1
h en 1
hi
+1
4kD
rh(en+1
h en 1
h);rh(en+1
h+en 1
h)E
; (3.22)
hen+1
h en 1
h;en+1
h en 1
hi=ken+1 en 1
hk2
20; (3.23)
hen+1
h en 1
h;en+1
h 2en
h+en 1
hi
=ken+1
h en
hk2
2 ken
h en 1
hk2
2; (3.24)
D
en+1
h en 1
h; h(en+1
h+en 1
h)E
=D
rh(en+1
h en 1
h);rh(en+1
h+en 1
h)E
=krhen+1
hk2
2 krhen 1
hk2
2
=(krhen+1
hk2
2+krhen
hk2
2) (krhen
hk2
2+krhen 1
hk2
2): (3.25)
This in turn leads to the following identity:
1
2khen+1
h en 1
h;~n
hi=1
k(En+1e;h Ene;h) +
4k2ken+1
h en 1
hk2
2; (3.26)
En+1e;h=
2ken+1
h en
h
kk2
2+1
4(krhen+1
hk2
2+krhen
hk2
2): (3.27)CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 11
The rst term on the right hand side of (3.21) vanishes, due to the fact that
mn+1
h+mn 1
h
2~n
his orthogonal to ~n
h, at a point-wise level:
(3.28) hmn+1
h+mn 1
h
2~n
h;~n
hi= 0:
The second term on the right hand side of (3.21) contains three parts:
hen+1
h+en 1
h
2n
h;~n
hi=I1+I2+I3; (3.29)
I1= hen+1
h+en 1
h
2n
h;en+1
h en 1
h
2ki; (3.30)
I2= hen+1
h+en 1
h
2n
h;en+1
h 2en
h+en 1
h
k2i; (3.31)
I3=hen+1
h+en 1
h
2n
h; hen+1
h+en 1
h
2
i: (3.32)
The rst inner product, I1, could be bounded in a straightforward way, with the
help of discrete H older inequality:
I1= hen+1
h+en 1
h
2n
h;en+1
h en 1
h
2ki
4ken+1
h+en 1
hk2kn
hk1ken+1
h en 1
h
kk2
Cken+1
h+en 1
hk2ken+1
h en 1
h
kk2
C(ken+1
hk2
2+ken 1
hk2
2+ken+1
h en 1
h
kk2
2): (3.33)
For the second inner product, I2, we denotegn
h:=en+1
h+en 1
h
2n
h. An application
of point-wise identity (3.3) (in lemma 3.2) reveals that
I2= hgn
h;en+1
h 2en
h+en 1
h
k2i
= hen
h en 1
h
k;gn
h gn 1
h
ki
+
k
hen+1
h en
h
k;gn
hi hen
h en 1
h
k;gn 1
hi
: (3.34)
Meanwhile, the following expansion is observed:
gn
h gn 1
h
k=1
4(en+1
h en
h
k+en 1
h en 2
h
k)(n
h+n 1
h)
+en+1
h+en
h+en 1
h+en 2
h
4n
h n 1
h
k: (3.35)12 JINGRUN CHEN, PANCHI LI, AND CHENG WANG
This in turn indicates the associated estimate:
kgn
h gn 1
h
kk21
4(ken+1
h en
h
kk2+ken 1
h en 2
h
kk2)(kn
hk1+kn 1
hk1)
+ken+1
hk2+ken
hk2+ken 1
hk2+ken 2
hk2
4kn
h n 1
h
kk1
C
ken+1
h en
h
kk2+ken 1
h en 2
h
kk2
+ken+1
hk2+ken
hk2+ken 1
hk2+ken 2
hk2
; (3.36)
in which the bound (3.16) has been applied. Going back to (3.34), we see that
hen
h en 1
h
k;gn
h gn 1
h
ki ken
h en 1
h
kk2kgn
h gn 1
h
kk2
C
ken+1
h en
h
kk2+ken 1
h en 2
h
kk2+ken+1
hk2
+ken
hk2+ken 1
hk2+ken 2
hk2
ken
h en 1
h
kk2
C
ken+1
h en
h
kk2
2+ken 1
h en 2
h
kk2
2+ken+1
hk2
2
+ken
hk2
2+ken 1
hk2
2+ken 2
hk2
2+ken
h en 1
h
kk2
2
; (3.37)
I2C
ken+1
h en
h
kk2
2+ken 1
h en 2
h
kk2
2+ken+1
hk2
2
+ken
hk2
2+ken 1
hk2
2+ken 2
hk2
2+ken
h en 1
h
kk2
2
+
k
hen+1
h en
h
k;gn
hi hen
h en 1
h
k;gn 1
hi
: (3.38)
For the third inner product part, I3, an application of summation by parts formula
gives
I3=hen+1
h+en 1
h
2n
h; hen+1
h+en 1
h
2
i
=hrhen+1
h+en 1
h
2n
h
;rhen+1
h+en 1
h
2
i: (3.39)
Meanwhile, we make use of the preliminary inequality (3.2) (in lemma 3.1) and get
krhen+1
h+en 1
h
2n
h
k2
C
ken+1
h+en 1
h
2k2krhn
hk1+kn
hk1krh(en+1
h+en 1
h
2)k2
C
ken+1
h+en 1
h
2k2+krh(en+1
h+en 1
h
2)k2
C
ken+1
hk2+ken 1
hk2+krhen+1
hk2+krhen 1
hk2
: (3.40)CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 13
Again, the bound (3.16) has been applied in the derivation. Therefore, the following
estimate is available for I3:
I3krhen+1
h+en 1
h
2n
h
k2krhen+1
h+en 1
h
2
k2
C
ken+1
hk2+ken 1
hk2+krhen+1
hk2+krhen 1
hk2
krhen+1
hk2+krhen 1
hk2
C
ken+1
hk2
2+ken 1
hk2
2+krhen+1
hk2
2+krhen 1
hk2
2
: (3.41)
The estimate of I3can also be obtained by a direct application of discrete H older
inequality:
I3=hen+1
h+en 1
h
2rhn
h
;rhen+1
h+en 1
h
2
i
1
4ken+1
h+en 1
hk2krhn
hk1krh(en+1
h+en 1
h)k2
C
ken+1
hk2
2+ken 1
hk2
2+krhen+1
hk2
2+krhen 1
hk2
2
: (3.42)
A substitution of (3.33), (3.38) and (3.42) into (3.29) yields the following bound:
hen+1
h+en 1
h
2n
h;~n
hi=I1+I2+I3
C
ken+1
h en
h
kk2
2+ken
h en 1
h
kk2
2+ken 1
h en 2
h
kk2
2
+ken+1
hk2
2+ken
hk2
2+ken 1
hk2
2+ken 2
hk2
2+krhen+1
hk2
2+krhen 1
hk2
2
+
k
hen+1
h en
h
k;gn
hi hen
h en 1
h
k;gn 1
hi
: (3.43)
The third term on the right hand side of (3.21) could be analyzed in a similar
fashion:
h n;~n
hi=I4+I5+I6; (3.44)
I4= h n;en+1
h en 1
h
2ki; I 5= h n;en+1
h 2en
h+en 1
h
k2i; (3.45)
I6=h n; hen+1
h+en 1
h
2
i; (3.46)
I4= h n;en+1
h en 1
h
2ki
2k nk2ken+1
h en 1
h
kk2
4(k nk2
2+ken+1
h en 1
h
kk2
2); (3.47)
I5= h n;en+1
h 2en
h+en 1
h
k2i
= hen
h en 1
h
k; n n 1
ki
+
k
hen+1
h en
h
k; ni hen
h en 1
h
k; n 1i
; (3.48)14 JINGRUN CHEN, PANCHI LI, AND CHENG WANG
hen
h en 1
h
k; n n 1
kiken
h en 1
h
kk2k n n 1
kk2
C(k2+h2)ken
h en 1
h
kk2C(k4+h4) +1
2ken
h en 1
h
kk2
2; (3.49)
I5C(k4+h4) +
2ken
h en 1
h
kk2
2
+
k
hen+1
h en
h
k; ni hen
h en 1
h
k; n 1i
; (3.50)
I6=h n; hen+1
h+en 1
h
2
i=hrh n;rhen+1
h+en 1
h
2
i
krh nk2krhen+1
h+en 1
h
2
k2C(k2+h2)krhen+1
h+en 1
h
2
k2
C(k4+h4) +1
2
krhen+1
hk2
2+krhen 1
hk2
2
: (3.51)
Notice that the truncation error estimate (3.12) has been repeatedly applied in the
derivation. Going back to (3.44), we obtain
h n;~n
hi=I4+I5+I6
C(k4+h4) +
2ken+1
h en
h
kk2
2+ ( + 1)
2ken
h en 1
h
kk2
2
+1
2
krhen+1
hk2
2+krhen 1
hk2
2
+
k
hen+1
h en
h
k; ni hen
h en 1
h
k; n 1i
: (3.52)
Finally, a substitution of (3.26)-(3.27), (3.28), (3.43) and (3.52) into (3.21) leads
to the following inequality:
1
k(En+1e;h Ene;h) +
4k2ken+1
h en 1
hk2
2
C(k4+h4) +C
ken+1
h en
h
kk2
2+ken
h en 1
h
kk2
2+ken 1
h en 2
h
kk2
2
+ken+1
hk2
2+ken
hk2
2+ken 1
hk2
2+ken 2
hk2
2+krhen+1
hk2
2+krhen 1
hk2
2
+
k
hen+1
h en
h
k;gn
h+ ni hen
h en 1
h
k;gn 1
h+ n 1i
: (3.53)
Subsequently, a summation in time yields
En+1e;hE2e;h+CT(k4+h4) +CknX
j=0kej+1
h ej
h
kk2
2+n+1X
j=0(kej
hk2
2+krhej
hk2
2)
+
hen+1
h en
h
k;gn
h+ ni he2
h e1
h
k;g1
h+ 1i
: (3.54)CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 15
For the term hen+1
h en
h
k;gn
h+ ni, the following estimate could be derived
hen+1
h en
h
k;gn
h+ ni
4ken+1
h en
h
kk2
2+ 2 (kgn
hk2
2+k nk2
2); (3.55)
kgn
hk2=ken+1
h+en 1
h
2n
hk2ken+1
h+en 1
h
2k2kn
hk1
Cken+1
h+en 1
h
2k2C(ken+1
hk2+ken 1
hk2); (3.56)
in which the bound (3.16) has been used again. Then we get
hen+1
h en
h
k;gn
h+ ni
4ken+1
h en
h
kk2
2+ 2 k nk2
2
+C(ken+1
hk2
2+ken 1
hk2
2)
1
2En+1e;h+ 2 k nk2
2+C(ken+1
hk2
2+ken 1
hk2
2); (3.57)
in which the expansion identity, En+1e;h=
2ken+1
h en
h
kk2
2+1
4(krhen+1
hk2
2+krhen
hk2
2)
(given by (3.27)), has been applied. Its substitution into (3.54) gives
En+1e;h2E2e;h+CT(k4+h4) +CknX
j=0kej+1
h ej
h
kk2
2+n+1X
j=0(kej
hk2
2+krhej
hk2
2)
+C(ken+1
hk2
2+ken 1
hk2
2) + 4 k nk2
2 2 he2
h e1
h
k;g1
h+ 1i: (3.58)
Moreover, an application of the preliminary error estimate (3.17) (in Lemma 3.3)
leads to
En+1e;h2E2e;h+CT(k4+h4) +C(T2+ 1)knX
j=0kej+1
h ej
h
kk2
2+CTke0
hk2
2
+Ckn+1X
j=0krhej
hk2
2+ 4 k nk2
2 2 he2
h e1
h
k;g1
h+ 1i; (3.59)
in which we have made use of the following fact:
kn+1X
j=0kej
hk2
2k(n+ 1)
2ke0
hk2
2+ 2TknX
j=0kej+1
h ej
h
kk2
2
2Tke0
hk2
2+ 2T2knX
j=0kej+1
h ej
h
kk2
2: (3.60)16 JINGRUN CHEN, PANCHI LI, AND CHENG WANG
In addition, for the initial error quantities, the following estimates are available:
E2e;h=
2ke2
h e1
h
kk2
2+1
4(krhe2
hk2
2+krhe1
hk2
2)C(k4+h4); (3.61)
ke0
hk2
2C(k4+h4); (3.62)
4 k nk2
2C(k4+h4); (3.63)
kg1
hk2=ke2
h+e0
h
21
hk2ke2
h+e0
h
2k2k1
hk1C(k2+h2); (3.64)
2 he2
h e1
h
k;g1
h+ 1i2 ke2
h e1
h
kk2(kg1
hk2+k 1k2)
C(k4+h4); (3.65)
which comes from the assumption in Theorem 3.1. Then we arrive at
En+1e;hC(T2+ 1)knX
j=0kej+1
h ej
h
kk2
2+Ckn+1X
j=0krhej
hk2
2+C(T+ 1)(k4+h4)
C(T+ 1)(k4+h4) +C(T2+ 1)knX
j=0Ej+1e;h; (3.66)
in which the fact that Ej+1e;h=
2kej+1
h ej
h
kk2
2+1
4(krhej+1
hk2
2+krhej
hk2
2), has been
used. In turn, an application of discrete Gronwall inequality results in the desired
convergence estimate:
En+1e;hCTeCT(k4+h4);for all (n+ 1) :n+ 1T
k
; (3.67)
ken+1
h en
h
kk2+krhen+1
hk2C(k2+h2): (3.68)
Again, an application of the preliminary error estimate (3.17) (in Lemma 3.3) im-
plies that
ken+1
hk2
22ke0
hk2
2+ 2TknX
j=0kej+1
h ej
h
kk2
2C(k4+h4);
so thatken+1
hk2C(k2+h2): (3.69)
A combination of (3.68) and (3.69) nishes the proof of Theorem 3.1.
4.A numerical solver for the nonlinear system
It is clear that Algorithm 2.1 is a nonlinear scheme. The following xed-point
iteration is employed to solve it.
Algorithm 4.1. Setmn+1;0
h= 2mn
h mn 1
handp= 0.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 17
(i)Computemn+1;p+1
hsuch that
(4.1)mn+1;p+1
h mn 1
h
2k= mn+1;p+1
h+mn 1
h
2h
mn+1;p
h+mn 1
h
2!
+ mn+1;p+1
h+mn 1
h
2
mn+1;p+1
h mn 1
h
2k+ mn+1;p+1
h 2mn
h+mn 1
h
k2!
:
(ii)Ifkmn+1;p+1
h mn+1;p
hk2, then stop and set mn+1
h=mn+1;p+1
h.
(iii) Setp p+ 1and go to (i).
Denote the operator
(4.2)Lp=I mn 1
h 2
kmn
h k
2h(mn+1;p
h+mn 1
h);
and make the xed-point iteration solve the following equation
(4.3)Lpmn+1;p+1
h=mn 1
h+2
kmn
hmn 1
h k
2mn 1
hh(mn+1;p
h+mn 1
h);
in its inner iteration. Under the condition kCh2withCa constant, the following
lemma con rms the convergence of Algorithm 4.1. For any l2Land owing to the
property ofjmh(xl)j= 1, it is clear that 0 <kmhk11. For the discretized `2
norm of ^m2C3([0;T]; [C0(
)]3)\C2([0;T]; [C2(
)]3)\L1([0;T]; [C4(
)]3), we
have
(4.4) krh^mk22h 1k^mk2:
Then
(4.5)kh^mk2
2= hrh^m;rhh^mi
krh^mk2krhh^mk22h 1krh^mk2kh^mk2;
which in turn implies the following inverse inequality:
(4.6) kh^mk24h 2k^mk2:
Lemma 4.1. Letjmn 1
hj=jmn
hj= 1, there exists a constant c0such thatkmn 1
hk1,
kmn
hk1c0. The solution mn+1;p
hcalculated by (4.1) satis esjmn+1;p
hj=jmn 1
hj
forp= 1;2;, which means that we can still nd the constant c01satisfying
kmn+1;p
hk1c0. Then, for all p1, there exists a unique solution mn+1;p
hin
(4.1) and the following inequality is valid:
(4.7)kmn+1;p+1
h mn+1;p
hk24c0kh 2kmn+1;p mn+1;p 1k2:
Proof. For anymh2S2, the following identity is clear:
hmh;Lpmhi= 1;
for allp1. Thus the operator Lpis positive de nite for all p1, which
provides the unique solvability of (4.1). Taking the discrete inner product with
(4.1) bymn+1;p+1
h+mn 1
h, we havejmn+1;p+1
hj= 1 in a point-wise sense, which
means that the length of the magnetization is preserved at each step in the inner18 JINGRUN CHEN, PANCHI LI, AND CHENG WANG
iteration. Thus, we can nd a constant c01 to control the `1norm ofmn 1
h,
mn
handmn+1;p
hforp= 1;2;simultaneously.
Subtraction of two subsequent equations in the xed-point iteration yields
1
2k(mn+1;p+1
h mn+1;p
h) = 1
4(mn+1;p+1
h mn+1;p
h)hmn+1;p
h
1
4mn+1;p
hh(mn+1;p
h mn+1;p 1
h)
1
4mn
hh(mn+1;p mn+1;p 1)
1
4(mn+1;p+1
h mn+1;p)hmn 1
h
+
2kmn 1
h(mn+1;p+1
h mn+1;p
h) +
2k2mn
h(mn+1;p+1
h mn+1;p
h):
Taking the inner product with ( mn+1;p+1
h mn+1;p
h) by the above equation produces
kmn+1;p+1
h mn+1;p
hk2k
2kmn+1;p
hk1kh(mn+1;p
h mn+1;p 1
h)k2
+k
2kmn
hk1kh(mn+1;p
h mn+1;p 1
h)k2
c0kkh(mn+1;p mn+1;p 1)k2:
In turn, the convergence result becomes
(4.8)kmn+1;p+1
h mn+1;p
hk24c0kh 2kmn+1;p
h mn+1;p 1
hk2;
which completes the proof of Lemma 4.1.
5.Numerical experiments
5.1.Accuracy tests. Consider the 1-D iLLG equation
@tm= m@xxm+ m(@tm+ @ttm) +f:
The exact solution is chosen to be me= (cos(x) sin(t2);sin(x) sin(t2);cos(t2))T
with x=x2(1 x)2, and the forcing term is given by f=@tme+me@xxme
me(@tme+ @ttme). Fixing the tolerance = 1.0e-07 for the xed-point
iteration, we record the discrete `2and`1errors between the exact solution and
numerical solution with a sequence of temporal step-size and spatial mesh-size. The
parameters in the above 1-D equation are set as: = 0:1, = 10:0, and the nal
timeT= 0:01. The temporal step-sizes and spatial mesh-sizes are listed in the
Table 1 and Table 2.
Table 1. The discrete `2and`1errors in terms of the temporal
step-size. The spatial mesh-size is xed as h= 0:001 over
=
(0;1) and the nal time is T= 0:01.
kkmh mek2kmh mek1
T/40 1.2500e-11 1.2584e-11
T/60 5.6887e-12 5.6024e-12
T/80 3.2008e-12 3.1525e-12
T/100 2.0487e-12 2.0174e-12
order 1.97 2.00CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 19
Table 2. The discrete `2and`1error in terms of the spatial
mesh-size. The parameters are set as: the temporal step-size k=
2.0e-06,
= (0 ;1) and the nal time T= 0:5.
hkmh mek2kmh mek1
1/20 1.9742e-05 2.5320e-05
1/40 4.9846e-06 6.3459e-06
1/60 2.2340e-06 2.8201e-06
1/80 1.2720e-06 1.5853e-06
order 1.98 2.00
In addition, the 3-D iLLG equation is also considered:
@tm= mm+ m(@tm+ @ttm) +f:
The exact solution is chosen to be me= (cos(xyz) sin(t2);sin(xyz) sin(t2));cos(t2))T
with y=y2(1 y)2and z=z2(1 z)2, and the forcing term f=@tme+me
me me(@tme+ @ttme). Similarly, we record the discrete `2and`1er-
rors between exact and numerical solutions with a sequence of temporal step-sizes
and spatial mesh-sizes. The corresponding parameters are set as: = 0:01 and
= 1000:0. Besides, the nal time of this simulation is given by T= 0:01, with
the temporal step-size and spatial mesh-size listed in Table 3 and Table 4.
Table 3. The discrete `2and`1errors in terms of the temporal
step-size. The spatial mesh-size is xed as h= 0:001 and nal time
isT= 0:01.
kkmh mek2kmh mek1
T/100 1.2678e-05 1.2765e-05
T/120 8.8067e-06 8.8830e-06
T/140 6.4725e-06 6.5419e-06
T/160 4.9576e-06 5.0224e-06
order 2.00 1.98
Table 4. The discrete `2and`1errors in terms of spatial mesh-
size. The temporal step-size is xed as k= 2.0e-06.
hkmh mek2kmh mek1
1/8 1.4392e-07 3.4940e-07
1/10 9.6832e-08 2.2864e-07
1/12 6.9825e-08 1.6079e-07
1/14 5.2828e-08 1.1895e-07
order 1.79 1.92
5.2.Micromagnetics tests. The inertial e ect can be observed during the relax-
ation of a system with a non-equilibrium initialization. To visualize this, we conduct
micromagnetics simulations for both the LLG equation and the iLLG equation.20 JINGRUN CHEN, PANCHI LI, AND CHENG WANG
In the following simulations, a 3-D domain
= [0 ;1][0;1][0;0:4] is uniformly
discretized into 10 104 cells, with uniform initialization m0= (p
2=2;p
2=2;0)T.
For comparison, the LLG equation is discretized by the mid-point scheme proposed
in [4] with the xed-point iteration solver proposed in this work. The damping pa-
rameter is = 0:5 and the eld is xed as He= (10;0;0)T, which indicates that the
system shall converge to m= (1;0;0)T. Here the relaxation of the magnetization
behavior controlled by the LLG equation is visualized in Figure 1.
Time0 1 2 3 4 5/angbracketleftm/angbracketright
-0.500.51
/angbracketleftm1/angbracketright
/angbracketleftm2/angbracketright
/angbracketleftm3/angbracketright
Figure 1. The relaxation of the spatially averaged magnetization
controlled by the LLG equation. The nal time is T= 5:0 with
k= 0:001, and the damping parameter is = 0:5.
As for the counterpart of the LLG equation, wtih a given reference eld He, the
discrete energy of the iLLG equation becomes
(5.1) E[mn+1;mn] =1
4(krhmn+1
hk2
2+krhmn
hk2
2)+
2
mn+1
h mn
h
k
2
2 1
2hmn+1
h+mn
h;Hei:
Setting the inertial parameter = 1:0, the spatially averaged magnetization is
recorded to depict the inertial e ect in Figure 2(a). Meanwhile, the energy decay is
also numerically veri ed by Figure 2(b). The inertial e ect is observed at shorter
timescales for magnetization dynamics during the relaxation of the system with a
non-equilibrium initialization.
Furthermore, the inertial e ect can also be activated by an external perturbation
applied to an equilibrium state. Here we set the damping parameter = 0:02 and
= 0:5, then the time step-size must be reduced to 0.001 with T= 3:0. For the
equilibrium state m0= (1;0;0)T, the perturbation 4 :0sin(2ft) is applied along
ydirection over the time interval [0 ;0:05], withf= 20. The relaxation of the
iLLG equation, revealed by the evolution of the spatially averaged magnetization,
is visualized in Figure 3.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 21
Time0 5 10 15 20/angbracketleftm/angbracketright
-0.500.51
/angbracketleftm1/angbracketright
/angbracketleftm2/angbracketright
/angbracketleftm3/angbracketright
(a)Spatially averaged magnetization
Time0 2 4 6 8 10 12 14 16 18 20E[mn+1,mn]
-4-3.8-3.6-3.4-3.2-3-2.8
(b)Energy decay
Figure 2. The spatially averaged magnetization (A) and the en-
ergy evolution (B) in the iLLG equation. Parameters setting:
T= 20,k= 0:02, = 1:0 and = 0:5.
6.Conclusion
In this work, an implicit mid-point scheme, with three time steps, is proposed to
solve the inertial Landau-Lifshitz-Gilbert equation. This algorithm preserves the
properties of magnetization dynamics, such as the energy decay and the constant
length of magnetization and is proved to be second-order accurate in both space
and time. In the convergence analysis, we rst construct a second-order approxi-
mationmof the exact solution. It is found that mproducesO(h5) accuracy at
the mesh points around the boundary sections, which simpli es the estimation at
boundary points. Then, by analyzing the error function between the numerical22 JINGRUN CHEN, PANCHI LI, AND CHENG WANG
Time0 0.5 1 1.5 2 2.5 3/angbracketleftm/angbracketright
-0.200.20.40.60.8/angbracketleftm1/angbracketright
/angbracketleftm2/angbracketright
/angbracketleftm3/angbracketright
Figure 3. The response of the spatially averaged magnetization
for the magnetic perturbation in the presence of the inertial e ect.
For the equilibrium initialization m0= (1;0;0)T, a perturbation
4:0sin(2ft) is applied along ydirection during time interval
00:05, withf= 20. The basic simulation parameters are:
= 0:02, = 0:5,T= 3:0 andk= 0:001.
solution and the constructed solution mh, we derive the convergence result in the
H1(
T) norm. Furthermore, a xed-point iteration method is proposed to solve
this implicit nonlinear scheme under the time-step restriction kCh2. Numerical
results con rm the theoretical analysis and clearly show the unique inertial e ect
in micromagnetics simulations.
Acknowledgments
This work is supported in part by the grant NSFC 11971021 (J. Chen), the
program of China Scholarships Council No. 202106920036 (P. Li), the grant NSF
DMS-2012269 (C. Wang).
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School of Mathematical Sciences and Suzhou Institute for Advanced Research, Uni-
versity of Science and Technology of China, China
Suzhou Institute for Advanced Research, University of Science and Technology of
China, Suzhou, Jiangsu 215123, China
Email address :jingrunchen@ustc.edu.cn
School of Mathematical Sciences, Soochow University, Suzhou, 215006, China
Email address :lipanchi1994@163.com
Mathematics Department, University of Massachusetts, North Dartmouth, MA 02747,
USA
Email address :cwang1@umassd.edu |
2002.02686v1.Engineering_Co__2_MnAl__x_Si___1_x___Heusler_compounds_as_a_model_system_to_correlate_spin_polarization__intrinsic_Gilbert_damping_and_ultrafast_demagnetization.pdf | 1
Engineering Co 2MnAl xSi1-x Heusler compounds as a model system to
correlate spin polarization, intrinsic Gilbert damping and ultrafast
demagnetization
C. Guillemard1,2, W. Zhang1*, G. Malinowski1, C. de Melo1, J. Gorchon1, S. Petit -
Watelot1, J. Ghan baja1, S. Mangin1, P. Le Fèvre2, F. Bertran2, S. Andrieu1*
1 Institut Jean Lamour, UMR CNRS 7198, Université de Lorraine, 54500 Nancy France
2 Synchrotron SOLEIL -CNRS, Saint -Aubin, 91192 Gif -sur-Yvette, France
Abstract:
Engineering of magnetic materials f or developing better spintronic applications relies on
the control of two key parameters: the spin polarization and the Gilbert damping
responsible for the spin angular momentum dissipation. Both of them are expected to
affect the ultrafast magnetization dyna mics occurring on the femtosecond time scale.
Here, we use engineered Co2MnAl xSi1-x Heusler compounds to adjust the degree of spin
polarization P from 60 to 100% and investigate how it correlates with the damping. We
demonstrate experimentally that the damping decreases when increasing the spin
polarization from 1.1 10-3 for Co 2MnAl with 63% spin polarization to an ultra -low value
of 4.10-4 for the half -metal magnet Co 2MnSi. This allows us investigating the relation
between these two parameters and the ultrafast demagnetization time characterizing the
loss of magnetization occurring after femtosecond laser pulse excitation. The
demagnetization time is observed to be inversely proportional to 1 -P and as a consequence
to the magnetic damping, which can be attributed to the similarity of the spin angular
momentum dissipation processes responsible for these two effects. Altogether, our high
quality Heusler compounds allow controlling the band structure and therefore the channel
for spin angular momentum dissipation.
* corresponding authors :
wei.z hang @univ -lorraine.fr
stephane.andrieu@univ -lorraine.fr 2
I - INTRODUCTION
During the last decades, extensive magnetic materials research has strived to
engineer denser, faster and more energy efficient processing and data storage devices. On
the one hand, a high spin polarization has been one of the most important ingredients th at
have been seek [1]. For example, the spin polarization is responsible for a high readout
signal in magnetic tunnel junction based devices [2,3] . Additionally, a high spin
polarization results in a decrease of the threshold current for magnetization reve rsal by
spin torques [4] required for the development of spin-transfer -torque magnetic random
access memory devices [5] , for gyrotropic dynamics in spin -torque nano -oscillators [6]
and for magnetic domain wall motion [7]. On the other hand, the intrinsic m agnetic
energy dissipation during magnetization dynamics, which is determined by the Gilbert
damping constant, needs to be low in order to build an energy efficient device. Fortunately,
spin polarization and damping are usually closely related in magnetic materials.
Nowadays, manipulation of the magnetization on the femtosecond timescale has
become an outstanding challenge since the demonstration of subpicosecond
magnetization quenching [8] and magnetization reversal on the picosecond timescale [9].
Despite the theoretical and experimental work that has been reported up to now , the
relationship between the polarization at the Fermi level or the magnetic damping and the
ultrafast demagnetization excited by femtosecond lasers, remains unclear [10-15]. Indeed,
numerous mech anisms have been proposed but no consensus has yet been reached. In
particular, efforts have been undertaken to unify the magnetization dynamics on the
nanosecond timescale and the ultrafast demagnetization considering that the sp in-flip
mechanisms involved in both phenomena could be the same [10-11,16] . Regarding the
influence of the damping on the demagnetization time, different predictions have been
reported both experimentally and theoretically . In this situation, the need for engineered
samples in which the spin -polarization and magnetic damping are well controlled is of
utmost importance to unveil their role on the ultrafast magnetization dynamics.
Heusler compounds are a notable class of magnetic materials allowing for tunabl e
spin-polarization and magnetic damping [ 17]. The absence of available electronic states
in the minority band at the Fermi level leads to very high spin polarization and ultra -low
damping due to a strong reduction of spin scattering [ 18-23]. Recently, ultra-low damping 3
coefficient associated with full spin polarization at the Fermi energy was reported in
Co2Mn-based Heusler compounds , [22-23]. Among those alloys, Co 2MnSi has the
smallest damping down to 4.1 x 10-4 with 100% spin -polarization while Co 2MnAl , which
is not predicted to be a half -meta llic magnet, has a damping of 1.1 x 10-3 and a spin -
polarization of 60 %.
In the present work, we used Co 2MnAl xSi1-x quaternary Heusler compounds
grown by Molecular Beam Epitaxy (MBE) to tune the spin -polarization at the Fermi
energy . Controlling the amount of Al within the alloys allows tuni ng the spin -polarization
from 60 to 100 % as measured by spin resolved photoemission. We show that the
magnetic damping parameter for these alloys is among the lowest reported in the literature
and decreases when the spin -polarization increases. Ultrafast magnetization dynamics
experiments were thus performed on these prototype samples. This complete
experimental characterization allows us to directly correlat e the ultrafast magnetization
dynamics to these parameters and comparing our results to the different theory discussed
above.
The Co 2MnSi compound grows in the L2 1 structure whereas the Co 2MnAl com pound
grows in the B2 phase as shown by STEM -HAADF analysis [22]. Such different
structures are directly observable during the growth by Reflexion High Energy Electron
Diffraction (RHEED ) since the surface lattice is different for bot h compounds. Ind eed,
half streaks are observed along Co 2MnSi [110] azimuth due to the L2 1 chemical ordering
[24] which is not the case for Co2MnAl [22]. The RHEED analysis on Co2MnAl xSi1-x
films with x= 0, ¼ ,½ , ¾ ,1 reveals a regular decrease of these half -streaks intensity with
x (Figure 1 a). This information that concerns only the surface is confirmed in the entire
thickness of the films by using x -ray diffraction. Indeed, the (111) peak typical of the
chemical ordering in the L2 1 structure clearly decreases and disappears with x ( Figure
1b). 4
Figure 1 : a) RHEED patterns along [110] showing the progressive vanishing of the half -streaks
(observed on Co 2MnSi, x=0) at the surface with x. b) Confirmation of the transition from L2 1 to
B2 chemical ordering in the entire film by the vanishing of the (111) peak and displacement of
(220) peak with x as shown by x -ray diffraction. c) Spatial distribution of both chemical ordering
in the films deduced from STEM -HAADF experiments: as the L 21 structure is observed in the
entire Co 2MnSi film (x=0 ), and the B2 one i n Co 2MnAl (x=1 ), a mixing of both structure is clearly
observed for x=0.5 .
In addition, the displacement of the (220) peak with x allows us to extract a linear
variation of the lattice constant (Figure 1b ), as observed in the case of a solid solution.
This is an indication that the L2 1 chemical ordering progressively vanishes when
increasing the Al substitution rate 𝑥. However, the chemical disorder distribution in the
films cannot be easily determined by using the electron and x -ray diffraction analyses. To
address this point, a STEM HAADF analysis has been carried on the Co 2MnAl ½Si½ films
with a comparison with Co 2MnSi and Co 2MnAl. A clear mixing of both structures is
5
observ ed for x=½ where around 50% is L2 1 chemically ordered and 50%, B2, with typical
domains size around 10nm along the growth axis (001) and a few nm in the plane of the
film ( Figure 1c).
The electronic properties of the Co 2MnAl xSi1-x(001) series were studied using spin -
resolved photoemission (SR -PES) and ferromagnetic resonance (FMR). The SR -PES
spectra were obtained by using the largest slit acceptance of the detector (+/ - 8°) at an
angle of 8° of the normal axis of the surface. Such geometry allows us to analyze all the
reciprocal space as confirmed by similar experiments but performed on similar
polycrystalline films [23]. Getting the spin-polarization dependence with x using raw SR -
PES spectra is however not obvious due to the existence of surface states systematically
observed on Co 2MnSi but also on other Co 2Mn-based Heusler compounds [19, 22-23].
To get the bulk spin polarization, we thus used the S polarization of the photon beam.
Indeed , we have shown that the surface states are no more detected due to their symmetry
[19] without any loss of information on the bulk band structure [ 23]. The corresponding
SR-PES spectra are shown in figure 2 . As expec ted, we thus obtain a tunable spin
polarization at EF from 100% to 63% by substituting Si by Al, as shown in figure 3 .
Figure 2 : spin -resolved photoemission spectra using P photon polarization (left), S photon
polarization (middle) and resulting spin polarization curves (right) for the Co 2MnAl xSi1-x series,
6
The radiofrequency magnetic dynamics of the films were thus studied using
ferromagnetic resonance (FMR) . The magnetic damping coefficient , the effective
magnetic moment Ms (close to the true moment in our films due to very small anisotropy
– see [ 22]), and the inhomogeneous linewidth f0 were thus extracted from the
measurements performed on the Co 2MnAl xSi1-x(001) series. The results obtained on the
same series used for photoemission experiments are shown in table I . As shown in figure
3, a clear correlation is observed between the spin polarization at EF and the magnetic
damping coefficient , as theoretically expected. An ultra -low value was obtained for
Co2MnSi (x= 0) due to the large spin gap [ 22]. By substituting Al by Si, the magnetic
damping increase is explained by the decrease of the spin polarization.
Co2MnAl xSi1-x Spin polarization
(%) Ms
(µB/f.u.)
(x 10-3) f0
(MHz) g factor
(0.01)
x = 0 973 5.08 0.460.05 14.3 2.01
x = 0.25 903 4.85 0.730.15 21.7 1.99
x = 0.5 833 4.85 0.680.15 9 2.01
x = 0.75 703 4.8 1.000.05 81.5 2.00
x = 1 633 4.32 1.100.05 22 2.01
Table 1: data extracted from spin -resolved photoemission and ferromagnetic resonance
experiments performed on the Co 2MnAl xSi1-x series.
Figure 3: -top- spin polarization and magnetic damping dependence with Al content for the
Co2MnAl xSi1-x series and –bottom - magnetic damping versus spin polarization . The lines are
guide to the eyes.
7
In addition, t he magnetization is also observed to decrease with x in agreement with the
Slater -Pauling description of the valence band electrons in Heusler compounds [25].
Indeed, as a 5 µ B magnetic moment per cell is expected for Co2MnSi (type IV valence
electrons), it should decrease to 4 when replacing Si by Al (type III) as actually observed
(Table I ). Finally, the FMR susceptibilities reach extremely small inhomogeneous
linewidth f0, a proof of the excellent homogeneity of the magnetic properties (hence a
high crystal quality) in our films.
Figure 4 (a) shows the ultrafast demagnetization curves measured on the same
Co2MnAl xSi1-x series with a maximum magnetization quenching ~1 5%. The temporal
changes of the Kerr signals ∆𝜃𝑘(𝑡) were normalized by the saturation value 𝜃𝑘 just before
the pump laser excitation. The time evolution of magnetization on sub -picosecond
timescales c an be fitted according to Eq. (2 ) in terms of the three -Temperature M odel
(3TM) [26], which describes the energy distribution among electrons, phonons, and spins
after laser excitation.
−∆𝑀(𝑡)
𝑀={[𝐴1
(𝑡𝜏0+1 ⁄ )0.5−𝐴2𝜏𝐸−𝐴1𝜏𝑀
𝜏𝐸−𝜏𝑀𝑒−𝑡
𝜏𝑀−𝜏𝐸(𝐴1−𝐴2)
𝜏𝐸−𝜏𝑀𝑒−𝑡
𝜏𝐸]Θ(𝑡)}∗𝐺(𝑡,𝜏𝐺) (2)
where 𝐺(𝑡,𝜏𝐺) represents the convolution product with the Gaussian laser pulse profile,
G
is the full width at half maximum (FWHM) of the laser pulses. Θ(𝑡) is the Heavyside
function . The constant A1 represents the amplitude of demagnetization obtained after
equilibrium between the electrons, spins, and phonons is reestablished while A 2 is
proportional to the initial electron temperature raise . The two critical time parameters
𝜏𝑀,𝜏𝐸 are the ultrafast demagnetization time and magnetization recovery time,
respectively. In the low fluence regime, which corresponds to our measurements, 𝜏𝐸
becomes close to the electron -phonon relaxation time . A unique value of 𝜏𝐸=550 ±
20 𝑓𝑠 was used for fitt ing the demagnetization curves for all samples. T he ultrafast
demagnetization time 𝜏𝑀 decrease s from 380 ±10 fs for Co 2MnSi to 165 ±10 fs for
Co2MnAl (Figure 4b). The evolution of the demagnetization time with both spin
polarization P and Gilbert damping 𝛼 is presented in figure 4c and 4d . A clear linear
variation between 1𝜏𝑀⁄ and 1−𝑃 is observed in this series . As the magnetic damping 𝛼
is proportional to P here, this means that 1𝜏𝑀⁄ is proportional to 𝛼 too. A similar relation 8
between these two par ameters was proposed by Koopmans et al. [10]. However, they also
predicted an influence of the Curie temperature . As the Curie temperature in Heusler
compounds changes with the number of valence electron s and because the Co 2MnAl xSi1-
x behave as solid solutions as indicated by the lattice spacing variation ( Figure 1b), we
thus consider a linear decrease of 𝑇𝑐 with x going from 985 K to 697 K as exper imentally
measured for x =0 and x=1, respectively. To test this possi ble influence of the Curie
temperature on the ultrafast magnetization dynamics , we plot in figure 4d first the product
𝜏𝑀.𝛼 and second the product 𝜏𝑀.𝛼.𝑇𝑐(𝑥)𝑇𝑐(𝐶𝑜2𝑀𝑛𝑆𝑖 ) ⁄ . These results demonstrate that
the Curie temperature does not influence the ultrafas t demagnetization in our samples .
Figure 4 : (a) Ultrafast demagnetization curves obtained for different Al concentration x . The
curves have been shifted vertically for sake of clarity. The solid lines represent fitted curves
obtained using Eq. ( 2). (b) Ultrafast demagnetization time as a function of Al content x, (c) the
inverse of 𝜏𝑀 as a function of 1-P, P being the spin polarization at E F, and d) test of Koopmans
model with and without taking into account the Curie temperature of the films (see text).
9
One can now compare our experimental results with existing theoretica l models.
We first discuss the dependence between the magnetic damping and the spin polarization.
Ultra -low magnetic damping values are predicted in Half -Metal Magnet (HMM) Heusler
compounds and explained by the lack of density of state at the Fermi energy for minority
spin, or in other words by the full spin polarization [18,27,28] . Consequently, the
magnetic damping is expected to increase when creating some states in the m inority band
structure around the Fermi energy that is when decreasing the spin polarization [28]. If
we confirmed in previous experimental works that ultra -low magnetic damping
coefficients are actually observed especially on HMM Co2MnSi and Co 2MnGe [19,2 2-
23], we could not state any quantitative dependence between the damping values and the
spin polarization. As prospected, the Co 2MnAl xSi1-x alloys are shown here to be ideal
candidates to address this point . This allows us getting a clear experimental demonstration
of these theoretical expectations. Furthermore, a linear dependence between the magnetic
damping and the spin polarization is obtained. This behavior may be explained by the
mixing of both L2 1 and B2 phases in the films. To the best of our knowledge, this
experimental result is the first quantitative demonstration of the link between the
magnetic damping and spin polarization.
Second , the dependence between the magnetic damping and the demagnetization
time observed here is a clear opportunity to test the different theoretical explanations
proposed in the literature to explain ultrafast dynamics . In the last 15 years, t he influence
of the damping on the ultrafast dynamics has been explored, both theoreticall y and
experimentally. The first type of prediction we want to address is the link between the
demagnetization time and the electronic structure via the spin polarization P. Using a
basic approach considering the Fermi golden rule, several groups [12,13] proposed that
the demagnetization process is linked to the population of minority and majority spin
states at E F, leading to a dependence of the spin-scattering rate proportional to 1 -P [13].
As this spin scattering rate is linked to the inverse of the dem agnetization delay time , the
𝜏𝑀~(1−𝑃)−1 law was proposed . This law is clearly verified i n our samples series. One
should note that this is a strong experimental demonstration since we compare samples
grown in the same conditions , so with the same control of the stoichiometry and structural
properties . 10
However, one point is still not clear since much larger demagnetization times in
the picosecond timescale would be expected for large band gap and full spin -polarization.
In the case of small band ga p of the order of 0.1 eV, Mann et al [13] showed that thermal
effects from the heated electron system lead to a decrease of 𝜏𝑀. They calculated a
reduction of the spin -flip suppression factor from 104 for a gap of 1 eV to 40 for a gap of
0.3 eV. However, the band gap of our Co 2MnSi was calculated to be around 0.8 eV with
a Fermi energy in the middle of the gap [27,28] . This was corroborated by direct
measurement using SR -PES [19, 22 ]. Therefore, according to their model, we should
expect a much longer demagnetization time for Co 2MnSi. However, the largest values
reported by several groups [13, 29] all on HMM materials are of the same order of
magnitude, i.e. around 350 to 400fs . This probably means that a limitation exists due to
another physical reason . One hypothesis should be to consider the 1.5eV photon energy
which is much larger than the spin gap. During the excitation, the electrons occupying the
top minority spin valence band can be directly excited into the conduction band. In a
similar way, maj ority spin electrons are excited at energies higher than the spin band gap.
Both of these effects may allow for spin flips scattering and only the majority electrons
excited within the spin band gap energy range cannot flip their spins. Even if such photon
energy influence is not considered based on the argument that the timescale for photon
absorption followed by electronic relaxation is very fast compared to the magnetic
relaxation process [16 ], performing experiments by changing the excitation wavelength
to energies below the spin band gap would be very interesting to better understand
ultrafast magnetization dynamics.
Concerning the dependence between the demag netization time and the magnetic
damping , different theoretical models have been proposed and two opposite trends were
obtained; 𝛼 and 𝜏𝑀 being either directly [15] or inversely [10 ] proportional . From the
experimental side, the inverse proportionality between 𝜏𝑀 and 𝛼 proposed by Koopmans
et al. [10] could not be reproduced by doping a thin Permalloy film with rare -earth atoms
[14]. However, the introduction of these rare -earth elements strongly modifies the
magnetic relaxation properties and could induce different relaxation channels for 𝜏𝑀 and
𝛼 [30]. Zhang et al. performed a similar st udy using thin Co/Ni multilayers and observed
a direct proportionality between 𝜏𝑀 and 𝛼 [15]. However, the damping extracted in their 11
study should be strongly influenced by the heavy metal Pt capping and seed layers which
may induce strong spin pumping effect during the magnetization precession [30].
Furthermore, they did not take into account the influence of the Curie temperature.
Therefore, in these studies, extrinsic effects might influence the magnetization dynamics
in a different way on both time scales which makes more complex the comparison
between theory and experiments. Therefore, o ur results offer a nice opportunity to
disentangle the se different effects. According to different studies , the ultrafast
demagnetization slows down when approaching the Curie temperature [ 10,16, 32,33]. In
other words, a larger difference between the initial temperature and 𝑇𝑐 would lead to a
faster demag netization . In our samples, 𝑇𝑐 goes up from Co2MnAl to Co2MnSi, whereas
the demagnetization process becomes slower . Therefore, we conclude that, in the present
case, the Curie temperatures of our samples are too high to affect 𝜏𝑀 which only depends
on the intrinsic propertie s of the films, i.e. Gilbert damping and spin polarization. This
also clarifies some points reported by Müller et al. work [ 12]. In their paper, they first
reported a very fast demagnetization process in Co 2MnSi(110) and second a slow one in
CrO 2 and LaSrMnO 3 films with 𝑇𝑐 values close to room temperature (390 K 360 K
respectively). Therefore, it is not possible to state whether the very slow demag netization
process in these compounds is due to a low 𝑇𝑐 or a large spin polarization. Furthermore,
recent experimental results demonstrated a large decrease in the spin polarization at the
Fermi level in CrO 2 as function of the temperature, resulting in less than 50% at 300 K
[34]. In our samples we disentangle these two effects and the longest demagnetization
time is found for Co 2MnSi (𝜏𝑀=380 𝑓𝑠), a true half -metal magnet with a 0.8 eV spin
gap and a large 𝑇𝑐.
In summary, we first demonstrate experimentally that substituting Si by Al in
Co2MnAl xSi1-x Heusler compounds allows us to get a tunable spin polarization at E F from
60% in Co 2MnAl to 100% in Co 2MnSi, indicati ng the transition from metallic to half
metallic behaviors. Second, a strong correlation between the spin polarization and the
Gilbert magnetic damping is established in these films . This confirms the theoretical
justification of ultra -low magnetic damping in Ha lf-Metal -Magnet s as a consequence of
the spin gap. Third , the ultrafast spin dynamics results also nicely confirm that the spin
gap is at the origin of the increase of the relaxation time. Our experiments allow us to go 12
further by establishing clear relati onships between the spin polarization, the magnetic
damping and the demagnetization time. A n inverse relationship between demagnetization
time and Gil bert damping is established in these alloys , which agrees well with the model
proposed by Mann et al. [13] and with Koopmans et al. [10] but without considering any
influence of Curie temperature much larger than room temperature in these films .
Experimental section
Co2MnAl xSi1-x(001) quaternary Heusler compounds are grown by Molecular
Beam Epitaxy using an MBE machine e quipped with 24 materials. The s toichiometry is
accurately controlled during the growth by calibration of the Co, Mn, Si and Al atomic
fluxes using a quartz microbalance located at the pl ace of the sample. The error on each
elem ent concentration is less than 1 % [23]. The films are grown directly on MgO(001)
substrates, with the epitaxial relationship [100] (001) MgO // [110] (001) Heusler
compound. The thickness is fixed to 20nm.
The phot oemission experiments were done at the CASSIOPEE beamline at
SOLEIL synchrotron source. The films were grown in a MBE connected to the beamline
(see [19,22,35 ] for details). The SR -PES spectra were obtained by using the largest slit
acceptance of the detec tor (+/ - 8°) at an angle of 8° of the normal axis of the surface. Such
geometry allows us to analyze all the reciprocal space on similar polycrystalline films
[23].
The radiofrequency magnetic dynamics of the films were thus studied using
ferromagnetic resonance (FMR). A Vectorial Network Analyzor FMR set -up was used in
the perpendicular geometry (see [ 22] for experimental details) where the static magnetic
field is applied out of the plane of the film in order to avoid extrinsic bro adening of the
linewidth due to the 2 -magnons scattering [ 36,37].
Ultrafast magnetization dynamics were investigat ed using polar time -resolved
magneto -optical Kerr (TR -MOKE) experiments. An amplified Ti -sapphire laser
producing 35 fs pulses at 800 nm with a repetition rate of 5 KHz is used . The pump beam
is kept at the fundamental mode and is focused down to spot size of ~260 𝜇𝑚 while the
probe is frequency doubled to 400 nm and focused to a spot size of ~60 𝜇𝑚. Samples
were magnetically saturated alon g the out -of-plane axis by applying a 1T magnetic field.
13
Acknowledgement
This work was supported partly by the French PIA project “Lorraine Université
d’Excellence”, reference ANR -15-IDEX -04-LUE, and by the Agence Nationale de la
Recherche (France) under contract no. ANR -17-CE24 -0008 (CHIPMuNCS).
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2009.07777v1.Exponential_decay_for_semilinear_wave_equations_with_viscoelastic_damping_and_delay_feedback.pdf | arXiv:2009.07777v1 [math.AP] 16 Sep 2020Exponential decay for semilinear wave equations with visco elastic
damping and delay feedback
Alessandro Paolucci∗Cristina Pignotti†
Abstract
In this paperwestudy aclassofsemilinearwavetype equationswith v iscoelasticdamping
anddelayfeedbackwithtimevariablecoefficient.Bycombiningsemigro uparguments,careful
energy estimates and an iterative approach we are able to prove, u nder suitable assumptions,
a well-posedness result and an exponential decay estimate for solu tions corresponding to
smallinitialdata.Thisextendsandconcludestheanalysisinitiatedin[16]an dthendeveloped
in [13, 17].
1 Introduction
LetHbe a Hilbert space and let Abe a positive self-adjoint operator with dense domain D(A)
inHand compact inverse in H. Let us consider the system:
utt(t)+Au(t)−/integraldisplay+∞
0µ(s)Au(t−s)ds+k(t)BB∗ut(t−τ) =∇ψ(u(t)), t∈(0,+∞)
u(t) =u0(t), t∈(−∞,0],
ut(0) =u1
B∗ut(t) =g(t), t∈(−τ,0),(1.1)
whereτ >0 represents the time delay, Bis a bounded linear operator of Hinto itself, B∗
denotes its adjoint, and ( u0(·),u1,g(·)) are the initial data taken in suitable spaces. Moreover,
the delay damping coefficient k: [0,+∞)→IR is a function in L1
loc([0,+∞)) such that
/integraldisplayt
t−τ|k(s)|ds<C∗,∀t∈(0,+∞), (1.2)
for a suitable constant C∗,and the memory kernel µ: [0,+∞)→[0,+∞) satisfies the following
assumptions:
(i)µ∈C1(IR+)∩L1(IR+);
(ii)µ(0) =µ0>0;
∗Dipartimento di Ingegneria e Scienze dell’Informazione e M atematica, Universit` a di L’Aquila, Via Vetoio,
Loc. Coppito, 67010 L’Aquila Italy ( alessandro.paolucci2@graduate.univaq.it ).
†Dipartimento di Ingegneria e Scienze dell’Informazione e M atematica, Universit` a di L’Aquila, Via Vetoio,
Loc. Coppito, 67010 L’Aquila Italy ( pignotti@univaq.it ).
1(iii)/integraltext+∞
0µ(t)dt= ˜µ<1;
(iv)µ′(t)/lessorequalslant−δµ(t), for some δ>0.
Furthermore, ψ:D(A1
2)→IR is a functional having Gˆ ateaux derivative Dψ(u) at every
u∈D(A1
2).Moreover, in the spirit of [2], we assume the following hypot heses:
(H1) For every u∈D(A1
2), there exists a constant c(u)>0 such that
|Dψ(u)(v)|/lessorequalslantc(u)||v||H∀v∈D(A1
2).
Then,ψcan be extended to the whole Hand we denote by ∇ψ(u) the unique vector
representing Dψ(u) in the Riesz isomorphism, i.e.
/an}b∇acketle{t∇ψ(u),v/an}b∇acket∇i}htH=Dψ(u)(v),∀v∈H;
(H2) for all r>0 there exists a constant L(r)>0 such that
||∇ψ(u)−∇ψ(v)||H/lessorequalslantL(r)||A1
2(u−v)||H,
for allu,v∈D(A1
2) satisfying ||A1
2u||H/lessorequalslantrand||A1
2v||H/lessorequalslantr.
(H3)ψ(0) = 0,∇ψ(0) = 0 and there exists a strictly increasing continuous fun ctionhsuch that
||∇ψ(u)||H/lessorequalslanth(||A1
2u||H)||A1
2u||H, (1.3)
for allu∈D(A1
2).
We are interested in studying well–posedness and stability results, for small initial data, for the
above model. Our results extend the ones of [16, 17] where abs tract evolution equations are
analyzed and, in the specific case of memory damping, exponen tial decay is obtained essentially
only in the linear case. Indeed, in the nonlinear setting, an extra standard frictional damping,
not delayed, was needed in order to obtain existence and uniq ueness of global solutions with
exponentially decaying energy for suitably small initial d ata. Moreover, in [16, 17] the delay
damping coefficient k(t) is assumed to be constant and the results there obtained req uire a
smallness assumption on /ba∇dblk/ba∇dbl∞.The analysis of [16, 17] has been extended in [13] by consider ing
a time variable delay damping coefficient k(t) as in the present paper. However, also in [13] an
extra frictional not delayed damping was needed, in the case of wave type equation with memory
damping, when a locally Lipschitz continuous nonlinear ter m is included into the equation.
Then, here, we focus on wave type equations with viscoelasti c damping, delay feedback and
source term, obtaining well-posedness and stability resul ts for small initial data without adding
extra frictional not delayed damping. So, we here improve an d conclude the analysis developed
in [16, 17, 13] for the class of models at hand. Other models wi th viscoelastic damping and
time delay are studied in recent literature. The first result is due to [12], in the linear setting.
In that paper a standard frictional damping, not delayed, is included into the model in order
to compensate the destabilizing effect of the delay feedback. Actually, at least in the linear
case, the viscoelastic damping alone can counter the destab ilizing delay effect, under suitable
assumptions, without needing other dampings. This has been shown, e.g., in [11, 3, 9, 23]. The
case of viscoelastic wave equation with intermittent delay feedback has been studied in [20]
2while [14] deals with a model for plate equation with memory, source term, delay feedback and
standard not delayed frictional damping.
More extended is the literature in case of a frictional/stru ctural damping, instead of a
viscoelastic term, which compensates the destabilizing effe ct of a time delay and, for specific
models, mainly in the linear setting, several stability res ults have been quite recently obtained
under appropriate assumptions (see e.g. [1, 4, 5, 7, 15, 18, 2 1, 22]).
The paper is organized as follows. In Section 2 we give some pr eliminaries, writing system
(1.1) in an abstract way. In Section 3 we prove the exponentia l decay of the energy associated
to (1.1). Finally, in Section 4 some examples are illustrate d.
2 Preliminaries
As in Dafermos [8], we define the function
ηt(s) :=u(t)−u(t−s), s,t∈(0,+∞), (2.4)
so that we can rewrite (1.1) in the following way:
utt(t)+(1−˜µ)Au(t)+/integraldisplay+∞
0µ(s)Aηt(s)ds+k(t)BB∗ut(t−τ)
=∇ψ(u(t)), t∈(0,+∞),
ηt
t(s) =−ηt
s(s)+ut(t), t,s∈(0,+∞),
u(0) =u0(0),
ut(0) =u1,
B∗ut(t) =g(t), t∈(−τ,0),
η0(s) =η0(s) =u0(0)−u0(−s)s∈(0,+∞).(2.5)
LetL2
µ((0,+∞);D(A1
2)) be the Hilbert space of the D(A1
2)−valued functions in (0 ,+∞)
endowed with the scalar product
/an}b∇acketle{tϕ,ψ/an}b∇acket∇i}htL2µ((0,+∞);D(A1
2))=/integraldisplay∞
0µ(s)/an}b∇acketle{tA1
2ϕ,A1
2ψ/an}b∇acket∇i}htHds
and denote by Hthe Hilbert space
H=D(A1
2)×H×L2
µ((0,+∞);D(A1
2)),
equipped with the inner product
/angbracketleftBigg
u
v
w
,
˜u
˜v
˜w
/angbracketrightBigg
H:= (1−˜µ)/an}b∇acketle{tA1
2u,A1
2˜u/an}b∇acket∇i}htH+/an}b∇acketle{tv,˜v/an}b∇acket∇i}htH+/integraldisplay∞
0µ(s)/an}b∇acketle{tA1
2w,A1
2˜w/an}b∇acket∇i}htHds.(2.6)
SettingU= (u,ut,ηt) we can restate (1.1) in the abstract form
U′(t) =AU(t)−k(t)BU(t−τ)+F(U(t)),
BU(t−τ) = ˜g(t) fort∈[0,τ],
U(0) =U0,(2.7)
3where the operator Ais defined by
A
u
v
w
=
v
−(1−˜µ)Au−/integraltext+∞
0µ(s)Aw(s)ds
−ws+v
with domain
D(A) ={(u,v,w)∈D(A1
2)×D(A1
2)×L2
µ((0,+∞);D(A1
2)) :
(1−˜µ)u+/integraldisplay+∞
0µ(s)w(s)ds∈D(A), ws∈L2
µ((0,+∞);D(A1
2))},(2.8)
in the Hilbert space H,and the operator B:H → His defined by
B
u
v
w
:=
0
BB∗v
0
.
Moreover, ˜g(t) = (0,Bg(t−τ),0), U0= (u0(0),u1,η0) andF(U) := (0,∇ψ(u),0)T.From (H2)
and (H3) we deduce that the function Fsatisfies:
(F1)F(0) = 0;
(F2) for each r>0 there exists a constant L(r)>0 such that
||F(U)−F(V)||H/lessorequalslantL(r)||U−V||H (2.9)
whenever ||U||H/lessorequalslantrand||V||H/lessorequalslantr.
3 Stability result
In this section we want to prove an exponential stability res ult for the system (1.1) for small
initial data. It’s well–known (see e.g. [10]) that the opera torAin the problem’s formulation
(2.7) generates an exponentially stable semigroup {S(t)}t/greaterorequalslant0,namely there exist two costants
M,ω>0 such that
||S(t)||L(H)/lessorequalslantMe−ωt. (3.10)
Denoting
/ba∇dblB/ba∇dblL(H)=/ba∇dblB∗/ba∇dblL(H)=b, (3.11)
then/ba∇dblB/ba∇dblL(H)=b2.Our result will be obtained under an assumption on the coeffici entk(t)
of the delay feedback which includes as particular cases kintegrable and kinL∞with/ba∇dblk/ba∇dbl∞
sufficiently small. More precisely, we assume (cf. [13]) that there exist two constants ω′∈[0,ω)
andγ∈IR such that
b2Meωτ/integraldisplayt
0|k(s+τ)|ds/lessorequalslantγ+ω′t,for allt/greaterorequalslant0. (3.12)
Theorem 3.1. Assume (3.12). Moreover, suppose that
4(W) there exist ρ>0,Cρ>0, withL(Cρ)<ω−ω′
Msuch that if U0∈ Hand if˜g∈C([0,τ];H)
satisfy
||U0||2
H+/integraldisplayτ
0|k(s)|·||˜g(s)||2
Hds<ρ2, (3.13)
then the system (2.7)has a unique solution U∈C([0,+∞);H)satisfying ||U(t)||H/lessorequalslantCρ
for allt>0.
Then, for every solution Uof(2.7), with initial datum U0satisfying (3.13),
||U(t)||H/lessorequalslant˜M/parenleftbigg
||U0||H+/integraldisplayτ
0eωs|k(s)|·||˜g(s)||Hds/parenrightbigg
e−(ω−ω′−ML(Cρ))t, t/greaterorequalslant0,(3.14)
with˜M=Meγ.
Proof.By Duhamel’s formula, using (3.10), we have
||U(t)||H/lessorequalslantMe−ωt||U0||H+Me−ωt/integraldisplayt
0eωs|k(s)|·||BU(s−τ)||Hds+ML(Cρ)e−ωt/integraldisplayt
0eωs||U(s)||Hds
/lessorequalslantMe−ωt||U0||H+Me−ωt/integraldisplayτ
0eωs|k(s)|·||BU(s−τ)||Hds
+Me−ωt/integraldisplayt
τeωs|k(s)|·||BU(s−τ)||Hds+ML(Cρ)e−ωt/integraldisplayt
0eωs||U(s)||Hds.
Hence, we obtain
eωt||U(t)||H/lessorequalslantM||U0||H+M/integraldisplayτ
0eωs|k(s)|·||BU(s−τ)||Hds
+/integraldisplayt
0/parenleftBig
Meωτ|k(s+τ)|·||B|| L(H)+ML(Cρ)/parenrightBig
eωs||U(s)||Hds,
and then
eωt||U(t)||H/lessorequalslantM||U0||H+M/integraldisplayτ
0eωs|k(s)|·||˜g(s)||Hds
+/integraldisplayt
0/parenleftBig
Mb2eωτ|k(s+τ)|+ML(Cρ)/parenrightBig
eωs||U(s)||Hds.
Therefore, using Gronwall’s inequality,
eωt/ba∇dblU(t)/ba∇dblH/lessorequalslantM/parenleftbigg
/ba∇dblU0/ba∇dblH+/integraldisplayτ
0eωs|k(s)|·/ba∇dbl˜g(s)/ba∇dblHds/parenrightbigg
eMb2eωτ/integraltextt
0|k(s+τ)|ds+ML(Cρ)t
and so, from (3.12),
||U(t)||H/lessorequalslantMeγ/parenleftbigg
||U0||H+/integraldisplayτ
0eωs|k(s)|·||˜g(s)||Hds/parenrightbigg
e−(ω−ω′−ML(Cρ))t.
This gives (3.14) with ˜Mas in the statement.
5In order to prove the stability result we need then to show tha t the well-posedness assump-
tion (W) of Theorem 3.1 is satisfied for problem (1.1). For thi s, let us define the energy of the
model (1.1) as
E(t) :=E(u(t)) =1
2||ut(t)||2
H+1−˜µ
2||A1
2u(t)||2
H−ψ(u)
+1
2/integraldisplayt
t−τ|k(s+τ)|·||B∗ut(s)||2
Hds+1
2/integraldisplay+∞
0µ(s)||A1
2ηt(s)||2
Hds.(3.15)
The following lemma holds.
Lemma 3.2. Letu: [0,T)→IRbe a solution of (1.1). Assume that E(t)/greaterorequalslant1
4||ut(t)||2
Hfor all
t∈[0,T). Then,
E(t)/lessorequalslant¯C(t)E(0), (3.16)
for allt∈[0,T), where
¯C(t) =e2b2/integraltextt
0(|k(s)|+|k(s+τ)|)ds. (3.17)
Proof.Differentiating E(t), we obtain
dE
dt=/an}b∇acketle{tut,utt/an}b∇acket∇i}htH+(1−˜µ)/an}b∇acketle{tA1
2u,A1
2ut/an}b∇acket∇i}htH−/an}b∇acketle{t∇ψ(u),ut/an}b∇acket∇i}htH+1
2|k(t+τ)|·||B∗ut(t)||2
H
−1
2|k(t)|·||B∗ut(t−τ)||2
H+/integraldisplay+∞
0µ(s)/an}b∇acketle{tA1
2ηt(s),A1
2ηt
t(s)/an}b∇acket∇i}htHds.
Then, from (1.1),
dE
dt=−/integraldisplay+∞
0µ(s)/an}b∇acketle{tut(t),Aηt(s)/an}b∇acket∇i}htHds−k(t)/an}b∇acketle{tut,BB∗ut(t−τ)/an}b∇acket∇i}htH+1
2|k(t+τ)|·||B∗ut(t)||2
−1
2|k(t)|·||B∗ut(t−τ)||2+/integraldisplay+∞
0µ(s)/an}b∇acketle{tAηt(s),ηt
t(s)/an}b∇acket∇i}htds.
Using the second equation of (2.5), we have that
dE
dt=−k(t)/an}b∇acketle{tut,BB∗ut(t−τ)/an}b∇acket∇i}htH+1
2|k(t+τ)|·||B∗ut||2
H−1
2|k(t)|·||B∗ut(t−τ)||2
H
−/integraldisplay+∞
0µ(s)/an}b∇acketle{tAηt(s),ηt
s(s)/an}b∇acket∇i}htHds.
Now, we claim that/integraldisplay+∞
0µ(s)/an}b∇acketle{tηt
s,Aηt(s)/an}b∇acket∇i}htHds/greaterorequalslant0.
Indeed, integrating by parts and recalling assumption (iv) onµ(·),we deduce
/integraldisplay+∞
0µ(s)/an}b∇acketle{tηt
s,Aηt(s)/an}b∇acket∇i}htHds=−1
2/integraldisplay+∞
0µ′(s)||A1
2ηt(s)||2
Hds/greaterorequalslant0.
Therefore, we have that
dE(t)
dt/lessorequalslant−k(t)/an}b∇acketle{tB∗ut,B∗ut(t−τ)/an}b∇acket∇i}htH+1
2|k(t+τ)|·||B∗ut(t)||2
H−1
2|k(t)|·||B∗ut(t−τ)||2
H.
6Now, using Cauchy-Schwarz inequality, we obtain the follow ing estimate:
dE(t)
dt/lessorequalslant1
2(|k(t)|+|k(t+τ)|)||B∗ut(t)||2
H
/lessorequalslant1
2(|k(t)|+|k(t+τ)|)b2||ut||2
H
= 2b2(|k(t)|+|k(t+τ)|)1
4||ut||2
H
/lessorequalslant2b2(|k(t)|+|k(t+τ)|)E(t).
Hence, the Gronwall Lemma concludes the proof.
Before proving the well-posedness assumption (W) for solut ions to (2.7), we need the fol-
lowing two lemmas.
Lemma 3.3. Let us consider the system (2.7)with initial data U0∈ Hand˜g∈C([0,τ];H).
Then, there exists a unique local solution U(·)defined on a time interval [0,δ), withδ/lessorequalslantτ.
Proof.Sincet∈[0,τ], we can rewrite the abstract system (2.7) as an undelayed pr oblem:
U′(t) =AU(t)−k(t)˜g(t)+F(U(t)), t∈(0,τ),
U(0) =U0.
Then, we can apply the classical theory of nonlinear semigro ups (see e.g. [19]) obtaining the
existence of a unique solution on a set [0 ,δ), withδ/lessorequalslantτ.
Lemma 3.4. LetU(t) = (u(t),ut(t),ηt)be a solution to (2.7)defined on the interval [0,δ),with
δ/lessorequalslantτ.Then,
1. ifh(||A1
2u0(0)||H)<1−˜µ
2,thenE(0)>0;
2. ifh(||A1
2u0(0)||H)<1−˜µ
2andh/parenleftbigg
2
(1−˜µ)1
2¯C1
2(τ)E1
2(0)/parenrightbigg
<1−˜µ
2,with¯C(τ)defined in (3.17),
then
E(t)>1
4||ut||2
H+1−˜µ
4||A1
2u||2
H
+1
4/integraldisplayt
t−τ|k(s+τ)|·||B∗ut(s)||2
Hds+1
4/integraldisplay+∞
0µ(s)||A1
2ηt(s)||2
Hds,(3.18)
for allt∈[0,δ). In particular,
E(t)>1
4/ba∇dblU(t)/ba∇dbl2
H,for allt∈[0,δ). (3.19)
Proof.We first deduce by assumption (H3) on ψthat
|ψ(u)|/lessorequalslant/integraldisplay1
0|/an}b∇acketle{t∇ψ(su),u/an}b∇acket∇i}ht|ds
/lessorequalslant||A1
2u||2
H/integraldisplay1
0h(s||A1
2u||H)sds/lessorequalslant1
2h(||A1
2u||H)||A1
2u||2
H.(3.20)
7Hence, under the assumption h(/ba∇dblA1
2u0(0)/ba∇dblH)<1−˜µ
2,we have that
E(0) =1
2||u1||2
H+1−˜µ
2||A1
2u0(0)||2
H−ψ(u0(0))+1
2/integraldisplay0
−τ|k(s+τ)|·||B∗ut(s)||2
Hds
+1
2/integraldisplay+∞
0µ(s)||A1
2η0(s)||2
Hds
/greaterorequalslant1
2||u1||2
H+1−˜µ
4||A1
2u0(0)||2
H+1
2/integraldisplay0
−τ|k(s+τ)|·||B∗ut(s)||2
Hds
+1
2/integraldisplay+∞
0µ(s)||A1
2η0(s)||2
Hds>0,
obtaining 1 .
In order to prove the second statement, we argue by contradic tion. Let us denote
r:= sup{s∈[0,δ) : (3.18) holds ∀t∈[0,s)}.
We suppose by contradiction that r<δ. Then, by continuity, we have
E(r) =1
4||ut(r)||2
H+1−˜µ
4||A1
2u(r)||2
H+1
4/integraldisplayr
r−τ|k(s+τ)|·||B∗ut(s)||2
Hds
+1
4/integraldisplay+∞
0µ(s)||A1
2ηr(s)||2
Hds.(3.21)
Now, from (3.21) and Lemma 3.2 we can infer that
h(||A1
2u(r)||H)/lessorequalslanth/parenleftBigg
2
(1−˜µ)1
2E1
2(r)/parenrightBigg
<h/parenleftBigg
2
(1−˜µ)1
2¯C1
2(τ)E1
2(0)/parenrightBigg
<1−˜µ
2.
Hence, we have that
E(r) =1
2||ut(r)||2
H+1−˜µ
2||A1
2u(r)||2
H−ψ(u(r))+1
2/integraldisplayr
r−τ|k(s+τ)|·||B∗ut(s)||2
Hds
+1
2/integraldisplay+∞
0µ(s)||A1
2ηr(s)||2
Hds
>1
4||ut(r)||2
H+1−˜µ
4||A1
2u(r)||2
H+1
4/integraldisplayr
r−τ|k(s+τ)|·||B∗ut(s)||2
Hds
+1
4/integraldisplay+∞
0µ(s)||A1
2ηr(s)||2
Hds,
which contradicts the maximality of r. Hence,r=δand this concludes the proof.
Theorem 3.5. Problem (2.7), with initial data U0∈ Hand˜g∈C([0,τ];H),satisfies the well-
posedness assumption (W). Then, for solutions of (2.7)corresponding to sufficiently small initial
data the exponential decay estimate (3.14)holds.
Proof.Let us fixN∈IN such that
2M2/parenleftBig
1+e2ωτC∗/parenrightBig
e2γe−(ω−ω′)(N−1)τ<1
1+eωτb2C∗, (3.22)
8whereC∗is the constant defined in (1.2). Then, let ρbe a positive constant such that
ρ/lessorequalslant(1−˜µ)1
2
2¯C1
2(Nτ)h−1/parenleftbigg1−˜µ
2/parenrightbigg
. (3.23)
Now, let us assume that the initial data ( u0(0),u1,η0) andB∗ut(s), s∈[−τ,0],satisfy the
smallness assumption
(1−˜µ)||A1
2u0(0)||2
H+||u1||2
H+/integraldisplay0
−τ|k(s+τ)|||B∗ut(s)||2
Hds
+/integraldisplay+∞
0µ(s)||A1
2η0(s)||2
Hds<ρ2.(3.24)
Note that (3.24) is equivalent to
/ba∇dblU0/ba∇dbl2
H+/integraldisplayτ
0|k(s)|·/ba∇dbl˜g(s)/ba∇dbl2
Hds<ρ2. (3.25)
From Lemma 3.3 we know that there exists a local solution defin ed on a time interval [0 ,δ),
withδ/lessorequalslantτ. From (3.24) and (3.23) we have that
h(||A1
2u0(0)||H)<h/parenleftBigg
ρ
(1−˜µ)1
2/parenrightBigg
/lessorequalslanth/parenleftBigg
1
2¯C1
2(Nτ)h−1/parenleftbigg1−˜µ
2/parenrightbigg/parenrightBigg
<1−˜µ
2,(3.26)
where we have used the fact that ¯C(Nτ)>1.This implies, from Lemma 3.4, E(0)>0. Fur-
thermore, from (3.20) and (3.26) we get
E(0)/lessorequalslant1
2||u1||2
H+3
4(1−˜µ)||A1
2u0(0)||2
H+1
2/integraldisplay0
−τ|k(s+τ)|·||B∗ut(s)||2
Hds
+1
2/integraldisplay+∞
0µ(s)||A1
2η0(s)||2
Hds<ρ2,
which gives, recalling (3.23),
h/parenleftBigg
2
(1−˜µ)1
2¯C1
2(Nτ)E1
2(0)/parenrightBigg
<h/parenleftBigg
2
(1−˜µ)1
2¯C1
2(Nτ)ρ/parenrightBigg
/lessorequalslanth/parenleftbigg
h−1/parenleftbigg1−˜µ
2/parenrightbigg/parenrightbigg
=1−˜µ
2.(3.27)
Since¯C(Nτ)/greaterorequalslant¯C(τ), then
h/parenleftBigg
2
(1−˜µ)1
2¯C1
2(τ)E1
2(0)/parenrightBigg
/lessorequalslanth/parenleftBigg
2
(1−˜µ)1
2¯C1
2(Nτ)E1
2(0)/parenrightBigg
<1−˜µ
2.(3.28)
So, we can apply Lemma 3.4 and we obtain
E(t)>1
4||ut(t)||2
H+1−˜µ
4||A1
2u(t)||2
H
+1
4/integraldisplayt
t−τ|k(s+τ)|·||B∗ut(s)||2
Hds+1
4/integraldisplay+∞
0µ(s)||A1
2ηt(s)||2
Hds,
for allt∈[0,δ).In particular we have that
E(t)>1
4||ut(t)||2
H,fort∈[0,δ).
9Therefore, we can apply Lemma 3.2, obtaining
E(t)/lessorequalslant¯C(τ)E(0)<¯C(τ)ρ2,
for anyt∈[0,δ]. Since
0<1
4||ut(t)||2
H+1−˜µ
4||A1
2u(t)||2
H
+1
4/integraldisplayt
t−τ|k(s+τ)|·||B∗ut(s)||2
Hds+1
4/integraldisplay+∞
0µ(s)||A1
2ηt(s)||2
Hds/lessorequalslantE(t)/lessorequalslant¯C(τ)E(0),(3.29)
for allt∈[0,δ], then we can extend the solution to the entire interval [0 ,τ].
Now, observe that from (3.29) and (3.28) we have
h(||A1
2u(τ)||H)/lessorequalslanth/parenleftBigg
2
(1−˜µ)1
2¯C1
2(τ)E1
2(0)/parenrightBigg
<1−˜µ
2. (3.30)
By continuity, (3.30) implies that there exists δ′>0 such that
h(||A1
2u(t)||H)<1−˜µ
2,∀t∈[τ,τ+δ′).
From this, arguing as before, we deduce
E(t)>1
4||ut(t)||2
H+1−˜µ
4||A1
2u(t)||2
H+1
4/integraldisplayt
t−τ|k(s+τ)|·||B∗ut(s)||2
Hds
+1
4/integraldisplay+∞
0µ(s)||A1
2ηt(s)||2
Hds,
for anyt∈[τ,τ+δ′). In particular, also in such an interval we have E(t)>1
4||ut(t)||2
H. Hence,
we can apply again Lemma 3.2 on the time interval [0 ,τ+δ′) obtaining
0<E(t)/lessorequalslant¯C(2τ)E(0).
As before, we can then extend the solution the whole interval [0,2τ]. At timet= 2τwe have
that
h(/ba∇dblA1
2u(2τ)/ba∇dblH)/lessorequalslanth/parenleftBigg
2
(1−˜µ)1
2E1
2(2τ)/parenrightBigg
/lessorequalslanth/parenleftBigg
2
(1−˜µ)1
2¯C1
2(2τ)E1
2(0)/parenrightBigg
<1−˜µ
2,
where we have used (3.28). Moreover, if 3 /lessorequalslantN,
h/parenleftBigg
2
(1−˜µ)1
2¯C1
2(3τ)E1
2(0)/parenrightBigg
/lessorequalslanth/parenleftBigg
2
(1−˜µ)1
2¯C1
2(Nτ)E1
2(0)/parenrightBigg
<1−˜µ
2.
Thus, one can repeat again the same argument. By iteration, w e then find a unique solution to
the problem (2.7) on the interval [0 ,Nτ], whereNis the natural number fixed at the beginning
of the proof. Moreover, the definition (3.23) of ρ,ensures that
h(/ba∇dblA1
2u(Nτ)/ba∇dblH)/lessorequalslanth/parenleftBigg
2
(1−˜µ)1
2E1
2(Nτ)/parenrightBigg
/lessorequalslanth/parenleftBigg
2
(1−˜µ)1
2¯C1
2(Nτ)E1
2(0)/parenrightBigg
<1−˜µ
2,
10where we have used (3.27). Note that, by construction, (3.18 ) and (3.19) are satisfied in the
whole [0,Nτ).Then, from (3.19),
||U(t)||2
H/lessorequalslant4E(t)/lessorequalslant4¯C(Nτ)E(0)<4¯C(Nτ)ρ2
and so
||U(t)||H/lessorequalslant2¯C1
2(Nτ)ρ,∀t∈[0,Nτ].
Thus, we have proved that, under the assumption (3.25) on the initial data, there exists a unique
solutionU(·) to problem (2.7) defined on the time interval [0 ,Nτ].Moreover,
||U(t)||H/lessorequalslantCρ:= 2¯C1
2(Nτ)ρ.
So far we have fixed ρsatisfying (3.23); now, eventually choosing a smaller ρ,we assume that ρ
satisfies the additional assumption
L(Cρ) =L(¯C1
2(Nτ)ρ)<ω−ω′
2M.
Then, the well–posedness assumption (W) of Theorem 3.1 is sa tisfied on [0 ,Nτ].Therefore, we
obtain that Usatisfies the exponential decay estimate (3.14) and then
||U(t)||H/lessorequalslantM/parenleftbigg
||U(0)||H+/integraldisplayτ
0|k(s)|eωs||˜g(s)||Hds/parenrightbigg
eγe−ω−ω′
2t,∀t∈[0,Nτ].(3.31)
In particular,
||U(Nτ)||H/lessorequalslantM/parenleftbigg
||U(0)||H+/integraldisplayτ
0eωs|k(s)|·||˜g(s)||Hds/parenrightbigg
eγe−ω−ω′
2Nτ.(3.32)
Now, observe that
/integraldisplayτ
0eωs|k(s)|·||˜g(s)||2
Hds/lessorequalslanteωτ/parenleftBigg/integraldisplayτ
0|k(s)|ds/parenrightBigg1/2/parenleftBigg/integraldisplayτ
0|k(s)|·||˜g(s)||2
Hds/parenrightBigg1/2
.
Hence,
||U(t)||H/lessorequalslantMρ/parenleftBig
1+eωτC∗1/2/parenrightBig
eγe−ω−ω′
2t,∀t∈[0,Nτ],
and then
||U(t)||2
H/lessorequalslant2M2ρ2/parenleftbig
1+e2ωτC∗/parenrightbig
e2γe−(ω−ω′)t,∀t∈[0,Nτ], (3.33)
whereC∗is the constant defined in (1.2). From (3.33) we deduce
/ba∇dblU(Nτ)/ba∇dbl2
H+/integraldisplay(N+1)τ
Nτeω(s−Nτ)|k(s)|·/ba∇dblB∗ut(s−τ)/ba∇dbl2
Hds
/lessorequalslant2M2ρ2/parenleftbig
1+e2ωτC∗/parenrightbig
e2γe−(ω−ω′)Nτ+eωτb2/integraldisplay(N+1)τ
Nτ|k(s)|·/ba∇dblU(s−τ)/ba∇dbl2
Hds.(3.34)
Now, observe that, for s∈[Nτ,(N+1)τ],it resultss−τ∈[(N−1)τ,Nτ],then from (3.33)
we deduce
/ba∇dblU(s−τ)/ba∇dbl2
H/lessorequalslant2M2ρ2/parenleftbig
1+e2ωτC∗/parenrightbig
e2γe−(ω−ω′)(N−1)τ,∀s∈[Nτ,(N+1)τ].
11This last estimate, used in (3.34), gives
/ba∇dblU(Nτ)/ba∇dbl2
H+/integraldisplay(N+1)τ
Nτeω(s−Nτ)|k(s)|·/ba∇dblB∗ut(s−τ)/ba∇dbl2
Hds
/lessorequalslant2M2ρ2/parenleftbig
1+e2ωτC∗/parenrightbig
e2γe−(ω−ω′)(N−1)τ/parenleftbig
1+eωτb2C∗/parenrightbig
.(3.35)
From (3.35) and (3.22), we then deduce
||U(Nτ)||2
H+/integraldisplay(N+1)τ
Nτ|k(s)|·||B∗ut(s−τ)||2
Hds<ρ2.
Thus (cf. with (3.25)), one can argue as before on the interva l [Nτ,2Nτ] obtaining a solution
on [0,2Nτ].Iterating this procedure we get a global solution satisfyin g
/ba∇dblU(t)/ba∇dblH<Cρ.
Therefore, we have showed that the problem (2.7) satisfies th e well-posedness assumption (W)
of Theorem 3.1.
We have then proved that, for suitably small data, solutions to problem (2.7) are globally
defined and their energies satisfy an exponential decay esti mate. Therefore, one can state the
following theorem.
Theorem 3.6. Let us consider (2.5). Then, there exists δ>0such that if
(1−˜µ)||A1
2u0||2
H+||u1||2
H+/integraldisplay+∞
0µ(s)||A1
2η0||2
Hds+/integraldisplayτ
0|k(s)|·||Bg(s−τ)||2
Hds<δ, (3.36)
then the solution uis globally defined and it satisfies
E(t)/lessorequalslantCe−βt, (3.37)
whereCis a constant depending only on the initial data and β >0.
Proof.By adirect application ofTheorem3.1 andTheorem3.5wehave that thereexist K,γ >0
such that
||U(t)||2
H/lessorequalslantKe−γt, (3.38)
for allt/greaterorequalslant0, if the initial data are suitably small. Now, observe that t here exists a constant
C >0 such that /integraldisplayt
t−τ|k(s)|·||B∗ut(s)||2
Hds/lessorequalslantCC∗e−γ(t−τ). (3.39)
Then, from (3.38) and (3.39) we obtain (3.37).
124 Examples
4.1 The wave equation with memory and source term
Let Ω be a non-empty bounded set in IRn, with boundary Γ of class C2. Moreover, let O ⊂Ω
be a nonempty open subset of Ω. We consider the following wave equation:
utt(x,t)−∆u(x,t)+/integraldisplay+∞
0µ(s)∆u(x,t−s)ds+k(t)χOut(x,t−τ)
=|u(x,t)|σu(x,t),in Ω×(0,+∞),
u(x,t) = 0,in Γ×(0,+∞),
u(x,t) =u0(x,t) in Ω ×(−∞,0],
ut(x,0) =u1(x),in Ω,
ut(x,t) =g(x,t),in Ω×(−τ,0],(4.40)
whereτ >0 is the time-delay, µ: (0,+∞)→(0,+∞) is a locally absolutely continuous memory
kernel, which satisfies the assumptions (i)-(iv), σ >0 and the damping coefficient k(·) is a
function in L1
loc([0,+∞)) satisfying (1.2). Then, system (4.40) falls in the form (1 .1) withA=
−∆ andD(A) =H2(Ω)∩H1
0(Ω).
Definingηt
sas in (2.4), we can rewrite the system (4.40) in the following way:
utt(x,t)−(1−˜µ)∆u(x,t)−/integraldisplay+∞
0µ(s)∆ηt(x,s)ds+k(t)χOut(x,t−τ)
=|u(x,t)|σu(x,t),in Ω×(0,+∞),
ηt
t(x,s) =−ηt
s(x,s)+ut(x,t),in Ω×(0,+∞)×(0,+∞),
u(x,t) = 0,in Γ×(0,+∞),
ηt(x,s) = 0,in Γ×(0,+∞),fort/greaterorequalslant0,
u(x,0) =u0(x) :=u0(x,0),in Ω,
ut(x,0) =u1(x) :=∂u0
∂t(x,t)/vextendsingle/vextendsingle/vextendsingle
t=0,in Ω,
η0(x,s) =η0(x,s) :=u0(x,0)−u0(x,−s),in Ω×(0,+∞),
ut(x,t) =g(x,t),in Ω×(−τ,0).(4.41)
As before, we introduce the Hilbert space L2
µ((0,+∞);H1
0(Ω)) endowed with the inner product
/an}b∇acketle{tφ,ψ/an}b∇acket∇i}htL2µ((0,+∞);H1
0(Ω)):=/integraldisplay
Ω/parenleftbigg/integraldisplay+∞
0µ(s)∇φ(x,s)∇ψ(x,s)dx/parenrightbigg
ds,
and consider the Hilbert space
H=H1
0(Ω)×L2(Ω)×L2
µ((0,+∞);H1
0(Ω)),
equipped with the inner product
/angbracketleftBigg
u
v
w
,
˜u
˜v
˜w
/angbracketrightBigg
H:= (1−˜µ)/integraldisplay
Ω∇u∇˜udx+/integraldisplay
Ωv˜vdx+/integraldisplay
Ω/integraldisplay+∞
0µ(s)∇w∇˜wdsdx.
SettingU= (u,ut,ηt), we can rewrite (4.43) in the form (2.7), where
A
u
v
w
=
v
(1−˜µ)∆u+/integraltext+∞
0µ(s)∆w(s)ds
−ws+v
,
13with domain
D(A) ={(u,v,w)∈H1
0(Ω)×H1
0(Ω)×L2
µ((0,+∞);H1
0(Ω)) :
(1−˜µ)u+/integraldisplay+∞
0µ(s)w(s)ds∈H2(Ω)∩H1
0(Ω), ws∈L2
µ((0,+∞);H1
0(Ω))},
B(u,v,ηt)T:= (0,χOv,0)TandF(U(t)) = (0,|u(t)|σu(t),0)T. For anyu∈H1
0(Ω) consider the
functional
ψ(u) :=1
σ+2/integraldisplay
Ω|u(x)|σ+2dx.
By Sobolev’s embedding theorem, we know that if 0 < σ <4
n−2, thenψis well-defined, and
Gˆ ateaux differentiable at any point u∈H1
0(Ω), with Gˆ ateaux derivative given by
Dψ(u)(v) =/integraldisplay
Ω|u(x)|σu(x)v(x)dx,
for anyv∈H1
0(Ω). Moreover, as in [2], if 0 <σ/lessorequalslant2
n−2, thenψsatisfies the assumptions (H1),
(H2), (H3). Define the energy as follows:
E(t) :=1
2/integraldisplay
Ω|ut(x,t)|2dx+1−˜µ
2/integraldisplay
Ω|∇u(x,t)|2dx−ψ(u(x,t))
+1
2/integraldisplayt
t−τ/integraldisplay
O|k(s+τ)|·|ut(x,s)|2dxds+1
2/integraldisplay+∞
0µ(s)/integraldisplay
Ω|∇ηt(x,s)|2dxds.
Theorem 3.1 applies to this model giving well-posedness and exponential decay of the energy for
suitably small initial data, provided that the condition (3 .12) on the time delay holds for every
t/greaterorequalslant0.
4.2 The plate system with memory and source term
Let Ω be a non-empty bounded set in IRn, with boundary Γ of class C2. Let us denote ν(x)
the outward unit normal vector at any point x∈Γ.Moreover, let O ⊂Ω be a nonempty open
subset of Ω. We consider the following viscoelastic plate sy stem:
utt(x,t)+∆2u(x,t)−/integraldisplay+∞
0µ(s)∆2u(x,t−s)ds+k(t)χOut(x,t−τ)
=|u(x,t)|σu(x,t),in Ω×(0,+∞),
u(x,t) =∂u
∂ν(x,t) = 0,in Γ×(0,+∞),
u(x,t) =u0(x,t) in Ω ×(−∞,0],
ut(x,0) =u1(x),in Ω,
ut(x,t) =g(x,t),in Ω×(−τ,0],(4.42)
whereτ >0 is the time-delay, µ: (0,+∞)→(0,+∞) is a locally absolutely continuous memory
kernel, which satisfies the assumptions (i)-(iv), σ >0 and the damping coefficient k(·) is a
function in L1
loc([0,+∞)) satisfying (1.2). This system again falls in (1.1) for A= ∆2with
domainD(A) =H4(Ω)∩H1
0(Ω).Definingηt
sas in (2.4), we can rewrite the system (4.42) in the
14following way:
utt(x,t)+(1−˜µ)∆2u(x,t)+/integraldisplay+∞
0µ(s)∆2ηt(x,s)ds+k(t)χOut(x,t−τ)
=|u(x,t)|σu(x,t),in Ω×(0,+∞),
ηt
t(x,s) =−ηt
s(x,s)+ut(x,t),in Ω×(0,+∞)×(0,+∞),
u(x,t) =∂u
∂ν(x,t) = 0,in Γ×(0,+∞),
ηt(x,s) = 0,in Γ×(0,+∞),fort/greaterorequalslant0,
u(x,0) =u0(x) :=u0(x,0),in Ω,
ut(x,0) =u1(x) :=∂u0
∂t(x,t)/vextendsingle/vextendsingle/vextendsingle
t=0,in Ω,
η0(x,s) =η0(x,s) :=u0(x,0)−u0(x,−s),in Ω×(0,+∞),
ut(x,t) =g(x,t),in Ω×(−τ,0).(4.43)
Then, arguing analogously to the previous example, one can r ecast (4.43) in the form (2.7).
Moreover, for ( n−4)σ/lessorequalslant4 (cf. e.g. [14]) the nonlinear source satisfies the required assumptions.
Theorem 3.1 applies then to this model giving well-posednes s and exponential decay of the
energy for suitably small initial data, provided that the co ndition (3.12) on the time delay holds
for everyt/greaterorequalslant0.
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16 |
1701.09110v1.Lack_of_correlation_between_the_spin_mixing_conductance_and_the_ISHE_generated_voltages_in_CoFeB_Pt_Ta_bilayers.pdf | arXiv:1701.09110v1 [cond-mat.mes-hall] 31 Jan 2017Lack of correlation between the spin mixing conductance and the ISHE-generated
voltages in CoFeB/Pt,Ta bilayers
A. Conca,1,∗B. Heinz,1M. R. Schweizer,1S. Keller,1E. Th. Papaioannou,1and B. Hillebrands1
1Fachbereich Physik and Landesforschungszentrum OPTIMAS,
Technische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany
(Dated: June 21, 2021)
We investigate spin pumping phenomena in polycrystalline C oFeB/Pt and CoFeB/Ta bilayers
and the correlation between the effective spin mixing conduc tanceg↑↓
effand the obtained voltages
generated by the spin-to-charge current conversion via the inverse spin Hall effect in the Pt and Ta
layers. For this purpose we measure the in-plane angular dep endence of the generated voltages on
the external static magnetic field and we apply a model to sepa rate the spin pumping signal from the
one generated by the spin rectification effect in the magnetic layer. Our results reveal a dominating
role of anomalous Hall effect for the spin rectification effect with CoFeB and a lack of correlation
between g↑↓
effand inverse spin Hall voltages pointing to a strong role of th e magnetic proximity
effect in Pt in understanding the observed increased damping . This is additionally reflected on the
presence of a linear dependency of the Gilbert damping param eter on the Pt thickness.
INTRODUCTION
In spin pumping experiments,[1, 2] the magnetization
of a ferromagnetic layer (FM) in contact with a non-
magnetic one (NM) is excited by a microwave field. A
spin current is generated and injected into the NM layer
and its magnitude is maximized when the ferromagnetic
resonance (FMR) condition is fulfilled. The spin cur-
rent can be detected by using the inverse spin Hall effect
(ISHE) for conversion into a charge current in appropri-
ate materials. The injected spin current Jsin the NM
layer has the form[1]
Js=/planckover2pi1
4πg↑↓ˆm×dˆm
dt(1)
where ˆmis the magnetization unit vector and g↑↓is the
realpartofthe spinmixing conductancewhich iscontrol-
ling the intensity of the generated spin current. Its value
is sensitive to the interface properties. The generation of
the spin current opens an additional loss channel for the
magnetic system and consequently causes an increase in
the measured Gilbert damping parameter α:
∆αsp=γ/planckover2pi1
4πMsdFMg↑↓(2)
This expression is only valid for thick enough NM lay-
ers where no reflection of the spin current takes place
at the interfaces. In principle, it allows the estimation
ofg↑↓by measuring the increase in damping compared
to the intrinsic value. However, other phenomena, like
the magnetic proximity effect (MPE) in the case of Pt
or interface effects depending on the exact material com-
bination or capping layer material, can have the same
influence, [7, 8] which challenges the measurement of the
contribution from the spin pumping. In this sense, it
is preferable to use an effective value g↑↓
eff. Still, if thespin pumping is the main contribution to the increase
inα, a correlation between g↑↓
effand the measured ISHE
voltages is expected. A suitable approach in order to un-
derstand the weight of MPE on the value of g↑↓
effis the
use of FM/NM with varying NM metals, with presence
and absence of the MPE effect. The measurement of ∆ α
andg↑↓
efftogether with the ISHE voltages generated by
the spin current in the NM layer can bring clarity to the
issue.
However, the generation of an additional dc voltage
by the spin rectification effect,[3–6] which adds to the
voltagegeneratedbythe ISHE spin-to-chargeconversion,
deters the analysis of the obtained data. The spin recti-
fication originates from the precession of the magnetiza-
tion in conducting layers with magnetoresistive proper-
ties, mainly Anisotropic Magnetoresistance (AMR) and
Anomalous Hall Effect (AHE). Information about the
physics behind the measured voltage can only be ob-
tained after separation of the different contributions. For
this purpose, we made use of the different angulardepen-
denciesofthecontributionsunderin-planerotationofthe
external magnetic field.
EXPERIMENTAL DETAILS
Here, we report on results on polycrystalline
Co40Fe40B20/Pt,Ta bilayers grown by rf-sputtering on Si
substrates passivated with SiO 2. CoFeB is a material
choice for the FM layer due to its low damping proper-
ties and easy deposition.[9, 10] A microstrip-based VNA-
FMR setup was used to study the damping properties. A
more detailed description of the FMR measurement and
analysis procedure is shown in previous work.[7, 10] A
quadrupole-based lock-in setup described elsewhere[11]
was used in order to measure the ISHE generated volt-
age. The dependence of the voltage generated during the
spin pumping experiment on the in-plane static external2
field orientation is recorded for a later separation of the
pure ISHE signal from the spin rectification effect.
GILBERT DAMPING PARAMETER AND SPIN
MIXING CONDUCTANCE
Figure1showsthedependenceoftheeffectivedamping
parameter αeff(sum ofall contributions)onthe thickness
dof the NM metal for a CoFeB layer with a fixed thick-
ness of 11 nm. The case d= 0 nm represents the case of
reference layers with Al capping. From previous studies
it is known that the use of an Al capping layer induces
a large increase of damping in Fe epitaxial layers.[7] For
polycrystallineNiFe andCoFeBlayersthis is notthe case
and it allows the measurement of the intrinsic value α0.
[8]
The observed behavior differs strongly for Pt and
Ta. In the Pt case a large increase in damping is ob-
served with a sharp change around d= 1 nm and a
fast saturation for larger thicknesses. This is quali-
tatively very similar to our previous report on Fe/Pt
bilayers.[7] From the measured ∆ αwe extract the value
g↑↓
eff= 6.1±0.5·1019m−2. This value is larger than the
onereportedpreviouslyinourgroup[8]forthinnerCoFeB
layers with larger intrinsic damping 4 .0±1.0·1019m−2
andalsolargerthanthevaluereportedbyKim et al.[12],
5.1·1019m−2. The impact of the Ta layer on damping
is very reduced and, consequently, a low value for g↑↓
effof
0.9±0.3·1019m−2is obtained. This value is now smaller
than the one reported by Kim et al.1.5·1019m−2) in-
dicating that the difference between CoFeB/Pt and Ta
is larger in our case. A reference has also to be made
to the work of Liu et al.on CoFeB films thinner than
in this work. [13] There, no value for the spin mixing
conductance is provided, but the authors claim a vanish-
FIG. 1. (Color online) Dependence of the effective Gilbert
damping parameter αeffon the thickness of the NM metal.
A large increase in damping is observed for the Pt case while
a very small but not vanishing increase is observed for Ta.
From the change ∆ αthe effective spin mixing conductance
g↑↓
effis estimated using Eq. 2.ing impact on αfor the Ta case. On the contrary the
increase due to Pt is almost three times larger than ours,
pointing to a huge difference between both systems. In
any case, the trend is similar, only the relative difference
between Ta and Pt changes.
A closer look to the data allows to distinguish a region
in the Pt damping evolution prior to the sharp increase
where a linear behavior is recognized ( d <1 nm). A lin-
ear thickness dependence of αin spin-sink ferromagnetic
films and in polarized Pt has been reported. [14, 15] The
increasein damping due to spin currentabsorptionin the
Pt with ferromagnetic order can then be described by:
∆α= ∆αMPE·dPt/dPt
c (3)
where ∆αMPEis the total increase in damping due only
to the magnetic proximity effect in Pt, dPtis the thick-
ness of the Pt layer and dPt
cis a cutoff thickness which
is in the order of magnitude of the coherence length in
ferromagnetic layers.[15, 16]
The inset in Fig. 1 shows a fit of Eq. 3 from where
dPt
c= 0.8nm isobtained assumingavalue ∆ αMPE= 1.2.
The value is in qualitative agreement with the reported
thickness where MPE is present in Pt, ( dPt
MPE≤1 nm
[17, 18]) and is lower than the one reported for Py/Pt
systems.[14]
/s45/s52/s48/s48/s52/s48/s56/s48/s49/s50/s48
/s40/s98/s41
/s80/s116/s32/s68/s97/s116/s97
/s32/s70/s105/s116
/s32/s83/s121/s109/s109/s101/s116/s114/s105/s99
/s32/s65/s110/s116/s105/s115/s121/s109/s109/s101/s116/s114/s105/s99/s86/s111/s108/s116/s97/s103/s101/s32/s40/s181/s86/s41
/s84/s97
/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s45/s50/s48/s48/s50/s48
/s32/s32/s86/s111/s108/s116/s97/s103/s101/s32/s40/s181/s86/s41
/s181
/s48/s72/s32/s40/s109/s84/s41/s40/s97/s41
FIG. 2. (Color online) Voltage spectra measured for (a)
CoFeB/Ta and (b) CoFeB/Pt at 13 GHz. The solid line is a
fit to Eq. 4. The symmetric voltage Vsymand antisymmetric
voltageVantisymcontributions are separated and plotted inde-
pendently (dashed lines). The voltage signal is dominated b y
Vantisymin the Pt case and by Vsymin the Ta case.3
The increase of damping due to spin pumping is de-
scribed by an exponential dependence and explains the
sharp increase at dPt= 1 nm. However, the fast increase
does not allow for a deep analysis and it is pointing to a
spin diffusion length in Pt not larger than 1 nm.
In any case, this point has to been treated with care.
The contribution of MPE to damping can be easily un-
derestimated and consequently also the value for dPt
c. In
any case, the value can be interpreted as a lower limit
for ∆αMPE. If this is substracted, under the assumption
that the rest of increase is due to spin pumping, the spin
mixing conductance due only to the this effect would be
g↑↓
eff= 4.9±0.5·1019m−2.
ELECTRICAL DETECTION OF SPIN PUMPING
Figure 2(a),(b) shows two voltage measurements
recorded at 13 GHz for a NM thickness of 3 nm and
a nominal microwave power of 33 dBm. The measured
voltage is the sum of the contribution of the ISHE effect
and of spin rectification effect originating from the dif-
ferent magnetoresistive phenomena in the ferromagnetic
layer. While the spin rectification effect generates both a
symmetric and an antisymmetric contribution, [3–5] the
pure ISHE signal is only symmetric. For this reason a
separation of both is carried out by fitting the voltage
spectra (solid line) to
Vmeas=Vsym(∆H)2
(H−HFMR)2+(∆H)2+
+Vantisym−2∆H(H−HFMR)
(H−HFMR)2+(∆H)2(4)
where ∆ HandHFMRare the linewidth and the reso-
nance field, respectively. The dotted lines in Fig. 2 show
the two contributions. When comparing the data for Pt
and Ta some differences are observed. First of all, the
absolute voltage values are smaller for the Pt cases and,
moreimportant, therelativeweightofbothcontributions
is different. While the first point is related to the differ-
ent conductivity of Ta and Pt, the second one is related
to the intrinsic effect causing the voltage. We calculate
the ratio S/A = Vsym/Vantisymfor all the measurements
and the results are shown in Fig. 3(a) as a function of the
NM thickness. While the antisymmetric contribution is
dominating in the Pt samples with a S/A ratio smaller
than 1 for the samples with Pt, the opposite is true for
the Ta case. Since the ISHE signal is contributing only
toVsymit might be concluded that spin pumping is tak-
ing place stronger in the Ta system. However, since also
the spin rectificationeffect has a symmetric contribution,
this conclusion cannot be supported. Furthermore, since
the spin Hall angle θSHEhas opposite sign in these two
materials, also the ISHE signal should have it. In appar-ent contradiction to this, we observe that both symmet-
ric contributions have the same sign in (a) and (b). This
points to the fact that for Pt, Vsymis dominated by the
spin rectification effect, which does not change sign and
overcompensates a smaller ISHE signal. All these con-
siderations have the consequence that it is not possible
to extract complete information of the origin of the mea-
sured voltage by analyzing single spectra. For the same
reason, the large increase in S/A for Ta for d= 5 nm or
the change in sign for Pt with the same thickness cannot
be correctly explained until the pure ISHE signal is not
separated from the spin rectification effect. As already
pointed out in recent papers[3–5, 11, 19], an analysis of
the angular dependence (in-plane or out-of-plane) of the
measured voltages can be used to separate the different
contributions.
In any case, before proceeding it has to be proven that
allthe measurementswereperformed in the linearregime
with small cone angles for the magnetization precession.
The measurements performed out of this regime would
have a large impact on the linewidth and a Gilbert-like
dampingwouldnotbeguaranteed. Figure3(b) showsthe
dependence of the voltage amplitude on the microwave
nominal power proving indeed that the measurements
were carried on in the linear regime.
FIG. 3. (Color online) (a) Dependence of the ratio S/A
=Vsym/Vantisymon the thickness of the NM layer. (b)
Dependence of the total voltage on the applied microwave
power proving the measurements were carried out in the lin-
ear regime.4
SEPARATION OF THE ISHE SIGNAL FROM
THE SPIN RECTIFICATION VOLTAGE
We performed in-plane angular dependent measure-
ments of the voltage and Eq. 4 was used to extract
Vsym,antisymfor each value of the azimuthal angle φ
spanned between the direction of the magnetic field and
the microstrip antenna used to excite the magnetization.
We used a model based on the work of Harder et al.[3] to
fit the dependence. This model considers two sources for
the spin rectification, which are the Anisotropic Mag-
netoresistance (AMR) and the Anomalous Hall Effect
(AHE):
Vsym=Vspcos3(φ)+
+VAHEcos(Φ)cos( φ)+Vsym
AMR−⊥cos(2φ)cos(φ)
+Vsym
AMR−/bardblsin(2φ)cos(φ)
Vantisym=VAHEsin(Φ)cos( φ) +Vantisym
AMR−⊥cos(2φ)cos(φ)
+Vantisym
AMR−/bardblsin(2φ)cos(φ)
(5)
Here,VspandVAHEare the contributions from spin
pumping (pure ISHE) and from AHE, respectively. Φ
is the phase between the rf electric and magnetic fields
in the medium. The contribution from the AMR is di-
vided in one generating a transverse ⊥(with respect to
the antenna) or longitudinal /bardblvoltage. In an ideal case
with perfect geometry and point-like electrical contacts
Vsym,antisym
AMR−/bardblshould be close to zero.
Figure 4 shows the angular dependence of Vsym(top)
andVantisym(bottom) for the samples with NM thick-
ness of 3 nm. The lines are a fit to the model which
is able to describe the dependence properly. From the
data it can be clearly concluded that while the values
ofVantisymare comparable, with the difference resulting
from the different resistivity of Pt and Ta, the values
ofVsymare much larger for Ta. The values obtained
from the fits for the different contributions are plotted
in Fig. 5 as a function of the thickness of the NM layer.
The value of Φ is ruling the lineshape of the electrically
measured FMR peak[20] which is always a combination
of a dispersive ( D, antisymmetric) and a Lorentzian ( L,
symmetric) contribution in the form D+iL. In order
to compare the relative magnitudes of the different con-
tributions independently of Φ we compute the quantities
VAMR−/bardbl,⊥=/radicalbigg/parenleftBig
Vantisym
AMR−/bardbl,⊥/parenrightBig2
+/parenleftBig
Vsym
AMR−/bardbl,⊥/parenrightBig2
which it
is equivalent to√
D2+L2and we show them together
withVAHEandVsp. This step is important to allow for
comparison of the different contributions independent of
the value of Φ.
Several conclusions can be extracted from Fig. 5. First
of all, the spin rectification effect in CoFeB systems is al-
most fully dominated by the AHE. AMR plays a veryFIG. 4. (Color online) Angular dependence of Vsym(top)
andVantisym(bottom) for CoFeB/Pt,Ta samples with NM
thickness of 3 nm. The lines are a fit to the model described
in Eq. 5.
minor role. This is a difference with respect to NiFe
or Fe. [4, 11, 20] This is correlated with the very large
AHE reported in CoFeB films. [21, 22] Second, the volt-
ages generated by the spin pumping via the ISHE are
larger in the case of Ta and of opposite sign as expected
from the different sign of θSHEin both materials. This
solves the apparent contradiction observed by the posi-
tive symmetric contributions in both materials as shown
in Fig. 2(a) and (b) and confirms the interpretation than
inthecaseofPtthesymmetriccontributionisdominated
by the spin rectification effect with opposite sign to the
ISHE signal. Again, this shows that the interpretation
using single spectra may lead to confusion and that angle
dependent measurements are required.
The evolution of the spin rectification voltages with
NM thickness shows a saturation behavior in both cases
for small thicknesses and a decrease with the NM layer
thickness compatible with a dominant role of the re-
sistance of the CoFeB layer. This is expected from
the resistivity values for amorphous CoFeB layers, 300-
600µm·cm,[23] which are much larger than for β-Ta
(6-10µm·cm) or sputtered Pt (100-200 µm·cm).[24, 25]
However, the dependence does not completely agree with
the expected behavior[19] 1 /dNMpointing out to addi-
tional effects like a variation of the conductivity of Pt for
the thinner layers.
Concerning the correlation of the absolute values of
the ISHE-generated voltages and the spin Hall angles in
both materials, unfortunately the scatter in θSHEvalues
in the literature is very large.[26] Howeverthis is reduced
if we consider works were θSHEwas measured simultane-
ously for Pt and Ta in similar samples. In YIG/Pt,Ta5
systems[27, 28] it was determined that |θPt
SHE|>|θTa
SHE|
with a relative difference of around 30% which it is at
odds with our results. On the contrary, in CoFeB/Pt,Ta
bilayers|θTa
SHE|= 0.15>|θPt
SHE|= 0.07 is reported.[13]
However the difference is not large enough to cover com-
pletely the difference in our samples. In order to ex-
plain this point together with the absolute low value in
CoFeB/Pt we have to take into account the possibility of
a certain loss of spin current at the interface FM/Pt or at
the very first nanometer, the latter due to the presence
of a static magnetic polarization due to the proximity ef-
fect. With this the spin current effectively being injected
in Pt would be lower than in the Ta case.
The data does not allow for a quantitative estimation
of the spin diffusion length λsd, but in any case the evo-
lution is only compatible with a value for Pt not thicker
than 1 nm, similarto reportedvalues forsputtered Pt[25]
and a a value of a few nm for Ta, also compatible with
literature.[28]
An important point is the lack of correlation of g↑↓
eff
and the expected generated spin current using Eq. 1 with
the absolute measured ISHE voltage that results from
the spin-to-charge current conversion, obtained after the
separationfromthe overimposedspin rectificationsignal.
This is true even if we substract the MPE contribution
assumed for Eq. 3. The same non-mutually excluding
explanations are possible here: ∆ αin Pt in mainly due
to the MPE, or the spin current pumped into Pt van-
ishes at or close to the interface. The first alternative
would render Eq. 2 unuseful since most of the increase
in damping is not due to spin pumping as long as the
MPE is present. The second would reduce the validity
of Eq. 1 to estimate the current injected in Pt and con-
verted into a charge current by the ISHE. In any case,
CoFeB/Ta shows very interesting properties, with strong
spin pumping accompanied by only a minor impact on
α.
Let us discuss the limitations of the model defined in
Eq. 5 and the suitability to describe the measurements.
First of all, the model assumes a perfect isotropic mate-
rial. The anisotropy in CoFeB is known to be small but
not zero and a weak uniaxial anisotropy is present. The
effect onthe angulardependenceisnegligible. Themodel
assumes also a perfect geometry and point-like electrical
contacts to measure the voltages. Our contacts are ex-
tended (∼200µm) and a small misalignment is possible
(angle between the antenna and the imaginary line con-
necting the electrical contacts may not be exactly 90◦).
This is the most probable reason for the non-vanishing
small value for Vsym,antisym
AMR−/bardbl. Nevertheless, the angular
dependence of the measured voltage is well described by
the model and no large deviations are observed.
FIG. 5. (Color online) NM thickness dependence of the dif-
ferent contributions to the measured voltages extracted fr om
the angular dependence of VsymandVantisymfor Ta (top) and
Pt (bottom).
CONCLUSIONS
In summary, we made use of in-plane angular de-
pendent measurements to separate ISHE-generated from
spin rectification voltages and we compare the absolute
values and thickness dependence for Pt and Ta. Differ-
ently to other materials, the spin rectification signal in
CoFeB is almost fully dominated by AHE. No correlation
between the observed spin mixing conductance via FMR
measurement and the spin pumping signal is obtained
pointing to a dominant role of the magnetic proximity
effect in the increase in damping with Pt.
ACKNOWLEDGEMENTS
Financial support by M-era.Net through the
HEUMEM project and by the Carl Zeiss Stiftung
is gratefully acknowledged.
∗conca@physik.uni-kl.de
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2107.00982v3.Anomalous_Gilbert_Damping_and_Duffing_Features_of_the_SFS___boldmath___varphi_0___Josephson_Junction.pdf | arXiv:2107.00982v3 [cond-mat.supr-con] 25 Aug 2021Anomalous Gilbert Damping and Duffing Features of the SFS ϕ0Josephson Junction
Yu. M. Shukrinov1,2, I. R. Rahmonov1,3, A. Janalizadeh4, and M. R. Kolahchi4
1BLTP, JINR, Dubna, Moscow Region, 141980, Russia
2Dubna State University, Dubna, 141980, Russia
3Umarov Physical Technical Institute, TAS, Dushanbe 734063 , Tajikistan
4Department of Physics, Institute for Advanced Studies in Ba sic Sciences, P.O. Box 45137-66731, Zanjan, Iran
(Dated: August 26, 2021)
We demonstrate unusual features of phase dynamics, IV-char acteristics and magnetization dy-
namics of the ϕ0Josephson junction at small values of spin-orbit interacti on, ratio of Josephson to
magnetic energy and Gilbert damping. In particular, an anom alous shift of the ferromagnetic reso-
nance frequency with an increase of Gilbert damping is found . The ferromagnetic resonance curves
show the Duffing oscillator behaviour, reflecting the nonline ar nature of Landau-Lifshitz-Gilbert
(LLG) equation. Based on the numerical analysis of each term in LLG equation we obtained an
approximated equation demonstrated both damping effect and Duffing oscillator features. The re-
sulting Duffing equation incorporates the Gilbert damping in a special way across the dissipative
term and the restoring force. A resonance method for the dete rmination of spin-orbit interaction in
noncentrosymmetric materials which play the role of barrie r inϕ0junctions is proposed.
Introduction. The Josephson junctions (JJ) with the
current-phaserelation I=Icsin(ϕ−ϕ0), wherethephase
shiftϕ0is proportional to the magnetic moment of ferro-
magneticlayerdetermined bythe parameterofspin-orbit
interaction, demonstratea number ofunique featuresim-
portant for superconducting spintronics, and modern in-
formation technology [1–6]. The phase shift allows one
to manipulate the internal magnetic moment using the
Josephson current, and the reverse phenomenon which
leads to the appearance of the DC component in the su-
perconducting current [7–9].
Interactive fields can bring nonlinear phenomena of
both classical, and quantum nature. A basic example
is the magnons strongly interacting with microwave pho-
tons [10]. As a result we could name Bose-Einstein con-
densation of such quasiparticles, i.e. magnons [11, 12],
and synchronization of spin torque nano-oscillators as
they coherently emit microwave signals in response to
d.c. current [13]. It is interesting that (semi)classical an-
harmonic effects in the magnetodynamics described by
the Landau-Lifshitz-Gilbert (LLG) model in thin films or
heterostructures [14, 15], and the quantum anharmonic-
ity in the cavity mangnonics [16] can well be modeled
by so simple a nonlinear oscillator as Duffing. The cor-
responding Duffing equation contains a cubic term and
describesthe oscillationsofthe variousnonlinearsystems
[17].
Despite the fact that nonlinear features of LLG are
studied often during a long time and in different systems,
manifestation of the Duffing oscillator behavior in the
frameworkofthisequationisstill notcompletelystudied.
Closer to our present investigation, in the study of the
dynamics of antiferromagnetic bimeron under an alter-
natingcurrent,Duffingequationformsagoodmodel, and
this has applications in weak signal detection [14, 18, 19].
As another application with Duffing oscillator at work,
we can mention the ultra thin Co 20Fe60B20layer, andits largeangle magnetizationprecessionunder microwave
voltage. There are also ‘foldover’ features, characteris-
tic of the Duffing spring, in the magnetization dynamics
of the Co/Ni multilayer excited by a microwave current
[15, 20, 21]. But nonlinear features of ϕ0Josephson junc-
tions have not been carefully studied yet. In this Letter,
we show that the Duffing oscillator helps in the under-
standing of the nonlinear features of ϕ0Josephson junc-
tions at small values of system parameters.
Coupling of superconducting current and magnetiza-
tion and its manifestation in the IV-characteristics and
magnetizationdynamicsopensthedoorfortheresonance
method determination of spin-orbit intensity in noncen-
trosymmetric materials playing the role of barrier in ϕ0
junctions. As it is well known, the spin-orbit interaction
plays an important role in modern physics, so any novel
method for its determination in real materials would be
very important. There are a series of recent experiments
demonstrating the modification of Gilbert damping by
the superconducting correlations (see Ref.[22] and cita-
tionstherein). Inparticular, the pronouncedpeaksin the
temperature dependence of Gilbert damping have been
observed for the ferromagnetic insulator/superconductor
multilayers [23] which might be explained by the pres-
ence of spin relaxation mechanisms like the spin-orbit
scattering [22]. Here, we use the noncentrosymmetric
ferromagnetic material as a weak link in ϕ0junctions.
The suitable candidates may be MnSi or FeGe, where
the lack of inversion center comes from the crystalline
structure [8].
The Gilbert damping determines the magnetization
dynamics in ferromagnetic materials but its origin is not
well understood yet. Effect of nonlinearity on damp-
ing in the system is very important for application of
these materials in fast switching spintronics devices. Our
study clarifies such effects. In Ref.[24] the authors dis-
cuss the experimental study of temperature-dependent2
Gilbert damping in permalloy (Py) thin films of varying
thicknesses by ferromagnetic resonance, and provide an
important insight into the physical origin of the Gilbert
damping in ultrathin magnetic films.
In this Letter we demonstrate an anomalous depen-
dence of the ferromagnetic resonance frequency with an
increase of the Gilbert damping. We find that the reso-
nance curves demonstrate features of Duffing oscillator,
reflecting the nonlinear nature of LLG equation. The
damped precession of the magnetic moment is dynami-
cally driven by the Josephson supercurrent, and the res-
onance behavior is given by the dynamics of the Duffing
spring. The resonance methods for the determination of
spin-orbit interaction in the ϕ0junction are proposed.
Model and Methods. In the considered SFS ϕ0junc-
tion (see Fig.1) the superconducting phase difference ϕ
and magnetization Mof the F layer are two coupled dy-
namical variables. Based on the LLG equation for the
Figure 1: Schematic view of SFS ϕ0Josephson junction. The
external current applied along xdirection, ferromagnetic easy
axis is along zdirection.
magnetic moment Mwith effective magnetic field Heff,
resistively capacitively shunted junction (RCSJ) model,
and Josephson relation for the phase difference ϕ, we de-
scribe dynamics of the SFS ϕ0junction by the system of
equations in normalized variables
dm
dt=ωFheff×m+α/parenleftbigg
m×dm
dt/parenrightbigg
,
heff=Grsin(ϕ−rmy)/hatwidey+mz/hatwidez, (1)
dV
dt=1
βc[I−V+rdmy
dt−sin(ϕ−rmy)],
dϕ
dt=V,
wheremis vector of magnetization with components
mx,y,z, normalized to the M0=/bardblM/bardbland and satisfy-
ing the constraint/summationtext
i=x,y,zm2
i(t) = 1,ωF= ΩF/ωc,
ΩF=γK/νis ferromagnetic resonance frequency, γis
the gyromagnetic ratio, Kis an anisotropic constant, ν
is the volume of the ferromagnetic F layer, αis the phe-
nomenologicaldamping constant(Gilbert damping), heff
is the vector of effective magnetic field, normalized to
theK/M0(heff=HeffM0/K),G=EJ/(Kν) relation
of Josephson energy to magnetic one, ris a parameter
of spin-orbit coupling, ϕis phase difference of JJ, Vis
voltage normalized to the Vc=IcR,Iccritical current
of JJ,Rresistance of JJ, βc= 2eIcCR2//planckover2pi1is McCumberparameter, Cis capacitance of JJ, Iis bias current nor-
malized to the Ic. In this system of equation time tis
normalized to the ω−1
c, whereωc= 2eIcR//planckover2pi1is character-
istic frequency. In the chosen normalization, the average
voltage corresponds to the Josephson frequency ωJ.
Ferromagnetic resonance in ϕ0junction. The ferro-
magnetic resonance features are demonstrated by aver-
age voltage dependence of the maximal amplitude of the
mycomponent ( mmax
y), taken at each value of bias cur-
rent. To stress novelty and importance of our finding,
we first present the analytical results for average volt-
age dependence of mmax
yalong IV-characteristics in the
ferromagnetic resonance region. As it was discussed in
Refs.[8, 25, 26], in case Gr≪1,mz≈1, and neglecting
quadratic terms mxandmy, we get
/braceleftBigg
˙mx=ξ[−my+GrsinωJt−αmx]
˙my=ξ[mx−αmy],(2)
whereξ=ωF/(1 +α2). This system of equations can
be written as the second order differential equation with
respect to the my
¨my=−2αξ˙my−ξ2(1+α2)my+ξ2GrsinωJt.(3)
Corresponding solution for myhas the form
my(t) =ω+−ω−
rsinωJt−α++α−
rcosωJt,(4)
where
ω±=Gr2ωF
2ωJ±ωF
((ωJ±ωF)2+(αωJ)2),(5)
and
α±=Gr2ωF
2αωJ
((ωJ±ωF)2+(αωJ)2).(6)
So,mydemonstrates resonance with dissipation when
Josephson frequency is approaching the ferromagnetic
one (ωJ→ωF). The maximal amplitude mmax
yas a
function of voltage (i.e., Josephson frequency ωJ) at dif-
ferentα, calculated using (4), is presented in Fig.2 (a).
We see the usual characteristicvariation of the resonance
curve with an increase in dissipation parameter when the
maximal amplitude and position of resonance pick cor-
responds to the damped resonance. We note that the
analytical result (4) were obtained in the case Gr≪1.
Presented in Fig.2(b) results of numerical simulations
mmax
y(V) dependence at different values of dissipation
parameter αdemonstrate the essential differences with
the results followedfrom the analytical consideration(4).
We note also that the strong coupling of the supercon-
ducting phase difference ϕand magnetization Mof the
F layermanifests itself by appearanceof subharmonics of
the resonance at ω= 1/2,1/3,1/4 demonstrated in the
inset to Fig.2(b).3
Figure 2: (a) Analytical results for maximal amplitude mmax
y
in the ferromagnetic resonance region for different α; (b)
Numerical results for maximal amplitude of magnetization
my−component at each values of bias current and voltage
along IV-characteristics of the ϕ0junction in the ferromag-
netic resonance region for various α. Inset shows the man-
ifestation of the resonance subharmonics. Parameters are:
βc= 25, G=0.05, r=0.05, ωF= 0.5.
We stress two important features followed from the
presented results. First, the ferromagnetic resonance
curves show the foldover effect, i.e., the features of Duff-
ing oscillator. Different from a linear oscillator, the non-
linear Duffing demonstrates a bistability under external
periodic force [27]. Second, the ferromagnetic resonance
curves demonstrate an unusual dependence of the reso-
nance frequency as a function of Gilbert damping α. As
shown in Fig. 3(a), an increase in damping leads to a
nonuniform change in the resonant frequency, i.e., with
an increase in damping the resonance maximum shifts
toωFat small α, but then moves to the opposite side,
demonstrating the usual damped resonance. So, with
an increase in α, unusual dependence of the resonance
voltage transforms to the usual one. For the parameters
chosen, the critical value of this transformation is around
α= 0.02−0.03. We call this unusual behaviour of the
resonance maximum of mmax
yas an “α-effect”. Both the
α−effect and Duffing features in our system appear due
to the nonlinear features of the system dynamics at small
Figure 3: (a) α-dependence of the resonance curve mmax
y(V)
peak presented in Fig.2 in the damping parameter interval
[0.006 – 0.2]. Dashed line indicates ferromagnetic resonan ce
position; (b) Comparison of the resonance curves mmax
y(V)
calculated by full LLG equation (1) and the approximate
equation (8).
G,r,α≪1. To prove it, we have carried out the nu-
merical analysis of each term of LLG full equation (first
two equations in (1)) for the set of model parameters
G= 0.05,r= 0.05α= 0.005. After neglecting the
terms of order 10−6, we have
˙mx
ξ=−mymz+Grmzsin(ϕ−rmy)−αmxm2
z,
˙my
ξ=mxmz−αmym2
z, (7)
˙mz
ξ=−Grmxsin(ϕ−rmy)+αmz(m2
x+m2
y),
Inthisapproximationweobserveboththe“ α–effect”and
Duffing oscillator features. Neglecting here the last term
αmz(m2
x+m2
y) in third equationfor ˙ mz, which is orderof
10−4, leadstothe losingoftheDuffing oscillatorfeatures,
but still keeps alpha-effect. We note that equation (7)
keeps the time invariance of the magnetic moment, so
that term plays an important role for manifestation of
Duffing oscillator features by LLG equation.
The generalized Duffing equation for ϕ0junction.
The LLG is a nonlinear equation and in case of simple
effective field it can be transformed to the Duffing equa-
tion [14, 17]. Such transformation was used in Ref.[17]
to demonstrate the nonlinear dynamics of the magnetic
vortex state in a circular nanodisk under a perpendicular
alternating magnetic field that excites the radial modes
of the magnetic resonance. They showed Duffing-type
nonlinear resonance and built a theoretical model corre-
sponding tothe Duffing oscillatorfromthe LLG equation
to explore the physics of the magnetic vortex core polar-
ity switching for magnetic storage devices.
The approximated LLG system of equations (7)
demonstrates both α-effect and features of Duffing os-
cillator. As demonstrated in the Supplemental Materials
[28], the generalizedDuffing equation forthe ϕ0junction,
¨my+2ξα˙my+ξ2(1+α2)my
−ξ2(1+α2)m3
y=ξ2GrsinωJt.(8)4
can be obtained directly from the LLG system of equa-
tions.
As we see, for small enough Gandr, it is only the
dimensionless damping parameter αin LLG that plays a
role in the dynamics of the system. We can think of a
harmonic spring with a constant that is hardened or soft-
ened by the nonlinear term. For a usual Duffing spring,
with independent coefficients of the various terms, the
resonancepeak relative to the harmonic (linear) resonant
frequency folds over to the smaller (softening) or larger
(hardening) frequencies. In the frequency response, the
interplay of the specific dependence of each coefficient on
αplays an important role and as Fig.3(a) shows, there is
a particular αthat brings the resonant frequency closest
to ferromagnetic resonance.
Simulations of the mydynamics in the framework of
Duffing equation can explain observed foldover effect in
the frequency dependence of mmax
y. Comparison the re-
sults followed from analytical approximate equation (8)
and results of full equation (1) for maximal amplitude of
mmax
yin the ferromagnetic resonance region is presented
in Fig.3(b). So, the magnetization dynamics in the SFS
ϕ0-junction due to the voltage oscillations can effectively
be described by a scalar Duffing oscillator, synchronizing
the precession of the magnetic moment with the Joseph-
son oscillations.
Effect of spin-orbit interactions. As we mentioned
above, the spin-orbit interaction plays an important role
in different fields of modern physics. Here we have sug-
gested a novel method for its determination in real non-
centrosymmetric ferromagnetic materials like MnSi or
FeGe, where the lack of inversion center comes from
the crystalline structure Ref.[8] and which play role a
weak link in ϕ0junctions. Based on the obtained re-
sults, presented in Fig.4, we propose different versions of
the resonance method for the determination of spin-orbit
interaction in these materials. Particularly, in Fig.4(a)
we present the simulation results of maximal amplitude
mmax
ybased on (1) at G= 0.05,α= 0.01 at different
values of spin-orbit parameter rin the ferromagnetic res-
onance region. This case corresponds to the nonlinear
approximation leading to the Duffing equation (8). The
same characteristics calculated by equation (1) for larger
valueα= 0.1, i.e. corresponding to the linear approxi-
mation (3) are presented in Fig.4(b). As it was expected,
in caseα= 0.01 the foldover effect is more distinct.
In Fig.4(c) the r-dependence of the resonancepeak po-
sition, obtained from the simulation results of full equa-
tion atα= 0.01 andα= 0.1 for the same set of model
and simulation parameters is demonstrated. We stress
here that nonlinear features of LLG equation leading to
the Duffing’s shift of the mmax
ypeak of main harmonic
with r presented in Fig.4(c) show the manifestation of
nonlinearity.
Despite the noted differences between results for α=
0.01 andα= 0.1 , we see in both cases a monotonic
Figure 4: (a) Voltage dependence of mmax
yin the ferromag-
netic resonance region at different values of spin-orbit int er-
action based on (1) at G= 0.05,α= 0.01. Inset enlarges
the main harmonic; (b) The same as in (a) for α= 0.1; (c)
Shift ofmmax
ypeak as a function of spin-orbit interaction at
two values of Gilbert damping; (d) r-dependence of the main
harmonic and subharmonics peaks in case (a); (e) The same
as in (d) for the case (b).
linear increase of mmax
ypeak of main harmonic and sub-
harmonics with rdemonstrated in Fig.4(d) and Fig.4(e).
Such lineardependence canbe noted fromEq. (6) ofRef.
[14], but the authors did not discuss it. This dependence
might serve as a calibrated curve for spin-orbit interac-
tion intensity, thus creating the resonance methods for r
determination.
Conclusions. Based on the reported features of the
ϕ0Josephson junction at small values of spin-orbit in-
teraction, ratio of Josephson to magnetic energy and
Gilbert damping, we have demonstrated that the cou-
pled superconducting current and the magnetic moments
in theϕ0-junction result in the current phase relation in-5
tertwining with the ferromagnetic LLG dynamics. The
ferromagnetic resonance clearly shows this interplay. In
particular, an anomalous shift of the ferromagnetic res-
onance frequency with an increase of Gilbert damping
is found. The ferromagnetic resonance curves demon-
strate features of Duffing oscillator, reflecting the nonlin-
ear nature of LLG equation. The obtained approximated
equation demonstrates both damping effect and Duffing
oscillator features. We have shown that due to the non-
linearity, asmodeledbythe generalizedDuffing equation,
the parameters of the system can compensate each other
resulting in unusual response. The position of the maxi-
mum can shift towards and then away from the expected
resonant frequency, as the damping is decreased. There
are also foldover effects that was explained by the pro-
posed model. A resonance method for the determination
of spin-orbit interaction in noncentrosymmetric materi-
als which play the role of barrier in ϕ0junctions was
proposed.
The experimental testing of our results would in-
volve SFS structures with ferromagnetic material having
enough small value of Gilbert damping. Potential candi-
date for experimental realization could be ferromagnetic
metals or insulators which have small values of damping
parameter ( α∼10−3−10−4). In Ref.[29] the authors
report on a binary alloy of cobalt and iron that exhibits
a damping parameterapproaching10−4, which is compa-
rable to values reported only for ferrimagnetic insulators
[30, 31]. Using superconductor-ferromagnetic insulator-
superconductor on a 3D topological insulator might be
a way to have strong spin-orbit coupling needed for ϕ0
JJ and small Gilbert dissipation for α-effect [5]. We note
in this connection that the yttrium iron garnet YIG is
especially interesting because of its small Gilbert damp-
ing (α∼10−5). The interaction between the Joseph-
son current and magnetization is determined by the ra-
tio of the Josephson to the magnetic anisotropy energy
G=EJ/(Kν) and spin-orbit interaction r. The value of
the Rashba-type parameter rin a permalloy doped with
Pt[32] and in the ferromagnets without inversion sym-
metry, like MnSi or FeGe, is usually estimated to be in
the range 0 .1−1. The value of the product Grin the ma-
terialwith weakmagneticanisotropy K∼4×10−5KA−3
[33], and a junction with a relatively high critical current
density of (3 ×105−5×106)A/cm2[34] is in the range
1−100. It givesthe set offerromagneticlayerparameters
and junction geometry that make it possible to reach the
values used in our numerical calculations for the possible
experimental observation of the predicted effect.
Numerical simulations were funded by the project 18-
71-10095oftheRussianScientificFund. A.J.andM.R.K.
are grateful to IASBS for financial support.[1] Jacob Linder and W. A. Jason Robinson, Nature Physics
11, 307 (2015).
[2] Yu. M. Shukrinov, Accepted for UFN.
DOI:https://doi.org/10.3367/UFNe.2020.11.038894
[3] A.A. Mazanik, I.R. Rahmonov, A.E. Botha, and Yu.M.
Shukrinov, Phys. Rev. Applied 14, 014003 (2020).
[4] M. Nashaat and Yu. M. Shukrinov, Physics of Particles
and Nuclei Letters, 17, 79. (2020).
[5] I. V. Bobkova , A. M. Bobkov, I. R. Rahmonov, A. A.
Mazanik , K. Sengupta, and Yu. M. Shukrinov, Phys.
Rev. B102, 134505 (2020).
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and M. G. Blamire, Scientific Report 2, 699 (2012).arXiv:2107.00982v3 [cond-mat.supr-con] 25 Aug 2021Supplemental Material to “Anomalous Gilbert Damping and Du ffing Features of the
SFSϕ0Josephson Junction”
Yu. M. Shukrinov1,2, I. R. Rahmonov1,3, A. Janalizadeh4, and M. R. Kolahchi4
1BLTP, JINR, Dubna, Moscow Region, 141980, Russia
2Dubna State University, Dubna, 141980, Russia
3Umarov Physical Technical Institute, TAS, Dushanbe 734063 , Tajikistan
4Department of Physics, Institute for Advanced Studies in Ba sic Sciences, P.O. Box 45137-66731, Zanjan, Iran
(Dated: August 26, 2021)
Here, we demonstrate by numerical methods that a generalize d Duffing equation can be obtained
directly from LLG system of equations, for small system para meters of S/F/S junction.
Both the α−effect and Duffing features obtained by
LLG system of equations appear due to the nonlinear
features of its dynamics at small G,r,α≪1. To proveit,
we have carried out the numerical analysis of each term
of LLG full equation (first two equations in the equation
(1) of the main text) for the set of model parameters
G= 0.05,r= 0.05α= 0.005. After neglecting the
terms of order 10−6, we have
˙mx
ξ=−mymz+Grmzsin(ϕ−rmy)−αmxm2
z,
˙my
ξ=mxmz−αmym2
z, (1)
˙mz
ξ=−Grmxsin(ϕ−rmy)+αmz(m2
x+m2
y),
The procedure is as follows. Expanding mn
zin a series
with the degree of ( mz−1) we can find
mn
z=nmz−(n−1). (2)
From expression m2
x+m2
y+m2
z= 1 and (2), we obtain
mz=2−m2
y
2. (3)
Using approximation sin( ϕ−rmy) = sin(ωJt) in (1),
differentiatingsecondequationofthe system(1) andsub-
stituting ˙ mx,mxand ˙mzfrom first second and third
equations of the system (1), respectively and using the
expression (2), (3) and assuming mz= 1 only in denom-
inators, we come to a second order differential equation
with respect to my
¨my=a1˙m3
y+a2my˙m2
y+a3m4
y˙my+a4m2
y˙my+a5˙my
+a6m5
y+a7m3
y+a8my−c1˙m2
ysinωJt (4)
+c2m4
ysinωJt+c3m2
ysinωJt+AsinωJt.The numerical calculation for the used set of model
parameters allows us to estimate each of the terms in the
equation, as presented in Table I.
Now, if we neglect those terms smaller than 10−4, the
equation (4) takes on the form of Duffing equation with
Table I: Numerical analysis of equation (4) terms.
a1α
ξa1˙m3
y∼1.76×10−5
a2 α2a2my˙m2
y∼3.4×10−8
a3ξα3a3m4
y˙my∼7.7×10−12
a4ξ(3α−α3)a4m2
y˙my∼2×10−5
a52ξα a5˙my∼6×10−4
a6ξ2(α2+2α4)a6m5
y∼5.56×10−9
a7ξ2(1+α2−α4)a7m3
y∼3.7×10−3
a8ξ2(1+α2)a8my∼6.1×10−2
c1 Gr c1˙m2
ysinϕ∼3.6×10−5
c22ξ2α2Grc2m4
ysinϕ∼5.3×10−11
c3ξ2Gr(α2−2)c3m2
ysinϕ∼4.5×10−5
Aξ2Gr AsinωJt∼6.25×10−4
damping dependent coefficients, i.e., we have a general-
ization of the Duffing equation
¨my+2ξα˙my+ξ2(1+α2)my
−ξ2(1+α2)m3
y=ξ2GrsinωJt.(5) |
2009.10299v1.Magnon_mediated_spin_currents_in_Tm3Fe5O12_Pt_with_perpendicular_magnetic_anisotropy.pdf | The following article has been accepted by Applied Physics Letters from AIP.
Magnon -mediated spin currents in Tm 3Fe5O12/Pt with perpendicular magnetic anisotropy
G. L. S. Vilela1,2, J. E. Abrao3, E. Santos3, Y. Yao4,5, J. B. S. Mendes 6, R. L. Rodríguez -Suárez7, S. M. Rezende3, W. Han4,5, A. Azevedo3, and J. S. Moodera1,8
1 Plasma Science and Fusion Center, and Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology , Cambridge, Massachusetts 02139, USA
2 Física de Materiais , Escola Politécnica de Pernambuco , Universidade de Pernambuco , Recife, Pernambuco 50720 -001, Bra sil
3 Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670 -901, Brasil
4 International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China.
5 Collaborative Innovation Center of Quantum Matter, Beijing 100871, China.
6 Depart amento de Física, Universidade Federal de Viçosa , Viçosa, Minas Gerais 36570 -900, Brasil
7Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, Chile
8Department of Physics, Massachusetts Institute of Technology , Cambridge, Massachusetts 02139, USA
Electronic mail: gilvania.vilela@upe.br
Abstract
The control of pure spin currents carried by magnons in magnetic insulator (MI) garnet films with a robust perpendicular
magnetic anisotropy (PMA) is of great interest to spintronic technology as they can be used to carry, transport and process
information . Garnet films with PMA p resent labyrinth domain magnetic structures that enrich the magnetization dynamics, and
could be employed in more efficient wave -based logic and memory computing devices. In MI/NM bilayers, where NM being
a normal metal providing a strong spin -orbit coupli ng, the PMA benefits the spin -orbit torque (SOT) driven magnetization's
switching by lowering the needed current and rendering the process faster, crucial for developing magnetic random -access
memories (SOT -MRAM). In this work, w e investigated the magnetic anisotropies in thulium iron garnet (TIG) films with PMA
via ferromagnetic resonance measurements , followed by the excitation and detection of magnon -mediated pure spin currents
in TIG/Pt driven by microwave s and heat currents. TIG films presented a Gilbert damping constant 𝛼 ≈0.01, with resonance
fields above 3.5 kOe and half linewidth s broader than 60 Oe , at 300 K and 9.5 GHz . The spin-to-charge current conversion
through TIG/Pt was observed as a micro -voltage generated at the edges of the Pt film. The obtained spin Seebeck coefficient
was 0.54 𝜇𝑉/K, confirm ing also the high interfacial spin transparenc y.
Spin-dependent phenomena in systems composed by
layers of magnetic insulators (MI) and non-magnetic heavy
metals ( NM) with strong spin -orbit coupling have been
extensively explored in the insulator -based spintronics [ 1-6].
Among th e MI materials, YIG (Y3Fe5O12) is widely employed
in devices for generation and transmission of pure spin
currents. The main reason is its very low magnetic damping
with Gilbert parameter on the order of 10-5, and its large spin
decay length which permits spin waves to travel distances of
orders of centimeters inside it before they vanish [7-9]. When
combined with heavy metals such as Pt, Pd, Ta, or W, many
intriguing spin -current related phenomena emerge , such as the
spin pumping effect (SPE) [10-14], spin Seebeck effect (SSE)
[7, 15 -18], spin Hall effect (SHE) [19-21], and spin-orbit
torque (SOT) [22-25]. The origin of these effects relies mainly
on the spin diffusion length , and the quantum -mechanical
exchange and spin -orbit interaction s at the interface and inside
the heavy metal [26]. All of these effects turn out the MI/NM
bilayer into a fascinating playground for exploring spin-orbit
driven phenomena at interfaces [27-30].
Well investigated for many years, intrinsic YIG(111)
films on GGG(111) , (GGG = Gd 3Ga5O12) exhibit in -plane
anisotropy. To obtain YIG single -crystal films with
perpendicular magnetic anisotropy (PMA) it is necessary to
grow them on top of a different substra te or partially substitute
the yttrium ions by rare-earth ions , to cause strain -induced
anisotropy [31-33]. Even so, it is well -known that magnetic
films with PMA play an important role in spintronic
technology. The PMA enhances the spin-switching efficiency , which reduces the current density for observing the spin -orbit
torque ( SOT) effect , and it is useful for developing SOT based
magnetoresistive random access memory ( SOT-MRAM ) [34-
36]. Besides that, PMA increase s the information density in
hard disk drives and magnetoresistive random access
memories [37-39], and it is crucial for breaking the time -
reversal symmetry in topological insulators (TIs) aiming
towards quantized anomalous Hall state in MI/TI [ 40-42].
Recent ly, thin films of another rare-earth iron garnet,
TIG (Tm 3Fe5O12), have caught the attention of researchers due
to its large negative magnetostriction constant , which favors
an out -of-plane easy axis [ 4, 43, 44 ]. TIG is a ferrimagnetic
insulator with a critical temperature of 549 K, a crystal
structure similar to YIG , and a Gilbert damping parameter on
the order of 𝛼~ 10−2 [4, 45 ]. Investigations of spin transport
effects have been reported in TIG/Pt [45, 46 ] and TIG/TI [42,
47], where the TIG was fabricated by pulsed laser deposition
(PLD) technique. The results showed a strong spin mixing
conductance at the interface of these materials that made it
possible to observe spin Hall magnetoresistance, spin
Seebeck , and spin -orbit tor que effects.
In this paper, we first present a study of the
magnetocrystalline and uni axial anisotropies , as well as the
magnetic damping of sputtered epitaxial TIG thin films using
the ferromagnetic resonance (FMR) technique . For obtaining
the cubic and uniaxial anisotropy fields, w e analyze d the
2
dependence of the FMR spectr a on the film thickness and the
orientation of the dc applied magnetic field at room
temperature and 9.5 GHz . Then, we swept the microwave
frequency for getting the ir magnetic damping at different
temperatures . Subsequently , we focused this investigation on
the excitation of magnon -mediated pure spin currents in
TIG/Pt via the spin pumping and spin Seebeck mechanisms
for different orientations of the dc applied magnetic field at
room temperature . Pure spin currents transport spin angular
momentum without carrying charge currents . They are free of
Joule heating and could lead to spin -wave based devices that
are energetically more efficient . Employing the inverse spin
Hall effect (ISHE) [12], we observed th e spin-to-charge
conversion of these currents inside the Pt film which was
detected as a developed micro -voltage .
TIG films with thickness ranging from 15 to 60 nm
were deposited by rf sputtering from a commercial target with
the same nominal composition o f Tm 3Fe5O12, and a purity of
99.9 %. The deposition process was performed at room
temperature, in pure argon working pressure of 2.8 mTorr , at
a deposition rate of 1.4 nm/min. To improve the crystallinity
and the magnetic ordering, the films were post -growth
annealed for 8 hrs at 800 ℃ in a quartz tube in flowing
oxygen. After the thermal treatment, the films yield ed a
magnetization saturation of 100 emu/cm3, and an RMS
roughness below 0.1 nm confirmed using a superconducting
quantum interference device ( SQUID) and high -resolution X -
ray diffraction measurements, as detailed in our recent article
[44]. Moreover, the out-of-plane hysteresis loops showed
curved shapes which might be related with labyrinth domain structures very common in garnet films with PMA [48]. The
next step of sample preparation consist ed of an ex -situ
deposition of a 4 nm -thick Pt film over the post -annealed TIG
films using the dc sputtering technique . Platinum films were
grown under a n Ar gas pressure of 3. 0 mTorr , at room
temperature , and a deposition rate of 10 nm/min . The Pt films
were not patterned.
Ferromagnetic resonance (F MR) is a well -
established technique for study of basic magnetic properties
such as saturation magnetization, anisotropy energies and
magnetic relaxation mechanisms . Furthermore , FMR has been
central to the investigat ion of microwave -driven spin-
pumping phenomena in FM/NM bilayers [11, 12, 49 ]. First,
we used a homemade FMR spectrometer running at a fixed
frequency of 9.5 GHz, at room temperature , where t he sample s
were placed in the middle of the back wall of a rectangular
microwave cavity operating in the TE102 mode with a Q factor
of 2500. Field scan spectra of the derivative of the absorption
power (𝑑𝑃𝑑𝐻⁄) were acquired by modulating the dc applied
field 𝐻⃗⃗ 0 with a small sinusoidal field ℎ⃗ at 100 kHz and using
lock-in amplifier detection . The resonance field 𝐻𝑅 was
obtained as a function of the polar and azimuthal angles
(θH,ϕ𝐻) of the applied magnetic field 𝐻⃗⃗ , as illustrated in Fig.
1(d), where 𝐻⃗⃗ =𝐻⃗⃗ 0+ℎ⃗ and ℎ≪ 𝐻0.
The FMR spectra for TIG(t) films are shown in Figs.
1 (a, b and c) for thickness es t = 15, 30 and 60 nm respectively.
The spectra were measured for 𝐻 applied along three different
polar angles: θ𝐻=0° (blue), θ𝐻≅45° (green ) and θ𝐻=
90° (red) . The complete dependenc e of 𝐻𝑅, for each sample,
Fig. 1. FMR absorption derivative spectra vs. field scan H for (a) TIG( 15 nm ), (b) TIG( 30 nm ), and (c) TIG( 60 nm ), at room T and 9.5 GHz .
The half linewidths ( ∆𝐻) for TIG( 15 nm ) with 𝐻 applied along 𝜃𝐻=0°,50°, and 90° are 112 Oe, 74 Oe, and 72 Oe, r espectively. For TIG(30
nm), ∆𝐻 is 82 Oe, 72 Oe, and 65 Oe for 𝜃𝐻=0°,50°, and 90°, respectively. For TIG(60 nm), ∆𝐻 is 72 Oe, 75 Oe, and 61 Oe for 𝜃𝐻=
0°,45°, and 90°, respectively. These values were extracted from the fits using the Lorentz function. (d) Illustration of the FMR experiment
where the magnetization ( 𝑀) under an applied magnetic field (H) is driven by a microwave . (e), (f) and ( g) show the dependence of the
resonance field 𝐻𝑅 with 𝜃𝐻 for different thickness of TIG. The red solid lines are theoretical fits obtained for the FMR condition.
Magnetization curves are given in reference [44].
3
as a function of the polar angle ( 0°≤𝜃𝐻≤90°) are shown in
Figs. 1(e, f and g ). For all samples, 𝐻𝑅 was minimum for 𝜃𝐻=
0°, confirming that the perpendicular anisotropy field was
strong enough to overcome the demagnetization field. While
the films with t = 15 nm and 30 nm exhibited the maximum
value of 𝐻𝑅 for 𝜃𝐻=90° (in-plane), the sample with t = 60
nm showed a maximum 𝐻𝑅at 𝜃𝐻~60°. To explain the
behavior of 𝐻𝑅 as a function of the out -of-plane angle 𝜃𝐻, it is
necessary to normalize the FMR data to compare with the
theory described as follows .
The most relevant contributions to the free magnetic
energy density 𝜖 for GGG(111) / TIG(111) films , are:
𝝐=𝝐𝒁+𝝐𝑪𝑨+𝝐𝑫+𝝐𝑼, (1)
where 𝝐𝒁 is the Zeeman energy density , 𝝐𝑪𝑨 is the cubic
anisotropy energy density for (111) oriented thin films , 𝝐𝑫 is
the demagnetization energy density , and 𝝐𝑼 is the uniaxial
energy density . Taking into consideration the reference frame
shown in Fig. 1(d), each energy density terms can be written
as [50]:
𝝐𝒁=−𝑴𝑺𝑯(𝒔𝒊𝒏𝜽𝒔𝒊𝒏𝜽𝑯𝒄𝒐𝒔(𝝓−𝝓𝑯)+𝒄𝒐𝒔𝜽𝒄𝒐𝒔𝜽𝑯),
(2)
𝝐𝑪𝑨=𝑲𝟏𝟏𝟐⁄(𝟑−𝟔𝒄𝒐𝒔𝟐𝜽+𝟕𝒄𝒐𝒔𝟒𝜽+
𝟒 √𝟐𝒄𝒐𝒔𝜽𝒔𝒊𝒏𝟑𝝓𝒔𝒊𝒏𝟑𝜽),
(3)
𝝐𝑫+𝝐𝑼=𝟐𝝅(𝑴⃗⃗⃗ ∙𝒆̂𝟑)𝟐−𝑲𝟐⊥(𝑴⃗⃗⃗ ∙𝒆̂𝟑𝑴𝑺 ⁄)𝟐
−𝑲𝟒⊥(𝑴⃗⃗⃗ ∙𝒆̂𝟑𝑴𝑺 ⁄)𝟒, (4)
where 𝜽 and 𝝓 are the polar and azimuthal angles of the
magnetization vector 𝑴⃗⃗⃗ , 𝑴𝑺 is the saturation magnetization ,
𝑲𝟏 is the first order cubic anisotropy constant, and 𝑲𝟐⊥ and 𝑲𝟒⊥
are the first and second order uniaxial anisotropy constants.
The uniaxial anisotropy terms come from two sources: growth
induced and stress induced anisotropy. The relation between
the resonance field and the excitation frequency 𝝎 can be
obtained from [51, 52 ]: (𝝎𝜸⁄)𝟐=𝟏
𝑴𝟐𝒔𝒊𝒏𝟐𝜽[𝝐𝜽𝜽𝝐𝝓𝝓−(𝝐𝜽𝝓)𝟐], (5)
where 𝜸 is the gyromagnetic ratio. The subscripts indicate
partial derivatives with respect to the coordinates, 𝝐𝜽𝜽=
𝝏𝟐𝝐𝝏𝜽𝟐⁄|𝜽𝟎,𝝓𝟎, 𝝐𝝓𝝓=𝝏𝟐𝝐𝝏𝝓𝟐⁄|𝜽𝟎,𝝓𝟎 and 𝝐𝜽𝝓=
𝝏𝟐𝝐𝝏𝜽𝝏𝝓⁄|𝜽𝟎,𝝓𝟎, where 𝜽𝟎,𝝓𝟎 are the equilibrium angles of
the magnetization determined by the energy density minimum
conditions, 𝝏𝝐𝝏𝜽⁄|𝜽𝟎,𝝓𝟎=𝟎 and 𝝏𝝐𝝏𝝓⁄|𝜽𝟎,𝝓𝟎=𝟎. The
best fits to the data obtained with the Eq. (5) are shown in Figs.
1 (e, f and g) by the solid red lines. The main physical
parameters extracted from the fits , including the effective
magnetization 4 𝝅𝑴𝒆𝒇𝒇, are summarized in Table 1 . Here
4𝝅𝑴𝒆𝒇𝒇= 4𝝅𝑴−𝟐𝑲𝟐⊥/𝑴𝑺 , where the second term is the
out-of-plane uniaxial anisotropy field 𝑯𝑼𝟐=𝟐𝑲𝟐⊥/𝑴, also
named as 𝑯⊥. It is important to notice that the large negative
values of 𝑯𝑼𝟐 were sufficiently strong to saturate the
magnetization along the direction perpendicular to the TIG
film’s plane , thus overcoming the shape anisotropy . We used
the saturation magnetization as the nominal value of 𝑴𝑺=
𝟏𝟒𝟎.𝟎 𝑮. As the thickness of the TIG film increase d, the
magnitude of the perpendicular magnetic anisotropy field ,
𝑯𝑼𝟐, decrease d due to the relaxation of the induced growth
stresses as expected .
Table 1. Physical parameters extracted from the theoretical fits of
the FMR response of the TIG thin films with thickness 𝑡, performed
at room 𝑇 and 9.5 GHz . 4𝜋𝑀𝑒𝑓𝑓 is the effective magnetization, H 1C
is the cubic anisotropy f ield, H U2 and HU4 are the first and second
order uniaxial anisotropy fields, respectively. H U2 is the out -of-
plane uniaxial anisotropy field, also named as 𝐻⊥.
TIG film’s thickness t 15 nm 30 nm 60 nm
4𝜋𝑀𝑒𝑓𝑓(G) -979 -799 - 383
𝐻1𝐶=2𝐾1𝑀𝑆⁄ (Oe) 31 26 -111
𝐻𝑈2=4𝜋𝑀𝑒𝑓𝑓−4𝜋𝑀𝑆 (Oe) -2,739 -2,559 -2,143
𝐻𝑈4=2𝐾4⊥𝑀𝑆⁄(Oe) 311 168 432
Fig. 2. (a) Ferromagnetic resonance spectra vs. in -plane applied field 𝐻 for a 30 nm -thick TIG film at frequencies ranging from 2 GHz to 14
GHz and temperature of 300 K, after normalization by background subtraction. (b) Half linewidth ∆𝐻 versus frequency for T IG(30 nm) at
300 K. The Gilbert damping parameter 𝛼 was extracted from the linear fitting of the data. (c) Damping 𝛼 versus temperature 𝑇 for TIG films
with 30 nm and 60 nm of thickness.
4
To obtain the Gilbert damping parameter (𝜶) of the
TIG thin films, we used the coplanar waveguide technique in
the variable temperature insert of a physical property
measurement system (PPMS) . A vector network analyzer
measure d the amplitude of the forward complex transmission
coefficients (𝑺𝟐𝟏) as a function of the in -plane magnetic field for different microwave frequencies (𝒇) and temperatures (𝑻).
Figure 2(a) shows the FMR spectra (𝑺𝟐𝟏 versus 𝑯) for
TIG(30 nm) corresponding to frequencies ranging from 2 GHz
to 14 GHz at 300K , with a microwave power of 0 dBm , after
normalization by background subtraction . Fitting each FMR
spectra using the Lorentz function, we were able to extract the
half linewidth ∆𝑯 for each frequency , as shown in Fig. 2(b).
Then, 𝜶 was estimated based on the linear approximation
∆𝑯=∆𝑯𝟎+(𝟒𝝅𝜶𝜸⁄)𝒇, where ∆𝑯𝟎 reflects the
contribution of magnetic inhomogeneities , the linear
frequency part is caused by the intrinsic Gilbert damping
mechanism , and 𝜸 is the gyromagnetic ratio [40]. The same
analysis was performed for lower temperature data, and it was
extended to TIG(60 nm). Due to the weak magnetization of
the thinnest TIG (15 nm ) the coplanar waveguide setup was
not able to detect its FMR signals. Figure 2(c) shows the
Gilbert damping dependence with 𝑻. At 300 K , 𝜶=𝟎.𝟎𝟏𝟓
for TIG(60 nm) which is in agree ment with the values reported
in the literature [4, 45 ], and it increases by 130 % as 𝑻 goes
down to 150 K [54].
Next, this work focused on the generation of pure
spin currents carried by spin waves in TIG at room 𝑻, followed
by their propagati on through the interface between TIG and
Pt, and their spin -to-charge conversion inside the Pt film.
Initially , we explored the FMR -driven spin -pumping effect in
TIG(60 nm)/Pt(4 nm) , where the coherent magnetization
precession of the TIG inject ed a pure spin current 𝑱𝒔 into the
Pt layer, which convert ed as a transverse charge current 𝑱𝒄 by
means of the inverse spin Hall effect , expressed as 𝑱 𝒄 =
𝜽𝑺𝑯(𝛔̂ × 𝑱 𝒔), where 𝜽𝑺𝑯 is the spin Hall angle and 𝜎̂ is the
spin polarization [55]. As the FMR was excited using the
homemade spectrometer at 9.5 GHz , a spin pumping voltage
(𝐕𝐒𝐏) was detected between the two silver painted electrodes
Fig. 4. Spin Seebeck voltage (V SSE) excited by a thermal gradient in the longitudinal configuration ( 𝛻𝑇∥𝐽 𝑆) at room 𝑇, as shown in (a). (b)
Field scan of V SSE for ∆𝑇=20𝐾 and different field polar angles 𝜃𝐻. (c) Field scan of V SSE for ∆𝑇=12,𝜃𝐻=90° and different azimuthal
angles 𝜙𝐻. Spin voltage amplitude ∆𝑉𝑆𝑆𝐸 versus (d) 𝜃𝐻, (e) 𝜙𝐻, and (f) ∆𝑇. The solid red lines are theoretical fits of the sine (d), cosine (e)
and linear (f) dependence of ∆𝑉𝑆𝑆𝐸 with 𝜃𝐻, 𝜙𝐻 and ∆𝑇, respectively .
Fig. 3. Spin pumping voltage (V SP) excited by a FMR microwave of
9.5 GHz , at room 𝑇, in TIG(60 nm)/Pt(4 nm). (a) Illustration of the
spin pumping setup . (b) In-plane field scan of V SP for different
microwave powers. (c) Linear dependence of the maximum VSP
with the microwave power. ( d) 𝜃𝐻 scan of the charge current (I SP)
generated by means of the inverse spin Hall effect in the Pt film. ( e)
In-plane field s can of the FMR absorption derivative spectrum for 5
mW.
5
placed on the edges of the Pt film, as illustrated in Fig. 3(a). It
is important to not e that when the magnetization vector was
perpendicular to the sample ’s plane no V SP was detected. The
sample TIG(60 nm)/Pt(4 nm) ha d dimension s of 3 x 4 mm2,
and a resistance between the silver electrodes of 𝟒𝟖 𝛀 at zero
field. The 𝑽𝑺𝑷 show ed a peak value of 𝟎.𝟖𝟓 𝛍𝐕 in the
resonance magnetic field for an incident power of 185 mW ,
and an in -plane dc magnetic field ( 𝛉𝑯=𝟗𝟎°) as shown in Fig.
3(b). The signal rever sed when the field direction went
through a 𝟏𝟖𝟎° rotation . The dependence of 𝑽𝑺𝑷 with the
microwave incident power was linear , as shown in Fig. 3(c),
whereas the spin pumping charge current ( 𝑰𝑺𝑷=𝑽𝑺𝑷𝑹⁄) had
the dependence of 𝑽𝑺𝑷∝𝐬𝐢𝐧𝜽𝑯, showed in Fig. 3(d), for a
fixed microwave power of 100 mW. The ratio between the
microwave -driven voltage and the microwave power was
4µV/W.
We also excited pure spin current s via the spin
Seebeck effect (SSE) in TIG(60 nm) /Pt(4 nm) at room 𝑻. SSE
emerges from the interplay between the spin and heat currents,
and it has the potential to harvest and reduce power
consumption in spintronic devices [ 16, 18 ]. When a magnetic
material is subject ed to a temperature gradient, a spin current
is thermally driven into the adjacent non -magnetic (NM) layer
by means of the spin -exchange interaction. The spin
accumulation in the NM layer can be detected by measuring a
transversal charge current due to the I SHE. To observe the
SSE in our samples , the uncovered GGG surface was placed
over a copper plate, acting as a thermal bath at room 𝑻, while
the sample’s top was in thermal contact with a 𝟐×
𝟐 𝐦𝐦𝟐 commercial Peltier module through a thermal paste,
as illustrate d in Fig. 4(a). The Peltier module was responsible
for creating a controllable temperature gradient across the
sample . On the other hand, the temperature difference ( ∆𝑻)
between the bottom and top of the sample was measured by a
differential thermocouple. The ISHE voltage due to the SSE
(𝑽𝑺𝑺𝑬) was detected between the two silver painted electrodes
placed on the e dges of the Pt film .
The behaviour of 𝑽𝑺𝑺𝑬 by sweeping the dc applied
magnetic field ( 𝑯), while ∆𝑻, 𝜽𝑯 and 𝝓𝑯 were kept fixed was
investigated . Fixing 𝛟𝑯=𝟎° and varying the magnetic field
from out -of-plane (𝜽𝑯=𝟎°) to in -plane along x -direction
(𝜽𝑯=𝟗𝟎°), 𝑽𝑺𝑺𝑬 went from zero to its maximum value of
5.5 𝝁𝑽 for ∆𝑻=𝟐𝟎 𝑲, as shown in F ig. 4(b). Around zero
field, no matter the value of 𝜽𝑯, the TIG’s film magnetization
tended to rely along its out -of-plane easy axis which zeroes
𝑽𝑺𝑺𝑬. For in -plane fields ( 𝜽𝑯=𝟗𝟎°) with ∆𝑻=𝟏𝟐 𝑲, 𝑽𝑺𝑺𝑬
was maximum when 𝛟𝑯=𝟎°, and it was zero for 𝛟𝑯=𝟗𝟎°.
The reason 𝑽𝑺𝑺𝑬 went to zero for 𝛟𝑯=𝟗𝟎°, may be
attributed to the generated charge flow along the x -direction
while the silver electrodes were placed along y -direction , thus
not enabling the current detection (see F ig. 4(c)). The analysis
of the spin Seebeck amplitude ∆𝑽𝑺𝑺𝑬 versus 𝜽𝑯, 𝛟𝑯 and ∆𝑻
show ed a sine, cosine and linear dependence, respectively as
can be seen in Fig. 4(d)-(e), where the red solid lines are
theoretical fits . The Spin Seebeck coefficient (SSC) extracted
from the linear fit of ∆𝑽𝑺𝑺𝑬 vs. ∆𝑻 was 0.54 𝝁𝑽/K.
In conclusion , we used the FMR technique to probe
the magnetic anisotropies and the Gilbert damping parameter of the sputtered TIG thin films with perpendicular magnetic
anisotropy. The results showed higher resonance fields (> 3.5
kOe) and broader linewidth s (> 60 Oe) when comparing with
YIG films at room 𝑇. Thinner TIG films (t = 15 nm and 30
nm) presented a well -defined PMA; on the other hand, the
easy axis of thicker TIG film (60 nm) showed a de viation of
30 degrees from normal to the film plane . By numerically
adjusting the FMR field dependence with the polar angle , we
extracted the effective magnetization , the cubic (H1C) and the
out-of-plane uniaxial anisotropy (HU2=H⊥) fields for the
three TIG films . The thinnest film presented the highest
intensity for H⊥ as expected, even so H⊥ was strong enough to
overcome the shape anisotropy and g ave place to a
perpendicular magnetic anisotropy in all the three thickness of
TIG films . The Gilbert damping parameter (𝛼) for TIG(30
nm) and TIG(60 nm) films were estimated to be ≈ 10-2, by
analyzing a set of FMR spectra using the coplanar waveguide
technique at various microwave frequencies and temperatures.
As 𝑇 went down to 150 K the damping increased
monotonically 130 % .
Furthermore , spin waves (magnons) were excited in
TIG(60 nm)/Pt(4 nm) heterostructure through the spin
pumping and spin Seebeck effects , at room 𝑇 and 9.5 GHz .
The generated pure spin currents carried by the magnons were
converted into charge current s once they reached the Pt film
by means of the inverse spin Hall effect . The charge currents
were detected as a micro -voltage measured at the edges of the
Pt film , and they showed sine and cosine dependence with the
polar and azimuthal angles , respectively, of the dc applied
magnetic field . This voltage was linearly dependent on the
microwave pow er for the SPE, and on the temperature
gradient for the SSE. These results confirmed a good spin -
mixing conductance in the interface TIG/Pt , and an efficient
conversion of pure spin currents into charge currents inside the
Pt film , which is crucial for the employment of TIG films with
a robust PMA in the development of magnon -based spintronic
devices for computing technologies.
ACKNOWLEDGEME NTS
This research is supported in the USA by Army Research
Office (ARO W911NF -19-2-0041 and W911NF -19-2-0015 ),
NSF (DMR 1700137), ONR (N00014 -16-1-2657), in Brazil
by CAPES (Gilvania Vilela/POS -DOC -88881.120327/2016 -
01), FACEPE (APQ -0565 -1.05/14 and APQ -0707 -1.05/14),
CNPq , UPE (PFA/PROGRAD/UPE 04/2017) and FAPEMIG
- Rede de Pesquisa em Materiais 2D and Rede de
Nanomagnetismo , in Chile by Fondo Nacional de Desarrollo
Cientí fico y Tec nológico (FONDECYT) No. 1170723, and in
China by the National Natural Science Foundation of China
(11974025).
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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|
2109.05901v1.Control_of_magnetization_dynamics_by_substrate_orientation_in_YIG_thin_films.pdf | 1
Control of magnetization dynamics by substrate orientation in YIG thin films
Ganesh Gurjar1, Vinay Sharma3, S. Patnaik1*, Bijoy K. Kuanr2,*
1School of Physical Sciences, Jawaharlal Nehru University, New Delhi, INDIA 110067
2Special Centre for Nanosciences, Jawaharlal Nehru University, New Delhi, INDIA 110067
3Morgan State University, Department of Physics, Baltimore, MD, USA 21251
Abstract
Yttrium Iron Garnet (YIG) and b ismuth (Bi) substituted YIG (Bi 0.1Y2.9Fe5O12, BYG) films are
grown in-situ on single crystalline Gadolinium Gallium Garnet (GGG) substrates [with (100) and
(111) orientation s] using pulsed laser deposition (PLD ) technique . As the orientation of the Bi-
YIG film changes from (100) to (111) , the lattice constant is enhanced from 12.384 Å to 12.401 Å
due to orientation dependent distribution of Bi3+ ions at dodecahedral sites in the lattice cell.
Atomic force microscopy (AFM) images show smooth film surfaces with roughness 0.308 nm in
Bi-YIG (111) . The change in substrate orientation leads to the modification of Gilbert damping
which , in turn, gives rise to the enhancement of ferromagnetic resonance (FMR) line width . The
best value s of Gilbert damping are found to be (0.54±0.06 )×10-4, for YIG (100) and
(6.27±0.33) ×10-4, for Bi-YIG (111) oriented films . Angle variation ( ) measurements of the H r are
also performed, that shows a four -fold symmetry for the resonance field in the (100) g rown film.
In addition, the value of effective magnetization (4πM eff) and extrinsic linewidth (ΔH 0) are
observed to be dependent on substrate orientation . Hence PLD growth can assist single -crystalline
YIG and BY G films with a perfect interface that can be used for spintronics and related device
applications.
Keyword s: Pulse Laser Deposition, Epitaxial YIG thin films, lattice strain, ferromagnetic
resonance, Gilbert damping, inhomogeneous broa dening
Corresponding authors: bijoykuanr@mail.jnu.ac.in , spatnaik@mail.jnu.ac.in 2
1. Introduction
One of the most widely studied material s for the realization of spintronic devices appears to be the
iron garnets , particularly the yttrium iron garnet (YIG , Y3Fe5O12) [1,2] . In thin film form of YIG
several potential applications have been envisaged that include spin-caloritronics [3,4] , magneto -
optical (MO) devices, and microwave resonators, circulators, and filters [5–8]. The attraction of
YIG over other ferroic materials is primarily due to their strong magnet o-crystalline anisotropy
and low magnetization damping [2]. Furthermore, towards high frequency applications, YIG’s
main advantage s are its electrically insulating behavior along with low ferromagnetic resonance
line-width (H) and low Gilbert damping parameter [9–11]. These are important parameters for
potential use in high fr equency filters and actuators [12–14]. In this paper, we report optimal
growth parameters for pure and Bi -doped YIG on oriented subs trates and identify the conditions
suitable for their prospective applications.
In literature, YIG is known to be a room temperature ferrimagnetic insulator with a Tc near 560 K
[15]. It has a cubic structure (space group Ia3̅d). The y ttrium (Y) ions occupy the dodecahedral
24c sites ( in the Wyckoff notation), two Fe ions at octahedral 16a and three at tetrahedral 24d sites,
and oxygen the 96h sites [16,17] . The d site is resp onsible for the ferri magnetic nature of YIG. It
is already reported that substitution of Bi in place of Y in YIG leads to substantial improvement in
the magneto -optical response [7,18 –25]. It was also observed that MO performance increa ses
linearly with Bi/Ce doping concentration [22]. Furthermore, substitution of Bi in YIG (BYG) is
documented to provide growth -induced anisotropy that is useful in applications such as magnetic
memory and logic devices [26–30]. The study of basic properties of Bi -substituted YIG materials
is of great current interest due to their applications in magneto -optical devices , magnon -3
spintronics , and related fields such as caloritronics due to its high uniaxial anisotropy and faraday
rotation [21,31 –35]. The structural and magnetic pr operties can be changed via change in Bi3+
concentration in YIG or via choosing a proper substrate orientation. Therefore, t he choice of
perfect substrate orientation is crucial for the identification of the growth of Bi substituted YIG
thin films.
In this work, we have studied the structural and magnetic properties of Bi-substituted YIG
[Bi0.1Y2.9Fe5O12 (BYG)] and YIG thin films with two different single crystalline Gadolinium
Gallium Garnet (GGG) substrate orientation s: (100) and (111) . The YIG and BYG films of
thickness ~150 nm were grown by pulsed laser deposit ion (PLD) method [23,36,37] on top of
single -crystalline GGG substrates . The structural and magnetic properties of all grown films were
carried out using x -ray diffraction (XRD), surface morphology by atomic force microscopy
(AFM) , and magnetic properties via vibrating sample magnetometer (VSM) and ferromagnetic
resonance (FMR) techniques. The FMR is the most useful technique to study the magnetization
dynamics by measuring the properties of magnetic materials through evaluation of their damping
parameter and linewidth . Furthermore, it provides insightful information on the static magnetic
properties such as the saturation magnetization and the anisotropy field. FMR is also extremely
helpful to study fundamentals of spin wave dynami cs and towards characte rizing the relaxation
time and L ande g factor of magnetic material s [11].
2. Experiment
YIG a nd BY G target s were synthesized via the solid -state reaction method . Briefly, y ttrium oxide
(Y2O3) and iron oxide (Fe 2O3) powder s were ground for ~14 hours before calcination at 1100 oC. 4
The calcined powders were pressed into pellets and sintered at 1300 oC. Using thes e YIG and
BYG targets, thin films of thickness ~150 nm were grown in-situ on (100) - and (111) -oriented
GGG substrate s by the PLD technique . The prepared samples have been labeled as YIG (100) ,
YIG (111) , BYG (100), and BYG (111). GGG substrates were cleaned using acetone and
isopropanol. Before deposition, the deposition chamber was thoroughly cleaned and evacuated to
a base vacuum of 2 ×10-6 mbar. We have used KrF excimer laser (248 nm), with pulse frequency
10 Hz to ablate the target s at 300mJ energy . During deposition , target to substrate distance,
substrate temperature , and oxygen pressure w ere kept at ~4.8 cm, 825 oC, and 0.15 mbar ,
respectively. Best films were grown at a rate of 6 nm/min . The as -grown thin film s were annealed
in-situ for 2 hours at 825 oC and cooled down to 300 oC in the presence of oxygen (0.15 mbar)
throughout the process . The structural properties of the thin film were determined by XRD using
Cu-Kα radiation (1.5406 Å) and surface morphology as well as the thickness of the film were
calculated with atomic force microscopy by WITec Gmb H, Germany . Magnetic properties were
studied using a 14 tesla PPMS (Cryogenic) . FMR measurements were carried out by the Vector
Network Analyzer ( VNA ) (Keysight , USA) using a coplanar waveguide ( CPW ) in a flip -chip
geometry with dc magnetic field applied parallel to the film plan e.
3. Results and Discussion
3.1 Structural properties
The room temperature XRD data for the polycrystalline targets of YIG and BYG are plotted in
figure 1 (a) and 1 (b) respectively. Rietveld refinement patterns after fitting XRD data are also
included in the panel s. XRD peaks are indexed according to the JCPDS card no. ( # 43-0507) . Inset 5
of figure 1 (a) shows crystallographic sub -lattices of YIG that elucidates Fe13+ tetrahedral site,
Fe23+ octahedral site , and Y3+ dodecahedral site. Inset (i) of figure 1 (b) shows evidence for
successful incorporation of Bi into YIG ; the lattice constant increases when Bi is substituted into
YIG due to larger ionic radii of Bi (1.170 Å) as compared with Y (1.019 Å) [19]. From Rietveld
refinement we estimate the lattice constant of YIG and BYG to be 12.377 Å [38] and 12.401 Å
respectively .
Figure 2 (a) and 2 (b) show the XRD pattern of bare (100) and (111) oriented GGG substrates .
This is followed by figure 2 (c) & 2(d) for YIG and figure 2 (e) & 2 (f) for BYG as grown thin
films. XRD patterns confirm the single -crystalline grow th of YIG and BYG thin film s over GGG
substrates . The l attice constant, lattice mismatch (with respect to substrate) , and lattice volume
obtain ed from XRD data are listed in Table 1. Lattice cons tant a for the cubic structure is evaluated
using the [39].
𝒂=𝜆√ℎ2+𝑘2+𝑙2
2sin𝜃 (1)
Where 𝜆 is the wavelength of Cu -Kα radiation , 𝜃 is the diffraction angle , and [h , k, l ] are the
miller indices of the corresponding XRD peak. Lattice misfit (𝛥𝑎
𝑎) is evaluated using equation 2
[24,38] .
𝛥𝑎
𝑎=(𝑎𝑓𝑖𝑙𝑚− 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 )
𝑎𝑓𝑖𝑙𝑚 100 (2)
Where 𝑎𝑓𝑖𝑙𝑚 and 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 are the lattice constant of film and substrate respectively. Lattice
constant of pure YIG bulk is 12.377 Å, whereas we have observed a larger value of lattice constants
of YIG and BY G films than th at of bulk YIG as shown in T able 1 . Sim ilarly, to these obtained
results, a larger value of lattice constants than that of bulk YIG has been reported as well [40–44]. 6
The obtained values of the lattice constant are in agreement with the previous reports
[18,21,25,34,45] . In the case of BYG (111), the value of lattice constant slightly increases
compared to BYG (100) because the distribution of Bi3+ in the dodecahedral site depends on the
orientation of the substrate [28,46] . Inset (ii) of figure 1 (b) shows plane (111) has more
contribution of Bi3+ ions [(ionic radius of bismuth (1.170 Å) is larger as compared with YIG (1.019
Å) [19]]. This slight increase in the lattice constant (in 111 direction) implies a more lattice
mismatch (or strain ) in BYG films . Positive value of lattice mismatch indicates the slightly larger
lattice constant of films (YIG and BYG) were observed as compared to substrates (GGG). We
would like to emphasize that lattice plane dependence growth is important to signify the changes
in the struc tural and magnetic properties.
3.2 Surface morphology study
Room temperature AFM images with roughness are shown in figure 3 (a)-(d). Roughness plays
an important role from the application prospective as it is related to Gilbert damping factor α.
Lower roughness (root mean square height) is observed for the (111) oriented films of YIG and
BYG compared to those grown on (100) oriented substrates. Available literature [61] indicate that
roughness would depend more on variation on growth parameters ra ther than on substrate
orientation. In this sense further study is needed to clarify substrate dependence of roughness. No
significant change in the roughness is observed between YIG and BYG films [38,47] . Table 1
depicts a comparison between the roughne ss measured in YIG and the BY G thin films. 7
3.3 Static magnetization p roperties
VSM magnetization measurements were performed at 300 K with magnetic field appl ied parallel
to the film plane (in-plane) . Figure 3 (e) and 3 (f) show the magnetization plot s for YIG and BYG
respectively after careful subtraction of paramagnetic contr ibution that is assigned to the substrate.
The m easured saturation magnetization ( 4πM S) data are given in Table 1 which are in general
agreement with the reported values [11,40,48] . Not much change in the measured 4πM S value of
YIG and Bi -YIG films are observed . The ferrimagnetism nature of YIG arises from super -
exchange interaction between the non -equivalent Fe3+ ions at octahedral and tetrahedral sites [49].
Bismuth located at dodecahedral site does not affect the tetrahedral and octahedral Fe3+ ions. So,
Bismuth does not show a significant change in saturation magnetization at room temperature. I t is
reported in literature th at Bi addition leads to increase in Curie temperature, so in t hat sense there
is an decreasing trend in saturation magnetization in BYG films in contrast to YIG films [50,51] .
Error bars in saturation magnetization relate to uncertainty in sample volume.
3.4 Ferromagnetic r esonance properties
The FMR absorption spectroscopy is shown in figure 4. These measurements were performed at
room temperature . The external dc magnetic field was appli ed parallel to the plane of the film .
Lorentzian fit of the calibrated experimental data are used to calculate t he FMR linewidth (∆H)
and resonance magnetic field (H r). From the e nsemble of all the FMR data at different resonance
frequencies (f = 1 GHz -12 GHz ), we have calculated the gyromagnetic ratio (γ) , effective
magnetization field ( 4𝜋𝑀𝑒𝑓𝑓) from the fitting of Kittel’s in-plane equation [52].
8
In general, t he uniform precession of magnetization can be described by the Landau -Lifshitz -
Gilbert (LLG) equation of motion;
𝜕𝑀⃗⃗
𝜕𝑡=−𝛾(𝑀⃗⃗ ×𝐻⃗⃗ 𝑒𝑓𝑓)+𝐺
𝛾𝑀𝑠2[𝑀⃗⃗ ×𝜕𝑀⃗⃗
𝜕𝑡] (3)
Here, t he first term corresponds to the precessional torque in the effective magnetic field and the
second term is the Gilbert damping torque. The gyromagnetic ratio is given by 𝛾=𝑔𝜇𝐵/ℏ , where
𝑔 is the Lande’s factor, 𝜇𝐵 is Bohr magnetron and ℏ is the Planck’s constant. Similarly, 𝐺=𝛾𝛼𝑀𝑠
is related to the intrinsic relaxation rate in the nanocomposites and 𝛼 represents the Gilbert
damping constant. Ms (or 4πMs) is the saturation magnetization. It can be shown that t he solution
for in -plane resonance frequency can be written as;
𝑓𝑟=𝛾′√(𝐻𝑟)(𝐻𝑟+4𝜋𝑀𝑒𝑓𝑓) (4),
Where 𝛾′=𝛾/2𝜋, 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑎𝑛𝑖 is the effective field and 𝐻𝑎𝑛𝑖=2𝐾1
𝑀𝑠 is the anisotropy
field. Following through, we have obtained Gilbert damping parameter (α) and inhomogeneous
broadening (∆H 0) linewidth from the fitting of Landau –Lifshitz –Gilbert equation (LLG) [53]
𝛥𝐻(𝑓)=𝛥𝐻0+4𝜋𝛼
√3𝛾𝑓 (5)
Derived parameters from the FMR study are listed in T able 2 . The obtai ned Gilbert damping (α)
is in agreement with the reported thin films used for the study of spin-wave propagation
[2,27,54,55] . In the case of YIG no t much change in the value of α is seen . However , a substantial
increase is observed in case of BYG with (111) orientation. Qualitatively this could be assigned to 9
the presence of Bi3+ ions which induce s spin-orbit coupling (SOC) [56–58] and also due to electron
scattering inside the lattice as lattice mismatch (or strain ) increases [59]. We have seen more
distribution of Bi3+ ions along (111) planes ( see inset (ii) of figure 1 (b) ) and also slightly larger
lattice mismatch in BYG (111) from our XRD results. These results also explain higher value of
Gilbert damping and ΔH 0 in case of BYG (111). The change in 4𝜋𝑀𝑒𝑓𝑓 could be attributed to
uniaxial in -plane magnetic anisotropy . This is because no change in 4πM S is observed from
magnetization measurements and 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑎𝑛𝑖 [38,40,60] . The uniaxial inplane magnetic
anisotropy is induced due to lattice mismatch between films and GGG substrates [38,40] . The
calculated gyromagnetic ratio (γ) and ΔH 0 are also included in Table 2 . The magnitude of ΔH 0 is
close to reported values for same substrate orientation [38]. In summary we find that YIG with
(100) orientation yields lowest damping fact or and extrinsic contribution to linewidth. These are
the r equired optimal parameters for spintronic s application with high spin diffusion length.
However, MOKE signal is usual ly very low in bare YIG thin films because of its lower magnetic
anisotropy and strain [61]. But previous reports suggest that magnetic anisotropy and magnetic
domains formation can be achieved in YIG system by doping rare earth materials like Bi and Ce
[18,61]. We have shown that anisotropic characteristic with Bi doping in YIG is more pron ounced
along <111> direction which can lead to the enhanced MOKE signal in Bi -YIG films on <111>
substrate.
We have also recorded polar angle () data of resonance field ( Hr) versus magnetic field
(H) at frequency 12 GHz for the BYG (100) and BYG (111) films (figure 5 (c) & 5(d) respectively
where inset shows the azimuthal angle ( ) variation of Hr measured at frequency of 3 GHz ). The
data are fitted with modified Kittel equation . From figure 5 (c) & (d), we can see that Hr increases
up to 2.5 kOe in BYG (100) and 3.0 kOe in BYG (111) by varying the direction of H from 0 to 90 10
degree with respect to sample surface (inset of Fig 5 (a)) . Obtained parameters from angular
variation of FMR magnetic field H r (θH) are listed in the inset of figure 5 (c) & (d) . From variation
data (by varying the direction of H from 0 to 18 0 degree with respect to sample edge (Fig 5 (a)
Inset ), we see clear four -fold and two -fold in-plane anisotropy in BYG (100) and BYG (111) films
[61,62] . This further consolidates single -crystalline characteristics of our films. The change
observed in Hr with respect to variation is 79.52 Oe in BYG (100) (H=0 to 45) and 19.25 Oe in
BYG (111) ( H=0 to 45). Thus, during in-plane rotation, higher change in FMR field is observed
along (100) orientation .
4. Conclusion
In conclusion , we have grown high quality YIG and B i-YIG thin film s on GGG substrates with
(100) and (111) orientation . The films were gr own by pulsed laser deposition. The optimal
parameters i.e. target to substrate distance, substrate temperature, and oxygen pressure are
determined to be ~ 4.8 cm, 825 oC, and 0.15 mbar, respectively. The as grown thin films have
smooth surfaces and are found to be phase pure from AFM and XRD characterizations. From FMR
measurements , we have found lower value of damping parameter in (100) YIG that indicates
higher spin diffusion length for potential spintronics application. On the other -hand bismuth
incorporation to YIG leads to dominance of anisotropic characteristics that augers well for
application in magnetic bubble memory and magneto -optic devices . The enhanced value of α in
Bi-YIG films is ascribed to the spin orbit coupled Bi3+ ions. We also ta bulate the values of
magnetic parameters such as linewidth ( ∆H0), gyromagnetic ratio ( γ), and effective magnetization
4𝜋𝑀𝑒𝑓𝑓 with respect to substrate orientation. Unambiguous four-fold in -plane anisotropy is
observed in (100) oriented films. We find high-quality magnetization dynamics and lower Gilbert 11
damping parameter is possible in Bi-YIG grown on (111) GGG in conjunction with enhanced
magnetic anisotropy. The choice of perfect substrate orientation is therefore found to be crucial
for the growth of YIG and Bi-YIG thin films for high frequency applications.
Acknowledgments
This work is supported by the MHRD -IMPRINT grant, DST (SERB, AMT , and PURSE -II) grant
of Govt. of India. Ganesh Gurjar acknowledges CSIR, New Delhi for financial support . We
acknowledge AIRF, JNU for access of PPMS facility.
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20
List of Tables with caption
Table 1: Lattice and magnetic p arameters obtained from XRD , AFM and VSM.
S. No. Sample Lattice
constant
(Å) Lattice
Mismatch
(%) Lattice
volume
(Å3) Roughness
(nm) 4πM S
(Gauss)
1. YIG (100) 12.403 0.42 1907.81 0.801 1670.15±83.51
2. YIG (111) 12.405 0.40 1909.02 0.341 1654.06±82.70
3. BYG (100) 12.384 0.36 1899.11 0.787 1788.50±89.43
4. BYG (111) 12.401 0.65 1906.93 0.308 1816.31±90.82
Table 2: Damping and linewidth p arameters obtained from FMR
S. No. Sample α
(10-4) ΔH 0
(Oe) 4πM eff
(Oe) γ'
(GHz/kOe)
1. YIG (100) (0.54±0.06) 26.24±0.10 1938 .60±37.57 2.89±0.01
2. YIG (111) (1.05±0.13) 26.51±0.21 2331 .38±65.78 2.86±0.02
3. BYG (100) (1.66±0.10) 26.52 ±0.17 1701.67±31.87 2.89±0.11
4. BYG (111) (6.27±0.33) 29.28 ±0.62 2366 .85±62.60 2.86±0.02
21
Figure Captions
Figure 1: XRD with Rietveld refinement pattern of (a) YIG target ( inset shows crystallographic
sub-lattices, Fe 13+ tetrahedral site, Fe 23+ octahedral site and Y3+ dodecahedral site ) (b) BYG target
(inset (i) shows effect of Bi doping into YIG , inset (ii) shows contribution of the Bi3+ along the
(100) and (111) planes ).
Figure 2: XRD pattern of (a) GGG (100) , (b) GGG (111), (c) YIG (100) , (d) YIG (111) , (e) BYG
(100) , and (f) BYG (111) .
Figure 3: AFM images of (a) YIG (100), (b ) YIG (111) , (c) BYG (100), (d) BYG (111) and static
magnetization graph of (e ) YIG (100), YIG (111); and (f) BY G (100), BYG (111) .
Figure 4: FMR absorption spectra of (a) YIG (100), (b) YIG (111), (c) BYG (100), and (d) BYG
(111).
Figure 5: (a) FMR magnetic field Hr is plotted as a function of frequency f. Experiment data fitted
with Kittel equation for YIG and BYG oriented films. Inset shows how the applied field angle is
measured from sample surface (b) Frequency -dependent FMR linewidth data fitted with LLG
equation for YIG and BYG oriented films. Inset shows the magnified version to illustrate the effect
of Bi doping in YIG . (c) and (d) show angular variation of FMR magnetic field (Hr (θH)) fitted
with modified Kittel equation at 12 GHz frequency for BYG (100) and BYG (111) films . Insets
show the FMR magnetic field (H r) as a function of azimuthal angle ( ).
22
Figure 1
23
Figure 2
24
Figure 3
25
Figure 4
26
Figure 5
|
1211.3611v2.Spin_transport_and_tunable_Gilbert_damping_in_a_single_molecule_magnet_junction.pdf | Spin transport and tunable Gilbert damping in a single-molecule magnet junction
Milena Filipović,1Cecilia Holmqvist,1Federica Haupt,2and Wolfgang Belzig1
1Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany
2Institut für Theorie der Statistischen Physik, RWTH Aachen, D-52056 Aachen, Germany
(Dated: October 24, 2019)
We study time-dependent electronic and spin transport through an electronic level connected
to two leads and coupled with a single-molecule magnet via exchange interaction. The molecular
spin is treated as a classical variable and precesses around an external magnetic field. We derive
expressions for charge and spin currents by means of the Keldysh nonequilibrium Green’s functions
techniqueinlinearorderwithrespecttothetime-dependentmagneticfieldcreatedbythisprecession.
The coupling between the electronic spins and the magnetization dynamics of the molecule creates
inelastic tunneling processes which contribute to the spin currents. The inelastic spin currents, in
turn, generate a spin-transfer torque acting on the molecular spin. This back-action includes a
contribution to the Gilbert damping and a modification of the precession frequency. The Gilbert
damping coefficient can be controlled by the bias and gate voltages or via the external magnetic
field and has a nonmonotonic dependence on the tunneling rates.
PACS numbers: 73.23.-b, 75.76.+j, 85.65.+h, 85.75.-d
I. INTRODUCTION
Single-molecule magnets (SMMs) are quantum mag-
nets, i.e., mesoscopic quantum objects with a perma-
nent magnetization. They are typically formed by
paramagnetic ions stabilized by surrounding organic
ligands.1SMMs show both classical properties such
as magnetization hysteresis2and quantum properties
such as spin tunneling,3coherence,4and quantum phase
interference.2,5They have recently been in the center of
interest2,6,7in view of their possible applications as in-
formation storage8and processing devices.9
Currently, a goal in the field of nanophysics is to
control and manipulate individual quantum systems, in
particular, individual spins.10,11Some theoretical works
have investigated electronic transport through a molecu-
larmagnetcontactedtoleads.12–19Inthiscase, thetrans-
port properties are modified due to the exchange inter-
action between the itinerant electrons and the SMM,20
making it possible to read out the spin state of the
molecule using transport currents. Conversely, the spin
dynamics and hence the state of an SMM can also be
controlled by transport currents. Efficient control of the
molecule’s spin state can be achieved by coupling to fer-
romagnetic contacts as well.21
Experiments have addressed the electronic trans-
port properties through magnetic molecules such as
Mn12and Fe 8,22,23which have been intensively stud-
ied as they are promising candidates for memory
devices.24Various phenomena such as large conduc-
tance gaps,25switching behavior,26negative differen-
tial conductance, dependence of the transport on mag-
netic fields and Coulomb blockades have been exper-
imentally observed.22,23,27,28Experimental techniques,
including, for instance, scanning tunneling microscopy
(STM),22,23,29–31break junctions,32and three-terminal
devices,22,23,27have been employed to measure elec-
tronic transport through an SMM. Scanning tunnelingspectroscopy and STM experiments show that quantum
properties of SMMs are preserved when deposited on
substrates.29The Kondo effect in SMMs with magnetic
anisotropyhasbeeninvestigatedboththeoretically14and
experimentally.33,34It has been suggested35and experi-
mentally verified36that a spin-polarized tip can be used
to control the magnetic state of a single Mn atom.
In some limits, the large spin Sof an SMM can be
treated as a classical magnetic moment. In that case,
the spin dynamics is described by the Landau-Lifshitz-
Gilbert (LLG) equation that incorporates effects of ex-
ternal magnetic fields as well as torques originating from
damping phenomena.37,38In tunnel junctions with mag-
netic particles, LLG equations have been derived using
perturbativecouplings39,40andthenonequilibriumBorn-
Oppenheimer approximation.16Current-induced magne-
tization switching is driven by a generated spin-transfer
torque (STT)41–44as a back-action effect of the elec-
tronic spin transport on the magnetic particle.16,45–47
A spin-polarized STM (Ref. 36) has been used to
experimentally study STTs in relation to a molecular
magnetization.48This experimental achievement opens
new possibilities for data storage technology and appli-
cations using current-induced STTs.
The goal of this paper is to study the interplay be-
tween electronic spin currents and the spin dynamics of
an SMM. We focus on the spin-transport properties of
a tunnel junction through which transport occurs via a
single electronic energy level in the presence of an SMM.
The electronic level may belong to a neighboring quan-
tum dot (QD) or it may be an orbital related to the
SMM itself. The electronic level and the molecular spin
are coupled via exchange interaction, allowing for inter-
actionbetweenthespinsoftheitinerantelectronstunnel-
ing through the electronic level and the spin dynamics of
the SMM. We use a semiclassical approach in which the
magnetization of the SMM is treated as a classical spin
whose dynamics is controlled by an external magneticarXiv:1211.3611v2 [cond-mat.mes-hall] 22 Jul 20132
field, while for the electronic spin and charge transport
we use instead a quantum description. The magnetic
field is assumed to be constant, leading to a precessional
motion of the spin around the magnetic field axis. The
electronic level is subjected both to the effects of the
molecular spin and the external magnetic field, generat-
ingaZeemansplitofthelevel. Thespinprecessionmakes
additional channels available for transport, which leads
to the possibility of precession-assisted inelastic tunnel-
ing. During a tunnel event, spin-angular momentum may
be transferred between the inelastic spin currents and the
molecular spin, leading to an STT that may be used to
manipulate the spin of the SMM. This torque includes
the so-called Gilbert damping , which is a phenomenolog-
ically introduced damping term of the LLG equation,38
and a term corresponding to a modification of the pre-
cession frequency. We show that the STT and hence the
SMM’s spin dynamics can be controlled by the external
magnetic field, the bias voltage across the junction, and
the gate voltage acting on the electronic level.
The paper is organized as follows: We introduce our
model and formalism based on the Keldysh nonequilib-
rium Green’s functions technique49–51in Sec. II, where
we derive expressions for the charge and spin currents in
linear order with respect to a time-dependent magnetic
field and analyze the spin-transport properties at zero
temperature. In Sec. III we replace the general magnetic
field of Sec. II by an SMM whose spin precesses in an
external constant magnetic field, calculate the STT com-
ponents related to the Gilbert damping, and the modifi-
cationoftheprecessionfrequency, andanalyzetheeffects
of the external magnetic field as well as the bias and gate
voltages on the spin dynamics. Conclusions are given in
Sec. IV.
II. CURRENT RESPONSE TO A TIME
DEPENDENT MAGNETIC FIELD
A. Model and Formalism
For the sake of clarity, we start by considering a junc-
tion consisting of a noninteracting single-level QD cou-
pled with two normal, metallic leads in the presence of an
external, time-dependent magnetic field (see Fig. 1). The
leads are assumed to be noninteracting and unaffected
by the external field. The total Hamiltonian describing
the junction is given by ^H(t) = ^HL;R+^HT+^HD(t).
The Hamiltonian of the free electrons in the leads reads
^HL;R=P
k;;2fL;Rgk^cy
k^ck, wheredenotes the
left (L) or right ( R) lead, whereas the tunnel cou-
pling between the QD and the leads can be written as
^HT=P
k;;2L;R[Vk^cy
k^d+V
k^dy
^ck]. The spin-
independent tunnel matrix element is given by Vk. The
operators ^cy
k(^ck)and ^dy
(^d)are the creation (an-
nihilation) operators of the electrons in the leads and
the QD, respectively. The subscript =";#denotes
eVB(t)
FIG. 1: (Color online) A quantum dot with a single electronic
level0coupled to two metallic leads with chemical potentials
LandRinthepresenceofanexternaltime-dependentmag-
netic field~B(t). The spin-transport properties of the junction
aredeterminedbythebiasvoltage eV=L R, theposition
of the level 0, the tunnel rates Land R, and the magnetic
field.
the spin-up or spin-down state of the electrons. The
electronic level 0of the QD is influenced by an ex-
ternal magnetic field ~B(t)consisting of a constant part
~Bcand a time-dependent part ~B0(t). The Hamiltonian
of the QD describing the interaction between the elec-
tronic spin ^~ sand the magnetic field is then given by
^HD(t) = ^Hc
D+^H0(t), where the constant and time-
dependent parts are ^Hc
D=P
0^dy
^d+gB^~ s~Bcand
^H0(t) =gB^~ s~B0(t). The proportionality factor gis the
gyromagnetic ratio of the electron and Bis the Bohr
magneton.
The average charge and spin currents from the left lead
to the electronic level are given by
IL(t) =q d
dt^NL
=qi
~
^H;^NL
;(1)
where ^NL=P
k;;0^cy
kL(^)0^ck0Lis the charge and
spin occupation number operator of the left contact. The
index= 0corresponds to the charge current, while
=x;y;zindicates the different components of the spin-
polarized current. The current coefficients qare then
q0= eandq6=0=~=2. In addition, it is useful to
define the vector ^= (^1;^~ ), where ^1is the identity
operator and ^~ consists of the Pauli operators with ma-
trix elements (^~ )0. Using the Keldysh nonequilibrium
Green’s functions technique, the currents can then be ob-
tained as50,51
IL(t) = 2q
~Re
dt0Tr
^[^Gr(t;t0)^<
L(t0;t)(2)
+^G<(t;t0)^a
L(t0;t)]
;
where ^Gr;a;<are the retarded, advanced, and lesser
Green’s functions of the electrons in the QD with the ma-
trix elements Gr;a
0(t;t0) =i(tt0)hf^d(t);^dy
0(t0)gi
andG<
0(t;t0) =ih^dy
0(t0)^d(t)i, while ^r;a;<
L(t;t0)
are self-energies from the coupling between the QD
and the left lead. Their nonzero matrix elements3
are diagonal in the electronic spin space with re-
spect to the basis of eigenstates of ^sz, given by
r;a;<
L(t;t0) =P
kVkLgr;a;<
kL(t;t0)V
kL. The Green’s func-
tionsgr;a;<
kL(t;t0)are the retarded, advanced and lesser
Green’s functions of the free electrons in the left lead.
The retarded Green’s functions ^Gr
0of the electrons in
the QD, in the presence of the constant magnetic field
~Bc, are found using the equation of motion technique,52
while the lesser Green’s functions ^G<
0are obtained from
the Keldysh equation ^G<
0=^Gr
0^<^Ga
0, where multipli-
cation implies internal time integrations.51The time-
dependent part of the magnetic field can be expressed
as~B0(t) =P
!(~B!e i!t+~B
!ei!t), where~B!is a com-
plex amplitude. This magnetic field acts as a time-
dependent perturbation that can be expressed as ^H0(t) =P
!(^H!e i!t+^Hy
!ei!t), where ^H!is an operator in the
electronic spin space and its matrix representaton in the
basis of eigenstates of ^szis given by
^H!=gB
2
B!zB!x iB!y
B!x+iB!y B!z
:(3)
Applying Dyson’s expansion, analytic continuation rules
and the Keldysh equation,51one obtains a first-order ap-
proximation of the Green’s functions describing the elec-
trons in the QD that can be written as
^Gr^Gr
0+^Gr
0^H0^Gr
0; (4)
^G<^Gr
0^<^Ga
0+^Gr
0^H0^Gr
0^<^Ga
0+^Gr
0^<^Ga
0^H0^Ga
0:
The expression for the currents in this linear approxima-
tion is given by
IL(t) = 2q
~Re Tr
^[^Gr
0^<
L+^G<
0^a
L (5)
+^Gr
0^H0^Gr
0^<
L+^Gr
0^H0^G<
0^a
L+^G<
0^H0^Ga
0^a
L]
:
Eq. (5) is then Fourier transformed in the wide-
band limit, in which the level width function, () =
2 Imfr()g, is constant, Refr()g= 0, and one can
hence write the retarded self-energy originating from the
dot-lead coupling as r;a() =i =2. From this trans-
formation, one obtains
IL(t) =Idc
L+X
![IL(!)e i!t+I
L(!)ei!t]:(6)
Using units in which ~= 1, the dc part of the
currents51Idc
Land the time-independent complex com-
ponentsIL(!)are given by
Idc
L=qd
L R
[fL() fR()] Tr Imf^^Gr
0()g(7)
and
IL(!) = iqd
2 L R
n
[fL() fR()] (8)
Trf^[^Gr
0(+!)^H!^Gr
0() + 2iImf^Gr
0()g^H!^Ga
0( !)]g
+X
=L;R
R[f( !) fL()] Tr[^^Gr
0()^H!^Ga
0( !)]o
:In the above expressions, f() = [e( )=kBT+ 1] 1is
the Fermi distribution of the electrons in lead , where
kBis the Boltzmann constant. The retarded Green’s
function ^Gr
0()is given by ^Gr
0() = [ 0 r()
(1=2)gB^~ ~Bc] 1.16
The linear response of the spin current with respect
to the applied time-dependent magnetic field can be ex-
pressed in terms of complex spin-current susceptibilities
defined as
L
j(!) =@IL(!)
@B!j; j =x;y;z: (9)
The complex components IL(!)are conversely given by
IL(!) =P
j L
j(!)B!j. By taking into account that
@^H!=@B!j= (1=2)gB^jand using Eq. (8), the current
susceptibilities can be written as
L
j(!) = iqgBd
4 L R
n
[fL() fR()](10)
Trf^[^Gr
0(+!)^j^Gr
0() + 2iImf^Gr
0()g^j^Ga
0( !)]g
+X
R[f( !) fL()]Tr[^^Gr
0()^j^Ga
0( !)]o
:
The components obey L
j( !) = L
j(!). In other
words, they satisfy the Kramers-Kronig relations53that
can be written in a compact form as
L
j(!) =1
iP1
1 L
j()
!d; (11)
withPdenoting the principal value.
For anyi;j;k =x;y;zsuch thatj6=kandj;k6=
i, whereiindicates the direction of the constant part
of the magnetic field ~Bc=Bc~ ei, the complex current
susceptibilities satisfy the relations
L
jj(!) = L
kk(!) (12)
and L
jk(!) = L
kj(!); (13)
in addition to Eq. (11). The other nonzero compo-
nents are L
0i(!)and L
ii(!). In the absence of a constant
magnetic field, the only nonvanishing components obey
L
xx(!) = L
yy(!) = L
zz(!).
Finally, the average value of the electronic spin in the
QD reads~ s(t) =h^~ s(t)i= (1=2)P
0~ 0h^dy
(t)^d0(t)i=
(i=2)P
0~ 0^G<
0(t;t)and the complex spin suscep-
tibilities are defined as
s
ij(!) =@si(!)
@B!j: (14)
They represent the linear responses of the electronic spin
componentstotheappliedtime-dependentmagneticfield
and satisfy the relations similar to Eqs. (11), (12), and
(13) given above.4
B. Analysis of the spin and current responses
We start by analyzing the transport properties of the
junction at zero temperature in response to the exter-
nal time-dependent magnetic field ~B(t). The constant
component of the magnetic field ~Bcgenerates a Zeeman
split of the QD level 0, resulting in the levels ";#, where
";#=0gBBc=2in this section. The time-dependent
periodic component of the magnetic field ~B0(t)then cre-
ates additional states, i.e., sidebands, at energies "!
and#!(see Fig. 2). These Zeeman levels and side-
bandscontributetotheelastictransportpropertiesofthe
junction when their energies lie inside the bias-voltage
window ofeV=L R.
However, energylevelsoutsidethebias-voltagewindow
may also contribute to the electronic transport due to in-
elastic tunnel processes generated by the time-dependent
magnetic field. In these inelastic processes, an electron
transmitted from the left lead to the QD can change its
energy by!and either tunnel back to the left lead or
out into the right lead. If this perturbation is small, as is
assumed in this paper where we consider first-order cor-
rections, the transport properties are still dominated by
the elastic, energy-conserving tunnel processes that are
associated with the Zeeman levels.
The energy levels of the QD determine transport prop-
erties such as the spin-current susceptibilities and the
spin susceptibilities, which are shown in Fig. 3. The
imaginary and real parts of the susceptibilities are plot-
ted as functions of the frequency !in Figs. 3(a) and 3(c).
In this case, the position of the unperturbed level 0is
symmetric with respect to the Fermi surfaces of the leads
and a peak or step in the spin-current and spin suscepti-
bilities appears at a value of !, for which an energy level
is aligned with one of the lead Fermi surfaces. In Figs.
3(b) and 3(d), the susceptibilities are instead plotted as
functions of the bias voltage, eV. Here, each peak or
step in the susceptibilities corresponds to a change in the
number of available transport channels. The bias volt-
age is applied in such a way that the energy of the Fermi
surface of the right lead is fixed at R= 0while the en-
ergy of the left lead’s Fermi surface is varied according
toL=eV.
III. SPIN-TRANSFER TORQUE AND
MOLECULAR SPIN DYNAMICS
A. Model with a precessing molecular spin
Now we apply the formalism of the previous section
to the case of resonant tunneling through a QD in the
presence of a constant external magnetic field and an
SMM [see Fig. 4(a)]. An SMM with a spin Slives in
a(2S+ 1)-dimensional Hilbert space. We assume that
the spinSof the SMM is large and neglecting the quan-
tum fluctuations, one can treat it as a classical vector
µLµR=00↑+ω↑↑−ω↓+ω↓−ω↓FIG. 2: (Color online) Sketch of the electronic energy levels of
the QD in the presence of a time-dependent magnetic field. In
a static magnetic field, the electronic level 0(solid black line)
splits into the Zeeman levels ";#(solid red and blue lines). If
the magnetic field in addition to the static component also in-
cludes a time-dependent part with a characteristic frequency
!, additionallevelsappearatenergies "!(dottedredlines)
and#!(dotted blue lines). Hence, there are six channels
available for transport.
whose end point moves on a sphere of radius S. In the
presence of a constant magnetic field ~Bc=Bc~ ez, the
molecular spin precesses around the field axis according
to~S(t) =S?cos(!Lt)~ ex+S?sin(!Lt)~ ey+Sz~ ez, where
S?is the projection of ~Sonto thexyplane,!L=gBBc
is the Larmor precession frequency and Szis the projec-
tion of the spin on the zaxis [see Fig. 4(b)]. The spins
of the electrons in the electronic level are coupled to the
spin of the SMM via the exchange interaction J. The
contribution of the external magnetic field and the pre-
cessional motion of the SMM’s spin create an effective
time-dependent magnetic field acting on the electronic
level.
The Hamiltonian of the system is now given by ^H(t) =
^HL;R+^HT+^HD(t) +^HS, where the Hamiltonians ^HL;R
and ^HTare the same as in Sec. II. The Hamiltonian
^HS=gB~S~Bcrepresents the interaction of the molecu-
lar spin~Swith the magnetic field ~Bcand consequently
does not affect the electronic transport through the junc-
tion but instead contributes to the spin dynamics of the
SMM. The Hamiltonian of the QD in this case is given by
^HD(t) =^Hc
D+^H0(t). Here, ^Hc
D=P
0^dy
^d+gB^~ s~Bc
e
is the Hamiltonian of the electrons in the QD in the pres-
ence of the constant part of the effective magnetic field,
given by~Bc
e =
Bc+J
gBSz
~ ez. The second term of the
QD Hamiltonian, ^H0(t) =gB^~ s~B0
e (t), represents the in-
teraction between the electronic spins of the QD, ^~ s, and
the time-dependent part of the effective magnetic field,
given by~B0
e (t) =JS?
gB
cos(!Lt)~ ex+ sin(!Lt)~ ey
. The
time-dependent effective magnetic field can be rewritten
as~B0
e (t) =~B!Le i!Lt+~B
!Lei!Lt, where~B!Lconsists
of the complex amplitudes B!Lx=JS?=2gB,B!Ly=5
0.00.51.01.52.02.5-6-4-2024
w@e0DcijL@10-3mBD
mL=e+wHaL
mL=e¯+w
ImczzLReczzLImcxyLRecxyLImcxxLRecxxL
0.00.51.01.52.0-1012
eV@e0DcijL@10-2mBD
e¯-wHbL
e¯
e¯+w
e-w
e
e+w
ImczzLReczzLImcxyLRecxyLImcxxLRecxxL
0.00.51.01.52.02.5-2-101
w@e0Dcijs@10-2mBe0-1DHcL
ImczzsReczzsImcxysRecxysImcxxsRecxxs
0.00.51.01.52.0-4-20
eV@e0Dcijs@10-1mBe0-1DHdL
ImczzsReczzsImcxysRecxysImcxxsRecxxs
FIG. 3: (Color online) (a) Frequency and (b) bias-voltage dependence of the spin-current susceptibilities. (c) Frequency and
(d) bias-voltage dependence of the spin susceptibilities. In (a) and (c), the chemical potential of the left lead is L= 20, while
in (b) and (d) the frequency is set to != 0:160. All plots are obtained at zero temperature with ~Bc=Bc~ ez, and the other
parameters set to R= 0; "= 1:480; #= 0:520; = 0:020, and L= R= 0:010.
iJS?=2gB, andB!Lz= 0. The time-dependent pertur-
bation can then be expressed as ^H0(t) = ^H!Le i!Lt+
^Hy
!Lei!Lt, where ^H!Lis an operator that can be written,
using Eq. (3) and the above expressions for B!Li, as
^H!L=JS?
2
0 1
0 0
: (15)
The time-dependent part of the effective magnetic field
creates inelastic tunnel processes that contribute to the
currents. The in-plane components of the spin current
fulfill
ILx(!L) = iILy(!L) (16)
=JS?
2gB[ L
xx(!L) +i L
xy(!L)];
where~Bcis replaced by ~Bc
e . Thezcomponent vanishes
to lowest order in H0(t).54Therefore, the inelastic spin
current has a polarization that precesses in the xyplane.
The inelastic spin-current components, in turn, exert an
STT (Refs. 41-44) on the molecular spin given by
~T(t) = [~IL(t) +~IR(t)]; (17)thus contributing to the dynamics of the molecular spin
through
_~S(t) =gB~Bc~S(t) +~T(t): (18)
Using expressions (6), (8), and (15), the torque of Eq.
(17) can be calculated in terms of the Green’s functions
^Gr
0()and ^Ga
0()as
Ti(t) = JS?
2d
2X
[f( !L) f()]
(19)
Imf(^i)#"Gr
0;""()Ga
0;##( !L)e i!Ltg;
with=L;R. Here (^i)#",Gr
0;""(), andGa
0;##()are
matrix elements of ^i,^Gr
0()and ^Ga
0()with respect to
the basis of eigenstates of ^sz. This STT can be rewritten
in terms of the SMM’s spin vector as
~T(t) =
S_~S(t)~S(t) + _~S(t) +
~S(t):(20)
The first term in this back-action gives a contribution to6
zBceVS⊥(t)(a) (b)
FIG. 4: (Color online) (a) Resonant tunneling in the pres-
ence of an SMM and an external, constant magnetic field.
The electronic level of Fig. 1 is now coupled with the spin
of an SMM via exchange interaction with the coupling con-
stantJ. The dynamics of the SMM’s spin ~Sis controlled by
the external magnetic field ~Bcthat also affects the electronic
level. (b)PrecessionalmotionoftheSMM’sspininaconstant
magnetic field ~Bcapplied along the zaxis.
the Gilbert damping, characterized by the Gilbert damp-
ing coefficient . The second term acts as an effective
constant magnetic field and changes the precession fre-
quency of the spin ~Swith the corresponding coefficient
. The third term cancels the zcomponent of the Gilbert
damping term, thus restricting the STT to the xyplane.
The coefficient of the third term
is related to by
= =!LS2
?=SSz. Expressing the coefficients and
in terms of the current susceptibilities
xx(!L)and
xy(!L)results in
= JSz
gB!LSX
[Ref
xx(!L)g Imf
xy(!L)g];
(21)
=J
gB!LX
[Imf
xx(!L)g+ Ref
xy(!L)g]:(22)
By inserting the explicit expressions for Gr
0;""()and
Ga
0;##( !L), one obtains
=J2S2
z
!LSd
8X
[f( !L) f()](23)
1
[(
2)2+ ( ")2][(
2)2+ ( # !L)2];
= J
!L d
4X
[f( !L) f()](24)
(
2)2+ ( ")( # !L)
[(
2)2+ ( ")2][(
2)2+ ( # !L)2];
where";#=0gBBc
e =2 =0(!L+JSz)=2are the
energies of the Zeeman levels in this section. In the small
precession frequency regime, !L kBT,
!0and in
the limit of Sz=S!1the expression for the coefficient
is in agreement with Ref. 16.
µLµR=0↑↓↑−ωL↓+ωLFIG. 5: (Color online) Sketch of the electronic energy levels
of the QD in the presence of a molecular spin precessing with
the frequency !Laround an external, constant magnetic field.
The corresponding Zeeman levels are ";#. The precessional
motion of the molecular spin results in emission (absorption)
of energy corresponding to a spin flip from spin up (down) to
spin down (up). Hence, there are only four channels available
for transport.
B. Analysis of the spin-transfer torque
In the case of resonant tunneling in the presence of
a molecular spin precessing in a constant external mag-
netic field, one also needs to take the exchange of spin-
angular momentum between the molecular spin and the
electronic spins into account in addition to the effects
of the external magnetic field. Due to the precessional
motion of the molecular spin, an electron in the QD emit-
ting (absorbing) an energy !Lalso undergoes a spin flip
from spin up (down) to spin down (up), as indicated
by the arrows in Fig. 5. As a result, the levels at en-
ergies";#!Lare forbidden and hence do not con-
tribute to the transport processes. Consequently, there
are only four transport channels, which are located at
energies";#!L. Also in this case, there are elastic
and inelastic tunnel processes. Some of the possible in-
elastic tunnel processes are shown in Fig. 6. These re-
strictions on the inelastic tunnel processes are also vis-
ible in Fig. 3(b), which identically corresponds to the
case of the presence of a precessing molecular spin with
!L= 0:160andJSz= 0:80. Namely, from Eq. (16),
which is equivalent to RefILx(!L)g= ImfILy(!L)g=
JS?
2gB[Ref L
xx(!L)g Imf L
xy(!L)g]andImfILx(!L)g=
RefILy(!L)g=JS?
2gB[Imf L
xx(!L)g+ Ref xy(!L)g],
and from the symmetries of the susceptibilities displayed
in Fig. 3(b), it follows that there are no spin currents at
eV=";#!L.
As was mentioned, the spin currents generate an STT
acting on the molecular spin. A necessary condition
for the existence of an STT, and hence finite values of
the coefficients and in Eqs. (23) and (24), is that
~IL(t)6= ~IR(t)[see Eq. (17)]. This condition is met
by the spin currents generated, e.g., by the inelastic tun-
nel processes shown in Figs. 6(b) and 6(c). These tunnel
processes occur when an electron can tunnel into the QD,
undergo a spin flip, and then tunnel off the QD into ei-7
µLµL
µLµLµL(a)
(f) (e) (d) (c) (b) µL
FIG. 6: (Color online) Sketch of the inelastic spin-tunneling
processes in the QD in the presence of the precessing molec-
ular spin in the field ~Bc=Bc~ ezfor different positions of
the energy levels with respect to the chemical potentials of
the leads,LandR. Only transitions between levels with
the same color (blue or red) are allowed. Different colored
curved arrows (magenta, brown, or green) represent different
processes.
ther lead. From these tunnel processes it is implied that
the Gilbert damping coefficient and the coefficient
can be controlled by the applied bias or gate voltage as
well as by the external magnetic field. If a pair of QD
energy levels, coupled via spin-flip processes, lie within
the bias-voltage window, the spin currents instead fulfill
~IL(t) =~IR(t), leading to a vanishing STT [see Fig. 6(d)].
In Figs. 6(e) and 6(f) the position of the energy levels of
the QD are symmetric with respect to the Fermi levels
of the leads, LandR. When the QD level with energy
"is aligned with L, this simultaneously corresponds to
the energy level #being aligned with R[see Fig. 6(f)].
As a result, a spin-up electron can now tunnel from the
left lead into the level ", while a spin-down electron in
the level#can tunnel into the right lead. These addi-
tional processes enhance the STT compared to that of
the case 6(e).
The two spin-torque coefficients and exhibit a non-
monotonic dependence on the tunneling rates , as can
be seen in Figs. 7, 8, and 9. For !0, it is obvious
that ; !0. In the weak coupling limit !L, the
coefficients and are finite if the Fermi surface energy
of the lead ,fulfills either of the conditions
##+!L (25)
or" !L" (26)
insuchawaythateachconditionissatisfiedbytheFermi
energy of maximum one lead. These conditions are re-
laxed for larger tunnel couplings as a consequence of the
broadening of the QD energy levels, which is also re-
sponsible for the initial enhancement of and with in-
creasing . Notice, however, that and are eventually
suppressed for !L, when the QD energy levels aresignificantly broadened and overlap so that spin-flip pro-
cesses are equally probable in each direction and there
is no net effect on the molecular spin. Physically, this
suppression of the STT can be understood by noticing
that for !La current-carrying electron perceives the
molecular spin as almost static due to its slow precession
compared to the electronic tunneling rates and hence the
exchange of angular momenta is reduced. With increas-
ing tunneling rates, the coefficient becomes negative
before it drops to zero, causing the torque _~Sto oppose
the rotational motion of the spin ~S.
In Fig. 7, the Gilbert damping coefficient and the
coefficient are plotted as functions of the applied bias
voltage at zero temperature. We analyze the case of the
smallestvalueof (redlines), assumingthat !L>0. For
smalleV, all QD energy levels lie outside the bias-voltage
window and there is no spin transport [see Fig. 6(a)].
Hence ; !0. AteV=#the tunnel processes in
Fig. 6(b) come into play, leading to a finite STT and the
coefficient increases while the coefficient has a local
minimum. In the voltage region specified by Eq. (25) for
L, the coefficient approaches a constant value while
the coefficient increases. By increasing the bias voltage
toeV=#+!Lthe tunnel processes in Fig. 6(c) occur,
leading to a decrease of and a local maximum of . For
#+!L<eV <" !L, the coefficients ; !0[see Fig
6(d)]. In the voltage region specified by Eq. (26) for L,
approaches the same constant value mentioned above
while decreases between a local maximum at eV=
" !Land a local minimum at eV=", which approach
the same values as previously mentioned extrema. With
further increase of eV, all QD energy levels lie within the
bias-voltage window and the STT consequently vanishes.
Figure 8 shows the spin-torque coefficients and as
functions of the position of the electronic level 0. An
STT acting on the molecular spin occurs if the electronic
level0is positioned in such a way that the inequalities
(25) and (26) may be satisfied by some values of eV,0
and!L. Again, we analyze the case of the smallest value
of (red curve). For the particular choice of parameters
in Fig. 8, there are four regions in which the inequalities
(25) and (26) are satisfied. Within these regions, ap-
proaches a constant value while has a local maximum
as well as a local minimum. These local extrema occur
when one of the Fermi surfaces is aligned with one of the
energy levels of the QD. For other values of 0, both
and vanish.
The coefficients and are plotted as functions of the
precession frequency !Lin Fig. 9. Here, 0=eV=2and
therefore the positions of the energy levels of the QD are
symmetric with respect to the Fermi levels of the leads,
LandR. Once more, we focus first on the case of the
smallest value of (indicated by the red curve). The
energies of all four levels of the QD depend on !L, i.e.,
~Bc. For!L>0, when the magnitude of the external
magnetic field is large enough, the tunnel processes in
Fig. 6(f) take place due to the above-mentioned sym-
metries. These tunnel processes lead to a finite STT,8
0.00.51.01.52.01234567
eV@e0Da@10-4D
e¯+wLHaL
e¯
ee-wLG=5e0G=3e0G=0.2e0G=0.02e0
0.00.51.01.52.02.5-8-6-4-2024
eV@e0Db@10-4D
e¯+wLHbL
e¯
ee-wLG=5e0G=3e0G=0.2e0G=0.02e0
FIG. 7: (Color online) (a) Gilbert damping coefficient and (b) coefficient as functions of the applied bias voltage eV=
L R, withR= 0, for different tunneling rates at zero temperature. Other parameters are L= R= =2,"= 1:480,
#= 0:520,S= 100,J= 0:010,JSz= 0:80, and!L= 0:160. In the case of the smallest value of (red lines), approaches
a constant value when Llies within the energy range specified by Eqs. (25) and (26). The coefficient has one local minimum
and one local maximum for the same energy range.
-0.50.00.51.01.52.01234567
e0@eVDa@10-4D
mR=eHaL
mR=e-wL
mR=e¯mR=e¯+wL
mL=e-wLmL=e
mL=e¯+wL
mL=e¯G=2.5eVG=1.5eVG=0.1eVG=0.01eV
-0.50.00.51.01.52.0-6-4-2024
e0@eVDb@10-4D
mR=eHbL
mR=e-wLmR=e¯+wL
mR=e¯
mL=e
mL=e-wL
mL=e¯+wL
mL=e¯G=2.5eVG=1.5eVG=0.1eVG=0.01eV
FIG. 8: (Color online) (a) Gilbert damping coefficient and (b) coefficient as functions of the position of the electronic
level0for different tunneling rates at zero temperature. The applied bias voltage is eV=L R, withR= 0. Other
parameters are L= R= =2," 0= 0:24eV,S= 100,J= 0:005eV,JSz= 0:4eV, and!L= 0:08eV. In the case of the
smallest value of (red lines), there are four regions in which the Gilbert damping and the change of the precession frequency
occur. In each of these regions 0satisfies the inequalities (25) and (26), and approaches a constant value, while has one
local maximum and one local minimum.
a maximum for the Gilbert damping coefficient , and
a negative minimum value for the coefficient. As !L
increases, the inequalities of Eqs. (25) and (26) are sat-
isfied and the tunnel processes shown in Fig. 6(e) may
occur. Hence, there is a contribution to the STT, but
as is shown in Eq. (23), the Gilbert damping decreases
with increasing precession frequency. At larger values of
!L, resulting in #+!L=L, the Gilbert damping co-
efficient has a step increase towards a local maximum,
while the coefficient has a local maximum, as a conse-
quence of the enhancement of the STT due to additional
spin-flip processes occurring in this case. For even larger
value of!L, the conditions (25) and (26) are no longerfulfilled and both coefficients vanish. It is energetically
unfavorable to flip the spin of an electron against the
antiparallel direction of the effective constant magnetic
fieldBc
e . Hence, as !Lincreases, more energy is needed
to flip the electronic spin to the direction of the field.
This causes to decrease with increasing !L. Addition-
ally, the larger the ratio !L= , the less probable it is that
spin-angular momentum will be exchanged between the
molecular spin and the itinerant electrons. For !L= 0,
the molecular spin is static, i.e.,_~S= 0. In this case
~T(t) =~0. The coefficient then drops to zero while the
coefficient reaches a negative local maximum which is
close to 0. Both and reach an extremum value for9
-4-20240123
wL@e0Da@10-4DHaL
mL=e-wL
mL=emL=e¯
mL=e¯+wL
G=5e0G=3e0G=0.2e0G=0.02e0
-4-2024-6-4-20
wL@e0Db@10-4DHbL
mL=e¯+wLmL=emL=e¯
mL=e-wL
G=5e0G=3e0G=0.2e0G=0.02e0
FIG. 9: (Color online) (a) Gilbert damping coefficient and (b) coefficient as functions of the precession frequency !L=
gBBcof the spin ~Sof the SMM, with ~Bc=Bc~ ez, for different tunneling rates at zero temperature. The applied bias voltage
iseV=L R= 20, withR= 0. The other parameters are the same as in Fig. 7. In the case of the smallest (red lines),
the coefficient has a step increase towards a local maximum while the coefficient has a local maximum or minimum at a
value of!Lcorresponding to a resonance of Lwith one of the levels in the QD.
large values of at this point. For !L<0and j!Lj
(red lines), at the value of !Lfor whichL=" !L, the
coefficient hasastepincreasetowardsalocalmaximum
whilethecoefficient hasanegativelocalminimum. The
coefficient then decreases with a further decrease of !L
as long as#L" !L. At the value of !Lfor
whichL=#, has another step increase towards a
local maximum while has a maximum value. Accord-
ing to Eq. (23), the Gilbert damping also does not occur
if~Sis perpendicular to ~Bc. In this case .0and the
only nonzero torque component _~S(t)acts in the oposite
direction than the molecular spin’s rotational motion.
IV. CONCLUSIONS
In this paper we have first theoretically studied time-
dependent charge and spin transport through a small
junction consisting of a single-level quantum dot cou-
pled to two noninteracting metallic leads in the pres-
ence of a time-dependent magnetic field. We used the
Keldysh nonequilibrium Green’s functions method to de-
rive the charge and spin currents in linear order with
respect to the time-dependent component of the mag-
netic field with a characteristic frequency !. We then
focused on the case of a single electronic level coupled
via exchange interaction to an effective magnetic field
created by the precessional motion of an SMM’s spin in
a constant magnetic field. The inelastic tunneling pro-
cesses that contribute to the spin currents produce an
STT that acts on the molecular spin. The STT con-
sists of a Gilbert damping component, characterized bythe coefficient , as well as a component, characterized
by the coefficient , that acts as an additional effective
constant magnetic field and changes the precession fre-
quency!Lof the molecular spin. Both and depend
on!Land show a nonmonotonic dependence on the tun-
neling rates . In the weak coupling limit !L,
can be switched on and off as a function of bias and gate
voltages. The coefficient correspondingly has a local
extremum. For !0, both and vanish. Taking
into account that spin transport can be controlled by the
bias and gate voltages, as well as by external magnetic
fields, our results might be useful in spintronic applica-
tions using SMMs. Besides a spin-polarized STM, it may
be possible to detect and manipulate the spin state of an
SMM in a ferromagnetic resonance experiment56–59and
thus extract information about the effects of the current-
induced STT on the SMM. Our study could be com-
plemented with a quantum description of an SMM in a
single-molecule magnet junction and its coherent prop-
erties, as these render the SMM suitable for quantum
information storage.
Acknowledgments
We gratefully acknowledge discussions with Mihajlo
VanevićandChristianWickles. Thisworkwassupported
by Deutsche Forschungsgemeinschaft through SFB 767.
We are thankful for partial financial support by an ERC
Advanced Grant, project UltraPhase of Alfred Leiten-
storfer.10
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0809.2910v1.Spin_transfer_torque_induced_reversal_in_magnetic_domains.pdf | arXiv:0809.2910v1 [cond-mat.other] 17 Sep 2008Spin-transfer torque induced reversal in
magnetic domains
S. MurugeshaM. Lakshmananb
aDepartment of Physics & Meteorology, IIT-Kharagpur, Kharagpu r 721 302, India
bCentre for Nonlinear Dynamics, School of Physics, Bharathid asan University,
Tiruchirappalli 620024, India
Abstract
Using the complex stereographic variable representation f or the macrospin, from
a study of the nonlinear dynamics underlying the generalize d Landau-Lifshitz(LL)
equation with Gilbert damping, we show that the spin-transf er torque is effectively
equivalent to an applied magnetic field. We study the macrosp in switching on a
Stoner particle due to spin-transfer torque on application of a spin polarized cur-
rent. We find that the switching due to spin-transfer torque i s a more effective
alternative to switching by an applied external field in the p resence of damping. We
demonstrate numerically that a spin-polarized current in t he form of a short pulse
can be effectively employed to achieve the desired macro-spin switching.
Key words: Nonlinear spin dynamics, Landau-Lifshitz equation, Spin- transfer
torque, Magnetization reversal
PACS:75.10.Hk, 67.57.Lm, 75.60.Jk, 72.25.Ba
1 Introduction
In recent times the phenomenon of spin-transfer torque has gained much at-
tention in nanoscale ferromagnets[1,2,3]. Electromigration refers to the recoil
linearmomentumimpartedontheatomsofametalorsemiconductor asalarge
current is conducted across. Analogously, if the current is spin-p olarized, the
transfer of a strong current across results in a transfer of spin angular momen-
tum to the atoms. This has lead to the possibility of current induced s witch-
ing of magnetization in nanoscale ferromagnets. With the success o f GMR,
∗Corresponding author. Tel: +91 431 2407093, Fax:+91 431 240 7093
Email address: lakshman@cnld.bdu.ac.in (M. Lakshmanan).
Preprint submitted to Chaos, Solitons and Fractals 7 Septem ber 2021this has immense application potential in magnetic recording devices s uch as
MRAMs[3,4,5,6]. The phenomenon has been studied in several nanomag netic
pile geometries. The typical set up consists of a nanowire[3,7,8,9,10,11 ], or a
spin-valve pillar, consisting of two ferromagnetic layers, one a long f erromag-
neticpinnedlayer, and another small ferromagnetic layer or film, separated
by a spacer conductor layer (see Figure 1). The pinned layer acts a s a reser-
voir for spin polarized current which on passing through the conduc tor and
on to the thin ferromagnetic layer induces an effective torque on th e spin
magnetization in the thin film ferromagnet. A number of experiments have
been conducted on this geometry and the phenomenon has been co nvincingly
confirmed [12,13,14,15]. Although the microscopic quantum theory of the phe-
nomenon is fairly well understood, interestingly the behavior of the average
spin magnetization vector can be described at the semi-classical lev el by the
LL equation with an additional term[16].
j
xyz
Pinned layer Conductor Thin film ConductorS S p
Fig. 1. A schematic diagram of the spin-valve pillar. A thin fi lm ferromagnetic layer
with magnetization Sis separated from long ferromagnetic layer by a conductor. ˆSp
is the direction of magnetization in the pinned region, whic h also acts as a reservoir
for spin polarized current.
From a different point of view, several studies have focused on mag netic pulse
induced switching of the macro-magnetization vector in a thin nanod ot un-
der different circumstances [17,18,19,20]. Several experimental st udies have
also focussed on spin-current induced switching in the presence of a magnetic
field, switching behavior for different choices of the angle of the app lied field,
variation in the switching time, etc., [12,21,22,23,24,25]. A numerical stu dy
on the switching phenomenon induced by a spin current in the presen ce of
a magnetic field pulse has also been investigated very recently in [26]. A s an
extension to two dimensional spin configurations, the switching beh avior on a
vortex has been studied in [27].
In this article, by investigating the nonlinear dynamics underlying the gener-
alized Landau-Lifshitz equation with Gilbert damping, we look at the ex citing
possibility of designing solid state memory devices at the nanoscale, w herein
memory switching is induced using a spin polarized current alone, witho ut
the reliance on an external magnetic field. We compare earlier studie d switch-
ing behavior for the macro-magnetization vector in a Stoner partic le [17] in
the presence of an external magnetic field, and the analogous cas e wherein
the applied field is now replaced by a spin polarized current induced spin -
2transfer torque, i.e., with the thin film in the first case replaced by a s pin
valve pillar. It will be shown that a pulse of spin polarized current is mor e
effective in producing a switching compared to an applied field. In doing so we
rewrite the system in terms of a complex stereographic variable inst ead of the
macro-magnetization vector. This brings a significant clarity in unde rstanding
the nonlinear dynamics underlying the macrospin system. Namely, it w ill be
shown that, in the complex system, the spin-transfer torque is eff ectively an
imaginary applied magnetic field. Thus the spin-transfer term can ac complish
the dual task of precession of the magnetization vector and dissip ation.
The paper is organized as follows: In Section 2 we discuss briefly the m odel
system and the associated extended LL equation. In Section 3 we in troduce
the stereographic mapping of the constant spin magnetization vec tor to a
complex variable, and show that the spin-transfer torque is effect ively an
imaginary applied magnetic field. In Section 4 we present results from our
numerical study on spin-transfer torque induced switching pheno menon of the
macro-magnetization vector, for a Stoner particle. In particular , we study two
different geometries for the free layer, namely, (a) an isotropic sp here and (b)
an infinite thin film. In applications to magnetic recording devices, the typ-
ical read/write time period is of the order of a few nano seconds. We show
that, in order to achieve complete switching in these scales, the spin -transfer
torque induced by a short pulse of sufficient magnitude can be affirma tively
employed. We conclude in Section 5 with a discussion of the results and their
practical importance.
2 The extended LL equation
The typical set up of the spin-valve pillar consists of a long ferromag netic
element, or wire, with magnetization vector pinned in a direction indica ted by
ˆSp, as shown in Figure 1. It also refers to the direction of spin polarizat ion of
the spin current. A free conduction layer separates the pinned ele ment from
the thin ferromagnetic film, or nanodot, whose average spin magne tization
vectorS(t) (of constant magnitude S0) is the dynamical quantity of interest.
The cross sectional dimension of the layers range around 70 −100nm, while
the thickness of the conduction layer is roughly 2 −7nm[3,20]. The free layer
thus acts as the memory unit, separated from the pinned layer cum reservoir
by the thin conduction layer. It is well established that the dynamics of the
magnetization vector Sin the film in the semiclassical limit is efficiently de-
scribed by an extended LL equation[16]. If ˆ m(={m1,m2,m3}=S/S0) is the
unit vector in the direction of S, then
dˆ m
dt=−γˆ m×/vectorHeff+λˆ m×dˆ m
dt−γag(P,ˆ m·ˆSp)ˆ m×(ˆ m׈Sp),(1)
3a≡ℏAj
2S0Ve. (2)
Here,γis the gyromagnetic ratio (= 0 .0176Oe−1ns−1) andS0is the satura-
tion magnetization (Henceforth we shall assume 4 πS0= 8400, the saturation
magnetization value for permalloy). The second term in (1) is the phe nomeno-
logical dissipation term due toGilbert[28] with damping coefficient λ. The last
term is the extension to the LL equation effecting the spin-transfe r torque,
whereAis the area of cross section, jis the current density, and Vis the
volume of the pinned layer.′a′, as defined in (2), has the dimension of Oe, and
is proportional to the current density j.g(P,ˆ m·ˆSP) is given by
g(P,ˆ m·ˆSp) =1
f(P)(3+ˆ m·ˆSp)−4;f(P) =(1+P)3
(4P3/2),(3)
wheref(P)isthepolarizationfactorintroducedbySlonczewski [1],and P(0≤
P≤1) is the degree of polarization of the pinned ferromagnetic layer. F or
simplicity, we take this factor gto be a constant throughout, and equal to 1.
/vectorHeffisthe effective fieldacting onthespin vector due toexchange intera ction,
anisotropy, demagnetization and applied fields:
/vectorHeff=/vectorHexchange+/vectorHanisotropy+/vectorHdemagnetization +/vectorHapplied,(4)
where
/vectorHexchange=D∇2ˆ m, (5)
/vectorHanisotropy=κ(ˆ m·ˆ e/bardbl)ˆ e/bardbl, (6)
∇·/vectorHdemagnetization =−4πS0∇·ˆ m. (7)
Here,κis the strength of the anisotropy field. ˆ e/bardblrefers to the direction of
(uniaxial) anisotropy, In what follows we shall only consider homogen eous
spin states on the ferromagnetic film. This leaves the exchange inte raction
term in (4) redundant, or D= 0, while (7) for /vectorHdemagnetization is readily solved
to give
/vectorHdemagnetization =−4πS0(N1m1ˆ x+N2m2ˆ y+N3m3ˆ z), (8)
whereNi,i= 1,2,3 are constants with N1+N2+N3= 1, and {ˆ x,ˆ y,ˆ z}are the
orthonormal unit vectors. Equation (1) now reduces to a dynamic al equation
for a representative macro-magnetization vector ˆ m.
In this article we shall be concerned with switching behavior in the film p urely
induced by the spin-transfer torque term, and compare the resu lts with earlier
studies on switching due to an applied field [17] in the presence of dissip ation.
Consequently, it will be assumed that /vectorHapplied= 0 in our analysis.
43 Complex representation using stereographic variable
It proves illuminating to rewrite (1) using the complex stereographic variable
Ω defined as[29,30]
Ω≡m1+im2
1+m3, (9)
so that
m1=Ω+¯Ω
1+|Ω|2;m2=−i(Ω−¯Ω)
1+|Ω|2;m3=1−|Ω|2
1+|Ω|2.(10)
For the spin valve system, the direction of polarization of the spin-p olarized
currentˆSpremains a constant. Without loss of generality, we chose this to be
the direction ˆ zin the internal spin space, i.e., ˆSp=ˆ z. As mentioned in Sec.
2, we disregard the exchange term. However, for the purpose of illustration,
we choose /vectorHapplied={0,0,ha3}for the moment but take ha3= 0 in the later
sections. Defining
ˆ e/bardbl={sinθ/bardblcosφ/bardbl,sinθ/bardblsinφ/bardbl,cosθ/bardbl} (11)
and upon using (9) in (1), we get
(1−iλ)˙Ω =−γ(a−iha3)Ω+im/bardblκγ/bracketleftBig
cosθ/bardblΩ−1
2sinθ/bardbl(eiφ/bardbl−
Ω2e−iφ/bardbl)/bracketrightBig
−iγ4π S0
(1+|Ω|2)/bracketleftBig
N3(1−|Ω|2)Ω−N1
2(1−Ω2−|Ω|2)Ω
−N2
2(1+Ω2−|Ω|2)Ω−(N1−N2)
2¯Ω/bracketrightBig
,(12)
wherem/bardbl=ˆ m·ˆ e/bardbl. Using (10) and (11), m/bardbl, and thus (12), can be written
entirely in terms of Ω.
It is interesting to note that in this representation the spin-trans fer torque
(proportional to the parameter a) appears only in the first term in the right
hand side of (12) as an addition to the applied magnetic field ha3but with a
prefactor −i. Thus the spin polarization term can be considered as an effective
applied magnetic field. Letting κ= 0, and N1=N2=N3in (12), we have
(1−iλ)˙Ω =−γ(a−iha3)Ω, (13)
which on integration leads to the solution
Ω(t) = Ω(0) exp( −(a−iha3)γt/(1−iλ))
= Ω(0) exp( −a+λha3
1+|λ|2γt) exp(−iaλ−ha3
1+|λ|2γt). (14)
5The first exponent in (14) describes relaxation, or switching, while t he second
term describes precession. From the first exponent in (14), we no te that the
time scale ofswitching is given by 1 /(a+λha3).λbeing small, thisimplies that
the spin-torque term is more effective in switching the magnetization vector.
Further, letting ha3= 0, we note that in the presence of the damping term
the spin transfer produces the dual effect of precession and diss ipation.
To start with we shall analyze the fixed points of the system for the two cases
which we shall be concerned with in this article: (i) the isotropic spher e char-
acterized by N1=N2=N3= 1/3, and (ii) an infinite thin film characterized
byN1= 0 =N3,N2= 1.
(i) First we consider the case when the anisotropy field is absent, or κ= 0.
From (12) we have
(1−iλ)˙Ω =−aγΩ−iγ4πS0
1+|Ω|2/bracketleftBig
N3(1−|Ω|2)Ω−N1
2(1−Ω2−|Ω|2)Ω
−N2
2(1+Ω2−|Ω|2)Ω−(N1−N2)
2¯Ω/bracketrightBig
.(15)
In the absence of anisotropy ( κ= 0), we see from (15) that the only fixed
point is Ω 0= 0.To investigate the stability of this fixed point we expand (15)
up to a linear order in perturbation δΩ around Ω 0. This gives
(1−iλ)δ˙Ω =−aγδΩ−iγ4πS0[N3−1
2(N1+N2)]δΩ+iγ2πS0(N1−N2)δ¯Ω.(16)
For the isotropic sphere, N1=N2=N3= 1/3, (16) reduces to
(1−iλ)δ˙Ω =−aγδΩ. (17)
We find the fixed point is stable since a >0. For the thin film, N1= 0 =
N3,N2= 1. (16) reduces to
(1−iλ)δ˙Ω =−aγδΩ+iγ2πS0δΩ−iγπS0δ¯Ω. (18)
This may be written as a matrix equation for Ψ ≡(δΩ,δ¯Ω)T,
˙Ψ =MΨ, (19)
whereMis a matrix obtained from (18) and its complex conjugate, whose
determinant and trace are
|M|=(a2+3π2S2
0)γ2
1+λ2;Tr(M) =(−2a−4πS0λ)γ
1+λ2.(20)
Since|M|is positive, the fixed point Ω 0= 0 is stable if Tr|M|<0, or,
(a+2πS0λ)>0.
6The equilibrium point (a), Ω 0= 0, corresponds to ˆ m=ˆ z. Indeed this holds
true even in the presence of an applied field, though we have little to d iscuss
on that scenario here.
(ii) Next we consider the system with a nonzero anisotropy field in the ˆ z
direction. (12) reduces to
(1−iλ)˙Ω =−aγΩ+iκγ(1−|Ω|2)
(1+|Ω|2)Ω−iγ4πS0
(1+|Ω|2)/bracketleftBig
N3(1−|Ω|2)Ω
−N1
2(1−Ω2−|Ω|2)Ω−N2
2(1+Ω2−|Ω|2)Ω−(N1−N2)
2¯Ω/bracketrightBig
.(21)
Here again the only fixed point is Ω 0= 0.As in (i), the stability of the
fixed point is studied by expanding (21) about Ω 0to linear order. Following
the same methodology in (i) we find the criteria for stability of the fixe d
point for the isotropic sphere is ( a+λκ)>0, while for the thin film it is
(a+λ(κ+2πS0))>0.
(iii) With nonzero κin an arbitrary direction the fixed point in general moves
away from ˆ z.
Finally, it is also of interest to note that a sufficiently large current lea ds to
spin wave instabilities induced through spin-transfer torque [31,32]. In the
present investigation, however, we have assumed homogeneous m agnetization
over the free layer, thus ruling out such spin wave instabilities. Rece ntly we
haveinvestigated spinwave instabilitiesoftheSuhltypeinduced bya napplied
alternating field in thin film geometries using stereographic represen tation[30].
It will be interesting to investigate the role of a spin-torque on such instabil-
ities in the spin valve geometry using this formulation. This will be pursu ed
separately.
4 Spin-transfer torque induced switching
Wenowlookattheinterestingpossibilityofeffectingcompleteswitchin g ofthe
magnetization using spin-transfer torque induced by a spin curren t. Numerical
studies on switching effected on a Stoner particle by an applied magne tic field,
or in the presence of both a spin-current and applied field, in the pre sence of
dissipation and axial anisotropy have been carried out recently and switching
has been demonstrated [17,26]. However, the intention here is to ind uce the
same using currents rather than the applied external fields. Also, achieving
such localized magnetic fields has its technological challenges. Spin-t ransfer
torque proves to be an ideal alternative to accomplish this task sinc e, as we
have pointed out above, it can be considered as an effective (albeit c omplex)
7magnetic field. In analogy with ref. [17], where switching behavior due to an
applied magnetic field has been studied we investigate here switching b ehav-
ior purely due to spin-transfer torque, on a Stoner particle. Nume rical results
in what follows have been obtained by directly simulating (12) and makin g
use of the relations in (10), for appropriate choice of parameters . It should
be remembered that (12) is equivalent to (1), and so the numerical results
have been further confirmed by directly numerically integrating (1) also for
the corresponding parameter values. We consider below two sample s differing
in their shape anisotropies, reflected in the values of ( N1,N2,N3) in the de-
magnetization field: a) isotropic sphere, N1=N2=N3= 1/3 and b) a thin
filmN1= 0 =N3,N2= 1. The spin polarization ˆSpof the current is taken
to be in the ˆ zdirection. The initial orientation of ˆ mis taken to be close to
−ˆ z. In what follows this is taken as 170◦fromˆ zin the (z−x) plane. The
orientation of uniaxial anisotropy ˆ e/bardblis also taken to be the initial direction
ofˆ m. With these specified directions for ˆSpandˆ e/bardblthe stable fixed point is
slightly away from ˆ z, the direction where the magnetization ˆ mis expected to
switch in time. A small damping is assumed, with λ= 0.008. The magnitude
of anisotropy κis taken to be 45 Oe. As stated earlier, for simplicity we have
considered the magnetization to be homogeneous.
4.1 Isotropic sphere
It is instructive to start by investigating the isotropic sphere, whic h is char-
acterized by the demagnetization field with N1=N2=N3= 1/3. With these
values for ( N1,N2,N3), (12) reduces to
(1−iλ)˙Ω =−aγΩ+im/bardblκγ/bracketleftBig
cosθ/bardblΩ−1
2sinθ/bardbl(eiφ/bardbl−Ω2e−iφ/bardbl)/bracketrightBig
(22)
A constant current of a= 10Oeis assumed. Using (2), for typical dimen-
sions, this equals a current density of the order 108A/cm2. We notice that
for the isotropic sample the demagnetization field does not play any r ole in
the dynamics of the magnetization vector. In the absence of aniso tropy and
damping the spin-transfer torque term leads to a rapid switching of Sto the
ˆ zdirection. This is evident from (22), which becomes
˙Ω =−aγΩ, (23)
with the solution Ω = Ω 0e−aγt, and the time scale for switching is given by
1/aγ. Figure 2.a shows the trajectory traced out by the magnetization vector
S, for 5 ns, initially close to the −ˆ zdirection, switching to the ˆ zdirection.
Figure 2.b depicts the dynamics with anisotropy but no damping, all ot her
parameters remaining same. While the same switching is achieved, this is
8more smoother due to the accompanying precessional motion. Not e that with
nonzero anisotropy, ˆ zis not the fixed point any more. The dynamics with
damping but no anisotropy (Figure 2.c) resembles Figure 2.a, while Figu re
2.d shows the dynamics with both anisotropy and damping.
-1
-0.5
0
0.5
1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(d)
xyz
-1
-0.5
0
0.5
1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(c)
xyz-1
-0.5
0
0.5
1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b)
xyz
-1
-0.5
0
0.5
1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a)
xyz
Fig. 2. Trajectory of the magnetization vector m, obtained by simulating (12) for
the isotropic sphere ( N1=N2=N3= 1/3), and using the relations in (10), for
a= 10Oe(a)withoutanisotropyanddamping,(b)withanisotropybut nodamping,
(c) without anisotropy but nonzero damping and (d) with both anisotropy and
damping nonzero. The results have also been confirmed by nume rically integrating
(1). The arrows point in the initial orientation (close to −ˆ z) and the direction of the
spincurrent ˆ z. Evolution shownis for aperiodof 5 ns.Note that thefinal orientation
is not exactly ˆ zin the case of nonzero anisotropy ((b) and (d)).
It may be noticed that Figures 2.c and 2.d resemble qualitatively Figure s
2.a and 2.b, respectively, while differing mainly in the time taken for the
switching. It is also noticed that switching in the absence of anisotro py is
faster. Precession assisted switching has been the favored reco rding process in
magnetic memory devices, as it helps in keeping the exchange interac tion at
a minimum[18,19]. The sudden switching noticed in the absence of anisot ropy
essentially refers to a momentary collapse of order in the magnetic m edia.
9This can possibly lead to strong exchange energy and a breakdown o f our
assumption regarding homogeneity of the magnetization field. Howe ver, such
rapid quenching assisted by short high intensity magnetic pulses has in fact
been achieved experimentally [33].
A comparison with reference [17] is in order. There it was noted that with an
applied magnetic field, instead of a spin torque, a precession assiste d switch-
ing was possible only in the presence of a damping term. In Section 3 we
pointed out how the spin transfer torque achieves both precessio n and damp-
ing. Consequently, all four scenarios depicted in Figure 2 show switc hing of
the magnetization vector without any applied magnetic field.
4.2 Infinite thin film
Next we consider an infinite thin film, whose demagnetization field is give n by
N1= 0 =N3andN2= 1. With these values (12) becomes
(1−iλ)˙Ω =−aγΩ+im/bardblκγ/bracketleftBig
cosθ/bardblΩ−1
2sinθ/bardbl(eiφ/bardbl−Ω2e−iφ/bardbl)/bracketrightBig
−iγ4πS0/parenleftbigg1−|Ω|2
1+|Ω|2/parenrightbigg
Ω.(24)
Here againin the absence of anisotropy Ω = 0 is the only fixed point. Th us the
spin vector switches to ˆ zin the absence of damping and anisotropy (Figure
3a). In order to achieve this in a time scale of 5 ns, we find that the value of a
has to be of order 50 Oe. Again the behavior is in stark contrast to the case
induced purely by an applied field[17], wherein the spin vector traces o ut a
distorted precessional trajectory. As in Sec. 4.1, the trajecto ry traced out in
the presence of damping is similar to that without damping (Figure 3c) . The
corresponding trajectories traced out in the presence of anisot ropy are shown
in Figures 3b and 3d.
4.3 Switching of magnetization under a pulsed spin-polariz ed current
We noticed that in the absence of uniaxial anisotropy, the constan t spin polar-
ized current can effect the desired switching to the orientation of ˆSp(Figure
2). This is indeed the fixed point for the system (with no anisotropy) . Fig-
ure 2 traces the dynamics of the magnetization vector in a period of 5 ns, in
the presence of a constant spin-polarized current. However, fo r applications in
magnetic media we choose a spin-polarized current pulseof the form shown
in Figure 4. It may be recalled here that, as was observed in 4.2, with a spin
10-1
-0.5
0
0.5
1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(d)
xyz
-1
-0.5
0
0.5
1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(c)
xyz-1
-0.5
0
0.5
1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b)
xyz
-1
-0.5
0
0.5
1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a)
xyz
Fig. 3. Trajectory of the magnetization vector Sin a period of 5 ns, obtained as
earlier by numerically simulating (12), and also confirming with (1), with demagne-
tization field with by N1= 0 =N3,N2= 1 and a= 50Oe, and all the parameter
values are as earlier. As in Figure 1, the ˆSand initial orientation are indicated
by arrows. (a) Without anisotropy or damping, (b) nonzero an isotropy but zero
damping, (c) without anisotropy but nonzero damping and (d) both anisotropy and
damping nonzero. As earlier, in the presence of nonvanishin g anisotropy, the fixed
point is not the ˆ zaxis.
polarized current of sufficient magnitude, the switching time can inde ed be
reduced. We choose a pulse, polarized as earlier along the ˆ zdirection, with
rise time and fall time of 1 .5ns, and a pulse width, defined as the time interval
between half maximum, of 4 ns. We assume the rise and fall phase of the pulse
to be of a sinusoidal form, though, except for the smoothness, t he switching
phenomenon is independent of the exact form of the rise or fall pha se.
In Figures 5 and 6, we show trajectories of the spin vector for a pe riod of
25ns, for the two different geometries, the isotropic sphere and a thin fi lm.
The action of the spin torque pulse, as in Figure 4, is confined to the fi rst
5ns. We notice that, with the chosen value of a, this time period is enough
11Rise time Fall time
Pulse width
0 20 40 60 80 100 120 140 160
0 1 2 3 4 5a (Oe)
time (ns)
Fig. 4. Pulse form showing the magnitude of a, or effectively the spin-polarized
current. The rise and fall phase are assumed to be of a sinusoi dal form. The rise
and fall time are taken as 1 .5ns, and pulse width 4 ns. The maximum magnitude
ofais 150Oe.
to effect the switching. In the absence of anisotropy, the directio n ofˆSpis
the fixed point. Thus a pulse of sufficient magnitude can effect a switc hing in
the desired time scale of 5 ns. From our numerical study we find that in order
for this to happen, the value of ahas to be of order 150 Oe, or, from (2),
a current density of order 109A/cm2, a magnitude achievable experimentally
(see for example [34]). Comparing with sections 4.1 and 4.2, we note th at the
extra oneorder ofmagnitude in current density isrequired due to t he duration
of the rise and fall phases of the pulse in Figure(4). Here again we co ntrast
the trajectories with those induced by an applied magnetic field [17], w here
the switching could be achieved only in the presence of a uniaxial aniso tropy.
In Figure 5b for the isotropic sphere with nonzero crystal field anis otropy,
we notice that the spin vector switches to the fixed point near ˆ zaxis in the
first 5ns. However the magnetization vector precesses around ˆ zafter the
pulse has been turned off. This is because in the absence of the spin- torque
term, the fixed point is along ˆ e/bardbl, the direction of uniaxial anisotropy. Due to
the nonzero damping term, the spin vector relaxes to the direction ofˆ e/bardblas
time progresses. The same behavior is noticed in Figure 6b for the th in film,
although the precessional trajectory is a highly distorted one due to the shape
anisotropy.
12-1
-0.5
0
0.5
1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b)
xyz
-1
-0.5
0
0.5
1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a)
xyz
Fig. 5. Evolution of the magnetization vector Sin a period of 25 nsinduced by the
spin-polarized current pulse in Figure 4, (a) with and (b) wi thout anisotropy for
the isotropic sample, with N1=N2=N3= 1/3 all other parameters remaining
same. A nonzero damping is assumed in both cases. The current pulse acts on the
magnetization vector for the first 5 ns. In both cases switching happens in the first
5ns. In the presence of nonzero anisotropy field, (b), the magnet ization vector
precesses to the fixed point near ˆ z.
-1
-0.5
0
0.5
1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b)
xyz
-1
-0.5
0
0.5
1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a)
xyz
Fig. 6. Evolution of the magnetization vector Sin a period of 25 nsinduced by
the spin-polarized current pulse in Figure 4, for a infinite t hin film sample, with
N1= 0 =N3, andN2= 1, all other parameters remaining same, along with a
nonzero damping. (a) Without anisotropy and (b) with anisot ropy. As in Figure 5,
switching happens in the first 5 ns.
5 Discussion and conclusion
We have shown using analytical study and numerical analysis of the n onlinear
dynamics underlying the magnetization behavior in spin-valve pillars th at a
very effective switching of macro-magnetization vector can be ach ieved by a
spin transfer-torque, modeled using an extended LL equation. Re writing the
13extended LL equation using the complex stereographic variable, we find the
spin-transfer torque term indeed acts as an imaginary applied field t erm, and
can lead to both precession and dissipation. It has also been pointed out why
the spin-torque term is more effective in switching the magnetization vector
compared to the applied field. On application of a spin-polarized curre nt the
average magnetization vector in the free layer was shown to switch to the
direction of polarization of the spin polarized current. For a consta nt current,
the required current density was found to be of the order of 108A/cm2. For
recording in magnetic media, switching is achieved using a stronger po larized
current pulse of order 109A/cm2. Currents of these magnitudes have been
achieved experimentally.
Acknowledgements
The work forms part of a research project sponsored by the Dep artment of
Science andTechnology, Government ofIndia anda DSTRamannaFe llowship
to M. L.
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16 |
1506.01303v3.Antidamping_spin_orbit_torque_driven_by_spin_flip_reflection_mechanism_on_the_surface_of_a_topological_insulator__A_time_dependent_nonequilibrium_Green_function_approach.pdf | Antidamping spin-orbit torque driven by spin-
ip re
ection mechanism on the surface
of a topological insulator: A time-dependent nonequilibrium Green function approach
Farzad Mahfouzi,1,Branislav K. Nikoli c,2and Nicholas Kioussis1
1Department of Physics, California State University, Northridge, CA 91330-8268, USA
2Department of Physics and Astronomy, University of Delaware, Newark, DE 19716-2570, USA
Motivated by recent experiments observing spin-orbit torque (SOT) acting on the magnetization
~ mof a ferromagnetic (F) overlayer on the surface of a three-dimensional topological insulator (TI),
we investigate the origin of the SOT and the magnetization dynamics in such systems. We predict
that lateral F/TI bilayers of nite length, sandwiched between two normal metal leads, will generate
a large antidamping-like SOT per very low charge current injected parallel to the interface. The
large values of antidamping-like SOT are spatially localized around the transverse edges of the F
overlayer. Our analysis is based on adiabatic expansion (to rst order in @~ m=@t ) of time-dependent
nonequilibrium Green functions (NEGFs), describing electrons pushed out of equilibrium both by the
applied bias voltage and by the slow variation of a classical degree of freedom [such as ~ m(t)]. From it
we extract formulas for spin torque and charge pumping, which show that they are reciprocal e ects
to each other, as well as Gilbert damping in the presence of SO coupling. The NEGF-based formula
for SOT naturally splits into four components, determined by their behavior (even or odd) under the
time and bias voltage reversal. Their complex angular dependence is delineated and employed within
Landau-Lifshitz-Gilbert simulations of magnetization dynamics in order to demonstrate capability
of the predicted SOT to eciently switch ~ mof a perpendicularly magnetized F overlayer.
PACS numbers: 72.25.Dc, 75.70.Tj, 85.75.-d, 72.10.Bg
I. INTRODUCTION
The spin-orbit torque (SOT) is a recently discovered
phenomenon1{4in ferromagnet/heavy-metal (F/HM)
lateral heterostructures involves unpolarized charge cur-
rent injected parallel to the F/HM interface induces
switching or steady-state precession5of magnetization
in the F overlayer. Unlike conventional spin-transfer
torque (STT) in spin valves and magnetic tunnel junc-
tion (MTJs),6{8where one F layer acts as spin-polarizer
of electrons that transfer torque to the second F layer
when its free magnetization is noncollinear to the direc-
tion of incoming spins, heterostructures exhibiting SOT
use a single F layer. Thus, in F/HM bilayers, spin-orbit
coupling (SOC) at the interface or in the bulk of the
HM layer is crucial to spin-polarized injected current
via the Edelstein e ect (EE)9,10or the spin Hall e ect
(SHE),11,12respectively.
The SOT o ers potentially more ecient magnetiza-
tion switching than achieved by using MTJs underlying
present STT-magnetic random access memories (STT-
MRAM).13Thus, substantial experimental and theo-
retical e orts have been focused on identifying physi-
cal mechanisms behind SOT whose understanding would
pave the way to maximize its value by using optimal
materials combinations. For example, very recent ex-
periments14{16have replaced HM with three-dimensional
topological insulators (3D TIs).17The TIs enhance18{20
(by a factor ~vF= R, wherevFis the Fermi velocity on
the surface of TI and Ris the Rashba SOC strength22?
at the F/HM interface) the transverse nonequilibrium
spin density driven by the longitudinal charge current,
which is responsible for the large eld-like SOT compo-
nent20,23observed experimentally.14{16
FIG. 1. (Color online) Schematic view of F/TI lateral bilayer
operated by SOT. The F overlayer has nite length LF
xand~ m
is the unit vector along its free magnetization. The TI layer
is attached to two N leads which are semi-in nite in the x-
direction and terminate into macroscopic reservoirs. We also
assume that F and TI layers, as well as N leads, are in nite in
they-direction. The unpolarized charge current is injected by
the electrochemical potential di erence between the left and
the right macroscopic reservoirs which sets the bias voltage,
L R=eVb. We mention that the results do not change if
the TI surface is covered by the F overlayer partially or fully.
Furthermore, recent experiments have also observed
antidamping-like SOT in F/TI heterostructures with
surprisingly large gure of merit (i.e., antidamping
torque per unit applied charge current density) that sur-
passes14{16those measured in a variety of F/HM het-
erostructures. This component competes against the
Gilbert damping which tries to restore magnetization
to equilibrium, and its large gure of merit is, there-
fore, of particular importance for increasing eciency of
magnetization switching. Theoretical understanding ofarXiv:1506.01303v3 [cond-mat.mes-hall] 24 Jan 20162
the physical origin of antidamping-like SOT is crucial to
resolve the key challenge for anticipated applications of
SOT generated by TIs|demonstration of magnetization
switching of the F overlayer at room temperature (thus
far, magnetization switching has been demonstrated only
at cryogenic temperature15).
However, the microscopic mechanism behind its large
magnitude14{16and ability to eciently (i.e., using as lit-
tle dc current density as possible) switch magnetization15
remains under scrutiny. For example, TI samples used
in these experiments are often unintentionally doped, so
that bulk charge carriers can generate antidamping-like
SOT via rather large24SHE (but not sucient to explain
all reported values14,15). The simplistic picture,14in
which electrons spin-polarized by the EE di use into the
F overlayer14to deposit spin angular momentum within
it, cannot operate in technologically relevant F overlayers
of'1 nm thickness16or explain complex angular depen-
dence2,15,25typically observed for SOT. The Berry cur-
vature mechanism25,26for antidamping-like SOT applied
to lateral F/TI heterostructures predicts its peculiar de-
pendence on the magnetization orientation,27vanishing
when magnetization ~ mis parallel to the F/TI interface.
This feature has thus far not been observed experimen-
tally,15and, furthermore, it makes such antidamping-like
SOT less ecient27(by requiring larger injected currents
to initiate magnetization switching) than standard SHE-
driven3,4antidamping-like SOT.
We note that the recent experimental14{16and theo-
retical14,27studies of SOT in lateral F/TI bilayer have
focused on the geometry where an in nite F overlayer
covers an in nite TI layer. Moreover, they assume14,27?
purely two-dimensional transport where only the top sur-
face of the TI layer is explicitly taken into account by the
low-energy e ective (Dirac) Hamiltonian supplemented
by the Zeeman term due to the magnetic proximity ef-
fect. On the other hand, transport in realistic TI-based
heterostructures is always three-dimensional, with unpo-
larized electrons being injected from normal metal con-
tacts, re
ected from the F/TI edge to
ow along the sur-
face of the TI in the yz-plane and then along the bottom
TI surface in Fig. 1. In fact, electrons also
ow within a
thin layer (of thickness .2 nm in Bi 2Se3as the prototyp-
ical TI material) underneath the top and bottom surfaces
due to top and bottom metallic surfaces of the TI doping
the bulk via evanescent wave functions.18Therefore, in
this study we consider more realistic and experimentally
relevant28F/TI bilayer geometries, illustrated in Fig. 1,
where the TI layer of nite length LTI
xand nite thick-
nessLTI
zis (partially or fully) covered by the F overlayer
of lengthLF
x. The two semi-in nite ideal N leads are di-
rectly attached to the TI layer. we should mention that
the result does not depend on the length of TI layer that
is covered by the FM.
Our principal results are twofold and are summarized
as follows:
(i)Theoretical prediction for SOT: We predict that the
geometry in Fig. 1 will generate large antidamping-likeSOT per low injected charge current. By studying spatial
dependence of the SOT (see Fig. 4), we show that in a
clean FM/TI interface the electrons exert anti-damping
torque on the FM as they enter into the interface and un-
less interfacial roughness or impurities are included the
torque remains mainly concentrated around the edge of
the interface. Although the exact results show strong
nonperturbative features, based on second order pertur-
bation we present two di erent interpretations showing
that the origin of the antidamping SOT relies on the
spin-
ip re
ection of the chiral electrons injected into
the FM/TI interface. Its strong angular dependence (see
Fig. 2), i.e., dependence on the magnetization direction
~ m, o ers a unique signature that can be used to distin-
guish it from other possible physical mechanisms. By
numerically solving the Landau-Lifshitz-Gilbert (LLG)
equation in the macrospin approximation, we demon-
strate (see Figs. 5 and 6) that the obtained SOT is ca-
pable of switching of a single domain magnetization of a
perpendicularly magnetized F overlayer with bias voltage
in the oder of the Magneto-Crystaline Anisotropy (MCA)
energy.
(ii)Theoretical formalism for SOT: The widely used
quantum (such as the Kubo formula25{27,30) and semi-
classical (such as the Boltzmann equation31) transport
approaches to SOT are tailored for geometries where an
in nite F layer covers an in nite TI or HM layer. Due to
translational invariance, the nonequilibrium spin density
~Sinduced by the EE on the surface of TI or HM layer
has uniform orientation ~S= (0;Sy;0) [in the coordinate
system in Fig. 1], which then provides reference direc-
tion for de ning eld-like, f~ m^y, and antidamping-
like, ad~ m(~ m^y), components of SOT. In order to
analyze spatial dependence of SOT in the device geome-
try of Fig. 1, while not assuming anything a priori about
the orientation of eld-like and antidamping-like compo-
nents of SOT, we employ adiabatic expansion32of time-
dependent nonequilibrium Green functions (NEGFs)33,34
to derive formulas for torque, charge pumping35,36and
Gilbert damping37in the presence of SOC. The NEGF-
based formula for SOT naturally splits into four compo-
nents, determined by their behavior (even or odd) under
the time and bias voltage reversal. This gives us a general
framework in quantum mechanics to analyze the dissi-
pative (antidamping-like) and nondissipative ( eld-like)
force (torque) vector elds for a set of canonical variables
(magnetization directions). Their angular (see Fig. 2)
and spatial (see Fig. 4) dependence shows that although
eld-like and antidamping-like SOTs are predominantly
along the~ m^yand~ m(~ m^y) directions, respectively,
they are not uniform and can exhibit signi cant devia-
tion from the trivial angular dependence de ned by these
cross products [see Fig. 2(h)].
The paper is organized as follows. In Sec. II, we present
the adiabatic expansion of time-dependent NEGFs, in a
representation that is alternative to Wigner representa-
tion34(usually employed for this type of derivation32),
and derive expressions for torque, charge pumping and3
Gilbert damping. In Sec. III, we decompose the NEGF-
based expression for SOT into four components, deter-
mined by their behavior (even or odd) under the time and
bias voltage reversal, and investigate their angular de-
pendence. Section IV discusses the angular dependence
of the zero-bias transmission function which identi es the
magnetization directions at which substantial re
ection
occurs. In Sec. V, we study spatial dependence of SOT
components and discuss their physical origin. Section VI
presents LLG simulations of magnetization dynamics in
the presence of predicted SOT, as well as a switching
phase diagram of the magnetization state as a function
of the in-plane external magnetic eld and SOT. We con-
clude in Sec. VII.
II. THEORETICAL FORMALISM
We rst describe the time-dependent Hamiltonian
model, H(t) =H0+U(t), of the lateral F/TI heterostruc-
ture in Fig. 1. Here H0is the minimal tight-binding
model for 3D TIs like Bi 2Se3on a cubic lattice of spacing
awith four orbitals per site.38The thickness, LTI
z= 8a
of the TI layer is sucient to prevent hybridization be-
tween its top and bottom metallic surface states.18The
time-dependent potential
U(t) = surf1m~ m(t)~=2; (1)
depends on time through the magnetization of the F over-
layer which acts as the slowly varying classical degree of
freedom. Here ~ m(t) is the unit vector along the direc-
tion of magnetization, surf= 0:28 eV is the proximity
induced exchange- eld term and 1mis a diagonal matrix
with elements equal to unity for sites within the F/TI
contact region in Fig. 1 and zero elsewhere. The semi-
in nite ideal N leads in Fig. 1 are taken into account
through the self-energies33,34L;Rcomputed for a tight-
binding model with one spin-degenerate orbital per site.
The details of how to properly couple L;RtoH0, while
taking into account that the spin operators for electrons
on the Bi and Se sublattices of the TI are inequivalent,39
can be found in Ref. 40.
Within the NEGF formalism33,34the advanced
and lesser GFs matrix elements of the tight-binding
Hamiltonian, H0, are de ned by Gii0;oo0;ss0(t;t0) =
i(t t0)hf^cios(t);^cy
i0o0s0(t0)gi, andG<
ii0;oo0;ss0(t;t0) =
ih^cy
i0o0s0(t0)^cios(t)i, respectively. Here, ^ cy
ios(^cios) is the
creation (annihilation) operator for an electron on site, i,
with orbital, o, and spins, respectively,h:::idenotes the
nonequilibrium statistical average, and ~= 1 to simplify
the notation. These GFs are the matrix elements of the
corresponding matrices GandG<used throughout the
text.
Under stationary conditions, the two GFs depend on
the di erence of the time arguments, t t0, and can
be Fourier transformed to energy. In the strictly adi-
abatic limit one can employ41the same retarded GF,Gt(E) = [E H(t) L R] 1, as under stationary
conditions, but where the GF depends parametrically on
time (denoted by the subscript t) and is computed for the
frozen-in-time con guration of U(t). However, even for
slow evolution of ~ m(t) corrections32to the adiabatic GF
are needed to describe dissipation e ects such as Gilbert
damping or the charge current which can be pumped35
by the dynamics of ~ m(t).
The so-called adiabatic expansion, which yields correc-
tions beyond the strictly adiabatic limit, is traditionally
performed using the Wigner representation34in which
the fast and slow time scales are easily identi able.32The
slow motion implies that the NEGFs vary slowly with the
central time tc= (t+t0)=2 while they change fast with
the relative time tr=t t0. By expanding the Wigner
transformation of NEGFs
G(<)
W(E;tc) =Z1
1dtreiEtrG(<)
tc+tr
2;tc tr
2
;
(2)
in the central time tcwhile keeping only terms contain-
ing rst-order derivatives @=@tc(due to the slow varia-
tion withtc) gives the rst-order correction beyond the
strictly adiabatic limit.32This route requires to han-
dle complicated expressions resulting from the Wigner
transform applied to convolutions of the type C(t1;t2) =R
dt3C1(t1;t3)C2(t3;t2).
Here we provide an alternative derivation of the rst-
order nonadiabatic correction. Namely, we consider t
(observation time) and t t0(relative time) as the nat-
ural variables to describe the time evolution of NEGFs
and then perform the following Fourier transform42
G(t;t0) =Z1
1dE
2eiE(t t0)G(E;t): (3)
The standard equations of motion for G(t;t0) and
G<(t;t0) are cumbersome to manipulate42,43or solve
numerically,44so they are usually transformed to some
other representation.35Here we replace G(t;t0) in the
standard equations of motion with the rhs of Eq. (3) to
arrive at:
E i@
@t
1 H0 U(t)
E i@
@t
G(E;t) =1;
(4)
and
G<(t;t0) =ZdE
2G(E;t)<(E)Gy(E;t0)eiE(t t0):(5)
For the two-terminal heterostructure in Fig. 1 in the
elastic transport regime,33,34(E) =L(E) +R(E)
and<(E) =ifL(E) L(E) +ifR(E) R(E), where
L;R=i(L;R y
L;R). The Fermi-Dirac distribution
functions of electrons in the macroscopic reservoirs into
which the left and right N leads terminate are fL;R(E) =
f(E L;R), where the di erence between the electro-
chemical potentials, L;R=EF+eVL;R, de nes the bias
voltageeVb=L R.4
Using the following identity
X
=L;RiG Gy= (G Gy) +i@
@E
G@U
@tGy
+O@2U
@t2
; (6)
the lesser GF to rst order in small @U(t)=@tand in the low bias Vbregime (i.e., the linear-response transport regime)
can be expressed as,
G<(t;t)'ZdE
20
@[G(E;t) Gy(E;t)]f(E) +X
=L;Rf0eV Gt Gy
t+if0Gt@U(t)
@tGy
t1
A; (7)
where, the rst term corresponds to the density matrix of the equilibrium electrons occupying the time dependent
single particle states, while the second and third terms describe the density matrix of the excited (nonequilibrium)
electrons occupying the states close to the Fermi energy due to the bias voltage and time dependent term in the
Hamiltonian, respectively. The retarded GF in the rst term in Eq. (7) expanded to rst order in @U(t)=@tis of the
form
G(E;t)'Gt+i@Gt
@E@U(t)
@tGt: (8)
The lesser GF determines the time-dependent nonequilibrium density matrix
(t) =1
iG<(t;t); (9)
from which we determine the time-dependent expectation values of physical observables, A(t) = Tr [(t)^A]. In
particular, the relevant quantities for the heterostructure in Fig. 1 are the charge current
I (t) =eZdE
2iTr
(E)G<(t;t) +f (E) (E)[G(E;t) Gy(E;t)]
=e2
2Z
dEf0(E)8
<
:X
(V V )T (E) surf
2eX
i@mi
@tT i(E)9
=
;; (10)
and the spin density
si(t) =ZdE
2iTr
i1mG<
=ZdE
28
<
:X
f (E)Ti (E) surf
2X
jf0(E)@mj
@tTij(E)9
=
;; (11)
where ; 2fL;Rgandi;j2fx;y;zg.
The \trace-formulas" in Eqs. (10) and (11)
T (E) = Trh
Gt Gy
ti
; (12a)
T i(E) = Trh
1miGy
t Gti
; (12b)
Ti (E) = Trh
1miGt Gy
ti
; (12c)
Tij(E) = Trh
1mi(Gy
t Gt)1mj(Gt Gy
t)i
;(12d)
determine charge current45due toVb, charge current
pumped35by the dynamics of ~ m(t) in the presence of
SOC, the spin torque, and Gilbert damping tensor, re-
spectively. In the expression for the spin density we ig-
nore the antisymmetric part of Tijwhich corresponds to
the renormalization of the precession frequency of mag-netization dynamics.32Note thatT iin Eq. (12)(b) and
its time-reversal Ti in Eq. (12)(c) reveal a reciprocal46
relation between charge pumping by magnetization dy-
namics in the presence of SOC and current-driven SOT
ateach instant of time t.
The spin density in Eq. (11) enters into the LL equa-
tion for the magnetization dynamics
@~ m
@t= surf
2~ m~ s(t): (13)
The rst term in Eq. (11) generates the spin torque in
Eq. (13) which has three contributions: ( i) an equilib-
rium component responsible for the interlayer exchange
interaction in the presence of a second F layer and/or
magneto-crystalline anisotropy (MCA) in the presence of
SOC; ( ii) a bias-induced eld-like torque modifying the5
equilibrium interlayer exchange and MCA elds; and ( iii)
a damping (antidamping)-like torque describing angular
momentum loss (gain) due to the
ux of Fermi surface
electrons.
III. ANGULAR DEPENDENCE OF SOT
COMPONENTS
In order to understand the di erent contributions to
Ti (E), we decompose it into even (e) or odd (o) terms
under time-reversal Gt7!Gy
t:
Ti
e(E) = [Ti (E) +T i(E)]=2; (14)
Ti
o(E) = [Ti (E) T i(E)]=2: (15)
SinceP
Ti
o(E) = 0, the contribution of the odd com-
ponent to the equilibrium spin density in Eq. (11) van-
ishes identically, while its nonzero values appear only for
Ewithin the bias window around EF. This motivates
further splitting of Ti (E) into four components for the
case of two-terminal devices
Ti
e;(E) =TiL
(E) +TiR
(E)
2; (16a)
Ti
o;(E) =TiL
(E) TiR
(E)
2; (16b)
where the rst and second subscripts denote their be-
havior (even or odd) under bias reversal Vb7! Vband
time reversal, respectively. The corresponding four com-
ponents of torque are determined by
~Te;=ZdE
2[fL(E) +fR(E)]~ e;(E); (16c)
~To;=ZdE
2[fL(E) fR(E)]~ o;(E);(16d)
where the energy-resolved torque is given by
~ ;(E) = surf
2~ m~T;(E); (17)
and;2fe;og. The terms ~To;oand~To;eare non-zero
only in nonequilibrium driven by Vb6= 0, and depend on
electronic states in the bias voltage window around the
Fermi energy (or on the Fermi surface states in the linear-
response regime where integrals are avoided by multiply-
ing integrand by eVb). The term ~Te;o0 is zero, while
~Te;eis nonzero also in equilibrium and, therefore, depends
on all occupied electronic states.
Figures 2(a-c) show the netvector eld (summed over
all sites of the F overlayer) of ~ ;(EF) at zero temper-
ature for di erent directions of ~ mon the unit sphere.
Their angular behavior reveals that: ( i)~ o;oshown in
Fig. 4(a) is the eld-like SOT generated by the EE, with
predominant orientation along the ~ m^y-direction; ( ii)
~ o;ein Fig. 4(b) is the antidamping-like SOT with pre-
dominant orientation along the ~ m(~ m^y)-direction;
and ( iii)~ e;ein Fig. 4(c) along the ~ m^zdirection isthe eld-like component whose angular dependence be-
haves approximately as 2( ~ m^z)j~ m^zjsin(2) typical
for torque components generated by the MCA eld.2
The corresponding angular dependence of the net
i
o;o(i2fx;y;zg),j~ o;ej, and i
e;ealong the solid trajecto-
ries shown in Figs. 2(a-c) are plotted in Figs. 2(d-f),
respectively. We nd that the magnitude of the max-
imum antidamping-like SOT is about a factor of four
larger than that of the eld-like SOT. Additionally, the
eld-like SOT peaks when the magnetization is in-plane.
In contrast, the antidamping-like SOT peaks when the
magnetization is out of plane, which can be attributed to
the gap opening of the Dirac cone which in turn enhances
the re
ection (see Sec. IV) at the lateral boundaries of
the F overlayer.
Note that the magnitude of the netSOT components
shown in Figs. 2(g-i) exhibits strong angular dependence
because of the large SOC on the TI surface similar to that
found in F/HM heterostructures when the Rashba SOC
at the interface is suciently strong.25,26In particular,
the signi cant deviation of the angular dependence of the
antidamping-like SOT from the trivial j~ m(~ m^y)jbe-
havior in the limit of ~ m!^y, shown quantitatively in
Fig. 2(h), indicates its nonperturbative variation with
respect to the magnetization direction. A similar nonper-
turbative angular behavior (i.e., strong deviation from
the standard/sin2dependence on the precession cone
angle) has been found for adiabatic charge pumping
from a precessing F overlayer attached to the edge of 2D
TIs47,48or to the surface of 3D TIs.36
IV. ANGULAR DEPENDENCE OF
TRANSMISSION FUNCTION
Figure 3 shows the transmission function TRL(EF) in
Eq. (12a) for the heterostructure in Fig. 1 versus the ori-
entation of ~ mon the unit sphere. We nd that the charge
current determined by TRL(E) is smallest49when~ mk^z
or~ mk^x. This is due to the re
ection of Dirac elec-
trons on the TI surface from the lateral boundaries of
the F overlayer. Underneath the F overlayer, the ex-
change eld, surf~ m~=2, induced by the magnetic
proximity e ect is superimposed on the Dirac cone sur-
face dispersion. This opens an energy gap surfwhen
~ mk^z(or smaller gap surfcosformz6= 0) at the DP
of the TI region underneath the F overlayer, which in
turn gives rise to strong electronic re
ection when ~ mk^z.
For~ mk^x, there is no energy gap at the DP and the
Dirac cone e ectively shifts away from the center of the
Brillouin zone due to proximity exchange eld. Neverthe-
less electrons polarized by the EE along the y-axis re
ect
from the magnetization pointing along the x-axis.6
(a) (b) (c)
(d) (e) (f)
(g) (h) ( i)
-0.20.00.2i
e,e(EF)(1/a) x
y
z
0 45 90 135 1800.00.10.20.3|e,e(EF)|/sin 2 (1/a)
Angle (deg)
-5-4-3-2-1010-2i
o,o(EF)(1/a) x
y
z
0 45 90 135 180024681010-2|o,o(EF)|/sin (1/a)
Angle (deg)
0.00.10.2|o,e(EF)|(1/a) =0o
=45o
=90o
0 45 90 135 1800.00.10.20.3|o,e(EF)|/sin (1/a)
Angle (deg)=90o=45o=0o
FIG. 2. (Color online) (a){(c) The vector eld of SOT components ~ ;(EF), de ned by Eq. (16), for di erent directions of
~ mon the unit sphere. The angular behavior in (a) and (c) shows that ~ o;oand~ e;ebehave as eld-like torques, while that in
(b) shows that ~ o;ebehaves as antidamping-like torque. Cartesian components [(d)-(f)] and magnitude of ~ ;[(g){(i)] along
the trajectories denoted by solid lines in the corresponding panels (a)-(c). The magnitude of ~ ;is divided by:j~ m^yjin (g);
j~ m(~ m^y)jin (h) and 2( ~ m^z)j~ m^zjin (i).
V. SPATIAL DEPENDENCE OF SOT
COMPONENTS AND PHYSICAL ORIGIN OF
ANTIDAMPING-LIKE SOT
Figure 4(a) demonstrates that large values of
antidamping-like SOT from Fig. 2 are spatially localized
around the transverse edges of the F overlayer, for Fermi
energy inside and outside of the surface state gap induced
by out-of-plane magnetic exchange coupling. While con-
ductance in this system is close to zero at the Dirac point,
we observe that the anti-damping torque does not de-
pend strongly on the Fermi energy. This suggests a high
eciency of SOT per injected current for the Fermi en-
ergy close to the Dirac point. For the eld-like SOT
in Fig. 4(b) we see the torque independent of the co-
ordinate in the entire F/TI contact region, as expected
from the phenomenology of the EE. In Fig. 4(c) we plot
the contribution of the Fermi energy electrons to the
FM/TI interface induced MCA eld. Even though to-
tal i
e;o(EF)0, its spatially-resolved value plotted in
Fig. 4(d) is nonzero which can be removed by perform-
ing a proper gauge transformation.
To understand the origin of the anti-damping SOT let
us nd an expression for the average of SOT around a xed axes~ m=~ m0+(~ m?cos( )+~ m0~ m?sin( ))with
small cone angle . By applying a rotation operator we
can align the xed axes along z-axis such that ~ m0= ^ez
and~ m?= ^ex. In this case we have
hTzi =surf
2=Zd
2ZdE
2X
f ei T ; ;(18)
where,T ; =Tx iTy and=refers to the imaginary
part. Expanding the GFs in Eq. 12(b) to the lowest order
with respect to e i , we obtain,
T ; =surf
4e i Trh
1m Gt1m+Gt Gy
t(19)
+1m Gt Gy
t1m+Gy
ti
:
Plugging this expression into Eq.(18), and using the iden-
tity,Gt Gy
t=iP
Gt Gy
t, in linear bias voltage
regime we obtain,
hTzi =Vb2
surf2
16Tr[""
L1m##
R1m ""
R1m##
L1m];(20)
where, = iG<
=Gt Gy
t, corresponds to the
density matrix inside the F overlayer for the electrons7
(a)
TRL(EF)×10−2
4.5
4
3.5
3
𝑥𝑦𝑧
FIG. 3. (Color online) The transmission function TRL(EF)
in Eq. (12a) for two-terminal heterostructure shown in Fig. 1
at di erent directions of magnetization ~ mon the unit sphere.
The Fermi energy is set at EF= 3:1 eV (which is 0 :1 eV above
the DP).
02 04 0-1010
2 04 0-505-10-
20τ
x,ze
,e(EF)=0×10-4×10-2τye
,e(EF) (1/a)L
ongitudinal Coordinate xF/TI (a)τxe
,o(EF) (1/a)m=(0,0,1)τ
y,ze
,o(EF)=0τy,zo
,o(EF)=0τ
x,zo
,e(EF)=0 EF=3.3eV
EF=3.1eV(
d)( c)(b)( a)×
10-2τyo
,e(EF) (1/a)×10-2τxo
,o(EF) (1/a)
FIG. 4. (Color online) (a){(d) Spatial dependence of SOT
components, i
;(EF) (i2fx;y;zgand;2fe;og), per
unit length, for Fermi energy inside the magnetization in-
duced gap around Dirac point ( EF= 3:1eV) and outside
the gap (EF= 3:3eV), for~ mk^zin Fig. 1. Their physical
meaning is explained in Fig. 2. The range of x-coordinate
corresponds to the length LF
x= 40aof the F overlayer, while
the results are independent of the length of the TI layer un-
derneath,LTI
x.
(holes) at the Fermi surface being injected from the lead
( 6= ). The electron-hole analogy can be understood
by de ning the hole density matrix, iG>
, from the iden-
tity i(G<
G>
) = 2=(G) = +P
6= . By con-
sidering left-lead induced holes instead of right-lead in-
duced electrons, we can interpret Eq.(20) as spin-resolved
electron-hole recombination rate, where opposite spins
have opposite contributions to the antidamping-like
SOT. This picture focuses on the energy anti-dissipative
aspect of the phenomena and, since
L(
R) cor-
FIG. 5. (Color online) SOT-induced magnetization trajecto-
ries~ m(t) under di erent Vband~Bext= 0. Higher color inten-
sity denotes denser bundle of trajectories which start from all
possible initial conditions ~ m(t= 0) on the unit sphere. Solid
curves show examples of magnetization trajectories, while the
white circles denote attractors of trajectories.
responds to the spin- right (left) moving electrons,
Eq. (20) suggests that spin-momentum locking natu-
rally has a signi cant e ect on the enhancement of
the antidamping-like SOT magnitude. In particular,
in the case of F/TI interface, the enhancement of the
antidamping-like SOT occurs when the spin-up/down is
along they-axis (~ m0k^y) which is the spin-polarization
direction of electrons passing through the surface of the
TI induced by the EE. Additionally, in this case the
antidamping-like SOT gets smaller away from the F/TI
transverse edge because the contribution of both of the
leads to the spin density become identical. Therefore the
anti-damping torque in this case is more localized around
the edge. This e ect is more signi cant when the magne-
tization is out of the plane and the Fermi energy is inside
the surfcosgap on the TI surface.
A alternative interpretation of the results can be
achieved by considering Gt Gy
t=iP
Gy
t Gt. In
this case, the average of the antidamping-like SOT is ex-
pressed by
hTzi =Vb
4Tr[T"#
LR T"#
RL]; (21)
where the F overlayer induced spin-
ip transmission ma-
trix is de ned as
T"#
= (t"#
)yt"#
; (22)
and
t"#
=surf
2p
Gt+Gtp
: (23)
Although Eq. (21) is obtained from perturbative con-
siderations, it looks identical to the Eq. (8) of Ref. 48
where a spin-
ip re
ection mechanism at the edge of the
F/2D-TI interface was recognized to be responsible for
the giant charge pumping (i.e., anti-damping torque) ob-
served in the numerical simulation.48Eq. (23) describes a
transmission event in which electrons injected from lead
, get spin-
ipped (from up to down) by the FM and
then transmit to the lead . The path of the electrons
describing this process is shown in Fig. 1. From the k-
resolved results of the anti-damping torque (not shown8
here) we observe that while for the in-plane magnetiza-
tion electrons moving in the same transverse direction
(same sign for ky) on both left and right edges of the
FM/TI interface contribute to the torque, in the case of
out-of-plane magnetization for the left (right) edge of the
interface the local anti-damping torque is induced mostly
by the electrons with ky>0 (ky<0).
It is worth mentioning that due to nonperturbative na-
ture of the SOT induced by the chiral electrons, the ap-
proximation presented in this section which can as well
be obtained from the self energy corresponding to the
vacuum polarization Feynman diagrams of the electron-
magnon coupled system59, does not capture the phenom-
ena accurately. This is evident in the angular dependence
of the anti-damping torque which in the current section
is considered up to second order e ect ( 2), while the
divergence-like behavior in Figs.2(h) suggest a linear de-
pendence when the magnetization direction is close to the
y-axis. This signi es the importance of the higher order
terms with respect to that can not be ignored. The ap-
proximation presented in this section also suggests that
blocking the lower surface leads to the reduction of the
anti-damping torque. However, in this case an electron
experiences multiple spin-
ip re
ections before transmit-
ting to the next lead and in fact it turns out that the ex-
act results stay intact even if the lower surface is blocked.
This is similar to the conclusion made in Ref. 48 which
shows the redundancy of blocking the lower edge of the
2D-TI to obtain a nonzero pumped charge current from
precessing FM as proposed in Ref. 47.
Although spin-momentum locking of the surface state
of the TI resembles the 2D Rashba plane, in the case of
TI surface state the cones with opposite spin-momentum
locking reside on opposite surface sides of the TI slab
while in the case of a Rashba plane they are only sepa-
rated by the SOC energy. This means one can expect a
smaller SOT for a FM on top of a 2D Rashba plane due
to cancellation of the e ects of the two circles with op-
posite spin-momentum locking, where the nonzero anti-
damping torque originates from the electron-hole asym-
metry.
VI. LLG SIMULATIONS OF MAGNETIZATION
DYNAMICS IN THE PRESENCE OF SOT
In order to investigate ability of predicted
antidamping-like SOT to switch the magnetization
direction of a perpendicularly magnetized F overlayer in
the geometry of Fig. 1, we study magnetization dynamics
in the macrospin approximation by numerically solving
LLG equation at zero temperature supplemented by
SOT components analyzed in Sec. III
@~ m
@t=1
2[~ o;e(~ m;EF) +~ o;o(~ m;EF)]eVb+
~Bext~ m
+~ m
(~ m)@~ m
@t
+ (~ m^z)(~ m^z)MCA:(24)
FIG. 6. (Color online) Phase diagram of the magnetization
state in lateral F/TI heterostructure from Fig. 1 as a function
of an in-plane external magnetic Bextk^xandVb(i.e., SOT
/Vb). Thick arrows on each of the panels (a){(d) show the
direction of sweeping of Bext
xorVbparameter. The small-
ness of central hysteretic region along the Vb-axis, enclosed
by white dashed line in panel (b) and (d), shows that low
currents are required to switch magnetization from mz>0
tomz<0 stable states.
Here
is the gyromagnetic ratio, (~ m)ij =
2
surfTij(~ m;EF)=8is the dimensionless Gilbert damp-
ing tensor, and MCA = 0
MCA +j~Te;ej=j(~ m^z)(~ m^z)j,
where 0
MCA represents the intrinsic MCA energy of the
FM. We solve Eq. (24) by assuming that the Gilbert
damping is a constant (its dependence on ~ mis relegated
to future studies) and ignore the dependence of MCA
on~ mandVbwhile retaining its out-of-plane direction.
Figure 5 shows the magnetization trajectories for all
possible initial conditions ~ m(t= 0) on the unit sphere
under di erent Vb. AtVb= 0, the two attractors are
located as the north and south poles of the sphere. At
niteVb, the attractors shift away from the poles along
thez-axis within the xz-plane, while additional attractor
appears on the positive (negative) y-axis under negative
(positive)Vb. Note that the applied bias voltage Vbdrives
dc current and SOT proportional to it in the assumed
linear-response transport regime.
Figure 6 shows the commonly constructed3,4,15,27
phase diagram of the magnetization state in the pres-
ence of an external in-plane magnetic eld Bextk^xand
the applied bias voltage Vb(i.e., SOT/Vb). The thick
arrows in each panel of Fig. 6 denote the direction of the
sweeping variable|in Fig. 6(a) [6(b)] we increase [de-
crease]Vbslowly in time, and similarly in Fig. 6(c) [6(d)]
we increase [decrease] the external magnetic eld gradu-
ally. The size of hysteretic region in the center of these
diagrams, enclosed by white dashed line in Figs. 6(b) and
6(d), measures the eciency of switching.3,4,15,27Since
this region, where both magnetization states mz>0 and
mz<0 are allowed, is relatively small in Figs. 6(a) and9
6(b), magnetization can be switched by low Bext
xand
smallVb(or, equivalently, small injected dc current), akin
to the phase diagrams observed in recent experiments.15
Although we considered the FM a single domain, the
fact that the anti-damping component of the SOT is
mainly peaked around the edge of the FM/TI interface
suggests that it is be more feasible in realistic cases to
have the local magnetic moments at the edge of the FM
switch rst and then the total magnetization switches
by the propagation of the domain walls formed at the
edge throughout the FM60,61. Therefore, a micromag-
netic simulation of the system is required to investigate
switching phenomena in large size systems which we rel-
egate to future works.
VII. CONCLUSIONS
In conclusion, by performing adiabatic expansion of
time-dependent NEGFs,33,34we have developed a frame-
work which yields formulas for spin torque and charge
pumping as reciprocal e ects to each other connected by
time-reversal, as well as Gilbert damping due to SOC. It
also introduces a novel way to separate the SOT com-
ponents, based on their behavior (even or odd) under
time and bias voltage reversal, and can be applied to
arbitrary systems dealing with classical degrees of free-
dom coupled to electrons out of equilibrium. For the
geometry28proposed in Fig. 1, where the F overlayercovers (either partially or fully) the top surface of the TI
layer, we predict that low charge current
owing solely
on the surface of TI will induce antidamping-like SOT on
the F overlayer via a physical mechanism that requires
spin-momentum locking on the surface of TIs|spin-
ip
re
ection at the lateral edges of a ferromagnetic island
introduced by magnetic proximity e ect onto the TI sur-
face. This mechanism has been overlooked in e orts to
understand why SO-coupled interface alone (i.e., in the
absence of SHE current from the bulk of SO-coupled non-
ferromagnetic materials) can generate antidamping-like
SOT, where other explored mechanisms have included
spin-dependent impurity scattering at the interface,55
Berry curvature mechanism,25,26as well as their com-
bination.56
The key feature for connecting experimentally ob-
served SOT and other related phenomena in F/TI het-
erostructures (such as spin-to-charge conversion28,36,58)
to theoretical predictions is their dependence2,24on the
magnetization direction. The antidamping-like SOT pre-
dicted in our study exhibits complex angular dependence,
exhibiting \nonperturbative" change with the magnetiza-
tion direction in Fig. 2(h), which should make it possible
to easily di erentiate it from other competing physical
mechanisms.
ACKNOWLEDGMENTS
F. M. and N. K. were supported by NSF PREM Grant
No. 1205734, and B. K. N. was supported by NSF Grant
No. ECCS 1509094.
farzad.mahfouzi@gmail.com
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1907.07470v2.Inhomogeneous_domain_walls_in_spintronic_nanowires.pdf | arXiv:1907.07470v2 [math.AP] 10 Dec 2019Inhomogeneous domain walls in
spintronic nanowires
L. Siemer∗I. Ovsyannikov†J.D.M. Rademacher‡
December 12, 2019
In case of a spin-polarized current, the magnetization dynamics in n anowires
are governed by the classical Landau-Lifschitz equation with Gilbertdamp-
ing term, augmented by a typically non-variational Slonczewski term. Tak-
ing axial symmetry into account, we study the existence of domain w all
type coherent structure solutions, with focus on one space dimen sion and
spin-polarization, but our results also apply to vanishing spin-torqu e term.
Using methods from bifurcation theory for arbitrary constant ap plied fields,
we prove the existence of domain walls with non-trivial azimuthal pro file,
referred to as inhomogeneous . We present an apparently new type of do-
main wall, referred to as non-flat, whose approach of the axial magnetiza-
tion has a certain oscillatory character. Additionally, we present th e leading
order mechanism for the parameter selection of flatandnon-flat inhomoge-
neous domain walls for an applied field below a threshold, which depends on
anisotropy, damping, and spin-transfer. Moreover, numerical c ontinuation
results of all these domain wall solutions are presented.
1 Introduction
Magnetic domain walls (DWs) are of great interest both from a theor etical perspective
and for applications, especially in the context of innovative magnetic storages [1]. Re-
cent developments in controlled movement of DWs via spin-polarized c urrent pulses in
nanomagnetic structures, in particular in nanowires, are thought to lead to a new class
of potential non-volatile storage memories, e.g. racetrack memor y [1, 2, 3, 4]. These
devices make use of the fact that spin-transfer driven effects ca n change the dynamics
in sufficiently small ferromagnetic structures (e.g. nanowires), wh ere regions of uniform
∗Universit¨ at Bremen, lars.siemer@uni-bremen.de ; Corresponding author
†Universit¨ at Hamburg, Lobachevsky State University of Nizhny No vgorod
‡Universit¨ at Bremen
1magnetization, separated by DWs, can appear [5, 6, 7]. This motivat es further studies of
the existence of magnetic domains and their interaction with spin-po larized currents as
a building block for the theory in this context. In this paper we take a mathematical per-
spective and, in a model for nanomagnetic wires, rigorously study t he existence of DWs.
This led us to discover an apparently new kind of DWs with a certain inho mogeneous
and oscillatory structure as explained in more detail below.
The description of magnetization dynamics in nanomagnetic structu res, governed by the
Landau-Lifschitz-Gilbert (LLG) equation, is based on works by Berger and Slonczewski
assuming a spin-polarized current [8, 9]. In the presence of a const ant applied field and
a spin-polarized current, the dynamics driven by the joint action of magnetic field and
spin torque can be studied by adding a spin-transfer term in the dire ction of the current
(current-perpendicular-to-plane (CPP) configuration). In cas e of a spatially uniform
magnetization, the resulting Landau-Lifschitz-Gilbert-Slonczewski (LLGS) equation for
unit vector fields ( m1,m2,m3) =m=m(x,t)∈S2(cf. Figure 1) reads
∂tm−αm×∂tm=−m×heff+m×(m×J). (LLGS)
with effective field heff, Gilbert damping factor α>0, and the last term is the so-called
polarized spin transfer pseudotorque.
Note that the above equation reduces to the LLG equation for J≡0, see§2 for more
details.
In this paper we consider the axially symmetric case and set
heff:=∂2
xm+h−µm3e3,J:=β
1+ccpm3e3, (1)
whereh=he3with a uniform and time-independent field strength h∈R, and m 3=
/an}bracketle{tm,e3/an}bracketri}ht,e3∈S2. This effective field heffalso includes the diffusive exchange term ∂2
xm,
the uniaxial anisotropy and demagnetization field. The specific here with parameter
µ∈Rderives from a first order approximation in the thin film/wire limit for a u niformly
magnetized body [6, 10]. In the axially symmetric structure, β≥0 andccp∈(−1,1)
describe the strength of the spin-transfer and the ratio of the p olarization [7, 11]. The
spin-transfer torque term may provide energy to the system und er certain conditions
and counterbalance dissipation associated to the Gilbert damping te rm, which gives rise
to coherent non-variational dynamics, see e.g. [12].
Notably, for β= 0 one obtains the LLG-equation that does not account for spin tr ansfer
effects. Moreover, as shown in [12], this also holds up to parameter change in case
ccp= 0. Hence, solutions to the LLGS equation for β= 0 orccp= 0 are also solutions
to the LLG equation, so that all the analytical as well as numerical r esults forccp= 0
in this paper directly transfer to the LLG equation.
A key ingredient for the separation of uniformly magnetized states in space are interfaces
betweentwomagneticdomains. Themostcoherentformofsuchint erfacesintheuniaxial
setting are relative equilibria with respect to translation and rotatio n symmetry of the
form
m(ξ,t) =m0(ξ)eiϕ(ξ,t),whereξ=x−standϕ(ξ,t):=φ(ξ)+Ωt,
2(a)
(b)-5 5-101
xm1,m2
(c)
Figure 1: Homogeneous DW profile ( q≡0) withα= 0.5,β= 0.1,µ=−1,h= 50,ccp= 0.
(a) (m2,m3)-profile. (b) Projection onto S2. (c) Zoom-in on m1(blue solid) and m2
(red dashed).
withspeedsandfrequency Ω. Here the complex exponential acts on m0∈S2by
rotation about the e3-axis, i.e., the azimuth, and in spherical coordinates we can choose
m0(ξ) = (sin(θ(ξ)),0,cos(θ(ξ))) with altitude angle θ.
We refer to such solutions with m0(ξ)→ ±e3asξ→ ±∞orξ→ ∓∞asdomain walls .
A first classification of DWs is based on the local wavenumber q:=φ′, which determines
φuniquely due to the axial rotation symmetry and satisfies
q(ξ) =/an}bracketle{t(m′
1,m′
2),(−m2,m1)/an}bracketri}ht
1−m2
3(ξ). (2)
Definition 1. We call a DW with constant φhomogeneous (hom) , i.e.,q≡0, and
inhomogeneous otherwise.
Inhomogeneous DWs have a spatially inhomogeneous varying azimuth al angle, compare
Figures 1 and 2.
In the case of uniaxial symmetry and the LLG case β= 0, an explicit family of homo-
geneous DWs was discovered in [13] for applied fields with arbitrary st rength and time
dependence, cf. Figure 1. Furthermore, for constant applied fie lds and in case of ccp/ne}ationslash= 0
it was shown in [12] that DWs cannot be homogeneous, and the existe nce of inhomoge-
neous DWs was proven, whose spatial profile slowly converges to ±e3and where |s| ≫1.
This latter type of DWs is ‘weakly localized’ and has large ‘width’ in the se nse that the
inverse slope of m3atm3(0) = 0 tends to infinity as |s| → ∞.
An apparently thus far unrecognized distinction of DWs is based on t he convergence
behavior of qasξ→ ±∞.
Definition 2. We call a DW flatif|q(ξ)|has a limit on R∪ {∞}as|ξ| → ∞and
non-flatotherwise.
Note that homogeneous DWs are flat ones by definition (recall φ′=q). Moreover, for
all DWsm0(ξ) converges to e3or−e3as|ξ| → ∞.
3small applied field
(a)large applied field
(b)
-5 5-101
ξm1
(c)-5 5-101
ξm1,m2
(d)
Figure 2: Shown are profiles of inhomogeneous DWs m(ξ) computed by parameter continua-
tion, cf.§4, inccptoccp= 0.5 with fixed α= 0.5,β= 0.1,µ=−1. (a,c) codim2case
h= 0.5,s= 0.112027,Ω = 0.447173, (b,d) codim 0 case h= 50,s= 19.92,Ω = 40.4.
(c) magnification of the m1-component; note the change of frequency for small vs.
largeξ. (d) Magnification of m1(blue solid) as well as m2(red dashed) component.
The main result of this paper is an essentially complete understanding of the existence
and the type of DWs near the aforementioned explicit solution family f or a nanowire
geometry, i.e., µ <0. This includes the LLG case β·ccp= 0, but our focus is on the
spintronic case β·ccp/ne}ationslash= 0 for which these results pertain 0 <|ccp| ≪1 and any value of
the (constant) applied field h.
The different types of DWs occur in parameter regimes close to ccp= 0 in the (spatial)
ODE which results from the coherent structure ansatz. Since the parameters αandµ
are material-dependent we take the applied field strength has the primary parameter.
In brief, organized by stability properties of the steady states ±e3in the ODE, this leads
to the following cases and existence results for localized DWs in nanow ires (µ<0):
•‘codim-2’ (h∗<h<h∗) : existence of flat inhomogeneous DWs with s,Ω selected
by the other parameters,
•‘center’ (h=h∗orh=h∗) : existence of flat and non-flat inhomogeneous DWs,
•‘codim-0’ (h<h∗orh>h∗) : existence of flat inhomogeneous DWs,
whereh∗:=β/α+2µ
α2(1+α2) as well as h∗:=β/α−2µ
α2(1+α2). Note that h∗<h∗always
(recallα>0 andµ<0). Due to symmetry reasons, we mainly discuss the existence of
4Figure 3: Stability diagram of homogeneous states ±e3inhandccpforα= 1,β= 0.5, and
µ=−1. State + e3unstable to the left and stable to the right of Γ+,−e3stable
to the left and unstable to the right of Γ−. Homogeneous DWs (hom) exist only on
theh-axis, i.e., ccp≡0. See text for further explanations. Note that also negativ e
applied fields are shown.
right-moving DWs close to the explicit solution family and thus focus on an applied field
β/α≤h(cf.§3). The main results can be directly transferred to the case of left -moving
DWs (h≤β/α). Notably, the codim-0 case occurs for ‘large’ magnetic field habove a
material dependent threshold. In the center and codim-2 cases t here is a selection of s
and Ω by the existence problem.
Thebasicrelationbetween thePDEandtheODEstabilitypropertiesw .r.t.handccpare
illustrated in Figure 3 for α= 1,β= 0.5,µ=−1 fixed. Due to the fact that sand Ω are
ODE parameters only, the diagram illustrates a slice in the four dimens ional parameter
space with axes h,ccp,s, and Ω. Note that homogeneous DWs (hom) can occur only
on the line ccp≡0 (see [12, Theorem 5] for details). The stability regions are defined
as follows. monostable−(blue): + e3unstable and −e3stable,bistable(shaded blue):
both +e3and−e3stable,monostable+(red): +e3stable and −e3unstable, unstable
(shaded red): both + e3and−e3unstable. For a more detailed stability discussion, see
Remark 5. Note that the transition from bistable to monostable in th e PDE does not
coincide with the transition, of the homogeneous family, from codim- 2 to codim-0 in
the ODE. In contrast, the analogous transitions occur simultaneo usly for example in the
well-known Allen-Cahn orNagumo equation.
In Figure 4 below, we present numerical evidence that inhomogeneo us DWs are indeed
also dynamically selected states, especially for large applied fields, als o in the LLG case
(β,ccp= 0).
The understanding of DW selection by stability properties generally d epends on the exis-
tence problem discussed in this paper, which is therefore a prerequ isite for the dynamical
selection problem, cf. Remark 5.
5(a)
(b)
(c)
Figure 4: Direct simulation of full PDE (LLGS) for α= 0.5,β= 0.1,µ=−1,h= 50, and
ccp= 0 with dynamical selection of an inhomogeneous DW. Initial condition near
homogeneous DW (9) in codim-0 regime ( h∗= 10.2,s0= 19.92, and Ω 0= 40.04).
(a) Profile at t= 20 projected onto the sphere. (b) Speed and frequency of DW
over time with asymptotic (selected) values s= 12.5 and Ω = 78 .28. (c) Space-time
plots of DW components (without co-moving frame), range as i n black box in (b).
Final profile is heteroclinic connection in (7), cf. Proposi tion 2.
To our knowledge, existence results of DWs for ccp/ne}ationslash= 0 are new. In more detail, the
existence of localised inhomogeneous, i.e. flat as well as non-flat, DW s forccp/ne}ationslash= 0 and
especially for ccp= 0 are new results. Indeed, the existence proof of non-flat DWs is
the most technical result and entails an existence proof of hetero clinic orbits in an ODE
between an equilibrium and a periodic orbit. These solutions indicate th e presence of
DWs in other regimes of spin driven phenomena and may be of interest for spin-torque
transfer MRAM (Magnetoresistive random-access memory) syst ems [14].
This paper is organized as follows. In §2, the LLGS equation and coherent structures as
well as first properties are discussed. Section 3 more precisely intr oduces homogeneous
and inhomogeneous as well as flat and non-flat DWs and it also includes the main
results of this paper (Theorem 1, 2, and 3). The technical proofs of Theorem 2 as
well as Theorem 3 are deferred to Appendix 6.1 and 6.2. Section 4 pre sents results of
6numerical continuation in parameter ccpfor the three regimes of the applied field (codim-
2, center, and codim-0), where the center case is studied in more d etail. We conclude
with discussion and outlook in §5.
Acknowledgements
L.S. and J.R. acknowledge support by the Deutsche Forschungsge meinschaft (DFG, Ger-
manResearchFoundation)-Projektnummer 281474342/GRK222 4/1. J.R.alsoacknowl-
edges support by DFG grant Ra 2788/1-1. I.O. acknowledges fund ing of a previous
position by Uni Bremen, where most of this paper was written, as we ll as support by
the recent Russian Scientific Foundation grant 19-11-00280.
2 Model equations and coherent structure form
The classical model for magnetization dynamics was proposed by La ndau and Lifschitz
based on gyromagnetic precession, and later modified by Gilbert [15, 16]. See [17] for
an overview. The Landau-Lifschitz-Gilbert equation for unit vector fields m(x,t)∈S2
in one space dimension x∈Rand in terms of normalized time in dimensionless form is
∂tm−αm×∂tm=−m×heff. (LLG)
Herem=M/MSrepresents thenormalizedmagnetization, heff=Heff/MStheeffective
field, i.e.thenegativevariationalderivativeofthetotalmagneticf reeenergywithrespect
tom, both normalized by the spontaneous magnetization MS. For gyromagnetic ratio
γand saturation magnetization MSthe time is measured in units of ( γMS)−1, and it
is assumed that the temperature of the magnetic body is constant and below the Curie
temperature [5]. Finally, Gilbertdampingα>0 turnsmtowardsheffand both vectors
are parallel in the static solution.
In modern spin-tronic applications, e.g. Spin-Transfer Torque Mag netoresistive Random
Access Memories (MRAM), the spin of electrons is flipped using a spin- polarized current.
To take these effects into account, the LLG equation is supplement ed by an additional
spin transfer torque term. Using a semiclassical approach, Sloncz ewski derived an ex-
tended effective field
Heff=heff−m×J,
whereJ=J(m) depends on the magnetization and the second term is usually called
Slonczewski term [9]. In contrast to the LLG equation, which can be written as the
gradient of free ferromagnetic energy, this generalized form is no longer variational and
the energy is no longer a Lyapunov functional.
As to the specific form of Heff, including a leading order form of exchange interaction,
uniaxial crystal anisotropy in direction e3, andZeemanas well as stray-field interactions
with an external magnetic field, see e.g. [6], gives the well known form (1).
In this paper we consider a constant applied magnetic field h∈Ralonge3and uniaxial
anisotropy with parameter µ∈R, for which the anisotropy energy density is rotationally
7symmetric w.r.t. e3. According to the energetically preferred direction in the uniaxial
case, minima of the anisotropy energy density correspond to easydirections, whereas
saddles or maxima correspond to medium-hard orharddirections, respectively. There-
fore, one refers to µ <0 aseasy-axis anisotropy and µ >0 aseasy-plane , both with
regard to e3.
As mentioned before, the LLG equation with its variational structu re appears as a
special case of (LLGS) for β= 0 orccp= 0. While our main focus is the non-variational
spintronic case β·ccp/ne}ationslash= 0, all results contain the case β·ccp= 0 and thus carry over
to (LLG).
It iswell-known that(LLGS) admitsanequivalent formasanexplicit ev olution equation
of quasilinear parabolic type in the form, see e.g. [12],
∂tm=∂x(A(m)∂xm)+B(m,∂xm).
As a starting point, we briefly note the existence of spatially homoge neous equilibrium
solutions of (LLGS) for which m(x,t) is constant in xandt.
Remark 1. The only (spatially)homogeneous equilibria of (LLGS)forβ >0are the
constant up- and down magnetization states ±e3. Indeed, setting ∂tm=∂2
xm= 0in
(LLGS), forβ/ne}ationslash= 0the last equation implies that m1=m2= 0and thus the only solutions
m∗
±∈S2arem∗
±= (0,0,±1)T.
Remark 2. In caseβ= 0as well as |h/µ|<1there exist a family of additional
homogeneous solutions of (LLGS) given by m∗= (m1,m2,h/µ)T,withm2
1+m2
2=
1−(h/µ)2. Note that similar cases occur for symmetry axis being e1ande2, respectively
(cf. Brown’s equations ).
2.1 Coherent structure ODE
Duetotherotationsymmetryaroundthe e3-axisof (LLGS), itisnaturaltousespherical
coordinates
m=
cos(ϕ)sin(θ)
sin(ϕ)sin(θ)
cos(θ)
,
whereϕ=ϕ(x,t) andθ=θ(x,t). This changes (LLGS) to
/parenleftbigg
α−1
1α/parenrightbigg/parenleftbigg
∂tϕsin(θ)
−∂tθ/parenrightbigg
=/parenleftbigg
2∂xϕ∂xθcos(θ)
−∂2
xθ/parenrightbigg
+sin(θ)/parenleftbigg
∂2
xϕ+β/(1+ccpcos(θ))
(∂xϕ)2cos(θ)+h−µcos(θ)/parenrightbigg (3)
Note that the rotation symmetry has turned into the shift symmet ry in the azimutal
angleϕ, as (3) depends on derivatives of ϕonly.
8Recall that DW solutions spatially connect the up and down magnetiza tion states ±e3
in a coherent way as relative equilibria with respect to the translation symmetry in x
andφ, which yields the ansatz
ξ:=x−st, θ=θ(ξ), ϕ=φ(ξ)+Ωt. (4)
Such solutions are generalized travelling waves that move with const ant speeds∈R
in space and rotate pointwise with a constant frequency Ω ∈Raround the e3-axis;
solutions with Ω = 0 are classical travelling waves.
As in [12], applying ansatz (4) to (3) leads to the so-called coherent structure ODE
/parenleftbiggα−1
1α/parenrightbigg/parenleftbigg(Ω−sφ′)sin(θ)
sθ′/parenrightbigg
=/parenleftbigg2φ′θ′cos(θ)
−θ′′/parenrightbigg
+sin(θ)/parenleftbiggφ′′+β/(1+ccpcos(θ))
(φ′)2cos(θ)+h−µcos(θ)/parenrightbigg
,(5)
where′=d/dξ. This system of two second-order ODEs does not depend on φand thus
reduces to three dynamical variables ( θ,ψ=θ′,q=φ′). Following standard terminology
for coherent structures, we refer to qas thelocal wavenumber .
Writing (5) as a first-order three-dimensional system gives
θ′=ψ
ψ′= sin(θ)[h−Ω+sq+(q2−µ)cos(θ)]−αsψ
q′=αΩ−β/(1+ccpcos(θ))−αsq−s+2qcos(θ)
sin(θ)ψ, (6)
and DWs in the original PDE are in 1-to-1-correspondence with the O DE solutions
connecting θ= 0 andθ=π.
2.1.1 Blow-up charts and asymptotic states
As in [12], the singularities at zeros of sin( θ) in (6) can be removed by the singular
coordinate change ψ:=psin(θ), which is a blow-up transformation mapping the poles
of the sphere ±e3to circles thus creating a cylinder. The resulting desingularized system
reads
θ′= sin(θ)p
p′=h−Ω−αsp+sq−(p2−q2+µ)cos(θ)
q′=αΩ−β/(1+ccpcos(θ))−sp−αsq−2pqcos(θ).(7)
The coherent structure system (6) is equivalent to the desingular ized system (7) for
θ/ne}ationslash=nπ,n∈Zand therefore also for maway from ±e3. Furthermore, the planar blow-
up chartsθ= 0 andθ=πare invariant sets of (7), which are mapped to the single
pointse3and−e3, respectively by the blow-down transformation. System (7) has a
special structure (cf. Figure 6) that will be relevant for the subs equent DW analysis. In
the remainder of this section we analyze this in some detail.
9Lemma 1. Consider the equations for pandqin(7)for an artificially fixed value of θ.
In terms of z:=p+iqthis subsystem can be written as the complex (scalar) ODE
z′=Az2+Bz+C, (8)
whereA:=−cos(θ),B:=−(α+i)s,andCθ:=h−Ω+Aµ+i/parenleftig
αΩ−β
1−Accp/parenrightig
.
ForA/ne}ationslash= 0the solution with z0=z(ξ0)away from the equilibria zθ
+=−B
2A+ iγθ
2Aand
zθ
−=−B
2A−iγθ
2A, withγθ=γ(θ):=√
4ACθ−B2, reads
z(ξ) =γθ
2Atan/parenleftbiggγθ
2ξ+δ0/parenrightbigg
−B
2A, (9)
where
δ0= arctan/parenleftbigg2Az0+B
γθ/parenrightbigg
−γθ
2ξ0.
ForA= 0, the solution away from the equilibrium zπ/2=−Cπ/2/Bis given by
z(ξ) =/parenleftbigg
z0+Cπ/2
B/parenrightbigg
eB(ξ−ξ0)−Cπ/2
B.
Clearly, the solution of (8) relates only to those solutions of (7) for whichθis constant,
i.e.,θ= 0,π. Although we are mostly interested in the dynamics on the blow-up ch arts,
we consider θas a parameter in order to demonstrate the special behaviour of ( 7) forθ
artificially fixed. Notably, the equilibria zθ
±of (8) forθ/ne}ationslash= 0,πare not equilibria in the
full dynamics, due to the fact that (7) is only invariant for θat the blow-up charts.
Proof.We readily verify the claimed form of the ODE and directly check the cla imed
solutions. /squaresolid
Remark 3. Lemma 1 states in particular that the desingularized ODE sys tem(7)can
be solved explicitly on the invariant blow-up charts, where θ= 0,πand thusA=−1,1,
respectively. System (7)possesses two real equilibria on each blow-up chart, Z0
±:=
(0,p0
±,q0
±)TandZπ
±:= (π,pπ
±,qπ
±)T. Herepθ
σ:= Re(zθ
σ),qθ
σ:= Im(zθ
σ)forθ= 0,π,
σ=±and
z0
+:= 1/2(B−iγ0), z0
−:= 1/2(B+iγ0)
and analogously
zπ
+:= 1/2(−B+iγπ), zπ
−:= 1/2(−B−iγπ),
where we set
γ0:=γ/vextendsingle/vextendsingle
A=−1=√
−4C0−B2andγπ:=γ/vextendsingle/vextendsingle
A=1=√
4Cπ−B2
withC0:=C/vextendsingle/vextendsingle
A=−1andCπ:=C/vextendsingle/vextendsingle
A=1.
Due to the analytic solution (9), we obtain the following more detailed r esult in case
θ/ne}ationslash=π/2 (cf. Figure 6).
10Lemma 2. For each given 0≤θ≤πwithθ/ne}ationslash=π/2as a parameter, the fibers of (7)with
constantθconsists entirely of heteroclinic orbits between zθ
−andzθ
+in caseIm(γθ)/ne}ationslash= 0,
orγθ/ne}ationslash= 0, except for the equilibrium states. In case Im(γθ) = 0andRe(γθ)/ne}ationslash= 0, the
fiber at fixed θis filled with periodic orbits away from the invariant line {q=s
2A}, for
which the period of solutions close to it tends to infinity.
Proof.Forθfixed in (8), consider the case Re( γθ) = 0 and also Im( γθ)/ne}ationslash= 0 forA/ne}ationslash= 0
which leads to
z(ξ) = iIm(γθ)
2A·tan/parenleftbigg
i/parenleftigIm(γθ)
2ξ+Im(δ0)
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
=:ˇξ/parenrightig
+Re(δ0)/parenrightbigg
−B
2A
=Im(γθ)
2A·isin(2Re(δ0))−sinh/parenleftbig
2ˇξ/parenrightbig
cos(2Re(δ0))+cosh/parenleftbig
2ˇξ/parenrightbig−B
2A.
For Re(γθ)/ne}ationslash= 0 as well as Im( γθ)/ne}ationslash= 0, we obtain
z(ξ) =γθ
2Atan/parenleftbiggRe(γθ)
2ξ+Re(δ0)
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
=:˜ξ+iIm(γθ)
2ξ+iIm(δ0)/parenrightbigg
−B
2A
=γθ
2A·sin(2˜ξ)+isinh/parenleftig
2/parenleftig
Im(γθ)
Re(γθ)˜ξ−Im(γθ)
Re(γθ)Re(δ0)+Im(δ0)/parenrightig/parenrightig
cos(2˜ξ)+cosh/parenleftig
2/parenleftig
Im(γθ)
Re(γθ)˜ξ−Im(γθ)
Re(γθ)Re(δ0)+Im(δ0)/parenrightig/parenrightig−B
2A,
The asymptotic states are
Im/parenleftbig
γθ/parenrightbig
>0 : lim
ξ→−∞z(ξ) =−iγθ
2A−B
2A,lim
ξ→+∞z(ξ) = iγθ
2A−B
2A,
as well as
Im/parenleftbig
γθ/parenrightbig
<0 : lim
ξ→−∞z(ξ) = iγθ
2A−B
2A,lim
ξ→+∞z(ξ) =−iγθ
2A−B
2A,
which simplify in case Re( γθ) = 0 to
Im/parenleftbig
γθ/parenrightbig
>0 : lim
ξ→−∞z(ξ) =Im(γθ)−B
2A,lim
ξ→+∞z(ξ) =−Im(γθ)+B
2A,
as well as
Im/parenleftbig
γθ/parenrightbig
<0 : lim
ξ→−∞z(ξ) =−Im(γθ)+B
2A,lim
ξ→+∞z(ξ) =Im(γθ)−B
2A.
Note that the asymptotic states coincide if γθ= 0.
11The last case to consider is Re/parenleftbig
γθ/parenrightbig
/ne}ationslash= 0 and Im/parenleftbig
γθ/parenrightbig
= 0, where the solutions are
z(ξ) =Re/parenleftbig
γθ/parenrightbig
2A·sin(2ˆξ)+isinh(2Im( δ0))
cos(2ˆξ)+cosh(2Im( δ0))−B
2A,
withˆξ:=Re(γθ)
2ξ+Re(δ0) and which leads to periodic solutions of (8) iff
Im(δ0)/ne}ationslash= 0⇔2AIm(z0)+Im(B)/ne}ationslash= 0⇔q0/ne}ationslash=s
2A,
wherez0=p0+iq0and recall that B=−(α+i)s. /squaresolid
Based on Lemma 2, explicitly onthe blow-up chart θ= 0 the heteroclinic orbits are from
z0
−toz0
+in case Im( −4C0−B2)>0, or Im( −4C0−B2) = 0 and Re( −4C0−B2)≤0,
and fromz0
+toz0
−if Im(−4C0−B2)<0 . Forθ=π, if Im(4Cπ−B2)>0, or
Im(4Cπ−B2) = 0 and Re(4 Cπ−B2)≤0 they are connections from zπ
−tozπ
+, and if
Im(4Cπ−B2)<0 fromzπ
+tozπ
−.
ForA/ne}ationslash= 0, the case s= 0 is a special situation, which will be also discussed in the
context of DWs in §3 later. It turns out that on the blow-up charts θ= 0 (orθ=π),
the solution with appropriate initial conditions has a limit as |ξ| → ∞if and only if
Im(√
−C0)/ne}ationslash= 0 (Im(√
Cπ)/ne}ationslash= 0). In terms of the parameters in (7) and with
β−:=β
1−ccpandβ+:=β
1+ccp,
this leads to the conditions for θ= 0 given by:
Ω/ne}ationslash=β+
αor Ω =β+
αand Ω≤h−µ, (10)
and forθ=πgiven by:
Ω/ne}ationslash=β−
αor Ω =β−
αand Ω≥h+µ, (11)
In caseccp= 0, i.e. for the LLG equation, the conditions in (10) and (11) reduce to
Ω/ne}ationslash=β
αor Ω =β
αand 2µ≤β
α,
where the latter inequality always holds in case of a nanowire geometr y (µ<0). Hence
standing domain walls in nanowires in case ccp= 0 can only connect equilibria, if they
exist.
Lemma 2 also states that the equilibria on the blow-up charts θ∈ {0,π}are surrounded
by periodic orbits in case Im( γ0) = 0 and Re( γ0)/ne}ationslash= 0 (Im(γπ) = 0 and Re( γπ)/ne}ationslash= 0).
In fact, system (7) is Hamiltonian (up to rescaling) on the blow-up ch arts for certain
frequencies Ω, as follows
122p
-11q
(a)-5 5ξ
-1
2
p q
(b)
Figure 5: (a) Phase plane streamplot with Mathematica of (14) around the equilibrium zπ
−,
i.e., (7) at θ=π, forα= 0.5,β= 0.1,µ=−1,h= 10.2,s= 4,Ω = 8.2 andccp= 0,
which leads to/parenleftbig
pπ
−,qπ
−/parenrightbigT= (1,0)T. The red solid line marks the trajectory with
initial condition ( p0,q0) = (7/4,0) (cf. plot of solutions in b). (b) Plot of the profile
for the solution highlighted in (a), where p(solid blue line) and q(dashed red line)
are given by (9) for the parameter set as in (a).
Proposition 1. The dynamics of (7)on the invariant blow-up chart θ= 0in case
Ω =β+
α−s2
2possesses the invariant line {q=−s
2}and, after time-rescaling, for q/ne}ationslash=−s
2
the Hamiltonian
H0(p,q) =−p2+q2+αsp+sq−h+β+/α+µ
q+s
2
along solutions of (8). Analogously on the chart θ=π, in case
Ω =β−
α+s2
2(12)
possesses the invariant line {q=s
2}and forq/ne}ationslash=s/2the Hamiltonian
Hπ(p,q) =p2+q2−αsp−sq+h−β−/α+µ
q−s
2. (13)
Moreover, each half plane {θ= 0,q≤ −s
2},{θ= 0,q≥ −s
2}/parenleftbig
{θ=π,q≤s
2}and
{θ=π,q≥s
2}/parenrightbig
is filled with periodic orbits encircling the equilibria at z0
±/parenleftbig
zπ
±/parenrightbig
if addi-
tionallyΩ>h−µ+s2
4(α2−1)/parenleftig
Ω<h+µ+s2
4(1+α2)/parenrightig
.
Proof.For the sake of clarity, we will only present the computation for the blow-up chart
θ=π; the computation on θ= 0 can be done in the same manner. With respect to the
parameters of equation (7), the condition Im(4 C−B2) = 0 is equivalent to Ω =β−
α+s2
2
and in this case the system (7) on {θ=π}is given by
p′=p2−q2−αsp+sq+h+µ−β−
α−s2
2
q′= 2pq−sp−αsq+αs2
2(14)
13We readily compute that for solutions of this
dHπ
dξ=∂Hπ
∂pp′+∂Hπ
∂qq′=q′·p′−p′·q′
(q−s
2)2= 0,
which shows the canonical Hamiltonian structure of (14) up to time r escaling. If addi-
tionally Ω< h+µ+s2
4(1+α2), it follows from Lemma 2 that each half plane is filled
with periodic orbits. /squaresolid
Proposition 1 concerns the special case that ccp∈(−1,1) andβ,Ω are such that (12)
holds, whichishenceforthreferredtoasthe center-case . Inparticular, eachorbitexcept
the lineq≡s/2 on the blow-up chart θ=πcan by identified via the quantity (13),
and each equilibrium zπ
±has a neighborhood filled with periodic orbits if additionally
h >β−
α−µ+s2
4(1−α2) (cf. Figure 5). Note the relation between the conditions (10)
and (11) and the conditions in Proposition 1 in case s= 0.
Based on Lemma 2, we also state the following uniqueness result.
Proposition 2. ForΩ<β+
α−s2
2/bracketleftig
Ω>β+
α−s2
2/bracketrightig
, orΩ =β+
α−s2
2andΩ≤h−µ+
s2
4(α2−1)there is a unique orbit with (θ,p,q)T(ξ)withθ(ξ)→0asξ→ −∞, and it
holds that (p+iq)(ξ)→z0
−/bracketleftbig
(p+iq)(ξ)→z0
+/bracketrightbig
asξ→ −∞.
Proof.The conditions on Ω are equivalent to those in Lemma 2. If the statem ent were
false, it nevertheless follows from Lemma 2 that ( p+iq)(ξ)→z0
−asξ→ −∞. However,
transverse to the blow-up chart, the equilibrium state Z0
−is stable for increasing ξand
thus repelling for decreasing ξ. This contradicts the requirement θ(ξ)→0 asξ→ −∞.
Together with the fact that Z0
−has a one-dimensional unstable manifold uniqueness
follows. Analogously in case Ω >β+
α−s2
2. /squaresolid
Domain walls are heteroclinic orbits between the blow-up charts and d ecisive for their
bifurcation structure are the dimensions (and directions) of un/s table manifolds of the
equilibria on these charts. Hence, we next discuss the equilibria Z0
±andZπ
±and their
stability.
Transverse to the blow-up charts in θ-direction we readily compute the linearization
∂θ(sin(θ)p) = cos(θ)p, i.e., the transverse eigenvalue is −Aθ·Re(zθ
±) atθ= 0 andπ,
respectively. The eigenvalues within the blow-up charts are determ ined by±iγ. With
σ=±, respectively, the eigenvalues for Z0
σare
ν0
1,σ=−σiγ0, ν0
2,σ=ν0
1,σ, ν0
3,σ= Re(z0
σ) (15)
and forZπ
σ
νπ
1,σ=σiγπ, νπ
2,σ=νπ
1,σ, νπ
3,σ=−Re(zπ
σ). (16)
Therefore, the signs of the real parts within each blow-up chart a re opposite at Zπ
+
compared to Zπ
−and determined by the sign of Re/parenleftbig
ν0,π
1,+/parenrightbig
. Hence, within the blow-up
charts each equilibrium is either two-dimensionally stable, unstable or a linearly neutral
center point.
142p
-11q
(a)2p
-11q
(b)2p
-11q
(c)
2p
-11q
(d)2p
-11q
(e)2p
-11q
(f)
Figure 6: Phase plane streamplots (with Mathematica ) in blow-up charts near the equilib-
riumzπ
−= (1,0) forα= 0.5,β= 0.1,µ=−1,ccp= 0, i.e., the second and third
equation of (7). (a-c) θ= 0 and (d-f) θ=π. (a,d) codim-2 regime, (b,e) center
case, where Ω = β/α+s2/2 holds on the chart θ=π, and (c,f) codim-0 regime.
The remaining parameters and equilibria in (a,d): h= 0.5,s0= 0.12, Ω0= 0.44,
andz0
+=−1.06−0.12i,zπ
+=−0.94+0.12i. In (b,e): h= 10.2,s0= 4, Ω 0= 8.2,
andz0
+=−3−4i,zπ
+= 1 + 4i. In (c,f): h= 50.0,s0= 19.92, Ω0= 40.04, and
z0
+=−10.96−19.92i,zπ
+= 8.96+19.92i.
For completeness, we next notethat the equilibria onbothblow-up c harts can beneutral
centers simultaneously (cf. Figure 3). However, this requires a ne gative spin polarization
and a small Gilbert damping factor, and is not further studied in this p aper.
Remark 4. The equilibria of both blow-up charts are centers simultane ously, if and only
ifIm(±γ0,π) = 0andγ0,π/ne}ationslash= 0(compare Lemma 2). For example if α= 0.5,β= 0.1,µ=
−1,ccp=−0.99,h= 10
s2=3960
199,Ω =β/α
1−ccp+s2
2=2000
199,
we obtain
γ0= 3.33551, γπ= 3.27469.
3 Domain Walls
All domain walls between ±e3that we are aware of are of coherent structure type,
and thus in one-to-one correspondence to heteroclinic connectio ns between the blow-up
15charts{θ= 0}and{θ=π}in(7). Typically we expect these to beheteroclinics between
equilibria within the charts, but this is not necessary. Based on the p revious analysis,
there are three options for heteroclinics between the charts: po int-to-point, point-to-
cycle, and cycle-to-cycle. We study the first two in this section, fo r which Proposition 2
implies uniqueness oftheDW(uptotranslations/rotations) foragiv enset ofparameters.
The third case can occur at most in a relatively small set of paramete rs (see Remark 4).
Its analysis is beyond the scope of this paper.
Note that in case of an existing connection between an equilibrium and a periodic orbit
(see Proposition 1), the domain wall is automatically an inhomogeneou s non-flat one.
Moreover, via the singular coordinate change any such heteroclinic solution is hetero-
clinic between θ= 0,πin (6) and through the spherical coordinates it is a heteroclinic
connection between ±e3in the sphere, possibly with unbounded ϕ.
3.1 Homogeneous Domain Walls
It is known from [13] in case β= 0 and from [12] in case ccp= 0 (and arbitrary β)
that (7) admits for µ<0 a family of explicit homogeneous DWs m0given by
θ0
p0
q0
=
2arctan/parenleftbig
eσ√−µξ/parenrightbig
σ√−µ
0
(17)
and parameterized by Ω =h+αβ
1+α2,s2=−(β−αh)2
µ(1+α2)2>0, andσ= 1 for positive speed s
andσ=−1 for negative s; the family extends to s= 0 in the limit h→β
αwith scaling
of the frequency by Ω =β
α+√−µ
αs. Fors= 0 (standing) fronts with both orientations
exist simultaneously ccp= 0 and are given by
θ0
p0
q0
=
2arctan/parenleftbig
e±√−µξ/parenrightbig
±√−µ
0
.
Hence, the branches of left and right moving walls as parametrized b yseach have
termination point at s= 0 (cf. Figure 13).
The family of explicit DWs (17) have domain wall width√−µ, a profile independent
of the applied field hand propagate along a nanowire ( µ <0) with velocity swhile
precessing with azimuthal velocity Ω. Since these are unique up to sp atial reflection
symmetry, the direction of motion is related to the spatial direction of connecting ±e3
throughσ,
θ(−∞) = 0θ(+∞) =π⇔s>0 (wall moves to the right)
θ(−∞) =π θ(+∞) = 0⇔s<0 (wall moves to the left) .(18)
To simplify some notation we will focus on the case of right-moving walls including
standing walls ( s≥0) and thus make the standing assumptions that h≥β/αas well as
16µ <0. We therefore have a 1-to-1 relation of parameters ( α,β,h,µ) and right-moving
DWs from
m(ξ,t) =m0(ξ,t;α,β,h,µ)
with speed and frequency given by
s0=s0(α,β,h,µ) :=αh−β√−µ(1+α2),Ω0= Ω0(α,β,h,µ) :=h+αβ
1+α2(19)
where the subindex 0 emphasizes that ccp= 0. Sinces0is surjective on R≥0any velocity
can be realised. Spatial reflection covers the case h≤β/α.
Based on Lemma 1 as well as Remark 3 for ccp= 0 and (homogeneous) speed and
frequency (19), one readily finds the asymptotic states of (7) giv en by
E0:=Z0
−/vextendsingle/vextendsingle
(s0,Ω0)=/parenleftbig
0,√−µ,0/parenrightbigTandEπ:=Zπ
−/vextendsingle/vextendsingle
(s0,Ω0)=/parenleftbig
π,√−µ,0/parenrightbigT,
with (spatial) eigenvalues (15), (16) given by
ν0
k,−:=−αs0−2√−µ−(−1)kis0, ν0
3,−=√−µ,
νπ
k,−:=−αs0+2√−µ−(−1)kis0, νπ
3,−=−√−µ,(20)
wherek= 1,2. Note that the above equilibria cannot be centers simultaneously ( recall
µ <0), hence a cycle-to-cycle connection can not exist close to it (see Remark 4 for
details). For this reason, we focus on point-to-point as well as poin t-to-cycle connections.
3.2 Inhomogeneous Domain Walls
Homogeneous DWs exist only in case ccp= 0 [12, Theorem 5], are explicitly given
by (17) and completely characterized by (19). By [12, Theorem 6], f ast inhomogeneous
DW solutions with |s| ≫1 exist for any ccp∈(−1,1), but in contrast to (17), the
gradient of these profiles is of order 1 /|s|and thus have a large ‘width’. The natural
question arises what happens for any sin caseccp/ne}ationslash= 0.
This section contains the main results of this paper: the existence, parameter selection
and structure of inhomogeneous DW solutions in case of small |ccp|for any value of the
applies field h, and thus any speed s. This will be achieved by perturbing away from the
explicit solution m0given by (17), where the bifurcation structure is largely determine d
by comparing the dimensions of the un/stable eigenspaces at the as ymptotic equilibrium
states, which are determined by (20).
LetW0
s/uandWπ
s/udenote the stable and unstable manifolds associated to E0and
Eπ, respectively, and w0
s/uas well aswπ
s/ube the dimension of these manifolds so that
w0
s+w0
u=wπ
s+wπ
u= 3. Notably w0
s= 2 andw0
u= 1 for all values of the parameters,
andwπ
sis either 1 or 3. Recall the standing assumption s0≥0.
Ifwπ
s= 1, the heteroclinic connection of E0andEπgenerically has codimension-2 ,
while forwπ
s= 3 it has codimension-0 , and we refer to the transition point between
17these cases, following the discussion in §2.1.1, as the center case . From (20) we have
wπ
s= 1⇔0≤s0<2√−µ
α,wπ
s= 3⇔s0>2√−µ
αand the center case at s0=2√−µ
α.
Hence, within the family of homogeneous DWs given by (17) and satisf ying (19), the
different bifurcation cases have speed and frequency relations
codim-0:s0>2√−µ
αand Ω 0>β
α−2µ
α2,
center:s0=2√−µ
αand Ω 0=β
α−2µ
α2,
codim-2: 0 ≤s0<2√−µ
αandβ
α≤Ω0<β
α−2µ
α2.
Using (19) these can be written in terms of the parameters of (LLG S), which gives the
characterization mentioned in the introduction §1.
Remark 5. The case distinction is also related to the spectral stabili ty of the asymptotic
statesm=±e3in the dynamics of the full PDE (LLGS)which is beyond the scope
of this paper, but see Figure 3 for an illustration. In short, it follows from, e.g., [12,
Lemma 1] that e3isL2-stable forh>β/α, while−e3isL2-stable forh<β/α −µand
unstable for h>β/α −µ. Based on this, the stability curves in Figure 3 are defined as
follows
Γ+:=β/α
h−µ−1,Γ−:= 1−β/α
h+µ,
which intersect at
h=β
2α+/radicalbigg
β2
4α2+µ2.
Since the destabilisation of −e3ifβ >0corresponds to a Hopf-instability of the (purely
essential)spectrum, it is effectively invisible in the coherent struct ure ODE, which detects
changes in the linearization at zero temporal eigenvalue on ly. Visible from the PDE
stability viewpoint is a transition of absolute spectrum th rough the origin in the complex
plane of temporal eigenmodes, cf. [18]. Now in the center cas e, the state −e3is already
L2-unstable since α>0as well asµ<0andh>β/α implies
h=h∗=β
α−2µ
α2(1+α2)>β
α−µ
and therefore that Γ−never intersects the line ccp≡0ath=h∗.
Moreover, it was shown in [19] that the family of explicit hom ogeneous DWs (9)is
(linearly) stable for sufficiently small applied fields, actually for h <−µ/2, in case
β= 0, hence in the bi-stable case where ±e3areL2-stable. As mentioned before, β= 0
is equivalent to ccp= 0in the LLGS equation with an additional shift in handβ, which
leads to the (LLG)case. We expect these DWs are also stable for small perturbat ions in
ccp, due to the properties of the operator established in [19], b ut further analysis also on
the transition from convective/transient to absolute inst ability will be done elsewhere.
18With these preparations, we next state the mainresults, which con cern existence of DWs
in the three regimes.
Theorem 1. For any parameter set (α0,β0,h0,µ0)in the codim-0 case, i.e., µ0<0and
h0> β0/α0−2µ0−2µ0/α2
0, the following holds. The explicit homogeneous DWs m0
in(9)lies in a smooth family mccpof DWs parameterized by (ccp,α,β,h,µ,s, Ω)near
(0,α0,β0,h0,µ0,s0,Ω0)withs0,Ω0from(19)evaluated at (α0,β0,h0,µ0). Moreover,
in caseccp= 0and(s,Ω)/ne}ationslash= (s0,Ω0)evaluated at (α,β,h,µ), orccp/ne}ationslash= 0, these are
inhomogeneous flat DWs.
Proof.As mentioned, in the codim-0 case we have wπ
s= 3 and for all parameters w0
u=
1. Due to the existence of the heteroclinic orbit (17), this means W0
uintersectsWπ
s
transversely and non-trivially for ccp= 0 in a unique trajectory. Therefore, this DW
perturbs to a locally unique family by the implicit function theorem for p erturbations
of the parameters in (7).
Forccp/ne}ationslash= 0 sufficiently small these are inhomogeneous DWs since the derivativ e of the
third equation, the q-equation, in (7) with respect to ccpis nonzero in this case; hence
already the equilibrium states move into the inhomogeneous regime.
Forccp= 0 but (s,Ω)/ne}ationslash= (s0,Ω0) at (α,β,h,µ), it follows from [12, Theorem 5] that
these DWs cannot be homogeneous. /squaresolid
Next we consider the center case, where h=h∗=β/α−2µ−2µ/α2. We start with
a result that follows from the same approach used in the codim-2 cas e and give refined
results below.
Corollary 1. The statement of Theorem 1 also holds for a parameter set in th e center
case if the perturbed parameters (ccp,α,β,h,µ,s, Ω)satisfyΩ> s2/2 +β−/α. IfΩ =
s2/2 +β−/αandΩ< h+µ+s2
4(1 +α2)the same holds except the DW is possibly
non-flat.
Proof.It follows from Proposition 1 and the discussion before that Ω >s2/2+β−/αfor
the parameter perturbation implies that the eigenvalues of the per turbed equilibrium
Zπ
−≈Eπsatisfy Re(νπ
k,−)<0,k= 1,2. Hence, the stable manifold at the target equi-
librium is two-dimensional and lies in a smooth family with the center-sta ble manifold
at the transition point. Then the proof is the same as in the codim-0 c ase. If the pertur-
bation has Ω = s2/2+β−/αthen we consider as target manifold the three-dimensional
stable manifold of a neighborhood of Zπ
−within the blow-up chart θ= 0. This neigh-
borhood consists of periodic orbits by Proposition 1 if Ω < h+µ+s2
4(1 +α2). By
dimensionality the intersection with the unstable manifold of Z0
−persists and yields a
heteroclinic orbit from the perturbed equilibrium at θ= 0 to the blow-up chart at θ=π.
Perturbing ccpaway from zero moves the left-asymptotic state into the inhomoge neous
regime and thus generates an inhomogeneous DWs. Note from Prop osition 1 that the
right-asymptotic state is either an equilibrium with q/ne}ationslash= 0 or a periodic orbit along which
q= 0 happens at most at two points. /squaresolid
19Next we present a refined result in which we show that typical pertu rbations indeed give
non-flat DWs, i.e., heteroclinic connections with right-asymptotic st ate being a periodic
orbit. The existence of flat DWs for ccp/ne}ationslash= 0 is severely constrained, but not ruled out
by this result. Our numerical results, such as those presented in §4, always lead to a
selected solution with a periodic asymptotic state.
In addition, attempts to perform numerical continuation (see §4) of flat DWs to ccp/ne}ationslash= 0
failed. Here we added the constraint /tildewideH= 0 and allowed adjustment the parameters h
ands, but the continuation process did not converge, which confirms nu merically the
generic selection of a periodic orbit.
As mentioned before, the right asymptotic state is e3in either case in the PDE coordi-
nates; the difference between flat and non-flat lies in the finer deta ils of how the profile
approaches e3in term ofpand alsoq, which relates to mthrough (2).
Theorem 2. Consider the smooth family of DWs from Corollary 1 with ccp= 0for
parameters satisfying (12)with fixedα >0,β≥0, andµ <0. Then there is a
neighborhood (ccp,s,h)of(0,s0,h∗)suchthat the followingholds. FlatDWs occurat most
on a surface in the (ccp,s,h)-parameter space and, for β/ne}ationslash= 0, satisfy|h−h0|2+|s−s0|2=
O(|ccp|3), more precisely (31)holds, where h0=h∗ands0= 2√−µ/α. Otherwise DWs
are non-flat, in particular all DWs not equal to m0forccp= 0orβ= 0are non-flat.
Due to its more technical nature, the proof of this theorem is defe rred to Appendix 6.1.
It remains to consider the codim-2 case.
Theorem 3. For any parameter set (α0,β0,h0,µ0)in the codim-2 case, i.e., µ0<0and
β0/α0≤h0< β0/α0−2µ0−2µ0/α2
0, the following holds. The explicit homogeneous
DWsm0in(9)lies in a smooth family of DWs parameterized by (ccp,α,β,h,µ )near
(0,α0,β0,h0,µ0). Here the values of (s,Ω)are functions of the parameters (ccp,α,β,h,µ )
and lie in a neighbourhood of (s0,Ω0)from(19). This family is locally unique near m0
and forccp/ne}ationslash= 0consists of inhomogeneous flat DWs.
The proof of Theorem 3 is presented in Appendix 6.2 and is based on th e Melnikov
method for perturbing from m0. As the unperturbed heteroclinic orbit has codimension
two, thebifurcationisstudied inathree-parametricfamilywithpert urbationparameters
η:= (ccp,s,Ω)T, which yields a two-component splitting function M(η) that measures
the mutual displacement of the manifolds W0
uandWπ
s.
Due to the fact that Re( νπ
j,−)<0 also forβ= 0 in the codim-0 regime and the case
β= 0 is included in Theorem 2 as well as Theorem 3, we immediately get the f ollowing
result.
Corollary 2. Inhomogeneous flat DWs also exist in the LLG equation (β= 0), which
can be flat or non-flat, respectively.
Theorem3 completes theexistence study ofDWs. Therefore, for any valueoftheapplied
fieldhthere exists a heteroclinic connection between the blow-up charts withccp/ne}ationslash= 0
andq/ne}ationslash≡0, thus an inhomogeneous (typically flat) DW. Recall that we have fo cused on
20right moving DWs, but all results are also valid for left moving walls due t o symmetry.
Therefore inhomogeneous DWs exist with ccp/ne}ationslash= 0 for any value of the applied field
h∈R.
4 Numerical Results
Numerical continuation for ordinary differential equations is an est ablished tool for bi-
furcation analysis in dynamical system. In this section we present c ontinuation results
to illustrate the analytical results discussed in §3. In particular, we will focus on con-
tinuation in the parameter ccpin the range of ( −0.5,0.5) as this perturbs away from the
known family m0from (17) (cf. Figure 1b) with speed and frequency determined by (19)
for a given applied field. Note that we also focus only on right-moving f ronts in this
section for reasons of clarity. All results were produced by contin uation in AUTO-07P
and graphics were created with Mathematica as well as MATLAB .
Heteroclinic orbits were detected as solutions to the boundary valu e problem given by
the desingularized system (7) plus a phase condition andboundary c onditions at ξ=−L
andξ=Ltaken from the analytic equilibrium states in pandqon the blow-up charts
(Remark 3). In the codim-2 case, the four required conditions are thep,qvalues at the
charts. In the center case, the three required conditions are: ( 1,2) the two p,qvalues
at the left chart and (3) the energy difference determined by the f unction (13). In the
codim-0 case, the two required conditions are the pvalues at both charts. Moreover, we
foundL= 50 was sufficiently large.
In order to relate to (LLGS), we plot most of the profiles after blow ing down to the
sphere rather than using the ODE phase space.
(a)-5 5-0.04-0.02
ξq
- 5 -101
ξ
m1
(b)
Figure 7: DWs obtained from continuation of m0in system (7) in the codim-2 regime h= 0.5
(h∗= 10.2) with initial speed and frequency s0= 0.12 as well as Ω 0= 0.44, and
(ccp,s,Ω) = (−0.5,0.11221,0.44077). (a) Projection onto the sphere. (b) Zoom-in
of corresponding q-profile (red) and m1component (blue).
Following the standing assumption on positive speeds and using ccpas well ashas the
main parameters, we keep the other parameters fixed with values
α= 0.5,β= 0.1,µ=−1.
21(a)
(b)
Figure 8: DWs obtained from continuation of m0in system (7) projected onto the sphere in
the codim-2 regime h= 10.1 (h∗= 10.2) with initial speed and frequency s0=
3.96 and Ω 0= 8.12. (a) ( ccp,s,Ω) = (−0.5,3.99541,8.05973). (b) ( ccp,s,Ω) =
(0.5,4.08089,8.22402).
The value of the applied field for the center case, given the fixed par ameters, is h∗= 10.2
(cf.§3.2), which leads to s0= 4.0 as well as Ω 0= 8.2 (cf. (19)).
4.1 Codim-2 case
The lower boundary for values of the applied field hlies in the codim-2 regime and
is given by h=β/α= 0.2. As a first numerical example we consider the slightly
larger value h= 0.5. The results upon continuation in the negative as well as positive
direction of ccpare presented in Figures 2a, 2c, and 7. The inhomogeneous nature of
these solutions ( ccp/ne}ationslash= 0) is reflected in the significantly varying azimuthal angles, also
visible in the oscillatory nature of the m1component in Figures 2c as well as 7b.
The linear part of the splitting function (33) (see Theorem 3), which predicts the direc-
tion of parameter variation for the existence of inhomogeneous DW s (ccp/ne}ationslash= 0) to leading
order, reads in this example
M(ccp,s,Ω) =/parenleftbigg
−0.00147567 −0.499245 0.245945
−0.000577908 −0.245945 −0.499245/parenrightbigg
·
ccp
s
Ω
,
sothatM= (0,0)Tfor(s,Ω) = (−0.00283744 ·ccp,0.000240252 ·ccp). Fortheparameter
values in Figure 7 and 2a we obtain, respectively,
M(−0.5,−0.007788,0.000771) = (0 .00481558,0.00181945)T,
M(0.5,−0.007973,0.007173) = (0 .00648248,−0.00133122)T.
Note that here the splitting of the (1-dimensional) unstable manifold of the left equilib-
rium and the (1-dimensional) stable manifold of the right equilibrium diffe r, i.e., are in
opposite directions (signs) in frequency and speed for variations in ccp.
22In addition note the decrease in frequency in the m1component, and thus also in the
m2as a result of the increase of the qcomponent towards zero, cf. Figure 7b. Here, the
azimuthal angle decreases since φ=/integraltext
qandq <0.
As a further example in the codim-2 regime, we consider h= 10.1<10.2 =h∗near the
upper boundary of the codim-2 regime in terms of the applied field h. The results of the
continuation in ccpare presented in Figure 8. The linear approximation of the splitting
in this case is given by M(−0.5,0.03541,−0.06027) = ( −0.000175537,−0.00104378)Tas
wellasM(0.5,0.12089,0.10402) = ( −0.00149519,0.0014975)Tin(a)and(b),respectively.
Note that the direction of splitting of the two components in this cas e is also dependent
on the polarity sign, as in the previous example. In both cases ( h= 0.5 andh= 10.1),
thecontinuationresultslookbasicallythesameinthecodim-2regime, wherethesolution
is, roughly speaking, constantly spiraling down from the north to th e south pole.
4.2 Center case
(a)-20 200.0450.055
ξq
(b)
Figure 9: DWs obtained from continuation of m0in system (7) in the center regime h=h∗=
10.2 withs=s0= 4,Ω = Ω 0= 8.2, andccp= 0.5. (a) Projection onto the sphere.
(b) Profile of corresponding q-component.
We perform computations in the center case with applied field h=h∗= 10.2 and fixed
frequency Ω = β−/α+s2/2 (see Proposition 1 and its discussion details). Theorem 2
shows that the right asymptotic state is generically a periodic orbit a nd more precisely
that in case ccp= 0, no constellation of handsexists, both not equal to zero, for which
the right asymptotic state is the (shifted) equilibrium. The results o f continuation in ccp
projected on the sphere look quite the same, which is why only the re sult forccp= 0.5
is presented in Figure 9. The fact that the right asymptotic state is a periodic orbit on
the blow-up chart θ=πis reflected by the nearly constant oscillations in the q-profile
forξclose to the right boundary (cf. Figure 9b).
That the right state is not the equilibrium in the blow-up chart is furth er corroborated
by computing the difference in energy /tildewideHbetween this equilibrium state (see Remark 3)
and the approximate right asymptotic state obtained from continu ation. The analytic
prediction of this difference up to second order is given by (31), whic h reads, for the
23-0.5 0 0.5
ccp-2-1010-6
(a)9.2 10.2 11.2
h-1010-4
(b)3.5 4 4.5
s-2010-3
(c)
Figure 10: Continuation of m0insystem(7)inthecenter casewithappliedfield h=h∗= 10.2
and fixed frequency Ω = β−/α+s2/2; heres0= 4. Shown is the energy difference
(solid blue line) between the equilibrium and asymptotic st ate from continuation
on right boundary against the continuation parameter ccpin (a),hin (b), and sin
(c). The red dashed curve in (b) and (c) is the quadratic appro ximation (21).
chosen parameters,
−0.006612+0.00673s−0.00183s2−0.00134h−0.000086h2+0.00077hs.(21)
As this analytic prediction is independent of ccpthe dependence of /tildewideHonccp≈0 is of
cubic or higher power. Indeed, the results plotted in Figure 10a sug gest an at least
quartic dependence since a maximum lies at ccp= 0. The asymmetric nature of the
graph suggests that odd powers appear in the expansion beyond o ur analysis, but also
note the order of 10−6in/tildewideH. In addition to the dependence on ccp, continuations for
ccp= 0 of/tildewideHinhwith fixeds=s0= 4 and in swith fixedh=h∗= 10.2 are plotted in
Figure 10b and 10c, respectively. Here we also plot the quadratic pr ediction (21).
4.3 Codim-0 case
Next, we consider an applied field h= 10.3 in the codim-0 regime, just above the applied
field value for the center case h∗= 10.2. The results of continuation in ccpare plotted
on the sphere in Figure 11. The azimuthal profile in φand hence in qare non-trivial as
predicted for inhomogeneous DWs.
In the ODE, qpossesses an oscillating profile and has a monotonically decreasing am -
plitude in both cases. This is a consequence of the proximity to the ce nter case and the
convergence to equilibria (see §3.2 for details). Recall that the speed and frequency are
not selected by the existence problem during continuation in ccp, but are taken as the
fixed parameters ( s0,Ω0) defined in (19).
The final example is for a relatively large applied field h= 50 in the codim-0 regime,
far away from the center case, and the results of continuation in ccpprojected on the
sphere are presented in Figure 2b as well as Figure 12a. Moreover, the corresponding
m1andm2profiles for ccp= 0.5 are presented in Figure 2d, and for ccp=−0.5, in
Figure 12b. As in the previous example, the inhomogeneous nature is visible in the
non-trivial azimuthal profile.
In summary, switching on the parameter ccpleads to a variety of inhomogeneous flat as
well as non-flat DW solutions, but also in case ccp= 0 there exist inhomogeneous DWs
24(a)
(b)
Figure 11: DWs obtained from continuation of m0in system (7) projected onto the sphere in
the codim-0 regime h= 10.3 (h∗= 10.2) and initial speed and frequency s0= 4.04
and Ω0= 8.28. (a)ccp=−0.5. (b)ccp= 0.5.
(a)-5 5-101
ξm1,m2
(b)
Figure 12: DWs obtained from continuation of m0in system (7) in the codim-0 regime h= 50
(h∗= 10.2) withinitial speedandfrequency s0= 19.92,Ω0= 40.04, andccp=−0.5.
(a) Projection onto the sphere. (b) Zoom-in of the correspon dingm1(solid blue)
andm2(dashed red) component.
(cf. Figure 4) which are much more complex than the homogeneous o ne given by (9) (cf.
Figure 1b).
Finally, recall from §3.1 that in the explicit family (9), the right moving DWs terminate
ats= 0. The question arises what happens for ccp/ne}ationslash= 0 along the parameter s. To
study this, we performed a continuation in the parameter sfor different values of ccp.
For decreasing swe found that numerical continuation failed at some s>0 forccp/ne}ationslash= 0
(cf. Figure 13). The details of this apparent existence boundary a re beyond the scope of
this paper. Note that the special role of sis reflected in the splitting function (34). In
cases0=s= 0, the first column of (32) is zero (see (34)) and thus the parame terization
ofccpcan not be written as a function in s.
In detail, we continued the analytic solution (9) in ccpaway from zero for different initial
values ofβ/α < h < h∗. This led to inhomogeneous ( ccp/ne}ationslash= 0) DWs, which we in turn
continued in the parameter stowards zero for different fixed values of ccpuntil the con-
tinuation process fails to converge. Based on this, there is numeric al evidence that DWs
with opposite speed sign (counter-propagating fronts) can only e xist simultaneously for
25s= 0 (standing fronts). We took a polynomial fit on these points as an approximation of
the existence boundary (cf. blue curve in Figure 13a). The continu ation process towards
the boundary is indicated by red arrows in Figure 13a for positive ccp. Additionally, the
corresponding results in the parameter space Ω and ccpis presented in 13b.
0 0.10.20.40.6
sc
c
(a)0.2 0.50.20.40.6
Ωc
(b)
Figure 13: Continuation results in sand Ω for fixed ccpin the codim-2 parameter regime.
Further parameters are α= 0.5,β= 0.1,µ=−1, andhfree. Blue solid line
represents interpolated termination boundaryfor sin (a) and Ω in (b), respectively.
Red arrows indicate continuation approach towards boundar y.
As a last point, we briefly describe the numerical method for time-int egration near DWs,
including freezing of speed and frequency (cf. Figure 4). All calcula tions were done
with the (free) software package pde2path which is based on a Finite Element Method
(FEM), cf. [20] and the references therein. Time-integration in pde2path with the so-
called ‘freezing’ method is discussed in [21]. In addition to the phase co ndition for the
speedweaddedaphaseconditionfortherotationandtime-integra tedvia asemi-implicit
Euler scheme.
5 Discussion and Outlook
We have presented results pertaining the existence of different ty pes of domain walls for
the LLG as well as LLGS equation. Our main focus has been on a nonze ro polarisation
parameterccp/ne}ationslash= 0 for any value of the applied field, including the high-field case, and
thus for any domain wall speed. These results extend what is known in particular for
inhomogeneously structured DWs, and we have discovered an appa rently new type of
DWs with certain oscillatory tails, referred to as non-flat here.
In detail, we have provided a classification of DWs based on co-dimens ion properties
in a reduced (spatial) coherent structure ODE, which relates to st ability and selection
properties that we review next. First, we have proven the existen ce of inhomogeneous
flat DWs in case ccp= 0 as well as ccp/ne}ationslash= 0 for an applied field above a certain threshold,
whichismainlymaterialdepending. Toourknowledge, theonlypreviou sexistence result
forccp/ne}ationslash= 0 with ’large’ applied fields concerns less relevant non-localized DWs [ 12]. Here
the existence problem does not select speed and frequency.
Second, we have discussed the so-called center case, which is char acterized by non-
hyperbolic equilibria in the underlying coherent structure ODE. In th is case, we have
26shown the existence of inhomogeneous DWs including the leading orde r selection mech-
anism. These solutions are non-flat in case ccp= 0 and generically also non-flat for ccp
away from zero, which was substantiated by numerical results. Th e fundamental obser-
vation has been the existence of a Hamiltonian function in a certain pa rameter regime
in the corresponding coherent structure ODE.
Third, we have proven the existence of inhomogeneous DWs in the so -called codim-2
regime, which is a range of values for the applied field in which the speed sis between
zero and the center case speed. In this regime, each solution in cas eccp/ne}ationslash= 0 is uniquely
determined by its speed as well as frequency. Here we have also pre sented the leading
order selection function in the coherent structure ODE variables pandq, which depends
on the speed s, the frequency Ω, and is independent of ccpfor standing fronts.
We believe that these results are not only interesting and relevant f rom a theoretical and
mathematical viewpoint, but also from an application viewpoint. They could help to
better understand the interfaces between different magnetic do mains in nanostructures,
e.g. in the development of racetrack memories, which are a promising prospective high
density storage unit that utilize a series of DWs by shifting at high spe ed along magnetic
nanowires through nanosecond current pulses.
In order to illustrate and corroborate these theoretical results , we have presented numer-
ical computations for a variety of values for the applied field in §4. On the one hand,
the examples in essence show that large applied fields lead to more com plex profiles of
the DWs in case ccp/ne}ationslash= 0. On the other hand, while in the center case the DWs projected
on the sphere appear similar to those for small applied fields, these s olutions approach
the poles in a qualitatively different ‘non-flat’ manner – as predicted b y our analysis.
Moreover, we compared the numerical and analytical results of th e selection mechanism
in the center case, showing that the analytical leading order appro ximation predicts the
effect of small perturbations in the parameters. Notably, for app lied fields above a cer-
tain threshold, where the existence analysis does not provide a sele ction of speed and
frequency, numerically the DWs selected in the PDE dynamics are in th e center case,
both forccp= 0 as well as ccp/ne}ationslash= 0. Hence, it might be possible to detect these solutions
in a high-field regime in real materials.
One question concerning existence beyond our analysis is whether in homogeneous (flat
or non-flat) solutions exist for any value of ccp∈(−1,1), and whether this class could
be utilized in applications.
A natural step towards the understanding of domain wall motion in n anowires beyond
the question of existence concerns the dynamic stability. For inhom ogeneous solutions
there appears to be no rigorous result in this direction. In particula r for larger applied
fields, stability results would be an essential step towards underst anding the selection
mechanismofsolutionsintermsofspeedandfrequency; ourfirstn umericalinvestigations
show that solutions inthe center parameter regime areselected, i.e ., inhomogeneous non-
flat DWs.
Moreover, preliminary analytic results, for ccp= 0 as well as ccp/ne}ationslash= 0, show that selection
mechanism is mainly determined by the value of the applied field, where in the bi-stable
27case (±e3linearly stable) homogeneous DWs are selected, and in the mono-sta ble case
inhomogeneous non-flatDWsareselected, which will bestudied indet ail inanupcoming
work.
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6 Appendix
6.1 Proof of Theorem 2
We use the notation
u=u(ξ;η,α,β,µ)= (θ(ξ;η,α,β,µ),p(ξ;η,α,β,µ),q(ξ;η,α,β,µ))T
and bifurcation parameters η= (ccp,s,h)T, wheres0andh0=h∗are defined below (see
§3.2 for details). The starting point for our perturbation analysis ar e the unperturbed
parameters and explicit heteroclinic solution in the center case (12) , where the frequency
is Ω0=s2
0/2+β/α. These are given by
η0:=
ccp0
s0
h0
:=
0
2√−µ
αβ
α−2µ−2µ
α2
29as well as
u0=u0(ξ;η0,α,β,µ) :=
θ0(ξ;η0,α,β,µ)
p0(ξ;η0,α,β,µ)
q0(ξ;η0,α,β,µ)
:=
2arctan(exp(√−µξ))√−µ
0
.
Unless stated otherwise, we suppress the explicit dependence of uonα,β, andµin the
following discussion. Let us write Zπ:=Zπ
−with the notation from Remark 3 so that
the unperturbed right asymptotic state is given by
Zπ(η0) =/parenleftbigg
π,αs0
2,s0
2−/radicalbigg
−µ
α2/parenrightbiggT
=/parenleftbig
π,√−µ,0/parenrightbigT
and its derivative with respect to ηis given by
Zπ
η(η0) =
0 0 0
0α
20
β
2√−µ2+α2
2−α
2√−µ
.
We write system (7) for brevity as
u′=f(u;η), (22)
sof(u;η) denotes the right side of (7). The linearization w.r.t. ηin the unperturbed
heteroclinic connection u0, given by (17), is the non-autonomous linear equation
u′
η=fu(u0;η0)uη+fη(u0;η0)η, (23)
whereuη= (θη,pη,qη)T. Its homogeneous part is
θ′
η=√−µcos(θ0)θη+pηsin(θ0)
p′
η=−(αs0+2√−µcos(θ0))pη+s0qη
q′
η=−s0pη−(αs0+2√−µcos(θ0))qη, (24)
withθ0(ξ) = 2arctan(exp(√−µξ)) due to (17). We next solve (24) and determine its
fundamental solution matrix.
The first obvious vector-solution of it is U1=u′
0= (θ′
0,0,0) since the second and the
third equation of (24) do not depend on θη. The other solutions can be obtained from
U1and the result of Lemma 1. Changing to polar coordinates
pη=rcosϕ, q η=rsinϕ,
the equations for pηandqηbecome
r′=−(αs0+2√−µcos(θ0))r
ϕ′=−s0,
30whose general solution can be written as
pη=r0r(ξ)cos(−s0ξ+ϕ0)
qη=r0r(ξ)sin(−s0ξ+ϕ0)
where
r(ξ) = exp
−αs0ξ−2√−µ/integraldisplay
ξcos(θ0(τ))dτ
=/parenleftig
1+e2√−µξ/parenrightig2
e(−2√−µ−αs0)ξ,
andr0,ϕ0arearbitrary integrationconstants corresponding to suitable init ial conditions.
Note that lim
ξ→±∞r(ξ) =∞for 0≤s0<2√−µ/α.
Next, the values of the integration constants have to be selected in order for the second
and the third vector-solutions
U2=
θ1
1
r1r(ξ)cos(−s0ξ+ϕ1)
r1r(ξ)sin(−s0ξ+ϕ1)
, U3=
θ2
1
r2r(ξ)cos(−s0ξ+ϕ2)
r2r(ξ)sin(−s0ξ+ϕ2)
(25)
tobelinearlyindependent. Here θ1
1,θ2
1arenotrelevantforwhatfollows. Thedeterminant
of the fundamental matrix reads
detΦ(ξ) = det(U1(ξ),U2(ξ),U3(ξ)) =r1r2r2(ξ)θ′
0(ξ)sin(ϕ2−ϕ1),
which is non-zero for r1=r2= 1,ϕ1= 0 andϕ2=π/2, i.e. detΦ( ξ) =r2(ξ)θ′
0(ξ).
Together, we get the fundamental solution matrix of the homogen eous part as
Φ(ξ) =
θ′
0(ξ)θ1
1(ξ) θ2
1(ξ)
0r(ξ)cos(−s0ξ)−r(ξ)sin(−s0ξ)
0r(ξ)sin(−s0ξ)r(ξ)cos(−s0ξ)
. (26)
The derivative of (22) with respect to ηis given by (23) and from the variation of
constants formula we get for some ξ0that
uη(ξ) = Φξ,ξ0uη(ξ0)+ξ/integraldisplay
ξ0Φξ,τfη(u0(τ);η0)dτ,
where Φ ξ,τ= Φ(ξ)·Φ−1(τ) is the evolution operator. Using (26) we find
Φξ,τ(ξ,τ;η) =
Θ1 Θ2 Θ3
0r(ξ)
r(τ)cos(−s0(ξ−τ))−r(ξ)
r(τ)sin(−s0(ξ−τ))
0r(ξ)
r(τ)sin(−s0(ξ−τ))r(ξ)
r(τ)cos(−s0(ξ−τ))
,(27)
where the explicit forms of the functions Θ 1,2,3(ξ) are not relevant for the remainder
of this proof. Since uη(ξ) tends to∂ηZ0
−forξ→ −∞the hyperbolicity of Z0
−(more
31precisely the resulting exponential dichotomy) implies Φ ξ,ξ0uη(ξ0)→0 asξ0→ −∞and
so
uη(ξ) =ξ/integraldisplay
−∞Φξ,τfη(u(τ;η0);η0)dτ. (28)
Regarding the limiting behavior as ξ→ ∞, recall that Corollary 1 states that the right
asymptotic limit of the perturbed heteroclinic orbit is either the pert urbed equilibrium
Zπ(η) or a periodic orbit around it in the blow-up chart at θ=π. The integral (28)
distinguishes these case in the sense that either it has a limit as ξ→+∞so the
heteroclinic orbit connects the two equilibria, or it does not and the h eteroclinic orbit
connects to a periodic solution.
We next determine uη(ξ) componentwise
uη(ξ) =v:=
v11v12v13
v21v22v23
v31v32v33
,
wherevijare the components of (28) and index i= 1,2,3 relates to θ,p,qas well as
j= 1,2,3 toccp,s,h.
Towards this, we compute
fη(u0(τ),η0) =
0 0 0
−β/α−√−µ
α(2+α2) 1
2β
1+e2√−µ τ√−µ0
,
and together with (28) and (27) we obtain
v21=−β
αIC−2βJS, v22=−√−µ
α(2+α2)IC−√−µIS, v23=IC,
v31=−β
αIS+2βJC, v32=−√−µ
α(2+α2)IS+√−µICv33=IS,
where
IC=IC(ξ):=ξ/integraldisplay
−∞(1+exp( −2√−µξ))2
(1+exp( −2√−µτ))2cos(−s0(ξ−τ))dτ,
IS=IS(ξ):=ξ/integraldisplay
−∞(1+exp( −2√−µξ))2
(1+exp( −2√−µτ))2sin(−s0(ξ−τ))dτ,
JC=JC(ξ):=ξ/integraldisplay
−∞exp(−2√−µτ)(1+exp( −2√−µξ))2
(1+exp( −2√−µτ))3cos(−s0(ξ−τ))dτ,
32ξ ξ0iπ
2√−µ
I1I2I3
I4
Figure 14: Contour CinCfor the integrals IandJ.
JS=JS(ξ):=ξ/integraldisplay
−∞exp(−2√−µτ)(1+exp( −2√−µξ))2
(1+exp( −2√−µτ))3sin(−s0(ξ−τ))dτ.
Note that we do not provide explicit formulas for v11,v12andv13, because they are
not needed for further computations. This is the reason why we ne glected the explicit
expressions of Θ 1,Θ2, and Θ 3before. We now introduce the following complex-valued
integrals for further computations:
I(ξ):=IC(ξ)+iIS(ξ) =ξ/integraldisplay
−∞/parenleftbig
1+e−2√−µξ/parenrightbig2
/parenleftbig
1+e−2√−µτ/parenrightbig2exp/parenleftbigg
−i2√−µ
α(ξ−τ)/parenrightbigg
dτ,
forICandISas well as
J(ξ):=JC(ξ)+iJS(ξ) =ξ/integraldisplay
−∞e−2√−µτ/parenleftbig
1+e−2√−µξ/parenrightbig2
/parenleftbig
1+e−2√−µτ/parenrightbig2exp/parenleftbigg
−i2√−µ
α(ξ−τ)/parenrightbigg
dτ
forJCandJS. We extend the above integrals to the complex plane and integrate a long
the counter-clockwise oriented rectangular contour Cas illustrated in Figure 14, and
letξ0→ −∞. We will provide the details of the computation of Ionly, asJcan be
calculated in a fully analogous way.
The complex integrand of Iis
g(z;ξ):=/parenleftbig
1+e−2√−µξ/parenrightbig2
/parenleftbig
1+e−2√−µz/parenrightbig2exp/parenleftbigg
−i2√−µ
α(ξ−z)/parenrightbigg
,
with singularities in Cat the pointsi(π+2kπ)
2√−µ,k∈Z, one of which lies in the interior of C,
namelyz0:=iπ
2√−µ. The contour integral Ican now be written via the residue theorem
as
I1(ξ)+I2(ξ)+I3(ξ)+I4(ξ) = 2πi/summationdisplay
intCResg(z;ξ),
whereI1,...,I 4are given by
33I1:z=x,
I1(ξ0,ξ) =/parenleftig
1+e−2√−µξ/parenrightig2
exp/parenleftbigg
−i2√−µ
αξ/parenrightbiggξ/integraldisplay
ξ0exp/parenleftig
i2√−µ
αx/parenrightig
/parenleftbig
1+e−2√−µx/parenrightbig2dx
I(ξ) = lim
ξ0→−∞I1(ξ0,ξ) =/parenleftig
1+e−2√−µξ/parenrightig2
exp/parenleftbigg
−i2√−µ
αξ/parenrightbiggξ/integraldisplay
−∞exp/parenleftig
i2√−µ
αx/parenrightig
/parenleftbig
1+e−2√−µx/parenrightbig2dx,
I2:z=ξ+iy,
I2(ξ) =/parenleftig
1+e−2√−µξ/parenrightig2π√−µ/integraldisplay
0iexp/parenleftig
−2√−µ
αy/parenrightig
/parenleftbig
1+e−2√−µ(ξ+iy)/parenrightbig2dy,
I3:z=x+π√−µi,
I3(ξ0,ξ) =/parenleftig
1+e−2√−µξ/parenrightig2
exp/parenleftbigg
−i2√−µ
αξ/parenrightbigg
e−2π
αξ0/integraldisplay
ξexp/parenleftig
i2√−µ
αx/parenrightig
/parenleftbig
1+e−2√−µξ/parenrightbig2dx
=−e−2π
αI1(ξ0,ξ),
I4:z=ξ0+iy,
I4(ξ0,ξ) =/parenleftig
1+e−2√−µξ/parenrightig2
exp/parenleftbigg
i2√−µ
α(ξ0−ξ)/parenrightbigg0/integraldisplay
π√−µiexp/parenleftig
−2√−µ
αy/parenrightig
/parenleftbig
1+e−2√−µ(ξ0+iy)/parenrightbig2dy,
lim
ξ0→−∞I4(ξ0,ξ) = 0.
Utilizing the Laurent series of gwe obtain
Resg(z;ξ)|z=z0=α+i
2α√−µe−π
α/parenleftig
1+e−2√−µξ/parenrightig2
exp/parenleftbigg
−i2√−µ
αξ/parenrightbigg
,
which leads to
I(ξ) =/parenleftig
1−e−2π
α/parenrightig−1/parenleftbiggπi(α+i)
α√−µe−π
α/parenleftig
1+e−2√−µξ/parenrightig2
exp/parenleftbigg
−i2√−µ
αξ/parenrightbigg
−I2(ξ)/parenrightbigg
.
Now we can write
IC(ξ) = ReI(ξ)
=π/parenleftbig
1+e−2√−µξ/parenrightbig2
α√−µ/parenleftbig
eπ
α−e−π
α/parenrightbig/bracketleftbigg
−cos/parenleftbigg
−2√−µ
αξ/parenrightbigg
−αsin/parenleftbigg
−2√−µ
αξ/parenrightbigg/bracketrightbigg
−1
1−e−2π
αIr
2(ξ)
34as well as
IS(ξ) = ImI(ξ)
=π/parenleftbig
1+e−2√−µξ/parenrightbig2
α√−µ/parenleftbig
eπ
α−e−π
α/parenrightbig/bracketleftbigg
αcos/parenleftbigg
−2√−µ
αξ/parenrightbigg
−sin/parenleftbigg
−2√−µ
αξ/parenrightbigg/bracketrightbigg
−1
1−e−2π
αIi
2(ξ),
whereIr
2(ξ) andIi
2(ξ) are the real and imaginary part of I2(ξ), respectively.
Studying the integral Jin a similar fashion, we obtain
JC(ξ) = ReJ(ξ)
=π/parenleftbig
1+e−2√−µξ/parenrightbig2
2α2√−µ/parenleftbig
eπ
α−e−π
α/parenrightbig/bracketleftbigg
αcos/parenleftbigg
−2√−µ
αξ/parenrightbigg
−sin/parenleftbigg
−2√−µ
αξ/parenrightbigg/bracketrightbigg
−1/parenleftig
1−e−2π
α/parenrightigJr
2(ξ)
as well as
JS(ξ) = ImJ(ξ)
=π/parenleftbig
1+e−2√−µξ/parenrightbig2
2α2√−µ/parenleftbig
eπ
α−e−π
α/parenrightbig/bracketleftbigg
cos/parenleftbigg
−2√−µ
αξ/parenrightbigg
+αsin/parenleftbigg
−2√−µ
αξ/parenrightbigg/bracketrightbigg
−1/parenleftig
1−e−2π
α/parenrightigJi
2(ξ),
where also here Jr
2(ξ) andJi
2(ξ) are the real and imaginary part of J2(ξ). Direct compu-
tations show also that
lim
ξ→+∞Ii
2(ξ) =α
2√−µ/parenleftbigg
1−exp/parenleftbigg
−2π
α/parenrightbigg/parenrightbigg
and
lim
ξ→+∞Ir
2(ξ) = lim
ξ→+∞Jr
2(ξ) = lim
ξ→+∞Ji
2(ξ) = 0.
Summing up, the second and third component of uη(ξ)·ηfor sufficiently large ξare
/parenleftigg
α
2s
β
2√−µccp+2+α2
2s−α
2√−µh/parenrightigg
+
π
ρ
/parenleftig
−1
α√−µh+2
α2s/parenrightig
cos/parenleftig
−2√−µ
αξ/parenrightig
+/parenleftig
−1√−µh+(3+α2)
αs/parenrightig
sin/parenleftig
−2√−µ
αξ/parenrightig
/parenleftig
1√−µh−(3+α2)
αs/parenrightig
cos/parenleftig
−2√−µ
αξ/parenrightig
+/parenleftig
−1
α√−µh+2
α2s/parenrightig
sin/parenleftig
−2√−µ
αξ/parenrightig
+
+O(e−2√−µξ),
whereρ:= exp(π/α)−exp(−π/α). One readily verifies that the oscillatory part in the
expression above vanishes if and only if sandhare zero and thus we infer that the
heteroclinic connection cannot be between equilibria to first order in the parameters.
In order to detect cancellations of these oscillatory parts for high er orders of sandh, we
next consider the behavior of the quantity (13) with respect to pa rameter perturbations.
With slight abuse of notation, for u= (θ,p,q)Twe writeH(u;η) :=H(p,q) evaluated
35at parameters η, and other parameters at some fixed value, and we always consider the
heteroclinic solutions from Corollary 1.
Our strategy in the following steps is as follows: we utilize the quantity Hbecause
limξ→+∞Halways exists along these solutions. In order to distinguish whether this
limit is an equilibrium or a periodic orbit, we consider
/tildewideH(u;η):=H(u;η)−H(Zπ;η),
i.e., the difference of the H-values of the (parameter dependent) equilibrium Zπand the
limit ofuasξtends to infinity. Expanding /tildewideHin the limit ξ→ ∞with respect to the
parameterηyields conditions for periodic asymptotics. In the following, subindice s of
Hdenote partial derivatives, e.g. Hu=∂uH.
Clearly,H(u0;η0) =H(Zπ(η0);η0), thus/tildewideH0= 0 and, since equilibria are critical points
ofH, we have/tildewideHu(u0;η0) =/tildewideHη(u0;η0) = (0,0,0)T. The second derivative is given by
d2
dη2/tildewideH=/an}bracketle{tuη,/tildewideHuuuη/an}bracketri}ht+/an}bracketle{tuη,/tildewideHuη/an}bracketri}ht+/an}bracketle{t/tildewideHηu,uη/an}bracketri}ht, (29)
since/tildewideHηηis the zero matrix, /tildewideHuthe zero vector, and /tildewideHuη=/tildewideHT
ηu. Thus
/tildewideH(u0+uηη;η) =1
2(uηη)THuu(u0;η0)(uηη)+(uηη)THuη(u0;η0)η
−1
2/parenleftbig
Zπ
η(η0)η/parenrightbigTHuu(Zπ(η0);η0)/parenleftbig
Zπ
η(η0)η/parenrightbig
−/parenleftbig
Zπ
η(η0)η/parenrightbigTHuη(Zπ(η0);η0)η+O/parenleftbig
/bardblη/bardbl3/parenrightbig
.(30)
With the derivatives Huu,Huηin (30) given by
Huu(u;η) =
0 0 0
02
q−s/2−2p−αs
(q−s/2)2
0−2p−αs
(q−s/2)22p2−αsp+h−β−/α+µ−s2/4
(q−s/2)3
,
Huη(u;η) =
0 0 0
0p−αq
(q−s/2)2 0
β/α
(1−ccp)2(q−s/2)2−p2−αpq−αs
2p−s
2q+h−β−/α+µ
(q−s/2)3 −1
(q−s/2)2
,
for the right hand side of (30) in the limit ξ→+∞we obtain
1
2(uηη)THuu(u0;η0)(uηη)+(uηη)THuη(u0;η0)η=−αβ2
4µ√−µc2
cp
+(4+5α2+α4)(α2ρ2−4(1+α2)π2)
4α3ρ2√−µ(s−s0)2−α4ρ2−4(1+α2)π2
4αρ2µ√−µ(h−h0)2
−αβ(2+α2)
2µccp(s−s0)+α2β
2µ√−µccp(h−h0)
+(2+α2)(α4ρ2−4(1+α2)π2)
2α2ρ2µ(s−s0)(h−h0),
36as well as
1
2/parenleftbig
Zπ
η(η0)η/parenrightbigTHuu(Zπ(η0);η0)/parenleftbig
Zπ
η(η0)η/parenrightbig
+/parenleftbig
Zπ
η(η0)η/parenrightbigTHuη(Zπ(η0);η0)η=
−αβ2
4µ√−µc2
cp+α(4+5α2+α4)
4√−µ(s−s0)2−α3
4µ√−µ(h−h0)2
−αβ(2+α2)
2µccp(s−s0)+α2β
2µ√−µccp(h−h0)+α2(2+α2)
2µ(s−s0)(h−h0).
Therefore, the expansion in the limit ξ→+∞is independent of ccpand reads
lim
ξ→∞/tildewideH(u0+uηη;η) =−(1+α2)2(4+α2)π2
α3ρ2√−µ(s−s0)2
−2(1+α2)(2+α2)π2
α2ρ2µ(s−s0)(h−h0)
+(1+α2)π2
αρ2µ√−µ(h−h0)2+O/parenleftbig
/bardblη−η0/bardbl3/parenrightbig
.(31)
Recallρ= exp(π/α)−exp(−π/α). One readily verifies that the resulting (binary)
quadratic form of (31) is negative definite for all α >0 so the only solution to the
leading order problem
d2
dη2/tildewideH(u0;η0) = 0
is thetrivial one( s,h) = (0,0), and anynon-trivial solution satisfies |s−s0|2+|h−h0|2=
O(|ccp|3).
In particular, for ccp= 0 there is a neighborhood of ( s0,h0) such that the only solution
is the trivial one, which is therefore also the case in the LLG equation . In caseccp/ne}ationslash= 0,
higher orders may lead to a solution with non-zero sand/orh, but there is numerical
evidence that such solutions do not exist (see §4 for details).
6.2 Proof of Theorem 3
The idea of the proof is to apply Lyapunov-Schmidt reduction, i.e., to determine a bi-
furcation equation whose solutions are in one-to-one correspond ence with heteroclinic
connections between equilibria (7) near one of the explicit solutions u0from (17) con-
necting the equilibria Z0
−andZπ
−. In the present context this is known as Melnikov’s
method, see for example [22].
Recallu0corresponds to a homogeneous DW for ccp= 0 with speed s0and rotation
frequency Ω 0given by (19). In the present codim-2 parameter regime we will show
that the bifurcation equation defines a codimension two bifurcation curve in the three-
dimensional parameter space ( ccp,s,Ω), which passes through the point (0 ,s0,Ω0). The
main part of the proof is to show the existence of certain integrals f or the considered
parameter set. These integrals are almost identical to the ones st udied within the proof
of Theorem 2 and we use the same approach.
37In this section we denote the parameter vector by η:= (ccp,s,Ω)T∈R3, with initial
valueη0= (0,s0,Ω0)Tcorresponding to the unperturbed values. The solutions of the
perturbedsystemcloseto u0hastheform u(ξ;η) =u0(ξ)+uη(ξ;η0)(η−η0)+O(/bardblη−η0/bardbl2),
whereuη= (θη,pη,qη)T=O(/bardblη−η0/bardbl).
As discussed in Appendix 6.1, the linearization (23) of system (7) aro und the unper-
turbed heteroclinic connection u0has the fundamental solution matrix Φ( ξ) as defined
in(26). Inthepresent codim-2casewithdim( W0
u) = dim(Wπ
s) = 1inR3, thebifurcation
equationM(η) = 0 entails two equations. Here M(η) measures the displacement of the
manifoldsW0
uandWπ
s, andwe willchoose thistobenearthepoint u0(0) =/parenleftbigπ
2,√−µ,0/parenrightbigT
in the directions given by vectors v1(0) andv2(0) from adjoint solutions as detailed be-
low. From the Taylor expansion M(η) =Mη(η0)(η−η0)+O(/bardblη−η0/bardbl2) we infer by the
implicit function theorem that a full rank of Mη(η0) implies a one-to-one correspondence
of solutions to the bifurcation equation with elements in the kernel o fM(η0).
In order to compute Mη(η0) and its rank, we project onto the transverse directions to
u0, which means to project the inhomogeneous part of equation (23) onto two linearly
independent bounded solutions v1,v2of the adjoint variational equation v′=−AT·v,
where
AT=
√−µcos(θ0) 0 0
sin(θ0)−αs0−2√−µcos(θ0) −s0
0 s0 −αs0−2√−µcos(θ0)
.
The solutions are given in terms of (25) by
v1=U1×U2
detΦ=
0
−1
r(ξ)sin(−s0ξ)
1
r(ξ)cos(−s0ξ)
andv2=U3×U1
detΦ=
0
1
r(ξ)cos(−s0ξ)
1
r(ξ)sin(−s0ξ)
.
Implementing the projection onto these, we obtain the so-called Melnikov integral
Mη(η0):=+∞/integraldisplay
−∞(v1,v2)T·fη(u0;η0)dξ
=/parenleftbigg
βICCα√−µIS−√−µICIS+αIC
βICS−α√−µIC−√−µIS−IC+αIS/parenrightbigg
,(32)
where
ICC:=+∞/integraldisplay
−∞/parenleftbig
1−e2√−µξ/parenrightbig
eαs0ξ+2√−µξcos(−s0ξ)
/parenleftbig
1+e2√−µξ/parenrightbig3dξ,
ICS:=+∞/integraldisplay
−∞/parenleftbig
1−e2√−µξ/parenrightbig
eαs0ξ+2√−µξsin(−s0ξ)
/parenleftbig
1+e2√−µξ/parenrightbig3dξ,
38IC:=+∞/integraldisplay
−∞eαs0ξcos(−s0ξ)/parenleftbig
1+e2√−µξ/parenrightbig
·/parenleftbig
1+e−2√−µξ/parenrightbigdξ,
IS:=+∞/integraldisplay
−∞eαs0ξsin(−s0ξ)/parenleftbig
1+e2√−µξ/parenrightbig
·/parenleftbig
1+e−2√−µξ/parenrightbigdξ.
We next show that the second and thirdcolumns in (32) have non-va nishing determinant
so that the rank is always 2, in particular also for β= 0.
For brevity, we present the calculations of ICCandICSonly, which are based on the
same idea as the computations in Appendix 6.1. The solutions for ICandIScan be
computed in an analogous way.
From Appendix 6.1 we know that the following integral would not exist in cases0=
2√−µ/αand one readily verifies the existence for s0= 0. Therefore, we first assume
0<s0<2√−µ/αfor the moment and discuss the case s0= 0 later. We set
I:=+∞/integraldisplay
−∞g(ξ)dξ, g(ξ):=/parenleftbig
1−e2√−µξ/parenrightbig
e(αs0+2√−µ)ξe−is0ξ
/parenleftbig
1+e2√−µξ/parenrightbig3,
so thatICC= Re(I) andICS= Im(I). Utilizing the same idea for the contour integral
as in Appendix 6.1 (cf. Figure 14) and the residue theorem, we obtain
I−eαs0iπ√−µ·es0π√−µ·I= 2πi/summationdisplay
Res(g).
The function ghas a pole of order three atiπ
2√−µand theLaurentseries gives
Res(g) =−(α−i)2s2
0
8µ√−µexp/parenleftbigg
(α−i)iπs0
2√−µ/parenrightbigg
,
and therefore
I=2πs2
0eπs0
2√−µ/bracketleftig
−2αcos/parenleftig
παs0
2√−µ/parenrightig
+(α2−1)sin/parenleftig
παs0
2√−µ/parenrightig/bracketrightig
8µ√−µ/parenleftig
1−eπs0√−µcos/parenleftig
παs0√−µ/parenrightig
−ieπs0√−µsin/parenleftig
παs0√−µ/parenrightig/parenrightig
+i2πs2
0eπs0
2√−µ/bracketleftig
(1−α2)cos/parenleftig
παs0
2√−µ/parenrightig
−2αsin/parenleftig
παs0
2√−µ/parenrightig/bracketrightig
8µ√−µ/parenleftig
1−eπs0√−µcos/parenleftig
παs0√−µ/parenrightig
−ieπs0√−µsin/parenleftig
παs0√−µ/parenrightig/parenrightig.
Separating the real and imaginary parts of Iwe get
ICC=πs2
0√−µeπs0
2√−µ
4µ2·2α/parenleftig
1−eπs0√−µ/parenrightig
cos/parenleftig
παs0
2√−µ/parenrightig
+(1−α2)/parenleftig
1+eπs0√−µ/parenrightig
sin/parenleftig
παs0
2√−µ/parenrightig
1+e2πs0√−µ−2eπs0√−µcos/parenleftig
παs0√−µ/parenrightig
39ICS=πs2
0√−µeπs0
2√−µ
4µ2·(1−α2)/parenleftig
1−eπs0√−µ/parenrightig
cos/parenleftig
παs0
2√−µ/parenrightig
+2α/parenleftig
1+eπs0√−µ/parenrightig
sin/parenleftig
παs0
2√−µ/parenrightig
1+e2πs0√−µ−2eπs0√−µcos/parenleftig
παs0√−µ/parenrightig .
As mentioned before, one analogously gets
IC=πs0eπs0
2√−µ
2µ·/parenleftig
1−eπs0√−µ/parenrightig
cos/parenleftig
παs0
2√−µ/parenrightig
−α/parenleftig
1+eπs0√−µ/parenrightig
sin/parenleftig
παs0
2√−µ/parenrightig
1+e2πs0√−µ−2eπs0√−µcos/parenleftig
παs0√−µ/parenrightig
IS=πs0eπs0
2√−µ
2µ·α/parenleftig
1−eπs0√−µ/parenrightig
cos/parenleftig
παs0
2√−µ/parenrightig
+/parenleftig
1+eπs0√−µ/parenrightig
sin/parenleftig
παs0
2√−µ/parenrightig
1+e2πs0√−µ−2eπs0√−µcos/parenleftig
παs0√−µ/parenrightig .
Having the explicit expressions for IC,IS,ICC, as well as ICS, we can study the rank
of (32) in case 0 < s0<2√−µ/α. The determinant of the second and third column
of (32) simplifies to
(αIS−IC)2+(IS+αIC)2=/parenleftbig
1+α2/parenrightbig2π2s2
0eπs0√−µ/ne}ationslash= 0,∀s0/ne}ationslash= 0.
Therefore,Mη(η0) has full rank and we obtain
M(η) =/parenleftbiggβICCα√−µIS−√−µICIS+αIC
βICS−α√−µIC−√−µIS−IC+αIS/parenrightbigg
·(η−η0)+O/parenleftbig
/bardblη−η0/bardbl2/parenrightbig
.(33)
In the remaining case s0= 0, which means h=β/α, we haveICC=ICS=IS= 0,
IC=1
2√−µ, and thus
Mη(η0) =/parenleftigg
0−1
2α
2√−µ
0−α
2−1
2√−µ/parenrightigg
, (34)
whose rank is always 2. Thus the splitting directions are independent ofccpin first order
in caseh=β/α. Moreover, note that the splitting in sis independent of the anisotropy
µto first order in case h=β/α. This completes the proof of Theorem 2.
40 |
2211.08048v2.Nonlinear_sub_switching_regime_of_magnetization_dynamics_in_photo_magnetic_garnets.pdf | 1
Nonlinear s ub-switching regime of magnetization dynamics in photo -magnetic garnets
A. Frej, I. Razdolski, A. Maziewski, and A. Stupakiewicz
Faculty of Physics, University of Bialystok, 1L Ciolkowskiego, 1 5-245 Bialystok, Poland
Abstract. We analyze, both experimentally and numerically, the nonlinear regime of the
photo -induced coherent magnetization dynamics in cobalt -doped yttrium iron garnet films.
Photo -magnetic excitation with femtosecond laser pulses reveals a strongly nonlinear
respo nse of the spin subsystem with a significant increase of the effective Gilbert damping. By
varying both laser fluence and the external magnetic field, we show that this nonlinearity
originates in the anharmonicity of the magnetic energy landscape. We numer ically map the
parameter workspace for the nonlinear photo -induced spin dynamics below the photo -
magnetic switching threshold. Corroborated by numerical simulations of the Landau -Lifshitz -
Gilbert equation, our results highlight the key role of the cubic sy mmetry of the magnetic
subsystem in reaching the nonlinear spin precession regime. These findings expand the
fundamental understanding of laser -induced nonlinear spin dynamics as well as facilitate the
development of applied photo -magnetism.
1. INTRODUCTION
Recently, a plethora of fundamental mechanisms for magnetization dynamics induced by
external stimul i at ultrashort time scale s has been actively d iscussed [1-5]. The main interest
is not only in the excit ation of spin precession but in the switching of ma gnetization between
multiple stable states, as it open s up rich possibilities for non-volatile magnetic data storage
technology . One of t he most intriguing example s is the phenomenon of ultrafast switching of
magnetization with laser pulses. Energy -efficie nt, non -thermal mechanisms of laser -induced
magnetization switching require a theoretical understanding of coherent magnetization
dynamics in a strongly non -equilibrium environment [6]. This quasiperiodic motion of
magnetization is often mode led as an oscillator where the key parameters , such as frequency
and damping , are considered within the framework of the Landau -Lifshit z-Gilbert (LLG)
equation [1, 7] . Although it is inhe rently designed to describe small -angle spin precession
with in the linear approximation, there are attempts to extend this formalism into the
nonlinear regime where the precession parameters become angle -dependent [8]. This is
particularly important in light of the discovery of the so -called p recessional switching , where
magnetization , having been impulsively driven out of equilibrium, ends its precessional
motion in a different minimum of the potential energy [6, 9 -11]. Obviously, such
magnetization trajectories are characterized by very large precession angles (usually on the
order of tens of degrees ). It is, however, generally believed that the magnetization excursion
from the equilibrium of about 10 -20 degrees is already sufficient for the violat ion of the linear
LLG approach [12, 13] . Thus, an intermediate regime under the switching stimulus threshold
exists, taking a large area in the phase space and presenting an intriguing c hallenge in
understanding fundamental spin dynamics.
An impulsive optical stimulus often results in a thermal excitation mechanism, inducing
concomitant temperature variations , which can impact the parameters of spin precession [14-
16]. This highlights the special role of the non -thermal optical mechanisms of switching [17-2
19]. Among those , we outline photo -magnetic excitation , which has been recently
demonstrated in dielectric Co -doped YIG (YIG:Co) films [6, 11] . There, laser photons at a
wavelength of 1300 nm resonantly excite the 5E → 5T2 electron transition s in Co -ions, resulting
in an emerging photo -induced magnetic anisotropy and thus in a highly efficient excitation of
the magnetic subsystem [6]. This photo -induced effective anisotropy field features a near ly
instant aneous rise time (within the femtosecond pump laser pulse duration), shifting the
equilibrium direction for the magnetization and thus triggering its l arge -amplitude precession.
In the sub -switching regime (at excitation strengths just below the switching threshold ), the
frequency of the photo -induced magnetization precession has been shown to depend on the
excit ation wavelength [20]. However, nonlinearities in magnetization dynamics in the sub-
switching regime have not yet been described in detail, and the underlying mechanism for the
frequency variations is not understood.
In this work , we systematically examine the intermediate sub -switching regime characterized
by large angles of magnetization precession and the nonlinear response of the spin system to
photo -magnetic excitations. We show a strong increase of the effective Gilbert damping at
elevated lase r-induced excitation levels and quantify its nonlinearity within the existing
phenomenological formalism [8]. We further map the nonlinear regime in the phase space
formed by the effective photo -induced anisotropy field and the external magnetic field.
Fig. 1. Sketch of m agnetization dynamics at various stimulus levels . Owing to the highly nonlinear
magnetization dynamics in the switching regime, the nonlinearity onset manifests in the sub -switching
regime too .
This paper is organized in the following order: in the first part, we describe the details of the
experiment for laser -induced large -amplitude magnetization precession. Next, we present the
experimental results, followed by the fitting analysis . Then, we complement our findings with
the results of numerical simulation of the photo -magnetic spin dynamics. Afterward , we
discuss the workspace of parameters for the sub-switching regime of laser -induced
magnetization precession. The paper ends with c onclusions.
3
2. EXPERIMENTAL DETAILS
The experiments were done on a 7.5 μm -thick YIG:Co film with a composition of
Y2CaFe 3.9Co0.1GeO 12. The Fe ions at the tetrahedral and octahedral sites are replaced by Co -
ions [21]. The sample was grown by liquid -phase epitaxy on a 400 μm-thick gadolinium gallium
garnet (GGG) substrate. It exhibits eight possible magnetization states along the garnet’s cubic
cell diagonals due to its cubic magnetocrystalline anisotropy ( 𝐾1=−8.4×103 𝑒𝑟𝑔/𝑐𝑚3)
dominating the energy landscape over the uniaxial anisotropy ( 𝐾𝑢=−2.5×103 𝑒𝑟𝑔/𝑐𝑚3).
Owing to the 4 ° miscut, additional in -plane anisotropy is introduced, tilting the magnetization
axes and resulting in slightly lower energy of half of the magnetiz ation states in comparison
to the others. In the absence of the external magnetic field, the equilibrium magnetic state
corresponds to the magnetization in the domains close to the <111> -type directions in YIG:Co
film. Measurements of the Gilbert damping 𝛼 using the fe rromagnetic resonance technique
resulted in 𝛼≈0.2. This relatively high damping is inextricably linked to the C o dopants [22-
24].
The n onlinearity of an oscillator is usually addressed by varying the intensity of the stimulus
and comparing the response of the system under study. Here , we investigated the nonlinear
magnetization dynamics by varying the optical pump fluence and , thus , the strength of the
photo -magnetic effective field driving the magnetization out of the equilibrium. We
perfor med systematic studies in various magnetic states of YIG:Co governed by the magnitude
of the external magnetic field s. The magnetic field 𝐻⊥ was applied perpendicular to the sample
plane and in -plane magnetic field 𝐻 was applied along the [110] direction of the YIG:Co crystal
by means of an electromagnet. Owing to the introduced miscut, the studied YIG:Co exhibits
four magnetic domains at 𝐻=0 [25]. The large jump at an in-plane magnetic f ield close to
zero shows the magnetization switching in the domain structures between four magnetic
phases. The optical spot size in this experiment was around 100 μm while the size of smaller
domains was around 5 μm, resulting in the spatial averaging of the domains in the
measurements. This behavior of magnetic domains was dis cussed and visualized in detail by
magneto -optical Faraday effect in our previous papers [6, 25] . With an increase of the
magnetic field up to a round 𝐻=0.4 kOe, larger and smaller domains are formed due to the
domain wall motion, eventually resulting in a formation of a single domain in a noncollinear
state. Upon further increase, the magnetization rotates towards the direction of the applied
field until a collinear state with in -plane magnetization orientation is reached at about 2 kOe
(see Fig. 2) . 4
Fig. 2. Magnetization reversal using static magneto -optical Faraday effect under perpendicular H (a)
and in -plane H (b) magnetic fields. The grey area indicates the magnetization switching in magnetic
domain structure [25]. The green area shows the saturation range with a collinear state of
magnetization.
Dynamic nonlinearities in the magne tic response were studied employing the pump -probe
technique relying on the optical excitation of the spin precession in YIG :Co film. The pumping
laser pulse at 1300 nm , with a duration of 50 fs and a repetition rate of 500 Hz , induce d spin
dynamics through the photo -magnetic mechanism [6]. The transient Faraday rotation of the
weak probe beam at 625 nm was used to monitor the dynamics of the out-of-plane
magnetization component Mz. The diameter of the pump spot was around 1 40 μm , while the
probe beam was focused within the pump spot with a size of around 50 μm . The fluence of
the pump beam was varied in the range of 0.2 6.5 mJ/cm2, below the switching threshold of
about 39 mJ/cm2 [20]. At 1300 nm pump wavelength, the optical absorption in our garnet is
about 12%. An estimation of the temperature increase ΔT due to the heat load for the laser
fluence of 6.5 mJ/cm2 results in ΔT <1 K (see Methods of Ref. 6). The polarization of both
beams was linear and set along the [100] crystallographic direction in YIG:Co for the pump and
the [010] direction for the probe pulse . The experiments were done at room temperature . At
each magnetic field, we performed a series of laser fluence -dependent pump -probe
experiments measuring the transients of an oscillating magnetization component normal to
the sample plane . We then used a phenomenological damped oscillator response function to
5
fit the experimental data and retrieve the fit parameters such as a mplitude, frequency,
lifetime and effective damping. In what follows, we analyze the obtained nonlinearities in the
response of the magnetic system and employ numerical simulations to reproduce the
experimental findings.
3. RESULTS
A. Time -resolved photo -magnetic dynamics
In order to determine the characteristics of the photo -magnetic precession , we carried out
time -resolved measurements of a transient Faraday rotation ∆𝜃𝐹 in YIG:Co film. Fig. 3(a-d)
exemplifies a few typical datasets obtained for four v arious pump fluences (between 1.7 and
6.5 mJ/cm2) in magnetic fields of various strength s. A general trend demonstrating a decrease
of the precession amplitude and an increase of its frequency is seen upon the magnetic field
increase . To get further insights into the magnetization dynamics, these datasets were fitted
with a damped sine function on top of a non -oscillatory, exponentially decaying background :
∆𝜃𝐹(∆𝑡)=𝐴𝐹sin(2𝜋𝑓∆𝑡+𝜙)exp (−∆𝑡
𝜏1)+𝐵exp (−∆𝑡
𝜏2), (1)
where 𝛥𝑡 is pump and probe time difference, 𝐴𝐹 is the amplitude, 𝑓 is the frequency , 𝜙 is the
phase, 𝜏1 is the decay time of precession, and 𝜏2 is the decay time of the background with an
amplitude 𝐵.
Fig. 3. Time -resolved Faraday rotation at different magnetic fields H (a-d) and laser fluence s (I1-I4
correspond to 1.7, 3.2, 5.0, and 6.5 mJ/cm2, respectively) . The normalized MZ on the vertical axis is
defined as ΔF/max, where max is obtained for saturation magnetization rotation at H (see. Fig. 2a).
The curves are offset vertically without rescaling. The s olid lines are fittings with the damped sine
function (Eq. 1) .
6
Fig. 4. Photo -magnetic precession parameters as a function of p ump fluence in different external
magnetic field H: a) amplitude of the Faraday rotation AF, b) frequency of the precession , and c)
effective damping. Different colors correspond to different external magnetic fields. The s olid lines are
the linear fits where applicable , while the dashed lines are the visual guides. Some of the error bars
are smaller than the data point symbols.
At low applied fields 𝐻<1 kOe, where the photo -magnetic anisotropy field ( 𝐻𝐿) contribution
to the total effective magnetic field is the strongest, the largest magnetization precession
amplitude is observed. Figure 4 show s the most important parameters of the magnetiza tion
precession, that is, amplitude, frequency and effective damping (Fig. 4a -c). The latter is
obtained from the frequency and the lifetime as (2𝜋𝑓𝜏1)−1. Although the amplitude
dependence on the pump fluence is mostly linear, the other two parameters exhibit a more
complicated dependence, which is indicative of the noticeable nonlinearity in the magnetic
system. In particular, at 𝐻=0.4 and 0.5 kOe, we observe d an increase in the effective
damping with laser fluence, resulting in a faster decay of the magnetic precession. This is
further corroborated by the frequency decrease seen in Fig. 4b. It is seen that the behavior of
the magnetic subsystem is noticeably dissimilar at low ( below 1 kOe) and high (above 2 kOe)
magnetic fields. At higher magnetic fie lds 𝐻>1 kOe we were unable to observe nonlinear
magnetization response at pump fluences up to 10 mJ/cm2. This is indicative of a significant
difference in the dynamic response in the collinear and noncollinear states of the magnetic
subsystem.
4. Nonlinear precession of magnetization in anisotropic cubic crystal s
The data shown in Fig. 4c clearly indicates the nonlinearity in the magnetic response
manifesting in the increase of the effective damping with the excitation (laser) fluence.
Previously, similar behavior was found in a number of metallic systems [26-29] and quickly
attributed to laser heating. Interestingly, Chen et al . [30] found a decrease of the effective
damping with laser fluence in FePt, while invoking the temper ature dependence of magnetic
inhomogeneities to explain the results. There, the impact of magnetic inhomogeneity -driven
damping contribution exhibits a similar response to laser heating and an increase in the static
magnetic field. A more complicated mecha nism relying on the temperature -dependent
7
competition between the surface and bulk anisotropy contributions and resulting in the
modification of the effective anisotropy field has been demonstrated in ultrathin Co/Pt
bilayers [31, 32] .
Nonlinear spin dynamics is a rapidly developing subfield enjoying rich prospects for ultrafast
spintronics [33]. Importantly, all those works featured thermal excitation of magnetization
dynamics in metallic, strongly absorptive systems. In stark contrast, we argue that the
mechanism in the Co-doped YIG studied here is essentially non -thermal. This negligible
temperature change ΔT is unable to induce significant variations of the parameters in the
magnetic syst em of YIG:Co (T N=450 K), thus ruling out the nonlinearity mechanism discussed
above. Rather, we note the work by M üller et al. [34], where the non -thermal nonlinear
regime of magnetization dynamics in CrO 2 at high laser fluences was ascribed to the spin -wave
instabilities at large precession amplitudes [35]. We also note the recently debated and
physically rich mechanisms of magnetic nonlinearities, such as spin inertia [36-39] and
relativistic effects [40, 41] . Yet, we argue that in our case of a cubic magnetic anisotropy -
dominated energy landscape, a much simpler explanation for the nonlinear spin dynamics can
be suggested. In particular, we attribute the amplitude -dependent effective dampin g to the
anharmonicity of the p otential well for magnetization .
Fig. 5. Energy landscape as a function of the polar angle 𝜃𝜑=45°in the linear (𝐻=2.5 kOe, green)
and nonlinear (𝐻=0.4 kOe, red) precession regime s. The d ashed lines are the parabolic fits in the
vicinity of the minima . 𝜃 is the polar angle of magnetization orientation measured from the normal to
the sample plane along the [001] axis in YIG:Co .
We performed numerical calculations of the energy density landscape 𝑊(𝜃,𝜑):
𝑊(𝜃,𝜑)=𝑊𝑐+𝑊𝑢+𝑊𝑑+𝑊𝑧 (2)
taking into account the following terms in the free energy of the system: the Zeeman energy
𝑊𝑧=−𝑴∙𝑯, demagnetizing field term 𝑊𝑑=−2𝜋𝑀𝑠2sin2𝜃, cubic 𝑊𝑐=𝐾1∙
(sin4𝜃sin2𝜑cos2𝜃+sin2𝜃cos2𝜃cos2𝜑+sin2𝜃cos2𝜃sin2𝜑) and uniaxial anisotropy
𝑊𝑢=𝐾𝑢sin2𝜃 (𝜃 and 𝜑 are the polar and azimuthal angles, respectively ). In the calculations,
we assume 𝐾1=−9∙ 103 erg/cm3, 𝐾𝑢=−3∙103 erg/cm3, and 𝑀𝑠 is the saturation
8
magnetization of 7.2 Oe [25]. Then, following [8] and [42], we calculate the precession
frequency 𝑓 and the effective damping 𝛼𝑒𝑓𝑓:
𝑓=𝛾
2𝜋𝑀𝑠sin𝜃√𝛿2𝑊
𝛿𝜃2𝛿2𝑊
𝛿𝜑2−(𝛿2𝑊
𝛿𝜃𝛿𝜑)2
, (3)
𝛼𝑒𝑓𝑓=𝛼0𝛾(𝛿2𝑊
𝛿𝜃2+𝛿2𝑊
𝛿𝜑2sin−2𝜃)
8𝜋2𝑓𝑀𝑠, (4)
where the 𝛾 is gyromagnetic ratio , and 𝛼0 is the Gilbert damping in YIG:Co [23, 24] . In Fig. 5,
we only show the total energy as a function of the polar angle 𝜃, to illustrate the
anharmonicity of the potential at small external in -plane magnetic fields. Experimental data
and calculations of the energy 𝑊(𝜃,𝜑) have been published in Refs. [25, 43] . There, it is seen
that at relative ly small external magnetic fields canting the magnetic state , the proximity of a
neighboring energy minimum (to the right) effectively modifies the potential well for the
corresponding o scillator (on the left) , introducing an anharmonicity . On the other hand, at
sufficiently large magnetic fields, whic h, owing to the Zeeman energy term, modify the
potential such that a single minimum emerges (shown in Fig. 5 in green), no nonlinearity is
expected. This is also in line with the decreas ing impact of the cubic symmetry in the magnetic
system, which is res ponsible for the anharmonicity of the energy potential.
To get yet another calculated quantity that can be compare d to the experiment, we
introduced the photo -magnetically in duced effective anisotropy term 𝐾𝐿. This contribution
depends on the laser fluence I through the effective light -induced field 𝐻𝐿∝𝐼 as:
𝐾𝐿=−2𝐻𝐿𝑀𝑠cos2𝜃 (5)
The presence of this term displaces the equilibrium for net magnetization. The equilibrium
direction s can be obtained by minimizing the total energy with and without the photo -
magn etic anisotropy term. Then, k nowing the angle between the perturbed and unperturbed
equilibrium directions for the magnetization, we calculate d the precession amplitude 𝐴. We
note the difference between the amplitudes 𝐴𝐹, which refers to the Faraday rotation of the
probe beam, and 𝐴 standing for the opening angle of magnetization precession. Alth ough both
are measured in degrees, their meaning is different.
Having repeated this for a few levels of optical excitation, we obtain ed a linear slope of the
amplitude vs excitation strength dependence. Figure 6 (a-c) illustrates the amplitude,
frequency , and (linear ) effective damping as a function of the external magnetic field. The
agreement between the calculated parameters and those obtained from fitting the
experimental data is an impressive indication of the validity of our total energy approach.
Further, the linear effective damping value of 𝛼≈0.2 obtained in the limit of strong field s, is
in good agreement with the values known for our Co -doped YIG from previous works [6, 24] .
In principle, the effective damping in garnets can increase towards lower magnetic fields.
Conventionally attributed to the extrinsic damping contributions, this behavior has been
observed in rare -earth iron garnets before as well and ascribed to the generation of the
backward volume spin w ave mode by ultrashort laser pulses [44]. It is worth noting that there
is no nonlinearity phenomenologically embedded in the approach given above. 9
Fig. 6. Photo -magnetic precession parameters at various magnetic fields: amplitude (a), frequency (b) ,
and (linear) effective damping (c). The points are from the experimental data, the solid lines are
calculated as described in the text. The dark rectangular points are obtained in the FMR experimen ts.
The g rey shaded area indicates the presence of a domain st ate (DS). The g reen shaded area show s the
magnetization saturation state .
Yet, the data presented in Fig. 4c indicates the persistent nonlinear behavior of the effective
damping. To clarify the role of the potential anharmonicity, we fitted the potentials 𝑊(𝜃,𝜑)
using a parabolic function with an anharmonic term :
𝑊(𝑥)=𝑊0+𝑘[(𝑥−𝑥0 )2+𝛽𝑥(𝑥−𝑥0)4] (6)
Here 𝑥=𝜃 or 𝜑, and 𝛽𝑥 is the anharmonicity parameter. We calculated it independently for
𝜃 and 𝜑 for each dataset of 𝑊(𝜃,𝜑) obtained at different values of the external magnetic
field 𝐻 by fitting the total energy with Eq. (6) in the vicinity of the energy minimum (Fig. 5) .
This anharmonicity should be examined on equal footing with the no nlinear damping
contribution. To quantify the latter, we follow the approach by Tiberkevich & Slavin [8] and
analyze the effective damping dependencies on the precession amplitude by means of fitting
a second -order polynomial to them :
𝛼=𝛼0+𝛼2𝐴2. (7)
The examples of th e fit curves are shown in Fig. 7 a, demonstrating a good quality of the fit
within a certain range of the amplitudes 𝐴 (below 45 ). It should, however, be noted that the
model in Ref. [8] has been developed for the in -plane magnetic anisotropy, and thus its
applicability for our case is limited. This is the re ason why we do not go beyond the amplitude
dependence of the effective damping and do not analy ze the frequency dependence on 𝐴 in
10
the limit of strong effective fields. We note that the amplitude 𝐴, the opening angle of the
precession, should be understood as a mathematical parameter only, and not as a true
excursion angle of magnetization obtained in the real experimental conditions. There, large
effective Gilbert damping values and a short decay t ime of the photo -magnetic anisotropy
preclude the excursion of magnetization from its equilibrium to reach these 𝐴 values.
Fig. 7. a) Effective damping in the linear and nonlinear precession regimes of the precession amplitude
𝐴. The lines are the second -order polynomial fits with Eq. (7). b ) Magnetic field dependence of the
nonlinearity parameters: n onlinear damping coefficient 𝛼2 (points, obtained from experiments) and
the 𝑊(𝜃) potential anharmonicity normalized 𝛽𝜃 (red line , calculated ).
We note that the anharmonicity parameter 𝛽𝑥 calculated for the W(θ) profiles was found to
be a few orders of magnitude larger than that obtained for W(𝜑). This difference in the
anharmonicity justifies our earlier decision to focus on the shape of W(θ) potential only (cf.
Fig. 5). This means that the potential for magnetization in the azimuthal plane is muc h closer
to the parabolic shape and much larger amplit udes of the magnetization precession are
required for it to start manifesting nonlinearities in dynamics. As such, we only consider the
anharmonicity 𝛽𝑥 originating in the W(θ) potential energy. I n Fig. 7b, we compare the 𝛽𝜃 (red
line) and 𝛼2 (points) dependencies on the external in -plane magnetic field. It is seen that its
general shape is very similar, corroborating our assumption that the potential anharmonicity
is the main driving force behind the obse rved nonlinearity. We argue that thanks to the c ubic
magnetic anisotropy in YIG:Co film, the potential anharmonicity -related mechanism of
nonlinearity allows for reaching the nonlinear regime at moderate excitation levels.
5. Simulation s of laser -induced magnetization dynamics
11
To further prove that the ob served nonlinearities in magnetization dynamics do not require
introducing additional inertial or relativistic terms [33], we complemented our experimental
findings with numerical simulations of the LLG equation:
𝑑𝐌
𝑑𝑡=−𝛾[𝐌×𝐇eff(𝑡)]+𝛼
𝑀𝑠(𝐌×𝑑𝐌
𝑑𝑡), (8)
where 𝐻𝑒𝑓𝑓 is the effective field derived from Eq. (2) as :
𝐇eff(𝑡)=−∂𝑊𝐴
∂𝑴+𝐇L(𝑡), (9)
We employ ed the simulation model from Ref. [11] and added a term corresponding to the
external magnetic field 𝐻. Calculations performed for a broad range of laser fluence s and
external field values allowed us to obtain a set of traces of the magnetization dynamics . Figure
8 show s a great deal of similarity between simulations and experimen tal data (cf. Fig. 3). It is
seen that t he frequency increases with increasing external field 𝐻 while the amplitude
decreases (see Fig. 8a). The simulations for various stimulus strengths show the expected
growth of the precession amplitude (see Fig. 8b).
Fig. 8. Photo -magnetic precession obtained in numerical simulations of the LLG equation for: a) field
dependence at moderate excitation level and b) power dependence (I=4, 10, 16, and 22 arb. units ) at
𝐻=0.4 kOe.
We further repeated our fit procedure with Eq.(1) to obtain the precession parameters from
these data. Figure 9 show s the values of the amplitude and frequency of the precession in the
power regime. At a low field 𝐻=0.4 kOe (red) , the nonlinearity is clearly visible and
comparable with experimental data, as seen in Fig. 4. Similarly, at high field s (green) , the
behavior is mostly linear. Figure 9a shows a great deal of similarity between simulations
(amplitude parameter) and experi mental data (normalize d value AF/max) (cf. Fig. 4a). The
analysis of the damping parameter (Fig. 9c) also confirms the exp erimental findings (as in Fig.
7a), revealing the existence of two regimes, linear and nonlinear . The results of the s imulations
confirm that the observation of the nonlinear response of the magnetic system can be
attributed to the anharmonicity of the energy landscape.
12
Notably, in the simulations , as well as in the experimental data, we not only observe a second -
order co rrection to the effective damping 𝛼2, but also a deviation from Eq.(7) at even larger
amplitudes (cf. Fig. 7 a and Fig. 9c). The latter manifests as a reduction of the effective damping
compared to the expected 𝛼0+𝛼2𝐴2 dependence shown with dashed lines. This higher -order
effect is unlikely to originate in the multi -magnon scattering contribution since the latter
would only further increase the effective damping [8]. We rather believe that th is is likely an
artifact of the used damped oscillator model where in the range of 𝛼𝑒𝑓𝑓≈1 the quasiperiodic
description of magnetization precession ceases to be physically justified.
Fig. 9. Power dependence of the a) amplitude and b) frequency as obtained in the simulations for low
(red dataset) and high (green dataset) external magnetic field s. c) Effective damping in the linear and
nonlinear precession regimes .
6. Photo -induced phase diagram of sub -switching regime
It is seen from both experimental and numerical results above that the cubic symmetry of the
magnetic system is key for the observed nonlinear magnetization dynamics. To quantify the
parameter space for the nonlinearity, we first estimate the realistic values of the effective
light -induced magnetic field 𝐻𝐿. Throughout a number of works on photo -magnetism in Co -
doped garnets, a single -ion approach to magnetic anisotropy is consistent ly utilized. We note
13
that in YIG:Co, it is the Co ions at tetrahedral sites that are predominantly responsible for the
cubic anisotropy of the magnetic energy landscape [22]. In the near -IR range, these ions are
resonantly excited at the 1300 nm wavelength, resulting in improved efficiency of the photo -
magnetic stimulus , as compared to previous works [45]. Further, we note that at the
magnetization switching threshold, about 90% of the Co3+ ions with a concentration on the
order of 1020 cm-3 are excited with incident photons [11, 46] . Taking into account the single -
ion contribution to the anisotropy 𝛥𝐾1~105 erg/cm3 [47], and assuming a linear relation
between the absorbed laser power (or fluence) and the effective photo -magnetic field 𝐻𝐿, for
the latter we find that 𝐻𝐿~1 kOe is sufficient for the magnetization switching. This means that
the sub -switching regime of magnetiz ation dynamics (cf. Fig. 1) refers to the laser fluences (as
well as wavelengths) , resulting in smaller effective fields.
We reiterate that in previous works, the impact of the external magnetic field on the photo -
magnetically driven magnetization precess ion has not been given detailed attention. To
address this gap , we plotted the amplitude of the precession 𝐴 calculated in the same way as
above in the sub-switching regime (Fig. 10) . As expected, the amplitude generally increases
with 𝐻𝐿. However, we n ote a critical external field of about 0.5 kOe at which the desired
amplitudes can be reached at smaller light -induced effective fields 𝐻𝐿. At this field, where the
system enters a single domain state, the potential curvature around the energy minimum
decreases, thus facilitating the large -angle precession. In other words, external magnetic field s
can a ct as leverage for the effective field of the photo -induced anisotropy, thus reducing the
magnetization switching threshold. An exhaustive study of magneti zation switching across the
parameter space shown in Fig. 10 remains an attractive perspective for future studies.
Fig. 10. Calculated amplitude m ap of the photo -induced magnetization precession in YIG:Co film.
In our analysis, we only considered a truly photo -magnetic excitation and neglected the laser -
induced effects of thermal origi n. It is, however, known that laser -driven heating can introduce
an additional, long -lasting modification of magnetic anisotropy in iron garnets [48, 49] . The
14
relatively long relaxation times associated with cooling are responsible for the concomitant
modulation of the precession parameters and thus facilitate nonlinearities in the response of
the magnetic system. Yet, 1300 nm laser excitation of magnetization dynamics in YIG :Co film
was shown to be highly polarization -dependent [6], thus indicating the dominant role of the
non-thermal excitation mechanism. On the other hand, the unavoid able laser -induced heating
with experi mental values of laser fluence in YIG:Co film has been estimated to not exceed 1 K
[6]. As such, we do not expect modification of the Gilbert damping associated with the
proximity of the ma gnetization compensation or N éel temperature in the ferromagnetic
garnet [50]. However, a detailed investigation of the temperature -dependent nonlinear
magnetization dynamics in the vicinity of the compensation point or a magnetic phase
transition [51, 52] represents another promising research direction. Further , exploring the
nonlinear regime in the response of the magnetic system to intense THz stimul i along the lines
discussed in [33] enjoys a rich potentia l for spintronic applications.
7. CONCLUSIONS
In summary, we studied, both experimentally and numerically, the nonlinear regime of
magnetization dynamics in photo -magn etic Co -doped YIG film. After excitation with
femtosecond laser pulses at fluences below the magnetization switching threshold, there is a
range of external magnetic field where the magnetic system demonstrates strongly non linear
precession characterized by a significant increase of t he effective Gilbert damping. We
attribute this nonlinearity to the anharmonicity of the potential for the magnetic oscillator
enhanced by the dominant role of the cubic magnetocrystalline anisotropy. The effective
damping and its nonlinear contribution, a s obtained from numerical simulations, both
demonstrate a very good agreement with the experimental findings. Simulations of the
magnetization dynamics by means of the LLG equation further confirm the nonlinearity in the
magnetic response below the switchi ng limit. Finally, we provide estimations for the realistic ,
effective photo -magnetic fields 𝐻𝐿 and map the workspace of the parameters in the sub -
switching, nonlinear regime of photo -induced magnetization dynamics.
ACKNOWLEDGMENTS
This work has been fu nded by the Foundation for Polish Science ( Grant No. POIR.04.04.00 -00-
413C/17) and the National Science Centre Poland (Grant No. DEC -2017/25/B/ST3/01305) .
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|
1903.05415v2.Higher_order_linearly_implicit_full_discretization_of_the_Landau__Lifshitz__Gilbert_equation.pdf | arXiv:1903.05415v2 [math.NA] 20 Mar 2020HIGHER-ORDER LINEARLY IMPLICIT FULL DISCRETIZATION
OF THE LANDAU–LIFSHITZ–GILBERT EQUATION
GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
Abstract. For the Landau–Lifshitz–Gilbert (LLG) equation of micromagnetics
we study linearly implicit backward difference formula (BDF) time discre tizations
up to order 5 combined with higher-order non-conforming finite elem ent space
discretizations, which are based on the weak formulation due to Alou ges but use
approximate tangent spaces that are defined by L2-averaged instead of nodal or-
thogonality constraints. We prove stability and optimal-order erro r bounds in the
situation of a sufficiently regular solution. For the BDF methods of or ders 3 to 5,
this requires that the damping parameter in the LLG equations be ab ove a posi-
tive threshold; this condition is not needed for the A-stable method s of orders 1
and 2, for which furthermore a discrete energy inequality irrespec tive of solution
regularity is proved.
1.Introduction
1.1.Scope.In this paper we study the convergence of higher-order time and s pace
discretizations of the Landau–Lifshitz–Gilbert (LLG) equation, wh ich is the basic
model for phenomena in micromagnetism, such as in recording media [ 26, 36].
The main novelty of the paper lies in the construction and analysis of w hat is
apparently the first numerical method for the LLG equation that is second-order
convergent in both space and time to sufficiently regular solutions an d that satisfies,
as an important robustness property irrespective of regularity, a discrete energy
inequality analogous to that of the continuous problem.
We study discretization in time by linearly implicit backward difference fo rmu-
lae (BDF) up to order 5 and discretization in space by finite elements o f arbitrary
polynomial degree. For the BDF methods up to order 2 we prove opt imal-order
error bounds in the situation of a sufficiently regular solution and a dis crete energy
inequality irrespective of solution regularity under very weak regula rity assumptions
on the data. For the BDF methods of orders 3 to 5, we prove optima l-order error
bounds in the situation of a sufficiently regular solution under the add itional condi-
tionthatthedampingparameter intheLLGequationbeaboveameth od-dependent
positive threshold. However, no discrete energy inequality irrespe ctive of solution
regularity is obtained for the BDF methods of orders 3 to 5.
The discretization in space is done by a higher-order non-conformin g finite ele-
ment method based on the approach of Alouges [4, 5], which uses a p rojection to
Date: March 23, 2020.
2010Mathematics Subject Classification. Primary 65M12, 65M15; Secondary 65L06.
Key words and phrases. BDF methods, non-conforming finite element method, Landau–
Lifshitz–Gilbert equation, energy technique, stability.
12 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
an approximate tangent space to the normality constraint. Contr ary to the point-
wise orthogonality constraints in the nodes, which define the appro ximate tangent
space in those papers and yield only first-order convergence also f or finite elements
with higher-degree polynomials, we here enforce orthogonality ave raged over the
finite element basis functions. With these modified approximate tang ent spaces we
proveH1-convergence of optimal order in space and time under the assump tion of
a sufficiently regular solution.
Key issues in the error analysis are the properties of the orthogon al projection
onto the approximate tangent space, the higher-order consiste ncy error analysis,
and the proof of stable error propagation, which is based on non-s tandard energy
estimates and uses both L2and maximum norm finite element analysis.
1.2.The Landau–Lifshitz–Gilbert equation. The standard phenomenological
model for micromagnetism is provided by the Landau–Lifshitz (LL) e quation
(1.1) ∂tm=−m×Heff−αm×(m×Heff)
where the unknown magnetization field m=m(x,t) takes values on the unit
sphereS2,α >0 is a dimensionless damping parameter, and the effective mag-
netic fieldHeffdepends on the unknown m. The Landau–Lifshitz equation (1.1)
can be equivalently written in the Landau–Lifshitz–Gilbert form
(1.2) α∂tm+m×∂tm= (1+α2)/bracketleftbig
Heff−/parenleftbig
m·Heff/parenrightbig
m/bracketrightbig
.
Indeed, in view of the vector identity a×(b×c) = (a·c)b−(a·b)c,fora,b,c∈R3,
we have−m×/parenleftbig
m×Heff/parenrightbig
=Heff−/parenleftbig
m·Heff/parenrightbig
m,and taking the vector product of
(1.1) withmand adding αtimes (1.1) then yields (1.2).
Sincem×ais orthogonal to m,for anya∈R3,it is obvious from (1.1) that
∂tmis orthogonal to m:m·∂tm= 0; we infer that the Euclidean norm satisfies
|m(x,t)|= 1 for allxand for allt, provided this is satisfied for the initial data.
The term in square brackets on the right-hand side in (1.2) can be re written as
P(m)Heff, where (with Ithe 3×3 unit matrix)
P(m) =I−mmT
is the orthogonal projection onto the tangent plane to the unit sp hereS2atm.
In this paper we consider the situation
(1.3) Heff=1
1+α2/parenleftbig
∆m+H/parenrightbig
,
whereH=H(x,t) is a given external magnetic field. The factor 1 /(1 +α2) is
chosen for convenience of presentation, but is inessential for th e theory; it can be
replaced by any positive constant factor.
Withthischoiceof Heff, we arriveattheLandau–Lifshitz–Gilbert (LLG)equation
in the form
(1.4) α∂tm+m×∂tm=P(m)(∆m+H).
Weconsiderthisequationasaninitial-boundaryvalueproblemonabou ndeddomain
Ω⊂R3and a time interval 0 /lessorequalslantt/lessorequalslant¯t, with homogeneous Neumann boundaryHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 3
conditions and initial data m0taking values on the unit sphere, i.e., the Euclidean
norm|m0(x)|equals 1 for all x∈Ω.
We consider the following weak formulation, first proposed by Alouge s [4, 5]:
Find the solution m:Ω×[0,¯t]→S2withm(·,0) =m0by determining, at
m(t)∈H1(Ω)3, the time derivative ∂tm(omitting here and in the following the
argumentt) as that function in the tangent space
T(m) :=/braceleftbig
ϕ∈L2(Ω)3:m·ϕ= 0 a.e./bracerightbig
=/braceleftbig
ϕ∈L2(Ω)3:P(m)ϕ=ϕ}
that satisfies, for all ϕ∈T(m)∩H1(Ω)3,
(1.5) α/parenleftbig
∂tm,ϕ/parenrightbig
+/parenleftbig
m×∂tm,ϕ/parenrightbig
+/parenleftbig
∇m,∇ϕ/parenrightbig
=/parenleftbig
H,ϕ/parenrightbig
,
where the brackets ( ·,·) denote the L2inner product over the domain Ω. The
numerical methods studied in this paper are based on this weak form ulation.
1.3.Previous work. There is a rich literature on numerical methods for Landau–
Lifshitz(–Gilbert) equations; for the numerical literature up to 20 07 see the review
by Cimr´ ak [17].
Alouges & Jaisson [4, 5] propose linear finite element discretizations in space and
linearly implicit backward Euler in time for the LLG equation in the weak fo rmula-
tion (1.5) and prove convergence withoutrates towards nonsmooth weak solutions,
using a discrete energy inequality and compactness arguments. Co nvergence of this
type was previously shown by Bartels & Prohl [11] for fully implicit meth ods that
are based on a different formulation of the Landau–Lifshitz equatio n (1.1). In [6],
convergence without rates towards weak solutions is shown for a m ethod that is
(formally) of “almost” order 2 in time, based on the midpoint rule, for the LLG
equation with an effective magnetic field of a more general type than (1.3).
In a complementary line of research, convergence withrates has been studied
under sufficiently strong regularity assumptions, which can, howev er, not be guar-
anteed over a given time interval, since solutions of the LLG equation may develop
singularities. A first-order error bound for a linearly implicit time discr etization
of the Landau–Lifshitz equation (1.1) was proved by Cimr´ ak [16]. Op timal-order
error bounds for linearly implicit time discretizations based on the bac kward Euler
and Crank–Nicolson methods combined with finite element full discret izations for
a different version of the Landau–Lifshitz equation (1.1) were obta ined under suf-
ficient regularity assumptions by Gao [23] and An [7], respectively. In contrast to
[4, 5, 6, 11], these methods do not satisfy an energy inequality irres pective of the
solution regularity.
Numerical discretizations for the coupled system of the LLG equat ion (1.5) with
the eddy current approximation of the Maxwell equations are stud ied by Feischl &
Tran [21], with first-order error bounds in space and time under suffi cient regularity
assumptions. This also yields thefirst result offirst-order conver gence ofthemethod
of Alouges & Jaisson [4, 5].
There are several methods for the LLG equations that are of for mal order 2 in
time (thoughonlyoforder 1 inspace), e.g., [35, 31, 19], but noneof t hemcomes with
an error analysis. Fully implicit BDF time discretizations for LLG equatio ns have4 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
been used successfully in the computational physics literature [37 ], though without
giving any error analysis.
To the authors’ knowledge, the second-order linearly implicit metho d proposed
and studied here is thus the first numerical method for the LLG (or LL) equation
thathasrigorousapriorierrorestimatesoforder2inbothspace andtimeunderhigh
regularity assumptions and that satisfies a discrete energy inequa lity irrespective of
regularity.
We conclude this brief survey of the literature with a remark: The ex isting con-
vergence results either give convergence of a subsequence witho ut rates to a weak
solution(withoutimposingstrongregularityassumptions), orthey showconvergence
with rates towards sufficiently regular solutions (as we do here). Bo th approaches
yield insight into the numerical methods and have their merits, and th ey comple-
ment each other. Clearly, neither approach is fully satisfactory, b ecause convergence
without rates of some subsequence is nothing to observe inactual computations, and
on the other hand high regularity is at best provable for close to con stant initial con-
ditions [22] or over short time intervals. We regard the situation as a nalogous to
the development of numerical methods and their analysis in other fie lds such as
nonlinear hyperbolic conservation laws: second-order methods ar e highly popular
in that field, even though they can only be shown to converge with ve ry low order
(1/2 or less or only without rates) for available regularity properties; s ee, e.g., [32,
Chapter 3]. Nevertheless, second-order methods arefavoredo ver first-order methods
in many applications, especially if they enjoy some qualitative propert ies that give
them robustness in non-regular situations. A similar situation occur s with the LLG
equation, where the most important qualitative property appears to be the energy
inequality.
1.4.Outline. InSection2we describe thenumerical methodsstudied inthispaper .
They use time discretization by linearly implicit BDF methods of orders u p to 5 and
space discretization by finite elements of arbitrary polynomial degr ee in a numerical
scheme that is based on the weak formulation (1.5), with an approxim ate tangent
space that enforces the orthogonality constraint approximately in anL2-projected
sense.
In Section 3 we state our main results:
•For the full discretization of (1.5) by linearly implicit BDF methods of or ders 1
and 2 and finite element methods of arbitrary polynomial degree we g ive optimal-
order error bounds in the H1norm, under very mild mesh conditions, in the case
of sufficiently regular solutions (Theorem 3.1). For these methods w e also show a
discrete energy inequality that requires only very weak regularity a ssumptions on
the data (Proposition 3.1). This discrete energy inequality is of the s ame type as
the one used in [5, 11] for proving convergence without rates to a weak solution.
•For the linearly implicit BDF methods of orders 3 to 5 and finite element m ethods
with polynomial degree at least 2, we have optimal-order error boun ds in theH1
norm only if the damping parameter αis larger than some positive threshold, which
depends on the order of the BDF method (Theorem 3.2). Moreover , a stronger (but
still mild) CFL condition τ/lessorequalslantchis required. A discrete energy inequality underHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 5
very weak regularity conditions is not available for the BDF methods o f orders 3
to 5, in contrast to the A-stable BDF methods of orders 1 and 2.
In Section 4 we prove a perturbation result for the continuous pro blem by energy
techniques, as a preparation for the proofs of our error bounds for the discretization.
In Section 5 we study properties of the L2-orthogonal projection onto the discrete
tangent space, which are needed to ensure consistency of the fu ll order and stability
of the space discretization with the higher-order discrete tangen t space.
In Section 6 we study consistency properties of the methods and p resent the error
equation.
In Sections 7 and 8 we prove Theorems 3.1 and 3.2, respectively. The higher-
order convergence proofs are separated into consistency (Sec tion 6) and stability
estimates. The stability proofs use the technique of energy estima tes, in an unusual
version where the error equation is tested with a projection of the discrete time
derivative of the error onto the discrete tangent space. These p roofs are different
for the A-stable BDF methods of orders 1 and 2 and for the BDF met hods of orders
3 to 5. For the control of nonlinearities, the stability proofs also re quire pointwise
error bounds, which are obtained with the help of finite element inver se inequalities
from theH1error bounds of previous time steps.
In Section 9 we illustrate our results by numerical experiments.
In an Appendix we collect basic results on energy techniques for BDF methods
that are needed for our stability proofs.
2.Discretization of the LLG equation
We now describe the time and space discretization that is proposed a nd studied
in this paper.
2.1.Time discretization by linearly implicit BDF methods. We shall dis-
cretize the LLG equation (1.5) in time by the linearly implicit k-step BDF methods,
1/lessorequalslantk/lessorequalslant5, described by the polynomials δandγ,
δ(ζ) =k/summationdisplay
ℓ=11
ℓ(1−ζ)ℓ=k/summationdisplay
j=0δjζj, γ(ζ) =1
ζ/bracketleftbig
1−(1−ζ)k/bracketrightbig
=k−1/summationdisplay
i=0γiζi.
We lettn=nτ, n= 0,...,N,be a uniform partition of the interval [0 ,¯t] with
time stepτ=¯t/N.For thek-step method we require kstarting values mifor
i= 0,...,k−1. Forn/greaterorequalslantk, we determine the approximation mntom(tn) as
follows. We first extrapolate the known values mn−k,...,mn−1to a preliminary
normalized approximation /hatwidermnattn,
(2.1) /hatwidermn:=k−1/summationdisplay
j=0γjmn−j−1/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay
j=0γjmn−j−1/vextendsingle/vextendsingle/vextendsingle.
To avoid potentially undefined quantities, we define /hatwidermnto be an arbitrary fixed
unit vector if the denominator in the above formula is zero.6 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
The derivative approximation ˙mnand the solution approximation mnare related
by the backward difference formula
(2.2) ˙mn=1
τk/summationdisplay
j=0δjmn−j,i.e.,mn=/parenleftig
−k/summationdisplay
j=1δjmn−j+τ˙mn/parenrightig
/δ0.
We determine mnby requiring that for all ϕ∈T(/hatwidermn)∩H1(Ω)3,
(2.3)α/parenleftbig
˙mn,ϕ/parenrightbig
+/parenleftbig
/hatwidermn×˙mn,ϕ/parenrightbig
+/parenleftbig
∇mn,∇ϕ/parenrightbig
=/parenleftbig
H(tn),ϕ/parenrightbig
˙mn∈T(/hatwidermn),i.e.,/hatwidermn·˙mn= 0.
Here we note that on inserting the formula in (2.2) for mnin the third term of (2.3),
we obtain a linear constrained elliptic equation for ˙mn∈T(/hatwidermn)∩H1(Ω)3of the
form
α/parenleftbig
˙mn,ϕ/parenrightbig
+/parenleftbig
/hatwidermn×˙mn,ϕ/parenrightbig
+τ
δ0/parenleftbig
∇˙mn,∇ϕ/parenrightbig
=/parenleftbig
fn,ϕ/parenrightbig
∀ϕ∈T(/hatwidermn)∩H1(Ω)3,
wherefnconsistsofknownterms. Thebilinearformontheleft-handsideis H1(Ω)3-
coercive on T(/hatwidermn)∩H1(Ω)3, and hence the above linear equation has a unique
solution ˙mn∈T(/hatwidermn)∩H1(Ω)3by the Lax–Milgram lemma. Once this elliptic
equation is solved for ˙mn, we obtain the approximation mn∈H1(Ω)3tom(tn)
from the second formula in (2.2).
2.2.Full discretization by BDF and higher-order finite elements .For a
familyofregularandquasi-uniformfiniteelement triangulationsof Ωwithmaximum
meshwidth h >0 we form the Lagrange finite element spaces Vh⊂H1(Ω) with
piecewise polynomials of degree r/greaterorequalslant1. We denote the L2-orthogonal projections
onto the finite element space by Πh:L2(Ω)→VhandΠh=I⊗Πh:L2(Ω)3→V3
h.
With a function m∈H1(Ω)3that vanishes nowhere on Ω, we associate the discrete
tangent space
(2.4)Th(m) ={ϕh∈V3
h: (m·ϕh,vh) = 0∀vh∈Vh}
={ϕh∈V3
h:Πh(m·ϕh) = 0}.
This space is different from the discrete tangent space used in [4, 5 ], where the
orthogonality constraint m·ϕh= 0 is required to hold pointwise at the finite
element nodes. Here, the constraint is enforced weakly on the finit e element space,
as is done in various saddle point problems for partial differential equ ations, for
example forthedivergence-free constraint inthe Stokes problem [14, 25]. Incontrast
to that example, here the bilinear form associated with the linear con straint, i.e.,
b(m;ϕh,vh) = (m·ϕh,vh), depends on the state m. This dependence substantially
affects both the implementation and the error analysis.
Following the general approach of [4, 5] with this modified discrete ta ngent space,
we discretize (1.5) in space by determining the time derivative ∂tmh(t)∈Th(mh(t))
such that (omitting the argument t)
(2.5)α/parenleftbig
∂tmh,ϕh/parenrightbig
+/parenleftbig
mh×∂tmh,ϕh/parenrightbig
+/parenleftbig
∇mh,∇ϕh/parenrightbig
=/parenleftbig
H,ϕh/parenrightbig
∀ϕh∈Th(mh),
where the brackets ( ·,·) denote again the L2inner product over the domain Ω.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 7
The full discretization with the linearly implicit BDF method is then readily
obtained from (2.3): determine ˙mn
h∈Th(/hatwidermn
h) such that
(2.6)α/parenleftbig˙mn
h,ϕh/parenrightbig
+/parenleftbig/hatwidermn
h×˙mn
h,ϕh/parenrightbig
+/parenleftbig
∇mn
h,∇ϕh/parenrightbig
=/parenleftbig
Hn,ϕh/parenrightbig
∀ϕh∈Th(/hatwidermn
h),
where/hatwidermn
hand˙mn
hare related to mn−j
hforj= 0,...,kin the same way as in (2.1)
and (2.2) above with mn−j
hin place ofmn−j, viz.,
(2.7) ˙mn
h=1
τk/summationdisplay
j=0δjmn−j
h,/hatwidermn
h=k−1/summationdisplay
j=0γjmn−j−1
h/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay
j=0γjmn−j−1
h/vextendsingle/vextendsingle/vextendsingle.
To avoid potentially undefined quantities, we define /hatwidermn
hto be an arbitrary fixed
unit vector if the denominator in the above formula is zero. (We will, ho wever, show
that this does not occur in the situation of sufficient regularity.)
To implement the discrete tangent space Th(/hatwidermn
h), there are at least two options:
using the constraints Πh(m·ϕh) = 0 or constructing a local basis of Th(m).
(a)Constraints : Letφifori= 1,...,N:= dimVhdenote the nodal basis of
Vhand denote the basis functions of V3
hbyφi=ek⊗φifori= (i,k), where
ekfork= 1,2,3 are the standard unit vectors of R3. We denote by MandA
the usual mass and stiffness matrices, respectively, with entries mij= (φi,φj)L2(Ω)
andaij= (∇φi,∇φj)L2(Ω)3. We further introduce the sparse skew-symmetric matrix
Sn= (sn
i,j)∈R3N×3Nwithentries sn
i,j= (/hatwidermn
h×φi,φj)L2(Ω)3andthesparseconstraint
matrixCn= (cn
i,j)∈R3N×Nbycn
i,j= (/hatwidermn
h·φi,φj)L2(Ω). Finally, we denote the
matrix of the unconstrained time-discrete problem as
Kn=αI⊗M+τ
δ0I⊗A+Sn.
Let ˙mn∈R3Ndenote the nodal vector of ˙mn
h∈Th(/hatwidermn
h). In this setting, (2.6) yields
a system of linear equations of saddle point type
Kn˙mn+(Cn)Tλn=fn,
Cn˙mn= 0,
whereλn∈RNis the unknown vector of Lagrange multipliers and fn∈R3Nis a
known right-hand side.
(b)Local basis : It is possible to compute a local basis of Th(m) by solving small
local problems. To see that, let ω⊂Ωdenote a collection of elements of the mesh
and letω⊃ωdenote the same set plus the layer of elements touching ω(the patch
ofω). A sufficient (and necessary) condition for ϕh∈V3
hwith supp(ϕh)⊆ωto
belong toTh(m) is
(2.8) ( m·ϕh,ψh) = 0 for all ψh∈Vhwith supp(ψh)⊆ω.
If we denote by # ωthe number of generalized hat functions of Vhsupported in ω,
the space of functions in V3
hwith support in ωis 3#ω-dimensional. On the other
hand, the space of test functions in (2.8) is # ω-dimensional. We may choose ω
sufficiently large (depending only on shape regularity) such that 3# ω >#ωand
hence (2.8) has at least one solution which is then a local basis functio n ofTh(m).
Choosing different ωto coverΩyields a full basis of Th(m).8 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
Let us denote the so obtained basis of Th(/hatwidermn
h) by (ψn
ℓ), given via ψn
ℓ=/summationtext
iφibn
iℓ,
and the sparse basis matrix by Bn= (bn
iℓ). Then, the nodal vector ˙ mn=Bnxnis
obtained by solving the linear system
(Bn)TKnBnxn= (Bn)Tfn.
An advantage of this approach is that the dimension is roughly halved compared
to the formulation with constraints. However, the efficiency of one approach versus
the other depends heavily on the numerical linear algebra used. Suc h comparisons
are outside the scope of this paper.
Remark 2.1. The algorithm described above does not enforce the norm constra int
|m|= 1 at the nodes. The user might add a normalization step in the definit ion
ofmnin (2.2). However, here we do not consider this normalized variant of the
method, whose convergence properties are not obvious to derive .
Remark 2.2. Differently to [4], we do not use the pointwise discrete tangent space
Tpw
h(m) ={ϕh∈V3
h:m·ϕ= 0 in every node }
={ϕh∈V3
h:Ih(m·ϕh) = 0}=IhP(m)V3
h,
whereIh:C(¯Ω)→Vhdenotesfiniteelementinterpolationand Ih=I⊗Ih:C(¯Ω)3→
V3
h. It is already reported in [4, Section 4] that an improvement of the o rder with
higher-degree finite elements could not be observed in numerical ex periments when
using the pointwise tangent spaces in the discretization (2.5). Our a nalysis shows
a lack of consistency of optimal order in the discretization with Tpw
h(m), which
originates from the fact that IhP(m) is not self-adjoint. The order reduction can,
however, be cured by adding a correction term: in the nth time step, determine
˙mn
h∈Tpw
h(/hatwidermn
h) such that for all ϕh∈Tpw
h(/hatwidermn
h),
(2.9)α/parenleftbig˙mn
h,ϕh/parenrightbig
+/parenleftbig/hatwidermn
h×˙mn
h,ϕh/parenrightbig
+/parenleftbig
∇mn
h,∇ϕh/parenrightbig
−/parenleftbig
∇/hatwidermn
h,∇(I−P(/hatwidermn
h))ϕh/parenrightbig
=/parenleftbig
P(/hatwidermn
h)H(tn),ϕh/parenrightbig
,
with notation /hatwidermn
hand˙mn
has in (2.7). With the techniques of the present paper, it
can be shown that like (2.6), also this discretization converges with o ptimal order
in theH1norm under sufficient regularity conditions. Since this paper is alread y
rather long, we do not include the proof of this result. In contrast to (2.6) for the
first- and second-order BDF methods, the method (2.9) does not admit anh- and
τ-independent bound of the energy that is irrespective of the smoo thness of the
solution.
3.Main results
3.1.Error bound and energy inequality for BDF of orders 1 and 2. For
the full discretization with first- and second-order BDF methods a nd finite elements
of arbitrary polynomial degree r/greaterorequalslant1 we will prove the following optimal-order error
bound in Sections 5 to 7.
Theorem 3.1 (Error bound for orders k= 1,2).Consider the full discretization
(2.6)of the LLG equation (1.4)by the linearly implicit k-step BDF time discretiza-
tion fork/lessorequalslant2and finite elements of polynomial degree r/greaterorequalslant1from a family ofHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 9
regular and quasi-uniform triangulations of Ω. Suppose that the solution mof the
LLG equation is sufficiently regular. Then, there exist ¯τ >0and¯h >0such that
for numerical solutions obtained with step sizes τ/lessorequalslant¯τand meshwidths h/lessorequalslant¯h, which
are restricted by the very mild CFL-type condition
τk/lessorequalslant¯ch1/2
with a sufficiently small constant ¯c(independent of handτ), the errors are bounded
by
(3.1) /ba∇dblmn
h−m(tn)/ba∇dblH1(Ω)3/lessorequalslantC(τk+hr)fortn=nτ/lessorequalslant¯t,
whereCis independent of h,τandn(but depends on αand exponentially on ¯t),
provided that the errors of the starting values also satisfy such a bound.
The precise regularity requirements are as follows:
(3.2)m∈Ck+1([0,¯t],L∞(Ω)3)∩C1([0,¯t],Wr+1,∞(Ω)3),
∆m+H∈C([0,¯t],Wr+1,∞(Ω)3).
Remark 3.1 (Discrepancy from normality ).Sincem(x,tn) are unit vectors, an
immediate consequence of the error estimate (3.1) is that
(3.3) /ba∇dbl1−|mn
h|/ba∇dblL2(Ω)/lessorequalslantC(τk+hr) fortn=nτ/lessorequalslant¯t,
with a constant Cindependent of n,τandh. The proof of Theorem 3.1 also shows
that the denominator in the definition of the normalized extrapolate d value/hatwidermn
h
satisfies
/vextenddouble/vextenddouble/vextenddouble1−/vextendsingle/vextendsinglek−1/summationdisplay
j=0γjmn−j−1
h/vextendsingle/vextendsingle/vextenddouble/vextenddouble/vextenddouble
L∞(Ω)/lessorequalslantCh−1/2(τk+hr)/lessorequalslant1
2fortn=nτ/lessorequalslant¯t,
which in particular ensures that /hatwidermn
his unambiguously defined.
Testing with ϕ=∂tm∈T(m) in (1.5), we obtain (only formally, if ∂tmis not
inH1(Ω)3)
α(∂tm,∂tm)+(∇m,∂t∇m) = (H,∂tm),
which, by integration in time and the Cauchy–Schwarz and Young ineq ualities, im-
plies the energy inequality
/ba∇dbl∇m(t)/ba∇dbl2
L2+1
2α/integraldisplayt
0/ba∇dbl∂tm(s)/ba∇dbl2
L2ds/lessorequalslant/ba∇dbl∇m(0)/ba∇dbl2
L2+1
2α/integraldisplayt
0/ba∇dblH(s)/ba∇dbl2
L2ds.
Similarly, wetestwith ϕh=˙mn
h∈Th(/hatwidermn
h)in(2.6). Thenwecanprovethefollow-
ing discrete energy inequality, which holds under very weak regularit y assumptions
on the data.
Proposition 3.1 (Energy inequality for orders k= 1,2).Consider the full dis-
cretization (2.6)of the LLG equation (1.4)by the linearly implicit k-step BDF time
discretization for k/lessorequalslant2and finite elements of polynomial degree r/greaterorequalslant1. Then, the10 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
numerical solution satisfies the following discrete energy inequality :forn/greaterorequalslantkwith
nτ/lessorequalslant¯t,
γ−
k/ba∇dbl∇mn
h/ba∇dbl2
L2+1
2ατn/summationdisplay
j=k/ba∇dbl˙mj
h/ba∇dbl2
L2/lessorequalslantγ+
kk−1/summationdisplay
i=0/ba∇dbl∇mi
h/ba∇dbl2
L2+τ
2αn/summationdisplay
j=k/ba∇dblH(tj)/ba∇dbl2
L2,
whereγ±
1= 1andγ±
2= (3±2√
2)/4.
This energy inequality is an important robustness indicator of the nu merical
method. In [5, 11], such energy inequalitys are used to prove conve rgence with-
out rates (for a subsequence τn→0 andhn→0) to a weak solution of the LLG
equation for the numerical schemes considered there (which have γ±= 1, but this
is inessential in the proofs).
As the proof of Proposition 3.1 is short, we give it here.
Proof.TheproofreliesontheA-stabilityofthefirst-andsecond-orderB DFmethods
via Dahlquist’s G-stabilitytheoryasexpressed inLemma 10.1ofthe Ap pendix, used
withδ(ζ) =/summationtextk
ℓ=1(1−ζ)ℓ/ℓandµ(ζ) = 1. The positive definite symmetric matrices
G= (gij)k
i,j=1are known to be G= 1 fork= 1 and (see [27, p.309])
G=1
4/parenleftbigg
1−2
−2 5/parenrightbigg
fork= 2,
which has the eigenvalues γ±= (3±2√
2)/4.
We test with ϕh=˙mn
h∈Th(/hatwidermn
h) in (2.6) and note/parenleftbig
/hatwidermn
h×˙mn
h,˙mn
h/parenrightbig
= 0, so that
α/ba∇dbl˙mn
h/ba∇dbl2
L2+(∇mn
h,∇˙mn
h) = (Hn,˙mn
h).
The right-hand side is bounded by
(Hn,˙mn
h)/lessorequalslantα
2/ba∇dbl˙mn
h/ba∇dbl2
L2+1
2α/ba∇dblHn/ba∇dbl2
L2.
Recalling the definition of ˙mn
h, we have by Lemma 10.1
(∇mn
h,∇˙mn
h)/greaterorequalslant1
τk/summationdisplay
i,j=1gij(∇mn−i+1
h,∇mn−j+1
h)−1
τk/summationdisplay
i,j=1gij(∇mn−i
h,∇mn−j
h).
We fix ¯nwithk/lessorequalslant¯n/lessorequalslant¯t/τand sum from n=kto ¯nto obtain
k/summationdisplay
i,j=1gij(∇m¯n−i+1
h,∇m¯n−j+1
h)+1
2ατ¯n/summationdisplay
n=k/ba∇dbl˙mn
h/ba∇dbl2
L2
/lessorequalslantk/summationdisplay
i,j=1gij(∇mk−i
h,∇mk−j
h)+τ
2α¯n/summationdisplay
n=k/ba∇dblHn/ba∇dbl2
L2.
Noting that
γ−/ba∇dbl∇m¯n
h/ba∇dbl2
L2/lessorequalslantk/summationdisplay
i,j=1gij(∇m¯n−i+1
h,∇m¯n−j+1
h),HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 11
k/summationdisplay
i,j=1gij(∇mk−i
h,∇mk−j
h)/lessorequalslantγ+k−1/summationdisplay
i=0/ba∇dbl∇mi
h/ba∇dbl2
L2,
we obtain the stated result. /square
3.2.Error bound for BDF of orders 3to5.For the BDF methods of orders
3 to 5 we prove the following result in Section 8. Here we require a stro nger, but
still moderate stepsize restriction in terms of the meshwidth. More importantly, we
must impose a positive lower bound on the damping parameter αof (1.1).
Theorem 3.2 (Error bound for orders k= 3,4,5).Consider the full discretization
(2.6)of the LLG equation (1.4)by the linearly implicit k-step BDF time discretiza-
tion for3/lessorequalslantk/lessorequalslant5and finite elements of polynomial degree r/greaterorequalslant2from a family of
regular and quasi-uniform triangulations of Ω. Suppose that the solution mof the
LLG equation has the regularity (3.2), and that the damping parameter αsatisfies
(3.4)α>α kwith
αk= 0.0913,0.4041,4.4348,fork= 3,4,5,respectively.
Then, for an arbitrary constant ¯C >0, there exist ¯τ >0and¯h >0such that for
numerical solutions obtained with step sizes τ/lessorequalslant¯τand meshwidths h/lessorequalslant¯hthat are
restricted by
(3.5) τ/lessorequalslant¯Ch,
the errors are bounded by
/ba∇dblmn
h−m(tn)/ba∇dblH1(Ω)3/lessorequalslantC(τk+hr)fortn=nτ/lessorequalslant¯t,
whereCis independent of h,τandn(but depends on αand exponentially on ¯C¯t),
provided that the errors of the starting values also satisfy such a bound.
Theorem 3.2 limits the use of the BDF methods of orders higher than 2 (and more
severely for orders higher than 3) to applications with a large dampin g parameter α,
such as cases described in [24, 39]. We remark, however, that in man y situations
αis of magnitude 10−2or even smaller [10]. A very small damping parameter α
affects not only the methods considered here. To our knowledge, t he error analysis
of any numerical method proposed in the literature breaks down as α→0, as does
the energy inequality.
It is not surprising that a positive lower bound on αarises for the methods of
ordersk/greaterorequalslant3, since they are not A-stable and a lower bound on αis required also for
the simplified linear problem ( α+i)∂tu=∆u, which arises from (1.4) by freezing m
in the termm×∂tmand diagonalizing this skew-symmetric linear operator (with
eigenvalues ±i and 0) and by omitting the projection P(m) on the right-hand side
of (1.4).
The proof of Theorem 3.2 uses a variant of the Nevanlinna–Odeh mult iplier tech-
nique [34], which is described in the Appendix for the convenience of th e reader.
While for sufficiently large αwe have an optimal-order error bound in the case of
a smooth solution, there is apparently no discrete energy inequality under weak
regularity assumptions similar to Proposition 3.1 for the BDF methods of orders 3
to 5.12 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
As in Remark 3.1, the error bounds also allow us to bound the discrepa ncy from
normality.
4.A continuous perturbation result
In this section we present a perturbation result for the continuou s problem, be-
cause we will later transfer the arguments of its proof to the discr etizations to prove
stability and convergence of the numerical methods.
Letm(t) be a solution of (1.4) for 0 /lessorequalslantt/lessorequalslant¯t, and letm⋆(t), also of unit length,
solve the same equation up to a defect d(t) for 0/lessorequalslantt/lessorequalslant¯t:
(4.1)α∂tm⋆+m⋆×∂tm⋆=P(m⋆)(∆m⋆+H)+d
=P(m)(∆m⋆+H)+r,
with
r=−/parenleftbig
P(m)−P(m⋆)/parenrightbig
(∆m⋆+H)+d.
Then,m⋆also solves the perturbed weak formulation
α(∂tm⋆,ϕ)+(m⋆×∂tm⋆,ϕ)+(∇m⋆,∇ϕ) = (r,ϕ)∀ϕ∈T(m)∩H1(Ω)3,
and the error e=m−m⋆satisfies the error equation
(4.2)α(∂te,ϕ)+(e×∂tm⋆,ϕ)+(m×∂te,ϕ)+(∇e,∇ϕ) =−(r,ϕ)
∀ϕ∈T(m)∩H1(Ω)3.
Before we turn to the perturbation result, we need Lipschitz-typ e bounds for the
orthogonal projection P(m) =I−mmTapplied to sufficiently regular functions.
Lemma 4.1. The projection P(·)satisfies the following estimates, for functions
m,m⋆,v:Ω→R3, wheremandm⋆take values on the unit sphere and m⋆∈
W1,∞(Ω)3:
/ba∇dbl(P(m)−P(m⋆))v/ba∇dblL2(Ω)3/lessorequalslant2/ba∇dblv/ba∇dblL∞(Ω)3/ba∇dblm−m⋆/ba∇dblL2(Ω)3,/vextenddouble/vextenddouble∇/parenleftbig
(P(m)−P(m⋆))v/parenrightbig/vextenddouble/vextenddouble
L2(Ω)3×3/lessorequalslant2/ba∇dblm⋆/ba∇dblW1,∞(Ω)3/ba∇dblv/ba∇dblW1,∞(Ω)3/ba∇dblm−m⋆/ba∇dblL2(Ω)3
+6/ba∇dblv/ba∇dblL∞(Ω)3/ba∇dbl∇(m−m⋆)/ba∇dblL2(Ω)3×3.
Proof.Settinge=m−m⋆, we start by rewriting
(P(m)−P(m⋆))v=−(mmT−m⋆mT
⋆)v=−(meT+emT
⋆)v.
The first inequality then follows immediately by taking the L2norm of both sides
of the above equality, using the fact that mandm⋆are of unit length. The second
inequality is proved similarly, using the product rule
∂i(P(m)−P(m⋆))v=−∂i(eeT+m⋆eT+emT
⋆)v
=−(∂ieeT+e∂ieT+∂im⋆eT+m⋆∂ieT+∂iemT
⋆+e∂imT
⋆)v
+(meT+emT
⋆)∂iv,
theL∞bound of∂im⋆, and the fact that /ba∇dble/ba∇dblL∞/lessorequalslant/ba∇dblm/ba∇dblL∞+/ba∇dblm⋆/ba∇dblL∞/lessorequalslant2./square
We have the following perturbation result.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 13
Lemma 4.2. Letm(t)andm⋆(t)be solutions of unit length of (1.5)and(4.1),
respectively, and suppose that, for 0/lessorequalslantt/lessorequalslant¯t, we have
(4.3)/ba∇dblm⋆(t)/ba∇dblW1,∞(Ω)3+/ba∇dbl∂tm⋆(t)/ba∇dblW1,∞(Ω)3/lessorequalslantR
and/ba∇dbl∆m⋆(t)+H(t)/ba∇dblL∞(Ω)3/lessorequalslantK.
Then, the error e(t) =m(t)−m⋆(t)satisfies, for 0/lessorequalslantt/lessorequalslant¯t,
(4.4) /ba∇dble(t)/ba∇dbl2
H1(Ω)3/lessorequalslantC/parenleftig
/ba∇dble(0)/ba∇dbl2
H1(Ω)3+/integraldisplayt
0/ba∇dbld(s)/ba∇dbl2
L2(Ω)3ds/parenrightig
,
where the constant Cdepends only on α,R,K, and¯t.
Proof.Let us first assume that ∂tm(t)∈H1(Ω)3for allt. Following [21], we test in
the error equation (4.2) with ϕ=P(m)∂te∈T(m). By the following argument,
this test function is then indeed in H1(Ω)3and can be viewed as a perturbation
of∂te:
ϕ=P(m)∂te=P(m)∂tm−P(m)∂tm⋆
=P(m)∂tm−P(m⋆)∂tm⋆−(P(m)−P(m⋆))∂tm⋆
=∂tm−∂tm⋆−(P(m)−P(m⋆))∂tm⋆,
and so we have
(4.5)ϕ=P(m)∂te=∂te+qwithq=−(P(m)−P(m⋆))∂tm⋆.
By Lemma 4.1 and using (4.3) we have
(4.6) /ba∇dblq/ba∇dblL2/lessorequalslant2R/ba∇dble/ba∇dblL2and/ba∇dbl∇q/ba∇dblL2/lessorequalslantCR/ba∇dble/ba∇dblH1.
Testing the error equation (4.2) with ϕ=∂te+q, we obtain
α(∂te,∂te+q)+(e×∂tm⋆,∂te+q)+(m×∂te,∂te+q)
+(∇e,∇(∂te+q)) =−(r,∂te+q),
where, by (4.1) and Lemma 4.1 with (4.3), ris bounded as
(4.7)/ba∇dblr/ba∇dblL2/lessorequalslant/ba∇dbl/parenleftbig
P(m)−P(m⋆)/parenrightbig
(∆m⋆+H)/ba∇dblL2+/ba∇dbld/ba∇dblL2
/lessorequalslant2K/ba∇dble/ba∇dblL2+/ba∇dbld/ba∇dblL2.
By collecting terms, and using the fact that ( m×∂te,∂te) vanishes, we altogether
obtain
α/ba∇dbl∂te/ba∇dbl2
L2+1
2d
dt/ba∇dbl∇e/ba∇dbl2
L2=−α(∂te,q)−(e×∂tm⋆,∂te+q)−(m×∂te,q)
−(∇e,∇q)−(r,∂te+q).
For the right-hand side, the Cauchy–Schwarz inequality and /ba∇dblm/ba∇dblL∞= 1 yield
α/ba∇dbl∂te/ba∇dbl2
L2+1
2d
dt/ba∇dbl∇e/ba∇dbl2
L2/lessorequalslantα/ba∇dbl∂te/ba∇dblL2/ba∇dblq/ba∇dblL2+R/ba∇dble/ba∇dblL2(/ba∇dbl∂te/ba∇dblL2+/ba∇dblq/ba∇dblL2)
+/ba∇dbl∂te/ba∇dblL2/ba∇dblq/ba∇dblL2+/ba∇dbl∇e/ba∇dblL2/ba∇dbl∇q/ba∇dblL2+/ba∇dblr/ba∇dblL2(/ba∇dbl∂te/ba∇dblL2+/ba∇dblq/ba∇dblL2).14 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
Young’s inequality and absorptions, together with the bounds in (4.6 ) and (4.7),
yield
α1
2/ba∇dbl∂te/ba∇dbl2
L2+1
2d
dt/ba∇dbl∇e/ba∇dbl2
L2/lessorequalslantc/ba∇dble/ba∇dbl2
H1+c/ba∇dbld/ba∇dbl2
L2.
Here, we note that
1
2d
dt/ba∇dble/ba∇dbl2
L2= (∂te,e)/lessorequalslant1
2/ba∇dbl∂te/ba∇dbl2
L2+1
2/ba∇dble/ba∇dbl2
L2,so that /ba∇dbl∂te/ba∇dbl2
L2/greaterorequalslantd
dt/ba∇dble/ba∇dbl2
L2−/ba∇dble/ba∇dbl2
L2.
Combining these inequalities and integrating in time, we obtain
/ba∇dble(t)/ba∇dbl2
H1/lessorequalslantc/ba∇dble(0)/ba∇dbl2
H1+c/integraldisplayt
0/ba∇dble(s)/ba∇dbl2
H1ds+c/integraldisplayt
0/ba∇dbld(s)/ba∇dbl2
L2ds.
By Gronwall’s inequality, we then obtain the stated error bound.
Finally, if∂tm(t) is not inH1(Ω)3for somet, then a regularization and density
argument, which we do not present here, yields the result, since th e error bound
does not depend on the H1norm of∂tm. /square
5.Orthogonal projection onto the discrete tangent space
For consistency and stability of the full discretization, we need to s tudy properties
of theL2(Ω)-orthogonal projection onto the discrete tangent space Th(m), which
we denote by
Ph(m):V3
h→Th(m).
We do not have an explicit expression for this projection, but the pr operties stated
in Lemmas 5.1 to 5.3 will be used for proving consistency and stability. W e recall
that we consider a quasi-uniform, shape-regular family Thof triangulations with
Lagrange finite elements of polynomial degree r.
The first lemma states that the projection Ph(m) approximates the orthogonal
projection P(m) =I−mmTonto the tangent space T(m) with optimal order. It
will be used in the consistency error analysis of Section 6.
Lemma 5.1. Form∈Wr+1,∞(Ω)3with|m|= 1almost everywhere we have
/ba∇dbl(Ph(m)−P(m))v/ba∇dblL2(Ω)3/lessorequalslantChr+1/ba∇dblv/ba∇dblHr+1(Ω)3,
/ba∇dbl(Ph(m)−P(m))v/ba∇dblH1(Ω)3/lessorequalslantChr/ba∇dblv/ba∇dblHr+1(Ω)3,
for allv∈Hr+1(Ω)3, whereCdepends on a bound of /ba∇dblm/ba∇dblWr+1,∞(Ω)3.
The second lemma states that the projection Ph(m) has Lipschitz bounds of the
same type as those of the orthogonal projection P(m) given in Lemma 4.1. It will
be used in the stability analysis of Sections 7 and 8.
Lemma 5.2. Letm∈W1,∞(Ω)3and/tildewiderm∈H1(Ω)3with|m|=|/tildewiderm|= 1almost
everywhere and /ba∇dblm/ba∇dblW1,∞/lessorequalslantR. There exist CR>0andhR>0such that for
h/lessorequalslanthR, for allvh∈V3
h,
(i)/ba∇dbl(Ph(m)−Ph(/tildewiderm))vh/ba∇dblL2(Ω)3/lessorequalslantCR/ba∇dblm−/tildewiderm/ba∇dblLp(Ω)3/ba∇dblvh/ba∇dblLq(Ω)3,
for(p,q)∈ {(2,∞),(∞,2)}, and
(ii)/ba∇dbl(Ph(m)−Ph(/tildewiderm))vh/ba∇dblH1(Ω)3/lessorequalslantCR/ba∇dblm−/tildewiderm/ba∇dblH1(Ω)3/ba∇dblvh/ba∇dblL∞(Ω)3HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 15
+CR/ba∇dblm−/tildewiderm/ba∇dblL2(Ω)3/ba∇dblvh/ba∇dblW1,∞(Ω)3.
The next lemma shows the Ws,p-stability of the projection. It is actually used for
p= 2 in the proof of Lemmas 5.1 and 5.2 and will be used for p= 2 in Section 6
and forp=∞in Sections 7 and 8.
Lemma 5.3. There exists a constant depending only on p∈[1,∞]and the shape
regularity of the mesh such that for all m∈W1,∞(Ω)3with|m|= 1almost every-
where,
/ba∇dblPh(m)vh/ba∇dblWs,p(Ω)3/lessorequalslantC/ba∇dblm/ba∇dbl2
W1,∞(Ω)3/ba∇dblvh/ba∇dblWs,p(Ω)3
for allvh∈V3
hands∈ {−1,0,1}.
These three lemmas will be proved in the course of this section, in whic h we
formulate also three more lemmas that are of independent interest but will not be
used in the following sections.
In the following, we use the dual norms
/ba∇dblv/ba∇dblW−1,q:= sup
w∈W1,p(v,w)
/ba∇dblw/ba∇dblW1,pfor 1/p+1/q= 1.
The space W−1,1(Ω) is not the dual space of W1,∞(Ω) but rather defined as the
closure ofL2(Ω) withrespect to thenorm /ba∇dbl·/ba∇dblW−1,1. Wealso recall that Πh:Ws,p(Ω)
→Ws,p(Ω) is uniformly bounded for s∈ {0,1}andp∈[1,∞] (see, e.g., [20]
for proofs in a much more general setting). By duality, we also obta in uniform
boundedness for s=−1 andp∈[1,∞]. A useful consequence is that for vh∈Vh,
/ba∇dblvh/ba∇dblW−1,q= sup
w∈W1,p(vh,Πhw)
/ba∇dblw/ba∇dblW1,p
/lessorequalslantsup
w∈W1,p(vh,Πhw)
/ba∇dblΠhw/ba∇dblW1,psup
w∈W1,p/ba∇dblΠhw/ba∇dblW1,p
/ba∇dblw/ba∇dblW1,p/lessorsimilarsup
wh∈Vh(vh,wh)
/ba∇dblwh/ba∇dblW1,p.
Lemma 5.4. There holds /ba∇dblv/ba∇dblWs,p(Ω)≃supw∈W−s,q(Ω)(v,w)
/bardblw/bardblW−s,q(Ω)with1/p+1/q= 1
forp∈[1,∞]ands∈ {−1,0,1}.
Proof.The interesting case is ( s,p) = (1,∞) since all other cases follow by duality.
Forv∈W1,∞(Ω), thereexists asequence offunctions qn∈C∞
0(Ω)3with/ba∇dblqn/ba∇dblL1= 1
such that
/ba∇dbl∇v/ba∇dblL∞= lim
n→∞(∇v,qn) = lim
n→∞−(v,divqn)/lessorequalslantsup
q∈W1,1(v,divq)
/ba∇dblq/ba∇dblL1.
Moreover, there holds
/ba∇dbldivq/ba∇dblW−1,1/lessorequalslantsup
w∈W1,∞(q,∇w)
/ba∇dbl∇w/ba∇dblL∞/lessorequalslant/ba∇dblq/ba∇dblL1.
Combining the last two estimates shows
/ba∇dbl∇v/ba∇dblL∞/lessorequalslantsup
w∈W−1,1(v,w)
/ba∇dblw/ba∇dblW−1,1.16 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
Since
/ba∇dblv/ba∇dblL∞= sup
w∈L1(v,w)
/ba∇dblw/ba∇dblL1/lessorequalslantsup
w∈W−1,1(v,w)
/ba∇dblw/ba∇dblW−1,1,
we conclude the proof. /square
Letthediscretenormalspace Nh(m) :=V3
h⊖Th(m)begivenasthe L2-orthogonal
complement of Th(m) inV3
h. We note that
(5.1) Nh(m) ={Πh(mψh) :ψh∈Vh}
by the definition of Th(m). The functions in the discrete normal space are bounded
from below as follows.
Lemma 5.5. For everyR >0, there exist hR>0andc >0such that for all
m∈W1,∞(Ω)3with|m|= 1almost everywhere and /ba∇dblm/ba∇dblW1,∞(Ω)/lessorequalslantRand for all
h/lessorequalslanthR,
/ba∇dblΠh(mψh)/ba∇dblWs,p(Ω)3/greaterorequalslantc/ba∇dblψh/ba∇dblWs,p(Ω)
for allψh∈Vhand(s,p)∈ {−1,0,1}×[1,∞].
Proof.(a) We first prove the result for s∈ {−1,0}. LetIh:C(Ω)→V3
hdenote the
nodal interpolation operator and define mh:=Ihm∈V3
h.
There holds
/ba∇dblΠh(mhψh)/ba∇dblLp/greaterorequalslant/ba∇dblmhψh/ba∇dblLp−/ba∇dbl(I−Πh)(mhψh)/ba∇dblLp.
Moreover, stability of ΠhinLp(Ω)3, for 1/lessorequalslantp/lessorequalslant∞, see [20], implies the estimate
/ba∇dbl(I−Πh)(mhψh)/ba∇dblLp/lessorequalslant(1+C) inf
vh∈V3
h/ba∇dblmhψh−vh/ba∇dblLp.
In turn, this implies
/ba∇dbl(I−Πh)(mhψh)/ba∇dblLp/lessorsimilar/ba∇dbl(I−Ih)(mhψh)/ba∇dblLp
=/parenleftig/summationdisplay
T∈Th/ba∇dbl(I−Ih)(mhψh)/ba∇dblp
Lp(T)3/parenrightig1/p
.
For each element, the approximation properties of Ihshow
/ba∇dbl(I−Ih)(mhψh)/ba∇dblLp(T)3/lessorsimilarhr+1/ba∇dbl∇r+1(mhψh)/ba∇dblLp(T)3
/lessorequalslanthr+1/summationdisplay
i+j=r+1/ba∇dbl∇min{i,r}mh/ba∇dblL∞(T)3/ba∇dbl∇min{j,r}ψh/ba∇dblLp(T)3.
Thus, multiple inverse estimates yield
/ba∇dbl(I−Ih)(mhψh)/ba∇dblLp(T)3/lessorsimilarh/ba∇dblmh/ba∇dblW1,∞/ba∇dblψh/ba∇dblLp(T)3.
Moreover, we have
/ba∇dblmhψh/ba∇dblLp/greaterorequalslant/ba∇dblmψh/ba∇dblLp−/ba∇dbl(m−mh)ψh/ba∇dblLp/greaterorequalslant1
2/ba∇dblψh/ba∇dblLp
provided that /ba∇dblm−mh/ba∇dblL∞/lessorequalslant1
2, which in view of
/ba∇dblm−mh/ba∇dblL∞=/ba∇dbl(I−Ih)m/ba∇dblL∞/lessorsimilarh/ba∇dbl∇m/ba∇dblL∞
is satisfied for h/lessorequalslanthRwith a sufficiently small hR>0 that depends only on R.
Altogether, this shows
/ba∇dblΠh(mhψh)/ba∇dblLp/greaterorsimilar/ba∇dblψh/ba∇dblLpHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 17
forh/lessorequalslanthR. Similarly we estimate
/ba∇dblΠh((m−mh)ψh)/ba∇dblLp/lessorsimilar/ba∇dblm−mh/ba∇dblL∞/ba∇dblψh/ba∇dblLp/lessorsimilarh/ba∇dbl∇m/ba∇dblL∞/ba∇dblψh/ba∇dblLp.
Altogether, we obtain
/ba∇dblΠh(mψh)/ba∇dblLp/greaterorsimilar/ba∇dblΠh(mhψh)/ba∇dblLp−/ba∇dblΠh((mh−m)ψh)/ba∇dblLp/greaterorsimilar/ba∇dblψh/ba∇dblLp
forh/lessorequalslanthR. This concludes the proof for s= 0. Finally, for s=−1 we note that by
using the result for s= 0 and an inverse inequality,
/ba∇dbl(I−Πh)(mψh)/ba∇dblW−1,p/lessorsimilarh/ba∇dblψh/ba∇dblLp
/lessorsimilarh/ba∇dblΠh(mψh)/ba∇dblLp/lessorsimilar/ba∇dblΠh(mψh)/ba∇dblW−1,p.
Since/ba∇dblmψh/ba∇dblW−1,p/greaterorsimilar/ba∇dblm/ba∇dbl−1
W1,∞/ba∇dblψh/ba∇dblW−1,p, this concludes the proof for s∈ {−1,0}.
(b) It remains to prove the result for s= 1. Note that the result follows from
duality if we show
(5.2) /ba∇dblΠh(m·wh)/ba∇dblW−1,q/greaterorsimilar/ba∇dblwh/ba∇dblW−1,q
for allwh∈Nh(m). To see this, note that (5.2) implies
/ba∇dblΠh(mψh)/ba∇dblW1,p/greaterorequalslantsup
wh∈Nh(m)(ψh,Πh(m·wh))
/ba∇dblwh/ba∇dblW−1,q
/greaterorsimilarsup
wh∈Nh(m)(ψh,Πh(m·wh))
/ba∇dblΠh(m·wh)/ba∇dblW−1,q= sup
ωh∈Vh(ψh,ωh)
/ba∇dblωh/ba∇dblW−1,q≃ /ba∇dblψh/ba∇dblW1,p,
whereweusedinthesecondtolastequalitythatpart(a)for s= 0alreadyshowsthat
dim(Nh(m)) = dim(Vh) and since (5.2) implies that the map Nh(m)→Vh,wh/ma√sto→
Πh(m·wh) is injective, it is already bijective. It remains to prove (5.2). To tha t
end, we first show for wh=Πh(mωh)∈Nh(m) for someωh∈Vh, using the reverse
triangle inequality, that
/ba∇dblm·wh/ba∇dblW−1,q/greaterorequalslant/ba∇dblωh/ba∇dblW−1,q−/ba∇dblm·(I−Πh)(mωh)/ba∇dblW−1,q
/greaterorsimilar/ba∇dblm/ba∇dbl−1
W1,∞/ba∇dblwh/ba∇dblW−1,q−/ba∇dblm·(I−Πh)(mωh)/ba∇dblW−1,q.
Withmh:=Ih(m)∈V3
h, the last term satisfies
/ba∇dblm·(I−Πh)(mωh)/ba∇dblW−1,q/lessorsimilarh/ba∇dblm/ba∇dblW1,∞/ba∇dbl(I−Πh)(mωh)/ba∇dblLq
/lessorsimilarh/ba∇dblm/ba∇dblW1,∞(/ba∇dblm−mh/ba∇dblL∞/ba∇dblωh/ba∇dblLq+h/ba∇dblmh/ba∇dblW1,∞/ba∇dblωh/ba∇dblLq),
where we used the same arguments as in the proof of part (a) to ge t the estimate
/ba∇dbl(I−Πh)(mhωh)/ba∇dblLq/lessorsimilarh/ba∇dblmh/ba∇dblW1,∞/ba∇dblωh/ba∇dblLq. The fact /ba∇dblmh/ba∇dblW1,∞/lessorsimilar/ba∇dblm/ba∇dblW1,∞, the
approximation property /ba∇dblm−mh/ba∇dblL∞/lessorsimilarh/ba∇dblm/ba∇dblW1,∞, and an inverse inequality con-
clude
(5.3) /ba∇dblm·wh/ba∇dblW−1,q/greaterorsimilar/ba∇dblwh/ba∇dblW−1,q
with (hidden) constants depending only on /ba∇dblm/ba∇dblW1,∞and shape regularity of the
mesh.18 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
Toprove(5.2), itremainstoboundtheleft-handsideaboveby /ba∇dblΠh(m·wh)/ba∇dblW−1,q.
To that end, we note
/ba∇dbl(I−Πh)(m·wh)/ba∇dblW−1,q/lessorsimilarh/ba∇dblwh/ba∇dblLq=hsup
v∈Lp(wh,v)
/ba∇dblv/ba∇dblLp
/lessorsimilarhsup
v∈Nh(m)(wh,v)
/ba∇dblv/ba∇dblLp=hsup
v∈Vh(Πh(m·wh),v)
/ba∇dblΠh(mv)/ba∇dblLp/lessorsimilarh/ba∇dblΠh(m·wh)/ba∇dblLq,
where we used part (a) for s= 0 for the last inequality. An inverse inequality
and the combination with (5.3) imply (5.2) for h >0 sufficiently small in terms of
/ba∇dblm/ba∇dbl−1
W1,∞. This concludes the proof. /square
Lemma 5.6. Define the matrix M∈RN×N, whereNdenotes the dimension of Vh,
byMij:=h−3(Πh(mφj),Πh(mφi)). Under the assumptions of Lemma 5.5, there
existsC >0such that for h/lessorequalslanthR,
/ba∇dblM/ba∇dblp+/ba∇dblM−1/ba∇dblp/lessorequalslantCfor1/lessorequalslantp/lessorequalslant∞,
whereCdepends only on the shape regularity.
Proof.Lemma 5.5 shows for x∈RN
(5.4) Mx·x=h−3/ba∇dblΠh(mN/summationdisplay
i=1xiφi)/ba∇dbl2
L2/greaterorsimilarh−3/ba∇dblN/summationdisplay
i=1xiφi/ba∇dblL2≃ |x|2,
where|·|denotes the Euclidean norm on RN. Letd(i,j) := dist(zi,zj)h−3denote
the metric which (approximately) measures the number of elements between the
supports of φiandφj, corresponding to the nodes ziandzj, and letBd(z) denote
the corresponding ball. In the following, we use a localization propert y of theL2-
projection, i.e., there exist a,b>0 such that for all ℓ∈N,
(5.5) /ba∇dblΠh(mφi)/ba∇dblL2(Ω\Bℓ(zi))3/lessorequalslantae−bℓ/ba∇dblmφi/ba∇dblL2.
The proof of this bound is essentially contained in the proof of [9, Le mma 3.1].
Since we use the very same arguments below, we briefly recall the st rategy: First,
one observes that the mass matrix /tildewiderM∈RN×Nwith entries /tildewiderMij:=h−3(φj,φi) is
bandedinthesense that d(i,j)/greaterorsimilar1implies /tildewiderMij= 0, anditsatisfies /tildewiderMx·x/greaterorsimilar|x|2. As
shown below, this implies that the inverse matrix /tildewiderM−1satisfies|(/tildewiderM−1)ij|/lessorsimilare−bd(i,j)
for someb >0 independent of h >0. Note that each entry of the vector field
Πh(mφi)∈V3
hcan be represented by/summationtextN
j=1xk,jφj,k= 1,2,3,and is computed by
solving/tildewiderMxk=gk∈RNwithm= (m1,m2,m3)Tandgk,j:= (mkφi,φj). Hence, the
exponential decay of /tildewiderM−1directly implies (5.5).
From the decay property (5.5), we immediately obtain
|Mij|/lessorequalslant/tildewideae−/tildewidebd(i,j)
for all 1/lessorequalslanti,j/lessorequalslantNand some /tildewidea,/tildewideb>0. This already proves /ba∇dblM/ba∇dblp/lessorequalslantC. We follow
the arguments from [28] to show that also M−1decays exponentially. To that end,HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 19
note that (5.4) implies the existence of c >0 such that /ba∇dblI−cM/ba∇dbl2=:q <1 and
hence
(5.6) M−1=c(I−(I−cM))−1=c∞/summationdisplay
k=0(I−cM)k.
Clearly,I−cMinherits the decay properties from Mand therefore
|((I−cM)k+1)ij|/lessorequalslant/tildewideak+1N/summationdisplay
r1,...,rk=1e−/tildewideb(d(i,r1)+···+d(rk,j))
/lessorequalslant/tildewideak+1/parenleftig
max
s=1,...,NN/summationdisplay
r=1e−/tildewidebd(s,r)/2/parenrightigk
e−/tildewidebd(i,j)/2.
The value of max s=1,...,N/summationtextN
r=1e−/tildewidebd(s,r)/2depends only on the shape regularity of the
triangulation and on /tildewideb, but is independent of h(it just depends on the number of
elements contained in an annulus of thickness ≈h). This implies the existence of
/tildewidec/greaterorequalslant1 such that
|((I−cM)k+1)ij|/lessorequalslantmin{qk+1,/tildewideck+1e−/tildewidebd(i,j)/2}.
Thus, for /tildewideck+1/lessorequalslante/tildewidebd(i,j)/4, we have |((I−cM)k+1)ij|/lessorequalslante−/tildewidebd(i,j)/4, whereas for /tildewideck+1>
e/tildewidebd(i,j)/4, we have |((I−cM)k+1)ij|/lessorequalslantqk+1<q/tildewidebd(i,j)/(4log(/tildewidec)). Altogether, we find some
/tildewideb>0 (we reuse the symbol), independent of hsuch that
|((I−cM)k+1)ij|/lessorequalslantq(k+1)/2|((I−cM)k+1)ij|1/2/lessorsimilarq(k+1)/2e−/tildewidebd(i,j).
Plugging this into (5.6), we obtain
|(M−1)ij|/lessorsimilar∞/summationdisplay
k=0q(k+1)/2e−/tildewidebd(i,j)/lessorsimilare−/tildewidebd(i,j).
This yields the stated result. /square
We are now in a position to prove Lemma 5.3.
Proof of Lemma 5.3. (a) We first consider the case s= 0. In view of (5.1), we write
(I−Ph(m))vh∈Nh(m) as
(I−Ph(m))vh=h−3/2N/summationdisplay
i=1xiΠh(mφi)
for some coefficient vector x∈RNand letbi:=h−3/2(vh,mφi) fori= 1,...,N.
Then, there holds Mx=bwith the matrix Mfrom Lemma 5.6. This lemma and
theLp-stability of the L2-orthogonal projection Π h[20] imply that for p∈[1,∞],
/ba∇dbl(I−Ph(m))vh/ba∇dblLp=/ba∇dblΠhh−3/2N/summationdisplay
i=1ximφi/ba∇dblLp/lessorsimilar/ba∇dblh−3/2N/summationdisplay
i=1ximφi/ba∇dblLp
/lessorsimilarh−3/2/parenleftigN/summationdisplay
i=1h3|xi|p/parenrightig1/p
=h3/p−3/2|x|p=h3/p−3/2|M−1b|p/lessorsimilarh3/p−3/2|b|p.20 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
With|bi|/lessorequalslanth−3/2/ba∇dblvh/ba∇dblLp(supp(φi))3h3(1−1/p)=/ba∇dblvh/ba∇dblLp(supp(φi))3h3/2−3/p, this shows
/ba∇dblPh(m)vh/ba∇dblLp/lessorsimilar/ba∇dblvh/ba∇dblLp.
(b) We now turn to the cases s=±1. Define the operator
/tildewideP⊥
h(m)vh:=Πh(mΠh(m·vh))
andnotethat /tildewideP⊥
h(m)vh∈Nh(m)aswellasker /tildewideP⊥
h(m) =Th(m)(duetoLemma5.5).
However, /tildewideP⊥
h(m) is no projection. We observe for vh=Πh(mψh)∈Nh(m) that
/ba∇dbl(I−/tildewideP⊥
h(m))vh/ba∇dblW−1,p=/ba∇dblΠhmψh−Πh(mΠh(m·Πh(mψh)))/ba∇dblW−1,p
/lessorsimilar/ba∇dblm/ba∇dblW1,∞/ba∇dblψh−m·Πh(mψh)/ba∇dblW−1,p
=/ba∇dblm/ba∇dbl2
W1,∞/ba∇dbl(I−Πh)(mψh)/ba∇dblW−1,p
/lessorsimilar/ba∇dblm/ba∇dbl2
W1,∞h/ba∇dblψh/ba∇dblLp.
With Lemma 5.5 we conclude
/ba∇dbl(I−/tildewideP⊥
h(m))vh/ba∇dblW−1,p/lessorsimilar/ba∇dblm/ba∇dbl2
W1,∞h/ba∇dblvh/ba∇dblLp.
Since/tildewideP⊥
h(m)Ph(m) = 0 by definition of Th(m), we obtain with part (a) and an
inverse inequality that for all vh∈V3
h,
/ba∇dbl(I−Ph(m)−/tildewideP⊥
h(m))vh/ba∇dblW−1,p=/ba∇dbl(I−/tildewideP⊥
h(m))(I−Ph(m))vh/ba∇dblW−1,p
/lessorsimilar/ba∇dblm/ba∇dbl2
W1,∞h/ba∇dbl(I−Ph(m))vh/ba∇dblLp
/lessorsimilar/ba∇dblm/ba∇dbl2
W1,∞h/ba∇dblvh/ba∇dblLp
/lessorsimilar/ba∇dblm/ba∇dbl2
W1,∞/ba∇dblvh/ba∇dblW−1,p.
TheW−1,p(Ω)-stability of Πhimplies/ba∇dbl/tildewideP⊥
h(m)vh/ba∇dblW−1,p/lessorsimilar/ba∇dblm/ba∇dbl2
W1,∞/ba∇dblvh/ba∇dblW−1,pand
the triangle inequality concludes the proof for s=−1. The case s= 1 follows by
duality. /square
Proof of Lemma 5.2. (a) (s= 0) The projection vh:=Ph(m)vis given by the
equation
(vh,ϕh) = (v,ϕh)∀ϕh∈Th(m),
which in view of the definition of Th(m) is equivalent to the solution of the saddle
point problem (with the Lagrange multiplier λh∈Vh)
(vh,wh)+(m·wh,λh) = (v,wh)∀wh∈V3
h,
(m·vh,µh) = 0 ∀µh∈Vh.
By the first equation, we also obtain the identity Πh(mλh) = (I−Ph(m))vh, which
will be used below. Furthermore, /tildewidevh:=Ph(/tildewiderm)vis given by the same system with
/tildewidermin place ofm, yielding a corresponding Lagrange multiplier /tildewideλh. Hence, the
differenceseh:=vh−/tildewidevhandδh:=λh−/tildewideλhsatisfy
(eh,wh)+(m·wh,δh) =−(wh,(m−/tildewiderm)/tildewideλh)∀wh∈V3
h,
(m·eh,µh) = −((m−/tildewiderm)·/tildewidevh,µh)∀µh∈Vh.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 21
The classical results on saddle-point problems (see [13, Proposition 2.1]) require two
inf-sup conditions to be satisfied. First,
inf
qh∈Vhsup
vh∈V3
h(m·vh,qh)
/ba∇dblvh/ba∇dblHs/ba∇dblqh/ba∇dblH−s>0
holds uniformly in hdue to Lemma 5.5. Second,
inf
wh∈Th(m)sup
vh∈Th(m)(vh,wh)
/ba∇dblvh/ba∇dblHs/ba∇dblwh/ba∇dblH−s>0
holdsuniformlyin hduetothestabilityestimatesfromLemma5.3(notingthat vh=
Ph(m)vhandwh=Ph(m)whforvh,wh∈Th(m)). For the above saddle-point
problems, these bounds for s= 0 give us an L2bound foreh=Ph(m)v−Ph(/tildewiderm)v:
From [13] we obtain
/ba∇dbl/tildewidevh/ba∇dblL2+/ba∇dbl/tildewideλh/ba∇dblL2/lessorsimilar/ba∇dblv/ba∇dblL2
and
/ba∇dbleh/ba∇dblL2+/ba∇dblδh/ba∇dblL2/lessorsimilar/ba∇dbl(m−/tildewiderm)/tildewideλh/ba∇dblL2+/ba∇dbl(m−/tildewiderm)·/tildewidevh/ba∇dblL2.
With the stability from Lemma 5.3 and Lemma 5.5, we also obtain
/ba∇dbl/tildewidevh/ba∇dblL∞+/ba∇dbl/tildewideλh/ba∇dblL∞/lessorsimilar/ba∇dblPh(/tildewiderm)v/ba∇dblL∞+/ba∇dbl(I−Ph(/tildewiderm))v/ba∇dblL∞/lessorsimilar/ba∇dblv/ba∇dblL∞.
Altogether, this implies
/ba∇dbleh/ba∇dblL2+/ba∇dblδh/ba∇dblL2/lessorsimilar/ba∇dblm−/tildewiderm/ba∇dblLp/ba∇dblv/ba∇dblLq
for (p,q)∈ {(2,∞),(∞,2)}.
(b) (s= 1) For the H1(Ω)-estimate, we introduce the Riesz mapping Jhbetween
Vh⊂H1(Ω) and its dual Vh⊂H1(Ω)′, i.e., the isometry defined by
(vh,Jhψh)H1=/a\}b∇acketle{tvh,ψh/a\}b∇acket∇i}ht ∀vh∈Vh, ψh∈Vh.
ByJh:=I⊗Jhwe denote the corresponding vector-valued mapping on V3
h. We
consider the bilinear form on V3
h×V3
hdefined by
ah(vh,wh) =/a\}b∇acketle{tvh,J−1
hwh/a\}b∇acket∇i}ht,vh,wh∈V3
h,
and reformulate the saddle-point problem for ( vh,λh)∈V3
h×Vh⊂H1(Ω)3×H1(Ω)′
as
ah(vh,wh)+/a\}b∇acketle{tm·J−1
hwh,λh/a\}b∇acket∇i}ht=a(v,wh)∀wh∈V3
h,
/a\}b∇acketle{tm·vh,J−1
hµh/a\}b∇acket∇i}ht = 0 ∀µh∈Vh.
As in the case s= 0 (algebraically it is the same system), we have vh=Ph(m)v
andΠh(mλh) = (I−Ph(m))v. The system for eh=vh−/tildewidevhandδh=λh−/tildewideλh
reads
ah(eh,wh)+/a\}b∇acketle{tm·J−1
hwh,δh/a\}b∇acket∇i}ht=−/a\}b∇acketle{t(m−/tildewiderm)·J−1
hwh,/tildewideλh/a\}b∇acket∇i}ht ∀wh∈V3
h,
/a\}b∇acketle{tm·eh,J−1
hµh/a\}b∇acket∇i}ht =−/a\}b∇acketle{t(m−/tildewiderm)·/tildewidevh,J−1
hµh/a\}b∇acket∇i}ht ∀µh∈Vh.
The above inf-sup bounds for s= 1 ands=−1 are precisely the inf-sup condi-
tions that need to be satisfied for these generalized saddle-point p roblems (see [15,
Theorem 2.1]), whose right-hand sides are bounded by
|ah(v,wh)|/lessorequalslant/ba∇dblv/ba∇dblH1/ba∇dblJ−1
hwh/ba∇dblH−1≃ /ba∇dblv/ba∇dblH1/ba∇dblwh/ba∇dblH122 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
and
|/a\}b∇acketle{t(m−/tildewiderm)·J−1
hwh,/tildewideλh/a\}b∇acket∇i}ht|/lessorsimilar/ba∇dbl(m−/tildewiderm)/tildewideλh/ba∇dblH1/ba∇dblwh/ba∇dblH1,
|/a\}b∇acketle{t(m−/tildewiderm)·/tildewidevh,J−1
hµh/a\}b∇acket∇i}ht|/lessorequalslant/ba∇dbl(m−/tildewiderm)·/tildewidevh/ba∇dblH1/ba∇dblµh/ba∇dblH1.
As in the case s= 0, we obtain from Lemma 5.3 and Lemma 5.5 that
/ba∇dbl/tildewidevh/ba∇dblW1,∞+/ba∇dbl/tildewideλh/ba∇dblW1,∞/lessorsimilar/ba∇dblPh(/tildewiderm)v/ba∇dblW1,∞+/ba∇dbl(I−Ph(/tildewiderm))v/ba∇dblW1,∞
/lessorsimilar/ba∇dblv/ba∇dblW1,∞.
Hence, we obtain from [15, Theorem 2.1], for ( p,q)∈ {(2,∞),(∞,2)},
/ba∇dbleh/ba∇dblH1/lessorsimilar/ba∇dbl(m−/tildewiderm)/tildewideλh/ba∇dblH1+/ba∇dbl(m−/tildewiderm)·/tildewidevh/ba∇dblH1
/lessorsimilar1/summationdisplay
s′=0/parenleftig
/ba∇dblm−/tildewiderm/ba∇dblH1/ba∇dbl/tildewideλh/ba∇dblW1−s′,q+/ba∇dblm−/tildewiderm/ba∇dblWs′,p/ba∇dbl/tildewidevh/ba∇dblW1−s′,q/parenrightig
/lessorsimilar1/summationdisplay
s′=0/ba∇dblm−/tildewiderm/ba∇dblWs′,p/ba∇dblv/ba∇dblW1−s′,q.
This implies the H1(Ω)3estimate and hence concludes the proof. /square
Proof of Lemma 5.1. SincePh(m)vis the Galerkin approximation of the saddle
point problem for P(m)v(as in the previous proof), the C´ ea lemma for saddle-
point problems (see [13, Theorem 2.1]) shows in L2
/ba∇dbl(Ph(m)−P(m))v/ba∇dblL2
/lessorsimilarinf
(wh,µh)∈V3
h×Vh/parenleftig
/ba∇dblP(m)v−wh/ba∇dblL2+/ba∇dblm·v−µh/ba∇dblL2/parenrightig
/lessorsimilarhr+1/ba∇dblm/ba∇dblWr+1,∞/ba∇dblv/ba∇dblHr+1
and similarly in H1, using [15, Theorem 2.1],
/ba∇dbl(Ph(m)−P(m))v/ba∇dblH1
/lessorsimilarinf
(wh,µh)∈V3
h×Vh/parenleftig
/ba∇dblP(m)v−wh/ba∇dblH1+/ba∇dblm·v−µh/ba∇dblH1/parenrightig
/lessorsimilarhr/ba∇dblm/ba∇dblWr+1,∞/ba∇dblv/ba∇dblHr+1.
This concludes the proof. /square
6.Consistency error and error equation
To study the consistency errors, we find it instructive to separat e the issues of
consistency for the time and space discretizations. Therefore, w e first show defect
estimates for the semidiscretization in time, and then turn to the fu ll discretization.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 23
6.1.Consistency error of the semi-discretization in time. The order of both
the fully implicit k-step BDF method, described by the coefficients δ0,...,δ kand 1,
andtheexplicit k-step BDFmethod, thatis themethoddescribed by thecoefficients
δ0,...,δ kandγ0,...,γ k−1,isk,i.e.,
(6.1)k/summationdisplay
i=0(k−i)ℓδi=ℓkℓ−1=ℓk−1/summationdisplay
i=0(k−i−1)ℓ−1γi, ℓ= 0,1,...,k.
We first rewrite the linearly implicit k-step BDF method (2.3) in strong form,
(6.2) α˙mn+/hatwidermn×˙mn=P(/hatwidermn)(∆mn+Hn),
with Neumann boundary conditions.
The consistency error dnof the linearly implicit k-step BDF method (6.2) for the
solutionmis the defect by which the exact solution misses satisfying (6.2), and is
given by
(6.3) dn=α˙mn
⋆+/hatwidermn
⋆×˙mn
⋆−P(/hatwidermn
⋆)(∆mn
⋆+Hn)
forn=k,...,N, where we use the notation mn
⋆=m(tn) and
(6.4)/hatwidermn
⋆=k−1/summationdisplay
j=0γjmn−j−1
⋆/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay
j=0γjmn−j−1
⋆/vextendsingle/vextendsingle/vextendsingle,
˙mn
⋆=P(/hatwidermn
⋆)1
τk/summationdisplay
j=0δjmn−j
⋆∈T(/hatwidermn
⋆).
Notethat thedefinition of ˙mn
⋆contains theprojection P(/hatwidermn
⋆), while ˙mnwasdefined
without a projection (see the first formula in (2.2)), since ˙mn=P(/hatwidermn)˙mnis
automatically satisfied due to the constraint in (2.3).
The consistency error is bounded as follows.
Lemma 6.1. If the solution of the LLG equation (1.4)has the regularity
m∈Ck+1([0,¯t],L2(Ω)3)∩C1([0,¯t],L∞(Ω)3)and∆m+H∈C([0,¯t],L∞(Ω)3),
then the consistency error (6.3)is bounded by
/ba∇dbldn/ba∇dblL2(Ω)3/lessorequalslantCτk
forn=k,...,N.
Proof.We begin by rewriting the equation for the defect as
(6.5)dn=α˙mn
⋆+/hatwidermn
⋆×˙mn
⋆−P(mn
⋆)(∆mn
⋆+Hn)
−/parenleftbig
P(/hatwidermn
⋆)−P(mn
⋆)/parenrightbig
(∆mn
⋆+Hn).
In view of (1.4), we have
P(mn
⋆)(∆mn
⋆+Hn) =α∂tm(tn)+mn
⋆×∂tm(tn),
and can rewrite (6.5) as
dn=α/parenleftbig
˙mn
⋆−∂tm(tn)/parenrightbig
+/parenleftbig
/hatwidermn
⋆×˙mn
⋆−mn
⋆×∂tm(tn)/parenrightbig
−/parenleftbig
P(/hatwidermn
⋆)−P(mn
⋆)/parenrightbig
(∆mn
⋆+Hn),24 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
i.e.,
dn=α/parenleftbig˙mn
⋆−∂tm(tn)/parenrightbig
+(/hatwidermn
⋆−mn
⋆)×˙mn
⋆+mn
⋆×/parenleftbig˙mn
⋆−∂tm(tn)/parenrightbig
−/parenleftbig
P(/hatwidermn
⋆)−P(mn
⋆)/parenrightbig
(∆mn
⋆+Hn).
Therefore,
(6.6)dn=α˙dn+/hatwidedn×˙mn
⋆+mn
⋆×˙dn−/parenleftbig
P(/hatwidermn
⋆)−P(mn
⋆)/parenrightbig
(∆mn
⋆+Hn),
with
(6.7) ˙dn:=˙mn
⋆−∂tm(tn),/hatwidedn:=/hatwidermn
⋆−mn
⋆.
Now, in view of the first estimate in Lemma 4.1, we have
/ba∇dbl/parenleftbig
P(/hatwidermn
⋆)−P(mn
⋆)/parenrightbig
(∆mn
⋆+Hn)/ba∇dblL2/lessorequalslantC/ba∇dbl/hatwidermn
⋆−mn
⋆/ba∇dblL2,
i.e.,
(6.8) /ba∇dbl/parenleftbig
P(/hatwidermn
⋆)−P(mn
⋆)/parenrightbig
(∆mn
⋆+Hn)/ba∇dblL2/lessorequalslantC/ba∇dbl/hatwidedn/ba∇dblL2.
Therefore, it suffices to estimate ˙dnand/hatwidedn.
To estimate /hatwidedn, we shall proceed in two steps. First we shall estimate the extrap-
olation error
(6.9)k−1/summationdisplay
j=0γjmn−j−1
⋆−mn
⋆
and then /hatwidedn.
By Taylor expanding about tn−k,the leading terms of order up to k−1 cancel,
due to the second equality in (6.1), and we obtain
(6.10)k−1/summationdisplay
i=0γimn−i−1
⋆−mn
⋆=1
(k−1)!/bracketleftiggk−1/summationdisplay
j=0γj/integraldisplaytn−j−1
tn−k(tn−j−1−s)k−1m(k)(s)ds
−/integraldisplaytn
tn−k(tn−s)k−1m(k)(s)ds/bracketrightigg
,
withm(ℓ):=∂ℓm
∂tℓ,whence
(6.11)/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay
i=0γimn−i−1
⋆−mn
⋆/vextenddouble/vextenddouble/vextenddouble
L2/lessorequalslantCτk.
Now, for a normalized vector aand a non-zero vector b,we have
a−b
|b|= (a−b)+1
|b|(|b|−|a|)b,
whence/vextendsingle/vextendsinglea−b
|b|/vextendsingle/vextendsingle/lessorequalslant2|a−b|.
Therefore, (6.11) yields
(6.12) /ba∇dbl/hatwidedn/ba∇dblL2/lessorequalslantCτk.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 25
To bound ˙dn,we use the fact that P(m(tn))∂tm(tn) =∂tm(tn)∈T(m(tn)), so
that we have
˙dn=P(/hatwidermn
⋆)1
τk/summationdisplay
j=0δjm(tn−j)−∂tm(tn)
=P(/hatwidermn
⋆)/parenleftig1
τk/summationdisplay
j=0δjm(tn−j)−∂tm(tn)/parenrightig
+/parenleftbig
P(/hatwidermn
⋆)−P(m(tn))/parenrightbig
∂tm(tn).
By Lemma 4.1 and (6.12), we have for the last term
/ba∇dbl/parenleftbig
P(/hatwidermn
⋆)−P(m(tn))/parenrightbig
∂tm(tn)/ba∇dblL2/lessorequalslantCτk.
By Taylor expanding the first term about tn−k,we see that, due to the order condi-
tions of the implicit BDF method, i.e., the first equality in (6.1), the leadin g terms
of order up to k−1 cancel, and we obtain
(6.13)1
τk/summationdisplay
j=0δjm(tn−j)−∂tm(tn) =1
k!/bracketleftigg
1
τk/summationdisplay
j=0δj/integraldisplaytn−j
tn−k(tn−j−s)km(k+1)(s)ds
−k/integraldisplaytn
tn−k(tn−s)k−1m(k+1)(s)ds/bracketrightigg
,
whence
(6.14) /ba∇dbl˙dn/ba∇dblL2/lessorequalslantCτk,
provided the solution mis sufficiently regular. Now, (6.6), (6.8), (6.14), and (6.12)
yield
(6.15) /ba∇dbldn/ba∇dblL2/lessorequalslantCτk.
This isthe desired consistency estimate, which isvalidfor BDFmethod s of arbitrary
orderk. /square
6.2.Consistency error of the full discretization. Wedefine theRitzprojection
Rh:H1(Ω)→Vhcorresponding to the Poisson–Neumann problem via/parenleftbig
∇Rhϕ,∇ψ/parenrightbig
+/parenleftbig
Rhϕ,1/parenrightbig/parenleftbig
ψ,1/parenrightbig
=/parenleftbig
∇ϕ,∇ψ/parenrightbig
+/parenleftbig
ϕ,1/parenrightbig/parenleftbig
ψ,1/parenrightbig
for allψ∈Vh, and we denote Rh=I⊗Rh:H1(Ω)3→V3
h. We denote again
theL2-orthogonal projections onto the finite element space by Πh:L2(Ω)→Vh
andΠh=I⊗Πh:L2(Ω)3→V3
h. As in the previous section, we write Ph(m) for
theL2-orthogonal projection onto the discrete tangent space at m. We insert the
following quantities, which are related to the exact solution,
mn
⋆,h=Rhm(tn),
/hatwidermn
⋆,h=k−1/summationdisplay
j=0γjmn−j−1
⋆,h/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay
j=0γjmn−j−1
⋆,h/vextendsingle/vextendsingle/vextendsingle, (6.16)
˙mn
⋆,h=Ph(/hatwidermn
⋆,h)1
τk/summationdisplay
j=0δjmn−j
⋆,h∈Th(/hatwidermn
⋆,h),26 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
intothelinearlyimplicit k-stepBDFmethod(2.6)andobtainadefect dn
h∈Th(/hatwidermn
⋆,h)
from
(6.17)α/parenleftbig
˙mn
⋆,h,ϕh/parenrightbig
+/parenleftbig
/hatwidermn
⋆,h×˙mn
⋆,h,ϕh/parenrightbig
=−/parenleftbig
∇mn
⋆,h,∇ϕh/parenrightbig
+/parenleftbig
Hn,ϕh/parenrightbig
+/parenleftbig
dn
h,ϕh/parenrightbig
for allϕh∈Th(/hatwidermn
⋆,h). By definition, there holds ( Rhϕ,1) = (ϕ,1) (this can be seen
by testing with ψ= 1) and hence/parenleftbig
∇mn
⋆,h,∇ϕ/parenrightbig
=/parenleftbig
∇m(tn),∇ϕ/parenrightbig
=−/parenleftbig
∆m(tn),ϕ/parenrightbig
.
Thus, we obtain the consistency error for the full discretization b y
(6.18)dn
h=Ph(/hatwidermn
⋆,h)Dn
hwithDn
h=α˙mn
⋆,h+/hatwidermn
⋆,h×˙mn
⋆,h−∆m(tn)−H(tn)
forn=k,...,N. The consistency error is bounded as follows.
Lemma 6.2. If the solution of the LLG equation (1.4)has the regularity
m∈Ck+1([0,¯t],L2(Ω)3)∩C1([0,¯t],Wr+1,∞(Ω)3)and
∆m+H∈C([0,¯t],Wr+1,∞(Ω)3),
then the consistency error (6.18)is bounded by
/ba∇dbldn
h/ba∇dblL2(Ω)3/lessorequalslantC(τk+hr)
fornwithkτ/lessorequalslantnτ/lessorequalslant¯t.
Proof.We begin by defining
Dn:=α∂tm(tn)+m(tn)×∂tm(tn)−∆m(tn)−H(tn)
and note that P(mn
⋆)Dn= 0. Here we denote again mn
⋆=m(tn) and in the
following we use also the notations ˙mn
⋆and/hatwidermn
⋆as defined in (6.4). With this, we
rewrite the equation for the defect as
dn
h=Ph(/hatwidermn
⋆,h)Dn
h−P(mn
⋆)Dn
=Ph(/hatwidermn
⋆,h)/parenleftbig
Dn
h−Dn/parenrightbig
+/parenleftbig
Ph(/hatwidermn
⋆,h)−Ph(/hatwidermn
⋆)/parenrightbig
Dn
+/parenleftbig
Ph(/hatwidermn
⋆)−P(/hatwidermn
⋆)/parenrightbig
Dn+/parenleftbig
P(/hatwidermn
⋆)−P(mn
⋆)/parenrightbig
Dn
≡I+II+III+IV.
For the term IVwe have by Lemma 4.1
/ba∇dblIV/ba∇dblL2/lessorequalslant2/ba∇dbl/hatwidermn
⋆−mn
⋆/ba∇dblL2/ba∇dblDn/ba∇dblL∞,
where the last term /hatwidermn
⋆−mn
⋆has been bounded in the L2norm byCτkin the proof
of Lemma 6.1.
The term IIIis estimated using the first bound from Lemma 5.1, under our
regularity assumptions, as
/ba∇dblIII/ba∇dblL2/lessorequalslantChr.
For the bound on IIwe use Lemma 5.2 ( i) (withp= 2 andq=∞), to obtain
/ba∇dblII/ba∇dblL2/lessorequalslantCR/ba∇dbl/hatwidermn
⋆,h−/hatwidermn
⋆/ba∇dblL2/ba∇dblDn/ba∇dblL∞,
where, using (7.11), we obtain
/ba∇dbl/hatwidermn
⋆,h−/hatwidermn
⋆/ba∇dblL2/lessorequalslant2/ba∇dbl/summationtextk
i=1γi(Rh−I)mn−i
∗/ba∇dblL2
min/vextendsingle/vextendsingle/summationtextk
i=1γimn−i
∗/vextendsingle/vextendsingle/lessorequalslantChr.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 27
The denominator is bounded from below by 1 −Cτk, because |mn
∗|= 1 and
|/summationtextk
i=1γimn−i
∗−mn
∗|/lessorequalslantCτk. For the first term we have
/ba∇dblI/ba∇dblL2/lessorequalslant/ba∇dblDn−Dn
h/ba∇dblL2
/lessorequalslantα/ba∇dbl∂tm(tn)−˙mn
⋆,h/ba∇dblL2+/ba∇dblm(tn)×∂tm(tn)−/hatwidermn
⋆,h×˙mn
⋆,h/ba∇dblL2.
The terms /ba∇dbl∂tm(tn)−˙mn
⋆/ba∇dblL2and/ba∇dblmn
⋆×∂tm(tn)−/hatwidermn
⋆×˙mn
⋆/ba∇dblL2can be handled
as in the proof of Lemma 6.1. Standard error estimates for the Ritz projectionRh
(we do not exploit the Aubin–Nitsche duality here) imply
/ba∇dbl(I−Rh)˙mn
⋆/ba∇dblL2/lessorequalslantchr/ba∇dbl˙mn
⋆/ba∇dblHr+1.
Together this yields, under the stated regularity assumption,
/ba∇dblI/ba∇dblL2/lessorequalslantC(τk+hr),
and the result follows. /square
6.3.Error equation. We recall, from (2.6), the fully discrete problem with the
linearly implicit BDF method: find ˙mn
h∈Th(/hatwidermn
h) such that for all ϕh∈Th(/hatwidermn
h),
(6.19) α(˙mn
h,ϕh)+(/hatwidermn
h×˙mn
h,ϕh)+(∇mn
h,∇ϕh) = (H(tn),ϕh).
Then, similarly as we have done in Section 4, we first rewrite (6.17): fo r all
ϕh∈Th(/hatwidermn
h),
(6.20) α(˙mn
⋆,h,ϕh)+(/hatwidermn
⋆,h×˙mn
⋆,h,ϕh)+(∇mn
⋆,h,∇ϕh) = (rn
h,ϕh)
with
(6.21) rn
h=−(Ph(/hatwidermn
h)−Ph(/hatwidermn
⋆,h))(∆m⋆(tn)+H(tn))+dn
h.
The erroren
h=mn
h−mn
⋆,hsatisfies the error equation that is obtained by sub-
tracting (6.20) from (6.19). We use the notations
/hatwideen
h=/hatwidermn
h−/hatwidermn
⋆,h, (6.22)
˙en
h=˙mn
h−˙mn
⋆,h=1
τk/summationdisplay
j=0δjen−j
h+sn
h, (6.23)
withsn
h= (I−Ph(/hatwidermn
⋆,h))1
τk/summationdisplay
j=0δjmn−j
⋆,h.
We have the following bound for sn
h.
Lemma 6.3. Under the regularity assumptions of Lemma 6.2, we have
(6.24) /ba∇dblsn
h/ba∇dblH1(Ω)3/lessorequalslantC(τk+hr).
Proof.We use Lemmas 5.1 and 5.3, and the bounds in the proof of Lemma 6.2.
We start by subtracting ( I−P(/hatwidermn
⋆,h))∂tmn
⋆= 0, and obtain (with ∂τmn
⋆,h:=
1
τ/summationtextk
j=0δjmn−j
⋆,h)
sn
h= (I−Ph(/hatwidermn
⋆,h))∂τmn
⋆,h−(I−P(/hatwidermn
⋆,h))∂tmn
⋆
= (∂τmn
⋆,h−∂tmn
⋆)−/parenleftbig
Ph(/hatwidermn
⋆,h)∂τmn
⋆,h−P(/hatwidermn
⋆,h)∂tmn
⋆/parenrightbig
.28 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
The first term above is bounded as O(τk+hr) via the techniques of the consistency
proofs, Lemma 6.1 and 6.2. For the second term we have
Ph(/hatwidermn
⋆,h)∂τmn
⋆,h−P(/hatwidermn
⋆,h)∂tmn
⋆
=Ph(/hatwidermn
⋆,h)(∂τmn
⋆,h−∂tmn
⋆)+/parenleftbig
Ph(/hatwidermn
⋆,h)−P(/hatwidermn
⋆,h)/parenrightbig
∂tmn
⋆,
where the first term is bounded as O(τk+hr), using Lemma 5.3 and the previous
estimate, while the second term is bounded as O(hr) by theH1estimate from
Lemma 5.1. Altogether, we obtain the stated H1bound forsn
h. /square
We then have the error equation
(6.25)α(˙en
h,ϕh)+(/hatwideen
h×˙mn
⋆,h,ϕh)+(/hatwidermn
h×˙en
h,ϕh)+(∇en
h,∇ϕh) =−(rn
h,ϕh),
for allϕh∈Th(/hatwidermn
h), which is to be taken together with (6.21)–(6.23).
7.Stability of the full discretization for BDF of orders 1 and 2
For the A-stable BDF methods (those of orders 1 and 2) we obtain t he follow-
ing stability estimate, which is analogous to the continuous perturba tion result
Lemma 4.2.
Lemma 7.1 (Stability for orders k= 1,2).Consider the linearly implicit k-step
BDF discretization (2.6)fork/lessorequalslant2with finite elements of polynomial degree r/greaterorequalslant1.
Letmn
handmn
⋆,h=Rhm(tn)satisfy equations (2.6)and(6.17), respectively, and
suppose that the exact solution m(t)is bounded by (4.3)and/ba∇dblH(t)/ba∇dblL∞/lessorequalslantMfor
0/lessorequalslantt/lessorequalslant¯t. Then, for sufficiently small h/lessorequalslant¯handτ/lessorequalslant¯τ, the erroren
h=mn
h−mn
⋆,h
satisfies the following bound, for kτ/lessorequalslantnτ/lessorequalslant¯t,
(7.1)/ba∇dblen
h/ba∇dbl2
H1(Ω)3/lessorequalslantC/parenleftigk−1/summationdisplay
i=0/ba∇dblei
h/ba∇dbl2
H1(Ω)3+τn/summationdisplay
j=k/ba∇dbldj
h/ba∇dbl2
L2(Ω)3+τn/summationdisplay
j=k/ba∇dblsj
h/ba∇dbl2
H1(Ω)3/parenrightig
,
where the constant Cis independent of h,τandn, but depends on α,R,K,M , and¯t.
This estimate holds under the smallness condition that the r ight-hand side is bounded
byˆchwith a sufficiently small constant ˆc(note that the right-hand side is of size
O((τk+hr)2)in the case of a sufficiently regular solution ).
Combining Lemmas 7.1, 6.2 and 6.3 yields the proof of Theorem 3.1 : These lem-
mas imply the estimate
/ba∇dblen
h/ba∇dblH1(Ω)3/lessorequalslant/tildewideC(τk+hr)
in the case of a sufficiently regular solution. Since then /ba∇dblRhm(tn)−m(tn)/ba∇dblH1(Ω)3/lessorequalslant
Chrand because of mn
h−m(tn) =en
h+(Rhm(tn)−m(tn)), this implies the error
bound (3.1).
The smallness condition imposed in Lemma 7.1 is satisfied under the very mild
CFL condition, for a sufficiently small ¯ c>0 (independent of h,τandn),
τk/lessorequalslant¯ch1/2.
Taken together, this proves Theorem 3.1.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 29
Proof.(a)Preparations. The proof of this lemma transfers the arguments of the
proof of Lemma 4.2 to the fully discrete situation, using energy estim ates obtained
by testing with (essentially) the discrete time derivative of the erro r, as presented
in the Appendix, which is based on Dahlquist’s G-stability theory.
However, testing the error equation (6.25) directly with ˙en
his not possible, since
˙en
his not in the tangent space Th(/hatwidermn
h). Therefore, as in the proof of Lemma 4.2, we
again start by showing that the test function ϕh=Ph(/hatwidermn
h)˙en
h∈Th(/hatwidermn
h)∩H1(Ω)3
is a perturbation of ˙en
hitself:
ϕh=Ph(/hatwidermn
h)˙en
h=Ph(/hatwidermn
h)˙mn
h−Ph(/hatwidermn
h)˙mn
⋆,h
=Ph(/hatwidermn
h)˙mn
h−Ph(/hatwidermn
⋆,h)˙mn
⋆,h+(Ph(/hatwidermn
⋆,h)−Ph(/hatwidermn
h))˙mn
⋆,h.
Here we note that Ph(/hatwidermn
h)˙mn
h=˙mn
h∈Th(/hatwidermn
h) by construction of the method(2.6),
andPh(/hatwidermn
⋆,h)˙mn
⋆,h=˙mn
⋆,h∈Th(/hatwidermn
⋆,h) by the definition of ˙mn
⋆,hin (6.4). So we have
ϕh=˙mn
h−˙mn
⋆,h−(Ph(/hatwidermn
h)−P(/hatwidermn
⋆,h))˙mn
⋆,h,
and hence
(7.2)ϕh=Ph(/hatwidermn
h)˙en=˙en
h+qn
hwithqn
h=−(Ph(/hatwidermn
h)−P(/hatwidermn
⋆,h))˙mn
⋆,h.
Theproofnowtransferstheproofofthecontinuousperturbat ionresultLemma4.2
to the discrete situation with some notable differences, which are em phasized here:
(i) Instead of using the continuous quantities it uses their spatially d iscrete coun-
terparts, in particular the discrete projections Ph(/hatwidermn
h) andPh(/hatwidermn
⋆,h), defined and
studied in Section 5. In view of the definition (2.1) and (6.16) of /hatwidermn
hand/hatwidermn
⋆,h, re-
spectively, thisrequiresthat/summationtextk−1
j=0γjmn−j−1
h(x)and/summationtextk−1
j=0γjmn−j−1
⋆,h(x)arebounded
away from zero uniformly for all x∈Ω.
(ii) Instead of Lemma 4.1 we use Lemma 5.2 (with /hatwidermn
hand/hatwidermn
⋆,hin the role of /tildewiderm
andm, respectively) to bound the quantity qn
h. This requires that /hatwidermn
⋆,hand˙mn
⋆,h
are bounded in W1,∞independently of h.
Ad(i): In order to show that |/summationtextk−1
j=0γjmn−j−1
h(x)|stays close to 1 for all x∈Ω,
we need to establish an L∞bound for the errors en−j−1
h=mn−j−1
h−mn−j−1
⋆,h.
We use an induction argument and assume that for some time step nu mber ¯n
with ¯nτ/lessorequalslant¯twe have
(7.3) /ba∇dblen
h/ba∇dblL∞/lessorequalslantρ,for 0/lessorequalslantn<¯n,
where we choose ρsufficiently small independent of handτ. (In this proof it suffices
to chooseρ/lessorequalslant1/(4Cγ), whereCγ=/summationtextk−1
j=0|γj|= 2k−1.)
Note that the smallness condition of the lemma implies that (7.3) is satis fied
for ¯n=k, because for the L∞errors of the starting values we have by an inverse
inequality, for i= 0,...,k−1,
/ba∇dblei
h/ba∇dblL∞/lessorequalslantCh−1/2/ba∇dblei
h/ba∇dblH1/lessorequalslantCh−1/2(ˆch)1/2=Cˆc1/2/lessorequalslantρ,
provided that ˆ cis sufficiently small (independent of τandh), as is assumed.
We will show in part (b) of the proof that with the induction hypothes is (7.3) we
obtain also /ba∇dble¯n
h/ba∇dblL∞/lessorequalslantρso that finally we obtain (7.3) for all¯nwith ¯nτ/lessorequalslant¯t.30 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
Using reverse and ordinary triangle inequalities, the error bound of [12, Corol-
lary 8.1.12] (noting that m(t)∈W2,∞(Ω) under our assumptions) and the L∞
boundedness of ∂tm, and the bound (7.3), we estimate
(7.4)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay
j=0γjmn−j−1
h/vextendsingle/vextendsingle/vextendsingle−1/vextenddouble/vextenddouble/vextenddouble/vextenddouble
L∞=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay
j=0γjmn−j−1
h/vextendsingle/vextendsingle/vextendsingle−|mn
⋆|/vextenddouble/vextenddouble/vextenddouble/vextenddouble
L∞/lessorequalslant/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay
j=0γjmn−j−1
h−mn
⋆/vextenddouble/vextenddouble/vextenddouble/vextenddouble
L∞
/lessorequalslant/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay
j=0γjen−j−1
h/vextenddouble/vextenddouble/vextenddouble
L∞+/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay
j=0γj(Rhmn−j−1
⋆−mn−j−1
⋆)/vextenddouble/vextenddouble/vextenddouble/vextenddouble
L∞+/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay
j=0γj(mn−j−1
⋆−mn
⋆)/vextenddouble/vextenddouble/vextenddouble/vextenddouble
L∞
/lessorequalslant/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay
j=0γjen−j−1
h/vextenddouble/vextenddouble/vextenddouble
L∞+Ch+Cτ/lessorequalslantk−1/summationdisplay
j=0|γj| ·ρ+Ch+Cτ/lessorequalslant1
2,
provided that handτare sufficiently small. The same argument also yields that/vextenddouble/vextenddouble|/summationtextk−1
j=0γjmn−j−1
⋆,h|−1/vextenddouble/vextenddouble
L∞/lessorequalslant1
2, and so we have
(7.5)1
2/lessorequalslant/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay
j=0γjmn−j−1
h(x)/vextendsingle/vextendsingle/vextendsingle/lessorequalslant3
2and1
2/lessorequalslant/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay
j=0γjmn−j−1
⋆,h(x)/vextendsingle/vextendsingle/vextendsingle/lessorequalslant3
2
for allx∈Ω. In particular, it follows that /hatwidermn
hand/hatwidermn
⋆,hare unambiguously defined.
Ad(ii): The required W1,∞bound formn
⋆,h=Rhm(tn) follows from the W1,∞-
stability of the Ritz projection: by [12, Theorem 8.1.11] and by the as sumedW1,∞
bound (4.3) for m(t),
(7.6) /ba∇dblmn
⋆,h/ba∇dblW1,∞/lessorequalslantC/ba∇dblm(tn)/ba∇dblW1,∞/lessorequalslantCR.
The bounds (7.5) and (7.6) for n/lessorequalslant¯nimply that also
(7.7) /ba∇dbl/hatwidermn
⋆,h/ba∇dblW1,∞/lessorequalslantCR
forn/lessorequalslant¯n(with a different constant C). Using this bound in Lemma 5.3 and the
assumedW1,∞bound (4.3) for ∂tm(t), we obtain with δ(ζ)/(1−ζ) =/summationtextk
ℓ=1(1−
ζ)ℓ−1/ℓ=:/summationtextk−1
j=0µjζjthat
/ba∇dbl˙mn
⋆,h/ba∇dblW1,∞=/ba∇dblPh(/hatwidermn
⋆,h)1
τk/summationdisplay
j=0δjmn−j
⋆/ba∇dblW1,∞
=/ba∇dblPh(/hatwidermn
⋆,h)k−1/summationdisplay
j=0µj1
τ(mn−j
⋆−mn−j−1
⋆)/ba∇dblW1,∞
=/ba∇dblPh(/hatwidermn
⋆,h)k−1/summationdisplay
j=0µj1
τ/integraldisplaytn−j
tn−j−1∂tm(t)dt/ba∇dblW1,∞
/lessorequalslantCR/ba∇dblk−1/summationdisplay
j=0µj1
τ/integraldisplaytn−j
tn−j−1∂tm(t)dt/ba∇dblW1,∞HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 31
/lessorequalslantCRk−1/summationdisplay
j=0|µj|R.
We can now establish a bound for qn
has defined in (7.2), using Lemma 5.2 together
with the above W1,∞bounds for /hatwidermn
⋆,hand˙mn
⋆,hto obtain
(7.8) /ba∇dblqn
h/ba∇dblL2/lessorequalslantc/ba∇dbl/hatwideen
h/ba∇dblL2and/ba∇dbl∇qn
h/ba∇dblL2/lessorequalslantc/ba∇dbl/hatwideen
h/ba∇dblH1.
With theW1,∞bound of /hatwidermn
⋆,hwe also obtain a bound of rn
hdefined in (6.21). Using
Lemma 5.2 ( i) and recalling the L∞bound of∆m+Hof (4.3), we find that rn
his
bounded by
(7.9)/ba∇dblrn
h/ba∇dblL2/lessorequalslant/ba∇dbl(Ph(/hatwidermn
h)−Ph(/hatwidermn
⋆,h))(∆mn
⋆+Hn)/ba∇dblL2+/ba∇dbldn
h/ba∇dblL2
/lessorequalslantc/ba∇dbl/hatwideen
h/ba∇dblL2+/ba∇dbldn
h/ba∇dblL2.
(b)Energy estimates. Forn/lessorequalslant¯nwith ¯nof (7.3), we test the error equation (6.25)
withϕh=˙en
h+qn
hand obtain
α(˙en
h,˙en
h+qn
h)+(/hatwideen
h×˙mn
⋆,h,˙en
h+qn
h)+(/hatwidermn
h×˙en
h,˙en
h+qn
h)
+(∇en
h,∇(˙en
h+qn
h)) =−(rn
h,˙en
h+qn
h).
By collecting the terms, and using the fact that ( /hatwidermn
h×˙en
h,˙en
h) = 0, we altogether
obtain
α/ba∇dbl˙en
h/ba∇dbl2
L2+(∇en
h,∇˙en
h) =−α(˙en
h,qn
h)−(/hatwideen
h×˙mn
⋆,h,˙en
h+qn
h)
−(/hatwidermn
h×˙en
h,qn
h)−(∇en
h,∇qn
h)−(rn
h,˙en
h+qn
h).
We now estimate the term ( ∇en
h,∇˙en
h) on the left-hand side from below using
Dahlquist’s Lemma 10.1, so that the ensuing relation (10.2) yields
(∇en
h,∇˙en
h)/greaterorequalslant1
τ/parenleftig
/ba∇dbl∇En
h/ba∇dbl2
G−/ba∇dbl∇En−1
h/ba∇dbl2
G/parenrightig
+(∇en
h,∇sn
h),
whereEn
h= (en−k+1
h,...,en
h) and theG-weighted semi-norm is given by
/ba∇dbl∇En
h/ba∇dbl2
G=k/summationdisplay
i,j=1gij(∇en−k+i
h,∇en−k+j
h).
This semi-norm satisfies the relation
(7.10) γ−k/summationdisplay
j=1/ba∇dbl∇en−k+j
h/ba∇dbl2
L2/lessorequalslant/ba∇dbl∇En
h/ba∇dbl2
G/lessorequalslantγ+k/summationdisplay
j=1/ba∇dbl∇en−k+j
h/ba∇dbl2
L2,
whereγ−andγ+are the smallest and largest eigenvalues of the positive definite
symmetric matrix G= (gij) from Lemma 10.1.
The remaining terms are estimated using the Cauchy–Schwarz inequ ality and
/ba∇dbl/hatwidermn
h/ba∇dblL∞= 1; we altogether obtain
α/ba∇dbl˙en
h/ba∇dbl2
L2+1
τ/parenleftig
/ba∇dbl∇En
h/ba∇dbl2
G−/ba∇dbl∇En−1
h/ba∇dbl2
G/parenrightig
/lessorequalslantα/ba∇dbl˙en
h/ba∇dblL2/ba∇dblqn
h/ba∇dblL2+/ba∇dbl/hatwideen
h/ba∇dblL2(/ba∇dbl˙en
h/ba∇dblL2+/ba∇dblqn
h/ba∇dblL2)
+/ba∇dbl˙en
h/ba∇dblL2/ba∇dblqn
h/ba∇dblL2+/ba∇dbl∇en
h/ba∇dblL2(/ba∇dbl∇qn/ba∇dblL2+/ba∇dbl∇sn
h/ba∇dblL2)+/ba∇dblrn
h/ba∇dblL2(/ba∇dbl˙en
h/ba∇dblL2+/ba∇dblqn
h/ba∇dblL2).32 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
We now show an L2error bound for /hatwideen
hin terms of (en−j−1
h)k−1
j=0. Using the fact that
fora,b∈R3\{0},
(7.11)/vextendsingle/vextendsingle/vextendsingle/vextendsinglea
|a|−b
|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(|b|−|a|)a+|a|(a−b)
|a| |b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslant2|a−b|
|b|,
and the lower bounds in (7.5) for both |/summationtextk−1
j=0γjmn−j−1
h|and|/summationtextk−1
j=0γjmn−j−1
⋆,h|, we
can estimate
(7.12)/ba∇dbl/hatwideen
h/ba∇dblL2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationtextk−1
j=0γjmn−j−1
h/vextendsingle/vextendsingle/vextendsingle/summationtextk−1
j=0γjmn−j−1
h/vextendsingle/vextendsingle/vextendsingle−/summationtextk−1
j=0γjmn−j−1
⋆,h/vextendsingle/vextendsingle/vextendsingle/summationtextk−1
j=0γjmn−j−1
⋆,h/vextendsingle/vextendsingle/vextendsingle/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
L2/lessorequalslantCk−1/summationdisplay
j=0/ba∇dblen−j−1
h/ba∇dbl2
L2.
To show a similar bound for /ba∇dbl∇/hatwideen
h/ba∇dblL2we need the following two observations: First,
theW1,∞bounds formn−j−1
⋆,hfrom (7.6) imply W1,∞boundedness for /hatwidermn
⋆,hby
/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂j/parenleftbiggb
|b|/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂jb
|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingleb(∂jb,b)
|b|3/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
Second, similarly we have/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂j/parenleftbigga
|a|−b
|b|/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ja
|a|−∂jb
|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsinglea(∂ja,a)|b|3−b(∂jb,b)|a|3
|a|3|b|3/vextendsingle/vextendsingle/vextendsingle/vextendsingle
/lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ja
|a|−∂jb
|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+||a|3−|b|3||∂jb|
|a|3|b|+|a(∂ja,a)−b(∂jb,b)|
|b|3
/lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ja
|a|−∂jb
|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+|a−b|(|b|2+|b||a|+|a|2)|∂jb|
|a|3|b|
+|a|2|∂ja−∂jb|
|b|3+|a||∂jb||a−b|
|b|3+|a−b||∂jb|
|b|2.
Combining these two observations, again with mhandm⋆,hin the role of aandb,
respectively, and the upper and lower bounds from (7.5) altogethe r yield
(7.13) /ba∇dbl∇/hatwideen
h/ba∇dbl2
L2/lessorequalslantCk−1/summationdisplay
j=0/ba∇dblen−j−1
h/ba∇dbl2
H1.
We estimate further using Young’s inequality and absorptions into th e term
/ba∇dbl˙en/ba∇dbl2
L2, together with the bounds in (7.8) and (7.9), to obtain
α1
2/ba∇dbl˙en
h/ba∇dbl2
L2+1
τ/parenleftig
/ba∇dbl∇En
h/ba∇dbl2
G−/ba∇dbl∇En−1
h/ba∇dbl2
G/parenrightig
/lessorequalslantck/summationdisplay
j=0/ba∇dblen−j
h/ba∇dbl2
H1+c/ba∇dbldn
h/ba∇dbl2
L2+c/ba∇dbl∇sn
h/ba∇dbl2
L2.
Multiplying both sides by τ, summing up from kton/lessorequalslant¯n, and using an absorption
yield
α1
2τn/summationdisplay
j=k/ba∇dbl˙ej
h/ba∇dbl2
L2+/ba∇dbl∇En
h/ba∇dbl2
G
/lessorequalslant/ba∇dbl∇Ek−1
h/ba∇dbl2
G+cτn/summationdisplay
j=k/ba∇dblej
h/ba∇dbl2
H1+cτn/summationdisplay
j=k/parenleftbig
/ba∇dbldj
h/ba∇dbl2
L2+/ba∇dblsj
h/ba∇dbl2
H1/parenrightbig
+ck−1/summationdisplay
i=0/ba∇dblei
h/ba∇dbl2
L2.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 33
We then arrive, using (7.10), at
(7.14)α1
2τn/summationdisplay
j=k/ba∇dbl˙ej
h/ba∇dbl2
L2+/ba∇dbl∇en
h/ba∇dbl2
L2/lessorequalslantcτn/summationdisplay
j=k/ba∇dblej
h/ba∇dbl2
H1+cτn/summationdisplay
j=k/parenleftbig
/ba∇dbldj
h/ba∇dbl2
L2+/ba∇dblsj
h/ba∇dbl2
H1/parenrightbig
+ck−1/summationdisplay
i=0/ba∇dblei
h/ba∇dbl2
L2,
withcdepending on α.
Similarly as in the time continuous case in the proof of Lemma 4.2, we con nect
/ba∇dblen
h/ba∇dbl2
L2andτ/summationtextn
j=k/ba∇dbl˙ej
h/ba∇dbl2
L2. We rewrite the identity
1
τk/summationdisplay
j=0δjen−j
h=˙en
h−sn
h, n/greaterorequalslantk,
as
1
τn/summationdisplay
j=kδn−jej
h=˙ehn−sn
h−gn
h, n/greaterorequalslantk,
withδℓ= 0 forℓ>kand where
gn
h:=1
τk−1/summationdisplay
i=0δn−iei
h
depends only on the starting errors and satisfies gn
h= 0 forn/greaterorequalslant2k. With the inverse
power series of δ(ζ),
κ(ζ) =∞/summationdisplay
n=0κnζn:=1
δ(ζ),
we then have, for n/greaterorequalslantk,
en
h=τn/summationdisplay
j=kκn−j(˙ej
h−sj
h−gj
h).
By the zero-stability of the BDF method of order k/lessorequalslant6, the coefficients κnare
uniformly bounded: |κn|/lessorequalslantcfor alln/greaterorequalslant0. Therefore we obtain via the Cauchy–
Schwarz inequality
/ba∇dblen
h/ba∇dbl2
L2/lessorequalslant2τ2/vextenddouble/vextenddouble/vextenddoublen/summationdisplay
j=kκn−j(˙ehj−sj
h)/vextenddouble/vextenddouble/vextenddouble2
L2+2τ2/vextenddouble/vextenddouble/vextenddouble2k−1/summationdisplay
j=kκn−jgj
h/vextenddouble/vextenddouble/vextenddouble2
L2
/lessorequalslant(2nτ)τc2n/summationdisplay
j=k/ba∇dbl˙ehj−sj
h/ba∇dbl2
L2+2τ2c2k2k−1/summationdisplay
j=k/ba∇dblgj
h/ba∇dbl2
L2
/lessorequalslantCτn/summationdisplay
j=k/ba∇dbl˙ej
h/ba∇dbl2
L2+Cτn/summationdisplay
j=k/ba∇dblsj
h/ba∇dbl2
L2+Ck/summationdisplay
i=0/ba∇dblei
h/ba∇dbl2
L2.34 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
Inserting this bound into (7.14) then yields
α/ba∇dblen
h/ba∇dbl2
L2+/ba∇dbl∇en
h/ba∇dbl2
L2/lessorequalslantcτn/summationdisplay
j=k/ba∇dblej
h/ba∇dbl2
H1+cτn/summationdisplay
j=k/parenleftbig
/ba∇dbldj
h/ba∇dbl2
L2+/ba∇dblsj
h/ba∇dbl2
H1/parenrightbig
+ck−1/summationdisplay
i=0/ba∇dblei
h/ba∇dbl2
L2,
and a discrete Gronwall inequality implies the stated stability result fo rn/lessorequalslant¯n. It
then follows from this stability bound, the smallness condition of the le mma and the
inverse estimate from H1toL∞that (7.3) is satisfied also for ¯ n+1. This completes
the induction step for (7.3) and proves the stated error bound. /square
8.Stability of the full discretization for BDF of orders 3 to 5
Stability for full discretizations using the BDF methods of orders 3 t o 5 can be
shown under additional conditions on the damping parameter αand the stepsize τ.
Lemma 8.1 (Stability for orders k= 3,4,5).Consider the linearly implicit k-step
BDF discretization (2.6)for3/lessorequalslantk/lessorequalslant5with finite elements of polynomial degree
r/greaterorequalslant2. Letmn
handmn
⋆,hsatisfy(2.6)and(6.17), respectively, and suppose that the
regularity assumptions of Lemma 7.1 hold. Furthermore, ass ume that the damping
parameterαsatisfies
(8.1) α>α k:=ηk
1−ηk
with the multiplier ηkof Lemma 10.2, and that τandhsatisfy the mild CFL-type
condition, for some ¯c>0,
(8.2) τ/lessorequalslant¯ch.
Then, for sufficiently small h/lessorequalslant¯handτ/lessorequalslant¯τ, the erroren
h=mn
h−mn
⋆,hsatisfies
the following bound, for kτ/lessorequalslantnτ/lessorequalslant¯t,
(8.3)/ba∇dblen
h/ba∇dbl2
H1(Ω)3/lessorequalslantC/parenleftigk−1/summationdisplay
i=0/ba∇dblei
h/ba∇dbl2
H1(Ω)3+τn/summationdisplay
j=k/ba∇dbldj
h/ba∇dbl2
L2(Ω)3+τn/summationdisplay
j=k/ba∇dblsj
h/ba∇dbl2
H1(Ω)3/parenrightig
,
where the constant Cis independent of τ,handn, but depends on α,R,K,M , and
exponentially on ¯c¯t. This estimate holds under the smallness condition that the
right-hand side is bounded by ˆch3with a constant ˆc(note that the right-hand side is
of sizeO((τk+hr)2)in the case of a sufficiently regular solution ).
Together with the defect bounds of Section 6, this stability lemma pr oves Theo-
rem 3.2. We remark that the thresholds αk>0 defined here are the same as those
appearing in Theorem 3.2.
Proof.The proof of this lemma combines the arguments of the proof of Lem ma 7.1
with a nonstandard variant of the multiplier technique of Nevanlinna a nd Odeh, as
outlined in the Appendix. Since the size of the parameter αdetermines which BDF
methods satisfy the stability estimate, the dependence on αwill be carefully traced
all along the proof.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 35
(a)Preparations. As in the previous proof, we make again the induction hypoth-
esis (7.3) for some ¯ nwith ¯nτ/lessorequalslant¯t, but this time with ρ=c0hfor some positive
constantc0:
(8.4) /ba∇dblen
h/ba∇dblL∞/lessorequalslantc0h, n< ¯n.
By an inverse inequality, this implies that /ba∇dblen
h/ba∇dblW1,∞has anh- andτ-independent
bound, and hence also /ba∇dblmn
h/ba∇dblW1,∞forn<¯n. Together with (7.5), this implies
(8.5) /ba∇dbl/hatwidermn
h/ba∇dblW1,∞/lessorequalslantC
and further
(8.6) /ba∇dbl/hatwideen
h/ba∇dblL∞/lessorequalslantCh.
As in the Appendix, we aim to subtract ηktimes the error equation for time
stepn−1 from the error equation for time step n, and then to test with ϕh=
Ph(/hatwidermn
h)˙en
h∈Th(/hatwidermn
h) (similarly as in the proof of Lemma 7.1). However, this is not
possible directly due to the different test spaces at different time st eps:
α(˙en
h,ϕh)+(/hatwideen
h×˙mn
⋆,h,ϕh)
+(/hatwidermn
h×˙en
h,ϕh)+(∇en
h,∇ϕh) =−(rn
h,ϕh),(8.7a)
for allϕh∈Th(/hatwidermn
h), and
α(˙en−1
h,ψh)+(/hatwideen−1
h×˙mn−1
⋆,h,ψh)
+(/hatwidermn−1
h×˙en−1
h,ψh)+(∇en−1
h,∇ψh) =−(rn−1
h,ψh),(8.7b)
for allψh∈Th(/hatwidermn−1
h).
As in (7.2), we have
(8.8)ϕh=Ph(/hatwidermn
h)˙en
h=˙en
h+qn
h,withqn
h=−(Ph(/hatwidermn
h)−Ph(/hatwidermn
⋆,h))˙mn
⋆,h,
whereqn
his bounded by (7.8).
In turn, the test function ψh=Ph(/hatwidermn−1
h)˙en
h∈Th(/hatwidermn−1
h) is a perturbation of
ϕh=˙en
h+qn
h, since using (8.8) we obtain
ψh=Ph(/hatwidermn−1
h)˙en
h
=Ph(/hatwidermn
h)˙en
h−(Ph(/hatwidermn
h)−Ph(/hatwidermn−1
h))˙en
h
=˙en
h+qn
h+pn
hwithpn
h=−(Ph(/hatwidermn
h)−Ph(/hatwidermn−1
h))˙en
h.
The perturbation pn
his estimated using the second bound in Lemma 5.2 ( i) with
p=∞,q= 2, and noting (8.5). We obtain
/ba∇dblpn
h/ba∇dblL2/lessorequalslant/ba∇dbl(Ph(/hatwidermn
h)−Ph(/hatwidermn−1
h))˙en
h/ba∇dblL2
/lessorequalslantc/ba∇dbl˙en
h/ba∇dblL2/ba∇dbl/hatwidermn
h−/hatwidermn−1
h/ba∇dblL∞
/lessorequalslantc/ba∇dbl˙en
h/ba∇dblL2/parenleftig
/ba∇dbl/hatwideen
h/ba∇dblL∞+/ba∇dbl/hatwidermn
⋆,h−/hatwidermn−1
⋆,h/ba∇dblL∞+/ba∇dbl/hatwideen−1
h/ba∇dblL∞/parenrightig
/lessorequalslantc/ba∇dbl˙en
h/ba∇dblL2/parenleftig
/ba∇dbl/hatwideen
h/ba∇dblL∞+k−1/summationdisplay
j=0|γj|/integraldisplaytn−j−1
tn−j−2/ba∇dblRh∂tm(t)/ba∇dblL∞dt+/ba∇dbl/hatwideen−1
h/ba∇dblL∞/parenrightig
.36 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
We have /ba∇dblRh∂tm(t)/ba∇dblL∞/lessorequalslantc/ba∇dbl∂tm(t)/ba∇dblW1,∞by [12, Theorem 8.1.11]. In view of (8.6)
we obtain, for τ/lessorequalslant¯Ch,
(8.9) /ba∇dblpn
h/ba∇dblL2/lessorequalslantCh/ba∇dbl˙en
h/ba∇dblL2,
and by an inverse estimate,
(8.10) /ba∇dbl∇pn
h/ba∇dblL2/lessorequalslantC/ba∇dbl˙en
h/ba∇dblL2.
We also recall the bound (7.9) for /ba∇dblrn
h/ba∇dblL2.
(b)Energy estimates. By subtracting (8.7a) −ηk(8.7b) with the above choice of
test functions, we obtain
(8.11)α(˙en
h−ηk˙en−1
h,˙en
h+qn
h)+(/hatwideen
h×˙mn
⋆,h−ηk/hatwideen−1
h×˙mn−1
⋆,h,˙en
h+qn
h)
+(/hatwidermn
h×˙en
h−ηk/hatwidermn−1
h×˙en−1
h,˙en
h+qn
h)+(∇en
h−ηk∇en−1
h,∇(˙en
h+qn
h))
−ηk/bracketleftbig
α(˙en−1
h,pn
h)+(/hatwideen−1
h×˙mn−1
⋆,h,pn
h)
+(/hatwidermn−1
h×˙en−1
h,pn
h)+(∇en−1
h,∇pn
h)/bracketrightbig
=−(rn
h−ηkrn−1
h,˙en
h+qn
h)−ηk(rn−1
h,pn
h).
We estimate the terms of the error equation (8.11) separately and track carefully
the dependence on ηkandα.
The termα(˙en
h−ηk˙en−1
h,˙en
h) is bounded from below, using Young’s inequality and
absorptions, by
α(˙en
h−ηk˙en−1
h,˙en
h)/greaterorequalslantα/parenleftbig
1−1
2ηk/parenrightbig
/ba∇dbl˙en
h/ba∇dbl2
L2−α
2ηk/ba∇dbl˙en−1
h/ba∇dbl2
L2,
while the term ( ∇en
h−ηk∇en−1
h,∇˙en
h) is bounded from below, via the relation (10.2)
and (6.23), by
(∇en
h−ηk∇en−1
h,∇˙en
h)/greaterorequalslant1
τ/parenleftig
/ba∇dbl∇En
h/ba∇dbl2
G−/ba∇dbl∇En−1
h/ba∇dbl2
G/parenrightig
+(∇en
h−ηk∇en−1
h,∇sn
h),
withEn
h= (en−k+1
h,...,en
h), and where the G-weighted semi-norm is generated by
the matrix G= (gij) from Lemma 10.1 for the rational function δ(ζ)/(1−ηkζ).
The remaining terms outside the rectangular bracket are estimate d using the
Cauchy–Schwarz and Young inequalities (the latter often with a suffi ciently small
but fixedh- andτ-independent weighting factor µ >0) and/ba∇dbl/hatwidermn
h/ba∇dblL∞= 1 and
orthogonality. We obtain, with varying constants c(which depend on αand are
inversely proportional to µ)
α(˙en
h−ηk˙en−1
h,qn
h)+(/hatwideen
h×˙mn
⋆,h−ηk/hatwideen−1
h×˙mn−1
⋆,h,˙en
h+qn
h)
+(/hatwidermn
h×˙en
h−ηk/hatwidermn−1
h×˙en−1
h,˙en
h+qn
h)+(∇en−ηk∇en−1
h,∇qn
h)
/lessorequalslant/parenleftbig
αµ+µ+1
2ηk/parenrightbig
/ba∇dbl˙en
h/ba∇dbl2
L2+/parenleftbig
αµηk+1
2ηk/parenrightbig
/ba∇dbl˙en−1
h/ba∇dbl2
L2
+c/parenleftbig
/ba∇dblqn
h/ba∇dblL2+/ba∇dbl/hatwideen
h/ba∇dbl2
L2+/ba∇dbl/hatwideen−1
h/ba∇dbl2
L2/parenrightbig
+1
2/parenleftbig
/ba∇dbl∇en
h/ba∇dbl2
L2+η2
k/ba∇dbl∇en−1
h/ba∇dbl2
L2+/ba∇dbl∇qn
h/ba∇dblL2/parenrightbig
/lessorequalslant/parenleftbig
αµ+µ+1
2ηk/parenrightbig
/ba∇dbl˙en
h/ba∇dbl2
L2+/parenleftbig
αµηk+1
2ηk/parenrightbig
/ba∇dbl˙en−1
h/ba∇dbl2
L2+ck/summationdisplay
j=0/ba∇dblen−j−1
h/ba∇dbl2
H1,
where in the last inequality we used (7.12) and (7.13) to estimate /hatwideen
h.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 37
The terms inside the rectangular bracket are bounded similarly, usin g (8.9) and
(8.10) and the condition τ/lessorequalslant¯Ch, by
α(˙en−1
h,pn
h)+(/hatwideen−1
h×˙mn−1
⋆,h,pn
h)+(/hatwidermn−1
h×˙en−1
h,pn
h)+(∇en−1
h,∇pn
h)
/lessorequalslantµ/ba∇dbl˙en
h/ba∇dbl2
L2+ch/ba∇dbl˙en−1
h/ba∇dbl2
L2+c/parenleftbig
/ba∇dbl/hatwideen−1
h/ba∇dbl2
L2+/ba∇dbl∇en−1
h/ba∇dbl2
L2/parenrightbig
/lessorequalslantµ/ba∇dbl˙en
h/ba∇dbl2
L2+ck/summationdisplay
j=0/ba∇dblen−j−1
h/ba∇dbl2
H1.
Hereµis an arbitrarily small positive constant (independent of τandh), andc
depends on the choice of µ.
In view of (7.9), the terms with the defects rn
hare bounded by
−(rn
h−ηkrn−1
h,˙en
h+qn
h)−ηk(rn−1
h,pn
h)
/lessorequalslantµ/ba∇dbl˙en
h/ba∇dbl2
L2+c/parenleftbig
/ba∇dblrn
h/ba∇dbl2
L2+/ba∇dblrn−1
h/ba∇dbl2
L2+/ba∇dblqn
h/ba∇dbl2
L2/parenrightbig
/lessorequalslantµ/ba∇dbl˙en
h/ba∇dbl2
L2+ck/summationdisplay
j=0/ba∇dblen−j−1
h/ba∇dbl2
L2+c1/summationdisplay
j=0/ba∇dbldn−j
h/ba∇dbl2
L2.
Combination of these inequalities yields
/parenleftig
α(1−1
2ηk)−1
2ηk−µ/parenrightig
/ba∇dbl˙en
h/ba∇dbl2
L2−/parenleftig
α
2ηk+1
2ηk+µαηk/parenrightig
/ba∇dbl˙en−1
h/ba∇dbl2
L2
+1
τ/parenleftig
/ba∇dbl∇En
h/ba∇dbl2
G−/ba∇dbl∇En−1
h/ba∇dbl2
G/parenrightig
/lessorequalslantck/summationdisplay
j=0/ba∇dblen−j−1
h/ba∇dbl2
H1+c1/summationdisplay
j=0/ba∇dbldn−j
h/ba∇dbl2
L2+c/ba∇dbl∇sn
h/ba∇dbl2
L2.
Under condition (8.1) we have
ω:=α(1−ηk)−ηk>0.
Multiplying both sides by τand summing up from ktonwithn/lessorequalslant¯nyields, for
sufficiently small µ,
1
2ωτn/summationdisplay
j=k/ba∇dbl˙ej
h/ba∇dbl2
L2+/ba∇dbl∇En
h/ba∇dbl2
G
/lessorequalslantcτ/ba∇dbl˙ek−1
h/ba∇dbl2
L2+/ba∇dbl∇Ek−1
h/ba∇dbl2
G+cτn−1/summationdisplay
j=0/ba∇dblej
h/ba∇dbl2
H1+cτn/summationdisplay
j=k/ba∇dbldj
h/ba∇dbl2
L2+cτn/summationdisplay
j=k/ba∇dbl∇sj
h/ba∇dbl2
L2.
The proof is then completed using exactly the same arguments as in t he last part of
theproofofLemma7.1,byestablishinganestimatebetween /ba∇dblen
h/ba∇dbl2
L2andτ/summationtextn
j=k/ba∇dbl˙ej
h/ba∇dbl2
L2
and using a discrete Gronwall inequality, and completing the induction step for
(8.4). /square
9.Numerical experiments
To obtain significant numerical results, we prescribe the exact solu tionmon
given three-dimensional domains Ω:= [0,1]×[0,1]×[0,L] withL∈ {1/100,1/4}.38 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
The discretizations of these domains will consist of a few layers of ele ments inz-
direction (one layer for L= 1/100 and ten layers for L= 1/4) and a later specified
number of elements in xandydirections. This mimics the common case of thin
film alloys as for example in the standard problems of the Micromagnet ic Modeling
ActivityGroupatNISTCenterforTheoreticalandComputational MaterialsScience
(ctcms.nist.gov ). Moreover, this mesh structure helps to keep the computationa l
requirements reasonable and allow us to compute the experiments o n a desktop
PC. We are aware that these experiments are only of preliminary nat ure and are
just supposed to confirm the theoretical results. A more thorou gh investigation
of the numerical properties of the developed method is needed. Th is will require
us to incorporate preconditioning, parallelization of the computatio ns, as well as
lower order energy contributions in the effective field (1.3) to be able to compare to
benchmark results from computational physics. This, however, is beyond the scope
of this paper, and will be the topic of a subsequent work.
We consider the time interval [0 ,¯t] with¯t= 0.2 and define two different exact
solutions. Since within our computational budget either the time disc retization
error or the space discretization error dominates, we construct the solutions such
that the first oneis harder to approximate inspace, while thesecon d oneis harder to
approximate in time. Both solutions are constant in z-direction as is often observed
in thin-film applications.
9.1.Implementation. The numerical experiments were conducted using the finite
element package FEniCS ( www.fenicsproject.org ) on a desktop computer. As al-
readydiscussed inSection2.2, thereareseveral ways toimplement thetangent space
restriction. We decided to solve a saddle point problem (variant (a) in Section 2.2)
for simplicity of implementation. For preconditioning, we used the blac k-box AMG
preconditioner that comes with FEniCS. Although this might not be th e optimal
solution, it keeps the number of necessary iterative solver steps w ithin reasonable
bounds. Assuming perfect preconditioning, the cost per time-ste p is then propor-
tional to the number of mesh-elements. We observed this behavior approximately,
although further research beyond the scope of this work is requir ed to give a definite
conclusion.
9.2.Exact solutions. We choose the damping parameter α= 0.2 and define
g(t) := (¯t+0.1)/(¯t+0.1−t) as well as d(x) := (x1−1/2)2+(x2−1/2)2, which is
the squared distance of the projection of xto [0,1]×[0,1] and the point (1 /2,1/2).
For some constant C= 400 (a choice made to have pronounced effects), define
(9.1)m(x,t) :=
Ce−g(t)
1/4−d(x)(x1−1/2)
Ce−g(t)
1/4−d(x)(x2−1/2)/radicalig
1−C2e−2g(t)
1/4−d(x)d(x)
ifd(x)/lessorequalslant1
4andm(x,t) :=
0
0
1
else.
It iseasy to check that |m(x,t)|= 1for all ( x,t)∈Ω×[0,¯t]. Moreover, ∂nm(x,t) =
0 for allx∈∂Ω. We may calculate the time derivative of min a straightforwardHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 39
fashion, i.e., ∂tm(x,t) = 0 ford(x)>1/4 and
∂tm(x,t) =
−g′(t)
1/4−d(x)Ce−g(t)
1/4−d(x)(x1−1/2)
−g′(t)
1/4−d(x)Ce−g(t)
1/4−d(x)(x2−1/2)
g′(t)
1/4−d(x)C2e−2g(t)
1/4−d(x)d(x)
m3(x,t)
ifd(x)/lessorequalslant1
4.
Here,m3denotes the third component of mas defined above.
The second exact solution is defined via
(9.2) /tildewiderm(x,t) :=
−(x3
1−3x2
1/2+1/4)sin(3πt/¯t)/radicalbig
1−(x3
1−3x2
1/2+1/4)2
−(x3
1−3x2
1/2+1/4)cos(3πt/¯t)
.
Due to the polynomial nature in the first and the third component, a nd the well-
behaved square-root, the space approximation error does not d ominate the time
approximation.
9.3.The experiments. We now may compute the corresponding forcings Hresp.
/tildewiderHto obtain the prescribed solutions by inserting into (1.4), i.e.,
H=α∂tm+m×∂tm−∆m.
(Note that we may disregard the projection P(m) from (1.4) since we solve in
the tangent space anyway.) We compute Hnumerically by first interpolating m
and∂tmand then computing the derivatives. This introduces an additional e rror
which is not accounted for in the theoretical analysis. However, th e examples below
confirm the expected convergence rates and hence conclude tha t this additional
perturbation is negligible. Figure 9.1 shows slices of the exact solution at different
time steps. Figure 9.2 shows the convergence with respect to the t ime step size τ,
while Figure 9.3 shows convergence with respect to the spatial mesh sizeh. All the
experiments confirm the expected rates for smooth solutions.
Finally, we consider an example with nonsmooth initial data and consta nt right-
hand side. The initial data are given by
(9.3)m0(x) :=
x1−1/2
x2−1/2/radicalbig
1−d(x)
ifd(x)/lessorequalslant1
4andm0(x) :=
0
0
1
else.
With the constant forcing field H:= (0,1,1)Twe compute a numerical approxima-
tion to the unknown exact solution. Note that we do not expect any smoothness of
the solution (even the initial data is not smooth). Figure 9.4 neverth eless shows a
physically consistent decay of the energy /ba∇dbl∇m(t)/ba∇dblL2(Ω)3over time as well as a good
agreement between different orders of approximation. Moreover , the computed ap-
proximation shows little deviation from unit length as would be expecte d for smooth
solutions.40 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
Figure 9.1. Thefirstrowshowstheexactsolution m(x,t)from(9.1)
forx∈[0,1]×[0,1]× {0}andt∈ {0,0.05,¯t}(from left to right),
whereas the second row shows the exact solution /tildewiderm(x,t) from (9.2)
forx∈[0,1]×[0,1]×{0}andt∈ {0,0.2/6,0.2/3}(from left to right).
While the problems are three-dimensional, the solutions are constan t
inz-direction and we only show one slice of the solution.PSfrag replacements
10−810−610−410−2100
10−310−210−1k= 1
k= 2
k= 3
k= 4
timestep τPSfrag replacements
10−610−510−410−310−210−1100101
10−310−210−1k= 1
k= 2
k= 3
k= 4
timestep τ
Figure 9.2. The plots show the error between computed solutions
and exact solution /tildewidermfor a given time stepsize with a spatial poly-
nomial degree of r= 2 and a spatial mesh size 1 /40 which results
in≈6·104degrees of freedom per time step in the left plot. In the
right plot we use a thicker domain D= [0,1]×[0,1]×[0,1/4] with 10
elements in z-direction. This results in ≈4·105degrees of freedom
per timestep. We use the k-step methods of order k∈ {1,2,3,4}and
observe the expected rates O(τk) indicated by the dashed lines. The
coarse levels of the higher order methods are missing because the kth
step is already beyond the final time ¯t.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 41
PSfrag replacements
r= 1
r= 2
r= 3
r= 4
meshsize h10−410−310−210−1100
10−1
Figure 9.3. The plot shows convergence in meshsize hwith respect
to the exact solution mfrom (9.1) on the domain D= [0,1]×[0,1]×
[0,1/100]withonelayerofelementsin z-direction. Weusedthesecond
order BDF method with τ= 10−3and spatial polynomial degrees
r∈ {1,2,3,4}. The mesh sizes range from 1 /2 to 1/32. We observe
the expected rates O(hr) indicated by the dashed lines. The finest
mesh-size for r= 4 does reach the expected error level. This is due to
the fact that the time-discretization errors start to dominate in t hat
region.
10.Appendix: Energy estimates for backward difference formula e
The stability proofs of this paper rely on energy estimates, that is, on the use
of positive definite bilinear forms to bound the error ein terms of the defect d.
This is, of course, a basic technique for studying the time-continuo us problem and
also for backward Euler and Crank–Nicolson time discretizations (se e, e.g., Thom´ ee
[38]), but energy estimates still appear to be not well known for bac kward difference
formula (BDF) time discretizations of order up to 5, which are widely u sed for
solving stiff ordinary differential equations. To illustrate the basic me chanism, we
here just consider the prototypical linear parabolic evolution equa tion in its weak
formulation, given by two positive definite symmetric bilinear forms ( ·,·) anda(·,·)
on Hilbert spaces HandVwith induced norms |·|and/ba∇dbl·/ba∇dbl, respectively, and with
Vdensely and continuously embedded in H. The problem then is to find u(t)∈V
such that
(10.1) ( ∂tu,v)+a(u,v) = (f,v)∀v∈V,
with initial condition u(0) =u0. Ifu⋆is a function that satisfies the equation up to
a defectd, that is,
(∂tu⋆,v)+a(u⋆,v) = (f,v)+(d,v)∀v∈V,42 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICHPSfrag replacements
0.15
0.14
0.13
0.12
0.11
0.1
0.09
0.08
0.07
0.06
0.050 0.05 0.1 0.15 0.2
timer=k= 1
r=k= 2
r=k= 3
r=k= 4r=k= 1
r=k= 2
r=k= 3
r=k= 4PSfrag replacements10−1
10−2
10−3
0 0 .05 0.1 0.15 0.2
time
Figure 9.4. Left plot: Decay of energies /ba∇dbl∇m(t)/ba∇dblL2(Ω)3for the ap-
proximations to the unknown solution with m0andHgiven in (9.3)
and one line after (9.3). We plot four approximations of the k-step
method with polynomial degree rforr=k∈ {1,2,3,4}. The spa-
tial mesh-size is 1 /40 and the size of the timesteps is 10−3(blue) and
10−2(red). Right plot: Deviation from unit length /ba∇dbl1−|m(t)|2/ba∇dblL∞(Ω)
plotted over time for step sizes τ= 10−2(blue),τ= 10−3(red), and
τ= 10−4(green). The solid lines indicate k= 1, whereas the dashed
lines indicate k= 2. The spatial mesh-size is 1 /40 withr= 1.
then the error e=u−u⋆satisfies, in this linear case, an equation of the same form,
(∂te,v)+a(e,v) = (d,v)∀v∈V,
with initial value e0=u0−u⋆
0. Testing with v=eyields
1
2d
dt|e|2+/ba∇dble/ba∇dbl2= (d,e).
Estimating the right-hand side by ( d,e)/lessorequalslant/ba∇dbld/ba∇dbl⋆/ba∇dble/ba∇dbl/lessorequalslant1
2/ba∇dbld/ba∇dbl2
⋆+1
2/ba∇dble/ba∇dbl2, with the dual
norm/ba∇dbl·/ba∇dbl⋆, and integrating from time 0 to tresults in the error bound
|e(t)|2/lessorequalslant|e(0)|2+/integraldisplayt
0/ba∇dbld(s)/ba∇dbl2
⋆ds.
On the other hand, testing with v=∂teyields
|∂te|2+1
2d
dt/ba∇dble/ba∇dbl2= (d,∂te),
which leads similarly to the error bound
/ba∇dble(t)/ba∇dbl2/lessorequalslant/ba∇dble(0)/ba∇dbl2+/integraldisplayt
0|d(s)|2ds.
This procedure is all-familiar, but it is not obvious how to extend it to tim e dis-
cretizations beyond the backward Euler and Crank–Nicolson metho ds. The use of
energy estimates for BDF methods relies on the following remarkable results.
Lemma 10.1. (Dahlquist [18]; see also [8] and [27, Section V.6]) Letδ(ζ) =δkζk+
···+δ0andµ(ζ) =µkζk+···+µ0be polynomials of degree at most k(and at leastHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 43
one of them of degree k)that have no common divisor. Let (·,·)be an inner product
with associated norm |·|.If
Reδ(ζ)
µ(ζ)>0for|ζ|<1,
then there exists a positive definite symmetric matrix G= (gij)∈Rk×ksuch that
forv0,...,v kin the real inner product space,
/parenleftigk/summationdisplay
i=0δivk−i,k/summationdisplay
j=0µjvk−j/parenrightig
/greaterorequalslantk/summationdisplay
i,j=1gij(vi,vj)−k/summationdisplay
i,j=1gij(vi−1,vj−1).
In combination with the preceding result for the multiplier µ(ζ) = 1−ηkζ,the
following property of BDF methods up to order 5 becomes important .
Lemma 10.2. (Nevanlinna & Odeh [34]) Fork/lessorequalslant5,there exists 0/lessorequalslantηk<1such
that forδ(ζ) =/summationtextk
ℓ=11
ℓ(1−ζ)ℓ,
Reδ(ζ)
1−ηkζ>0for|ζ|<1.
The smallest possible values of ηkare
η1=η2= 0, η3= 0.0836, η4= 0.2878, η5= 0.8160.
Precise expressions for the optimal multipliers for the BDF methods of orders 3,4
and 5 are given by Akrivis & Katsoprinakis [1].
An immediate consequence of Lemma 10.2 and Lemma 10.1 is the relation
(10.2)/parenleftigk/summationdisplay
i=0δivk−i,vk−ηkvk−1/parenrightig
/greaterorequalslantk/summationdisplay
i,j=1gij(vi,vj)−k/summationdisplay
i,j=1gij(vi−1,vj−1)
with a positive definite symmetric matrix G= (gij)∈Rk×k; it is this inequality that
plays a crucial role in our energy estimates, and the same inequality f or the inner
producta(·,·).
The errorequationfortheBDFtimediscretization ofthelinear para bolicproblem
(10.1) reads
(˙en,v)+a(en,v) = (dn,v)∀v∈V,where ˙en=1
τk/summationdisplay
j=0δjen−j,
with starting errors e0,...,ek−1. When we test with v=en−ηken−1, the first term
can be estimated from below by (10.2), the second term is bounded f rom below by
(1−1
2ηk)/ba∇dblen/ba∇dbl2−1
2ηk/ba∇dblen−1/ba∇dbl2, and the right-hand term is estimated from above by the
Cauchy-Schwarz inequality. Summing up from ktonthen yields the error bound
(10.3) |en|2+τn/summationdisplay
j=k/ba∇dblej/ba∇dbl2/lessorequalslantCk/parenleftigk−1/summationdisplay
i=0/parenleftbig
|ei|2+τ/ba∇dblei/ba∇dbl2/parenrightbig
+τn/summationdisplay
j=k/ba∇dbldj/ba∇dbl2
⋆/parenrightig
,
whereCkdepends only on the order kof the method. This kind of estimate for
the BDF error has recently been used for a variety of linear and non linear parabolic
problems [33, 3, 2, 30].44 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
On the other hand, when we first subtract ηktimes the error equation for n−1
from the error equation with nand then test with ˙ en, we obtain
(˙en−ηk˙en−1,˙en)+a(en−ηken−1,˙en) = (dn−ηkdn−1,˙en).
Here, thesecond termis boundedfrombelow by (10.2)withthe a(·,·)inner product,
the first term is bounded from below by (1 −1
2ηk)|˙en|2−1
2ηk|˙en−1|2, and the right-
hand term is estimated from above by the Cauchy–Schwarz inequalit y. Summing
up fromktonthen yields the error bound
(10.4) /ba∇dblen/ba∇dbl2+τn/summationdisplay
j=k|˙ej|2/lessorequalslantCk/parenleftigk−1/summationdisplay
i=0/ba∇dblei/ba∇dbl2+τn/summationdisplay
j=k|dj|2/parenrightig
.
It is this type of estimate that we use in the present paper for the n onlinear problem
considered here. It has previously been used in [29].
Acknowledgment. The work of Michael Feischl, Bal´ azs Kov´ acs and Christian Lu-
bichissupportedbyDeutscheForschungsgemeinschaft –projec t-id 258734477–SFB
1173.
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anisotropy and non-locality , New J. Phys. 16(2014) 013032.46 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH
Department of Computer Science & Engineering, University o f Ioannina, 45110
Ioannina, Greece, and Institute of Applied and Computation al Mathematics, FORTH,
70013 Heraklion, Crete, Greece
E-mail address :akrivis@cse.uoi.gr
Institute for Analysis and Scientific Computing (E 101), Te chnical University
Wien, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria
E-mail address :michael.feischl @kit.edu
E-mail address :michael.feischl @tuwien.ac.at
Mathematisches Institut, Universit ¨at T¨ubingen, Auf der Morgenstelle, D-72076
T¨ubingen, Germany
E-mail address :kovacs@na.uni-tuebingen.de
Mathematisches Institut, Universit ¨at T¨ubingen, Auf der Morgenstelle, D-72076
T¨ubingen, Germany
E-mail address :lubich@na.uni-tuebingen.de |
2306.13013v4.Gilbert_damping_in_metallic_ferromagnets_from_Schwinger_Keldysh_field_theory__Intrinsically_nonlocal_and_nonuniform__and_made_anisotropic_by_spin_orbit_coupling.pdf | Gilbert damping in metallic ferromagnets from Schwinger-Keldysh field theory:
Intrinsically nonlocal and nonuniform, and made anisotropic by spin-orbit coupling
Felipe Reyes-Osorio and Branislav K. Nikoli´ c∗
Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA
(Dated: March 1, 2024)
Understanding the origin of damping mechanisms in magnetization dynamics of metallic ferro-
magnets is a fundamental problem for nonequilibrium many-body physics of systems where quantum
conduction electrons interact with localized spins assumed to be governed by the classical Landau-
Lifshitz-Gilbert (LLG) equation. It is also of critical importance for applications as damping affects
energy consumption and speed of spintronic and magnonic devices. Since the 1970s, a variety of
linear-response and scattering theory approaches have been developed to produce widely used for-
mulas for computation of spatially-independent Gilbert scalar parameter as the magnitude of the
Gilbert damping term in the LLG equation. The largely unexploited for this purpose Schwinger-
Keldysh field theory (SKFT) offers additional possibilities, such as to rigorously derive an extended
LLG equation by integrating quantum electrons out. Here we derive such equation whose Gilbert
damping for metallic ferromagnets is nonlocal —i.e., dependent on all localized spins at a given
time—and nonuniform , even if all localized spins are collinear and spin-orbit coupling (SOC) is
absent. This is in sharp contrast to standard lore, where nonlocal damping is considered to emerge
only if localized spins are noncollinear—for such situations, direct comparison on the example of
magnetic domain wall shows that SKFT-derived nonlocal damping is an order of magnitude larger
than the previously considered one. Switching on SOC makes such nonlocal damping anisotropic , in
contrast to standard lore where SOC is usually necessary to obtain nonzero Gilbert damping scalar
parameter. Our analytical formulas, with their nonlocality being more prominent in low spatial
dimensions, are fully corroborated by numerically exact quantum-classical simulations.
I. INTRODUCTION
The celebrated Landau-Lifshitz equation [1] is the
foundation of standard frameworks, such as classical mi-
cromagnetics [2, 3] and atomistic spin dynamics [4], for
modelling the dynamics of local magnetization within
magnetic materials driven by external fields or currents
in spintronics [2] and magnonics [3]. It considers localized
spins as classical vectors M(r) of fixed length normalized
to unity whose rotation around the effective magnetic
fieldBeffis governed by
∂tM=−M×Beff+M×(D ·∂tM), (1)
where ∂t≡∂/∂t. Although spin is a genuine quan-
tum degree of freedom, such phenomenological equation
can be fully microscopically justified from open quantum
many-body system dynamics where M(r) tracks the tra-
jectories of quantum-mechanical expectation value of lo-
calized spin operators [5] in ferromagnets, as well as in
antiferromagnets as long as the spin value is sufficiently
large S >1. The presence of a dissipative environment in
such justification invariably introduces damping mecha-
nisms, which were conjectured phenomenologically in the
earliest formulation [1], as well as in later renderings us-
ing the so-called Gilbert form of damping [6, 7] written as
the second term on the right-hand side (RHS) of Eq. (1).
The Gilbert damping Dwas originally considered as a
spatially uniform scalar D ≡αG, or possibly tensor [8, 9],
∗bnikolic@udel.edudependent on the intrinsic properties of a material. Its
typical values are αG∼0.01 in standard ferromagnetic
metals [10], or as low as αG∼10−4in carefully designed
magnetic insulators [11] and metals [12]. Furthermore,
recent extensions [13–21] of the Landau-Lifshitz-Gilbert
(LLG) Eq. (1) for the dynamics of noncollinear magneti-
zation textures find Dto be a spatially nonuniform and
nonlocal tensor
Dαβ=αGδαβ+ηX
β′(M×∂β′M)α(M×∂β′M)β,(2)
where ∂β′≡∂/∂β′, and α, β, β′∈ {x, y, z}.
It is generally believed that αGisnonzero only
when SOC [22, 23] or magnetic disorder (or both) are
present [15, 24, 25]. For example, αGhas been ex-
tracted from a nonrelativistic expansion of the Dirac
equation [22, 23], and spin-orbit coupling (SOC) is vir-
tually always invoked in analytical (conducted for sim-
plistic model Hamiltonians) [26–28] or first-principles
calculations [24, 25, 29–33] of αGvia Kubo linear-
response [9, 30, 34–36] or scattering [8] theory-based for-
mulas.
The second term on the RHS of Eq. (2) is the
particular form [13] of the so-called nonlocal (i.e.,
magnetization-texture-dependent) and spatially nonuni-
form (i.e., position-dependent) damping [13–21, 37]. The
search for a proper form of nonlocal damping has a long
history [19, 37]. Its importance has been revealed by ex-
periments [10] extracting very different Gilbert damping
for the same material by using its uniformly precessing
localized spins versus dynamics of its magnetic domain
walls, as well as in experiments observing wavevector-
dependent damping of spin waves [38]. Its particulararXiv:2306.13013v4 [cond-mat.mes-hall] 29 Feb 20242
B
L lead R leadxyz MnJsd
e
ee
(a)
(b)
(c)
FIG. 1. Schematic view of (a) classical localized spins, mod-
eled by unit vectors Mn(red arrows), within an infinite metal-
lic ferromagnet defined on a cubic lattice in 1D–3D (1D is
used in this illustration); or (b) finite-size metallic ferromag-
net (central region) attached to semi-infinite NM leads termi-
nating in macroscopic reservoirs, whose difference in electro-
chemical potentials inject charge current as commonly done
in spintronics. The localized spins interact with conduction
electron spin ⟨ˆs⟩(green arrow) via sd-exchange of strength
Jsd, while both subsystems can experience external magnetic
fieldB(blue arrow). (c) Nonlocal damping λD
nn′[Eq. (10)]
obtained from SKFT vs. distance |rn−rn′|between two sites
nandn′of the lattice for different dimensionality Dof space.
form [13] in Eq. (2) requires only noncollinear and non-
coplanar textures of localized spins, so it can be nonzero
even in the absence of SOC, but its presence can greatly
enhance its magnitude [18] (without SOC, the nonlocal
damping in Eq. (2) is estimated [18] to be relevant only
for small size ≲1 nm noncollinear magnetic textures).
However, recent quantum-classical and numerically ex-
act simulations [39, 40] have revealed that αGcan be
nonzero even in the absence of SOC simply because ex-
pectation value of conduction electron spin ⟨ˆs⟩(r) isal-
ways somewhat behind M(r). Such retarded response of
electronic spins with respect to motion of classical lo-
calized spins, also invoked when postulating extended
LLG equation with phenomenological time-retarded ker-
nel [41], generates spin torque ∝ ⟨ˆs⟩(r)×M(r) [42] and,
thereby, effective Gilbert-like damping [39–41] that is
nonzero in the absence of SOC and operative even if
M(r) at different positions rarecollinear [40]. Including
SOC in such simulations simply increases [43] the an-
gle between ⟨ˆs⟩(r) and M(r) and, therefore, the effective
damping.
To deepen understanding of the origin of these phe-
nomena observed in numerical simulations, which are
analogous to nonadiabatic effects discussed in diversefields where fast quantum degrees of freedom interact
with slow classical ones [44–47], requires deriving an an-
alytical expression for Gilbert damping due to interac-
tion between fast conduction electrons and slow local-
ized spins. A rigorous path for such derivation is offered
by the Schwinger-Keldysh nonequilibrium field theory
(SKFT) [48] which, however, remains largely unexplored
for this problem. We note that a handful of studies have
employed SKFT to study small systems of one or two
localized spins [49–54] as they interact with conduction
electrons. While some of these studies [49, 53, 54] also
arrive at extended LLG equation with nonlocal damp-
ing, they are only directly applicable to small magnetic
molecules rather than macroscopic ferromagnets in the
focus of our study. It is also worth mentioning that an
early work [55] did apply SKFT to the same model we
are using—electrons whose spins interact via sdexchange
interaction with many Heisenberg-exchange-coupled lo-
calized spins representing metallic ferromagnet in self-
consistent manner—but they did not obtain damping
term in their extended Landau-Lifshitz equation, and in-
stead focused on fluctuations in the magnitude of Mn. In
contrast, the vectors Mnare of fixed length in classical
micromagnetics [2, 3] and atomistic spin dynamics [4], as
well as in our SKFT-derived extended LLG Eq. (9) and
all other SKFT-based analyses of one or two localized
spin problems [49–54].
In this study we consider either an infinite [Fig. 1(a)],
or finite [Fig. 1(b)] but sandwiched between two semi-
infinite normal metal (NM) leads terminating in macro-
scopic electronic reservoirs [8, 52, 53], metallic magnet
whose localized spins are coupled by ferromagnetic ex-
change in equilibrium. The setups in Fig. 1 are of di-
rect relevance to experiments [10, 38] on external field
[Fig. 1(a)] or current-driven dynamics [Fig. 1(b)] of lo-
calized spins in spintronics and magnonics. Our princi-
pal result is encapsulated by Fig. 1(c)—Gilbert damping,
due to conduction electron spins not being able to instan-
taneously follow changes in the orientation of classical
localized spins, is always nonlocal and inhomogeneous,
with such features becoming more prominent in low-
dimensional ferromagnets. This result is independently
confirmed [Fig. 2] by numerically exact simulations (in
one dimension) based on time-dependent nonequilibrium
Green’s function combined with LLG equation (TD-
NEGF+LLG) scheme [40, 43, 56, 57].
We note that conventional linear-response formulas [9,
30, 34–36] produce unphysical divergent Gilbert damp-
ing [33] in a perfectly crystalline magnet at zero tempera-
ture. In contrast to previously proposed solutions to this
problem—which require [58–60] going beyond the stan-
dard picture of electrons that do not interact with each
other, while interacting with classical localized spins—
our formulas are finite in the clean limit, as well as in
the absence of SOC. The scattering theory [8] yields a
formula for αGwhich is also always finite (in the absence
of SOC, it is finite due to spin pumping [61]). However,
that result can only be viewed as a spatial average of our3
nonlocal damping which cannot produce proper LLG dy-
namics of local magnetization [Fig. 3].
The paper is organized as follows. In Sec. II we for-
mulate the SKFT approach to the dynamics of local-
ized spins interacting with conduction electrons within
a metallic ferromagnet. Sections III A and III B show
how this approach leads to nonlocal and isotropic, or
nonlocal and anisotropic, damping in the presence or ab-
sence of SOC, respectively. The SKFT-derived analyt-
ical results are corroborated by numerically exact TD-
NEGF+LLG simulations [40, 43, 56, 57] in Sec. III C.
Then, in Secs. III D and III E we compare SKFT-derived
formulas with widely used scattering theory of conven-
tional scalar Gilbert damping [8, 61, 62] or spin-motive
force (SMF) theory [13, 19] of nonlocal damping, respec-
tively. Finally, in Sec. III F, we discuss how to com-
bine our SKFT-derived formulas to first-principles calcu-
lations on realistic materials via density functional theory
(DFT). We conclude in Sec. IV.
II. SCHWINGER-KELDYSH FIELD THEORY
FOR METALLIC FERROMAGNETS
The starting point of SKFT is the action [48] of metal-
lic ferromagnet, S=SM+Se,
SM=Z
CdtX
nh
∂tMn(t)·An− H[Mn(t)]i
,(3a)
Se=Z
CdtX
nn′h
¯ψn(t)
i∂t−γnn′
ψn′(t) (3b)
−δnn′JsdMn(t)·sn′(t)i
,
where SMis contribution from localized spins and Seis
contribution from conduction electrons. The integrationR
Cis along the Keldysh closed contour C[48]. Here the
subscript nlabels the site of a D-dimensional cubic lat-
tice;∂tMn·Anis the Berry phase term [63, 64]; H[Mn]
is the Hamiltonian of localized spins; ψn= (ψ↑
n, ψ↓
n)T
is the Grassmann spinor [48] for an electron at site
n;γnn′=−γis the nearest-neighbor (NN) hopping;
sn=¯ψnσψnis the electronic spin density, where σis
the vector of the Pauli matrices; and Jsdis the magni-
tude of sdexchange interaction between flowing spins of
conduction electrons and localized spins. For simplicity,
we use ℏ= 1.
The Keldysh contour C, as well as all functions defined
on it, can be split into forward (+) and backward ( −)
segments [48]. These functions can, in turn, be rewritten
asM±
n=Mn,c±1
2Mn,qfor the real-valued localized spins
field, and ψ±
n=1√
2(ψ1,n±ψ2,n) and ¯ψ±
n=1√
2(¯ψ2,n±
¯ψ1,n) for the Grassmann-valued fermion fields ψnand¯ψn.
The subscripts candqrefer to the classical and quantum
components of time evolution. This rewriting yields thefollowing expressions for the two actions
SM=Z
dtX
nMα
nq
ϵαβγ∂tMβ
n,cMγ
nc+Bα
eff[Mn,c]
,(4a)
Se=Z
dtdt′X
nn′¯ψσ
n ˇG−1
nn′δσσ′−JsdˇMα
nn′σα
σσ′
ψσ′
n′,(4b)
where subscript σ=↑,↓is for spin; summation over
repeated Greek indices is implied; ψ≡(ψ1, ψ2)T;
Beff=−δH/δMis the effective magnetic field; ϵαβγis
the Levi-Civita symbol; and ˇOare 2×2 matrices in the
Keldysh space, such as
ˇGnn′=
GRGK
0GA
nn′,ˇMα
nn′=
McMq
2Mq
2Mcα
nδnn′.
(5)
Here GR/A/K
nn′(t, t′) are electronic re-
tarded/advanced/Keldysh Green’s functions (GFs) [48]
in the real-space representation of sites n.
The electrons can be integrated out [49] up to the sec-
ond order in Jsdcoupling, thereby yielding an effective
action for localized spins only
Seff
M=Z
dtX
nMα
n,qh
ϵαβγ∂tMβ
n,cMγ
n,c+Bα
eff[Mn,c]
+Z
dt′X
n′Mα
n′,c(t′)ηnn′(t, t′)i
, (6)
where
ηnn′(t, t′) = iJ2
sd
GR
nn′(t, t′)GK
nn′(t′, t)
+GK
nn′(t, t′)GA
nn′(t′, t)
, (7)
is the non-Markovian time-retarded kernel. Note that
terms that are second order in the quantum fluctuations
Mn,qare neglected [48] in order to write Eq. (6). The
magnetization damping can be explicitly extracted by
analyzing the kernel, as demonstrated for different ferro-
magnetic setups in Secs. III A and III B.
III. RESULTS AND DISCUSSION
A. Nonlocality of Gilbert damping in metallic
ferromagnets in the absence of SOC
Since ηnn′(t−t′) depends only on the difference t−t′, it
can be Fourier transformed to energy ε. Thus, the kernel
can be written down explicitly for low energies as
ηnn′(ε) =J2
sdiε
2πX
k,qeik·(rn−rn′)eiq·(rn−rn′)Ak(µ)Aq(µ),
(8)
where Ak(µ)≡i[GR
k(µ)−GA
k(µ)] is the spectral func-
tion [52] evaluated at chemical potential µ;kis a4
wavevector; and rnandrn′are the position vectors of
sites nandn′. Equation (8) remains finite in the clean
limit and for low temperatures, so it evades unphysical
divergences in the linear-response approaches [58–60]. By
transforming it back into the time domain, we minimize
the effective action in Eq. (6) with respect to the quan-
tum fluctuations to obtain semiclassical equations of mo-
tion for classical localized spins. This procedure is equiv-
alent to the so-called large spin approximation [65, 66] or
a one loop truncation of the effective action. The higher
order terms neglected in Eq. (6) contribute a stochas-
tic noise that vanishes in the low temperature and large
spin limit. Although the fluctuating effect of this noise
can modify the exact dynamics [54, 65], the determinis-
tic regime suffices for a qualitative understanding and is
often the main focus of interest [66, 67].
Thus, we arrive at the following extended LLG equa-
tion
∂tMn=−Mn×Beff,n+Mn×X
n′λD
nn′∂tMn′,(9)
where the conventional αGMn×∂tMnGilbert term
is replaced by the second term on the RHS exhibit-
ing nonlocal damping λD
nn′instead of Gilbert damping
scalar parameter αG. A closed expression for λD
nn′can
be obtained for one-dimensional (1D), two-dimensional
(2D) and three-dimensional (3D) metallic ferromagnets
by considering quadratic energy-momentum dispersion of
their conduction electrons
λD
nn′=
2J2
sd
πv2
Fcos2(kF|rn−rn′|) 1D ,
k2
FJ2
sd
2πv2
FJ2
0(kF|rn−rn′|) 2D ,
k2
FJ2
sd
2πv2
Fsin2(kF|rn−rn′|)
|rn−rn′|2 3D.(10)
Here kFis the Fermi wavevector of electrons, vFis their
Fermi velocity, and J0(x) is the 0-th Bessel function of
the first kind.
B. Nonlocality and anisotropy of Gilbert damping
in metallic ferromagnets in the presence of SOC
Taking into account that previous analytical calcu-
lations [26–28] of conventional Gilbert damping scalar
parameter always include SOC, often of the Rashba
type [68], in this section we show how to generalize
Eq. (8) and nonlocal damping extracted in the presence
of SOC. For this purpose, we employ the Rashba Hamil-
tonian in 1D, with its diagonal representation given by,
ˆH=P
kσεkσˆc†
kσˆckσ, where ˆ c†
kσ/ˆckσcreates/annihilates
an electron with wavenumber kand spin σoriented along
they-axis, εkσ=−2γcosk+ 2σγSOsinkis the Rashba
spin-split energy-momentum dispersion, and γSOis the
strength of the Rashba SOC coupling. By switching
from second-quantized operators ˆ c†
kσ/ˆckσto Grassmann-
valued two-component fields [64] ¯cσ
n/cσ
n, where cσ
n=
FIG. 2. (a) Time evolution of two localized spins Mn, lo-
cated at sites n= 1 and n′= 3 within a chain of 19 sites
in the setup of Fig. 1(b), computed numerically by TD-
NEGF+LLG scheme [40, 43, 56, 57]. The two spins are
collinear at t= 0 and point along the x-axis, while mag-
netic field is applied along the z-axis. (b) The same infor-
mation as in panel (a), but for two noncollinear spins with
angle ∈ {0,45,90,135,180}between them. (c) and (d) Ef-
fective damping extracted from TDNEGF+LLG simulations
(red dashed line) vs. the one from SKFT [black solid line plots
1D case in Eq. (10)] as a function of the site n′of the second
spin. The two spins are initially parallel in (c), or antiparallel
in (d). The Fermi wavevector of conduction electrons is cho-
sen as kF=π/2a, where ais the lattice spacing.
(cσ
1,n, cσ
2,n)T, we obtain for the electronic action
Se=Z
dtdt′X
nn′¯cσ
n
(ˇGσ
nn′)−1δσσ′−JsdˇMα
nn′σβ
σσ′
cσ′
n′.
(11)
Here ˇGσ
nn′is diagonal, but it depends on spin through
εkσ. In addition, ˇMx,y,z
nn′, as the matrix which couples to
the same σx,y,zPauli matrix in electronic action without
SOC [Eq. (3b)], is coupled in Eq. (11) to a different Pauli
matrix σy,z,x.
By integrating electrons out up to the second order in
Jsd, and by repeating steps analogous to those of Sec. II
while carefully differentiating the spin-split bands, we
find that nonlocal damping becomes anisotropic
λ1D
nn′=
α⊥
nn′0 0
0α∥
nn′0
0 0 α⊥
nn′.
. (12)5
where
α⊥
nn′=J2
sd
πcos2(k↑
F|rn−rn′|)
v↑
F2+cos2(k↓
F|rn−rn′|)
v↓
F2
,
(13a)
α∥
nn′=J2
sd
π|v↑
Fv↓
F|
cos
(k↑
F+k↓
F)|rn−rn′|
(13b)
+ cos
(k↑
F−k↓
F)|rn−rn′|
,
andk↑/↓
Fandv↑/↓
Fare the Fermi wavevectors and veloc-
ities, respectively, of the Rashba spin-split bands. This
means that the damping term in Eq. (9) is now given by
Mn×P
n′λ1D
nn′·∂tMn′.
We note that previous experimental [69], numeri-
cal [9, 70], and analytical [26–28] studies have also found
SOC-induced anisotropy of Gilbert damping scalar pa-
rameter. However, our results [Eqs. (12) and (13)] ex-
hibit additional feature of nonlocality (i.e., damping at
sitendepends on spin at site n′) and nonuniformity (i.e.,
dependence on |rn−rn′|). As expected from Sec. III A,
nonlocality persists for γSO= 0, i.e., k↑
F=k↓
F=kF,
with λ1D
nn′properly reducing to contain αnn′three di-
agonal elements. Additionally, the damping component
α∥
nn′given by Eq. (13b) can take negative values, re-
vealing the driving capability of the conduction electrons
(see Sec. III C). However, for realistic small values of γSO,
the driving contribution of nearby localized spins is like-
wise small. Furthermore, the decay of nonlocal damping
with increasing distance observed in 2D and 3D, together
with the presence of intrinsic local damping from other
sources, ensures that the system tends towards equilib-
rium.
C. Comparison of SKFT-derived formulas with
numerically exact TDNEGF+LLG simulations
An analytical solution to Eq. (9) can be obtained in
few special cases, such as for two exchange-uncoupled lo-
calized spins at sites n= 1 and n′̸= 1 within 1D wire
placed in an external magnetic field Bext=Bextez, on
the proviso that the two spins are collinear at t= 0.
The same system can be simulated by TDNEGF+LLG
scheme, so that comparing analytical to such numeri-
cally exact solution for trajectories Mn(t) makes it pos-
sible to investigate accuracy of our derivation and ap-
proximations involved in it, such as: truncation to J2
sd
order; keeping quantum fluctuations Mn,qto first order;
and low-energy approximation used in Eq. (8). While
such a toy model is employed to verify the SKFT-based
derivation, we note that two uncoupled localized spins
can also be interpreted as macrospins of two distant ferro-
magnetic layers within a spin valve for which oscillatory
Gilbert damping as a function of distance between the
layers was observed experimentally [71]. Note that semi-
infinite NM leads from the setup in Fig. 1(b), always usedin TDNEGF+LLG simulations to ensure continuous en-
ergy spectrum of the whole system [40, 56], can also be
included in SKFT-based derivation by using self-energy
ΣR/A
k(ε) [52, 72] which modifies the GFs of the central
magnetic region in Fig. 1(b), GR/A
k= (ε−εk−ΣR/A
k)−1,
where εk=−2γcosk.
The TDNEGF+LLG-computed trajectory M1(t) of lo-
calized spin at site n= 1 is shown in Figs. 2(a) and
2(b) using two localized spins which are initially collinear
or noncollinear, respectively. For the initially parallel
[Fig. 2(a)] or antiparallel localized spins, we can ex-
tract Gilbert damping from such trajectories because
Mz
1(t) = tanh ¯λ1D
nn′Bextt/(1 + ( ¯λ1D
nn′)2)
[4, 40], where
the effective damping is given by ¯λ1D
nn′=λ1D
00±λ1D
nn′
(+ for parallel and −for antiparallel initial condition).
The nonlocality of such effective damping in Figs. 2(c)
and 2(d) manifests as its oscillation with increasing sep-
aration of the two localized spins. The same result
is predicted by the SKFT-derived formula [1D case in
Eq. (10)], which remarkably closely traces the numeri-
cally extracted ¯λ1D
nn′despite approximations involved in
SKFT-based analytical derivation. Note also that the
two localized spins remain collinear at all times t, but
damping remains nonlocal. The feature missed by the
SKFT-based formula is the decay of ¯λ1D
nn′with increasing
|rn−rn′|, which is present in numerically-extracted effec-
tive damping in Figs. 2(c) and 2(d). Note that effective
drastically reduced for antiparallel initial conditions, due
to the driving capabilities of the conduction electrons, in
addition to their dissipative nature. For noncollinear ini-
tial conditions, TDNEGF+LLG-computed trajectories
become more complicated [Fig. 2(b)], so that we can-
not extract the effective damping λ1D
nn′akin to Figs. 2(c)
and 2(d) for the collinear initial conditions.
D. Comparison of SKFT-derived formulas with the
scattering theory [8] of uniform local Gilbert
damping
The scattering theory of Gilbert damping αGwas
formulated by studying a single domain ferromagnet
in contact with a thermal bath [8]. In such a setup,
energy [8] and spin [61] pumped out of the system
by time-dependent magnetization contain information
about spin-relaxation-induced bulk [8, 62] and interfa-
cial [61] separable contributions to αG, expressible in
terms of the scattering matrix of a ferromagnetic layer
attached to two semi-infinite NM leads. For collinear lo-
calized spins of the ferromagnet, precessing together as
a macrospin, scattering theory-derived αGis a spatially-
uniform scalar which can be anisotropic [62]. Its expres-
sion is equivalent [62] to Kubo-type formulas [9, 34–36]
in the linear response limit, while offering an efficient al-
gorithm for numerical first-principles calculations [24, 25]
that can include disorder and SOC on an equal footing.
On the other hand, even if all localized spins are ini-
tially collinear, SKFT-derived extended LLG Eq. (9) pre-6
FIG. 3. (a) Comparison of trajectories of localized spins
Mz
n(t), in the setup of Fig. 1(b) whose central region is
1D metallic ferromagnet composed of 5 sites, using LLG
Eq. (9) with SKFT-derived nonlocal damping (solid red lines)
vs. LLG equation with conventional spatially-independent
αG= 0.016 (black dashed line). This value of αGis ob-
tained by averaging nonlocal damping over the whole ferro-
magnet. The dynamics of Mn(t) is initiated by an external
magnetic field along the z-axis, while all five localized spins
point along the x-axis at t= 0. (b) Comparison of spin cur-
rentISz
R(t) pumped [56, 57, 61] by the dynamics of Mn(t) for
the two cases [i.e., nonuniform Mn(t) for nonlocal vs. uniform
Mn(t) for conventional damping] from panel (a). The Fermi
wavevector of conduction electrons is chosen as kF=π/2a.
dicts that due to nonlocal damping each localized spin
will acquire a distinct Mn(t) trajectory, as demonstrated
by solid red lines in Fig. 3(a). By feeding these trajec-
tories, which are affected by nonlocal damping [1D case
in Eq. (10)] into TDNEGF+LLG simulations, we can
compute spin current ISz
R(t) pumped [56, 57] into the
right semi-infinite lead of the setup in Fig. 1(b) by the
dynamics of Mn(t). A very similar result for pumped
spin current is obtained [Fig. 3(b)] if we feed identical
Mn(t) trajectories [black dashed line in Fig. 3(a)] from
conventional LLG equation with Gilbert damping scalar
parameter, αG, whose value is obtained by averaging the
SKFT-derived nonlocal damping over the whole ferro-
magnet. This means that scattering theory of Gilbert
damping [8], which in this example is purely due to inter-
facial spin pumping [61] because of lack of SOC and dis-
order (i.e., absence of spin relaxation in the bulk), would
predict a constant αGthat can only be viewed as the
spatial average of SKFT-derived nonlocal and nonuni-
form λ1D
nn′. In other words, Fig. 3 reveals that different
types of microscopic magnetization dynamics Mn(t) can
yield the same total spin angular momentum loss into
the external circuit, which is, therefore, insufficient on
its own to decipher details (i.e., the proper form of ex-
tended LLG equation) of microscopic dynamics of local
magnetization.
1.5 2.0 2.5 3.0
w/a024vDW(aJ/¯ h)×10−2
αG= 0.1
Eq.(9)
Ref.[13] withη= 0.05
Ref.[19] withη= 0.05
Eq.(1) withη= 0(a)
0 25 50 75
Site i−1.0−0.50.00.51.0Mα(t)t = 410 ¯ h/J
α=x,y,z(b)
0 25 50 75
Site i−2−101(Mn×/summationtext
n/primeλd
nn/prime·∂tMn/prime)α×10−2
(c)
0 25 50 75
Site i−2−101(M×D·∂tM)α×10−3
(d)FIG. 4. (a) Comparison of magnetic DW velocity vDWvs.
DW width wextracted from numerical simulations using: ex-
tended LLG Eq. (9) with SKFT-derived nonlocal damping
[Eq. (10), red line]; extended LLG Eq. (1) with SMF-derived
in Ref. [13] nonlocal damping [Eq. (2), blue line] or SMF-
derived nonlocal damping (green line) in Ref. [19] [with ad-
ditional term when compared to Ref. [13], see Eq. (14)]; and
conventional LLG Eq. (1) with local Gilbert damping [i.e.,
η= 0 in Eq. (2), black line]. (b) Spatial profile of DW within
quasi-1D ferromagnetic wire at time t= 410 ℏ/J, where Jis
exchange coupling between Mnat NN sites, as obtained from
SKFT-derived extended LLG Eq. (9) with nonlocal damping
λ2D
nn′[Eq. (10)]. Panels (c) and (d) plot the corresponding spa-
tial profile of nonlocal damping across the DW in (b) using
SKFT-derived expression [Eqs. (9) and Eq. (10)] vs. SMF-
derived [13] expression [second term on the RHS of Eq. (2)],
respectively.
E. Comparison of SKFT-derived formulas with
spin motive force theory [13] and [19] of nonlocal
damping
The dynamics of noncollinear and noncoplanar magne-
tization textures, such as magnetic DWs and skyrmions,
leads to pumping of charge and spin currents assumed
to be captured by the spin motive force (SMF) the-
ory [16, 73, 74]. The excess angular momentum of dy-
namical localized spins carried away by pumped spin cur-
rent of electrons appears then as backaction torque [57]
exerted by nonequilibrium electrons onto localized spins
or, equivalently, nonlocal damping [13, 17–19]. From this
viewpoint, i.e., by using expressions for pumped spin cur-
rent [13, 17–19], a particular form for nonlocal damp-
ing [second term on the RHS of Eq. (2)] was derived in
Ref. [13] from the SMF theory, as well as extended in
Ref. [19] with an additional term, while also invoking a
number of intuitively-justified but uncontrolled approxi-
mations.
In this Section, we employ an example of a magnetic
field-driven DW [Fig. 4(b)] of width wwithin a quasi-7
1D ferromagnetic wire to compare its dynamics obtained
by solving extended LLG Eq. (1), which includes non-
local damping tensor [Eq. (2)] of Ref. [13], with the
dynamics obtained by solving SKFT-derived extended
LLG Eq. (9) whose nonlocal damping is different from
Ref. [13]. By neglecting nonlocal damping in Eq. (2),
the ferromagnetic domain wall (DW) velocity vDWis
found [75] to be directly proportional to Gilbert damping
αG,vDW∝ −BextwαG, assuming high external magnetic
fieldBextand sufficiently small αG. Thus, the value of αG
can be extracted by measuring the DW velocity. How-
ever, experiments find that αGdetermined in this fashion
can be up to three times larger than αGextracted from
ferromagnetic resonance linewidth measurement scheme
applied to the same material with uniform dynamical
magnetization [10]. This is considered as a strong evi-
dence for the importance of nonlocal damping in systems
hosting noncollinear magnetization textures.
In order to properly compare the effect of two different
expressions for the nonlocal damping, we use αG= 0.1
in Eq. (1) and we add the same standard local Gilbert
damping term, αGMn×∂tMn, into SKFT-derived ex-
tended LLG Eq. (9). In addition, we set λ2D
00=ηin
Eq. (10), so that we can vary the same parameter ηin all
versions of extended LLG Eqs. (1), and (9). Note that
we use λ2D
nn′in order to include realistic decay of nonlo-
cal damping with increasing distance |rn−rn′|, thereby
assuming quasi-1D wire. By changing the width of the
DW, the effective damping can be extracted from the DW
velocity [Fig. 4(a)]. Figure 4(a) shows that vDW∝wre-
gardless of the specific version of nonlocal damping em-
ployed, and it increases in its presence—compare red,
blue, and green data points with the black ones obtained
in the absence of nonlocal damping. Nevertheless, the
clear distinction between red, and blue or green data
points signifies that our SKFT-derived nonlocal damping
can be quite different from previously discussed SMF-
derived nonlocal damping [13, 19], which are compara-
ble regardless of the inclusion of the nonadiabatic terms.
For example, the effective damping extracted from blue
or green data points is D= 0.17 or D= 0.15, respec-
tively, while λ2D
nn′= 0.48. This distinction is further clar-
ified by comparing spatial profiles of SKFT-derived and
SMF-derived nonlocal damping in Figs. 4(c) and 4(d),
respectively, at the instant of time used in Fig. 4(b). In
particular, the profiles differ substantially in the out-
of-DW-plane or y-component, which is, together with
thex-component, an order of magnitude greater in the
case of SKFT-derived nonlocal damping. In addition,
the SKFT-derived nonlocal damping is nonzero across
the whole wire , while the nonlocal damping in Eq. (2)
is nonzero only within the DW width, where Mnvec-
tors are noncollinear [as obvious from the presence of
the spatial derivative in the second term on the RHS
of Eq. (2)]. Thus, the spatial profile of SKFT-derived
nonlocal damping in Fig. 4(c) illustrates how its nonzero
value in the region outside the DW width does not re-
quire noncollinearity of Mnvectors.Since SKFT-derived formulas are independently con-
firmed via numerically exact TDNEGF+LLG simula-
tions in Figs. 2(c) and 2(d), we conclude that previously
derived [13] type of nonlocal damping [second term on
the RHS of Eq. (2)] does not fully capture backaction of
nonequilibrium conduction electrons onto localized spins.
This could be due to nonadiabatic corrections [16, 19, 74]
to spin current pumped by dynamical noncollinear mag-
netization textures, which are present even in the ab-
sence of disorder and SOC [43]. One such correction was
derived in Ref. [19], also from spin current pumping ap-
proach, thereby adding a second nonlocal damping term
ηX
β′h
(M·∂β′∂tM)M×∂β′M−M×∂2
β′∂tMi
,(14)
into the extended LLG Eq. (1). However, combined us-
age [green line in Fig. 4(a)] of both this term and the one
in Eq. (2) as nonlocal damping still does not match the
effect of SKFT-derived nonlocal damping [compare with
red line in Fig. 4(a)] on magnetic DW. As it has been
demonstrated already in Fig. 3, the knowledge of total
spin angular momentum loss carried away by pumped
spin current [Fig. 3(b)], as the key input in the deriva-
tions of Refs. [13, 19], is in general insufficient to decipher
details of microscopic dynamics and dissipation exhibited
by localized spins [Fig. 3(a)] that pump such current.
F. Combining SKFT-derived nonlocal damping
with first-principles calculations
Obtaining the closed form expressions for the nonlocal
damping tensor λnn′in Secs. III A and III B was made
possible by using simplistic model Hamiltonians and ge-
ometries. For realistic materials and more complicated
geometries, we provide in this Section general formulas
which can be combined with DFT quantities and evalu-
ated numerically.
Notably, the time-retarded dissipation kernel in
Eq. (7), from which λnn′is extracted, depends on the
Keldysh GFs. The same GFs are also commonly used
in first-principles calculations of conventional Gilbert
damping scalar parameter via Kubo-type formulas [29–
33]. Specifically, the retarded/advanced GFs are ob-
tained from first-principles Hamiltonians ˆHDFTDFT as
ˆGR/A(ε) =
ε−ˆHDFT+ˆΣR/A(ϵ)−1. Here, ˆΣR/A(ε) are
the retarded/advanced self-energies [52, 72] describing es-
cape rate of electrons into NM leads, allowing for open-
system setups akin to the scattering theory-derived for-
mula for Gilbert damping [8, 62] and its computational
implementation with DFT Hamiltonians [24, 25]. Since
escape rates are encoded by imaginary part of the self-
energy, such calculations do not require iηimaginary pa-
rameter introduced by hand when using Kubo-type for-
mulas [29–33] (where η→0 leads to unphysical divergent
results [58–60]). Therefore, ˆHDFTcan be used as an
input to compute the nonlocal damping tensor, via the8
calculation of the GFs ˆGR/A(ε) and the spectral function
ˆA(ε) =iˆGR(ε)−ˆGA(ε)
.
For these purposes, it is convenient to separate the
nonlocal damping tensor into its symmetric and anti-
symmetric components, λαβ
nn′=λ(αβ)
nn′+λ[αβ]
nn′, where the
parenthesis (brackets) indicate that surrounded indices
have been (anti)symmetrized. They are given by
λ(αβ)
nn′=−J2
sd
2πZ
dε∂f
∂εTrspin
σαAnn′σβAn′n
, (15a)
λ[αβ]
nn′=−2J2
sd
πZ
dε∂f
∂εTrspin
σαReˆGR
nn′σβAn′n
−σαAnn′σβReˆGR
n′n
+J2
sd
2πZ
dε(1−2f)
×Trspin
σαReˆGR
nn′σβ∂An′n
∂ε−σα∂Ann′
∂εσβReˆGR
n′n
,
(15b)
where f(ε) is the Fermi function, and the trace is taken
in the spin space. The antisymmetric component either
vanishes in the presence of inversion symmetry, or is of-
ten orders of magnitude smaller than the symmetric one.
Therefore, it is absent in our results for simple models
on hypercubic lattices. As such, the nonlocal damping
tensors in Eqs. (10) and (13), are fully symmetric and
special case of Eq. (15a) when considering specific energy-
momentum dispersions and assuming zero temperature.
IV. CONCLUSIONS AND OUTLOOK
In conclusion, we derived a novel formula, displayed
as Eq. (15), for magnetization damping of a metallic fer-
romagnet via unexploited for this purpose rigorous ap-
proach offered by the Schwinger-Keldysh nonequilibrium
field theory [48]. Our formulas could open a new route for
calculations of Gilbert damping of realistic materials by
employing first-principles Hamiltonian ˆHDFTfrom den-
sity functional theory (DFT) as an input, as discussed
in Sec. III F. Although a thorough numerical exploration
of a small two-spin system based on SKFT was recently
pursued in Ref. [54], our Eqs. (15) are not only applica-
ble for large systems of many localized spins, but are also
refined into readily computable expressions that depend
on accessible quantities.While traditional, Kubo linear-response [9, 30, 34–
36] or scattering theory [8] based derivations produce
spatially uniform scalar αG, SKFT-derived damping in
Eqs. (15) is intrinsically nonlocal and nonuniform as it
depends on the coordinates of local magnetization at two
points in space rnandrn′. In the cases of model Hamil-
tonians in 1D–3D, we reduced Eqs. (15) to analytical ex-
pressions for magnetization damping [Eq. (10)], thereby
making it possible to understand the consequences of
such fundamental nonlocality and nonuniformity on lo-
cal magnetization dynamics, such as: ( i) damping in
Eq. (10) osc illates with the distance between xandx′
where the period of such oscillation is governed by the
Fermi wavevector kF[Figs. 1(c), 2(c), and 2(d)]; ( ii)
it always leads to nonuniform local magnetization dy-
namics [Fig. 3(a)], even though spin pumping from it
can appear [Fig. 3(b)] as if it is driven by usually an-
alyzed [8, 61] uniform local magnetization (or, equiv-
alently, macrospin); ( iii) when applied to noncollinear
magnetic textures, such as DWs, it produces an order
of magnitude larger damping and, therefore, DW wall
velocity, than predicted by previously derived [13] non-
local damping [second term on the RHS of Eq. (2)].
Remarkably, solutions of SKFT-based extended LLG
Eq. (9) are fully corroborated by numerically exact TD-
NEGF+LLG simulations [40, 43, 56, 57] in 1D, despite
the fact that several approximations are employed in
SKFT-based derivations. Finally, while conventional un-
derstanding of the origin of Gilbert damping scalar pa-
rameter αGrequires SOC to be nonzero [22, 23], our non-
local damping is nonzero [Eq. (10)] even in the absence
of SOC due to inevitable delay [39, 40] in electronic spin
responding to motion of localized classical spins. For
typical values of Jsd∼0.1 eV [76] and NN hopping pa-
rameter γ∼1 eV, the magnitude of nonlocal damping is
λD
nn′≲0.01, relevant even in metallic magnets with con-
ventional local damping αG∼0.01 [10]. By switching
SOC on, such nonlocal damping becomes additionally
anisotropic [Eq. (13)].
ACKNOWLEDGMENTS
This work was supported by the US National Science
Foundation (NSF) Grant No. ECCS 1922689.
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1201.1218v1.Damped_bead_on_a_rotating_circular_hoop___a_bifurcation_zoo.pdf | Damped bead on a rotating circular hoop - a bifurcation zoo
Shovan Dutta
Department of Electronics and Telecommunication Engineering,
Jadavpur University, Calcutta 700 032, India.
Subhankar Ray
Department of Physics, Jadavpur University, Calcutta 700 032, India.
The evergreen problem of a bead on a rotating hoop shows a multitude of bifurca-
tions when the bead moves with friction. This motion is studied for di erent values
of the damping coecient and rotational speeds of the hoop. Phase portraits and
trajectories corresponding to all di erent modes of motion of the bead are presented.
They illustrate the rich dynamics associated with this simple system. For some range
of values of the damping coecient and rotational speeds of the hoop, linear stability
analysis of the equilibrium points is inadequate to classify their nature. A technique
involving transformation of coordinates and order of magnitude arguments is pre-
sented to examine such cases. This may provide a general framework to investigate
other complex systems.arXiv:1201.1218v1 [physics.class-ph] 5 Jan 20122
I. INTRODUCTION
The motion of a bead on a rotating circular hoop1shows several classes of xed points
and bifurcations2{4. It also exhibits reversibility, symmetry breaking, critical slowing down,
homoclinic and heteroclinic orbits and trapping regions. It has been shown to provide a
mechanical analogue of phase transitions5. It can also operate as a one-dimensional pon-
deromotive particle trap6. The rigid pendulum, with many applications, can be considered
a special case of this system7,8.
In this article we examine the motion of a damped bead on a rotating circular hoop.
Damping alters the nature of the xed points of the system, showing rich nonlinear fea-
tures. The overdamped case of this model3,9and a variant involving dry friction10has been
previously studied.
For certain values of the damping coecient and the rotational speed of the hoop, linear
stability analysis predicts a line of xed points and some of the xed points appear as
degenerate nodes. However, such xed points are borderline cases, sensitive to nonlinear
terms. By transforming to polar coordinates and employing order of magnitude arguments
we analyze these borderline cases to determine the exact nature of these xed points. To
our knowledge, such analytical treatment does not appear in literature. The basic equations
obtained for this system are quite generic and arise in other systems (e.g. electrical systems)
as well. Hence, our technique may serve as a framework for investigating other more complex
nonlinear systems.
II. THE PHYSICAL SYSTEM
A bead of mass m, moves on a circular hoop of radius a. The hoop rotates about its
vertical diameter with a constant angular velocity !. The position of the bead on the hoop
is given by angle , measured from the vertically downward direction ( zaxis), and is the
angular displacement of the hoop from its initial position on the x-axis (Figure 1).
The Lagrangian of the system with no damping is,
L(;_) =ma2
2(_2+!2sin2) +mga cos;where!=_ is a constant : (1)
Using the Euler-Lagrange equation, the equation of motion is obtained as,
= sin(!2cos g=a): (2)3
Φ
ΘΩ
xz
y
FIG. 1. Schematic gure of a bead sliding on a rotating hoop showing the angles and .
To include friction, a term b_is introduced in (2) as,
= sin(!2cos g=a) b_; (3)
wherebis the damping coecient. We identify !2
c=g=aas the critical speed of rotation of
the hoop, and write k=!2=!2
c,=b=!c. De ning =!ct, (3) may be made dimensionless
by changing from tto ,
00= sin(kcos 1) 0;whered2
d 2=00: (4)
For phase plane analysis, we de ne a new variable 1=0, and write (4) as,
0=1 (5)
10= sin(1 kcos) 1: (6)
The parameter kcan take only positive values whereas may be either positive or negative.
Due to the symmetry of the hoop about its vertical axis, (5) and (6) remain invariant
under the transformations ! ; 1! 1. This implies that alternate quadrants of the
1plane have similar trajectories. Similarly, it is easily veri ed that if ( (t);1(t)) is a
solution for positive damping ( >0), then for negative damping ( <0), (( t); 1( t))
and ( ( t);1( t)) are two solutions. The phase portrait of the system for negative
damping will just be the re
ection of the positive damping phase portrait with the arrows
reversed. Hence we con ne our attention to 2[0;] and>0.4
When there is no damping11, the xed points are at (0 ;0) and (;0) for 0k1,
whereas for k > 1, an additional xed point appears at (
1= cos 1(1=k);0). Damping
changes the nature of xed points and not their number or location.
III. NATURE OF THE FIXED POINT (0;0)
The Jacobian matrix at (0 ;0) is obtained by Taylor expanding (5) and (6) about (0 ;0)
and retaining the linear terms.
J(0;0) =0
@0 1
k 1 1
A: (7)
Let and denote the trace and determinant of the above matrix.
1. Whenk>1, both and are negative. The xed point is a saddle with eigenvalues
and eigenvectors given by,
1;2= 1
2;v1;2=0
@1
( 1)=21
A; (8)
where1=p
2+ 4(k 1). Saddles are robust and do not get perturbed by non-
linearities. Thus, (0 ;0) will remain a saddle even if nonlinear terms are taken into
account (see Figures 8, 9 and 16).
For= 0, both1and2equalp
k 1. Ask!1+,1!0+and2! , which
means that the saddle will start looking like a line of xed points along the direction
ofv1with solutions decaying along v2.
2. For 0k <1, = is negative, whereas = 1 kis positive. When there is no
damping, the point (0 ;0) is a center. As is increased, the center transforms into a
stable spiral for <2p
1 kas shown in Figure 2(a). The frequency of spiralling is
p
1 k 2=4. As!2p
1 k ,!0+. For > 2p
1 k, the xed point
(0;0) transforms to a stable node (Figure 2(b)). For = 2p
1 k, it is a degenerate
node. However, degenerate nodes are borderline cases and are sensitive to nonlinear
terms.5
3. Fork= 1, (5) and (6) simplify to,
0=1 (9)
0
1= sin(1 cos) 1: (10)
In the linearized dynamics, 1decays exponentially as e t. In the phase plane, all
trajectories move along a straight line with slope and stop on reaching the axis.
However, the inclusion of nonlinear terms changes this situation.
(a) stable spiral, <2p
1 k
(b) stable node, >2p
1 k
FIG. 2. Phase trajectories around (0 ;0) fork= 0:91 showing (a) stable spiral for = 0:3 and (b)
stable node for = 1.
A. Nature of (0;0)with nonlinearities
1. 0k<1and<2p
1 k:
To include the e ect of nonlinear terms, let us de ne two new variables,
=rcos ; 1=rsin (11)
Equations (5) and (6) then may be written as,
r0=r[cos sin 2p
1 ksin2 ] sin sin(rcos )[1 kcos(rcos )] (12)
0= p
1 ksin(2 ) sin2 cos
rsin(rcos )[1 kcos(rcos )] (13)
We wish to examine the xed point(s) in the r plane corresponding to (0 ;0) in the 1
plane, to determine their true nature. Strictly speaking, (12) and (13) are meaningful only6
whenr >0. Neither nor 0have any meaning when r= 0. Hence, we may assign any
arbitrary function f( ) to 0atr= 0 without altering physical predictions. However, (13)
describes accurately the approach to r= 0 (if any) in the r plane at arbitrarily small
scales. We therefore set f( ) equal to the limiting value of 0asr!0.
f( ) = lim
r!0 0= p
1 ksin(2 ) +k
2cos(2 )
1 k
2
(14)
Equations (12) and (13) are periodic in with period . Hence, the phase portrait in the
r plane is periodic along the -axis with period . This means that in the 1plane,
the phase space is symmetric about (0 ;0).
Using the identity (p
1 k)2+ (k=2)2= (1 k=2)2, one can write f( ) in the form,
f( ) =
1 k
2
[cos(2 + ) 1]; (15)
where = 2 tan 1p
1 k,n= 0;1;2;:::. Therefore, xed points (0 ; ) in ther
plane, where f( ) = 0, are given by
n=n tan 1p
1 kwithn= 0;1;2;:::. These
correspond to the point (0 ;0) in the 1plane.
The xed points (0 ;
n) in ther plane are separated by n(wherenis any integer).
Hence, in the 1plane, there are no trajectories that can approach (0 ;0) along two
independent directions. So we can say that (0 ;0) in the 1plane cannot be a stable
node. In the r plane, close to some xed point (0 ;
n), if there exist trajectories that
approach this point and stop there, the corresponding xed point (0 ;0) in the 1plane
cannot be a stable spiral. For a spiral, !1 asr!0. As (12) and (13) are periodic
in with period , the nature of all xed points on the -axis separated by is identical.
So we may choose to investigate (0 ; = tan 1p
1 k). Linearization about this point
incorrectly predicts the whole axis to be a line of xed points. So, we must include the
e ects of the nonlinear terms. Let = . For small r, we may write (12) and (13) as,
r0= rhp
1 k
1 k
2
sin(2)i
+O(r3) (16)
0=
1 k
2
[1 cos(2)] +O(r2) (17)
Consider an initial condition, r( = 0) =r0and( = 0) =0, where 0 < 0<
tan 1(p
1 k=2). For any nite positive value of 1 k,0is nite and <1=2. There-
fore,r0may be chosen suciently small, 0 <r 0 0, so as to make all terms of O(r2) and7
higher, negligible compared to the leading terms in (16) and (17). When these terms are
neglected, (16) and (17) may be solved to yield,
( ) = tan 1h1
(2 k) + cot0i
(18)
r( ) =r0sin0p
[(2 k) + cot0]2+ 1 exp( p
1 k ) (19)
According to this solution, the trajectory monotonically approaches the point (0 ;0) in the
r plane as !1 . This behaviour will hold even with inclusion of nonlinear terms,
provided, the terms independent of rand ofO(r) remain dominant over the entire trajectory.
The following arguments establish that it is indeed so.
First, the trajectory cannot reach the axis at a positive value of . This is because on
the axis,r0= 0 and 0=f( )<0 in between two xed points. Thus, there is already a
trajectory running along the axis directed towards = 0.
For the initial point ( r0;0), with the choice 0 <r 0 0<1=2, bothrandwill start to
decrease as per (16) and (17). Hence r2will become more negligible compared to r. Also,
for all 00, theindependent term, namely, p
1 kr, in (16), will be dominant.
However, if decays more rapidly, such that at some stage r, then theO(r2) term will
contribute on the same scale as the rst term in (17), which is O(2). Similarly, the O(r3)
term will contribute on the same scale as the 2nd term in (16) if in the course of decay, at
some point r2. However, such situations will never arise as is shown below.
Let us assume that r0=20
0 1 and that decreases very rapidly, such that, at some
instance,r=2. Asrhas decreased monotonically from its initial value, we must have
=r1=2< r1=2
0. However, r1=2
0=10
0 1. Hence, along the entire trajectory, up to this
instance, terms of O(r2) and higher are negligible compared to the leading terms in (16) and
(17). Thus the solutions (18) and (19) are valid and give the correct orders of magnitudes
of the dynamical quantities.
As 0< < 10
0 1 and 0< 0<1=2, we may write, tan and tan00. Hence,
0<tan <tan100, which implies tan <(p
1 k=2)10 1. Combining this with (18)
and using the fact that (2 k)1, we get &(cot0)10, a very large quantity. Meanwhile
(18) and (19) together imply,
r
2r0sin0h 3
exp(p
1 k )i
(20)
Bothr0sin0and the quantity within brackets are 1, which implies that r 2. This is8
in contradiction to the initial assumption that r=2. Therefore, we conclude that starting
with the prescribed initial condition, rwill never become equal to 2. This ensures that
along the entire trajectory, the terms independent of rand ofO(r) remain the dominant
terms in (16) and (17). Both randwill decrease monotonously toward their respective
zero values. Neither can the trajectory cross the curve r=2nor reach the axis before
becomes zero. Thus, the trajectories in the r plane must approach (0 ;0) tangential to
the axis and slow to a halt there.
In the 1plane, the above arguments imply that, trajectories exist which start at a
nite distance from (0 ;0) and reach this point along a line of slope p
1 k. Also, no other
such line with a di erent slope exists. These facts clearly establish that (0 ;0) is a stable
degenerate node (Figure 3).
(a) stable degenerate node, = 0:6
(b) central region magni ed
FIG. 3. Phase trajectories around (0 ;0) fork= 0:91 and= 2p
1 k.
2. k =1:
Proceeding as before, (9) and (10) may be written in terms of rand as,
r0=r[cos sin sin2 ] sin sin(rcos )[1 cos(rcos )] (21)
0= cos sin sin2 cos
rsin(rcos )[1 cos(rcos )] (22)
f( ) is de ned as,
f( ) = lim
r!0 0= cos sin sin2 (23)
The phase portrait is periodic in with period . The xed points in the r plane of
the form (0, ) are given by,
1=n and
2=n tan 1 n= 0;1;2;:::: (24)9
For tan 1< < 0,f( )>0 whereas for 0 < < tan 1,f( )<0. The positive
and negative nature of f( ) repeats periodically along the axis.
The Jacobian at the points (0 ;n tan 1) given by,
J(0; 2) =0
@ 0
01
A (25)
is traceless and has a negative determinant = 2. So, this family of xed points are
saddles having stable manifold along raxis and unstable manifold along axis.
Linear analysis of the family of xed points (0 ;n), incorrectly predicts =nto be lines
of xed points. Let us examine the point (0 ;0) in ther plane for simplicity. Consider
the condition,
0j jr minf1;g (26)
If (26) holds, then neglecting terms of O(r3) and smaller in (21) and (22), we may write,
r0= r+1 (27)
0= r2
2 2+2 (28)
where1 2rand2 3or 2r2, whichever is larger. Note that as long as (26) is
satis ed, both 1and2can at most be of the order of r3.
Let us take the initial condition 0= 0 and 0< r 0 1. Then, 0(t= 0) = r2
0=2 and
r0(t= 0) = 0. Hence, will start decreasing and become negative. As a result, r0will
become negative and remain so until orrvanishes, provided (26) remains true. It is seen
that as long as the trajectory is above the curve = r2=2, (26) is satis ed and both r0
and 0are negative. Therefore, the trajectory approaches and eventually crosses this curve,
where 0is still negative, being of the order of 2.
Let us consider the `trapping region' in Figure 4, which shows the phase
ow on the
curves = rand = r2=2. At any point on the line = rfor whichr minf1;g,
0
r0=
r+O(1)
Asr ,j 0=r0j 1. Thus the phase
ow is almost vertically upward, as shown in Figure
4. Everywhere inside the region, (26) is satis ed and hence r0<0 and nite. Consequently,
after entering the region at point P, the trajectory must constantly move towards left.
Again, it cannot penetrate the curves APorAB, because other trajectories are actually10
FIG. 4. Trapping Region.
owing inward across them. Hence, we have trapped it. Upon arrival at any point on the
arcAP, a trajectory must inevitably land up at (0 ;0). Note that at any point on the line
= mr, (0<m1),
0
r0= m (m2+ 1=2)r+O(m3r2)
mr+O(m2r2):
For any nite value of m, this approaches 1 asr!0, meaning that the phase
ow is
almost vertically upward on any line of non-zero slope near (0 ;0). Therefore, the trajectory
must reach (0 ;0) along the raxis.
Thus, the xed points (0 ;n) in ther plane are stable nodes having slow eigenvector
alongraxis and fast eigenvector along axis. In between these, lie the saddle points
(0;n tan 1) (Figure 5).
(a) stable node and saddle points
(b) region near (0 ;0) magni ed
FIG. 5. Phase portrait in the r plane.
These results from the r plane mean that in the 1plane, two trajectories exist11
which reach (0 ;0) along the line of slope and all other neighbouring trajectories reach it
along theaxis. In other words, (0 ;0) is a stable node here.
(a) stable node
(b) central region magni ed
FIG. 6. Phase trajectories about (0 ;0) in the 1plane fork= 1 and= 0:5.
Figure 7a shows the nature of the xed point at (0 ;0) over the entire parameter space.
(a) (0;0), curve is = 2p
1 k
(b) (
1;0), curve is = 2p
k 1=k
FIG. 7. Nature of xed points (0 ;0) and (
1;0) over the k plane
IV. NATURE OF THE FIXED POINT (
1;0)
This xed point exists when k1. Linearization of (5) and (6) gives us the Jacobian at
(
1;0) as,
J(
1;0) =0
@0 1
1
k k
1
A (29)12
For 0<2q
k 1
k, (
1;0) is a stable spiral with eigenvalues given by,
=
2ir
(k 1
k) 2
4
Trajectories spiral in with an angular frequency p
(k 1=k) 2=4, while their radial
distance decreases as e t=2. As!0, this decay rate vanishes and (
1;0) turns into a
center (Figure 8). Also, vanishes as !2p
k 1=k , representing a smooth transition
to a stable node, similar to the behaviour of the xed point (0 ;0). When > 2q
k 1
k,
(a) center,= 0
(b) stable spiral, = 0:1
FIG. 8. Phase trajectories around (0 ;0) fork= 1:1 and 0<2q
k 1
k.
2 4>0, hence (
1;0) is a stable node (Figure 9), with eigenvalues and eigenvectors
given by,
1;2= 3
2;v1;2=0
@1
( 3)=21
A;
where3=p
2 4(k 1=k). Both1and2approach the value p
k 1=kas!
2p
k 1=k+, indicating a stable degenerate node. For = 2q
k 1
k, 2 4 = 0. In the
linear stability analysis, (
1;0) is a stable degenerate node with a single eigenvector,
v=0
@1
p
k 1=k1
A; (30)
corresponding to the eigenvalue = p
k 1=k. However, degenerate nodes can be trans-
formed into stable nodes or stable spirals due to perturbation introduced by nonlinear terms.13
(a) stable node, = 1
(b) stable node, = 2:5
FIG. 9. Phase trajectories around (0 ;0) fork= 1:1 and>2q
k 1
k.
A. Nature of (
1;0)with nonlinearities
As discussed in subsection III A, (5) and (6) may be transformed to equations in rand
and with the substitutions,
1=rcos and1=rsin . De ne,
f( ) = lim
r!0 0= r
k 1
ksin(2 ) 1
2(k 1
k 1) cos(2 ) 1
2(k 1
k+ 1) (31)
f( ) is negative at all points on the axis except at the xed points given by (0 ; ) with
=n , where = tan 1p
k 1=k, (n= 0;1;2;:::), where it is zero. These xed
points are separated by n. Then, by the same reasoning as used in III A, we can argue
that (0;0) cannot be a stable node. The remaining possibilities are a spiral or a degenerate
node.
Let us consider the xed point (0 ; ), and let= + . Then we may expand r0and
0uptoO(r),
r0= 1
2
k 1
k+ 1
sin(2 ) sin(2)
r+O(r2) (32)
0=
k 1
k+ 1
sin2 3
2rr
1 1
r2cos3
9
4r
1 1
k2cos sin(2 )r+O(r2) (33)
We choose an initial point, ( r0;0), and 0< r 0 0< minf1; g. This choice ensures
that bothrandwill start decreasing. In the course of this monotonic decay, rcannot
reach zero before becomes zero, as there is a straight line trajectory moving downward14
along the axis. From (32) and (33) we note that r0and0each contains an independent
term ofO(r). Therefore, as randdecrease toward their respective zero values, the 1st
order approximation gets even better. However, if at some stage, r, then the O(r2)
terms would contribute on the same scale as some of the terms of O(r) in (32) and (33).
But the following argument rules out such a possibility.
Let us consider a speci c case and choose r0=20
0. Ifis to become O(r), at some stage,
we must have r=2. However, for r=2, we have
r0= r
k 1
k2+O(3)
0= 2
4(k 1
k+ 1) +3
2q
1 1
k2
(k 1
k+ 1)3
23
52+O(3)
whereas along the curve r=2,dr=d = 2. Therefore, for a given value of k >1, we can
always select an 0, suciently small, for which, at all points on the curve r=2contained
between= 0 and=0, the ratior0=0>dr=d . This would guarantee that the trajectory
cannot penetrate down this curve, which means that cannot reach zero before rdoes. Thus,
for a suitable choice of initial conditions, the trajectory must slow to a halt at ( r= 0;= 0).
In the 1plane, this means that there exist trajectories which start at a nite distance
from (
1;0) and reach it along the line of slope . Also, there is no other such line with a
di erent slope. Hence, (
1;0) is a stable degenerate node (Figure 10). Figure 7b gives the
(a) stable degenerate node
(b) region around (
1;0) magni ed
FIG. 10. Phase trajectories for k= 1:1 and= 2q
k 1
k.
nature of the xed point at (
1;0) in di erent parts of the parameter space.15
V. NATURE OF THE FIXED POINT (;0)
The Jacobian matrix at ( ,0) is given as,
J(;0) =0
@0 1
k+ 1 1
A: (34)
The xed point is a saddle for all values of kand remains so even with the inclusion of
nonlinear terms. The eigenvalues and corresponding eigenvectors are given by,
1;2= 2
2;v1;2=0
@1
( 2)=21
A;
where2=p
2+ 4(k+ 1).
A. Trajectories
Damping of the bead leads to some qualitatively di erent trajectories in addition to
those observed for the frictionless case11. These are mainly the di erent kinds of damped
oscillations (underdamped, critically damped, overdamped) about the stable equilibrium
points. Some of these are illustrated with the following numerical plots.
(a)k= 0:75,= 0:05,0(0) = 0
(b)k= 4,= 0:5,0(0) = 0
FIG. 11. Underdamped oscillation about (a) = 0 and (b) =
1.
VI. PHASE PORTRAITS AND BIFURCATION
For 0k <1, the xed point at (0 ;0) transforms its nature as the damping coecient
is varied. It is a center at = 0, asincreases, it becomes a stable spiral. At = 2p
1 k,16
(a)k= 0:75,= 1:5,0(0) = 0
(b)k= 4,= 4,0(0) = 0
FIG. 12. Overdamped oscillation about (a) = 0 (b)=
1.
it turns into a stable degenerate node. It makes a smooth transition to a stable node as
damping is increased further. Thus, a spiral-node bifurcation takes place at this critical
condition (Figures 2 and 3). Physically, as damping is gradually increased from 0, the
system undergoes a continuous transition from undamped oscillations of the bead about
= 0 (center), to underdamped oscillations (stable spiral). At = 2p
1 k, the system is
critically damped (degenerate node) and becomes overdamped (stable node) as is increased
further.
(a) unstable spiral, = 1
(b) center, = 0
(c) stable spiral, = 1
FIG. 13. Phase portraits for k= 0 showing degenerate Hopf bifurcation.
For negative damping, (0 ;0) becomes an unstable spiral and changes to an unstable node
asis made more negative. Consequently, as one crosses = 0, the xed point (0 ;0),
undergoes a degenerate Hopf bifurcation (Figure 13).
With increase in the angular speed of the hoop (i.e., k), the stability of the origin degrades
continuously. When k= 1, (0;0) is a weak center. A special case of Hopf bifurcation occurs,
whenis swept from negative to positive values acroos 0, keeping k xed at 1 (Figure 14).17
Askis increased beyond 1, (0 ;0) transforms from a stable ( >0) or unstable ( <0)
node to a saddle. Two new stable nodes appear at
1=cos 1(1=k) and branch out in
opposite directions. Thus, a supercritical pitchfork bifurcation occurs at fk= 1g(Figure
15a).
(a) unstable node, = 0:2
(b) stable node, = 0:2
FIG. 14. Phase portraits for k= 1.
(a) Supercritical bifurcation at k= 1,= 0
(b) Supercritical bifurcation at k= 1,=p
2
FIG. 15. Section of the bifurcation diagram for (a) = 0 and (b) >0. In (a) solid curve represents
center, dashed curve represents saddle. In (b) solid curve represents stable node, densely dashed
curve represents saddle and sparsely dashed curve represents stable spiral. Supercritical pitchfork
bifurcation occurs at k= 1 and spiral-node bifurcation occurs at k= 0:5 andk= 1:28.
Askincreases from 1 to 1,
1varies from 0 to =2. The xed point (
1;0), is a
center for zero damping, a stable spiral in the region 0 < < 2p
k 1=k(underdamped
oscillation), and becomes a stable degenerate node at critical damping = 2p
k 1=k. For
the overdamped condition >2p
k 1=k, it is a stable node.18
For negative damping, we get just the unstable counterparts. Accordingly, a spiral-node
bifurcation is observed at =2p
k 1=k(Figures 8, 9, 10 and 15)b. A degenerate Hopf
bifurcation is observed for k>1 and= 0 (Figure 16). The xed points ( ;0) are saddles
(a) unstable spiral, = 1
(b) center, = 0
(c) stable spiral, = 1
FIG. 16. Phase portraits for k= 4 showing degenerate Hopf bifurcation.
for all values of k. They have saddle connections between them at = 0, which break in
opposite directions for positive and negative damping.
The above observations are summarized in Figure 17 and Table I below.
FIG. 17. The bifurcation diagram
In Fig 17, red denotes stable node, green denotes stable spiral, blue denotes unstable
spiral, yellow denotes unstable node, brown denotes saddle, pink denotes center, gray denotes
unstable degenerate node, and black denotes stable degenerate node.
Up to now, we have limited attention to those bifurcations resulting from a variation of k
or a variation of . From Figure 17, we see that the curves 2= 2(1 k),2= 2(k 1=k),
andk= 1 divide the parameter space into 8 distinct regions of di erent dynamics. All these19
TABLE I. Bifurcation Table.
Points in parameter space Bifurcation along k Bifurcation along Figure references
1) (k;0) ;k6= 1 { degenerate Hopf Figures 13, 16
2) (k;2p
1 k) ; 0k<1 spiral-node spiral-node Figures 2-3, 15b
3) (1;0) supercritical pitchfork Hopf Figures 13b, 8a, 14 , 15a
4) (1;) ;6= 0 supercritical pitchfork { Figures 2b, 9a, 14b, 15b
5) (k;2q
k 1
k) ;k>1 spiral-node spiral-node Figures 8-10, 15b
regions meet at the point fk= 1;= 0g. Traversing suitable curves in k space, one
can move from any one region to another, yielding new kinds of bifurcation. Following such
a curve amounts to keeping a certain function (k;) constant, while varying some other
function (k;). Mathematically, the possibilities are rich. But whether it is possible to
actually implement this in the bead-hoop system is subject to further inquiry. However, this
would attain physical signi cance if there exists another system where and themselves
are the control parameters.
CONCLUDING REMARKS
The simple introduction of damping to the bead-hoop system enriches its dynamics and
leads to various new modes of motion and di erent classes of bifurcations. We have studied
this system over the entire parameter space and presented phase portraits and trajectories.
This serves to illustrate the qualitative changes in the system's dynamics across di erent bi-
furcation curves. We have presented exact analytical treatment of the borderline cases where
linearization fails, for which no general methods are available in the literature. The method
of transforming to polar coordinates and using order of magnitude arguments, employed in
this article, can serve as a useful technique for other dynamical systems as well.20
REFERENCES
sray.ju@gmail.com
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1403.4996v1.The_effects_of_time_dependent_dissipation_on_the_basins_of_attraction_for_the_pendulum_with_oscillating_support.pdf | The e ects of time-dependent dissipation on the basins of
attraction for the pendulum with oscillating support
James A. Wright1, Michele Bartuccelli1, Guido Gentile2
1Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK
2Dipartimento di Matematica e Fisica, Universit a di Roma Tre, 00146 Roma, Italy
Abstract
We consider a pendulum with vertically oscillating support and time-dependent damping
coecient which varies until reaching a nite nal value. Although it is the nal value which
determines which attractors eventually exist, however the sizes of the corresponding basins of
attraction are found to depend strongly on the full evolution of the dissipation. In particular
we investigate numerically how dissipation monotonically varying in time changes the sizes of
the basins of attraction. It turns out that, in order to predict the behaviour of the system,
it is essential to understand how the sizes of the basins of attraction for constant dissipation
depend on the damping coecient. For values of the parameters where the systems can be
considered as a perturbation of the simple pendulum, which is integrable, we characterise
analytically the conditions under which the attractors exist and study numerically how the sizes
of their basins of attraction depend on the damping coecient. Away from the perturbation
regime, a numerical study of the attractors and the corresponding basins of attraction for
di erent constant values of the damping coecient produces a much more involved scenario:
changing the magnitude of the dissipation causes some attractors to disappear either leaving
no trace or producing new attractors by bifurcation, such as period doubling and saddle-node
bifurcation. Finally we pass to the case of an initially non-constant damping coecient, both
increasing and decreasing to some nite nal value, and we numerically observe the resulting
e ects on the sizes of the basins of attraction: when the damping coecient varies slowly
from a nite initial value to a di erent nal value, without changing the set of attractors, the
slower the variation the closer the sizes of the basins of attraction are to those they have for
constant damping coecient xed at the initial value. Furthermore, if during the variation
of the damping coecient attractors appear or disappear, remarkable additional phenomena
may occur. For instance it can happen that, in the limit of very large variation time, a xed
point asymptotically attracts the entire phase space, up to a zero measure set, even though
no attractor with such a property exists for any value of the damping coecient between the
extreme values.
Keywords: action-angle variables, attractors, basins of attraction, dissipative systems,
non-constant dissipation, periodic motions, simple pendulum.
Mathematical Subject Classi cation (2000) 34C60, 34C25, 37C60, 58F12, 70K40, 70K50.
1 Introduction
Consider the ordinary di erential equations
x+G(x;t) +
_x= 0; +F(;t) +
_= 0; (1.1)
1arXiv:1403.4996v1 [math.DS] 19 Mar 2014where (x;_x)2R2and (;_)2TR, with T=R=2Z. The functions FandGare smooth and
2-periodic in time t(Fis also 2-periodic in ); the dots denote derivatives with respect to time.
Equations to describe the motion of one-dimensional physical systems are often of this form, in
which case the functions G(x;t) andF(;t) can be considered as an external driving force and the
parameter
represents the damping coecient, which we shall assume positive.
First, for convenience, let us summarise some of the already known ideas regarding systems of
the form (1.1), which can be found in the literature [1, 3, 5]. When
is xed at zero, the system is
Hamiltonian and no attractors are present. For
>0 numerical experiments show that a nite set
of attractors exist: this is consistent with Palis' conjecture [31, 16, 33]. The number of attractors
present and the percentage of phase space covered by their basins of attraction depend upon the
chosen values of the parameters (perturbation parameter "and damping coecient
), but, for all
values of the parameters, the union of the corresponding basins of attraction completely ll the phase
space, up to a set of zero measure. Moreover if the system is a perturbation of an integrable system
(perturbation regime), all attractors found numerically turn out to be either xed points or periodic
solutions with periods that are rational multiples of the forcing period (subharmonic solutions); we
cannot exclude the presence of chaotic attractors [20, 15], but apparently they either do not arise or
seem to be irrelevant.
Generally in the literature the damping coecient is taken as constant, but in many physical
systems it changes non-periodically over time. This can be due to several factors, such as the
heating or cooling of a mechanical system and the wear out or rust on mechanical parts. Despite
this, usually models and numerical simulations of such systems only take the nal value of dissipation
into account when calculating basins of attraction. The recent paper [3] puts forward the idea that,
although the nal value of dissipation determines which attractors exist, the relative sizes of their
basins of attraction depend on the evolution of the dissipation. In particular the e ect of dissipation
increasing to some constant value over a given time span induces a signi cant change to the sizes of
the basins of attraction in comparison to those when dissipation is constant.
Let us illustrate in more detail the phenomenology. Suppose that for two values
0and
1of
the damping coecient, with
06=
1, the same set of attractors exists. Provided the di erence
between the two values is suciently large, the relative sizes of the basins of attraction under the
two coecients will in general be appreciably di erent. If we allow the damping coecient
to
depend on time,
=
(t), and vary from
0to
1over an initial period of time T0, after which it
remains constant at the value
1, then the sizes of the basins of attraction will be di erent from those
where the system has constant coecient
1throughout. Moreover if T0is taken larger, the sizes of
the basins of attraction tend towards those for the system under constant
=
0: this re
ects the
fact that the damping coecient remains close to
0for longer periods of time.
Now consider two values
0and
1of the damping coecient for which the corresponding sets of
attractorsA0andA1are not the same. As a system evolving under dissipation is expected to have
only nitely many attractors, there can only be a nite number of attractors which exist for one
of the two values and not for the other one. What happens is that, by varying
(t) from
0to
1,
an attractor can either appear or disappear, and in the latter case it can disappear either without
leaving any trace or being replaced by a new attractor by bifurcation. Suppose, for instance, that
the only di erence between A0andA1is that the attractor a02A 0simply disappears, that is
A0nA1=fa0g; then, if the time T0over which
(t) varies is large, each remaining attractor tends
to have a basin of attraction not smaller than that it has for
xed at
0: the reason being, again,
that the damping coecient remains close to
0for a long time and, moreover, the trajectories which
would be attracted by a0at
=
0will move towards some other attractor when a0disappears. If,
instead, the only di erence between the sets of attractors A0andA1is that the attractor a02A 0
is replaced by an attractor a1, say by period doubling bifurcation, then, letting
(t) vary from
0to
1over a suciently large time T0causes the size of the basin of attraction of a1to tend towards
that of the basin of attraction that a0has for
=
0.
We summarise our results by the following statements.
1. IfA0, the set of attractors at
=
0, is a subset ofA1, the set of attractors which exist at
=
1, that isA0A 1, then, as the time T0over which
(t) is varied from
0to
1is taken
larger, the basins of attraction tend towards those when
is kept constant at
=
0. In
2particular, if an attractor belongs to A1nA0, then the larger T0the more negligible is the
corresponding basin of attraction.
2. If the set of attractors at xed
=
1is a proper subset of those which exist at
=
0, that
isA1A 0, then, asT0is taken larger, the basins of attraction for the attractors which exist
at both
0and
1change so that for
(t) varying from
0to
1they tend to become greater
than or equal to those for constant
=
0.
3. If an attractor a0exists for
=
0but is destroyed as
(t) tends towards
1, and a new
attractora1is created from it by bifurcation (we will explicitly investigate the case of saddle-
node or period doubling bifurcations), then the size of the basin of attraction of a1, asT0is
taken larger, tends towards that of a0at constant
=
0.
4. IfA01is the set of attractors which exist at both
=
0and
=
1, that isA01=A0\A 1,
and none of the elements in A0nA01are linked by bifurcation to elements in A1nA01, then, as
T0is taken larger, the phase space covered by the basins of attraction of the attractors which
belong toA01tends towards 100%. Moreover, all such attractors have a basin of attraction
larger than or equal to that they have when the coecient of dissipation is xed at
=
0.
The main model used in [3] to convey some of the ideas above is a version of the forced cubic
oscillator, which is of the form of the rst equation in (1.1), with G(x;t) = (1 +"cost)x3. This
system, considered in the perturbation regime (both "and
small), apart from the xed point and
as far as the numerics fortells, exhibits only oscillatory attractors with di erent periods depending
on the parameter values. Also discussed in [3] is the relevance to the spin-orbit problem, describing
an asymmetric ellipsoidal satellite moving in a Keplerian elliptic orbit around a planet [28]: the
corresponding equations of motion are of the form of the second equation in (1.1), with the tidal
friction term
(t) (_ 1) instead of
_, with
(t) slowly increasing in time because of the the cooling
of the satellite.
In the present paper we wish to extend the discussion to the pendulum with periodically oscil-
lating support [25, 32]. The latter is a system which has been already extensively studied in the
literature (we refer to [5] for a list of references): it o ers a wide variety of dynamics and, because
of the separatrix of the unperturbed system, in the perturbation regime, unlike the cubic oscillator,
also includes rotatory attractors in addition to the oscillatory attractors. An important di erence
with respect to the results in [1] is the following. In [1], if an attractor exists for some value of
, it
is found to exist for smaller values of
too. This is not always true for the pendulum considered in
the present paper, where we will see that, at least for some values of the parameters, both increas-
ing and decreasing
can destroy attractors as well as create new ones. However, this occurs away
from the perturbation regime, where the system can no longer be considered as a perturbation of
an integrable one: the appearance and disappearance of attractors would occur also in the case of
the cubic oscillator for larger values of the forcing. In addition to the case of increasing dissipation
studied in [3], here we also include the case where the damping coecient decreases to a constant
value, which is appropriate for physical systems where joints are initially tight and require time to
loosen. In this case similar phenomena are expected. For instance, as the value of variation time
T0is taken larger, the amount of phase space covered by each of the basins of attraction should
tend towards that corresponding to original value
0of
, providing the set of attractors remains
the same.
The non-linear pendulum with vertically oscillating support is described by
+f(t) sin+
_= 0; f (t) =g
` b!2
`f0(!t)
; (1.2)
wheref0is a smooth 2 -periodic function and the parameters `,b,!andgrepresent the length,
amplitude and frequency of the oscillations of the support and the gravitational acceleration, re-
spectively, all of which remain constant; for the sake of simplicity we shall take f0(!t) = cos(!t)
in (1.2), as in [5, 6, 7]. As mentioned above, the parameter
represents the damping coecient,
3which, for analysis where it remains constant, we shall model as
=Cn"n, where"is small and n
is an integer. We shall consider (1.2) as a pair of coupled rst order non-autonomous di erential
equations by letting x=andy= _x, such that the phase space is TRand the system can be
written as
_x=y; _y= g
` b!2
`cos (!t)
sinx
y:
The system described by (1.2) can be non-dimensionalised by taking
=g
`!2; =b
`; =!t;
so that it becomes
00+f( ) sin+
0= 0; f ( ) = ( cos ); (1.3)
or, written as a system of rst order di erential equations,
x0=y; y0= f( ) sinx
y; (1.4)
where the dashes represent di erentiation with respect to the new time and
has been normalised
so as not to contain the frequency. Linearisation of the system about either xed point results in a
system of the form of Mathieu's equation, see for instance [27]. When the downwards xed point is
linearly stable, it is possible, for certain parameter values, to prove analytically the conditions for
which the xed point attracts a full measure set of initial conditions; see Appendix A.
In the Sections that follow we shall use the non-dimensionalised version of the system (1.3),
preferable for numerical implementation as it reduces the number of parameters in the system. In
Section 2 we detail the calculations of the threshold values for the attractors, that is the values of
constant
below which periodic attractors exist in the perturbation regime (small ). As we shall
see, because of the presence of the separatrix for the unperturbed pendulum, this will be of limited
avail for practical purposes: the persisting periodic solutions found to rst order are in general too
close to the separatrix for the perturbation theory to converge. In Section 3 we present numerical
results, in the case of both constant and non-constant (either increasing or decreasing) dissipation,
for values of the parameters in the perturbation regime. Since for such values the downwards position
turns out to be stable, we shall refer to this case as the downwards pendulum. Next, in Section 4 we
perform the numerical analysis for values of the parameters for which the upwards position is stable
(hence such a case will be referred to as the inverted pendulum). Of course such parameter values
are far away from the perturbation regime: as a consequence additional phenomena occur, including
period doubling and saddle-node bifurcations. In Section 5 we include a discussion of numerical
methods used. Finally in Section 6 we draw our conclusions and brie
y discuss some open problems
and possible directions for future investigation.
2 Thresholds values for the attractors
The method used below to calculate the threshold values of
below which given attractors exist
follows that described in [1, 3], where it was applied to the damped quartic oscillator and the spin-
orbit model. We consider the system (1.4), with ="and
=C1", where";C1>0. This approach
is well suited to compute the leading order of the threshold values. In general, it would be preferable
to write
as a function of "of the form
=C1"+C2"2+:::(bifurcation curve), and x the constants
Ckby imposing formal solubility of the equations to any perturbation order, see [18]; however this
only produces higher order corrections to the leading order value.
For"= 0 the system reduces to the simple pendulum 00+ sin= 0, which admits periodic
solutions inside the separatrix (librations or oscillations) and outside the separatrix (rotations). In
terms of the variables ( x;y) the equations (1.4) become x0=y,y0= sinx: the librations are
described by(
xosc( ) = 2 arcsin [ k1sn (p ( 0);k1)];
yosc( ) = 2k1p cn (p ( 0);k1);k1<1; (2.1)
4while the rotations are described by
(
xrot( ) = 2 arcsin [sn (p ( 0)=k2;k2)];
yrot( ) = 2k 1
2p dn (p ( 0)=k2:k2);k2<1; (2.2)
where cn (;k), sn(;k) and dn(;k) are the Jacobi elliptic functions with elliptic modulus k[8, 11,
26, 35], and k1andk2are such that k2
1= (E+ )=2 andk2
2= 1=k2
1, withEbeing the energy of
the pendulum. From (2.1) and (2.2) it can be seen that the solutions are functions of ( 0), so
that the phase of a solution depends on the initial conditions. We can x the phase of the solution
to zero without loss of generality by instead writing f( ) in equation (1.4) as f( 0). This moves
the freedom of choice in the initial condition to the phase of the forcing.
The dynamics of the simple pendulum can be conveniently written in terms of action-angle
variables (I;'), for which we obtain two sets of variables: for the librations inside the separatrix
one expresses the action as
I=8
p h
(k2
1 1)K(k1) +E(k1)i
; (2.3)
where K(k) and E(k) are the complete elliptic integrals of the rst and second kind, respectively,
and writes
x= 2 arcsin
k1sn2K(k1)
';k1
; y = 2k1p cn2K(k1)
';k1
; (2.4)
withk1obtained by inverting (2.3), while for the rotations outside the separatrix one expresses the
actions as
I=4
k2p E(k2); (2.5)
and writes
x= 2 arcsin
snK(k2)
';k2
; y =2
k2p dnK(k2)
';k2
; (2.6)
withk2obtained by inverting (2.5); further details can be found in Appendix B.
For"small, in order to compute the thresholds values, we rst write the equations of motion for
the perturbed system in terms of the action-angle coordinates ( I;') of the simple pendulum, then
we look for solutions in the form of power series expansions in ",
I( ) =1X
n=0"nI(n)( ); ' ( ) =1X
n=0"n'(n)( ); (2.7)
whereI(0)( ) and'(0)( ) are the solutions to the unperturbed system, that is, see Appendix B,
(I(0)( );'(0)( )) = (Iosc;'osc( )) and (I(0)( );'(0)( )) = (Irot;'rot( )), in the case of oscillations
and rotations, respectively, with
Iosc=8
p h
(k2
1 1)K(k1) +E(k1)i
; ' osc( ) =
2K(k1)p ;
Irot=4
k2p E(k2); ' rot( ) =
K(k2)p
k2;(2.8)
and with given k1=k(0)
1andk2=k(0)
2.
As the solution (2.7) is found using perturbation theory, its validity is restricted to the system
where"is comparatively small. In particular this limitation has the result that the calculations of
the threshold values are not valid for the inverted pendulum, where large "is required to stabilise
the system. On the other hand the regime of small "has the advantage that we can characterise
analytically the attractors and hence allows a better understanding of the dynamics with respect to
the case of large ", where only numerical results are available.
52.1 Librations
We rst write the equations of motion (1.4) in action-angle variables, see Appendix D, as
'0=p
2K(k1) "
2K(k1)p
sn2() +k2
1sn2() cn2()
1 k2
1 Z() sn() cn() dn()
1 k2
1
cos( 0)
+C1"cn()
2K(k1)sn()
dn()+k2
1sn() cn2()
(1 k2
1) dn() Z() cn()
1 k2
1
;
I0=8"k2
1K(k1)
cos( 0) sn() cn() dn() 8C1"k2
1p K(k1)
cn2();(2.9)
where Z() is the Jacobi zeta function, see [26]. Here and throughout Section 2.1 to save clutter we
de ne () =
2K(k1)
';k1
. Note that in (2.9), the dependence on Iof the vector eld is through
the variable k1, according to (2.3).
The coordinates for the unperturbed system ( "= 0) satisfy
'0=dE
dI:=
(I) =p
2K(k1); I0= 0: (2.10)
Linearising around ( '(0)( );I(0)( )) = (
(I(0)) ;I(0)), we have
'0=@
@I(I(0))I; I0= 0; (2.11)
where, see Appendix B,
(I) :=@
@I(I) = 2
16k2
1K3(k1)E(k1)
1 k2
1 K(k1)
: (2.12)
SinceI=I(k1), that is the action is a function of k1, settingI=I(0) xesk1=k(0)
1, yielding
(I(0)) =(0), with(0)given by (2.12) with k1=k(0)
1.
The linearised system (2.11) can by written in compact form as
'0
I0
=
0(0)
0 0
'
I
: (2.13)
The Wronskian matrix W( ) is de ned as the solution of the unperturbed linear system
W0( ) =
0(0)
0 0
W( ); W (0) =I;
where Iis the 22 identity matrix. Hence
W( ) =
1(0)
0 1
; (2.14)
with (1;0) and ((0) ;1) two linearly independent solutions to (2.13).
We now look for periodic solutions ( '( );I( )) to (2.9) with period T= 2q= 4K(k1)p=p ,
with p=q2Q, of the form (2.7); see also [18, 19] for a more general discussion. A solution of this
kind will be referred to as a p:qresonance.
The functions ( '(n)( );I(n)( )) are formally obtained by introducing the expansions (2.7) into
the equations (2.9) and equating the coecients of order n. This leads to the equations
('(n))0
(I(n))0
=
(0)I(n)
0
+
F(n)
1( )
F(n)
2( )!
(2.15)
6withF(n)
1( ) andF(n)
2( ) given by
F(n)
1( ) =p
2K(k1) (0)I(n)
+"
2K(k1)p
sn2() +k2
1
1 k2
1sn2() cn2()
Z()
1 k2
1sn() cn() dn()
cos( 0)
+C1cn()
2K(k1)sn()
dn()+k2
1sn() cn2()
(1 k2
1) dn() Z() cn()
1 k2
1#(n 1)
;
F(n)
2( ) ="
8k2
1K(k1)
cos( 0) sn() cn() dn() 8C1k2
1p K(k1)
cn2()#(n 1)
:
The notation [ :::](n)means that one has to take all terms of order nin"of the function inside
[:::]. By construction, F(n)
1( ) andF(n)
2( ) depend only on the coecients '(p)( ) andI(p)( ), with
p<n , so that (2.15) can be solved recursively.
Then, by using the Wronskian matrix (2.14), see [30], one can integrate (2.15) so as to obtain
'(n)( )
I(n)( )
=W( )'(n)
I(n)
+W( )Z
0dW 1()
F(n)
1()
F(n)
2()!
(2.16)
where '(n)and I(n)are thenthorder in the "-expansion of the initial conditions for 'andI,
respectively. In the last term of (2.16) we have
W( )Z
0dW 1()
F(n)
1()
F(n)
2()!
=Z
0dW( )
F(n)
1()
F(n)
2()!
:
This yields
'(n)( ) = '(n)+(0) I(n)+Z
0dF(n)
1() +(0)Z
0dZ
0d0F(n)
2(0);
I(n)( ) =I(n)( ) +Z
0dF(n)
2():(2.17)
For a periodic function g, let us denote the average of gwithhgiand the zero-average function
g hgiwith g. Suppose that
hF(n)
2i:=1
TZT
0d F(n)
2( ) = 0; (2.18)
whereT= 4K(k1)p; we will check later on the validity of (2.18). Then we may write
F(n)
1( ) =Z
0dF(n)
1() =hF(n)
1i +Z
0dF(n)
1();
F(n)
2( ) =Z
0F(n)
2() d=Z
0dF(n)
2();
and subsequently rewrite (2.17) as
'(n)( ) = '(n)+(0) I(n)+hF(n)
1i +Z
0dF(n)
1() +(0)hF(n)
2i +(0)Z
0dF(n)
2();
I(n)( ) =I(n)+Z
0dF(n)
2();
in which all the terms which are not linear in are periodic. If we choose our initial conditions I(n)
such that they satisfy
I(n)= 1
(0)hF(n)
1i hF(n)
2i;
7the above reduces to
'(n)( ) = '(n)( ) +Z
0dF(n)
1() +(0)Z
0dF(n)
2();
I(n)( ) =I(n)+Z
0dF(n)
2(~ );
so that both '(n)( ) andI(n)( ) are periodic functions with period T, provided (2.18) holds.
Lemma 1 Consider the series (2.7) . If p=q= 1=2m,m2NandC1is small enough, then it is
possible to x the initial conditions ( '(n);I(n))in such a way that (2.18) holds for all n1. If
p=q6= 1=2mfor allm2N, then (2.18) can be satis ed only for C1= 0.
Proof Forn= 1 we have
F(1)
2( ) =8k2
1K(k1)
cos( 0) sn(p ;k 1) cn(p ;k 1) dn(p ;k 1)
8C1k2
1p K(k1)
cn2(p ;k 1);
withk1=k(0)
1here and henceforth. Moreover set, see Appendix E,
:=p
4K(k1)Z4K(k1)=p
0d cn2(p ;k 1)
=1
2K(k1)Z2K(k1)
0d cn2( ;k1) =1
k2
11
2K(k1)E(2K(k1);k1) (1 k2
1)
;(2.19)
where E(u;k) is the incomplete elliptic integral of the second kind, and 1( 0;p;q) := sin( 0)G1(p;q),
with
G1(p;q) =1
TZT
0sn(p ;k 1) cn(p ;k 1) dn(p ;k 1) sin( )
=1
4K(k1)pZ4K(k1)p
0d sn( ;k1) cn( ;k1) dn( ;k1) sin( =p ):(2.20)
Under the resonance condition =2K(k1) = p=q, one has
sin( =p ) = sin
2K(k1)q
p
;
where pand qare relatively prime. By expanding the Jacobi elliptic functions in Fourier series, see
Appendix E, we nd that p;qmust also satisfy the condition
p
(2m1 1)(2m2 1)2m3
q= 0
forG1(p;q) to be non-zero. Thus q= 2mp,m2N, that is p= 1 and q22N, andhF(1)
2i= 0
providedC1and 0satisfy
C1=sin( 0)p G1(p;q): (2.21)
Note that the existence of a value of 0satisfying (2.21) is possible only if
jC1jC1(p=q) :=1p G1(p;q):
Some values of the constants C1(p=q) for = 0:5 are listed in Table 1.
8qk1G1(1=q) C1(1=q)
2 0.885201568846 0.172135 0.407121 0.597944
4 0.998888384493 0.077675 0.224342 0.489649
6 0.999986981343 0.051734 0.150043 0.487616
8 0.999999846887 0.038800 0.112539 0.487578
10 0.999999998199 0.031040 0.090032 0.487577
12 0.999999999979 0.025867 0.075026 0.487577
Table 1: Constants for the oscillating attractors with = 0:5.
For alln2 we can write F(n)
2( ) as
F(n)
2( ) =8k2
1K(k1)
@
@'
cos( 0) sn() cn() dn() p C1cn2()
'='(0)'(n 1)+R(n)( );
whereR(n)( ) is a suitable function which does not depend on '(n 1). It can be seen that hF(n)
2i= 0
if and only if
hR(n)i= 8k2
1K(k1)
1
TZT
0d 2K(k1)p @
@
sn(p ) cn(p ) dn(p )
cos( 0)
p C1
TZT
0d 2K(k1)p @
@
cn2(p )!
'(n 1):
This can be rewritten as
hR(n)i= 16k2
1K2(k1)p 2cos( 0)G1(p;q) '(n 1):
We refer the reader to Appendix E for more details on the evaluation of the integrals. The coecient
of '(n 1)is non-vanishing for 0chosen such that (2.21) is satis ed. Therefore it is possible to x
the initial conditions '(n 1)in such a way that one has hF(n)
2i= 0 at all orders, thus completing
the proof of the lemma. 2
Lemma 1 implies that the threshold values of the p:qresonances are
(p=q) =C1(p=q)"forp= 1
and qeven, with the constants C1(p=q) in Table 1, while the threshold values of the other resonances
are at least O("2).
2.2 Rotations
Similarly for the rotating scenario, again further details can be found in Appendix D, the perturbed
system can be written as
'0=p
k2K(k2)+"k2p K(k2)k2
2sn2() cn2()
1 k2
2 Z() sn() cn() dn()
1 k2
2
cos( 0)
C1"
K(k2)k2
2sn() cn() dn()
1 k2
2 Z() dn2()
1 k2
2
;
I0=4"K(k2)
cos( 0) sn() cn() dn() 4C1"p K(k2)
k2dn2():(2.22)
Here and throughout this subsection we set ( ) =
K(k2)
';k2
. In this scenario, the Wronskian
matrixW( ) can be written as in (2.14), with (0)given by, see Appendix B,
(I) = 2p
4K3(k2)E(k2)
1 k2
2 K(k2)
; (2.23)
9fork2=k(0)
2. We again look for solutions ( '( );I( )) with period T= 2q= 4k2K(k1)p=p
corresponding to a resonance p:q, of the form (2.7), the functions '(n)( ) andI(n)( ) being de ned
as in (2.16), with F(n)
1( ) andF(n)
2( ) de ned as
F(n)
1( ) =p
k2K(k2) (0)I(n)
+"
k2p K(k2)k2
2sn2() cn2()
1 k2
2
Z() sn() cn() dn()
1 k2
2
cos( 0)
C1
K(k2)k2
2sn() cn()
1 k2
2 Z() dn()
1 k2
2#(n 1)
;
F(n)
2( ) =4K(k2)
cos( 0) sn() cn() dn() 4C1p K(k2)
k2dn2()(n 1)
:
The theory goes through exactly as previously shown for the case of libration and we must show
thathF(n)
2i= 0.
Lemma 2 Consider the series (2.7) . If p=q= 1=2m,m2N, andC1is small enough, then it is
possible to x the initial conditions ( '(n);I(n))in such a way that hF(n)
2i= 0 for alln1. If
p=q6= 1=2mfor allm2NthenhF(n)
2i= 0only whenC1= 0.
Proof One has
F(1)
2( ) =4K(k2)
cos ( 0) snp
k2 ;k2
cnp
k2 ;k2
dnp
k2 ;k2
4C1p K(k2)
k2dn2p
k2 ;k2
;
withk2=k(0)
2here and henceforth. De ne, see Appendix E,
:=p
2k2K(k2)Z2k2K(k2)=p
0d dn2p
k2 ;k2
=1
2K(k2)E
2K(k2);k2
:
and 1( 0;p;q) := sin( 0)G1(p;q), where
G1(p;q) =1
TZT
0d snp
k2 ;k2
cnp
k2 ;k2
dnp
k2 ;k2
sin( )
=1
4K(k2)pZ4K(k2)p
0d sn( ;k2) cn( ;k2) dn( ;k2) sin(k2 =p );
then use the resonance condition to set
sink2 p
= sin
2K(k2)q
p
:
On inspection of the above we see that 1( 0;p;q) can be calculated similarly to the case inside the
separatrix. It follows that the same applies and p=q= 1=2mform2N. ThenhF(1)
2i= 0 if
C1=k2sin( 0)p G1(p;q); (2.24)
which requires
jC1jC1(p=q) :=k2p G1(p;q):
10Some values of the constants C1(p=q) for = 0:5 are listed in Table 2.
qk2G1(1=q) C1(1=q)
2 0.924397052341 0.156774 0.474414 0.432005
4 0.998899257272 0.077612 0.225808 0.485542
6 0.999986983601 0.051734 0.150063 0.487439
8 0.999999846887 0.038800 0.112540 0.487577
10 0.999999998199 0.031040 0.090032 0.487577
12 0.999999999978 0.025867 0.075026 0.487577
Table 2: Constants for the rotating attractors with = 0:5.
Forn2 one has
F(n)
2( ) =4K(k2)
@
@'
cos( 0) sn() cn() dn() p C1dn2()
'='(0)'(n 1)+R(n)( );
where again R(n)( ) will be a suitable function which does not depend on '(n 1). Similarly to the
case of libration, hF(n)
2i= 0 if and only if
hR(n)i= 4K(k2)
1
TZT
0k2K(k1)p @
@
sn(p ) cn(p ) dn(p )
cos( 0) d
p C1
TZT
0k2K(k1)p @
@
dn2(p )
d !
'(n 1)= 4k2K2(k2)p 2cos( 0)G1(p;q) '(n 1);
so that the coecient of '(n 1)turns out to be non-vanishing for 0chosen such that equation (2.24)
is satis ed. Therefore it is possible to x the initial conditions '(n 1)in such a way that one has
hF(n)
2( )i= 0 at all orders, thus completing the proof. 2
Lemma 2 implies that the threshold values of the p:qresonances are
(p=q) =C1(p=q)"forp= 1
and qeven, with the constants C1(p=q) in Table 2, while the threshold values of the other resonances
are at least O("2).
Note, in Tables 1 and 2 it is apparent that, for = 0:5, increasing qcauses the value of C1(1=q)
to converge to 0.487577 in both cases. However this does not mean that for
<0:487577"there
are in nitely many attracting solutions with increasing period: this would be a counter-example to
Palis' conjecture! The explanation for this seeming paradox is as follows: As qincreases the solutions
move closer and closer to the separatrix (this can be seen by the corresponding values of k1andk2),
where the power series expansions (2.7) for the solutions I( ) and'( ) which were constructed with
perturbation theory converge only for very small values of ": the larger q, the smaller must be ". In
particular, for any xed "there is only a nite number of periodic solutions which can be studied by
perturbation theory. In particular, for the chosen parameters the only periodic solution corresponds
to the resonance 1:2 inside the separatrix. We also note that the above analysis applies only to
periodic attractors with p= 1 and qeven. However we shall see that the numerical simulations
provide also rotating attractors with period 2 , that is the same period as the forcing: we expect
that continuing the analysis to second order and writing
=C2"2, see [1], would give the threshold
values for these periodic attractors.
3 Numerics for the downwards pendulum
We shall investigate the system (1.3) in the same region of phase space used in [5], namely 2[ ;],
02[ 4;4] and calculate the relative areas of the basins of attraction, that is the percentage of phase
space they cover relative to this region.
11Throughout this Section we x the parameters = 0:5 and = 0:1. These parameter values,
also investigated in [5], correspond to a stable region of the stability tongues for the system linearised
around= 0, see [22], so that the downwards con guration is stable. The chosen values for the
damping coecient span values between
= 0:002 and 0:06, of which only
= 0:03 was previously
investigated in [5]. For some values of
, the system exhibits three non- xed-point attractors,
examples of which are shown in Figure 1, as well as the downwards xed point attractor. Here
and henceforth, for brevity, we shall say that a solution has period nif it comes back to its initial
value after nperiods of the forcing. Of course the upwards xed point also exists as a solution to
the system, however it is unstable and thus does not attract any non-zero measure subset of phase
space. It can also be seen from Figure 1 that the attractive solutions are near the separatrix of the
unperturbed system: this is evident as the curves described by the two rotating attractors are close
to that of the oscillating attractor and the separatrix lies between them. This observation con rms
the reasoning as to why the computation of the threshold values can only produce valid results for
the period 2 oscillating attractor (see the conclusive remarks in Section 2).
Figure 1: Attracting solutions for the system (1.3) with = 0:5, = 0:1 and
= 0:02, namely two period
1 rotations and one period 2 solution which oscillates about the downwards xed point. Periods can be
deduced by circles corresponding to the Poincar e map.
For
<
0, for a suitable
02(0:4;0:5), the system has three periodic attractors, in addition
to the downwards xed point: one oscillating and two rotating attractors. For
0the two
rotating attractors no longer exist, leaving just the oscillatory attractor and the xed point. The
basins of attraction for
= 0:02, 0:03, 0:04 and 0:05 are shown in Figure 2, from which we can
see that the entire phase space is covered: this suggests that no other attractors exist, at least
for the values of the parameters considered. The corresponding relative areas, as estimated by the
numerical simulations, are given in Table 3 and plotted in Figure 3. The relative areas of the basins
of attraction for positive and negative rotations have been listed in the same column: numerical
simulations found a di erence in size no greater than 10 2% and, due to the symmetries of the
system, it is expected that this di erence is numerically induced by the selection of initial conditions
The basins of attraction were estimated using numerical simulations with 600 000 random initial
conditions in phase space. More notes on the numerics used can be found in Section 5.
It can be seen that the results in Table 3 are in agreement with the calculations for the threshold
value for the period 2 oscillatory attractor. The calculations in Appendix A predict that, for the
chosen values = 0:5 and = 0:1, taking
>
10:1021 ensures for the origin to capture a
full measure set of initial conditions. From Table 3 a stronger result emerges numerically: the xed
point attracts the full phase space, up to a zero-measure set, for
20:06. Upon comparing
results in Table 3, we see that, essentially, the basin of attraction for the xed point becomes smaller
with an increase in
, up to approximately 0 :035, after which it grows again. Similarly, by increasing
the value of
, the basins of attraction of the oscillating and rotating solutions attractors increase
initially, up to some value (about 0 :025 and 0:035, respectively), after which they become smaller.
12Furthermore, the variations of the relative areas of the basins of attraction are never monotonic, as
one observes slight oscillations for small variations of
. These features seem contrary to systems
such as the cubic oscillator, where decreasing dissipation seems to cause the relative area of the
basin of attraction of the xed point to decrease monotonically, while the basins of attraction of
the periodic attractors reach a maximum value, after which their relative areas slightly decrease,
see for instance Table III in [3]. We note, however, that a more detailed investigation shows that
oscillations occur also in the case of the cubic oscillator. This was already observed for some values
of the parameters (see Table IX in [3]), but the phenomenon can also be observed for the parameter
values of Table III, simply by considering smaller changes of the value of
with respect to the values
in [3]. For instance, by varying slightly
around 0:0005 (see Table III in [3] for notations), one nds
for the main attractors the relative areas in Table 4.
(a)
(b)
(c)
(d)
Figure 2: Basins of attraction for the system (1.3) with = 0:5, = 0:1 and (a)
= 0:02, (b)
= 0:03,
(c)
= 0:04 and (d)
= 0:05. The xed point (FP) is shown in blue, the positive and negative rotating
solutions (PR and NR) are shown in red and yellow, respectively. and the oscillating solution (OSC) in
green.
In conclusion, for the pendulum, apart from small oscillations, by decreasing the value of
from
0:06 to 0:002, the basins of attraction of the periodic attractors, after reaching a maximum value,
becomes smaller. A similar phenomenon occurs also in the cubic oscillator (albeit less pronounced).
However, a new feature of the pendulum, with respect to the cubic oscillator, is that the basin
of attraction of the origin after reaching a minimum value increases again: the increase seems to
be too large to be ascribed simply to an oscillation, even though this cannot be excluded. In the
case of the cubic oscillator the slight decrease of the sizes of the basins of attraction of the periodic
attractors was due essentially to the appearance of new attractors and their corresponding basins
13of attraction. It would be interesting to investigate further, in the case of the pendulum, how the
basins of attractions, in particular that of the xed point, change by taking smaller and smaller
values of
. We intend to come back to this in the future [36].
Basin of attraction %
FP PR/NR OSC
0.0020 84.57 3.35 8.73
0.0050 79.91 3.88 12.32
0.0100 72.24 4.60 18.57
0.0200 71.95 4.57 18.90
0.0230 70.73 5.18 18.90
0.0250 69.28 5.19 20.35
0.0300 69.94 4.42 21.23
0.0330 68.92 3.75 23.59
0.0350 68.77 3.16 24.90
0.0400 73.84 1.42 23.32
0.0500 85.61 0.00 14.39
0.0590 96.96 0.00 3.04
0.0597 98.59 0.00 1.41
0.0600 100:00 0.00 0.00
Table 3: Relative areas of the basins of
attraction with = 0:5, = 0:1 and con-
stant
.
Figure 3: Plot of the relative areas of the basins of
attraction with constant
as per Table 3.
Basin of attraction %
0 1/2 1/4 1a 1b 1/6 3a 3b
0.00052 39.03 41.73 14.72 1.22 1.22 1.59 0.25 0.25
0.00051 39.73 41.70 13.88 1.24 1.24 1.66 0.27 0.27
0.00050 38.72 41.85 14.65 1.28 1.28 1.65 0.29 0.29
0.00049 39.26 41.96 13.81 1.29 1.29 1.77 0.27 0.32
0.00048 38.48 41.87 14.60 1.30 1.30 1.75 0.34 0.34
Table 4: Relative areas of the basins of attraction of the main attractors for
the system x+ (1 +"cost)x3+
_x= 0, with"= 0:1 and
around 0:0005. The
basins of attraction were estimated using numerical simulations with 1000000
random initial conditions in the square [ 1;1][ 1;1] in phase space.
3.1 Increasing dissipation
In this section we shall investigate the case where dissipation increases with time, up to a time T0,
after which it remains constant. We will consider a linear increase in dissipation from a value
0at
timet= 0 up to
1at timeT0, that is (see Figure 4)
=
(t) =(
0+ (
1
0)
T0;0 <T 0;
1; T 0 :(3.1)
Although this is a greatly simpli ed model of what might take place in reality, it serves the purpose
of demonstrating the signi cant e ects of initially non-constant dissipation on the nal basins of
attraction. Below we will consider explicitly the cases
0= 0:2 and
1= 0:3, 0:4 and 0:5.
As previously mentioned, we expect that increasing the value of T0results in the relative areas
of the basins of attraction moving along the curves plotted for constant
. The movement along
these curves starts at
1and goes towards
0. In particular, any values of the relative area of a
basin of attraction for constant values of the damping coecient between
1and
0are traced as the
value ofT0varies for time-dependent dissipation. This movement along the curve is not linear with
14the value of T0but asymptotic, with the relative area of the basin of attraction tending towards
the value at
=
0asT0!1 , providing the attractors existing at
=
0also persist at
1.
When the attractors which persist at
=
1are a proper subset of those which exist at
0, we
expect the persisting attractors to absorb the remaining phase space left by the attractors which
have disappeared: thus their basins of attraction should be greater than or equal to those at
=
0.
For the values of the parameters in the chosen range, only these two cases may occur as the solutions
which exist for
=
1also exist at
=
0, see Table 3.
Figure 4: Plot of equation (3.1) with
0= 0:1,
1= 0:2 and varying T0.
The results in Tables 5, 6 and 7 are in agreement with the expectations above. It can be seen
from Tables 5 and 6 that the relative areas of the basins of attraction trace those of constant
. In
particular, the relative area of the basins of attraction of the rotating attractors tends towards that
at
=
0from above, despite having a smaller basin of attraction for the chosen values of
1. More
precisely, the longer T0, the closer is the relative area of the basin of attraction to the value it has
at
=
0. However, the convergence to the asymptotic value is rather slow: for instance in Table
5, evenT0= 2000 is not enough to reach the values corresponding to
= 0:02. The simulations for
time varying dissipation have in general taken 300 000 or 400 000 initial conditions in phase space.
In some cases more points were used for additional accuracy.
Basin of Attraction %
FP PR/NR OSC
T00 69.94 4.42 21.23
25 69.80 4.42 21.36
50 69.57 4.45 21.52
75 69.40 4.64 21.33
100 68.84 4.85 21.47
200 68.82 5.10 20.99
500 69.86 5.17 19.80
1000 70.65 5.18 18.99
2000 71.17 5.11 18.61
Table 5: Relative areas of the basins of
attraction with
0= 0:02,
1= 0:03 and
T0varying.
Figure 5: Plot of the relative areas of the basins of
attraction as per Table 5.
In Table 7, we see that for
0= 0:02 and
1= 0:05 only two attractors are present: indeed
the oscillating attractors no longer exist for
= 0:05. Hence, when
(t) increases and crosses the
value at which those attractors disappear, all the trajectories that up to this time were converging to
them, will fall into the basins of attraction of the persisting attractors, that is the xed point and the
oscillating solution. In particular the corresponding basins of attraction will acquire relative areas
larger than those they have at constant
=
0, because of the absorption of all these trajectories.
It is dicult to predict how such trajectories are distributed among the persisting attractors. In
15the case of Table 7 they seem to be attracted slightly more by the xed point, even though the
percentage increase is larger for the oscillating solution.
Basin of Attraction %
FP PR/NR OSC
T00 73.84 1.42 23.32
25 73.66 1.44 23.45
50 73.37 1.50 23.63
75 72.15 2.22 23.41
100 68.69 3.50 24.31
200 67.46 4.76 23.03
500 69.05 5.02 20.92
1000 69.85 5.17 19.81
2000 70.63 5.18 19.02
Table 6: Relative areas of the basins of
attraction with
0= 0:02,
1= 0:04 and
T0varying.
Figure 6: Plot of the relative areas of the basins of
attraction as per Table 6.
Basin of Attraction %
FP OSC
T00 85.61 14.39
25 86.01 13.99
50 86.18 13.82
75 84.19 15.87
100 80.42 19.58
200 75.47 24.53
500 77.95 22.06
1000 77.55 22.45
1500 78.03 21.97
Table 7: Relative areas of the basins
of attraction with
0= 0:02,
1= 0:05
andT0varying.
Figure 7: Plot of the relative areas of the basins of
attraction as per Table 7.
3.2 Decreasing dissipation
In this section we conversely look at the damping coecient decreasing from some value
0>
1,
with di erent rates of decrease, see Figure 8. We will consider the cases
0= 0:04 and
1= 0:02,
0= 0:04 and
1= 0:03,
0= 0:05 and
1= 0:02.
Figure 8: Plot of equation (3.1) with
0= 0:23,
1= 0:2 and varying T0.
16In this situation it is possible that more attractors exist at
1than at
0, see Table 3. We again
expect that increasing T0causes the relative areas of the basins of attraction to tend towards those
at
0. The result of this is that solutions which do not exist at
0will attract less and less of the
phase space as T0increases, and for T0large enough their basins of attraction will tend to zero.
Basin of Attraction %
FP PR/NR OSC
T00 71.95 4.57 18.90
25 71.85 4.60 18.94
50 72.36 4.48 18.69
75 73.64 4.43 17.51
100 74.10 4.32 17.27
200 72.31 2.99 21.71
500 71.51 2.09 24.31
1000 72.61 1.79 23.81
2000 73.11 1.63 23.64
Table 8: Relative areas of the basins of
attraction with
0= 0:04,
1= 0:02 and
T0varying.
Figure 9: Plot of the relative areas of the basins of
attraction as per Table 8.
Basin of Attraction %
FP PR/NR OSC
T00 69.94 4.42 21.23
25 69.73 4.50 21.28
50 70.78 4.23 20.77
75 72.03 3.45 21.07
100 71.77 2.95 22.33
500 72.61 1.79 23.81
1000 73.11 1.63 23.64
Table 9: Relative areas of the basins of
attraction with
0= 0:04,
1= 0:03 and
T0varying.
Figure 10: Plot of the relative areas of the basins of
attraction as per Table 9.
Basin of Attraction %
FP PR/NR OSC
T00 71.95 4.57 18.90
25 72.13 4.58 18.72
50 72.97 4.24 18.55
75 76.14 3.15 17.56
100 77.02 2.18 18.62
200 77.71 0.31 21.67
500 81.94 0.00 18.06
Table 10: Relative areas of the basins of
attraction with
0= 0:05,
1= 0:02 andT0
varying.
Figure 11: Plot of the relative areas of the basins of
attraction as per Table 10.
17Tables 8 and 9 illustrate cases in which the system admits the same set of attractors for both
values
0and
1of the damping coecient. An example of what happens when an attractor exists
at
1but not at
0can be seen in the results of Table 10, where
(t) varies from 0 :05 to 0:02. As
the damping coecient starts o at a larger value, then decreases to some smaller value, we also
expect the change in the basins of attraction to happen over shorter values of T0. The reasoning for
this is simply that larger values of dissipation cause trajectories to move onto attractors in less time.
IncreasingT0results in the system remaining at higher values of dissipation for more time and thus
trajectories land on the attractors in less time.
4 Numerics for the inverted pendulum
The upwards xed point of the inverted pendulum can be made stable for large values of , i.e when
the amplitude of the oscillations is large relative to the length of the pendulum. In this section we
numerically investigate the system (1.3) for parameter values for which this happens. For simplicity,
as mentioned in the introduction, we refer to this case as the inverted pendulum. It can be more
convenient to set x=+, so as to centre the origin at the upwards position of the pendulum.
Then the equations of motion become
00+f( ) sin+
0= 0; f ( ) = ( + cos ); (4.1)
=g
`!2; =b
`; =!t:
The di erence between equations (1.3) and (4.1) is that here the parameter has changed sign.
(a)
(b)
(c)
(d)
Figure 12: Attracting solutions for the system (4.1) with = 0:1 and = 0:545. Figure (a) shows an
example of the positive and negative rotating attractors with period 2, taken for
= 0:05. Figure (b) shows
the positive and negative attractors with period 1 when
= 0:23. Figures (c) and (d) show the oscillatory
attractors with periods 2 and 4 respectively when
is taken equal to 0.2725. These solutions oscillate about
the downwards xed point =and the axis has been shifted to show a connected curve in phase space.
The period of each solution can be deduced from the circles corresponding to the Poincar e map.
18The stability of the upwards xed point creates interesting dynamics to study numerically, how-
ever it means that the system is no longer a perturbation of the simple pendulum system. This
in turn has the result that the analysis in Section 2 to compute the thresholds of friction cannot
be applied. However, we shall see that the very idea that attractors have a threshold value below
which they always exist does not apply to the inverted pendulum: both increasing and decreasing
the damping coecient can create and destroy solutions.
For numerical simulations of the inverted pendulum throughout we shall take parameters = 0:1
and = 0:545, which are within the stable regime for the upwards position. For these parameter
values the function f( ) changes sign. As such, the analysis in Appendix A cannot be applied.
Again these particular parameter values were also investigated in [5], but with a small value for
the damping coecient, that is
= 0:08, where only three attractors appeared in the system: the
upwards xed point and the left and right rotating solutions. We have opted to focus on larger
dissipation because the range of values considered for
allows us to incorporate already a a wide
variety of dynamics, in which remarkable phenomena occur, and, at the same time, larger values of
are better suited to numerical simulation because of the shorter integration times. We note that,
for the values of the parameters chosen, no strange attractors arise: numerically, besides the xed
points, only periodic attractors are found.
For constant dissipation we provide results for
2[0:05;0:6]. These values of
are considered
to correspond to large dissipation, however non- xed-point solutions still persist due to the large
coecient of the forcing term. Some examples of the persisting non- xed-point solutions can be
seen in Figure 12; of course, the exact form of the curves depends on the particular choices of
. For
varying in the range considered the following attractors arise (we follow the same convention as in
Section 3 when saying that a solution has period n): the upwards xed point (FP), the downwards
xed point (DFP), a positively rotating period 1 solution (PR), a negatively rotating period 1
solution (NR), a positively rotating period 2 solution (PR2), a negatively rotating period 2 solution
(NR2), an oscillating period 2 solution (DO2) and an oscillating period 4 solution (DO4). However,
as we will see, the solution DO2 deserves a separate, more detailed discussion.
(a)
(b)
(c)
Figure 13: Basins of attraction for constant dissipation with
= 0:2, 0:23 and 0:2725 from left to right
respectively. The xed point (FP) is shown in blue, the positively rotating solution (PR) in red, the
negatively rotating solution (NR) in yellow, the downwards oscillation with period 2 (DO2) in green and
nally the downwards oscillation with period 4 (DO4) in orange.
The basins of attraction corresponding to the values
= 0:2, 0:23 and 0:2725 are shown in Figure
13. The relative areas of the basins of attraction for
2[0:05;0:6] are listed in Table 11. Again the
positive and negative rotations have been listed together as any di erence in the size of their basins
of attraction is expected to be due to numerical inaccuracies.
In Figure 14, there is a large jump in the relative area of the basin of attraction for the upwards
xed point (FP) between the values of
= 0:22 and 0:225, approximately: this is due to the
appearance of the oscillatory solution which oscillates about the downwards pointing xed point
(=). For values of
slightly larger than 0 :22 large amounts of phase space move close to the
solution DO2, where they remain for long periods of time; however they do not land on the solution
19and are eventually attracted to FP. The percentage of phase space which does this is marked in
Figure 14 by a dotted line, which becomes solid when the trajectories remain on the solution for all
time (however, see comments below).
Basin of Attraction %
FP DFP PR/NR PR2/NR2 DO2 DO4
0.0500 4.30 0.00 0.00 47.85 0.00 0.00
0.0750 5.08 0.00 0.00 47.46 0.00 0.00
0.0900 7.41 0.00 0.00 46.30 0.00 0.00
0.1000 8.51 0.00 45.74 0.00 0.00 0.00
0.1700 49.65 0.00 25.17 0.00 0.00 0.00
0.2000 64.31 0.00 17.84 0.00 0.00 0.00
0.2230 72.09 0.00 13.95 0.00 0.00 0.00
0.2250 27.60 0.00 13.59 0.00 45.22 0.00
0.2300 25.00 0.00 12.68 0.00 49.61 0.00
0.2500 15.87 0.00 8.49 0.00 67.16 0.00
0.2690 16.50 0.00 2.13 0.00 79.25 0.00
0.2694 17.26 0.00 0.00 0.00 82.74 0.00
0.2700 17.28 0.00 0.00 0.00 82.73 0.00
0.2725 17.21 0.00 0.00 0.00 79.44 3.35
0.2800 17.30 0.00 0.00 0.00 82.70 0.00
0.2900 17.30 0.00 0.00 0.00 82.70 0.00
0.3000 16.97 0.00 0.00 0.00 83.03 0.00
0.4600 9.61 0.00 0.00 0.00 90.39 0.00
0.4700 9.80 90.20 0.00 0.00 0.00 0.00
0.5000 10.06 89.94 0.00 0.00 0.00 0.00
0.5500 10.32 89.68 0.00 0.00 0.00 0.00
0.6000 8.79 91.21 0.00 0.00 0.00 0.00
Table 11: Relative areas of the basins of attraction with constant damping coecient
. The solutions are
named as per Figures 12 and 13 with the addition of the downwards xed point (DFP) and the rotating
period 2 solutions (PR2/NR2). For details on the DO2 solution we refer to the text.
Figure 14: Relative sizes of basins of attraction with constant
as per Table 11. The lines are labeled as in
Table 11 and regions in which a bifurcation takes place are marked with a dot. The basin of attraction for
the oscillatory solution with period 4 (DO4), has not been included due to its small size and the solutions
low range of persistence with respect to
. The broken lines for FP and DO2 represent areas of transition
just before DO2 (and the solutions created by the period doubling bifurcation) becomes stable, see text.
20The solution DO4 listed in Table 11 is found to persist only in the interval [0 :272;0:27422], where
it only attracts a small amount of the phase space. As such it has not been included in Figure 14.
(a)
(b)
(c)
(d)
Figure 15: The transition from the period 2 rotating solutions to the period 1 rotating solutions.
= 0:09,
0.094, 0.096 and 0.098 from (a) to (d) respectively.
Numerical simulations can be used to nd estimates for the values of
at which solutions
appear/disappear. This is done by starting with initial conditions on a solution, then allowing the
parameter
to be varied to see for which value that solution vanishes. In Figure 15 the transition
from the period 2 rotating solution to the period 1 rotating solution can be observed: by moving
towards smaller values of
, this corresponds to a period doubling bifurcation [21, 13] (period halving,
if we think of
as increasing). Similarly, starting on the period 1 rotating attractor and increasing
further
, the solution disappears at
0:2694.
The same analysis can be done for the oscillatory solutions. We nd that the downwards oscilla-
tory solution labeled DO2 persists for
in the interval [0 :224;0:46], approximately. However such a
solution is really a period 2 solution only for
greater than
0:24. In the interval [0 :224;0:24] the
trajectory is \thick", see Figure 16: only due to its similarity to the solution DO2 and to prevent
Table 11 having yet more columns, the basin of attraction of these solutions in that range has also
been listed under that of DO2. Nevertheless, by moving
backwards starting from 0 :24 we have
a sequence of period doubling bifurcations, corresponding to values of
closer and closer to each
other. A period doubling cascade is expected to lead to a chaotic attractor, which, however, may
survive only for a tiny window of values of
(at
= 0:223 it has already de nitely disappeared)
and has a very small basin of attraction (for
getting closer to the value 0.223 its relative area
goes to zero). The appearance of chaotic attractors for small sets of parameters and with small
basins of attraction has been observed in similar contexts of multistable dissipative systems close
to the conservative limit [15]. For the value
= 0:223 numerical simulations nd that trajectories
remain in the region of phase space occupied by DO2 for a long time, before eventually moving onto
the xed point. As
increases further towards
0:46, the amplitude of the period 2 oscillatory
solution gradually decreases and taking
larger causes a slow spiral into the downwards xed point,
which now becomes stable.
(a)
(b)
(c)
(d)
Figure 16: The solution DO2 for di erent values of time independent
. As the damping coecient is
increased, the solution becomes more clearly de ned: this is due to a period halving bifurcation which stops
when the period becomes 2. The damping coecient is
= 0:23, 0.235, 0.239 and 0.24 from (a) to (d),
respectively. The position of the trajectory at every 2 , i.e the period of the forcing, is shown by circles.
214.1 Increasing dissipation
As mentioned in Section 3.1, as
increases from
0to
1it is expected that taking T0larger causes
the sizes of the basins of attraction to tend towards the sizes of the corresponding basins when
=
0, when the set of attractors remains the same for all values in between. If an attractor is
replaced by a new attractor (by bifurcation), then the new attractor inherits the basin of attraction
of the old one.
We shall begin by xing
1= 0:2 and
02[0:05;0:2], as for
=
1the basins of attraction are
not so sensitive to initial conditions, see Figure 13(a), and for
in that range the set of attractors
consists only of the the upwards xed point and two rotating solutions; moreover the pro les of the
corresponding relative areas plotted in Figure 14 are rather smooth and do not present any sharp
jumps.
Basin of Attraction %
FP PR/NR
T00 64.31 17.84
25 51.41 24.30
50 32.09 33.95
75 23.48 38.26
100 17.09 41.45
200 6.85 46.58
500 4.35 47.82
1000 4.15 47.92
1500 4.09 47.96
Table 12: Relative areas of the basins of
attraction with
0= 0:05,
1= 0:2 andT0
varying.
Figure 17: Plot of the relative areas of the basins of
attraction as per Table 12.
Basin of Attraction %
FP PR/NR
T00 64.31 17.84
25 49.06 25.47
50 38.54 30.73
75 31.25 34.37
100 25.67 37.16
150 18.12 40.94
200 14.14 42.93
500 9.81 45.09
1000 9.45 45.27
Table 13: Relative areas of the basins of
attraction with
0= 0:1,
1= 0:2 andT0
varying.
Figure 18: Plot of the relative areas of the basins of
attraction as per Table 13.
Tables 12, 13 and 14 show the relative area of each basin of attraction as
increases from 0.05,
0.1 and 0.17, respectively, to 0.2 with varying T0. It can be seen from the results in Tables 13 to 14
that the numerical simulations are in agreement with the above expectation. With the exception of
Table 12 the relative areas of the basins of attraction tend towards those when
is kept constant at
0. The exception of the case of Table 12 is due to the fact that the set of attractors has changed as
passes from 0 :05 to 0:2: the period 2 rotating solutions have been destroyed and replaced by the
period 1 rotating solutions. However, when the transition occurs, the new attractors are located in
22phase space very close to the previous ones and we nd that the initial conditions which were heading
towards or had indeed landed on the period 2 rotating solutions move onto the now present period
1 rotating solutions. On the other hand, when the damping coecient crosses the value
0:1,
the attractor undergoes topological changes, but, apart from that, the transition is rather smooth:
the location in phase space and the basin of attraction change continuously. In conclusion, we nd
that the relative areas of the basins of attraction for the two period 1 rotating attractors (PR/NR)
tend towards those the now destroyed period 2 rotating attractors (PR2/NR2) had at
=
0. As in
Section 3 we expect that the sizes of the basin of attraction at
=
0are recovered asymptotically
asT0!1 . Nevertheless, once more, the larger T0the smaller is the variation in the relative area:
for instance in Table 12 for T0= 100 the relative area of the basin of attraction of the xed point
has become nearly 1 =4 of the value for
= 0:2 constant, while in order to have a further reduction
by a factor of 4 one has to take T0= 1000.
Basin of Attraction %
FP PR/NR
T00 64.31 17.84
25 58.54 20.73
50 56.78 21.62
100 55.41 22.29
200 53.69 23.15
500 51.93 24.04
1000 50.80 24.60
1500 50.62 24.69
Table 14: Relative areas of the basins of
attraction with
0= 0:17,
1= 0:2 andT0
varying.
Figure 19: Plot of the relative areas of the basins of
attraction as per Table 14.
Basin of Attraction %
FP PR/NR DO2
T00 25.00 12.68 49.61
10 24.86 13.81 47.52
15 24.34 14.84 45.98
20 24.43 15.57 44.43
25 24.77 16.04 43.15
50 27.60 16.98 38.45
75 30.12 17.28 35.32
100 33.08 17.42 32.08
150 38.29 17.56 26.58
200 42.60 17.66 22.08
300 49.36 17.73 15.18
400 54.20 17.75 10.30
500 57.37 17.77 7.08
1000 63.39 17.78 1.06
1500 64.26 17.79 0.16
2000 64.38 17.80 0.03
Table 15: Relative areas of the basins of
attraction with
0= 0:2,
1= 0:23 and
T0varying.
Figure 20: Plot of the relative areas of the basins of
attraction as per Table 15.
23We now consider the case where either
0= 0:2 and
1= 0:23 or 0:2725 or
0= 0:23 and
1= 0:2725. Such values for
o er more complexities as not only are there more attractors to
consider, but one may have attractors (PR and NR) that are destroyed without leaving any trace.
When this happens it is not obvious which persisting attractor will inherit their basins of attraction.
The result of this could cause the nal basins of attraction to be drastically di erent from those for
constant
and even not monotonically increasing or decreasing as the value of T0is increased.
In Table 15 we see that initially, for values of T0not too large, the basin of attraction of FP
slightly reduces in size, while those of the rotating solutions PR/NR increase substantially. Instead,
for larger values of T0, the basin of attraction of FP increases appreciably, while those of PR/NR
increase very slowly. Apparently, the rotating solutions react more quickly as
is varied, attracting
phase space faster, so that the relative areas of their basins of attraction tend towards the values at
=
0for shorter initial times T0. It would be interesting to study further this phenomenon.
When
(t) varies from
= 0:2 to
= 0:2725 and from
= 0:23 to
= 0:2725, the rotating
solutions PR/NR disappear, so that their basins of attractions are absorbed by the persisting at-
tractors. In Table 16 one sees a very slow movement towards global attraction of the upwards xed
point, which is the only attractor persisting for both
0and
1. However even taking T0= 5000
is not enough for the asymptotic behaviour to be approached. The results in Table 17 show that,
by takingT0larger and larger, the relative areas of the basins of attraction of FP and DO4 both
tend to the values corresponding to
0= 0:23 (in particular the basin of attraction of DO4 becomes
negligible). Nearly all trajectories which were converging towards the rotating solutions before the
latter disappeared are attracted by the period two oscillations. This could be due to the fact that
DO2 is the closest attractor in phase space which persists at both
0and
1.
Basin of Attraction %
FP DO2 DO4
T00 17.21 79.44 3.35
25 18.63 78.33 3.04
50 20.32 79.29 0.39
75 23.38 76.52 0.12
100 25.71 74.27 0.02
150 28.45 71.55 0.01
200 30.92 69.08 0.00
300 35.70 64.30 0.00
400 39.82 60.18 0.00
500 42.76 57.24 0.00
980 54.19 45.81 0.00
990 54.39 45.81 0.00
995 54.30 45.70 0.00
1000 90.11 9.89 0.00
1005 54.58 45.43 0.00
1010 54.52 45.48 0.00
1020 90.34 9.66 0.00
1030 54.81 45.19 0.00
1050 55.13 44.87 0.00
1500 59.78 40.22 0.00
2000 62.12 37.88 0.00
3000 63.89 36.11 0.00
5000 64.29 35.71 0.00
Table 16: Relative areas of the basins of
attraction with
0= 0:2,
1= 0:2725 and
T0varying.
Figure 21: Plot of the relative areas of the basins of
attraction as per Table 16.
However, the more striking feature of Figures 21 and 22 are the jumps corresponding T0= 1000
24in the prior and T0= 100 and T0= 500 in the latter. Moreover such jumps are very localised: for
instance in Figure 22 for T0= 99 andT0= 100 the basins of attraction of FP and DO2 are found
to be about 44% and 56%, respectively, whereas by slightly increasing or decreasing T0they settle
around 20% and 80%. The quantity of phase space exchanged in these instances is roughly equal
to that attracted to the rotating solutions for
0. For particular values of T0when the rotating
attractors disappear their trajectories move to the upwards xed point rather than the period 2
oscillations. The reason for this to happen is not clear. Moreover, note that in principle there could
be other jumps, corresponding to values of T0which have not been investigated: however, it seems
hard to make any prediction as far as it remains unclear how the disappearing basins of attractions
are absorbed by the persisting ones.
Basin of Attraction %
FP DO2 DO4
T00 17.21 79.44 3.35
25 17.96 79.45 2.60
50 16.04 83.36 0.60
75 18.39 80.27 1.34
90 19.56 80.38 0.05
95 20.05 79.91 0.04
97 20.08 79.89 0.04
98 20.01 79.96 0.03
99 44.14 55.83 0.03
100 43.96 56.01 0.03
101 20.54 79.42 0.03
105 20.40 79.57 0.03
110 20.79 79.19 0.02
125 21.55 78.45 0.01
150 22.46 77.54 0.00
200 23.33 76.67 0.00
300 24.06 75.94 0.00
400 24.36 75.64 0.00
490 24.54 75.46 0.00
500 49.45 50.55 0.00
510 24.42 75.58 0.00
1000 24.81 75.19 0.00
Table 17: Relative areas of the basins of
attraction with
0= 0:23,
1= 0:2725 and
T0varying.
Figure 22: Plot of the relative areas of the basins of
attraction as per Table 17.
4.2 Decreasing dissipation
Tables 18, 19, 20 and 21 and the corresponding Figures 23, 24, 25 and 26 illustrate the cases when
dissipation decreases over an initial period of time T0. We have considered the cases with
0= 0:23,
0:02725 and
1= 0:2, with
0= 0:2725 and
1= 0:23 and with
0= 0:3 and
1= 0:2725.
In particular they show that if the set of attractors at
=
1is a proper subset of the set of
attractors which exist at
=
0, then, asT0!1 , the basin of attraction of each attractor which
exists at
1turns out to have a relative area which tend to be greater than or equal to that found
for
=
0. In Table 18 we consider the situation in which the attractor DO2, which has a large
basin of attraction for
0= 0:23, is no longer present when
(t) has reached the nal value
1= 0:2:
as a consequence the trajectories which would be attracted by DO2 at
=
0end up onto the other
attractors: in fact most of them are attracted by the xed point.
25Basin of Attraction %
FP PR/NR
T00 64.31 17.84
25 70.69 14.65
50 72.57 13.72
75 73.24 13.38
100 73.57 13.22
200 74.10 12.95
500 74.42 12.79
Table 18: Relative areas of the basins of
attraction with
0= 0:23,
1= 0:2 andT0
varying.
Figure 23: Plot of the relative areas of the basins of
attraction as per Table 18.
Basin of Attraction %
FP PR/NR
T00 64.31 17.84
5 65.80 17.10
10 69.52 15.24
15 74.40 12.80
20 77.90 11.05
25 80.38 9.81
50 86.22 6.89
75 88.64 5.68
100 90.07 4.97
200 92.79 3.61
500 95.92 2.04
1000 99.34 0.33
1500 99.35 0.32
Table 19: Relative areas of the basins of
attraction with
0= 0:2725,
1= 0:2 and
T0varying.
Figure 24: Plot of the relative areas of the basins of
attraction as per Table 19.
Basin of Attraction %
FP PR/NR DO2
T00 25.00 12.68 49.61
25 24.73 7.44 60.40
50 19.67 5.35 69.63
75 17.13 4.44 73.99
100 16.42 3.88 75.83
200 16.29 2.72 78.28
500 17.03 0.76 81.45
Table 20: Relative areas of the basins of
attraction with
0= 0:2725,
1= 0:23 and
T0varying.
Figure 25: Plot of the relative areas of the basins of
attraction as per Table 20.
We also notice the interesting features in Table 19: as the xed point is the only attractor which
26exists for both
0and
1, we nd that as T0increases its basin of attraction tends towards 100%,
which corresponds to attraction of the entire phase space, up to a zero-measure set. This happens
despite the fact that
(t) does not pass through any value for which global attraction to the xed
point is satis ed. It also suggests that it is possible to provide conditions on the intersection of the
two setsA0andA1of the attractors corresponding to
0and
1, respectively, in order to obtain
that all trajectories move towards the same attractor when the time T0over which
(t) is varied is
suciently large. In particular, it is remarkable that it is possible to create an attractor for almost
all trajectories by suitably tuning the damping coecient as a function of time.
In Table 20, the relative areas of the rotating solutions, which are absent at
=
0, tend to
become negligible when T0is large. Similarly, in Table 21, the basin of attraction of the period
4 oscillating attractor, which exists only for the nal value
1of the damping coecient, tends
to disappear when T0is taken large enough. This con rms the general expectation: the basin of
attraction of the disappearing attractor is absorbed by the closer attractor, that is the solution DO2
in this case.
Basin of Attraction %
FP DO2 DO4
T00 17.21 79.44 3.35
25 17.15 80.99 1.86
50 16.83 83.03 0.14
75 16.75 83.24 0.01
100 16.79 83.21 0.00
200 16.81 83.19 0.00
500 16.80 83.20 0.00
Table 21: Relative areas of the basins of
attraction with
0= 0:3,
1= 0:2725 and
T0varying.
Figure 26: Plot of the relative areas of the basins of
attraction as per Table 21.
5 Numerical Methods
The two main numerical methods implemented for the simulations throughout were a variable order
Adams-Bashforth-Moulton method and the method of analytic continuation [10, 23, 12, 17, 34, 36];
the latter consists in a numerical implementation of the Frobenius method. Also used to check
the results was a Runge-Kutta method. The Adams-Bashforth-Moulton integration scheme used is
the built in integrator found in matlab , ODE113, whereas the programs based on the method of
analytic continuation and Runge-Kutta scheme were written in C. Of the three methods, the slowest
was the Adams-Bashforth-Moulton method, however it was found that the method worked well for
the system with the chosen parameter values and the results produced were reliable.
Both the Adams-Bashforth-Moulton and Runge-Kutta methods are standard methods for solving
ODEs of this type. When implementing these two integrators to calculate the basins of attraction,
two di erent methods for both choosing initial conditions in phase space and classifying attractors
were used. The rst method for picking initial conditions was to take a mesh of equally spaced points
in the phase space: this method ensures uniform coverage of the phase space. When taking this
approach a mesh of either 321 141 or 503 289 points was used depending on the desired accuracy. The
second method was to take random initial conditions: this can be done by choosing initial conditions
from a stream of random points, which allows the user to use the same random initial conditions
in each simulation if required. This method is often preferred as the accuracy of the estimates
of the relative areas of the basins of attraction compared to the number of initial conditions used
can be calculated [29]; also it is easier to run additional simulations for extra random points to
improve estimates later on, when needed. When using random points, the number of points used
to calculate the basins of attraction was 300 000 or 400 000 depending on the expected complexity
27of the system under given parameters. In some cases where extra accuracy was required due to
some attractors having particularly small basins of attraction, additional 200 000 or 300 000 random
points were used. This allowed us to obtain an error less than 0.20 on the relative areas of the basins
of attraction. In the most delicate cases, where more precise estimates were needed to distinguish
between values very close to each other (for instance in Table 4), the error was made smaller by
increasing the number of points. We decided to express in all cases the relative areas up to the
second decimal digit because often further increasing the number of points did not alter appreciably
that digit.
To detect and classify solutions, two methods can be used. The rst method consists in nding
all attractors as a rst step, before computing the corresponding basins of attraction: this required
a complete characterisation of both the period and the location in phase space of the attractors. In
principle, this works very well, but has the downside of having to nd initially all the attractors,
and di erentiate between those that occupy the same region in phase space. The second method
for classifying the solutions was to create a library of solutions. This was created as the program
ran and built up as new solutions were found. The solution of each integration was then checked
against the library and, if not already known, was added. In this way, the program nds solutions
as it goes, so has the advantage of the user not having to know the existing solutions in the system
prior to calculating the basins of attraction.
The method of analytic continuation was implemented and produced results very similar to those
of the Adams-Bashforth-Moulton method, but in general was much quicker to run. When imple-
menting this method of integration, we only used random initial conditions in the phase space and
the library method for classifying solutions. The reason for using the Adams-Bashforth-Moulton
method, despite being the slowest of the three integrators, was for comparison with the method of
analytic continuation. Analytic continuation has not previously been used for numerically integrat-
ing an ODE of the form (1.2), that is an equation with an in nite polynomial nonlinearity that
satis es an addition formula, thus it was good to have a reliable method to check results with. The
method for using analytic continuation for integrating ODE's that have in nite polynomial nonlin-
earity that satisfy an addition formula will be described more extensively in [36]. The similarity in
results of these completely di erent methods for integration, initial condition selection and solution
classi cation provides reassurance and con dence that the results produced are accurate.
6 Concluding remarks
In this work we have numerically shown the importance of not only the nal value of dissipation but
its entire time evolution, for understanding the long time behaviour of the pendulum with oscillating
support. This extends the work done in [3] to a system which, even for values of the parameters
in the perturbation regime, exhibits richer and more varied dynamics, due to the presence of the
separatrix in the phase space of the unperturbed system. In addition we have considered also values
of the parameters beyond the perturbation regime (the inverted pendulum), where the system cannot
be considered a perturbation of an integrable one. In particular this results in a more complicated
scenario, with bifurcation phenomena and the appearance of attractors which exist only for values
of the damping coecient
in nite intervals away from zero, say
2[
1;
2], with
1>0.
We have preliminarily studied the behaviour of the system in the case of constant dissipation.
Firstly, in the perturbation regime, we have analytically computed to rst order the threshold values
below which the periodic attractors exist. We have also discussed why this approach fails due to
intrinsic perturbation theory limitations, in particular why the method cannot be applied to stable
cases of the upwards con guration or to solutions too close to the unperturbed separatrix. Next,
we have studied numerically the dependence of the sizes of the basins of attraction on the damping
coecient.
Then we have explicitly considered the case of damping coecient varying monotonically between
two values and outlined a few expectations for the way in which the basins of attraction accordingly
change with respect to the case of constant dissipation. These expectations were later illustrated
and backed up with numerical simulations: in particular the relevance of the study of the dynamics
28at constant dissipation was argued at length. While the expectations account for many features
observed numerically, there are still some facts which are dicult to explain, even at an heuristic
level, and which would deserve further investigation, such as the relationship between the xed point
solution and the oscillating attractors, to better understand why in some cases they exchange large
areas of their basins of attraction when the damping coecient varies in time. More generally, an
in-depth numerical study of the system with constant dissipation, also for other parameter values,
would be worthwhile. In particular it would be interesting to perform a more detailed bifurcation
analysis with respect to the parameter
and to study the system for very small values of the damping
coecient
(which relates to the spin-orbit model in celestial mechanics), both in the perturbation
regime and for large values of the forcing amplitude. We think that, in order to study cases with
very small dissipation, the method of analytical continuation brie
y described in Section 5 could be
particularly fruitful.
Some interesting features appeared in our analysis which would deserve further consideration
are:
the increase in the basin of attraction of the xed point observed in Figure 3 when the damping
coecient becomes small enough;
the appearance of the period 4 solution for a thin interval of values of the damping coecient,
as emerges in Table 11;
the rate at which the values of the relative areas of the basins attraction corresponding to the
initial value
0of
(t) are approached as the variation time T0increases;
the way in which the basins of attraction of the disappearing attractors distribute among the
persisting ones;
the oscillations through which the relative areas of the basins of attraction approach the
asymptotic value when taking larger and larger values of the variation time T0, as observed
for instance in Tables 16 and 17;
the jumps corresponding to T0= 1000 in Figure 21 and T0= 100,T0= 500 in Figure 22;
the computation of the threshold values to second order, so as to include the rotating solutions
found numerically in the perturbation regime investigated in Section 3.
Finally, investigating analogous systems such as the pendulum with periodically varying length
to see if similar dynamics occur would also be fascinating in its own right. Another interesting model
to investigate further, especially in the case of very small dissipation, is the spin-orbit model already
considered in [3], which is expected to be of relevance to understand the locking into the resonance
3 : 2 of the system Mercury-Sun.
Acknowledgements
The Adams-Bashforth-Moulton method used was MATLAB 's ODE113. We thank Jonathan Deane
for helpful conversations on analytic continuation and support with coding in C. This research was
completed as part of an EPSRC funded PhD.
A Global attraction to the two xed points
To compute the conditions for attraction to the origin we use the method outlined in [4]; see also
[2]. We de ne f( ) as in (1.3) and require f( )>0: the consequences of this restriction are that
the method can only be applied to the downwards pointing pendulum when > . Then we apply
the Liouville transformation
~ =Z
0p
f(s)ds (A.1)
29and write our equation (1.3) in terms of the new time ~ as
~ ~ +0
@~f(~ )~
2~f(~ )+
q
~f(~ )1
A~ + sin= 0; (A.2)
where the subscript ~ represents derivative with respect to the new time ~ and ~f(~ ) :=f( ). This
can be represented as the two-dimensional system on TR, by setting x(~ ) =(~ ) and writing
x~ =y; y ~ = yq
~f0
@~f~
2q
~f+
1
A sinx; (A.3)
for which we have the energy E(x;y) = 1 cosx+y2=2. By setting H(~ ) =E(x(~ );y(~ )), one nds
H~ = y2
q
~f0
@~f~
2q
~f+
1
A; (A.4)
thusH~ 0, i.ex,yare bounded given that
satis es
> min
~ 0~f~
2q
~f= min
0f0
2f: (A.5)
Moreover we have that for all ~ >0
H(~ ) +Z~
0y2
q
~f0
@~f0
2q
~f+
1
Ads=H(0); (A.6)
so that, as ~ !1 , using the properties above we can arrive at
min
s02
41q
~f0
@~f~
2q
~f+
1
A3
5Z1
0y2(s)ds<1: (A.7)
Hencey!0 as time tends to in nity. There are two regions of phase space to consider. Any level
curve ofHstrictly inside the separatrix of the unperturbed pendulum is the boundary of a positively
invariant set Dcontaining the origin: since S=f(x(~ );y(~ )) :H~ = 0g[Dconsists purely of the
origin, we can apply the local Barbashin-Krasovsky-La Salle theorem [24] to conclude that every
trajectory that begins strictly inside the separatrix will converge to the origin as ~ !+1.
Outside of the separatrix we may use equation (A.4), which shows the energy to be strictly
decreasing while y6= 0, provided
is chosen large enough, coupled with y!0 as time tends to
in nity. The result is that all trajectories tend to the invariant points on the x-axis as time tends
to in nity. One of two cases must occur: either the trajectory moves inside the separatrix or it does
not. In the rst instance we have already shown that the limiting solution is the origin. In the latter
there is only one possibility. As all points on the x-axis are contained within the separatrix other
than the unstable xed point, the trajectory must move onto such a xed point and hence belongs
to its stable manifold, which is a zero-measure set. Therefore we conclude that a full measure set
of initial conditions are attracted by the origin. Reverting back to the original system with time ,
we conclude that for that system too the basin of attraction of the origin has full measure, provided
< and
satis es (A.5).
30B Action-angle variables
In this section we detail the calculation of the action-angle variables for the simple pendulum in
time . More details on calculating action-angle variables can be found in [14, 32, 9]. The simple
pendulum has equation of motion given by
00+ sin= 0; (B.1)
where the dashes represent derivative with respect to the scaled time . The Hamiltonian for the
simple pendulum in this notation is
E=H(;0) =1
2(0)2 cos (B.2)
or, in terms of the usual notation for Hamiltonian dynamics,
E=H(p;q) =1
2p2 cosq; (B.3)
whereq=andp=q0=0. Rearranging this for pwe obtainp=p(E;q), with
p(E;q) =p
2(E+ cosq) =p
2 (E0+ cosq); (B.4)
whereE0=E= . It is clear that there are two types of dynamics, oscillatory dynamics when
E0<1 and rotational dynamics when E0>1, separated at a separatrix when E0= 1, for which no
action-angle variables exist.
B.1 Librations
We rst consider the case E0<1. The action variable is
I=1
2I
pdq=2
p
2 Zq1
0p
E0+ cosqdq=8
p h
(k2
1 1)K(k1) +E(k1)i
; (B.5)
wherek2
1= (E0+1)=2 andq1= arccos( E0). The functions K(k) and E(k) are the complete elliptic
integrals of the rst and second kinds respectively.
The angle variable 'can be found as follows
'0=@H
@I=dE
dI=dI
dE 1
; (B.6)
so that
dI
dE=d
dE2
Zq1
0p
E+ cosqdq=2
p K(k1): (B.7)
Hence we have
'( ) =
2K(k1)p ( 0): (B.8)
Takes= sin (q=2); then using equation (B.2) it is easy to show that
(s0)2=g
l(1 s2)
k2
1 s2
: (B.9)
Integrating using the Jacobi elliptic functions
s( ) =k1snrg
l( 0);k1
; (B.10)
the expresssion can then be rearranged to achieve the following result:
q= 2 arcsin
k1sn2K(k1)
';k1
; p = 2k1p cn2K(k1)
';k1
; (B.11)
which coincide with equations (2.4). By using (E.3) in Appendix E, one obtains from (B.5)
@I
@k1=8
k1K(k1)p ; (B.12)
a relation which has been used to derive (2.12).
31B.2 Rotations
In the case of rotational dynamics we have
I=1
2Z2
0pdq=1
2p Z2
0p
E0+ cosqdq=4
k2p E(k2); (B.13)
where this time we let k2
2= 2=(E0+ 1) = 1=k2
1. The angle variable 'can similarly be found using
(B.6), where d I=dEcan be similarly calculated as
dI
dE=d
dE1
2Z2
0p
E+ cosqdq=k2
p K(k2); (B.14)
which hence gives
'( ) =
K(k2)p ( 0)
k2: (B.15)
Using (B.9) and the de nition of k2, for the rotating solutions we nd that
s( ) = snp ( 0)
k2;k2
; (B.16)
and similarly, by simple rearrangement, we nd that
q= 2 arcsin
snK(k2)
';k2
; p =2
k2p dnK(k2)
';k2
; (B.17)
which again yields equations (2.6) By using (E.3) in Appendix E, one obtains from (B.13)
@I
@k2= 4
k2
2K(k2)p : (B.18)
which has been used to derive (2.23).
C Jacobian determinant
Here we compute the entries of the Jacobian matrix Jof the transformation to action-angle variables,
which will be used in the next Appendix. As a by-product we check that Jdeterminant equal to 1,
that is@q
@'@p
@I @q
@I@p
@'= 1: (C.1)
For further details on the proof of (C.1) we refer the reader to [9], where the calculations are given
in great detail. The derivative with respect to 'is straightforward in both the libration and rotation
case, however the dependence of pandqon the action Iis less obvious. That said, the dependence of
pandqonk1in the oscillating case and k2in the rotating case is clear and we know the relationship
betweenIandkin both cases, hence the derivative of the Jacobi elliptic functions can be calculated
by using that
@
@I=@k
@I@
@k+@u
@I@
@u=@k
@I@
@k+@u
@k@
@u
; (C.2)
whereuis the rst argument of the functions, i.e sn( u;k), etc. Then for the oscillations we have
@q
@I=
4k1K(k1)p sn()
dn()+2E(k1)'cn()
k02
1+k2
1sn() cn2()
k02
1dn() E() cn()
k02
1
;
@p
@I=
4k1K(k1)
cn() 2E(k1)'sn() dn()
k02
1 k2
1sn2() cn()
k02
1+E() sn() dn()
k02
1
;
@q
@'=4k1K(k1) cn()
;
@p
@'= p 4k1K(k1) sn() dn()
;(C.3)
32where () =
2K(k1)'
;k1
andk0
1=p
1 k2
1. From the above it is easy to check that equation (C.1)
is satis ed. Similarly, for the rotations we have
@q
@I= k2
2
2K(k2)p 'E(k2) dn()
k2k02
2+k2sn() cn()
k02
2 E() dn()
k2k02
2
;
@p
@I=k2
2
2K(k2)dn()
k2
2+'E(k2) sn() cn()
k02
2+sn2() dn()
k02
2 E() sn() cn()
k02
2
;
@q
@'=2K(k2) dn()
;
@p
@'= p 2k2K(k2) sn() cn()
;(C.4)
where () =
K(k2)
';k2
andk0
2=p
1 k2
2. It is once again easily checked from the above that
(C.1) is satis ed.
D Equations of motion for the perturbed system
By (C.1) one has@'=@q @'=@p
@I=@q @I=@p
=@p=@I @q=@I
@p=@' @q=@':
(D.1)
We rewrite the equation (1.3) in the action-angle coordinates introduced in Appendix B as follows.
D.1 Librations
In this section we want to write (1.3) in terms of the action-angle introduced in Appendix B. By
taking into account the forcing term cos cosin (1.3) one nds
I0=@I
@qq0+@I
@pp0= @p
@'q0+@q
@'p0
=8 k2
1K(k1)
cos( 0) sn() cn() dn();
'0=@'
@qq0+@'
@pp0=@p
@Iq0 @q
@Ip0
=p
2K(k1)
2K(k1)p
sn2() +2E(k1)'sn() cn() dn()
k02
1
+k2
1sn2() cn2()
1 k2
1 E() sn() cn() dn()
1 k2
1
cos( 0);(D.2)
where we have used the properties of the Jacobi elliptic functions in Appendix E. As in Appendix
C, we are shortening ( ) =
2K(k1)'
;k1
.
We then wish to add the dissipative term
0. This results in the following equations:
I0=8 k2
1K(k1)
cos( 0) sn() cn() dn() 8
k2
1p K(k1)
cn2();
'0=p
2K(k1)
2K(k1)p
sn2() +2E(k1)'sn() cn() dn()
(1 k2
1)
+k2
1sn2() cn2()
1 k2
1 E() sn() cn() dn()
1 k2
1
cos( 0)
+
cn()
2K(k1)sn()
dn()+2E(k1)'cn()
(1 k2
1)+k2
1sn() cn2()
(1 k2
1) dn() E() cn()
1 k2
1
:(D.3)
Using the property that, see [26], E(u;k) =E(k)u=K(k) +Z(u;k) we arrive at equations (2.9). The
function Z(u;k) is the Jacobi zeta function, which is periodic with period 2 K(k) inu.
33D.2 Rotations
The presence of the forcing term leads to te equations
I0= @p
@'q0+@q
@'p0=4 K(k2)
cos( 0) sn() cn() dn();
'0=@p
@I_q @q
@I_p=p
k2K(k2)+k2 p K(k2)E(k2)'sn() cn() dn()
(1 k2
2)
+k2
2sn2() cn2()
1 k2
2 E() sn() cn() dn()
1 k2
2
cos( 0):(D.4)
Again, if we wish to add a dissipative term, we arrive at the equations
I0=4 K(k2)
cos( 0) sn() cn() dn() 4
p K(k2)
k2dn2();
'0=p
k2K(k2)+k2 p K(k2)E(k2)'sn() cn() dn()
(1 k2
2)
+k2
2sn2() cn2()
1 k2
2 E() sn() cn() dn()
1 k2
2
cos( 0)
K(k2)E(k2)'dn2()
(1 k2
2)+k2
2sn() cn() dn()
1 k2
2 E() dn2()
1 k2
2
:(D.5)
Again using that E(u;k) =E(k)u=K(k) +Z(u;k) we arrive at the equations (2.22).
E Useful properties of the elliptic functions
The complete integrals of the rst and second kind are, respectively,
K(k) =Z=2
0d p
1 k2sin2 ; E(k) =Z=2
0d q
1 k2sin2 ; (E.1)
whereas the incomplete elliptic integral of the second kind is
E(u;k) =Zsn(u;k)
0dxp
1 k2x2
p
1 x2: (E.2)
One has
@K(k)
@k=1
kE(k)
1 k2 K(k)
;@E(k)
@k=1
k(E(k) K(k)): (E.3)
The following properties of the Jacobi elliptic functions have been used in the previous sections.
The derivatives with respect to the rst arguments are
@
@usn(u;k) = cn(u;k) dn(u;k);
@
@ucn(u;k) = sn(u;k) dn(u;k);
@
@udn(u;k) = k2sn(u;k) cn(u;k);(E.4)
while the derivatives with respect to the elliptic modulus are
@
@ksn(u;k) =u
kcn(u;k) dn(u;k) +k
k02sn(u;k) cn2(u;k) 1
kk02E(u;k) cn(u;k) dn(u;k);
@
@kcn(u;k) = u
ksn(u;k) dn(u;k) k
k02sn2(u;k) cn(u;k) +1
kk02E(u;k) sn(u;k) dn(u;k);
@
@kdn(u;k) = kusn(u;k) cn(u;k) k
k02sn2(u;k) dn(u;k) +k
k02E(u;k) sn(u;k) cn(u;k);(E.5)
34wherek02= 1 k2.
Finding the value of for rotations in Section 2 requires use of
Zx1
0dn2(x;k) dx=Zsn(x1;k)
0p
1 k2^x2
p
1 ^x2d^x=E(x1;k): (E.6)
In the case of librations we also require the relation k2cn2() + (1 k2) = dn2().
The integral for 1( 0;p;q) in equation (2.20) is found by
1( 0;p;q) =1
TZT
0sn(p ) cn(p ) dn(p ) cos( 0) d
=cos( 0)
TZT
0sn(p ) cn(p ) dn(p ) cos( ) d
+sin( 0)
TZT
0sn(p ) cn(p ) dn(p ) sin( ) d
=sin( 0)
TZT
0sn(p ) cn(p ) dn(p ) sin( ) d ;(E.7)
whereT= 2q= 4K(k1)p.
The Jacobi elliptic functions can be expanded in a Fourier series as
sn(u;k) =2
kK(k)1X
n=1qn 1=2
1 q2n 1sin(2n 1)u
2K(k)
;
cn(u;k) =2
kK(k)1X
n=1qn 1=2
1 +q2n 1cos(2n 1)u
2K(k)
;
dn(u;k) =
2K(k)+2
K(k)1X
n=1qn
1 q2ncos2nu
2K(k)
;(E.8)
where qis the nome, de ned as
q= exp
K(k0)
K(k)
;
withk0=p
1 k2.
In the calculation of hR(n)iforn2, when the pendulum is in libration, we require the evaluation
of the integrals
1
TZT
02K(k1)p @
@
cn2(p )
d = 0; (E.9)
and
1
TZT
02K(k1)p @
@
sn(p ) cn(p ) dn(p )
cos( 0) d
=1
TZT
02K(k1)p sn(p ) cn(p ) dn(p ) sin( 0) d
=cos( 0)
TZT
02K(k1)p sn(p ) cn(p ) dn(p ) sin( ) d
sin( 0)
TZT
02K(k1)p sn(p ) cn(p ) dn(p ) cos( ) d ;(E.10)
whereT= 2q= 4K(k1)p. The integral multiplying sin( 0) vanishes due to parity and hence
2K(k1)p TZT
0@
@
sn(p ) cn(p ) dn(p )
cos( 0) d =2K(k1)p cos( 0)G1(p;q):(E.11)
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37 |
1012.4455v1.Global_attractors_for_the_one_dimensional_wave_equation_with_displacement_dependent_damping.pdf | arXiv:1012.4455v1 [math.AP] 20 Dec 2010GLOBAL ATTRACTORS FOR THE ONE DIMENSIONAL WAVE EQUATION
WITH DISPLACEMENT DEPENDENT DAMPING
A.KH.KHANMAMEDOV
Abstract. We study the long-time behavior of solutions of the one dimen sional wave equation
with nonlinear damping coefficient. We prove that if the dampi ng coefficient function is strictly
positive near the origin then this equation possesses a glob al attractor.
1.INTRODUCTION
In this paper, we consider the following Cauchy problem:
utt+σ(u)ut−uxx+λu+f(u) =g(x),(t,x)∈(0,∞)×R, (1.1)
u(0,x) =u0(x), u t(0,x) =u1(x), x ∈R, (1.2)
whereλis a positive constant, g∈L1(R)+L2(R) and nonlinear functions f(·) andσ(·) satisfy the
following conditions:
f∈C1(R),f(u)u≥0,/notturnstileu∈R, (1.3)
σ∈C(R),σ(0)>0,σ(u)≥0,/notturnstileu∈R. (1.4)
Applying standard Galerkin’s method and using techniques of [6, Pro position 2.2], it is easy to
prove the following existence and uniqueness theorem:
Theorem 1. Assume that the conditions (1.3)-(1.4) hold. Then for any T >0and(u0,u1)∈
H:=H1(R)×L2(R)the problem (1.1)-(1.2) has a unique weak solution u∈C([0,T];H1(R))∩
C1([0,T];L2(R))∩C2([0,T];H−1(R))on[0,T]×Rsuch that
/bardbl(u(t),ut(t))/bardblH≤c(/bardbl(u0,u1)/bardblH),/notturnstilet≥0,
wherec:R+→R+is a nondecreasing function. Moreover if v∈C([0,T];H1(R))∩C1([0,T];L2(R))∩
C2([0,T];H−1(R))is also weak solution to (1.1)-(1.2) with initial data (v0,v1)∈ H, then
/bardblu(t)−v(t)/bardblL2(R)+/bardblut(t)−vt(t)/bardblH−1(R)≤
≤/tildewidec(T,/tildewideR)/parenleftBig
/bardblu0−v0/bardblL2(R)+/bardblu1−v1/bardblH−1(R)/parenrightBig
,/notturnstilet∈[0,T],
where/tildewidec:R+×R+→R+is a nondecreasing function with respect to each variable an d/tildewideR=
max{/bardbl(u0,u1)/bardblH,/bardbl(v0,v1)/bardblH}.
Thus, by the formula ( u(t),ut(t)) =S(t)(u0,u1), the problem (1.1)-(1.2) generates a weak con-
tinuous semigroup {S(t)}t≥0inH, whereu(t,x) is a weak solution of (1.1)-(1.2), determined by
Theorem 1.1, with initial data ( u0,u1).
The attractors for equation (1.1) in the finite interval were studie d in [2], assuming positivity of
σ(·). For two dimensional case, the attractors for the wave equatio n with displacement dependent
damping were investigated in [7] under conditions
σ∈C1(R), 0< σ0≤σ(u)≤c(1+|u|q),/notturnstileu∈R, 0≤q <∞,
and
|σ′(u)| ≤c[σ(u)]1−ε,/notturnstileu∈R, 0< ε <1, (1.5)
on the damping coefficient. Recently, in [3], condition (1.5) has been im proved as
|σ′(u)| ≤cσ(u),/notturnstileu∈R.
2000Mathematics Subject Classification. 35B41, 35L05.
Key words and phrases. Attractors, wave equations.
12 A.KH.KHANMAMEDOV
For the three dimensional bounded domain case, the existence of a global attractor for the wave
equation with displacement dependent damping was proved in [6] when σ(·) is a strictly positive
and globally bounded function. In this case, when σ(·) is not globally bounded, but is equal to a
positive constant in a large enough interval, the existence of a globa l attractor has been established
in [4].
In the articles mentioned above, the existence of global attracto rs was proved under positivity or
strict positivity condition on the damping coefficient function σ(·). In this paper, we study a global
attractor for (1.1)-(1.2) under weaker conditions on σ(·) and prove the following theorem:
Theorem 2. Under conditions (1.3)-(1.4) a semigroup {S(t)}t≥0generated by (1.1)-(1.2) possesses
a global attractor in H.
2.Proof of Theorem 1.2
To prove this theorem we need the following lemma:
Lemma 1. Let conditions (1.3)-(1.4) hold and let Bbe a bounded subset of H. Then for any ε >0
there exist T0=T0(ε,B)>0andr0=r0(ε,B)>0such that
/bardblS(t)ϕ/bardblH1(R\(−r0,r0))×L2(R\(−r0,r0))< ε,/notturnstilet≥T0,/notturnstileϕ∈B. (2.1)
Proof.Let (u0,u1)∈BandS(t)(u0,u1) = (u(t),ut(t)). Multiplying (1.1) by utand integrating
over (0,t)×Rwe obtain
/bardblut(t)/bardbl2
L2(R)+/bardblu(t)/bardbl2
H1(R)+t/integraldisplay
0/integraldisplay
Rσ(u(τ,x))u2
t(τ,x)dxdτ≤c1,/notturnstilet≥0.(2.2)
Letη∈C1(R), 0≤η(x)≤1,η(x) =/braceleftbigg0,|x| ≤1
1,|x| ≥2,ηr(x) =η(x
r) and Σ( u) =u/integraltext
0σ(s)ds.
Multiplying (1.1) by η2
rΣ(u), integrating over (0 ,t)×Rand taking into account (2.2) we have
t/integraldisplay
0/integraldisplay
Rη2
r(x)σ(u(τ,x))u2
x(τ,x)dxdτ+λt/integraldisplay
0/integraldisplay
Rη2
r(x)Σ(u(τ,x))u(τ,x)dxdτ≤
≤c2(1+√
t+t
r+t/bardblg/bardblL1(R\(−r,r))+L2(R\(−r,r))),/notturnstilet≥0,/notturnstiler >0. (2.3)
By (1.4), there exists l >0, such that
σ(0)
2≤σ(s)≤2σ(0),/notturnstiles∈[−l,l]. (2.4)
Using embedding H1
2+ε(R)⊂L∞(R) and taking into account (2.2) and (2.4) we find
t/integraldisplay
0/integraldisplay
Rη2
r(x)u2(τ,x)dxdτ≤2
σ(0)t/integraldisplay
0/integraldisplay
{x:|u(τ,x)|≤l}η2
r(x)Σ(u(τ,x))u(τ,x)dxdτ+
+c3t/integraldisplay
0/integraldisplay
{x:|u(τ,x)|>l}η2
r(x)|u(τ,x)|dxdτ≤2
σ(0)t/integraldisplay
0/integraldisplay
{x:|u(τ,x)|≤l}η2
r(x)Σ(u(τ,x))u(τ,x)dxdτ+
+2c3
σ(0)lt/integraldisplay
0/integraldisplay
{x:|u(τ,x)|>l}η2
r(x)Σ(u(τ,x))u(τ,x)dxdτ
and consequently
t/integraldisplay
0/bardblηru(τ)/bardbl5
L∞(R)dτ≤c4t/integraldisplay
0/bardblηru(τ)/bardbl2
L2(R)dτ≤c5t/integraldisplay
0/integraldisplay
Rη2
r(x)Σ(u(τ,x))u(τ,x)dxdτ, (2.5)GLOBAL ATTRACTORS 3
forr≥1. So by (2.2), (2.3) and (2.5), we get
t/integraldisplay
0/bracketleftbigg/vextenddouble/vextenddouble/vextenddoubleη2rσ1
2(u(τ))ut(τ)/vextenddouble/vextenddouble/vextenddouble2
L2(R)+/vextenddouble/vextenddouble/vextenddoubleη2rσ1
2(u(τ))ux(τ)/vextenddouble/vextenddouble/vextenddouble2
L2(R)+λ/vextenddouble/vextenddouble/vextenddoubleη2rσ1
2(u(τ))u(τ)/vextenddouble/vextenddouble/vextenddouble2
L2(R)+
+/bardblηru(τ)/bardbl5
L∞(R)/bracketrightBig
dτ≤c6(1+√
t+t
r+t/bardblg/bardblL1(R\(−r,r))+L2(R\(−r,r))),/notturnstilet≥0,/notturnstiler≥1.(2.6)
Now denote Φ r(u(t)) :=1
2/bardblηrut(t)/bardbl2
L2(R)+1
2/bardblηrux(t)/bardbl2
L2(R)+µ/an}bracketle{tηrut(t), ηru(t)/an}bracketri}ht+λ
2/bardblηru(t)/bardbl2
L2(R)+
/an}bracketle{tηrF(u(t)), ηr/an}bracketri}ht+/an}bracketle{tηrg, ηru(t)/an}bracketri}ht, where µ= min/braceleftbigg/radicalBig
λ
2,σ(0)
5,λ
2σ(0)/bracerightbigg
,/an}bracketle{tu, v/an}bracketri}ht=/integraltext
Ru(x)v(x)dxand
F(u) =u/integraltext
0f(s)ds.By (2.4) and (2.6), it follows that for any δ >0 there exist Tδ=Tδ(B)>0,
r1,δ=r1,δ(B)>1 and for any r≥r1,δthere exists t∗
δ,r∈[0,Tδ] such that
Φr(u(t∗
δ,r))< δ,/notturnstiler≥r1,δ. (2.7)
Again by (2.2), we have
/bardblηru(t)/bardblL2(R)≤/vextenddouble/vextenddoubleηru(t∗
δ,r)/vextenddouble/vextenddouble
L2(R)+t/integraldisplay
t∗
δ,r/bardblηrut(s)/bardblL2(R)ds≤/vextenddouble/vextenddoubleηru(t∗
δ,r)/vextenddouble/vextenddouble
L2(R)+c7(t−t∗
δ,r)
and consequently
/bardblηru(t)/bardbl3
L∞(R)≤c8/bardblηru(t)/bardblL2(R)≤c9(Φ1
2r(u(t∗
δ,r))+/bardblg/bardbl1
2
L1(R\(−r,r))+L2(R\(−r,r))+t−t∗
δ,r)<
< c9(δ1
2+/bardblg/bardbl1
2
L1(R\(−r,r))+L2(R\(−r,r))+t−t∗
δ,r),/notturnstilet≥t∗
δ,r,/notturnstiler≥r1,δ.
Denoting T∗
δ,r=t∗
δ,r+l3
3c9and choosing δ∈(0,l6
9c2
9), by the last inequality, we can say that there
existsr2,δ≥2r1,δsuch that
/bardblu(t)/bardblL∞(R\(−r2,δ,r2,δ))< l,/notturnstilet∈[t∗
δ,r,T∗
δ,r]. (2.8)
Now multiplying (1.1) by η2
r(ut+µu), integrating over Rand taking into account (2.4) and (2.8) we
obtain
d
dtΦr(u(t))+c10Φr(u(t))≤c11(1
r+/bardblg/bardblL1(R\(−r,r))+L2(R\(−r,r))),/notturnstilet∈[t∗
δ,r,T∗
δ,r],
and consequently
Φr(u(t))≤Φr(u(t∗
δ,r))e−c10(t−t∗
δ,r)+c11(1
r+/bardblg/bardblL1(R\(−r,r))+L2(R\(−r,r)))1−e−c10(t−t∗
δ,r)
c10,(2.9)
forr≥r2,δ. By (2.7) and (2.9), there exists r3,δ≥r2,δsuch that
Φr(u(t))< δ,/notturnstiler≥r3,δ,/notturnstilet∈[t∗
δ,r,T∗
δ,r].
Hence denoting by nδthe smallest integer number which is not less than3c9Tδ
l3and applying above
procedure at most nδtime, we find
Φr(u(Tδ))< δ,/notturnstiler≥r4,δ,
for some r4,δ≥2nδr1,δ. From the last inequality it follows that for any ε >0 there exist /hatwideTε=
/hatwideTε(B)>0 and/hatwiderε=/hatwiderε(B)>0 such that/vextenddouble/vextenddouble/vextenddoubleS(/hatwideTε)ϕ/vextenddouble/vextenddouble/vextenddouble
H1(R\(−/hatwiderε,/hatwiderε))×L2(R\(−/hatwiderε,/hatwiderε))< ε,/notturnstileϕ∈B.
Since, by (2.2), B0=∪
t≥0S(t)Bis a bounded subset of H, for any ε >0 there exist T0=T0(ε,B)>0
andr0=r0(ε,B)>0 such that
/bardblS(T0)ϕ/bardblH1(R\(−r0,r0))×L2(R\(−r0,r0))< ε,/notturnstileϕ∈B0.
Taking into account positively invariance of B0, from the last inequality we obtain (2.1). /square4 A.KH.KHANMAMEDOV
By (2.1) and (2.4), for any bounded subset BofHthere exist /hatwideT0=/hatwideT0(B)>0 and/hatwider0=/hatwider0(B)>0
such that
σ(u(t,x))≥σ(0)
2,/notturnstilet≥/hatwideT0,/notturnstile|x| ≥/hatwider0. (2.10)
Hence using techniques of [5] one can prove the asymptotic compac tness of the semigroup {S(t)}t≥0,
which is included in the following lemma:
Lemma 2. Assume that conditions (1.3)-(1.4) hold and Bis bounded subset of H. Then every
sequence of the form {S(tn)ϕn}∞
n=1,{ϕn}∞
n=1⊂B,tn→ ∞, has a convergent subsequence in H.
By (2.10) and the unique continuation result of [8], it is easy to see tha t problem (1.1)-(1.2) has a
strict Lyapunov function (see [1] for definition). Thus according t o [1, Corollary 2.29] the semigroup
{S(t)}t≥0possesses a global attractor.
Remark 1. We note that, for the problem considered in [2], from compact embedding H1
0(0,π)⊂
C[0,π], it immediately follows that σ(u(t,x))≥σ(0)
2,/notturnstilet≥0,/notturnstilex∈[0,ε]∪[π−ε,π]for some
ε∈(0,π). So a global attractor still exists if one replaces the posit ivity condition on σ(·)by the
σ(0)>0.
References
[1] I. Chueshov, I. Lasiecka, Long-time behavior of second o rder evolution equations with nonlinear damping, Memoirs
of AMS, 195 (2008).
[2] S. Gatti, V. Pata, A one-dimensional wave equation with n onlinear damping, Glasg. Math. J. , 48 (2006), 419–430.
[3] A. Kh. Khanmamedov, Global attractors for 2-D wave equat ions with displacement dependent damping, Math.
Methods Appl. Sci. , 33 (2010) 177-187.
[4] A. Kh. Khanmamedov, A strong global attractor for 3-D wav e equations with displacement dependent damping,
Appl. Math. Letters , 23 (2010) 928-934.
[5] A. Kh. Khanmamedov, Global attractors for the plate equa tion with a localized damping and a critical exponent
in an unbounded domain, J. Diff. Eqs., 225 (2006) 528-548.
[6] V. Pata, S. Zelik, Global and exponential attractors for 3-D wave equations with displacement dependent damping,
Math. Methods Appl. Sci. , 29 (2006), 1291–1306.
[7] V. Pata, S. Zelik, Attractors and their regularity for 2- D wave equations with nonlinear damping, Adv. Math. Sci.
Appl., 17 (2007), 225–237.
[8] A. Ruiz, Unique continuation for weak solutions of the wa ve equation plus a potential, J. Math. Pures Appl. , 71
(5) (1992) 455–467.
Department of Mathematics, Faculty of Science, Hacettepe U niversity, Beytepe 06800 ,
Ankara, Turkey
E-mail address :azer@hacettepe.edu.tr |
1402.1787v1.One_dimensional_random_attractor_and_rotation_number_of_the_stochastic_damped_sine_Gordon_equation.pdf | arXiv:1402.1787v1 [math.DS] 7 Feb 2014One-Dimensional Random Attractor and Rotation Number
of the Stochastic Damped Sine-Gordon Equation
Zhongwei Shena,1, Shengfan Zhoua,1,∗, Wenxian Shenb,2
aDepartment of Applied Mathematics, Shanghai Normal Univer sity,
Shanghai 200234, PR China
bDepartment of Mathematics and Statistics, Auburn Universi ty,
Auburn 36849, USA
Abstract : This paper is devoted to the study of the asymptotic dynamic s
of the stochastic damped sine-Gordon equation with homogen eous Neumann
boundary condition. It is shown that for any positive dampin g and diffusion
coefficients, the equation possesses a random attractor, and when the damping
and diffusion coefficients are sufficiently large, the random att ractor is a one-
dimensionalrandomhorizontal curveregardlessofthestre ngthofnoise. Hence
its dynamics is not chaotic. It is also shown that the equatio n has a rotation
number provided that the damping and diffusion coefficients are sufficiently
large, whichimpliesthatthesolutionstendtooscillatewi ththesamefrequency
eventually and the so called frequency locking is successfu l.
Keywords : Stochastic damped sine-Gordon equation; random horizont al
curve; one-dimensional random attractor; rotation number ; frequency lock-
ing
AMS Subject Classification : 60H10, 34F05, 37H10.
1 Introduction
Let (Ω,F,P) be a probability space , where
Ω ={ω= (ω1,ω2,...,ω m)∈C(R,Rm) :ω(0) = 0},
the Borel σ-algebra Fon Ω is generated by the compact open topology (see [1]), and Pis the
corresponding Wiener measure on F. Define ( θt)t∈Ron Ω via
θtω(·) =ω(·+t)−ω(t), t∈R.
Thus, (Ω ,F,P,(θt)t∈R) is an ergodic metric dynamical system.
Consider the following stochastic damped sine-Gordon equa tion with additive noise:
dut+αdu+(−K∆u+sinu)dt=fdt+m/summationdisplay
j=1hjdWjinU×R+(1.1)
1The first two authors are supported by National Natural Scien ce Foundation of China under Grant 10771139,
and the Innovation Program of Shanghai Municipal Education Commission under Grant 08ZZ70
2The third author is partially supported by NSF grant DMS-090 7752
∗Corresponding Author: zhoushengfan@yahoo.com
1complemented with the homogeneous Neumann boundary condit ion
∂u
∂n= 0 on ∂U×R+, (1.2)
whereU⊂Rnis a bounded open set with a smooth boundary ∂U,u=u(x,t) is a real function
ofx∈Uandt≥0,α, K >0 are damping and diffusion coefficients, respectively, f∈H1(U),
hj∈H2(U) with∂hj
∂n= 0 on∂U,j= 1,...,m, and{Wj}m
j=1are independent two-sided real-
valued Wiener processes on (Ω ,F,P). We identify ω(t) with (W1(t),W2(t),...,W m(t)), i.e.,
ω(t) = (W1(t),W2(t),...,W m(t)), t∈R.
Sine-Gordon equations describe the dynamics of continuous Josephosn junctions (see [18])
and have been widely studied (see [3], [4], [5], [11], [13], [ 14], [15], [17], [18], [19], [25], [26], [29],
[30], [31], etc.). Various interesting dynamical scenario s such as subharmonic bifurcation and
chaotic behavior are observed in damped and driven sine-Gor don equations (see [3], [4], [19],
etc.). Note that interesting dynamics of a dissipative syst em occurs in its global attractor (if it
exists). It is therefore of great importance to study the exi stence and the structure/dimension
of a global attractor of a damped sine-Gordon equation.
As it is known, under various boundary conditions, a determi nistic damped sine-Gordon
equation possesses a finite dimensional global attractor (s ee [15, 16, 27, 29, 30, 31]). Moreover,
some upperboundsof thedimension of the attractor were obta ined in [15, 29, 30, 31]. In[26, 27],
the authors proved that under Neumann boundary condition, w hen the damping is sufficiently
large, the dimension of the global attractor is one, which ju stifies the folklore that there is no
chaotic dynamics in a strongly damped sine-Gordon equation .
Recently, the existence of attractors of stochastic damped sine-Gordon equations has been
studied by several authors (see [5], [13], [14]). For exampl e, for the equation (1.1) with Dirichlet
boundary condition considered in [13], the author proved th e existence of a finite-dimensional
attractor in the random sense. However, the existing works o n stochastic damped sine-Gordon
equations deal with Dirichlet boundary conditions only. Th e case of a Neumann boundary
condition is of great physical interest. It is therefore imp ortant to investigate both the existence
andstructureofattractors ofstochasticdampedsine-Gord onequationswithNeumannboundary
conditions. Observe that there is no bounded attracting set s in such case in the original phase
space due to the uncontrolled space average of the solutions , which leads to nontrivial dynamics
and also some additional difficulties. Nevertheless, it is st ill expected that (1.1)-(1.2) possesses
an attractor in the original phase space in proper sense.
The objective of the current paper is to provide a study on the existence and structure
of random attractors (see Definition 2.2 for the definition of random attractor) of stochastic
damped sine-Gordon equations with Neumann boundary condit ions, i.e. (1.1)-(1.2). We will do
so in terms of the random dynamical system generated by (1.1) -(1.2) (see Definition 2.1 for the
definition of random dynamical system).
The following are the main results of this paper.
(1) For any α >0 andK >0, (1.1)-(1.2) possesses a random attractor (see Theorem 4. 1 and
Corollary 4.2).
(2) When Kandαaresufficientlylarge, therandomattractorof (1.1)-(1.2)i saone-dimensional
random horizontal curve (and hence is one dimensional) (see Theorem 5.3 and Corollary
5.4).
2(3) When Kandαare sufficiently large, the rotation number of (1.1) exists (S ee Theorem 6.4
and Corollary 6.5).
The above results make an important contribution to the unde rstanding of the nonlinear
dynamics of stochastic damped sine-Gordon equations with N eumann boundary conditions.
Property (1) extends the existence result of random attract or in the Dirichlet boundary case
to the Neumann boundary case and shows that system (1.1)-(1. 2) is dissipative. By property
(2), the asymptotic dynamics of (1.1)-(1.2) with sufficientl y largeαandKis one dimensional
regardless of the strength of noise and hence is not chaotic. Observe that ρ∈Ris called
therotation number of (1.1)-(1.2) (see Definition 6.1 for detail) if for any solu tionu(t,x) of
(1.1)-(1.2) and any x∈U, the limit lim t→∞u(t,x)
texists almost surely and
lim
t→∞u(t,x)
t=ρfora.e. ω∈Ω.
Property (3) then shows that all the solutions of (1.1)-(1.2 ) tend to oscillate with the same fre-
quency eventually almost surely and hence frequency lockin g is successful in (1.1)-(1.2) provided
thatαandKare sufficiently large.
We remark that the results in the current paper also hold for s tochastic damped sine-Gordon
equations with periodic boundary conditions.
It should be pointed out that the dynamical behavior of varie ty of systems of the form (1.1)
have been studied in [22, 23, 24, 25] for ordinary differential equations, [26, 27] for partial
differential equations and [6, 21, 28] for stochastic (random ) ordinary differential equations.
In above literatures, two main aspects considered are the st ructure of the attractor and the
phenomenon of frequency locking. For example, in [28], the a uthors studied a class of nonlinear
noisy oscillators. They proved the existence of a random att ractor which is a family of horizontal
curves and the existence of a rotation number which implies t he frequency locking.
The rest of the paper is organized as follows. In section 2, we present some basic concepts
and properties for general random dynamical systems. In sec tion 3, we provide some basic
settings about (1.1)-(1.2) and show that it generates a rand om dynamical system in proper
function space. We prove in section 4 the existence of a uniqu e random attractor of the random
dynamical system φgenerated by (1.1)-(1.2) for any α,K >0. We show in section 5 that the
random attractor of φis a random horizontal curve provided that αandKare sufficiently large.
In section 6, we prove the existence of a rotation number of (1 .1)-(1.2) provided that αandK
are sufficiently large.
2 General Random Dynamical Systems
In this section, we collect some basic knowledge about gener al random dynamical systems
(see [1, 8] for details). Let ( X,/ba∇dbl ·/ba∇dblX) be a separable Hilbert space with Borel σ-algebra B(X)
and (Ω,F,P,(θt)t∈R) be the ergodic metric dynamical system mentioned in sectio n 1.
Definition 2.1. A continuous random dynamical system over (Ω,F,P,(θt)t∈R)is a(B(R+)×
F ×B(X),B(X))-measurable mapping
ϕ:R+×Ω×X→X,(t,ω,u)/ma√sto→ϕ(t,ω,u)
such that the following properties hold
3(1)ϕ(0,ω,u) =ufor allω∈Ωandu∈X;
(2)ϕ(t+s,ω,·) =ϕ(t,θsω,ϕ(s,ω,·))for alls,t≥0andω∈Ω;
(3)ϕis continuous in tandu.
For given u∈XandE,F⊂X, we define
d(u,E) = inf
v∈E/ba∇dblu−v/ba∇dblX
and
dH(E,F) = sup
u∈Ed(u,F).
dH(E,F) is called the Hausdorff semi-distance fromEtoF.
Definition 2.2. (1) A set-valued mapping ω/ma√sto→D(ω) : Ω→2Xis said to be a random set
if the mapping ω/ma√sto→d(u,D(ω))is measurable for any u∈X. IfD(ω)is also closed
(compact) for each ω∈Ω, the mapping ω/ma√sto→D(ω)is called a random closed (compact)
set. A random set ω/ma√sto→D(ω)is said to be bounded if there exist u0∈Xand a random
variable R(ω)>0such that
D(ω)⊂ {u∈X:/ba∇dblu−u0/ba∇dblX≤R(ω)}for allω∈Ω.
(2) A random set ω/ma√sto→D(ω)is called tempered provided for P-a.s.ω∈Ω,
lim
t→∞e−βtsup{/ba∇dblb/ba∇dblX:b∈D(θ−tω)}= 0for allβ >0.
(3) A random set ω/ma√sto→B(ω)is said to be a random absorbing set if for any tempered random
setω/ma√sto→D(ω), there exists t0(ω)such that
ϕ(t,θ−tω,D(θ−tω))⊂B(ω)for allt≥t0(ω), ω∈Ω.
(4) A random set ω/ma√sto→B1(ω)is said to be a random attracting set if for any tempered random
setω/ma√sto→D(ω), we have
lim
t→∞dH(ϕ(t,θ−tω,D(θ−tω),B1(ω)) = 0for allω∈Ω.
(5) A random compact set ω/ma√sto→A(ω)is said to be a random attractor if it is an random
attracting set and ϕ(t,ω,A(ω)) =A(θtω)for allω∈Ωandt≥0.
Theorem 2.3. Letϕbe a continuous random dynamical system over (Ω,F,P,(θt)t∈R). If there
is a tempered random compact attracting set ω/ma√sto→B1(ω)ofϕ, thenω/ma√sto→A(ω)is a random
attractor of ϕ, where
A(ω) =/intersectiondisplay
t>0/uniondisplay
τ≥tϕ(τ,θ−τω,B1(θ−τω)), ω∈Ω.
Moreover, ω/ma√sto→A(ω)is the unique random attractor of φ.
Proof.See [8, Theorem 1.8.1].
43 Basic Settings
In this section, we give some basic settings about (1.1)-(1. 2) and show that it generates a
random dynamical system. Define an unbounded operator
A:D(A)≡/braceleftBig
u∈H2(U) :∂u
∂n/vextendsingle/vextendsingle/vextendsingle
∂U= 0/bracerightBig
→L2(U), u/ma√sto→ −K∆u. (3.1)
Clearly,Ais nonnegative definite and self-adjoint. Its spectral set c onsists of only nonnegative
eigenvalues, denoted by λi, satisfying
0 =λ0< λ1≤λ2≤ ··· ≤λi≤ ···,(λi→+∞asi→ ∞). (3.2)
It is well known that −Agenerates an analytic semigroup of bounded linear operator s{e−At}t≥0
onL2(U) (andH1(U)). LetE=H1(U)×L2(U), endowed with the usual norm
/ba∇dblY/ba∇dblH1×L2=/parenleftbig
/ba∇dbl∇u/ba∇dbl2+/ba∇dblu/ba∇dbl2+/ba∇dblv/ba∇dbl2/parenrightbig1
2forY= (u,v)⊤, (3.3)
where/ba∇dbl·/ba∇dbldenotes the usual norm in L2(U) and⊤stands for the transposition.
The existence of solutions to problem (1.1)-(1.2) follows f rom [10]. We next transform the
problem (1.1)-(1.2) to a deterministic system with a random parameter, and then show that it
generates a random dynamical system.
Let(Ω,F,P,(θt)t∈R)betheergodicmetricdynamicalsysteminsection1. For j∈ {1,2,...,m},
consider the one-dimensional Ornstein-Uhlenbeck equatio n
dzj+zjdt=dWj(t).
Its unique stationary solution is given by
zj(θtωj) =/integraldisplay0
−∞es(θtωj)(s)ds=−/integraldisplay0
−∞esωj(s+t)ds+ωj(t), t∈R.
Note that the random variable |zj(ωj)|is tempered and the mapping t/ma√sto→zj(θtωj) isP-a.s.
continuous (see [2, 12]). More precisely, there is a θt-invariant Ω 0⊂Ω withP(Ω0) = 1 such that
t/ma√sto→zj(θtωj) is continuous for ω∈Ω0andj= 1,2,···,m. Putting z(θtω) =/summationtextm
j=1hjzj(θtωj),
which solves dz+zdt=/summationtextm
j=1hjdWj.
Now, let v=ut−z(θtω) and take the functional space Einto consideration, we obtain the
equivalent system of (1.1)-(1.2),
/braceleftBigg
˙u=v+z(θtω),
˙v=−Au−αv−sinu+f+(1−α)z(θtω).(3.4)
LetY= (u,v)⊤,C=/parenleftbigg0I
−A−αI/parenrightbigg
,F(θtω,Y) = (z(θtω),−sinu+f+(1−α)z(θtω))⊤, problem
(3.4) has the following simple matrix form
˙Y=CY+F(θtω,Y). (3.5)
We will consider (3.4) or (3.5) for ω∈Ω0and write Ω 0as Ω from now on.
5Clearly,Cis an unbounded closed operator on Ewith domain D(C) =D(A)×H1(U). It is
not difficult to check that the spectral set of Cconsists of only following points [27]
µ±
i=−α±√
α2−4λi
2, i= 0,1,2,...
andCgenerates a C0-semigroup of bounded linear operators {eCt}t≥0onE. Furthermore, let
Fω(t,Y) :=F(θtω,Y), it is easy to see that Fω(·,·) :R+×E→Eis continuous in tand globally
Lipschitz continuous in Yfor each ω∈Ω. By the classical theory concerning the existence and
uniqueness of the solutions, we obtain (see [20, 29])
Theorem 3.1. Consider (3.5). For each ω∈Ωand each Y0∈E, there exists a unique function
Y(·,ω,Y0)∈C([0,+∞);E)such that Y(0,ω,Y0) =Y0andY(t,ω,Y0)satisfies the integral
equation
Y(t,ω,Y0) =eCtY0+/integraldisplayt
0eC(t−s)F(θsω,Y(s,ω,Y0))ds. (3.6)
Furthermore, if Y0∈D(C), there exists Y(·,ω,Y0)∈C([0,+∞);D(C))∩C1((R+,+∞);E)
which satisfies (3.6)andY(t,ω,Y0)is jointly continuous in t,Y0, and is measurable in ω.
Then,Y:R+×Ω×E→E(orR+×Ω×D(C)→D(C))is a continuous random dynamical
system.
We now define a mapping φ:R+×Ω×E→E(orR+×Ω×D(C)→D(C)) by
φ(t,ω,φ0) =Y(t,ω,Y0(ω))+(0,z(θtω))⊤, (3.7)
whereφ0= (u0,u1)⊤andY0(ω) = (u0,u1−z(ω))⊤. It is easy to see that φis a continuous
random dynamical system associated with the problem (1.1)- (1.2) on E(orD(C)). We next
show a useful property of just defined random dynamical syste ms.
Lemma 3.2. Suppose that p0= (2π,0)⊤. The random dynamical system Ydefined in (3.6)is
p0-translation invariant in the sense that
Y(t,ω,Y0+p0) =Y(t,ω,Y0)+p0, t≥0, ω∈Ω, Y0∈E.
Proof.SinceCp0= 0 and F(t,ω,Y) isp0-periodic in Y,Y(t,ω,Y0) +p0is a solution of (3.5)
with initial data Y0+p0. Thus,Y(t,ω,Y0)+p0=Y(t,ω,Y0+p0).
Note that µ+
1→0 asα→+∞, which will cause some difficulty. In order to overcome it, we
introduce a new norm which is equivalent to the usual norm /ba∇dbl·/ba∇dblH1×L2onEin (3.3). Here, we
only collect some results about the new norm (see [27] for det ails). Since Chas at least two real
eigenvalues 0 and −αwith correspondingeigenvectors η0= (1,0)⊤andη−1= (1,−α)⊤, letE1=
span{η0},E−1= span{η−1}andE11=E1+E−1. For any u∈L2(U), define ¯ u=1
|U|/integraltext
Uu(x)dx,
i.e., the spatial average of u, let˙L2(U) ={u∈L2(U) : ¯u= 0},˙H1(U) =H1(U)∩˙L2(U) and
E22=˙H1(U)×˙L2(U). It’s easy to see that E=E11⊕E22andE1is invariant under C. We
now define two bilinear forms on E11andE22respectively. For Yi= (ui,vi)⊤∈E11,i= 1,2, let
/an}b∇acketle{tY1,Y2/an}b∇acket∇i}htE11=α2
4/an}b∇acketle{tu1,u2/an}b∇acket∇i}ht+/an}b∇acketle{tα
2u1+v1,α
2u2+v2/an}b∇acket∇i}ht, (3.8)
6where/an}b∇acketle{t·,·/an}b∇acket∇i}htdenotes the inner product on L2(U), and for Yi= (ui,vi)⊤∈E22, i= 1,2, let
/an}b∇acketle{tY1,Y2/an}b∇acket∇i}htE22=/an}b∇acketle{tA1
2u1,A1
2u2/an}b∇acket∇i}ht+(α2
4−δλ1)/an}b∇acketle{tu1,u2/an}b∇acket∇i}ht+/an}b∇acketle{tα
2u1+v1,α
2u2+v2/an}b∇acket∇i}ht,(3.9)
whereA1
2=√
K∇(see (3.1) for the definition of A) andδ∈(0,1]. By the Poincar´ e inequality
/ba∇dblA1
2u/ba∇dbl2≥λ1/ba∇dblu/ba∇dbl2,∀u∈˙H1(U),
(3.9) is then positive definite. Note that for any Y∈E,¯Y=/integraltext
UY(x)dx∈E11andY−¯Y∈E22,
thus we define
/an}b∇acketle{tY1,Y2/an}b∇acket∇i}htE=/an}b∇acketle{t¯Y1,¯Y2/an}b∇acket∇i}htE11+/an}b∇acketle{tY1−¯Y1,Y2−¯Y2/an}b∇acket∇i}htE22forY1,Y2∈E. (3.10)
Lemma 3.3 ([27]).(1)(3.8)and(3.9)define inner products on E11andE22, respectively.
(2)(3.10)defines an inner product on E, and the corresponding norm /ba∇dbl· /ba∇dblEis equivalent to
the usual norm /ba∇dbl·/ba∇dblH1×L2in(3.3), where
/ba∇dblY/ba∇dblE=/parenleftBigα2
4/ba∇dblu/ba∇dbl2+/ba∇dblα
2u+v/ba∇dbl2+/ba∇dblA1
2(u−¯u)/ba∇dbl2−δλ1/ba∇dblu−¯u/ba∇dbl2/parenrightBig1
2
=/parenleftBigα2
4/ba∇dblu/ba∇dbl2+/ba∇dblα
2u+v/ba∇dbl2+/ba∇dblA1
2u/ba∇dbl2−δλ1/ba∇dblu−¯u/ba∇dbl2/parenrightBig1
2(3.11)
forY= (u,v)⊤∈E.
(3) In terms of the inner product /an}b∇acketle{t·,·/an}b∇acket∇i}htE,E1andE11are orthogonal to E−1andE22, respec-
tively.
(4) In terms of the norm /ba∇dbl·/ba∇dblE, the Lipschitz constant LFofFin(3.5)satisfies
LF≤2
α. (3.12)
Now let E2=E−1⊕E22, thenE2is orthogonal to E1andE=E1⊕E2. Thus,E2is also
invariant under C. Denote by PandQ(=I−P) the projections from EintoE1andE2,
respectively.
Lemma 3.4. (1) For any Y∈D(C)∩E2,/an}b∇acketle{tCY,Y/an}b∇acket∇i}htE≤ −a/ba∇dblY/ba∇dbl2
E, where
a=α
2−/vextendsingle/vextendsingle/vextendsingleα
2−δλ1
α/vextendsingle/vextendsingle/vextendsingle. (3.13)
(2)/ba∇dbleCtQ/ba∇dbl ≤e−atfort≥0.
(3)eCtPY=PYforY∈E,t≥0.
Proof.See Lemma 3.3 and Corollary 3.3.1 in [27] for (1) and (2). We no w show (3). For Y∈
D(C)∩E1, sinced
dteCtY=eCtCY= 0, we have eCtY=eC0Y=Y. Then, by approximation,
eCtY=Yforu∈E1, t≥0, sinceD(A)∩E1is dense in E1. Thus,eCtPY=PYforY∈E,
t≥0.
7We will need the following lemma and its corollaries.
Lemma 3.5. For any ǫ >0, there is a tempered random variable r: Ω/ma√sto→R+such that
/ba∇dblz(θtω)/ba∇dbl ≤eǫ|t|r(ω)for allt∈R, ω∈Ω, (3.14)
wherer(ω),ω∈Ωsatisfies
e−ǫ|t|r(ω)≤r(θtω)≤eǫ|t|r(ω), t∈R, ω∈Ω. (3.15)
Proof.Forj∈ {1,2,...,m}, since|zj(ωj)|is a tempered random variable and the mapping
t/ma√sto→ln|zj(θtωj)|isP-a.s. continuous, it follows from Proposition 4.3.3 in [1] t hat for any ǫj>0
there is a tempered random variable rj(ωj)>0 such that
1
rj(ωj)≤ |zj(ωj)| ≤rj(ωj),
whererj(ωj) satisfies, for P-a.s.ω∈Ω,
e−ǫj|t|rj(ωj)≤rj(θtωj)≤eǫj|t|rj(ωj), t∈R. (3.16)
Takingǫ1=ǫ2=···=ǫm=ǫ, then we have
/ba∇dblz(θtω)/ba∇dbl ≤m/summationdisplay
j=1|zj(θtωj)|·/ba∇dblhj/ba∇dbl ≤m/summationdisplay
j=1rj(θtωj)/ba∇dblhj/ba∇dbl ≤eǫ|t|m/summationdisplay
j=1rj(ωj)/ba∇dblhj/ba∇dbl.
Letr(ω) =/summationtextm
j=1rj(ωj)/ba∇dblhj/ba∇dbl, (3.14) is satisfied and (3.15) is trivial from (3.16).
Corollary 3.6. For any ǫ >0, there is a tempered random variable r′: Ω/ma√sto→R+such that
/ba∇dblA1
2z(θtω)/ba∇dbl ≤eǫ|t|r′(ω)for allt∈R, ω∈Ω,
wherer′(ω) =/summationtextm
j=1rj(ωj)/ba∇dblA1
2hj/ba∇dblsatisfies
e−ǫ|t|r′(ω)≤r′(θtω)≤eǫ|t|r′(ω), t∈R, ω∈Ω.
Corollary 3.7. For any ǫ >0, there is a tempered random variable r′′: Ω/ma√sto→R+such that
/ba∇dblAz(θtω)/ba∇dbl ≤eǫ|t|r′′(ω)for allt∈R, ω∈Ω
wherer′′(ω) =/summationtextm
j=1rj(ωj)/ba∇dblAhj/ba∇dblsatisfies
e−ǫ|t|r′′(ω)≤r′′(θtω)≤eǫ|t|r′′(ω), t∈R, ω∈Ω.
84 Existence of Random Attractor
In this section, we study the existence of a random attractor . Throughout this section we
assume that p0= 2πη0= (2π,0)⊤∈E1andδ∈(0,1] is such that a >0, where ais as in (3.13).
We remark in the end of this section that such δalways exists.
The space D(C) can be endowed with the graph norm,
/ba∇dblY/ba∇dbl˜E=/ba∇dblY/ba∇dblE+/ba∇dblCY/ba∇dblEforY∈D(C).
SinceCis a closed operator, D(C) is a Banach space under the graph norm. We denote
(D(C),/ba∇dbl·/ba∇dbl˜E) by˜Eand let˜E1=˜E∩E1,˜E2=˜E∩E2.
By Lemma 3.2 and the fact that operator Chas a zero eigenvalue, we will define a random
dynamical system Ydefined on torus induced from Y. Then by properties of Yrestricted on
E2, we can prove the existence of a random attractor of Y. Thus, we can say that Yhas a
unbounded random attractor. Now, we define Y.
LetT1=E1/p0ZandE=T1×E2. ForY0∈E, letY0:=Y0(modp0) =Y0+p0Z⊂E
denotes the equivalence class of Y0, which is an element of E. And the norm on Eis denoted by
/ba∇dblY0/ba∇dblE= inf
y∈p0Z/ba∇dblY0+y/ba∇dblE.
Note that, by Lemma 3.2, Y(t,ω,Y0+kp0) =Y(t,ω,Y0) +kp0,∀k∈Zfort≥0,ω∈Ω and
Y0∈E. With this, we define Y:R+×Ω×E→Eby setting
Y(t,ω,Y0) =Y(t,ω,Y0) (modp0), (4.1)
whereY0=Y0(modp0). It is easy to see that Y:R+×Ω×E→Eis a random dynamical
system.
Similarly, the random dynamical system φdefined in (3.7) also induces a random dynamical
systemΦonE. By (3.7) and (4.1), Φis defined by
Φ(t,ω,Φ0) =Y(t,ω,Y0)+ ˜z(θtω) (modp0), (4.2)
whereΦ0=φ0(modp0), ˜z(θtω) = (0,z(θtω))⊤andY0=Φ0−˜z(ω) (modp0).
The main result of this section can now be stated as follows.
Theorem 4.1. The random dynamical system Ydefined in (4.1)has a unique random attractor
ω/ma√sto→A0(ω), where
A0(ω) =/intersectiondisplay
t>0/uniondisplay
τ≥tY(τ,θ−τω,B1(θ−τω)), ω∈Ω,
in which ω/ma√sto→B1(ω)is a tempered random compact attracting set for Y.
Corollary 4.2. The induced random dynamical system Φdefined in (4.2)has a random attrac-
torω/ma√sto→A(ω), whereA(ω) =A0(ω)+ ˜z(ω) (modp0)for allω∈Ω.
Proof.It follows from (4.2) and Theorem 4.1.
To prove Theorem 4.1, we first introduce the concept of random pseudo-balls and prove a
lemma on the existence of a pseudo-tempered random absorbin g pseudo-ball.
9Definition 4.3. LetR: Ω→R+be a random variable. A random pseudo-ball ω∈Ω/ma√sto→B(ω)⊂
Ewith random radius ω/ma√sto→R(ω)is a set of the form
ω/ma√sto→B(ω) ={b(ω)∈E:/ba∇dblQb(ω)/ba∇dblE≤R(ω)}.
Furthermore, a random set ω/ma√sto→B(ω)⊂Eis called pseudo-tempered provided ω/ma√sto→QB(ω)is a
tempered random set in E, i.e., for P-a.s.ω∈Ω,
lim
t→∞e−βtsup{/ba∇dblQb/ba∇dblE:b∈B(θ−tω)}= 0for allβ >0.
Notice that any random pseudo-ball ω/ma√sto→B(ω) inEhas the form ω/ma√sto→E1×QB(ω), where
ω/ma√sto→QB(ω) is a random ball in E2, which implies the measurability of ω/ma√sto→B(ω).
By Definition 4.3, if ω/ma√sto→B(ω) is a random pseudo-ball in E, thenω/ma√sto→B(ω) (modp0) is
random bounded set in E. And if ω/ma√sto→B(ω) is a pseudo-tempered random set in E, then
ω/ma√sto→B(ω) (modp0) is tempered random set in E.
Lemma 4.4. Leta >0. Then there exists a tempered random set ω/ma√sto→B0(ω) :=B0(ω) (modp0)
inEsuch that, for any tempered random set ω/ma√sto→B(ω) :=B(ω) (modp0)inE, there is a
TB(ω)>0such that
Y(t,θ−tω,B(θ−tω))⊂B0(ω)for allt≥TB(ω), ω∈Ω,
whereω/ma√sto→B0(ω)is a random pseudo-ball in Ewith random radius ω/ma√sto→R0(ω)andω/ma√sto→B(ω)
is any pseudo-tempered random set in E.
Proof.Forω∈Ω, we obtain from (3.6) that
Y(t,ω,Y0(ω)) =eCtY0(ω)+/integraldisplayt
0eC(t−s)F(θsω,Y(s,ω,Y0(ω)))ds. (4.3)
The projection of (4.3) to E2is
QY(t,ω,Y0(ω)) =eCtQY0(ω)+/integraldisplayt
0eC(t−s)QF(θsω,Y(s,ω,Y0(ω)))ds. (4.4)
By replacing ωbyθ−tω, it follows from (4.4) that
QY(t,θ−tω,Y0(θ−tω)) =eCtQY0(θ−tω)+/integraldisplayt
0eC(t−s)QF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))ds,
and it then follows from Lemma 3.4 and Q2=Qthat
/ba∇dblQY(t,θ−tω,Y0(θ−tω))/ba∇dblE
≤e−at/ba∇dblQY0(θ−tω)/ba∇dblE+/integraldisplayt
0e−a(t−s)/ba∇dblF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))/ba∇dblEds.(4.5)
10By (3.11), Lemma 3.5 and Corollary 3.6 with ǫ=a
2,
/ba∇dblF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))/ba∇dblE
=/parenleftBigα2
4/ba∇dblz(θs−tω)/ba∇dbl2+/ba∇dbl(1−α
2)z(θs−tω)−sin(Yu)+f/ba∇dbl2+/ba∇dblA1
2z(θs−tω)/ba∇dbl2
−δλ1/ba∇dblz(θs−tω)−z(θs−tω)/ba∇dbl2/parenrightBig1
2
≤/parenleftBig
(α2−3α+3)/ba∇dblz(θs−tω)/ba∇dbl2+3/ba∇dblsin(Yu)/ba∇dbl2+3/ba∇dblf/ba∇dbl2+/ba∇dblA1
2z(θs−tω)/ba∇dbl2/parenrightBig1
2
≤/parenleftBig
(α2−3α+3)ea(t−s)(r(ω))2+ea(t−s)(r′(ω))2+3|U|+3/ba∇dblf/ba∇dbl2/parenrightBig1
2
≤a1ea
2(t−s)r(ω)+ea
2(t−s)r′(ω)+a2,
whereYusatisfies Y(s,θ−tω,Y0(θ−tω)) = (Yu,Yv)⊤,a1=√
α2−3α+3,a2=/radicalbig
3|U|+3/ba∇dblf/ba∇dbl2
and|U|is the Lebesgue measure of U. We find from (4.5) that
/ba∇dblQY(t,θ−tω,Y0(θ−tω))/ba∇dblE≤e−at/ba∇dblQY0(θ−tω)/ba∇dblE+2
a(1−e−a
2t)(a1r(ω)+r′(ω))+a2
a(1−e−at).
Now for ω∈Ω, define
R0(ω) =4
a(a1r(ω)+r′(ω))+2a2
a.
Then, for any pseudo-tempered random set ω/ma√sto→B(ω) inEand any Y0(θ−tω)∈B(θ−tω), there
is aTB(ω)>0 such that for t≥TB(ω),
/ba∇dblQY(t,θ−tω,Y0(θ−tω))/ba∇dblE≤R0(ω), ω∈Ω,
which implies
Y(t,θ−tω,B(θ−tω))⊂B0(ω) for all t≥TB(ω), ω∈Ω,
whereω/ma√sto→B0(ω) is the random pseudo-ball centered at origin with random ra diusω/ma√sto→R0(ω).
In fact,ω/ma√sto→R0(ω) is a tempered random variable since ω/ma√sto→r(ω) andω/ma√sto→r′(ω) are tempered
random variables. Then the measurability of random pseudo- tempered ball ω/ma√sto→B0(ω) is
obtained from Definition 4.3 and ω/ma√sto→B0(ω) is a random pseudo-ball. Hence, ω/ma√sto→B0(ω) :=
B0(ω) (modp0) is a tempered random ball in E. It then follows from the definition of Ythat
Y(t,θ−tω,B(θ−tω))⊂B0(ω) for all t≥TB(ω), ω∈Ω,
whereTB(ω) =TB(ω) forω∈Ω. This complete the proof.
We now prove Theorem 4.1.
Proof of Theorem 4.1. By Theorem 2.3, it suffices to prove the existence of a random at tracting
set which restricted on E2is tempered and compact, i.e., there exists a random set ω/ma√sto→B1(ω)
such that ω/ma√sto→QB1(ω) is tempered and compact in E2and for any pseudo-tempered random
setω/ma√sto→B(ω) inE,
dH(Y(t,θ−tω,B(θ−tω)),B1(ω))→0 ast→ ∞, ω∈Ω,
11wheredHis the Hausdorff semi-distance. Since pseudo-tempered rand om sets in Eare absorbed
by the random absorbing set ω/ma√sto→B0(ω), it suffices to prove that
dH(Y(t,θ−tω,B0(θ−tω)),B1(ω))→0 ast→ ∞, ω∈Ω. (4.6)
Clearly, if such a ω/ma√sto→B1(ω) exists, then ω/ma√sto→B1(ω) :=B1(ω) (modp0) is a tempered random
compact attracting set for Y. We next show that (4.6) holds.
By the superposition principle, (3.5) with initial data Y0(ω) can be decomposed into
˙Y1=CY1+F(θtω,Y(t,ω,Y0(ω))), Y10(ω) = 0 (4.7)
and
˙Y2=CY2, Y20(ω) =Y0(ω), (4.8)
whereY(t,ω,Y0(ω)) is the solution of (3.5) with initial data Y0(ω)∈B0(ω). LetY1(t,ω,Y10(ω))
andY2(t,Y20(ω)) be solutions of (4.7) and (4.8), respectively. We now give some estimations of
Y1(t,ω,Y10(ω)) andY2(t,Y20(ω)), which ensure the existence of a random attracting set whi ch
restricted on E2is tempered and compact.
We first estimate Y2(t,Y20(ω)). Clearly, (4.8) is a linear problem. It is easy to see that
Y2(t,Y20(ω)) =eCtY20(ω),
which implies (with ωbeing replaced by θ−tω) that
/ba∇dblQY2(t,Y20(θ−tω))/ba∇dblE≤ /ba∇dbleCtQ/ba∇dbl·/ba∇dblQY20(θ−tω)/ba∇dblE≤e−atR0(θ−tω)→0 ast→ ∞.(4.9)
ForY1(t,ω,Y10(ω)), we show that it is bounded by a tempered random bounded clo sed set
in˜E, which then is compact in Esince˜Eis compactly imbedded in E. Note that
Y1(t,ω,Y10(ω)) =/integraldisplayt
0eC(t−s)F(θsω,Y(s,ω,Y0(ω)))ds, (4.10)
it then follows that
/ba∇dblQY1(t,θ−tω,Y10(θ−tω))/ba∇dblE≤2
a(1−e−a
2t)(a1r(ω)+r′(ω))+a2
a(1−e−at),(4.11)
wherea1=√
α2−3α+3 and a2=/radicalbig
3|U|+3/ba∇dblf/ba∇dbl2are the same as in the proof of Lemma 4.4,
|U|denotes the Lebbesgue measure of U.
We next estimate CQY1(t,θ−tω,Y10(θ−tω)). We find from (4.10) that
CQY1(t,θ−tω,Y10(θ−tω)) =/integraldisplayt
0eC(t−s)CQF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))ds
=/integraldisplayt
0eC(t−s)CF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))ds.
Then,
/ba∇dblCQY1(t,θ−tω,Y10(θ−tω))/ba∇dblE≤/integraldisplayt
0e−a(t−s)/ba∇dblCF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))/ba∇dblEds.(4.12)
12Obviously,
CF(θs−tω,Y(s,θ−tω,Y0(θ−tω))) =/parenleftbigg−sin(Yu)+f+(1−α)z(θs−tω)
αsin(Yu)−αf−α(1−α)−Az(θs−tω)/parenrightbigg
,
whereYusatisfiesY(s,θ−tω,Y0(θ−tω)) = (Yu,Yv)⊤. By (3.11), Lemma 3.5, Corollary 3.6 and
Corollary 3.7 with ǫ=a
2,
/ba∇dblCF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))/ba∇dbl2
E
≤7
4α2/ba∇dblsin(Yu)/ba∇dbl2+7
4α2/ba∇dblf/ba∇dbl2+7
4α2(1−α)2/ba∇dblz(θs−tω)/ba∇dbl2+4/ba∇dblAz(θs−tω)/ba∇dbl2
+3/ba∇dblA1
2sin(Yu)/ba∇dbl2+3/ba∇dblA1
2f/ba∇dbl2+3(1−α)2/ba∇dblA1
2z(θs−tω)/ba∇dbl2
≤a2
3+7
4α2(1−α)2ea(t−s)(r(ω))2+3(1−α)2ea(t−s)(r′(ω))2
+4ea(t−s)(r′′(ω))2+3/ba∇dblA1
2sin(Yu)/ba∇dbl2
≤/parenleftBig
a3+√
7
2α|1−α|ea
2(t−s)r(ω)+√
3|1−α|ea
2(t−s)r′(ω)
+2ea
2(t−s)r′′(ω)+√
3/ba∇dblA1
2sin(Yu)/ba∇dbl/parenrightBig2
,
wherea3=/radicalBig
7
4α2|U|+7
4α2/ba∇dblf/ba∇dbl2+3/ba∇dblA1
2f/ba∇dbl2. Then, (4.12) implies
/ba∇dblCQY1(t,θ−tω,Y10(θ−tω))/ba∇dblE
≤a3
a(1−e−at)+√
3/integraldisplayt
0e−a(t−s)/ba∇dblA1
2sin(Yu)/ba∇dblds
+2
a/parenleftBig√
7
2α|1−α|r(ω)+√
3|1−α|r′(ω)+2r′′(ω)/parenrightBig
(1−e−a
2t).(4.13)
For the integral on the right-hand side of (4.13), we note tha t
/ba∇dblA1
2sin(Yu)/ba∇dbl ≤ /ba∇dblA1
2Yu/ba∇dbl ≤a4/ba∇dblQY(s,θ−tω,Y0(θ−tω))/ba∇dblE,
wherea4=/radicalbig
2/(2−δ). Since
/ba∇dblQY(s,θ−tω,Y0(θ−tω))/ba∇dblE
≤e−as/ba∇dblQY0(θ−tω)/ba∇dblE+/integraldisplays
0e−a(s−τ)/ba∇dblF(θτ−tω,Y(τ,θ−tω,Y0(θ−tω)))/ba∇dblEdτ,
13we find that
/integraldisplayt
0e−a(t−s)/ba∇dblA1
2sin(Yu)/ba∇dblds
≤a4te−at/ba∇dblQY0(θ−tω)/ba∇dblE+a4/integraldisplayt
0/integraldisplays
0e−a(t−τ)/ba∇dblF(θτ−tω,Y(τ,θ−tω,Y0(θ−tω)))/ba∇dblEdτds
≤a4te−at/ba∇dblQY0(θ−tω)/ba∇dblE
+a4/integraldisplayt
0/parenleftBigg
2
a/parenleftBig
a1r(ω)+r′(ω)/parenrightBig
(e−a
2(t−s)−e−a
2t)+a2
a(e−a(t−s)−e−at)/parenrightBigg
ds
=a4te−at/ba∇dblQY0(θ−tω)/ba∇dblE+2a4
a/parenleftBig
a1r(ω)+r′(ω)/parenrightBig/parenleftBig2
a(1−e−a
2t)−te−a
2t/parenrightBig
+a2a4
a/parenleftBig1
a(1−e−at)−te−at/parenrightBig
.
(4.14)
Combining (4.11), (4.13) and (4.14), there is a T(ω)>0 such that for all t≥T(ω),
/ba∇dblQY1(t,θ−tω,Y10(θ−tω))/ba∇dbl˜E
=/ba∇dblQY1(t,θ−tω,Y10(θ−tω))/ba∇dblE+/ba∇dblCQY1(t,θ−tω,Y10(θ−tω))/ba∇dblE
≤R1(ω), (4.15)
whereR1(ω) =a5r(ω) +a6r′(ω) +8
ar′′(ω) +a7is a tempered random variable, in which
a5=4a1+2√
7α|1−α|
a+8√
3a1a4
a2,a6=4+4√
3|1−α|
a+8√
3a4
a2anda7=2a2+2a3
a+2√
3a2a4
a2.
Now, let ω/ma√sto→B1(ω) be the random pseudo-ball in ˜Ecentered at origin with random radius
ω/ma√sto→R1(ω), thenω/ma√sto→B1(ω) is tempered and measurable. By (4.9), (4.15) and
QY(t,θ−tω,φ0(θ−tω)) =QY1(t,θ−tω,Y10(θ−tω))+QY2(t,Y20(θ−tω)),
we have for ω∈Ω,
dH(Y(t,θ−tω,B0(θ−tω)),B1(ω))→0 ast→ ∞.
Then by the compact embedding of ˜EintoE,ω/ma√sto→QB1(ω) is compact in E2, which implies
thatω/ma√sto→B1(ω) :=B1(ω) (modp0) is a tempered random compact attracting set for Y. Thus
by Theorem 2.3, Yhas a unique random attractor ω/ma√sto→A0(ω), where
A0(ω) =/intersectiondisplay
t>0/uniondisplay
τ≥tY(τ,θ−τω,B1(θ−τω)), ω∈Ω.
This completes the proof.
Remark 4.5. (1) For any α >0andλ1=K˜λ1>0(see(3.2)), there is a δ∈(0,1]such that
a >0holds, where ais as in(3.13)and˜λ1is the smallest positive eigenvalue of −△and
a constant.
(2) We can say that the random dynamical Y(orφ) has a unique random attractor in the
sense that the induced random dynamical system Y(orΦ) has a unique random attractor,
14and we will say that Y(orφ) has a unique random attractor directly in the sequel. We
denote the random attractor of Yandφbyω/ma√sto→A0(ω)andω/ma√sto→A(ω)respectively. Indeed,
ω/ma√sto→A0(ω)andω/ma√sto→A(ω)satisfy
A0(ω) =A0(ω) (modp0),A(ω) =A(ω) (modp0), ω∈Ω.
(3) For the deterministic damped sine-Gordon equation with homogeneous Neumann boundary
condition, the authors proved in [27] that the random attrac tor is a horizontal curve pro-
vided that αandKare sufficiently large. Similarly, we expect that the random a ttractor
ω/ma√sto→A(ω)ofφhas the similar property, i.e., A(ω)is a horizontal curve for each ω∈Ω
provided that αandKare sufficiently large. We prove that this is true in next secti on.
(4) By (2), system (1.1)-(1.2)is dissipative (i.e. it possesses a random attractor). In sec tion
6, we will show that (1.1)-(1.2)with sufficiently large αandKalso has a rotation number
and hence all the solutions tend to oscillate with the same fr equency eventually.
5 One-dimensional Random Attractor
In this section, we apply the theory established in [7] to sho w that the random attractor
ofY(orφ) is one-dimensional provided that αandKare sufficiently large. This method has
been used by Chow, Shen and Zhou [6] to systems of coupled nois y oscillators. Throughout this
section we assume that p0= 2πη0= (2π,0)⊤∈E1anda >4LF(see (3.13) for the definition
ofaand see (3.12) for the upper bound of LF). We remark in the end of this section that this
condition can be satisfied provided that αandKare sufficiently large.
Definition 5.1. Suppose {Φω}ω∈Ωis a family of maps from E1toE2andn∈N. A family of
graphsω/ma√sto→ℓ(ω)≡ {(p,Φω(p)) :p∈E1}is said to be a random np0-periodic horizontal curve if
ω/ma√sto→ℓ(ω)is a random set and {Φω}ω∈Ωsatisfy the Lipshcitz condition
/ba∇dblΦω(p1)−Φω(p2)/ba∇dblE≤ /ba∇dblp1−p2/ba∇dblEfor allp1,p2∈E1, ω∈Ω
and the periodic condition
Φω(p+np0) = Φω(p)for allp∈E1, ω∈Ω.
Clearly, for any ω∈Ω,ℓ(ω) is a deterministic np0-periodic horizontal curve. When n= 1,
we simply call it a horizontal curve.
Lemma 5.2. Leta >4LF. Suppose that ω/ma√sto→ℓ(ω)is a random np0-periodic horizontal curve
inE. Then,ω/ma√sto→Y(t,ω,ℓ(ω))is also a random np0-periodic horizontal curve in Efor allt >0.
Moreover, ω/ma√sto→Y(t,θ−tω,ℓ(θ−tω))is a random np0-periodic horizontal curve for all t >0.
Proof.First, since Yis a random dynamical system and ω/ma√sto→ℓ(ω) is a random set in E,
ω/ma√sto→Y(t,ω,ℓ(ω)) andω/ma√sto→Y(t,θ−tω,ℓ(θ−tω)) are random sets in Efor allt >0. We next show
the Lipschitz condition and periodic condition.
It is sufficient to prove the Lipschitz condition and periodic condition valid for ω/ma√sto→ℓ(ω) in
D(C) sinceD(C) is dense in E. Clearly, for ω∈Ω andt >0,
Y(t,ω,ℓ(ω)) ={(PY(t,ω,p+Φω(p)),QY(t,ω,p+Φω(p))) :p∈E1∩D(C)}.
15Forp1,p2∈E1∩D(C),p1/ne}ationslash=p2, letYi(t,ω) =Y(t,ω,pi+Φω(pi)),i= 1,2,p(t,ω) =P(Y1(t,ω)−
Y2(t,ω)) andq(t,ω) =Q(Y1(t,ω)−Y2(t,ω)), where P,Qare defined as in section 1. We have
by Lemma 3.4
PYi(t,ω) =eCtP(pi+Φω(pi))+/integraldisplayt
0eC(t−s)PF(θsω,Yi(s,ω))ds
=P(pi+Φω(pi))+/integraldisplayt
0PF(θsω,Yi(s,ω))ds, i= 1,2,
and then,d
dtPYi(t,ω) =PF(θtω,Yi(t,ω)), i= 1,2, it then follows that
d
dtp(t,ω) =d
dtP(Y1(t,ω)−Y2(t,ω))
=P(F(θtω,Y1(t,ω))−F(θtω,Y2(t,ω))).(5.1)
Sincep(t,ω)+q(t,ω) =Y1(t,ω)−Y2(t,ω),
d
dt(p(t,ω)+q(t,ω)) =d
dt(Y1(t,ω)−Y2(t,ω))
=C(Y1(t,ω)−Y2(t,ω))+F(θtω,Y1(t,ω))−F(θtω,Y2(t,ω)),
then, by the orthogonal decomposition,
d
dtq(t,ω) =C(Y1(t,ω)−Y2(t,ω))+Q(F(θtω,Y1(t,ω))−F(θtω,Y2(t,ω)))
=Cq(t,ω)+Q(F(θtω,Y1(t,ω))−F(θtω,Y2(t,ω))).(5.2)
We find from (5.1) that
d
dt/ba∇dblp(t,ω)/ba∇dbl2
E= 2/angbracketleftbig
p(t,ω),d
dtp(t,ω)/angbracketrightbig
E
≥ −2/ba∇dblp(t,ω)/ba∇dblE·/ba∇dblP(F(θtω,Y1(t,ω))−F(θtω,Y2(t,ω)))/ba∇dblE
≥ −2LF(/ba∇dblp(t,ω)/ba∇dbl2
E+/ba∇dblp(t,ω)/ba∇dblE/ba∇dblq(t,ω)/ba∇dblE).
Similarly, by (5.2) and Lemma 3.4,
d
dt/ba∇dblq(t,ω)/ba∇dbl2
E≤ −2a/ba∇dblq(t,ω)/ba∇dbl2
E+2LF(/ba∇dblp(t,ω)/ba∇dblE/ba∇dblq(t,ω)/ba∇dblE+/ba∇dblq(t,ω)/ba∇dbl2
E).
Becausea >4LF, ifthereisa t0≥0suchthat /ba∇dblq(t0,ω)/ba∇dblE=/ba∇dblp(t0,ω)/ba∇dblEandsince /ba∇dblp(t,ω)/ba∇dblE/ne}ationslash= 0
fort≥0, then
d
dt/vextendsingle/vextendsingle/vextendsingle
t=t0/parenleftBig
/ba∇dblq(t,ω)/ba∇dbl2
E−/ba∇dblp(t,ω)/ba∇dbl2
E/parenrightBig
≤(8LF−2a)/ba∇dblq(t0,ω)/ba∇dbl2
E<0,
which means that there is a ¯t0> t0such that for t∈(t0,¯t0),
/ba∇dblq(t,ω)/ba∇dbl2
E−/ba∇dblp(t,ω)/ba∇dbl2
E</ba∇dblq(0,ω)/ba∇dbl2
E−/ba∇dblp(0,ω)/ba∇dbl2
E
=/ba∇dblΦω(p1)−Φω(p2)/ba∇dbl2
E−/ba∇dblp1−p2/ba∇dbl2
E
≤0,
16namely,/ba∇dblq(t,ω)/ba∇dblE</ba∇dblp(t,ω)/ba∇dblEfort∈(t0,¯t0).
If there is another t1≥¯t0such that /ba∇dblq(t1,ω)/ba∇dblE=/ba∇dblp(t1,ω)/ba∇dblE, then
d
dt/vextendsingle/vextendsingle/vextendsingle
t=t1/parenleftBig
/ba∇dblq(t,ω)/ba∇dbl2
E−/ba∇dblp(t,ω)/ba∇dbl2
E/parenrightBig
≤(8LF−2a)/ba∇dblq(t1,ω)/ba∇dbl2
E<0,
which means that there is a ¯t1> t1such that for t∈(t1,¯t1),/ba∇dblq(t,ω)/ba∇dblE</ba∇dblp(t,ω)/ba∇dblE. Continue
this process, we have for all t≥0,/ba∇dblq(t,ω)/ba∇dblE≤ /ba∇dblp(t,ω)/ba∇dblE, i.e.,
/ba∇dblQ(Y1(t,ω)−Y2(t,ω))/ba∇dblE≤ /ba∇dblP(Y1(t,ω)−Y2(t,ω))/ba∇dblE,
which shows that ω/ma√sto→Y(t,ω,ℓ(ω)) satisfies the Lipschitz condition in Definition 5.1.
We next show the periodic condition. We find from Lemma 3.2 tha t
Y(t,ω,p+Φω(p))+np0=Y(t,ω,p+np0+Φω(p))..
Since Φω(p) = Φω(p+np0),Y(t,ω,p+Φω(p))+np0=Y(t,ω,p+np0+Φω(p+np0)). It follows
that
QY(t,ω,p+Φω(p)) =QY(t,ω,p+np0+Φω(p+np0)).
Consequently, ω/ma√sto→Y(t,ω,ℓ(ω)) is a random np0-periodic horizontal curve for all t >0.
Moreover, for any fixed ω∈Ω andt >0, ¯ω=θ−tω∈Ω is fixed. Then, Y(t,¯ω,ℓ(¯ω)) is a
deterministic np0-periodic horizontal curve, which yields the assertion.
Chooseγ∈(0,a
2) such that
2
α/parenleftBigg
1
γ+1
a−2γ/parenrightBigg
<1, (5.3)
where2
αis the upper bound of the Lipschitz constant of F(see (3.12)). We remark in the end
of this section that such a γexists provided that αandKare sufficiently large. We next show
the main result in this section.
Theorem 5.3. Assume that a >4LFand that there is a γ∈(0,a
2)such that (5.3)holds. Then
the random attractor ω/ma√sto→A0(ω)of the random dynamical system Yis a random horizontal
curve.
Proof.By the equivalent relation between φandY, we mainly focus on equation (3.5), which
can be viewed as a deterministic system with a random paramet erω∈Ω. We write it here as
(3.5)ωfor some fixed ω∈Ω.
Observe that the linear part of (3.5) ω, i.e.
˙Y=CY (5.4)
has a one-dimensional center space Ec= span{(1,0)}=E1and a one co-dimensional stable
spaceEs=E2. We first show that (3.5) ωhas a one-dimensional invariant manifold, denoted by
W(ω), and will show later that W(ω) exponentially attracts all the solutions of (3.5) ω.
LetFω(t,Y) =F(θtω,Y),ω∈Ω. For fixed ω∈Ω, consider the following integral equation
˜Y(t) =eCtξ+/integraldisplayt
0eC(t−s)PFω(s,˜Y(s))ds+/integraldisplayt
−∞eC(t−s)QFω(s,˜Y(s))ds, t≤0,(5.5)
17whereξ=P˜Y(0)∈E1. Forg: (−∞,0]→Esuch that supt≤0/ba∇dbleγtg(t)/ba∇dblE<∞, define
(Lg)(t) =/integraldisplayt
0eC(t−s)Pg(s)ds+/integraldisplayt
−∞eC(t−s)Qg(s)ds, t≤0.
It is easy to see that
sup
t≤0/ba∇dbleγt(Lg)(t)/ba∇dblE≤/parenleftBigg
1
γ+1
a−γ/parenrightBigg
sup
t≤0/ba∇dbleγtg(t)/ba∇dblE≤/parenleftBigg
1
γ+1
a−2γ/parenrightBigg
sup
t≤0/ba∇dbleγtg(t)/ba∇dblE,
which means that /ba∇dblL/ba∇dbl ≤1
γ+1
a−2γ. Then, Theorem 3.3 in [7] shows that for any ξ∈E1, equation
(5.5) has a unique solution ˜Yω(t,ξ) satisfying supt≤0/ba∇dbleγt˜Yω(t,ξ)/ba∇dblE<∞. Let
h(ω,ξ) =Q˜Yω(0,ξ) =/integraldisplay0
−∞e−CsQFω(s,˜Yω(s,ξ))ds, ω∈Ω.
Let
W(ω) ={ξ+h(ω,ξ) :ξ∈E1}, ω∈Ω.
For anyǫ∈(0,γ) in Lemma 3.5 and Corollary 3.6, we have
/ba∇dblh(θ−tω,ξ)/ba∇dblE≤1
a−ǫ(a1r(ω)+r′(ω))eǫt+a2
a, t≥0. (5.6)
Observe that
˜Yω(t,ξ) =eCtξ+/integraldisplayt
0eC(t−s)PFω(s,˜Yω(s,ξ))ds+/integraldisplayt
−∞eC(t−s)QFω(s,˜Y(s,ω,ξ))ds
=eCt(ξ+h(ω,ξ))+/integraldisplayt
0eC(t−s)Fω(s,˜Yω(s,ξ))ds,
i.e.,˜Yω(t,ξ) is the solution of (3.5) with initial data ξ+h(ω,ξ) fort≤0. Thus, for Y0(ω) =
ξ+h(ω,ξ)∈W(ω), there is a negative continuation of Y(t,ω,Y0(ω)), i.e.,
Y(t,ω,Y0(ω)) =˜Yω(t,ξ), t≤0. (5.7)
Moreover, for t≥0, we obtain from (3.6) and (5.7) that
Y(t,ω,Y0(ω))
=eCt(ξ+h(ω,ξ))+/integraldisplayt
0eC(t−s)F(θsω,Y(s,ω,Y0(ω)))ds
=eCtξ+/integraldisplayt
0eC(t−s)F(θsω,Y(s,ω,Y0(ω)))ds+/integraldisplay0
−∞eC(t−s)QFω(s,˜Yω(s,ξ))ds
=eCtξ+/integraldisplayt
0eC(t−s)F(θsω,Y(s,ω,Y0(ω)))ds+/integraldisplay0
−∞eC(t−s)QF(θsω,Y(s,ω,Y0(ω)))ds
=eCtξ+/integraldisplayt
0eC(t−s)PF(θsω,Y(s,ω,Y0(ω)))ds+/integraldisplayt
−∞eC(t−s)QF(θsω,Y(s,ω,Y0(ω)))ds
=eCt/parenleftBig
ξ+/integraldisplayt
0e−CsPF(θsω,Y(s,ω,Y0(ω)))ds/parenrightBig
+/integraldisplay0
−∞e−CsQF(θt+sω,Y(t+s,ω,Y0(ω)))ds.
18Then by the uniqueness of solution of (5.5) for fixed ω∈Ω, we have
h/parenleftBig
θtω,eCt/parenleftBig
ξ+/integraldisplayt
0e−CsPF(θsω,Y(s,ω,Y0(ω)))ds/parenrightBig/parenrightBig
=/integraldisplay0
−∞e−CsQF(θt+sω,Y(t+s,ω,Y0(ω)))ds,
and then for t≥0,
Y(t,ω,W(ω)) =W(θtω). (5.8)
By (5.7) and (5.8), W(ω) is an invariant manifold of (3.5) ω.
Next, we show that W(ω) attracts the solutions of (3.5) ω, more precisely, for the given ω∈Ω,
we prove the existence of a stable foliation {Ws(ω,Y0) :Y0∈W(ω)}of the invariant manifold
W(ω) of (3.5) ω. Consider the following integral equation
ˆY(t) =eCtη+/integraldisplayt
0eC(t−s)Q/parenleftBig
Fω(s,ˆY(s)+Yω(s,ξ+h(ω,ξ)))
−Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig
ds
+/integraldisplayt
∞eC(t−s)P/parenleftBig
Fω(s,ˆY(s)+Yω(s,ξ+h(ω,ξ)))
−Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig
ds, t≥0,(5.9)
whereξ+h(ω,ξ)∈W(ω),η=QˆY(0)∈E2andYω(t,ξ+h(ω,ξ)) :=Y(t,ω,ξ+h(ω,ξ)),t≥0
is the solution of (3.5) with initial data ξ+h(ω,ξ) for fixed ω∈Ω. Theorem 3.4 in [7] shows
that for any ξ∈E1andη∈E2, equation (5.9) has a unique solution ˆYω(t,ξ,η) satisfying
supt≥0/ba∇dbleγtˆYω(t,ξ,η)/ba∇dblE<∞and for any ξ∈E1,η1, η2∈E2,
sup
t≥0eγt/ba∇dblˆYω(t,ξ,η1)−ˆYω(t,ξ,η2)/ba∇dblE≤M/ba∇dblη1−η2/ba∇dblE. (5.10)
whereM=1
1−2
α/parenleftbig
1
γ+1
a−2γ/parenrightbig. Let
ˆh(ω,ξ,η) =ξ+PˆYω(0,ξ,η)
=ξ+/integraldisplay0
∞e−CsP/parenleftBig
Fω(s,ˆYω(s,ξ,η)+Yω(s,ξ+h(ω,ξ)))
−Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig
ds.
Then,Ws(ω,ξ+h(ω,ξ)) ={η+h(ω,ξ)+ˆh(ω,ξ,η) :η∈E2}is the stable foliation of W(ω) at
ξ+h(ω,ξ).
Observe that
ˆYω(t,ξ,η)+Yω(t,ξ+h(ω,ξ))−Yω(t,ξ+h(ω,ξ))
=ˆYω(t,ξ,η)
=eCt(η+h(ω,ξ)+ˆh(ω,ξ,η)−ξ−h(ω,ξ))
+/integraldisplayt
0eC(t−s)/parenleftBig
Fω(s,ˆYω(s,ξ,η)+Yω(s,ξ+h(ω,ξ)))
−Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig
ds(5.11)
19and
Yω(t,η+h(ω,ξ)+ˆh(ω,ξ,η))−Yω(t,ξ+h(ω,ξ))
=eCt(η+h(ω,ξ)+ˆh(ω,ξ,η)−ξ−h(ω,ξ))
+/integraldisplayt
0eC(t−s)/parenleftBig
Fω(s,Yω(s,η+h(ω,ξ)+ˆh(ω,ξ,η)))
−Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig
ds.(5.12)
Comparing (5.11) with (5.12), we find that
ˆYω(t,ξ,η) =Yω(t,η+h(ω,ξ)+ˆh(ω,ξ,η))−Yω(t,ξ+h(ω,ξ)), t≥0.(5.13)
In addition, if η= 0, then by the uniqueness of solution of (5.9), ˆYω(t,ξ,0)≡0 fort≥0, which
associates with (5.10) and (5.13) show that
sup
t≥0eγt/ba∇dblYω(t,η+h(ω,ξ)+ˆh(ω,ξ,η))−Yω(t,ξ+h(ω,ξ))/ba∇dblE≤M/ba∇dblη/ba∇dblE(5.14)
for anyξ∈E1andη∈E2.
We now claim that ω/ma√sto→W(ω) is the random attractor of Y. Letω/ma√sto→B(ω) be any pseudo-
tempered random set in E. For any ω/ma√sto→Y0(ω)∈ω/ma√sto→B(ω), there is ω/ma√sto→ξ(ω)∈E1such
that
Y0(θ−tω)∈Ws(θ−tω,ξ(θ−tω)+h(θ−tω,ξ(θ−tω))).
Letη(θ−tω) =QY0(θ−tω)−h(θ−tω,ξ(θ−tω)). By (5.6), it is easy to see that
sup
Y0(θ−tω)∈B(θ−tω)/ba∇dblη(θ−tω)/ba∇dbl ≤ sup
Y0(θ−tω)∈B(θ−tω)/ba∇dblQY0(θ−tω)/ba∇dbl+1
a−ǫ(a1r(ω)+r′(ω))eǫt+a2
a.
It then follows from (5.14) and the fact that ω/ma√sto→QB(ω) is tempered that
sup
Y0(θ−tω)∈B(θ−tω)/ba∇dblY(t,θ−tω,Y0(θ−tω))−Y(t,θ−tω,ξ(θ−tω)+h(θ−tω,ξ(θ−tω)))/ba∇dblE
≤Me−γtsup
Y0(θ−tω)∈B(θ−tω)/ba∇dblη(θ−tω)/ba∇dblE
≤Me−γtsup
Y0(θ−tω)∈B(θ−tω)/ba∇dblQY0(θ−tω)/ba∇dbl+M
a−ǫ(a1r(ω)+r′(ω))e(ǫ−γ)t+a2M
ae−γt
→0 ast→ ∞,
which associates with (5.8) lead to
dH(Y(t,θ−tω,B(θ−tω)),W(ω))→0 ast→ ∞.
Therefore, A0(ω) =W(ω) forω∈Ω. Next, we show that ω/ma√sto→A0(ω) is a random horizontal
curve. Infact, forsomerandomhorizontalcurve ω/ma√sto→ℓ(ω)inE, forexample, ℓ(ω)≡ {(p,Φω(p)) :
20Φω(p) =c,p∈E1},ω∈Ω, where c∈E2is constant, it must be contained in some pseudo-
tempered random set, for example ω/ma√sto→B2/bardblc/bardblE(ω), where B2/bardblc/bardblE(ω) is a pseudo-ball with radius
2/ba∇dblc/ba∇dblE. Then, for ω∈Ω,
dH(Y(t,θ−tω,ℓ(θ−tω)),A0(ω))→0 ast→ ∞,
which means that lim t→∞Y(t,θ−tω,ℓ(θ−tω))⊂A0(ω). SinceA0(ω) is one-dimensional, we have
forω∈Ω,
A0(ω) = lim
t→∞Y(t,θ−tω,ℓ(θ−tω)).
It then follows from Lemma 5.2 that ω/ma√sto→A0(ω) is a random horizontal curve.
Corollary 5.4. Assume that a >4LFand that there is a γ∈(0,a
2)such that (5.3)holds.
Then the random attractor ω/ma√sto→A(ω)of the random dynamical system φis a random horizontal
curve.
Proof.It follows from Corollary 4.2, Remark 4.5 and Theorem 5.3.
Remark 5.5. At the beginning of this section, we assume that a >4LF. Sincea=α
2−|α
2−δλ1
α|
andLF≤2
α, we can take α,λ1satisfyingα
2−/vextendsingle/vextendsingle/vextendsingleα
2−δλ1
α/vextendsingle/vextendsingle/vextendsingle>8
α, whereλ1is the smallest positive
eigenvalue of Aand its value is determined by the diffusion coefficient K. On the other hand,
we need some γ∈(0,a
2)such that (5.3)holds. Note that
min
γ∈(0,a
2)/parenleftBigg
1
γ+1
a−2γ/parenrightBigg
=/parenleftBigg
1
γ+1
a−2γ/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
γ=(2−√
2)a
2=√
2
(3√
2−4)a,
which implies that there exist α,λ1satisfying
α
2−/vextendsingle/vextendsingle/vextendsingleα
2−δλ1
α/vextendsingle/vextendsingle/vextendsingle>2√
2
(3√
2−4)α>8
α. (5.15)
Indeed, let c=2√
2
3√
2−4, then for any α >√
2candλ1> c, there is a δ >0satisfying
c
λ1< δ <min/braceleftBigα2−c
λ1,1/bracerightBig
such that (5.15)holds.
6 Rotation Number
In this section, we study the phenomenon of frequency lockin g, i.e., the existence of a
rotation number of the stochastic damped sine-Gordon equat ion (1.1)-(1.2), which characterizes
the speed that the solution of (1.1)-(1.2) moves around the o ne-dimensional random attractor.
21Definition 6.1. The stochastic damped sine-Gordon equation (1.1)with boundary condition
(1.2)is said to have a rotation number ρ∈Rif, forP-a.e.ω∈Ωand each φ0= (u0,u1)⊤∈E,
the limit limt→∞Pφ(t,ω,φ0)
texists and
lim
t→∞Pφ(t,ω,φ0)
t=ρη0,
whereη0= (1,0)⊤is the basis of E1.
We remark that the rotation number of (1.1)-(1.2) (if exists ) is unique. In fact, assume that
ρ1andρ2are rotation numbers of (1.1)-(1.2). Then there is ω∈Ω such that for any φ0∈E,
ρ1η0= lim
t→∞Pφ(t,ω,φ0)
t=ρ2η0.
Therefore, ρ1=ρ2and then the rotation number of (1.1)-(1.2) (if exists) is un ique.
From (3.7), we have
Pφ(t,ω,φ0)
t=PY(t,ω,Y0(ω))
t+P(0,z(θtω))⊤
t, (6.1)
whereφ0= (u0,u1)⊤andY0(ω) = (u0,u1−z(ω))⊤. By Lemma 2.1 in [12], it is easy to prove
that lim t→∞P(0,z(θtω))⊤
t= (0,0)⊤. Thus, it sufficient to prove the existence of the rotation
number of the random system (3.5).
By the random dynamical system Ydefined in (4.1), we define the corresponding skew-
product semiflow Θt: Ω×E→Ω×Efort≥0 by setting
Θt(ω,Y0) = (θtω,Y(t,ω,Y0)).
Obviously, (Ω ×E,F ×B,(Θt)t≥0) is a measurable dynamical system, where B=B(E) is the
Borelσ-algebra of E.
Lemma 6.2. There is a measure µonΩ×Esuch that (Ω×E,F ×B, µ,(Θt)t≥0)becomes an
ergodic metric dynamical system.
Proof.LetPrΩ(E) be the set of all random probability measures on EandPrP(Ω×E) be the
set of all probability measures on Ω ×Ewith marginal P. We know from Proposition 3.3 and
Proposition 3.6 in [9] that PrΩ(E) andPrP(Ω×E) are isomorphism. Moreover, both PrΩ(E)
andPrP(Ω×E) are convex, and the convex structure is preserved by this is omorphism.
Let Γ = {ω/ma√sto→µω∈PrΩ(E) :P-a.s.µω(A0(ω)) = 1, ω/ma√sto→µωis invariant for Y}. Clearly, Γ
is convex. Since ω/ma√sto→A0(ω) is the random attractor of Y, we obtain from Corollary 6.13 in [9]
that Γ/ne}ationslash=∅. Letω/ma√sto→µωbe an extremal point of Γ. Then, by the isomorphism between PrΩ(E)
andPrP(Ω×E) and Lemma 6.19 in [9], the corresponding measure µon Ω×Eofω/ma√sto→µωis
(Θt)t≥0-invariant and ergodic. Thus, (Ω ×E,F×B, µ,(Θt)t≥0) is an ergodic metric dynamical
system.
We next show a simple lemma which will be used. For any pi= (si,0)⊤∈E1,i= 1,2, we
define
p1≤p2ifs1≤s2.
Then we have
22Lemma 6.3. Suppose that a >4LF. Letℓbe any deterministic np0-periodic horizontal curve
(ℓsatisfies the Lipschitz and periodic condition in Definition 5 .1). For any Y1, Y2∈ℓwith
PY1≤PY2, there holds
PY(t,ω,Y1)≤PY(t,ω,Y2)fort >0, ω∈Ω. (6.2)
Proof.Clearly, if PY1=PY2, then (6.2) holds. We now prove that (6.2) holds for PY1< PY2.
If not, then by the continuity of Ywith respect to t, there is a t0>0 such that PY(t0,ω,Y1) =
PY(t0,ω,Y2), which implies that Y(t0,ω,Y1) =Y(t0,ω,Y2) sinceY(t0,ω,Y1) andY(t0,ω,Y2)
belong to the same deterministic np0-periodic horizontal curve Y(t0,ω,ℓ), which leads to a
contradiction. The lemma is thus proved.
We now show the main result in this section.
Theorem 6.4. Assume that a >4LF. Then the rotation number of (3.5)exists.
Proof.Note that
PY(t,ω,Y0)
t=PY0
t+1
t/integraldisplayt
0PF(θsω,Y(s,ω,Y0))ds.
SinceF(θsω,Y(s,ω,Y0)+kp0) =F(θsω,Y(s,ω,Y0)),∀k∈Z, wecanidentify F(θsω,Y(s,ω,Y0))
withF(θsω,Y(s,ω,Y0)). Precisely, define h:E→ E,Y/ma√sto→ {Y}, whereEis the collection of all
singleton sets of E, i.e.E={{Y}:Y∈E}(see Remark 6.6 for more details of the space E).
Clearly,his a homeomorphism from EtoE. Then,
F(θsω,Y(s,ω,Y0)) =h−1(F(θsω,Y(s,ω,Y0))).
Thus,
PY(t,ω,Y0)
t=PY0
t+1
t/integraldisplayt
0Ph−1(F(θsω,Y(s,ω,Y0)))ds
=PY0
t+1
t/integraldisplayt
0F(Θs(ω,Y0))ds.(6.3)
whereF=P◦h−1◦F∈L1(Ω×E,F ×B, µ). Lett→ ∞in (6.3), lim t→∞PY0
t= (0,0)⊤and
by Lemma 5.2 and Ergodic Theorems in [1], there exist a consta ntρ∈Rsuch that
lim
t→∞1
t/integraldisplayt
0F(Θs(ω,Y0))ds=ρη0,
which means
lim
t→∞PY(t,ω,Y0)
t=ρη0
forµ-a.e.(ω,Y0)∈Ω×E. Thus, there is Ω∗⊂Ω withP(Ω∗) = 1 such that for any ω∈Ω∗, there
isY∗
0(ω)∈Esuch that
lim
t→∞PY(t,ω,Y∗
0(ω))
t=ρη0.
By Lemma 3.2, we have that for any n∈Nandω∈Ω∗,
lim
t→∞PY(t,ω,Y∗
0(ω)±np0)
t= lim
t→∞PY(t,ω,Y∗
0(ω))±np0
t=ρη0. (6.4)
23Now for any ω∈Ω∗and any Y0∈E, there is n0(ω)∈Nsuch that
PY∗
0(ω)−n0(ω)p0≤PY0≤PY∗
0(ω)+n0(ω)p0
and there is a n0(ω)p0-periodic horizontal curve l0(ω) such that Y∗
0(ω)−n0(ω)p0,Y0,Y∗
0(ω)+
n0(ω)p0∈l0(ω). Then by Lemma 6.3, we have
PY(t,ω,Y∗
0(ω)−n0(ω)p0)≤PY(t,ω,Y0)≤PY(t,ω,Y∗
0(ω)+n0(ω)p0),
which together with (6.4) implies that for any ω∈Ω∗and any Y0∈E,
lim
t→∞PY(t,ω,Y0)
t=ρη0.
Consequently, for any a.e. ω∈Ω and any Y0∈E,
lim
t→∞PY(t,ω,Y0)
t=ρη0.
The theorem is thus proved.
Corollary 6.5. Assume that a >4LF. Then the rotation number of the stochastic damped
sine-Gordon equation (1.1)with the boundary condition (1.2)exists.
Proof.It follows from (6.1) and Theorem 6.4.
Remark 6.6. We first note that the space E={{Y}:Y∈E}in the proof of Theorem 6.4 is a
linear space according to the linear structure defined by
α{X}+β{Y}={αX+βY},forα,β∈R,{X},{Y} ∈ E.
Also, for {X},{Y} ∈ E, we define
/an}b∇acketle{t{X},{Y}/an}b∇acket∇i}htE=/an}b∇acketle{tX,Y/an}b∇acket∇i}htE. (6.5)
It is easy to verify that the functional /an}b∇acketle{t·,·/an}b∇acket∇i}htE:E ×E → Rdefined by (6.5)is bilinear, symmetric
and positive, thus defining the scalar product in EoverR. Moreover, the completeness of Eis
from the completeness of E. Hence, Eis a Hilbert space.
Remark 6.7. In the proof of Theorem 6.4, we used an ergodic invariant measur eµof(Ω×
E,F ×B, µ,(Θt)t≥0). It should be pointed out that the measure µonΩ×Ethat makes (Ω×
E,F ×B, µ,(Θt)t≥0)becomes an ergodic metric dynamical system may not be unique , because
the convex set Γin the proof of Lemma 6.2 may have more than one extremal point s. However,
as mentioned above, the rotation number in Theorem 6.4 and Coro llary 6.5 are independent of
µand are unique.
Acknowledgement . We would like to thank the referee for carefully reading the manuscript
and making very useful suggestions.
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26 |
1608.00984v2.Ferromagnetic_Damping_Anti_damping_in_a_Periodic_2D_Helical_surface__A_Non_Equilibrium_Keldysh_Green_Function_Approach.pdf | arXiv:1608.00984v2 [cond-mat.mes-hall] 13 Aug 2016Ferromagnetic Damping/Anti-damping in a Periodic 2D Helic al surface; A
Non-Equilibrium Keldysh Green Function Approach
Farzad Mahfouzi1,∗and Nicholas Kioussis1
1Department of Physics, California State University, North ridge, California 91330-8268, USA
In this paper, we investigate theoretically the spin-orbit torque as well as the Gilbert damping for
a two band model of a 2D helical surface state with a Ferromagn etic (FM) exchange coupling. We
decompose the density matrix into the Fermi sea and Fermi sur face components and obtain their
contributions to the electronic transport as well as the spi n-orbit torque (SOT). Furthermore, we
obtain the expression for the Gilbert damping due to the surf ace state of a 3D Topological Insulator
(TI) and predicted its dependence on the direction of the mag netization precession axis.
PACS numbers: 72.25.Dc, 75.70.Tj, 85.75.-d, 72.10.Bg
I. INTRODUCTION
The spin-transfer torque (STT) is a phenomenon in
which spin current of large enough density injected into
a ferromagnetic layer switches its magnetization from
one static configuration to another [1]. The origin of
STT is absorption of itinerant flow of angular momen-
tum components normal to the magnetization direc-
tion. It represents one of the central phenomena of the
second-generation spintronics, focused on manipulation
of coherent spin states, since reduction of current den-
sities (currently of the order 106-108A/cm2) required
for STT-based magnetization switching is expected to
bring commercially viable magnetic random access mem-
ory (MRAM) [2]. The rich nonequilibrium physics [3]
arising in the interplay of spin currents carried by fast
conduction electrons and collective magnetization dy-
namics, viewed as the slow classical degree of freedom,
is of great fundamental interest.
Very recent experiments [4, 5] and theoretical stud-
ies[6] havesoughtSTT innontraditionalsetupswhich do
not involvetheusual two(spin-polarizingandfree) F lay-
ers with noncollinear magnetizations [3], but rely instead
on the spin-orbit coupling (SOC) effects in structures
lacking inversion symmetry. Such “SO torques” [7] have
been detected [4] in Pt/Co/AlO xlateral devices where
current flows in the plane of Co layer. Concurrently, the
recent discovery [8] of three-dimensional (3D) topologi-
cal insulators (TIs), which possess a usual band gap in
the bulk while hosting metallic surfaces whose massless
Dirac electrons have spins locked with their momenta
due to the strong Rashba-type SOC, has led to theoreti-
cal proposals to employ these exotic states of matter for
spintronics [9] and STT in particular [10]. For example,
magnetizationofa ferromagneticfilm with perpendicular
anisotropy deposited on the TI surface could be switched
by interfacial quantum Hall current [10].
In this paper, we investigate the dynamical properties
of a FM/3DTI heterostructure, where the F overlayer
∗farzad.mahfouzi@gmail.comcovers a TI surface and the device is periodic along in-
planex−ydirections. The effect of the F overlayer is a
proximityinduced exchangefield −∆surf/vector m·/vectorσ/2superim-
posed on the Dirac cone dispersion. For a partially cov-
ered FM/TI heterostructure, the spin-momentum-locked
Dirac electrons flip their spin upon entering into the in-
terface region, thereby inducing a large antidamping-like
SOT on the FM [15–17]. The antidamping-like SOT
driven by this mechanism which is unique to the sur-
face of TIs has been predicted in Ref. [17], where a time-
dependent nonequilibrium Green function [18] (NEGF)-
based framework was developed. The formalism made it
possible to separate different torque components in the
presenceofarbitraryspin-flipprocesseswithinthedevice.
Similaranti-dampingtorqueshasalsobeen predicted[19]
to exist due to the Berry phase in periodic structures
where the device is considered infinite in in-plane direc-
tions and a Kubo formula was used to describe the SOT
as a linear response to homoginiuos electric field at the
interface. However, the connection between the two ap-
proaches is not clear and one of the goals of the current
paper is to address the similarities and the differences
between the two. In the following we present the theo-
retical formalism of the SOT and damping in the regime
ofslowlyvaryingparametersofaperiodicsysteminspace
and time.
Generally, in a quantum system with slowly varying
parameters in space and/or time, the system stays close
toits equilibrium state ( i.e.adiabaticregime)and the ef-
fects of the nonadiabaticity is taken into account pertur-
batively using adiabatic expansion. Conventionally, this
expansion is performed using Wigner representation [20]
after the separation of the fast and slow variations in
space and/or time. [21] The slow variation implies that
the NEGFs vary slowly with the central space ( time ),
/vector xc= (/vector x+/vector x′)/2(tc= (t+t′)/2 ), while they changefast
with the relative space (time), /vector xr=/vector x−/vector x′(tr=t−t′).
Here we use an alternative approach, where we consider
(x,t) and (/vector xr,tr) as the natural variables to describe the
close to adiabatic apace-time evolution of NEGFs and
then perform the following Fourier transform
ˇG(/vector xt;/vector x′t′) =/integraldisplaydE
2πd/vectork
ΩkeiE(t−t′)+i/vectork·(/vector x−/vector x′)ˇG/vectorkE(/vector xt).(1)2
where, Ω kis the volume of the phase space that the
/vectork-integration is being performed. The standard Dyson
equation of motion for ˇG(/vector xt;/vector x′t′) is cumbersome to ma-
nipulate[22,23]orsolvenumerically,[24]sotheyareusu-
ally transformed to some other representation.[11] Gen-
eralizing the equation to take into account slowly varying
time and spatial dependence of the Hamiltonian we ob-
tain,
ˇG=/parenleftbigg
GrG<
0Ga/parenrightbigg
, (2)
=/parenleftbigg
Gr,−1
ad−iDxtΣ<
0 Ga,−1
ad−iDxt/parenrightbigg−1
,
where,
Gr,−1
ad= (E−iη)1−H(/vectork,t)−µ(/vector x),(3a)
Σ<=−2iηf(E−i∂
∂t−µ(/vector x)), (3b)
Dxt=∂
∂t+∂H
∂/vectork·/vector∇, (3c)
and,η=/planckover2pi1/2τis the phenomenological broadening pa-
rameter, where τis the relaxation time. It is worth
mentioning that for a finite ηthe number of particles
is not conserved, and a more accurate interpretation of
the introduced broadening might be to consider it as an
energy-independent scape rate of electrons to fictitious
reservoirs attached to the positions /vector x. Consequently, a
finite broadening could be interpreted as the existence
of an interface in the model between each atom in the
system and the reservoir that is spread homogeneously
along the infinite periodic system.
Eq. (2) shows that the effect of the space/time varia-
tion is to replace E→E−i∂/∂tand/vectork→/vectork−i/vector∇in the
equation of motion for the GFs in stationary state. To
the lowest order with respect to the derivatives we can
write,
ˇG=ˇGad−i∂ˇGad
∂E∂ˇG−1
ad
∂tˇGad−i∂ˇGad
∂/vectork·/vector∇ˇG−1
adˇGad,
(4)
where,
ˇG−1
ad=/parenleftbigg
Gr,−1
ad−2iηf(E−µ(/vector x))
0 Ga,−1
ad/parenrightbigg
.(5)
For the density matrix of the system, ρ(t) =1
iG<(t,t),
we obtain,
ρneq
/vectork,t≈ −/integraldisplaydE
2πℜ/parenleftbigg
[D(Gr
ad),Gr
ad]f+2iηD(Gr
ad)Ga
ad∂f
∂E/parenrightbigg
(6)
whereD=∂
∂t−/vector∇µ·∂
∂/vectorkis the differential operator act-
ing on the slowly varying parameters in space and time.The details of the derivation is presented in Appendix.A.
The density matrix in Eq. (6) is the central formula of
the paper and consists of two terms; the first term con-
tains the equilibrium Fermi distribution function from
the electrons bellow the Fermi surface occupying a slowly
(linearly) varying single particle states that has only in-
terband contributions and can as well be formulated in
terms of the Berry phase as we will show the following
sections, and; the second term corresponds to the elec-
trons with Fermi energy (at zero temperature we have,
∂f/∂E=δ(E−EF)) which are the only electrons al-
lowed to get excited in the presence of the slowly varying
perturbations. The fact that the first term originates
from the assumption that the electric field is constant
inside the metallic FM suggests that this term might dis-
appearoncethescreeningeffect isincluded. Onthe other
hand, duetothe factthat thesecondtermcorrespondsto
the nonequilibrium electrons injected from the fictitious
reservoirs attached to the device through the scape rate
η, it might capture the possible physical processes that
occur at the contact region and makes it more suitable
for the calculation of the relevant physical observables in
such systems.
Using the expression for the nonequilibrium density
matrix the local spin density can be obtained from,
/vectorSneq(t) =/angb∇acketleft/vector σ/angb∇acket∇ightneq=1
4π2/integraldisplay
d2/vectorkTr[ρneq
/vectork,t/vector σ],(7)
where/angb∇acketleft.../angb∇acket∇ightneqrefers to the ensemble average over many-
body states out of equilibrium demonstrated by the
nonequilibrium density matrix of the electrons and, Tr
refers to the trace. In this case the time derivative in the
differential operator Dleads to the damping of the dy-
namics of the ferromagnet while the momentum deriva-
tive leads to either damping or anti-damping of the FM
dynamics depending on the direction of the applied elec-
tric field. In the followingsection we apply the formalism
to a two band helical surface state model attached to a
FM.
II. SOT AND DAMPING OF A HELICAL 2D
SURFACE
A two band Hamiltonian model for the system can be
generally written as,
H(/vectork,t) =ε0(/vectork)1+/vectorh(/vectork,t)·/vectorσ (8)
where,/vectorh=/vectorhso(/vectork)+∆xc(/vectork)
2/vector m(t), with/vectorhso(/vectork) =−/vectorhso(−/vectork)
and ∆ xc(/vectork) = ∆ xc(−/vectork) being spin-orbit and magnetic
exchange coupling terms respectively. In particular in
the case of Rashba type helical states we have /vectorhso=
αsoˆez×/vectork. In this case for the adiabatic single particle
GF we have,
Gr
ad(E,t) =(E−ε0−iη)1+/vectorh·/vectorσ
(E−ε0−iη)2−|/vectorh|2(9)3
From Eq. (7) for the local spin density, we obtain (See
Appendix B for details),
/vectorSneq(t) =/integraldisplayd2/vectork
4π2/parenleftBigg/vectorh×D/vectorh
2|/vectorh|3(f1−f2)−(/vector∇µ·/vector v0)/vectorh
2η|/vectorh|(f′
1−f′
2)
+(/vectorh×D/vectorh
2|/vectorh|2+ηD/vectorh−1
η(/vectorh·D/vectorh)/vectorh
2|/vectorh|2)(f′
1+f′
2)/parenrightBigg
(10)
where,f1,2=f(ε0± |/vectorh|) and/vector v0=∂ε0/∂/vectorkis the group
velocity of electrons in the absence of the SOI. Here, we
assumeη≪ |/vectorh|which corresponds to a system close to
the ballistic regime. In this expression we kept the ηD/vectorh
because of its unique vector orientation characteristics.
As it becomes clear in the following, the first term in
Eq. (10) is a topological quantity which in the presence
of an electric field becomes dissipative and leads to an
anti-damping torque. The second term in this expression
leads to the Rashba-Edelstein field-like torque which is a
nondissipative observable. The third term has the exact
formasthe firstterm with the difference that it is strictly
a Fermi surface quantity. The fourth term, also leads to
a field like torque that as we will see in the following has
similar features as the Rashba-Edelstein effect. It is im-
portant to pay attention that unlike the first term, the
rest of the terms in Eq. (10) are solely due to the flow
of the non-equilibrium electrons on the Fermi surface.
Furthermore, we notice that the terms that lead to dissi-
pation in the presence of an electric field ( D ≡/vector∇µ·∂
∂/vectork)
become nondissipative when we consider D ≡∂/∂tand
vice versa.
A. Surface State of a 3D-TI
In the case of the surface state of a 3D-TI, as an ap-
proximation we can ignore ε0(/vectork) and consider the helical
term as the only kinetic term of the Hamiltonian. In this
casethelocalchargecurrentandthenonequilibriumlocal
spin density share a similar expression, /vectorI=/angb∇acketleft∂(/vectorh·/vector σ)/∂/vectork/angb∇acket∇ight.
For the conductivity, analogous to Eq. (10), we obtain,
σij=e/integraldisplayd2/vectork
4π2
/vectorh·∂/vectorh
∂ki×∂/vectorh
∂kj
2|/vectorh|2/parenleftBigg
f1−f2
|/vectorh|+f′
1+f′
2/parenrightBigg
δi/negationslash=j
+−η|∂/vectorh
∂ki|2+1
η(∂|/vectorh|2
∂ki)2
2|/vectorh|2(f′
1+f′
2)δij
(11)
This shows that the Fermi sea component of the density
matrixcontributesonlytotheanomalousHallconductiv-
ity which is in terms of a winding number. On the other
hand, the second term is finite only for the longitudinal
components of the conductivity and can be rewritten in
terms of the group velocity of the electrons in the system
which leads to the Drude-like formula.Should the linear dispersion approximation for the ki-
netic term in the Hamiltonian be valid in the range of
the energy scale corresponding to the magnetic exchange
coupling ∆ xc(i.e. when vF≫∆xc), the effect of the in-
plane component of the magnetic exchange coupling is to
shift the Dirac point (i.e. center of the k-space integra-
tion) which does not affect the result ofthe k-integration.
In this case after performing the partial time-momentum
derivatives, ( D(/vectorh) =∆xc
2∂/vector m
∂t−vFˆez×/vector∇µ), we use
/vectorh(/vectork,t) =vFˆez×/vectork+∆xc
2mz(t)ˆez, to obtain,
/vectorSneq(t) =/integraldisplaykdk
4π|/vectorh|2/parenleftBigg
/vectorS1f1−f2
|/vectorh|+(/vectorS1+/vectorS2)(f′
1+f′
2)/parenrightBigg
,
(12)
where,
/vectorS1(/vectork,t) =∆2
xc
4mz(t)ˆez×∂/vector m
∂t+∆xcvF
2mz(t)/vector∇µ(13)
/vectorS2(/vectork,t) =∆xc
4η(2η2−v2
F|k|2)(∂mx
∂tˆex+∂my
∂tˆey)
+∆xc
4η(2η2−∆2
xcm2
z
2)∂mz
∂tˆez
−vF
η(η2−v2
F|k|2
2)ˆez×/vector∇µ (14)
The dynamics of the FM obeys the LLG equation where
the conductions electrons insert torque on the FM mo-
ments through the magnetic exchange coupling,
∂/vector m
∂t=/vector m×
γ/vectorBext+∆xc
2/vectorSneq(t)−/summationdisplay
ijαij
0∂mi
∂tˆej
(15)
where,αij
0=αji
0, withi,j=x,y,z, is the intrinsic
Gilbert damping tensor of the FM in the absence of
the TI surface state and /vectorBextis the total magnetic field
applied on the FM aside from the contribution of the
nonequilibrium electrons.
While the terms that consist of /vector∇µare called SOT,
the ones that contain∂/vector m
∂tare generally responsible for
the damping of the FM dynamics. However, we no-
tice that ˆ ez×∂/vector m
∂tterm in Eq. (13) which arises from
the Berry curvature, becomes mz∂/vector m
∂tin the LLG equa-
tion that does not contribute to the damping and only
renormalizes the coefficient of the left hand side of the
Eq. (15). The second term in the Eq. (13), is the
anti-damping SOT pointing along ( ez×/vector∇µ)-axis. The
cone angle dependence of the anti-damping term can
be checked by assuming an electric field along the x-
axis when the FM precesses around the y-axis, (i.e.
/vector m(t) = cos(θ)ˆey+sin(θ)cos(ωt)ˆex+sin(θ)sin(ωt)ˆez). In
this case the average of the SOT along the y-axis in one
period of the precession leads to the average of the an-
tidamping SOT that shows a sin2(θ) dependence, which
is typical for the damping-like torques. Keeping in mind4
that in this section we consider vF≫∆xc, the first and
second terms in Eq. (14) show that the Gilbert damp-
ing increases as the precession axis goes from in-plane ( x
ory) to out of plane ( z) direction. Furthermore, when
the precession axis is in-plane (e.g. along y-axis), the
damping rate due to the oscillation of the out of plane
component of the magnetization ( ∂mz/∂t) has a sin4(θ)
dependence that can be ignored for low power measure-
ment of the Gilbert damping θ≪1. This leaves us with
the contribution from the in-plane magnetization oscilla-
tion (∂mx∂t) only. Therefore, the Gilbert damping for
in-plane magnetization becomes half of the case when
magnetization is out-of-plane. The anisotropic depen-
dence of the Gilbert damping can be used to verify the
existence of the surface state of the 3DTI as well as the
proximity induce magnetization at the interface between
a FM and a 3DTI. Finally, the third term in Eq. (14)
demonstrates a field like SOT with the same vector field
characteristics as the Rashba-Edelstein effect.
III. CONCLUSION
In conclusion, we have developed a linear response
NEGF framework which provides unified treatment of
both spin torque and damping due to SOC at interfaces.
We obtained the expressions for both damping and anti-
damping torques in the presence of a linear gradiance of
the electric field and adiabatic time dependence of the
magnetization dynamics for a helical state correspond-
ing to the surface state of a 3D topological insulator.
We present the exact expressions for the damping/anti-
damping SOT as well as the field like torques and showed
that, (i); Both Fermi surface and Fermi sea contribute
similarly to the anti-damping SOT as well as the Hall
conductivity and, ( ii); The Gilbert damping due to the
surface state of a 3D TI when the magnetization is in-
plane is less than the Gilbert damping when it is in the
out-of-plane direction. This dependence can be used as
a unique signature of the helicity of the surface states
of the 3DTIs and the presence of the proximity induced
magnetic exchange from the FM overlayer.
ACKNOWLEDGMENTS
We thank Branislav K. Nikoli´ c for the fruitful discus-
sions. F. M. and N. K. were supported by NSF PREM
Grant No. 1205734.Appendix A: Derivation of the Density Matrix
Using Eqs. (2) and . (4) it is straightforwardto obtain,
G<=(Gr
ad−Ga
ad)f−2ηf′∇µ·∂Gr
ad
∂kGa
ad
+i∂G<
ad
∂E∂H
∂tGa
ad+i∂Gr
ad
∂E∂H
∂tG<
ad
+i∂G<
ad
∂k·∇HGa
ad+i∂Gr
ad
∂k·∇HG<
ad(A1)
We plug in the expression for the adiabatic lesser GF in
equilibrium, G<
ad= 2iηfGr
adGa
ad= (Gr
ad−Ga
ad)f, and
obtain,
G<= (Gr
ad−Ga
ad)f−2ηf′∇µ·∂Gr
ad
∂kGa
ad
+if∂(Gr
ad−Ga
ad)
∂E∂H
∂tGa
ad+if∂Gr
ad
∂E∂H
∂t(Gr
ad−Ga
ad)
+if′(Gr
ad−Ga
ad)∂H
∂tGa
ad+if∂(Gr
ad−Ga
ad)
∂k·∇HGa
ad
+if∂Gr
ad
∂k·∇H(Gr
ad−Ga
ad). (A2)
Expanding the terms, leads to,
G<=/parenleftbigg
Gr
ad−Ga
ad+iGa
ad∂H(t)
∂t∂Ga
ad
∂E
−iGr
ad∂H
∂k·∇µ(x)∂Gr
ad
∂E−i∂Ga
ad
∂E∂H(t)
∂tGa
ad
+i∂Gr
ad
∂E∂H
∂k·∇µ(x)Gr
ad/parenrightbigg
f
+if′Gr
ad∇µ·∂H
∂k(Gr
ad−Ga
ad)
+if′(Gr
ad−Ga
ad)∂H
∂tGa
ad, (A3)
where,forthefirstandthirdlineswehaveusedtheiranti-
Hermitian forms instead. Since to calculate the density
matrix we integrate G<over energy, we can use integra-
tion by parts and obtain,
G<=/parenleftbigg
Gr
ad−Ga
ad+2iGa
ad∂H(t)
∂t∂Ga
ad
∂E
−2iGr
ad∂H
∂k·∇µ(x)∂Gr
ad
∂E/parenrightbigg
f
−if′/parenleftbigg
Gr
ad∇µ·∂H
∂kGa
ad−Gr
ad∂H
∂tGa
ad/parenrightbigg
= (Gr
ad−Ga
ad)f
+i/parenleftbigg
2Gr
adD(H)∂Gr
ad
∂Ef+Gr
adD(H)Ga
adf′/parenrightbigg
.(A4)
where we define, D=∂
∂t− ∇µ·∂
∂k. Finally, using the
identity,
2Gr
adD(H)∂Gr
ad
∂E=Gr
adD(H)∂Gr
ad
∂E−∂Gr
ad
∂ED(H)Gr
ad
+∂(Gr
adD(H)Gr
ad)
∂E, (A5)5
and performing the differential over the energy, we arrive
at Eq. (6).
Appendix B: Derivation of the Local Spin Density
From Eqs. (6) and (7), the local spin density can be written as, /vectorSneq=/vectorS(1)
neq+/vectorS(2)
neq, where /vectorS(1)
neqis due to the
nonequilibrium electrons at the Fermi surface and for a two band mo del can be calculated as the following,
/vectorS(1)
neq≈−2ηℑ/integraldisplaydE
2πTr
/parenleftBig
D(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig/parenleftBig
(E−ε0+iη)1+/vectorh·/vectorσ/parenrightBig
((E−ε0−iη)2−|/vectorh|2)((E−ε0+iη)2−|/vectorh|2)/vectorσ
+D(1
(E−ε0−iη)2−|/vectorh|2)((E−ε0+iη)1+/vectorh·/vectorσ)/vectorσ((E−ε0−iη)1+/vectorh·/vectorσ)
(E−ε0+iη)2−|/vectorh|2/bracketrightBigg
f′(E) (B1)
Preforming the trace over the Pauli matrix, we obtain,
/vectorS(1)
neq≈ −4η/integraldisplaydE
2π/parenleftBigg
ηD/vectorh+D(/vectorh)×/vectorh
((E−ε0−iη)2−|/vectorh|2)(E−ε0+iη)2−|/vectorh|2)
+4ℑ(/parenleftBig
/vectorh·D(/vectorh)+(E−ε0−iη)(/vector∇µ·/vector v0)/parenrightBig
(E−ε0)/vectorh
((E−ε0−iη)2−|/vectorh|2)2((E−ε0+iη)2−|/vectorh|2))
f′(E) (B2)
In the limit of small broadening, η≪ |/vectorh|, we obtain,
/vectorS(1)
neq≈−ηD/vectorh−/vectorh×D/vectorh+1
η/vectorh·D(/vectorh)/vectorh
2|/vectorh|2/parenleftBig
f′(ε0+|/vectorh|)+f′(ε0−|/vectorh|)/parenrightBig
−1
η(/vector∇µ·/vector v0)/vectorh
2|/vectorh|/parenleftBig
f′(ε0+|/vectorh|)−f′(ε0−|/vectorh|)/parenrightBig
(B3)
For the Fermi sea contribution to the nonequilibrium local spin densit y we have,
/vectorS(2)
neq≈ℜ/integraldisplaydE
2πTr/bracketleftBigg
∂
∂E/parenleftBigg
(E−ε0−iη)1+/vectorh·/vectorσ
(E−ε0−iη)2−|/vectorh|2/parenrightBigg/parenleftBig
D(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig(E−ε0−iη)1+/vectorh·/vectorσ
(E−ε0−iη)2−|/vectorh|2/vectorσ
−(E−ε0−iη)1+/vectorh·/vectorσ
(E−ε0−iη)2−|/vectorh|2/parenleftBig
D(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig∂
∂E/parenleftBigg
(E−ε0−iη)1+/vectorh·/vectorσ
(E−ε0−iη)2−|/vectorh|2/parenrightBigg
/vectorσ/bracketrightBigg
f(E). (B4)
Similarly, we obtain,
/vectorS(2)
neq≈ℜ/integraldisplaydE
2πTr
/bracketleftBig
D(/vectorh·/vectorσ)+/vector∇µ·/vector v01,(E−ε0−iη)1+/vectorh·/vectorσ/bracketrightBig
((E−ε0−iη)2−|/vectorh|2)2/vectorσ
f(E)
≈2ℑ/integraldisplaydE
πD(/vectorh)×/vectorh
((E−ε0−iη)2−|/vectorh|2)2f(E)
≈−1
2D(/vectorh)×/vectorh
|/vectorh|3/parenleftBig
f(ε0+|/vectorh|)−f(ε0−|/vectorh|)/parenrightBig
(B5)
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2309.08281v1.On_the_formation_of_singularities_for_the_slightly_supercritical_NLS_equation_with_nonlinear_damping.pdf | arXiv:2309.08281v1 [math.AP] 15 Sep 2023ON THE FORMATION OF SINGULARITIES FOR THE SLIGHTLY
SUPERCRITICAL NLS EQUATION WITH NONLINEAR DAMPING
PAOLO ANTONELLI AND BORIS SHAKAROV
Abstract. We consider the focusing, mass-supercritical NLS equation augmented with a nonlin-
ear damping term. We provide sufficient conditions on the nonl inearity exponents and damping
coefficients for finite-time blow-up. In particular, singula rities are formed for focusing and dissipa-
tive nonlinearities of the same power, provided that the dam ping coefficient is sufficiently small.
Our result thus rigorously proves the non-regularizing effe ct of nonlinear damping in the mass-
supercritical case, which was suggested by previous numeri cal and formal results.
We show that, under our assumption, the damping term may be co ntrolled in such a way that
the self-similar blow-up structure for the focusing NLS is a pproximately retained even within the
dissipative evolution. The nonlinear damping contributes as a forcing term in the equation for
the perturbation around the self-similar profile, that may p roduce a growth over finite time inter-
vals. We estimate the error terms through a modulation analy sis and a careful control of the time
evolution of total momentum and energy functionals.
1.Introduction
In this work, we consider the NLS equation with nonlinear damping
(1)/braceleftBigg
i∂tψ+∆ψ+|ψ|2σ1ψ+iη|ψ|2σ2ψ= 0,
ψ(0) =ψ0∈H1(Rd),
whereψ:R+×Rd→Candη>0 is the damping coefficient. More precisely, our goal is to investi-
gate the formation of singularities in finite time. Equation (1) arises a s an effective model in various
contexts, see for instance [13, 7, 6].
The nonlinear damping appearing in (1) is usually introduced as a regula rizing term of the singu-
lar dynamics provided by the focusing NLS equation. It is well known t hat the undamped NLS
equation (1) with η= 0 andσ1≥2
dmay experience the formation of singularity in finite time, see
[21, 24]. From the modeling point of view, this means that the NLS effec tive description fails to be
accurate close to the blow-up time and further effects, that were neglected in the derivation of the
NLS equation, become relevant near the singularity. Several phen omena may be taken into account
within the NLS description, see for instance [12] for a quite general overview. In this work, we focus
on nonlinear damping terms as in (1).
A relevant question in this perspective is to determine whether the n onlinear damping truly acts as
a regularization in the vanishing dissipation regime. More precisely, giv en the singular dynamics for
η= 0, we are interested in determining whether the regularized equat ion (1) has a global solution
for anyη>0, no matter how small.
The eventual (weak) limit as η→0 of such solutions (if it exists) may be seen as a possible criterion
to continue the solution beyond the singularity, in the same spirit of v anishing viscosity limits for
conservation laws [8]. To our knowledge, the only related rigorous re sults available in the literature
are due to Merle [30, 29], where Hamiltonian-type regularization is ado pted, see also [28] for a re-
lated result for the generalized KdV equation. On the other hand, s everal numerical simulations
12 P. ANTONELLI AND B. SHAKAROV
were performed to investigate this issue, see for instance [39, 14, 15, 17] and the references therein.
The regularizing property of nonlinear damping is already established in some cases. For2
d≤σ1<
σ2≤2
(d−2)+[4, 3] and for2
d=σ1=σ2[11], the Cauchy problem (1) is globally well-posed in H1
for anyη >0. On the other hand, there are also other cases where the dampin g does not act as a
regularization and the dynamics remains singular for sufficiently small values ofη >0. This is the
case for instance of a linear damping σ2= 0 [40, 37]. Moreover, in [11] it is also shown that, for
0≤σ2< σ1=2
d, it is possible to determine an open set of initial data that develop a sin gularity
in finite time. This is achieved by adapting the analysis developed in [33, 3 2, 31, 34, 38] by Merle
and Rapha¨ el for the mass-critical NLS, where they study the st ability of blow-up in a self-similar
regime.
In the physically relevant case2
d< σ1=σ2≤2
(d−2)+, this question remains unanswered in the
general case. In [3] the authors prove global well-posedness of ( 1) inH1only for sufficiently large η,
namely by requiring η≥min(σ1,√σ1). It is not clearwhether this result is sharp, howevernumerical
simulations [39, 14, 15] suggest that finite time blow-up still occurs f or small values of η.
This work provides a rigorous answer to this question in the slightly su percritical case, confirming
the numerical findings of [15]. In particular, we prove that for2
d<σ1=σ2<2
d+δ∗, whereδ∗>0
is sufficiently small, it is possible to provide an open set of initial data tha t develop a singularity in
finite time in a self-similar regime. Our result shows that the self-similar blow-up regime studied in
[36] remains unaltered even under the action of the dissipative effec ts encoded in (1). In fact, we are
going to prove a more general theorem on finite time blow-up for a lar ger class of nonlinear damping
terms, see Theorem 1.1 below.
We first introduce the following notations. Let sc=sc(σ1) be the Sobolev critical exponent associ-
ated withσ1,
(2) sc=d
2−1
σ1,
namelyscdetermines the critical regularity ˙Hscfor the well-posedness of equation (1) with η= 0.
We also define
(3) σ∗=2sc
d−2sc=scσ1, σ∗=2
d−2−σ1.
Moreover, we also set
(4) σ2,max=/braceleftbiggσ1ford≤3,
σ∗ford≥4.
Our main result is stated as follows.
Theorem 1.1. There exists σcrit>2
d, such that for any2
d< σ1< σcrit,σ∗< σ2≤σ2,maxthe
following holds true. There exists η∗=η∗(σ1)>0such that for any 0< η≤η∗, there exists an
open set O ⊂H1(Rd)such that if ψ0∈ Othen the corresponding solution ψ∈C([0,Tmax),H1(Rd))
to(1)develops a singularity in finite time, that is Tmax<∞and
lim
t→Tmax/ba∇dbl∇ψ(t)/ba∇dblL2=∞.
A more precise statement of our blow-up result is provided later in Th eorem 1.2. In particular,
the explicit blow-up rate is given there. As previously said, our proof follows the strategy developed
in [36] for the Hamiltonian dynamics, given by (1) with η= 0, which in turn exploited previous
results by the same authors related to the formation of singularitie s in the mass-critical case [31, 32].
More precisely, we construct a set of initial data whose evolution is a lmost self-similar. By a fine
control of the modulation parameters entering the description of the self-similar regime, it is thenBLOW-UP OF THE DAMPED NLS EQUATION 3
possible to show the occurrence of finite-time blow-up.
Let us emphasize that the introduction of a dissipative term in the dy namics introduces further
mathematicaldifficulties. First ofall, the self-similarprofileweconside rin ouranalysisisdetermined
by the undamped equations, see also (6) below, for instance. At pr esent, it is not even clear whether
it would be possible to determine a profile that takes into account also the dissipative term. A
generalized notion of dissipative solitons is present in the literature [ 1, 2, 23], where the profiles are
determined not only by the balance between dispersion and focusing effects but also between gain
and loss terms. In (1) the sole presence of a nonlinear damping cann ot be balanced by other effects.
The fact that the self-similar profile is determined by the Hamiltonian p art of the equation, yields
a non-trivial forcing term in the equation for the perturbation, giv en by the nonlinear damping
itself. Through a careful modulation analysis, we determine conditio ns under which this forcing can
be controlled, so that the perturbation is shown to be sufficiently sm all with respect to the self-
similar profile. In particular, in the case σ1=σ2, the control is determined by imposing a smallness
condition on η>0, as stated in the main theorem above.
Moreover,asecondmaindifficultyisthatinourcasethefunctionalsr elatedtothephysicalquantities
such as total mass, momentum and energy are - straightforward ly - not conserved in time anymore.
We thus need a suitable control on the time evolution of these quant ities that will in turn provide
the necessary bounds on the perturbation and the modulation par ameters.
Ford≥4, the restriction σ2≤σ∗<σ1prevents us to consider the case σ2=σ1. This is a technical
condition, needed to ensure the validity of the Sobolev embedding H1(Rd)֒→L2(σ1+σ2+1)(Rd) in
(17), (18) below, see also Section 5. On the other hand, the condit ionσ∗<σ2is motivated by the
fact that we cannot control Sobolev norms ˙Hsnorms with s < scin the self-similar regime, see
(103). We remark that for the undamped dynamics (1) with η= 0, it was proved in [35] that all
radially symmetric blowing-up solutions leave the critical space ˙Hscat blow-up time.
Let us further remark that the smallness condition η≤η∗is only necessary when σ2=σ1. In
the caseσ1> σ2, it is also possible to show that for any η, there exists a set of initial conditions
depending on η,σ1andσ2whose corresponding solutions blow-up in finite time. We will further
discuss this point in Section 6 and throughout this work.
Moreover, as it will be clear in our analysis, the smallness of η∗is determined in terms of the
smallness of sc, which is related to σ1through identity (2). Indeed, we will see that η∗∼s3
c. For
this reason, with some abuse of notation, in what follows we often wr iteη∗(sc) instead of η∗(σ1),
wherescandσ1are related by identity (2).
Finally, ourresultprovidesadditionalevidencethatthemasssuper criticalself-similarcollapseisvery
different from the one occurring in the mass critical case. In the lat ter case, indeed, any damping
coefficientη>0 regularizes the dynamics and prevents the formation of finite time blow-up [11, 4].
In Section 6 we will provide an argument explaining why the solution esc apes the self-similar regime
in the case σ1=σ2=2
dfor anyη>0. We will now present a road map of how Theorem 1.2 will be
proved.
1.1.Singularity formation. We start with an initial condition which can be decomposed as a
soliton profile and a small perturbation
(5) ψ0(x) =λ−1
σ1
0/parenleftbigg
Qb0/parenleftbiggx−x0
λ0/parenrightbigg
+ξ0/parenleftbiggx−x0
λ0/parenrightbigg/parenrightbigg
eiγ0,
where 0< b0,λ0≪1 are small parameters, and Qb0is roughly a localized radial solution to the
stationary equation
(6) ∆ Qb0−Qb0+ib0/parenleftbigg1
σ1Qb0+x·∇Qb0/parenrightbigg
+|Qb0|2σ1Qb0= 0.4 P. ANTONELLI AND B. SHAKAROV
It is known (see, for instance, [39, 36]) that self-similar blowing-up s olutions of the supercritical NLS
focus as a zero energy solution to (6) plus a non-focusing radiation . For any initial value Qb0(0) and
anyb0∈R, nontrivial solutions to (6) exist [41, 9], but any zero energy solutio n does not belong
toL2(Rd) [25] and thus we employ a suitable localization in space. Moreover, th e parameter b0is
chosen to be close to a value b∗=b∗(sc)>0. By continuity, we may find a time interval where the
solution can be decomposed as
(7) ψ(t,x) =1
λ1
σ1(t)/parenleftbigg
Qb(t)/parenleftbiggx−x(t)
λ(t)/parenrightbigg
+ξ/parenleftbigg
t,x−x(t)
λ(t)/parenrightbigg/parenrightbigg
eiγ(t)
where the parameter bis still close enough to b∗andλandξare still small enough. By perturbation
techniques, we choose the parameters b,λ,x,γ so thatξsatisfies four suitable orthogonality condi-
tions.
Next, we will use a local virial law and a suitable Lyapunov functional t o prove that the parame-
terb(t) is trapped around the value b∗for all times. This yields the following law for the scaling
parameter
λ(t)∼/radicalBig
−2b∗t+λ2
0.
In particular, there exists a time Tmax(sc,λ0)>0 such that λ(t)→0 ast→Tmax. Consequently,
the kinetic energy of the solution diverges
lim
t→Tmax/ba∇dbl∇ψ(t)/ba∇dblL2=∞.
We will use the coercivity property stated in Proposition2.12 below to provethe dynamical trapping
of the parameter b. In order to control the six negative directions on the right-hand side of (37), we
will use four orthogonality conditions implied by the selection of the pa rametersb,λ,x,γ and two
almost orthogonal conditions which come from suitable bounds on th e energy and the momentum
of the solution.
In ourcase, therearetwomoredifficulties thatdo notarisewhen η= 0. First, since Qbapproximates
a solution ofthe undamped NLS, one issue will be to show that the con tribution ofthe dampingterm
in (1) can be considered of a smaller order with respect to the rest o f the dynamics. in particular,
the damping term generates a forcing in the equation of the remaind er, which will be controlled
using the smallness of ηandλ. Second, the presence of the damping implies that the energy
E[ψ(t)] =/integraldisplay1
2|∇ψ(t)|2−1
2σ+2|ψ(t)|2σ+2dx
and the momentum
P[ψ(t)] =/parenleftbig
∇ψ(t),iψ(t)/parenrightbig
are not conserved. But to show the dynamical trapping of the par ameterbaroundb∗, we needE
andPto remain small enough to control two negative directions in the coe rcivity (37) below. In
comparison with the undamped case, this is not a trivial consequenc e implied by the smallness of the
energy and momentum of the initial condition. Thus, we will study the ir time evolution and show
that they remain sufficiently small until the blow-up time under the as sumptiond≤3,σ2≤σ1and
η≤η∗(sc) ord≥4 andσ2<σ∗. Finally, we recall that the localized profile Qbis close inH1(Rd)
to the ground state of the undamped NLS which is the unique real-va lued solution Q∈H1(Rd)
[27, 18] to the elliptic equation
(8) ∆ Q−Q+|Q|2σ1Q= 0.
In light of the discussion above, Theorem 1.1 will be the consequence of the following result.BLOW-UP OF THE DAMPED NLS EQUATION 5
Theorem 1.2. There exists s∗
c>0such that for any 0<sc<s∗
candσ∗<σ2≤σ1whend≤3or
σ∗<σ2<σ2,maxwhend≥4, there exists η∗(sc)>0such that if η≤η∗, then there exists an open
setO ⊂H1(Rd)such that if ψ0∈ O, then for any t∈[0,Tmax)the corresponding maximal solution
to(1)can be written as
ψ(t,x) =1
λ1
σ1(t)/parenleftbigg
Q/parenleftbiggx−x(t)
λ(t)/parenrightbigg
+ζ/parenleftbigg
t,x−x(t)
λ(t)/parenrightbigg/parenrightbigg
eiγ(t),
whereλ,γ∈C1([0,Tmax);R),x∈C1([0,Tmax),Rd),
lim
t→Tmaxλ(t) = 0,lim
t→Tmaxx(t) =x∞∈Rd,
there exists 0<δ(sc) =δ≪1such that
/ba∇dbl∇ζ/ba∇dblL∞([0,Tmax),L2)≤δ,
and the blow-up rate is given by
/ba∇dbl∇ψ(t)/ba∇dbl2
L2∼(Tmax−t)−(1−sc).
More precisely, we will show that if we define
(9) b∗=−π(1+δ)
ln(sc)
then the law of the scaling parameter will satisfy the following bounds
(10) λ2(0)−2(1+δ)b∗t≤λ2(t)≤λ2(0)−2(1−δ)b∗t.
Remark 1.3.Recently, in [5], the authors provided a complete description of a ze ro energy solution
to (6) forσ1∈/parenleftbig2
d,2
d+ε/parenrightbig
andεsmall enough.
2.Preliminaries
We start this section with a list of notations that we are going to use t hroughout this work.
We use the symbol A∼B, to indicate the fact that there exist two constants C1,C2>0 such that
C1B≤A≤C2B.
For anyf∈H1(Rd), we define the operators
(11) Λ f=1
σ1f+x·∇f, Df=d
2f+x·∇f.
We notice that
(12) Λf=Df−scf.
Moreover, by integrating by parts we have
(13) ( f,Λg) =−2sc(f,g)−(g,Λf)
and
(14) ∆Λ f= 2∆f+Λ∆f.
We use the following notation
1
(d−2)+=/braceleftBigg
∞,ford≤2,
1
(d−2),ford≥3.
Next, we recall some preliminary results.6 P. ANTONELLI AND B. SHAKAROV
Definition 2.1. We say that a pair ( q,r) is admissible if 2 ≤q,r≤ ∞, (q,r,d)/\e}atio\slash= (2,∞,2) and
2
q=d/parenleftbigg1
2−1
r/parenrightbigg
We will exploit the following Strichartz estimates [20, 26].
Theorem 2.2. For everyφ∈L2(Rd)and every (q,r)admissible, there exists a constant C >0
such that for any t>0,
/ba∇dbleit∆φ/ba∇dblLq((0,t),Lr)≤C/ba∇dblφ/ba∇dblL2.
Moreover, if f∈Lγ′((s,t),Lρ′(Rd))where(γ,ρ)is an admissible pair, and
N(f) =/integraldisplayt
sei(t−τ)∆f(τ)dτ,
then there exists a constant C >0such that for any (q,r)admissible, we have
/ba∇dblN(f)/ba∇dblLq((s,t),Lr)≤C/ba∇dblf/ba∇dblLγ′((s,t),Lρ′).
Byusingthe Strichartzestimates, itis possibletoprovethe localex istenceofsolutionstoequation
(1) (see [3, Proposition 2 .3]).
Theorem 2.3. Letη∈R,σ1,σ2<2/(d−2)ifd≥3. Then for any ψ0∈H1(Rd)there exists
Tmax>0and a unique solution ψto(1)such that for any (q,r)admissible, we have
(15) ψ,∇ψ∈C/parenleftbig
[0,Tmax),L2(Rd)/parenrightbig
∩Lq
loc/parenleftbig
(0,Tmax),Lr(Rd)/parenrightbig
.
Moreover, either Tmax=∞orTmax<∞and
lim
t→Tmax/ba∇dbl∇ψ(t)/ba∇dblL2=∞.
The usual physical quantities (the total mass, the total energy and the momentum) are governed
by the following time-dependent functions [10].
Theorem 2.4. Letψ0∈H1(Rd)andψ∈C/parenleftbig
[0,Tmax),H1(Rd)/parenrightbig
the corresponding solution to (1).
Then the total mass, the total energy and the momentum satisf y
(16) M[ψ(t)] =/integraldisplay
|ψ(t)|2dx=M[ψ0]−2η/integraldisplayt
0/ba∇dblψ(s)/ba∇dbl2σ2+2
L2σ2+2ds,
(17)E[ψ(t)] =/integraldisplay1
2|ψ(t)|2−1
2σ1+2|ψ(t)|2σ1+2dx
=E[ψ0]+η/integraldisplayt
0/integraldisplay
|ψ|2σ1+2σ2+2−|ψ|2σ2|∇ψ|2
−2σ2|ψ|2σ2−2Re/parenleftbig¯ψ∇ψ/parenrightbig2dxds,
and
(18) P[ψ(t)] =/parenleftbig
∇ψ(t),iψ(t)/parenrightbig
=P[ψ0]−2η/integraldisplayt
0/integraldisplay
|ψ|2σ2Im(¯ψ∇ψ)dxds,
respectively.
Remark 2.5.The dissipative terms appearing in (16), (17) and (18) are always we ll defined because
of Strichartz estimates in Theorem 2.3, see (15).BLOW-UP OF THE DAMPED NLS EQUATION 7
We also notice that if σ2<σ1<1
d−2, then it follows
4σ2+2≤2(σ1+σ2+1)≤4σ1+2<2d
(d−2)+.
Thus, by the Sobolev embedding theorem, the positive term on the r ight-hand side of (17) can be
bounded by/integraldisplayt
s/integraldisplay
|ψ|2(σ1+σ2+1)dxdτ≤(t−s)/ba∇dblψ/ba∇dbl2(σ1+σ2+1)
L∞((s,t),H1).
Analogously, we can also estimate the term on the right-hand side of (18) as
/integraldisplayt
s/integraldisplay
|ψ|2σ2Im(¯ψ∇ψ)dxdτ/lessorsimilar(t−s)/ba∇dblψ/ba∇dbl2σ2+1
L4σ2+2/ba∇dbl∇ψ/ba∇dblL2/lessorsimilar(t−s)/ba∇dblψ/ba∇dbl2σ2+2
H1.
These computations will be exploited in Section 5. Indeed, they will be crucial to control uniformly
in time the right-hand side of (17) and (18).
2.1.Construction of the Approximated Soliton Core. In this subsection, we are going to
constructasuitablelocalizedsolutiontothe stationaryequation(6 ), thatwillconstitutethe blowing-
up soliton core in the self-similar regime. This construction already ap peared in [36, Section 2],
however for the sake of completeness we are going to present the main related results and properties
here.
Let us notice that the damping term does not enter into the constr uction of the approximated blow-
up core.
Letρ∈(0,1),b>0, we define the following radii
Rb=2
b/radicalbig
1−ρ, R−
b=/radicalbig
1−ρRb.
Letφb∈C∞(Rd) be a radially symmetric cut-off function such that
(19) φb(x) =/braceleftBigg
0,for|x| ≥Rb,
1,for|x| ≤R−
b,0≤φb(x)≤1.
We also denote the open ball in Rdof radiusRband centered at the origin by
B(0,Rb) ={x∈Rd:|x|<Rb}.
We recall that scdenotes the Sobolev critical exponent defined in (2). We start with the following
lemma.
Lemma 2.6. There exists σ(1)
c>2
dandρ(1)>0, such that for any2
d< σ1< σ(1)
c,ρ∈(0,ρ(1)),
there exists b(1)(ρ)>0, such that for any 0< b≤b(1), there exists a unique radial solution
Pb∈H1
0(B(0,Rb))to the elliptic equation
/parenleftbigg
−∆+1−b2|x|2
4/parenrightbigg
Pb−P2σ1+1
b= 0,
withPb>0inB(0,Rb). Moreover Pb∈C3(B(0,Rb))and
/ba∇dblPb−Q/ba∇dblC3→0
asb→0, whereQthe unique positive solution to (8). Finallyb(1)(ρ)→0asρ→0.8 P. ANTONELLI AND B. SHAKAROV
This lemma was stated in [36, Proposition 2 .1] and its proof is a straightforward adaptation of
that given for [31, Proposition 1] in the mass-critical case. To prov e the existence, one shows that
Pbis a suitably scaled minimizer of the functional
F(u) =/integraldisplay
B(0,Rb)|∇u|2+/parenleftbigg
1−b2|x|2
4/parenrightbigg
|u|2dx
in the set
U=/braceleftbig
u∈H1
0,rad(B(0,Rb)) :/ba∇dblu/ba∇dblL2σ1+2= 1/bracerightbig
,
whereH1
0,rad(B(0,Rb))isthesubsetof H1
0(B(0,Rb))containingradiallysymmetricfunctions. Notice
that the functional Fis bounded from below in B(0,Rb) and−∆+1−b2|x|2
4is an uniformly elliptic
operator. In particular, both the maximum principle and standard r egularity results for elliptic
equations are satisfied, see [19] for instance.
Next, we define the function
ˆQb=φbe−ib|x|2
4Pb,
in the whole Rdspace, where the cut-off φbis defined in (19). Notice that the profile ˆQb∈H1(Rd)
satisfies the equation
(20) ∆ ˆQb−ˆQb+ib/parenleftbiggd
2ˆQb+x·∇ˆQb/parenrightbigg
+|ˆQb|2σ1ˆQb=−˜Ψb,
where the remainder term ˜Ψbcomes from the localization in space and is defined by
(21) ˜Ψb=/parenleftbig
2∇φb·Pb+Pb(∆φb)+(φ2σ1+1
b−φb)P2σ1+1
b/parenrightbig
e−ib|x|2
4.
We recall some properties of the profile ˆQbwhich were shown in [36, Proposition 2 .1].
Proposition 2.7. For any polynomial pand anyk= 0,1, there exists a constant C >0such that
(22)/vextenddouble/vextenddouble/vextenddouble/vextenddoublepdk
dxk˜Ψb/vextenddouble/vextenddouble/vextenddouble/vextenddouble
L∞≤e−C
|b|.
Moreover, the function ˆQbsatisfies
P[ˆQb] =/parenleftBig
∇ˆQb,iˆQb/parenrightBig
= 0,/parenleftBig
ΛˆQb,iˆQb/parenrightBig
=/parenleftBig
x·∇ˆQb,iˆQb/parenrightBig
=−b
2/ba∇dblxˆQb/ba∇dbl2
L2,
whereΛis defined in (11)and
d
db2/parenleftbigg/integraldisplay
|ˆQb|2dx/parenrightbigg
|b=0=C(σ1),
withC(σ1)→C >0asσ1→2
d.
Notice that heuristically ˆQbprovides an approximating solution to (6) in the ball B(0,Rb). In
the complementary region, Rd\B(0,Rb) nonlinear effects become negligible, hence it is sufficient
to study the outgoing radiation defined in the next lemma, that first appeared in [32, Lemma 15].
The results in the lemma below will be exploited, in particular, to estimat e the mass flux leaving
the collapsing core in Lemma 4.6 below.
Lemma 2.8. There exists ρ2>0such that for any 0<ρ<ρ 2there exists b2(ρ)>0such that for
any0<b<b 2, there exists a unique radial solution ζb∈˙H1(Rd)to
(23) ∆ ζb−ζb+ib/parenleftbiggd
2ζb+x·∇ζb/parenrightbigg
=˜Ψb,BLOW-UP OF THE DAMPED NLS EQUATION 9
where˜Ψbis defined in (21). Moreover there exists C >0such that
(24) Γ b= lim
|x|→+∞|x|d|ζb(x)|2/lessorsimilare−C
b,
and there exists c>0such that for |x| ≥R2
b
(25) e−(1+cρ)π
b≤4
5Γb≤ |x|d|ζb(x)|2≤e−(1−cρ)π
b.
Furthermore, the following estimates hold
(26) /ba∇dbl∇ζb/ba∇dbl2
L2≤Γ1−cρ
b,
/vextenddouble/vextenddouble/vextenddouble|y|d
2(|ζb|+|y||∇ζb|)/vextenddouble/vextenddouble/vextenddouble
L∞(|y|≥Rb)≤Γ1
2−cρ
b.
Finally, we have that/vextenddouble/vextenddouble/vextenddouble|ζb|e−|y|/vextenddouble/vextenddouble/vextenddouble
C2(|y|≤Rb)≤Γ3
5
b,/ba∇dbl∂bζb/ba∇dblC1≤Γ1
2−cρ
b.
Remark 2.9.We observe that the profile ζb∈˙H1
rad(Rd) is not inL2(Rd) because of the logarithmic
divergence implied by (24), (25).
We now want to suitably modify the profile ˆQbto obtain an approximating solution to
(27) i ∂tQb+∆Qb−Qb+ib/parenleftbigg1
σ1Qb+x·∇Qb/parenrightbigg
+|Qb|2σ1Qb= 0,
whereb=b(t) is a function depending on time. Notice that if we suppose that ˙b(t) = 0 anddσ1= 2,
thenˆQbis already an approximating solution to (27), see (20) and (22). In o ur case, we suppose
that there exists β(b)>0 such that ˙b(t) =βsc. In other words, we search for localized solutions to
the following equation
(28) i scβ∂bQb+∆Qb−Qb+ib/parenleftbigg1
σ1Qb+x·∇Qb/parenrightbigg
+|Qb|2σ1Qb= 0.
We find a solution to this equation as a suitable perturbation of the pr ofileˆQb. We make the
following ansatz on Qb,
Qb=ˆQb+scTb,
by requiring that it solves equation (28) inside the ball |x| ≤R−
b, up to an error of order s1+C
cfor
someC >0. Such a solution was already studied in [36, Proposition 2 .6].
Proposition 2.10. There exists ˜σc>2
dandρ3>0such that for any2
d< σ1<˜σc,ρ∈(0,ρ3)
there exists b3(ρ)>0such that for any 0<b<b 3there exists a radial function Tb∈C3(Rd)and a
constantβ >0such thatQb=e−ib|y|2
4(Pbφb+scTb)satisfies
(29) iscβ∂bQb+∆Qb−Qb+ibΛQb+|Qb|2σ1Qb=−Ψb
where
(30) Ψb=˜Ψb+Φb,
and we have
(31)/vextenddouble/vextenddouble/vextenddouble/vextenddoubledk
dxkΦb/vextenddouble/vextenddouble/vextenddouble/vextenddouble
L∞(|x|≤R−
b)/lessorsimilars1+C
c,
/vextenddouble/vextenddouble/vextenddouble/vextenddoubledk
dxkΦb/vextenddouble/vextenddouble/vextenddouble/vextenddouble
L∞(|x|≥R−
b)/lessorsimilarsc,10 P. ANTONELLI AND B. SHAKAROV
fork= 0,1andC >0. Moreover, Qbsatisfies
(32)|E[Qb]|/lessorsimilarΓ1−cρ
b+sc,
P[Qb] = (∇Qb,iQb) = 0,
(ΛQb,iQb) = (x·∇Qb,iQb) =−b
2/ba∇dblxQ/ba∇dbl2
L2(1+δ1(sc,b)),
/ba∇dblQb/ba∇dbl2
L2=/ba∇dblQ/ba∇dbl2
L2+O(sc)+O(b2),
whereδ1(sc,b)→0as asb,sc→0andcρ≪1is defined in (26)andQis a solution to (8). The
profileQbsatisfies also the uniform estimate
(33)/vextendsingle/vextendsingle/vextendsingle/vextendsingleP(x)dk
dxkQb(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−(1+c)|x|
for any polynomial P(x). Finally, we have that
(34)/vextenddouble/vextenddouble/vextenddoublee(1−ρ)π
4|x|Tb/vextenddouble/vextenddouble/vextenddouble
C3+/vextenddouble/vextenddouble/vextenddoublee(1−ρ)π
4|x|∂bTb/vextenddouble/vextenddouble/vextenddouble
C2+|∂bβ(b)|/lessorsimilar1.
We will now show a useful Pohozaev-type estimate for the profile Qb.
Proposition 2.11. LetQbbe a solution to (29). Then we have
(35) 2 E[Qb]−(iscβ∂bQb+Ψb,ΛQb) =sc/parenleftbig
2E[Qb]+/ba∇dblQb/ba∇dbl2
L2/parenrightbig
.
Proof.We take the scalar product of equation (29) with Λ Qb
0 = (∆Qb−Qb+ibΛQb+|Qb|2σ1Qb,ΛQb)+(iscβ∂bQb+Ψb,ΛQb).
Now by integration by parts, we observe that
(∆Qb+|Qb|2σ1Qb,1
σ1Qb+x·∇Qb) = 2(sc−1)E[Qb],
and
−(Qb,ΛQb) =sc/ba∇dblQb/ba∇dbl2
L2.
as a consequence, we obtain that
0 = 2(sc−1)E[Qb]+sc/ba∇dblQb/ba∇dbl2
L2+(iscβ∂bQb+Ψb,ΛQb)
which is equivalent to (35). /square
Finally, by applying the operator Λ to (29), we see that Λ Qbsatisfies the following equation
(36)iβscΛQb+Λ∆Qb−ΛQb+ibΛ(ΛQb)+|Qb|2σ1ΛQb
+2σ1Re(¯Qbx·∇Qb)|Qb|2σ1−2Qb=−ΛΨb,
that will be used in the Appendix.
2.2.Coercivity Property. We conclude the section by discussing the coercivity properties of t he
linearized operator around the ground state in the mass-critical c ase. Since our analysis deals with
the slightly mass-supercritical case, we derive a similar property by perturbative arguments. Let us
first define the following quadratic form associated with the linearize d operator around the ground
state in the mass-critical case,
(37)H(f,f) =/ba∇dbl∇f/ba∇dbl2
L2+2
d/parenleftbigg
1+4
d/parenrightbigg/integraldisplay
Q4
d−1
c(y·∇Qc)Re(f)2dy
+2
d/integraldisplay
Q4
d−1
c(y·∇Qc)Im(f)2dy.BLOW-UP OF THE DAMPED NLS EQUATION 11
whereQcis the ground state profile for the mass-critical NLS, namely is the u nique positive solution
to
(38) ∆ Qc−Qc+|Qc|4
dQc= 0.
We recall that
DQc=2
dQc+y·∇Qc.
We statethefollowingcoercivitypropertyforthe quadraticform H. Letusremarkthat, eventhough
we present it as a proposition, the following result was proved rigoro usly only in the one-dimensional
case [33], while for the multi-dimensional case d≤10 it was proved only numerically [16, 42].
Proposition 2.12. For anyf∈H1(Rd)there exists c>0such that
(39)H(f,f)≥c/parenleftbigg
/ba∇dbl∇f/ba∇dbl2
L2+/integraldisplay
|f|2e−|y|dy/parenrightbigg
−c/parenleftbigg
(f,Qc)2+(f,|y|2Qc)2+(f,yQc)2+(f,iDQc)2
+(f,iD(DQc))2+(f,i∇Qc)2/parenrightbigg
.
3.Setting up the Bootstrap
In this section, we are going to define the set of initial conditions tha t we consider for our analysis.
These data are suitable perturbations of the scaled self-similar solit on profile defined in Proposition
2.10 and are going to form singularities in finite time. In what follows, we also discuss how to
estimate the solution in such a way that it retains its self-similar struc ture along the evolution.
Finally, we are going to show that there exists a finite time when the sc aling parameter λ(t), related
to the size of the solution, goes to zero, thus exhibiting the format ion of a singularity. The idea is
to find a set that is almost invariant under the dynamics of equation ( 1) and to use a bootstrap
argument to show that the solution remains inside this set until the s caling parameter becomes zero
and consequently the solution blows up.
We use the quantity ρ3and the profile Qbdefined in Proposition 2.10, and the constant Γ bdefined
in (24). Moreover, we denote by
˜sc=d
2−1
˜σc,
where ˜σcis determined in Proposition 2.10. In the next definition and subseque nt propositions, it
will be useful to write the assumptions on σ1by exploiting identity (2). This means that every time
a condition on scis found, such as for instance 0 < sc< Scfor some constant Sc, it should be
interpreted as a condition on the exponent σ1<σ1(Sc) whereσ1(Sc) is given by
σ1(Sc) =2
d−2Sc.
Definition 3.1. Let 0<sc<˜scand 0< ρ<ρ 3. We define the set O ⊂H1(Rd) as the family of
all functions φ∈H1such that there exist λ0,b0>0,x0∈Rd,γ0∈Randξ0∈H1(Rd) such that
(40) φ(x) =λ−1
σ1
0/parenleftbigg
Qb0/parenleftbiggx−x0
λ0/parenrightbigg
+ξ0/parenleftbiggx−x0
λ0/parenrightbigg/parenrightbigg
eiγ0,
with the following conditions: there exists 0 <ν(ρ)≪1 such that
(41) Γ1+ν10
b0≤sc≤Γ1−ν10
b0,12 P. ANTONELLI AND B. SHAKAROV
the scaling parameter satisfies
(42) 0<λ0<Γ100
b0,
the initial momentum and energy are bounded as
(43) λ2−2sc|E[φ]|+λ1−2sc|P[φ]|<Γ50
b0,
there exists s=s(σ1,σ2) in the interval
(44) sc<s<min/parenleftbiggdσ1
2σ1+2,dσ2
2σ2+2,1
2/parenrightbigg
such that the following inequality is verified
(45)/integraldisplay
||∇|sξ0|2+|∇ξ0|2+|ξ0|2e−|y|dy<Γ1−ν
b0,
where
y=x−x0
λ.
Finally, the remainder ξ0satisfies also the orthogonality conditions
(46) ( ξ0,|y|2Qb0) = (ξ0,yQb0) = (ξ0,iΛ(ΛQb0)) = (ξ0,iΛQb0) = 0.
In this definition, the constant ν=ν(ρ) is chosen so that
(47) Γ1−cρ
b0≤Γ1−ν50
b0,
wherec>0 is the constant in (25). Note that from the definition of Γ bin (25) and from condition
(41), it follows that
sc∼Γb∼e−(1+cρ)π
b
that is
b0∼b∗=−π(1+δ)
ln(sc)
namelyb0is chosen to be close to the value b∗(sc)>0 defined in (9). The constant s=s(σ1,σ2) is
chosen so that the L2σ1+2andL2σ2+2norms ofξmay be controlled by interpolating between ˙Hs
and˙H1, see (78) and (103) below. Finally, we recall that the set Ois non-empty, see [36, Remark
2.10]. We remark that this is the same set defined in [36], whose evolution s develop a singularity in
finite time. In this work, we are going to prove that the same initial da ta produce singular solutions
also under the dissipative dynamics (1). Thus, our result implies that the nonlinear damping is not
able to regularize the dynamics and to prevent the formation of sing ularities, in the cases under
our consideration. This is not in contradiction with the global well-pos edness result proven in [3],
where the condition η≥min(σ1,√σ1) was needed. In fact, here we prove our main result under a
smallness assumption on η, namely we require
(48) η≤s3
c.
Moreover, in Section 6 we are going to provide an alternative definitio n of the set O, in the case
σ2< σ1. In particular, hypothesis (48) will no longer be needed, even thou gh the set will depend
onσ1,σ2andη.
Now, let us consider an initial datum ψ0∈ Oand letψ∈C([0,Tmax),H1(Rd)) be the corresponding
solution to (1), where Tmax≤ ∞is its maximal time of existence. By continuity of the solution,
there exists a time interval [0 ,T1), withT1≤Tmax, where the inequalities in Definition 3.1 are
still valid but with slightly larger bounds. Moreover, by standard per turbation techniques, we can
also preserve the orthogonal conditions in (46) for any t∈[0,T1) by modulational analysis, see forBLOW-UP OF THE DAMPED NLS EQUATION 13
instance [31, Lemma 2] where this was proved for the Hamiltonian dyn amics. The proof of a similar
property for solutions to the dissipative dynamics (1) follows straig htforwardly.
Proposition 3.2. There exists 0< T1≤Tmax,b,λ,γ∈C1([0,T1),R),x∈C1([0,T1),Rd)and
ξ∈C([0,T1),H1(Rd))such that for all t∈[0,T1), the solution ψto(1)may be decomposed as
(49) ψ(t,x) =λ(t)−1
σ1/parenleftbig
Qb(t)(y)+ξ(t,y)/parenrightbig
eiγ(t),
with
(50) y=x−x(t)
λ(t),
whereξsatisfies the following orthogonal conditions:
(51) ( ξ(t),|y|2Qb(t)) = (ξ(t),yQb(t)) = (ξ(t),iΛ(ΛQb(t))) = (ξ(t),iΛQb(t)) = 0,
and we have
Γ1+ν2
b(t)≤sc≤Γ1−ν2
b(t), (52)
0≤λ(t)≤Γ10
b(t), (53)
λ2−2sc|E[ψ(t)]|+λ1−2sc(t)|P[ψ(t)]| ≤Γ2
b(t), (54)
/integraldisplay
||∇|sξ(t)|2dy≤Γ1−50ν
b(t), (55)
/integraldisplay
|∇ξ(t)|2+|ξ(t)|2e−|y|dy≤Γ1−20ν
b(t). (56)
We notice that (52) and the smallness of νandscimply that Γ b(t)<1 for anyt∈[0,T1). Our
main goal is to prove that the solution remains in this self-similar regime untilTmax, that is the
decomposition of ψ, along with the properties of the modulation parameters as stated in Proposition
3.2 are valid until the maximal time of existence. This is achieved by usin g a bootstrap argument.
We show that the bounds in Proposition 3.2 can be improved and thus t he self-similar regime may
be extended in time. This amounts to finding a dynamical trapping of t he parameter bto improve
(52) and (56), to find a suitable differential equation for λto improve (53), and to prove that the
energy and the momentum of the solution remain small enough to impr ove (54). The trapping of the
control parameter bwill be achieved in Section 4 by finding a lower bound for ˙busing a virial-type
argument, and an upper bound with a monotonicity formula. The equ ation satisfied by the scaling
parameterλwill be a direct consequence of the dynamical trapping of b(t) aroundb∗, see Section 5.
Finally, the controls on EandPwill be obtained in Section 5 as a consequence of the choice of the
damping parameter ηin (48). In fact, we will show the following bootstrap result.
Proposition 3.3. There exists s∗
c>0, such that for any sc< s∗
c, there exist smax(sc)> sc, and
ν∗(sc)>0such that for any sc<s<s max(sc)and0<ν <ν∗and for any t∈[0,T1),the following
inequalities are true:
Γ1+ν4
b(t)≤sc≤Γ1−ν4
b(t), (57)
0≤λ(t)≤Γ20
b(t), (58)
λ2−2sc|E[ψ(t)]|+λ1−2sc|P[ψ(t)]| ≤Γ3−10ν
b(t), (59)
/integraldisplay
||∇|sξ(t)|2≤Γ1−45ν
b(t), (60)
/integraldisplay
|∇ξ(t)|2+|ξ(t)|2e−|y|dy≤Γ1−10ν
b(t). (61)14 P. ANTONELLI AND B. SHAKAROV
As a consequence of this proposition, we see that the solution satis fies improved bounds with
respect to those stated in Proposition 3.2. A standard continuity a rgument then implies that the
same bounds are satisfied in the whole interval [0 ,Tmax).
Let us remark that the same dynamical trapping argument to show self-similar blow-up was already
exploited in [36]. In particular, this implies that a similar argument to [36] works also in our case
for the damped NLS, under the assumptions of Theorem 1.2. On the other hand, dealing with the
dissipative dynamics (1) entails new mathematical difficulties with resp ect to [36]. First of all, we
notice that the self-similar profile Qb, given by equation (29), does not determine an approximate
solution to (1), not even close to the collapsing core. In particular, this implies that the equation
for the perturbation ξbears a forcing term of order one, depending on Qb. In the case under our
consideration here, we can see that the forcing term produces an error that becomes non-negligible
on a range of times of order/parenleftBig
ηλ2−2σ2
σ1/parenrightBig−1
.
We overcome this difficulty by choosing the parameters (and in partic ular, the damping coefficient
in the case σ1=σ2) in such a way that Tmax≪/parenleftBig
ηλ2−2σ2
σ1/parenrightBig−1
.
The second mathematical difficulty induced by the dissipative dynamic s is that the global quantities,
such as total momentum and total energy, are not conserved alo ng the flow of (1), see (18) and (17),
respectively. Consequently, while the bootstrap condition (54) is s traightforwardly satisfied in the
Hamiltonian case, a fundamental step in our analysis would be to cont rol the time evolution of these
quantities.
In Section 6 we will also show that in the case σ2< σ1, it is possible to deal with these issues by
choosing the scaling parameter λsmall enough and depending on σ1,σ2andηso that the initial
condition is very close to the blow-up point and the damping term is not strong enough to force the
solution out of the self-similar regime before the blow-up time.
As the last step, we will show that inside the self-similar regime, the sc aling parameter λ(t) goes to
zero.
In particular, the following Corollary can be inferred from Propositio n 3.2.
Corollary 3.4. For anysc<s∗
cwheres∗
cis defined in Proposition 3.3, the bounds in Proposition
3.2 are valid for any t∈[0,Tmax). Moreover, there exists a constant 0< C=C(ν,sc)≪1such
that for any t∈[0,Tmax),
(62) λ2
0−2(1+C)t≤λ2(t)≤λ2
0−2(1−C)t.
As a consequence of the corollary above, there exists Tmax(ν,sc,λ0)<∞such thatλ→0 as
t→Tmax(ν,sc,λ0) andλbehaves like
λ2(t)∼Tmax−t.
Moreover, the estimate (62) also provides a blow-up rate for the L2-norm of the gradient of the
solution, as we have
/vextenddouble/vextenddouble∇ψ(t)/vextenddouble/vextenddouble2
L2=λ2(sc−1)(t)/vextenddouble/vextenddouble∇(Qb(t)+ξ(t))/vextenddouble/vextenddouble2
L2∼(Tmax−t)−(1−sc).
4.Estimates on the Modulation Parameters
In this section, we start our analysis of solutions emanating from init ial conditions in the set O,
defined in Definition 3.1. Let ψ0∈ O, we denote by ψ∈C([0,Tmax),H1(Rd)) its corresponding
maximal solution, with Tmax≤ ∞. Then we decompose the solution as the sum of a soliton core
and remainder as in (49). By Proposition 3.2, the bounds (52) - (56) are satisfied for all t∈[0,T1],
together with the four orthogonal conditions in (51).
The purpose of this section is to obtain preliminary estimates on the p arameters that are requiredBLOW-UP OF THE DAMPED NLS EQUATION 15
to prove the dynamical trapping of bin the next section. A crucial step will be to use the coercivity
property stated in Proposition 2.12. Even though this proposition is related to the mass critical
case, we will use the smallness of scand property (12) to obtain a similar coercivity property in the
slightly super-critical case. We will then show that the negative par t of the right-hand side of (39)
is controlled by the positive part. When computing the quadratic for m (39) on the perturbation ξ,
we see that four of the negative terms appearing in are small enoug h because of the orthogonality
conditions (51), the smallness of scand property (12). We now show how to control the two
remaining negative terms.
By plugging the decomposition (49) into equation (1), we obtain the f ollowing equation for the
perturbation ξ,
i∂τξ+Lξ+i(˙b−βsc)∂bQb−i/parenleftBigg˙λ
λ+b/parenrightBigg
Λ(Qb+ξ)−i˙x
λ·∇(Qb+ξ)
−(˙γ−1)(Qb+ξ)+iηλ2−2σ2
σ1|Qb+ξ|2σ2(Qb+ξ)+R(ξ)−Ψb= 0,(63)
where the scaled time τis defined according to
(64) τ(t) =/integraldisplayt
01
λ2(v)dv,
and we recall that
Λf=1
σ1f+y·∇f.
In (63) the dot denotes the derivative with respect to τand the space derivatives are intended to be
performed with respect to the scaled space variable ywhere
(65) y=x−x(t)
λ(t).
Moreover, we used the notation Lfor the linearized operator around Qb,
Lξ= ∆ξ−ξ+ibΛξ+2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb+|Qb|2σ1ξ, (66)
andRcontains all nonlinear terms in ξ,
(67)R(ξ) =|Qb+ξ|2σ1(Qb+ξ)−|Qb|2σ1Qb−2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb−|Qb|2σ1ξ,
whereas Ψ bis defined in (30).
We notice that the damping term yields a forcing in the equation for ξ, given by
(68) i ηλ2−2σ2
σ1|Qb|2σ2Qb
depending on Qb,ηandλ. One of the main goals in our subsequent analysis is to exploit the
smallness of the product ηλ2−2σ2
σ1in order to obtain a suitable control on the nonlinear damping
term so that the self-similar regime is maintained along the evolution. T his task is achieved by the
selection of ηfor instance as in (48) or by the smallness of λforσ2<σ1, see Section 6. Let us notice
that the assumption σ∗<σ2≤σ2,max, see (3) and (4), implies that
0≤2−2σ2
σ1<2(1−sc)
ford≤3, while
0<4(d−2)σ1−1
(d−2)σ1≤2−2σ2
σ1<2(1−sc),
ford≥4.
In what follows, we will use the following estimates on scalar products .16 P. ANTONELLI AND B. SHAKAROV
Lemma 4.1. For anyt∈[0,T1), any polynomial P(y)and any integers k∈ {0,1,2,3}andn∈
{0,1}, we have
(69)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg
ξ,Pdk
dykQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/parenleftbigg/integraldisplay
|ξ|2e−|y|dy/parenrightbigg1
2
,
(70)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg
ξ,Pdk
dyk∂bQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/parenleftbigg/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1
2
,
(71)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbiggdk
dykQb,Pdn
dynΨb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg
ξ,Pdn
dynΨb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓ1−ν
b,
(72)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg
∂bQb,Pdk
dykQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg
|y|2Qb,Pdk
dykQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar1.
These inequalities have been proven in [31, Lemma 4]. They follow from t he Cauchy-Schwarz
inequalityand theuniform estimate(33). In the next twoLemmas4.2 and4.3weprovethe smallness
of the scalar products ( ξ(t),Qb(t)) and (ξ,i∇Qb(t)), so to conclude our control of the negative terms
appearing in (39) up to additional terms controlled by the smallness o fsc. This will be achieved by
exploiting the balance laws satisfied by the total momentum and ener gy (18) and (17), respectively,
together with the bounds (54).
Lemma 4.2. There exists ν1>0such that for any ν <ν1andt∈[0,T1), we have
(73) |(ξ(·,t),Qb(t))|/lessorsimilar/integraldisplay
|ξ|2e−|y|+|∇ξ|2dy+Γ1−2ν
b(t).
Let us remark that, by combining (73) with the bound (56), we would obtain the following
estimate
(74) |(ξ(·,t),Qb(t))|/lessorsimilarΓ1−20ν
b(t).
Although this rougher bound is sufficient to show the coercivity of th e linearized operator, the
estimate (73) will be crucial to prove Proposition 3.3.
Proof.We plug the decomposition of the solution (49) into the energy equat ion to obtain
(75)2λ2−2scE[ψ] = 2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2
L2−2(∆Qb,ξ)
−1
σ1+1/integraldisplay
|Qb+ξ|2σ1+2−|Qb|2σ1+2dy.
By using equation (29) satisfied by Qbin the expression above, we get
2λ2−2scE[ψ] = 2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2
L2
+2(iscβ∂bQb−Qb+ibΛQb+Ψb,ξ)−/integraldisplay
R(2)(ξ)dy,
whereR(2)(ξ) is defined by
(76) R(2)(ξ) =1
σ1+1/parenleftbig
|Qb+ξ|2σ1+2−|Qb|2σ1+2−(2σ1+2)|Qb|2σ1Re(Qb¯ξ)/parenrightbig
.
Equivalently, we write the identity above as
2(Qb,ξ) =−2λ2−2scE[ψ]+2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2
L2
+2(iscβ∂bQb+ibΛQb+Ψb,ξ)−/integraldisplay
R(2)(ξ)dy.(77)BLOW-UP OF THE DAMPED NLS EQUATION 17
Now we bound the terms on the right-hand side of (77). For the firs t term, we use the control on
the energy (54)/vextendsingle/vextendsingle−2λ2−2scE[ψ]/vextendsingle/vextendsingle≤2Γ2
b.
For the second term, we use the properties of Qblisted in (32), inequality (47) and the control (52)
onscto obtain that/vextendsingle/vextendsingleE[Qb]/vextendsingle/vextendsingle≤Γ1−cρ
b+sc≤Γ1−ν50
b+Γ1−ν2
b.
For the fourth term, we use again the control (52) on sc, estimates (70), (71) and (56) to get
|sc(iβ∂bQb,ξ)+(Ψ b,ξ)|/lessorsimilar/parenleftBigg
sc/parenleftbigg/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1
2
+Γ1−ν
b/parenrightBigg
/lessorsimilar/parenleftBig
Γ1−ν2
bΓ1
2−10ν
b+Γ1−ν
b/parenrightBig
.
Moreover, we also observe that (i bΛQb,ξ) = 0 from (51). Finally, we use estimates (69) and (56)
to control the remainder term R(2)(ξ) defined in (76) as follows,
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
R(2)(ξ)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy+/ba∇dblξ/ba∇dbl2σ1+2
L2σ1+2.
In order to bound the L2σ1+2-norm ofξ, we observe that since
sc<s<d
2−d
2σ1+2=dσ1
2σ1+2=s(σ1)
then by Sobolev embedding and subsequent interpolation between H1(Rd) andHs(Rd), we have
(78) /ba∇dblξ/ba∇dbl2(σ1+1)
L2(σ1+1)/lessorsimilar/ba∇dbl|∇|s(σ1)ξ/ba∇dbl2(σ1+1)
L2≤ /ba∇dblξ/ba∇dbl2θ(σ1+1)
˙H1/ba∇dblξ/ba∇dbl2(1−θ)(σ1+1)
˙Hs,
for someθ=θ(s)∈(0,1), wheresis determined in Definition 3.1, see (44). Now by using the
controls (56) and (55) we obtain
/ba∇dbl∇ξ/ba∇dbl2θ(σ1+1)
˙H1/ba∇dbl∇ξ/ba∇dbl2(1−θ)(σ1+1)
˙Hs ≤Γ(1−20ν)θ(σ1+1)
bΓ(1−50ν)(1−θ)(σ1+1)
b
≤Γ(1−50ν)(σ1+1)+30 νθ(1+σ1)
b.
By collecting everything together, we obtain that
|(ξ,Qb)|/lessorsimilarΓ2
b+Γ1−ν50
b+Γ1−ν2
b+/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy
+Γ1−ν2
bΓ1
2−10ν
b+Γ1−ν
b+Γ(1−50ν)(σ1+1)+30 νθ(1+σ1)
b.
By choosing νsmall enough, we have thus obtained estimate (73). /square
We also exploit the equation of the momentum (18) to derive a bound o n (ξ(·,t),i∇Qb(t)), as in
the following lemma.
Lemma 4.3. There exists ν(1)>0such that for any 0<ν <ν(1)and anyt∈[0,T1), we have
(79) |(ξ(·,t),i∇Qb(t))| ≤Γ1−50ν
b(t).
Proof.We plug decomposition (49) into the momentum (18) to obtain
2(ξ,i∇Qb) =−λ1−2scP[ψ]+P[Qb]+P[ξ].
The first term on the right-hand side is bounded by (54),
/vextendsingle/vextendsingle−λ1−2scP[ψ]/vextendsingle/vextendsingle≤Γ2
b.18 P. ANTONELLI AND B. SHAKAROV
Next, we observe that by the properties of Qb, we haveP[Qb] = 0, see (32). For the last term, since
s<1
2, we use (44) to estimate
|P[ξ]|/lessorsimilar/ba∇dblξ/ba∇dbl2
˙H1
2/lessorsimilar/ba∇dblξ/ba∇dbl2θ
˙H1/ba∇dblξ/ba∇dbl2(1−θ)
˙Hs
where
θ=2−4s
2−s>0.
Now we use estimates (56) and (55) to obtain
/ba∇dblξ/ba∇dbl2θ
˙H1/ba∇dblξ/ba∇dbl2(1−θ)
˙Hs≤Γ(1−20ν)θ
bΓ(1−50ν)(1−θ)
b= Γ(1−50ν)+30νθ
b.
By combining all previous estimates, we get
|(ξ,i∇Qb)|/lessorsimilarΓ2
b+Γ(1−50ν)+30νθ
b.
Inequality (79) thus follows by choosing νsufficiently small. /square
Our next step is to obtain suitable estimates on the modulational par ameters defined in the
decomposition (49). This is accomplished by exploiting the equation (6 3) satisfied by the remainder
ξ, and the orthogonality conditions listed in (51).
Lemma 4.4. For anyt∈[0,T1), we have
(80)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle˙λ(τ)
λ(τ)+b(τ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+|˙b(τ)|/lessorsimilarΓ1−20ν
b(τ),
and/vextendsingle/vextendsingle/vextendsingle/vextendsingle(˙γ(τ)−1)−1
/ba∇dblΛQb(τ)/ba∇dbl2
L2(ξ(τ),LΛ(ΛQb(τ))/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle˙x(τ)
λ(τ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
/lessorsimilarδ2/parenleftbigg/integraldisplay
|∇ξ(τ)|2+|ξ(τ)|2e−|y|dy/parenrightbigg1
2
+Γ1−20ν
b(τ),(81)
whereδ2=δ2(sc)>0. Moreover, δ2(sc)→0, assc→0.
Proof.The lemma is proved by taking the scalar product of equation (63) wit h suitable terms that
allow us to exploit the orthogonality conditions (51). The same appro ach was already used in [36],
see Lemma 3.1 and Proposition 3.3 therein, see also [38, Appendix A], to study the undamped
dynamics. For this reason we write equation (63) as
(82) U(ξ)+iηλ2−2σ2
σ1|Qb+ξ|2σ2(Qb+ξ) = 0,
so that, in what follows, we exploit the analysis already developed in [36 ].
Let us first consider the bound on |˙b|. We take the scalar product of (82) with Λ Qb. Following the
computations in Appendix A, we use estimates (69), (74), (79), (5 5) and (56) to obtain
|˙b|/lessorsimilar/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy+Γ1−11ν
b+ηλ2−2σ2
σ1/vextendsingle/vextendsingle/parenleftbig
i|Qb+ξ|2σ2(Qb+ξ),ΛQb/parenrightbig/vextendsingle/vextendsingle.
Similarly, we obtain the estimate for other parameters
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle˙λ
λ+b/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy+Γ1−11ν
b+ηλ2−2σ2
σ1/vextendsingle/vextendsingle/parenleftbig
|Qb+ξ|2σ2(Qb+ξ),|y|2Qb/parenrightbig/vextendsingle/vextendsingle.BLOW-UP OF THE DAMPED NLS EQUATION 19
and
/vextendsingle/vextendsingle/vextendsingle/vextendsingle(˙γ−1)−1
/ba∇dblΛQb/ba∇dbl2
L2(ξ,LΛ(ΛQb)/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle˙x
λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarδ2/parenleftbigg/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1
2
+/integraldisplay
|∇ξ|2dy+Γ1−11ν
b
+ηλ2−2σ2
σ1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbig
|Qb+ξ|2σ2(Qb+ξ),yQb/parenrightbig
+/parenleftbig
i|Qb+ξ|2σ2(Qb+ξ),Λ(ΛQb)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle.
Let us now control the contributions coming from the damping term . We write
|Qb+ξ|2σ2(Qb+ξ) =|Qb|2σ2Qb+R(1)(ξ).
We use (72) to obtain that
(83)/vextendsingle/vextendsingle/parenleftbig
i|Qb|2σ2Qb,ΛQb+Λ(ΛQb)+iyQb+i|y|2Qb/parenrightbig/vextendsingle/vextendsingle/lessorsimilar1.
On the other hand, by using (69) and (56), we also have that
/vextendsingle/vextendsingle/vextendsingle/parenleftBig
R(1)(ξ),ΛQb+Λ(ΛQb)+iyQb+i|y|2Qb/parenrightBig/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/parenleftbigg/integraldisplay
|∇ξ|2dy+/integraldisplay
|ξ|2e−|y|dy/parenrightbigg1
2
.
Thus it follows that
(84)ηλ2−2σ2
σ1/vextendsingle/vextendsingle/parenleftbig
i|Qb+ξ|2σ2(Qb+ξ),ΛQb+Λ(ΛQb)+iyQb+i|y|2Qb/parenrightbig/vextendsingle/vextendsingle
/lessorsimilarηλ2−2σ2
σ1/parenleftbigg
1+/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1
2
/lessorsimilarΓ3−3ν2
bΓ20−20σ2
σ1
b≤Γ2
b,
where we used the hypothesis on η(48), (52) and (53). /square
One can see that the estimates (81) and (80) are the same as thos e in [36, Lemma 3 .1] for the
undamped case η= 0. This is because our choice of ηandσ2≤σ1imply that the damping term is
of lower order with respect to other terms in equation (63). in part icular, the scalar products with
the forcing term defined in (68) are well-controlled by the smallness o fηλ2−2σ2
σ1.
4.1.Local Virial Law. We will now derive suitable inequalities on ˙bto prove that bis trapped
around the value b∗defined in (9). We first obtain a lower bound, see (85) below. This est imate
is connected with the local virial law for the remainder ξ. For details, see [33, Section 3] and [31,
Section 4].
Lemma 4.5. There exists s(2)
c>0such that for any sc<s(2)
cand for any t∈[0,T1), there exists
C >0such that
(85) ˙b(t)≥C/parenleftbigg
sc+/integraldisplay
|∇ξ(t)|2+|ξ(t)|2e−|y|dy−Γ1−ν6
b(t)/parenrightbigg
.20 P. ANTONELLI AND B. SHAKAROV
Proof.By taking the scalar product of (63) with Λ Qbwe obtain
0 = (i∂τξ,ΛQb)+˙b(i∂bQb,ΛQb)−(iβsc∂bQb+Ψb,ΛQb)+(Lξ+R(ξ),ΛQb)
−/parenleftBigg
i/parenleftBigg˙λ
λ+b/parenrightBigg
ΛQb+i˙x
λ·∇Qb+(˙γ−1)Qb,ΛQb/parenrightBigg
−/parenleftBigg
i/parenleftBigg˙λ
λ+b/parenrightBigg
Λξ+(˙γ−1)ξ+i˙x
λ·∇ξ,ΛQb/parenrightBigg
+ηλ2−2σ2
σ1(i|Qb+ξ|2σ2(Qb+ξ),ΛQb).
Notice that only the last term in this equation depends on the damping . The computation involving
the other terms will be shown in Appendix A following the exposition of [ 36, Proposition 3 .3]. These
computations yield the following inequality
˙b≥C/parenleftbigg
sc+/integraldisplay
|∇ξ|2dy+/integraldisplay
|ξ|2e−|y|dy−Γ1−ν6
b/parenrightbigg
−ηλ2−2σ2
σ1/vextendsingle/vextendsingle(i|Qb+ξ|2σ2(Qb+ξ),ΛQb)/vextendsingle/vextendsingle,
for someC >0. On the other hand, the contribution of the damping term is negligib le as we control
it with the calculations in the previous lemma (see (84))
ηλ2−2σ2
σ1/vextendsingle/vextendsingle(i|Qb+ξ|2σ2(Qb+ξ),ΛQb)/vextendsingle/vextendsingle/lessorsimilarΓ2
b.
/square
4.2.Refined virial estimate. In this subsection, we derive an upper bound for ˙b. We will study
the mass flux escaping the self-similar soliton core region. The outgo ing radiation ζbdefined in
Lemma 2.8 will play a central role in this task. Let φ∈C∞
c(Rd) be a radial cut-off defined by
φ(r)∈[0,1], φ(r) =/braceleftBigg
1 forr∈[0,1),
0 forr≥2,
We also define
φA(r) =φ/parenleftBigr
A/parenrightBig
whereA=A(t) is determined by
(86) A(t) = Γ−a
b(t)
anda=a(ν)>0 will be chosen later. We denote by ˜ζb=φAζbthe localization of the outgoing
radiation. We notice that the equation for the outgoing radiation (2 3) implies that ˜ζbsatisfies the
equation
(87) ∆ ˜ζb−˜ζb+ib/parenleftbiggd
2+y·∇/parenrightbigg
˜ζb=φA˜Ψb+F=˜Ψb+F
where
(88) F= (∆φA)ζb+2∇φA·∇ζb+iby·∇φAζb,
and˜Ψbwas defined in (21). In particular, by recalling that suppΨ b⊂B(0,Rb)⊂B(0,2
b) and by
the definition of A(t), we have that φA˜Ψb=˜Ψb. Moreover, from the properties of ζbestablished in
Lemma 2.8, we infer that ˜ζb∈H1
radand, by suitably choosing a=a(ν)>0, we also have
(89) /ba∇dbl˜ζb/ba∇dbl2
H1≤Γ1−cρ
b.BLOW-UP OF THE DAMPED NLS EQUATION 21
We define the following refined soliton core and remainder,
˜Qb=Qb+˜ζband˜ξ=ξ−˜ζb,
so that we decompose the solution as
(90) ψ(t,x) =λ(t)−1
σ1/parenleftbigg
˜Qb(t)/parenleftbiggx−x(t)
λ(t)/parenrightbigg
+˜ξ/parenleftbigg
t,x−x(t)
λ(t)/parenrightbigg/parenrightbigg
eiγ(t).
We now claim that also the new soliton core ˜Qbis an approximating solution to (28). By using (87)
and (29), we obtain
iβsc∂b˜Qb+∆˜Qb−˜Qb+ibΛ˜Qb+|˜Qb|2σ1˜Qb= iβsc∂bQb+∆Qb−Qb
+ibΛQb+|Qb|2σ1Qb
+∆˜ζb−˜ζb+i/parenleftbiggd
2˜ζb+y·∇˜ζb/parenrightbigg
−iscb˜ζb
−iscb˜ζb+iscβ∂b˜ζb
+|Qb+˜ζb|2σ1(Qb+˜ζb)−|Qb|2σ1Qb
=−Ψb+˜Ψb+F−iscb˜ζb+iscβ∂b˜ζb
+|Qb+˜ζb|2σ1(Qb+˜ζb)−|Qb|2σ1Qb.
We observe that by using (30), we get −Ψb+φA˜Ψb= Φb. Thus the new profile ˜Qbsatisfies
(91) i βsc∂b˜Qb+∆˜Qb−˜Qb+ibΛ˜Qb+|˜Qb|2σ1˜Qb=−˜Φb+F
where
(92) ˜Φb=−Φb−iscb˜ζb+iβ∂b˜ζb+|Qb+˜ζb|2σ1(Qb+˜ζb)−|Qb|2σ1Qb,
andFis defined in (88). By using the equation (91) for ˜Qband decomposition (90), we derive the
equation for ˜ξ,
i∂τ˜ξ+˜L˜ξ+i(˙b−βsc)∂b˜Qb−i/parenleftBigg˙λ
λ+b/parenrightBigg
Λ(˜Qb+˜ξ)−i˙x
λ·∇(˜Qb+˜ξ)
−(˙γ−1)(˜Qb+˜ξ)+iηλ2−2σ2
σ1|˜Qb+˜ξ|2σ2(˜Qb+˜ξ)+˜R(˜ξ)−˜Φb+F= 0,(93)
where
˜L˜ξ= ∆˜ξ−˜ξ+ibΛ˜ξ+2σ1Re(˜ξ˜Qb)|˜Qb|2σ1−2˜Qb+|˜Qb|2σ1˜ξ,
and
˜R(˜ξ) =|˜Qb+˜ξ|2σ1(˜Qb+˜ξ)−|˜Qb|2σ1˜Qb−2σ1Re(˜ξ˜Qb)|˜Qb|2σ1−2˜Qb−|˜Qb|2σ1˜ξ.
We recall again that the scaled time and space denoted by τandyare defined in (64) and (65),
respectively. With some abuse of notations, in what follows we often explicit the τ−dependence of
functions. For instance, we denote ˜ξ(τ), to actually mean ˜ξ(t−1(τ)).
Byexploitingagainthe equation(93) forthe remainder ˜ξ, the controls(81) and(80)and estimates
in Proposition 3.2, we obtain the following bounds.
Lemma 4.6. There exists C >0such that any t∈[0,T1), there exists C >0such that
C2/parenleftbigg/integraldisplay
|∇˜ξ(τ)|2+|˜ξ(τ)|2e−|y|dy+Γb(τ)/parenrightbigg
≤C/parenleftbiggd
dτf(τ)+sc/parenrightbigg
+/integraldisplay
{A(τ)≤|y|≤2A(τ)}|ξ(τ)|2dy,(94)22 P. ANTONELLI AND B. SHAKAROV
where
(95) f(τ) =−1
2(i˜Qb(τ),y·∇˜Qb(τ))−(i˜ξ(τ),Λ˜Qb(τ)).
Proof.As for Lemmas 4.4 and 4.5, we conveniently write equation (93) as
U(1)(˜ξ)+iηλ2−2σ2
σ1|˜Qb+˜ξ|2σ2(˜Qb+˜ξ) = 0,
so that forU(1)((˜ξ) we again exploit the analysis already presented in [36, Lemma 3 .5], [34, Lemma
6] and reported in Appendix B . By taking the scalar product of equa tion (93) with Λ ˜Qb, we obtain
(96) ( P(˜ξ),Λ˜Qb)+(iηλ2−2σ2
σ1|˜Qb+˜ξ|2σ2(˜Qb+˜ξ),Λ˜Qb) = 0.
Exploiting the computations in Appendix B yields the following inequality
d
dτf≥C/parenleftbigg/integraldisplay
|∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg
−1
C/parenleftBigg
sc+/integraldisplay2A
A|ξ|2dy/parenrightBigg
−ηλ2−2σ2
σ1/vextendsingle/vextendsingle/vextendsingle(i|˜Qb+˜ξ|2σ2(˜Qb+˜ξ),Λ˜Qb)/vextendsingle/vextendsingle/vextendsingle.
We will now show that in this inequality, the contribution of the damping term can be considered
negligible.
From (89) and (72) it follows that
(97)/vextendsingle/vextendsingle/vextendsingle/parenleftBig
i|˜Qb|2σ2˜Qb,Λ˜Qb/parenrightBig/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/parenleftBig
i|Qb+˜ζb|2σ2(Qb+˜ζb),Λ(Qb+˜ζb)/parenrightBig/vextendsingle/vextendsingle/vextendsingle
≤/vextendsingle/vextendsingle/parenleftbig
i|Qb|2σ2Qb,ΛQb/parenrightbig/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/parenleftBig
i|Qb|2σ2Qb,Λ˜ζb/parenrightBig/vextendsingle/vextendsingle/vextendsingle
+/vextendsingle/vextendsingle/vextendsingle/parenleftBig
i|Qb+˜ζb|2σ2(Qb+˜ζb)−i|Qb|2σ2Qb,Λ(Qb+˜ζb)/parenrightBig/vextendsingle/vextendsingle/vextendsingle.
We already know from (83) that/vextendsingle/vextendsingle/parenleftbig
i|Qb|2σ2Qb,ΛQb/parenrightbig/vextendsingle/vextendsingle/lessorsimilar1.
For the other terms in (97), we use (89) and (47) to obtain
/vextendsingle/vextendsingle/vextendsingle/parenleftBig
i|Qb|2σ2Qb,˜ζb/parenrightBig/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/parenleftBig
i|Qb+˜ζb|2σ2(Qb+˜ζb)−i|Qb|2σ2Qb,Λ(Qb+˜ζb)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓ1−ν
b.
Thus we obtain that
ηλ2−2σ2
σ1/vextendsingle/vextendsingle/vextendsingle(i|˜Qb+˜ξ|2σ2(˜Qb+˜ξ),Λ˜Qb)/vextendsingle/vextendsingle/vextendsingle/lessorsimilarηλ2−2σ2
σ1≤Γ2
b
where we used again (48), (52) and (53). Inequality (94) follows st raightforwardly from (166) and
(52). /square
Notice that the control (94) is the same as that in [36, Lemma 3 .5]. This is again because the
contribution of the damping term is negligible in the self-similar regime du e to the choice of ηand
the fact that σ2≤σ1.
We proceed by finding a suitable bound for the mass flux term/integraldisplay
{A(τ)≤|y|≤2A(τ)}|ξ|2dy
that appears on the right-hand side of (94). To do this, we introdu ce a further radial, smooth cut-off
χ∈[0,1] withχ(r) = 0 forr≤1
2, andχ(r) = 1 forr≥3, withχ′≥0 andχ′(r)∈/bracketleftbig1
4,1
2/bracketrightbig
for
1≤r≤2. Let
(98) χA(r) =χ/parenleftBigr
A/parenrightBigBLOW-UP OF THE DAMPED NLS EQUATION 23
whereA=A(t) is defined in (86). In the following lemma, we will choose s>0 depending also on
σ2in order to be able to control the L2σ2+2-norm of the remainder ξ, see (103) below.
Lemma 4.7. For anyt∈[0,T1), we have
(99)b(τ)/integraldisplay
{A(τ)≤|y|≤2A(τ)}|ξ(τ)|2dy/lessorsimilarλ−2sc(τ)d
dτ/parenleftbigg
λ2sc(τ)/integraldisplay
χA|ξ(τ)|2dy/parenrightbigg
+Γ3
2−10ν
b(τ)+Γ2a
b(τ)/ba∇dbl∇ξ(τ)/ba∇dbl2
L2.
Proof.We take the scalar product of equation (1) with i χA/parenleftBig
x−x(t)
λ(t)/parenrightBig
ψ(t,x) and obtain
1
2d
dt(ψ,χAψ)−(ψ,(∂tχA)ψ)−(∇xψ,i(∇xχA)ψ)+η(|ψ|2σ2ψ,χAψ) = 0.
By using decomposition (49) and the scaled space and time variables, we rewrite the equation above
as
(100)0 =1
2λ2scd
dτ/parenleftbig
λ2sc(ξ,χAξ)/parenrightbig
−(ξ,(∂τχA)ξ)−(∇ξ,i(∇χA)ξ)
+ηλ2−2σ2
σ1(|ξ|2σ2ξ,χAξ)+R(1)(Qb,ξ),
where the gradient and the scalar products are taken with respec t to the variable
y(τ) =x−x(τ)
λ(τ)
and
R(1)(Qb,ξ) =1
2λ2scd
dτ/parenleftbig
λ2sc((Qb,χAQb)+2(Qb,χAξ))/parenrightbig
−(Qb,(∂τχA)Qb)
−2(Qb,(∂τχA)ξ)−(∇Qb,i(∇χA)Qb)−2(∇Qb,i(∇χA)ξ)
+ηλ2−2σ2
σ1/parenleftbig
(|Qb+ξ|2σ2(Qb+ξ),χA(Qb+ξ))−(|ξ|2σ2ξ,χAξ)/parenrightbig
.
We now estimate the reminder term R(1)(Qb,ξ). Let us recall that for |y| ≥2
bwe haveQb=scTb,
whereTbis exponentially decreasing, see Proposition 2.10. We claim that, for t he terms depending
only onQb, we have
/vextendsingle/vextendsingle/vextendsingleR(2)(Qb)/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
2λ2scd
dτ/parenleftbig
λ2sc(Qb,χAQb)/parenrightbig
−(Qb,(∂τχA)Qb)−(∇Qb,i(∇χA)Qb)
+ηλ2−2σ2
σ1/parenleftbig
|Qb|2σ2Qb,χAQb)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle
/lessorsimilars2
c+ηλ2−2σ2
σ1s2σ2+2
c.
Here we used the bound
d
dτ/parenleftbig
λ2sc(Qb,χAQb)/parenrightbig
=s2
cd
dτ/parenleftbigg
λ2sc/integraldisplay
χA|Tb|2dy/parenrightbigg
/lessorsimilars3
cλ2sc/parenleftBigg˙λ
λ−b+b/parenrightBigg
/ba∇dblQb/ba∇dbl2
L2+s2
cλ2sc˙b∂b/ba∇dblQb/ba∇dbl2
L2
/lessorsimilars3
cλ2sc/parenleftbigg
Γ1
2−10ν
b−c
lnsc/parenrightbigg
+s2
cλ2scΓ1
2−10ν
b/lessorsimilars2
c.24 P. ANTONELLI AND B. SHAKAROV
that follows from the properties of Qb(32). All other terms may be estimated by
/vextendsingle/vextendsingle/vextendsingleR(1)(Qb,ξ)−R(2)(Qb)/vextendsingle/vextendsingle/vextendsingle/lessorsimilarsc/parenleftbigg/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1
2
.
To prove the two claims above, we use the following property
|(Tb,ξ)| ≤/parenleftbigg/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1
2
,
that can be proven similarly to (69) and (70). Thus by using (52), (5 3) and (48), we obtain that
/vextendsingle/vextendsingle/vextendsingleR(1)(Qb,ξ)/vextendsingle/vextendsingle/vextendsingle/lessorsimilars2
c+ηλ2−2σ2
σ1s2σ2+2
c+sc/parenleftbigg/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1
2
/lessorsimilarΓ3
2−10ν
b.
In particular, from (100) we get the following inequality
1
2λ2scd
dτ/parenleftbig
λ2sc(ξ,χAξ)/parenrightbig
≥(ξ,(∂τχA)ξ)+(∇ξ,i(∇χA)ξ)
−ηλ2−2σ2
σ1(|ξ|2σ2ξ,χAξ)−Γ3
2−10ν
b.
From straightforward computations, we see that
∂τχA(τ)/parenleftbiggx−x(τ)
λ(τ)/parenrightbigg
=−1
A/parenleftBigg
˙x
λ+/parenleftBigg˙λ
λ+˙A
A/parenrightBigg
y/parenrightBigg
·∇χ/parenleftbiggx−x(τ)
λ(τ)A(τ)/parenrightbigg
.
and consequently, we can rewrite the inequality above as
(101)1
2λ2scd
dτ/parenleftbig
λ2sc(ξ,χAξ)/parenrightbig
≥b(ξ,y·∇χAξ)
−/parenleftBigg
ξ,1
A/parenleftBigg
˙x
λ+/parenleftBigg˙λ
λ+b/parenrightBigg
y+˙A
Ay/parenrightBigg
·(∇χ)ξ/parenrightBigg
+(∇ξ,i(∇χA)ξ)−ηλ2−2σ2
σ1/integraldisplay
χA|ξ|2σ2+2dy−Γ3
2−10ν
b.
The definition of χA(98) implies the following chain of estimates
(102)1
8/integraldisplay
{A≤|y|≤2A}|ξ|2dy≤1
2/integraldisplay
{A≤|y|≤2A}χ′(|y|
A)|ξ|2dy≤1
2/integraldisplay
{A≤|y|≤2A}|y|
Aχ′(|y|
A)|ξ|2dy
=1
2/integraldisplay
{A≤|y|≤2A}y
A·∇χ(|y|
A)|ξ|2dy.
We will now exploit it to bound the terms in inequality (101). For the firs t term, we can easily infer
the following bound
b/integraldisplay
y·∇χA|ξ|2dy≥b
8/integraldisplay
{A≤|y|≤2A}|ξ|2dy.
For the second term, the definition of A(t) (86) and estimate (25) for Γ ballow us to infer
˙A
A=−ac˙b
b2,
which yields
1
A/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBigg˙λ
λ+b+˙A
A/parenrightBigg/integraldisplay
y·∇χ|ξ|2dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓa
bΓ1−20ν
b/integraldisplay
A≤|y|≤2A|ξ|2dy,BLOW-UP OF THE DAMPED NLS EQUATION 25
where we have used (80) and (102). By using (81) we may analogous ly estimate
1
A/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay˙x
λ·∇χ|ξ|2dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓa
b/parenleftbigg
(/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy)1/2+Γ1−20ν
b/parenrightbigg/integraldisplay
A≤|y|≤2A|ξ|2dy
/lessorsimilarΓa
bΓ1
2−10ν
b/integraldisplay
A≤|y|≤2A|ξ|2dy,
where we used (56) in the last inequality.
For the third term on the right-hand side of (101) we use Young’s ine quality and get
1
A|(∇ξ,i(∇χ)ξ)| ≤1
A/ba∇dbl∇ξ/ba∇dblL2/parenleftBigg/integraldisplay
A≤|y|≤2A|ξ|2dy/parenrightBigg1/2
≤40
bA2/ba∇dbl∇ξ/ba∇dbl2
L2
+b
40/integraldisplay
A≤|y|≤2A|ξ|2dy.
Finally, we consider the contribution coming from the nonlinear dampin g. By recalling that σ2>
σ∗= 2sc/(d−2sc), we have
sc<d
2−d
2σ2+2=dσ2
2σ2+2.
This implies that we can choose ssuch that
sc<s<dσ2
2σ2+2=s(σ2).
Consequently, we can interpolate the space ˙Hs(σ2)(Rd) between ˙H1(Rd) and˙Hs(Rd) and obtain
that
(103) /ba∇dblξ/ba∇dbl2σ2+2
L2σ2+2/lessorsimilar/ba∇dbl|∇|s(σ2)ξ/ba∇dbl2σ2+2
L2/lessorsimilar/ba∇dblξ/ba∇dblθ(2σ2+2)
˙H1/ba∇dblξ/ba∇dbl(1−θ)(2σ2+2)
˙Hs
for someθ(s)∈(0,1). Now from (56) and (55) it follows that
/ba∇dblξ/ba∇dblθ(2σ2+2)
˙H1/ba∇dblξ/ba∇dbl(1−θ)(2σ2+2)
˙Hs ≤Γ(1−50ν)(σ2+1)+30 νθ(1+σ2)
b.
Thus, by collecting everything, we have that
(104)1
2λ2scd
dτ/parenleftbig
λ2sc(ξ,χAξ)/parenrightbig
≥/parenleftbiggb
8−cΓa
bΓ1
2−10ν
b−b
40/parenrightbigg/integraldisplay
{A≤|y|≤2A}|ξ|2dy
−Γ2a
b/ba∇dbl∇ξ/ba∇dbl2
L2−Γ3
2−10ν
b
−ηλ2−2σ2
σ1Γ(1−50ν)(σ2+1)+30 νθ(1+σ2)
b.
Inequality (99) is a simple consequence of the choice of η(48). /square
Let us emphasize that the assumption σ2> σ∗, see (3), is required to obtain the bound (103).
As already remarked, this assumption is related to the fact that it is not possible to control Sobolev
norms ofξrougher than the critical norm ˙Hsc. Consequently, our argument cannot be applied for
instance to the linearly damped NLS equation (equation (1) with σ2= 0) since we would need to
estimate the term
ηλ2/integraldisplay
χA|ξ|2dy.
By using estimates (94) and (99), it is possible to define a Lyapunov f unctional that provides an
upper bound for ˙b. In the next lemma, it will be fundamental to exploit the decay of the total mass.
We notice that this fact suggests that a different regularization of the focusing NLS dynamics could26 P. ANTONELLI AND B. SHAKAROV
not yield the same result.
In what follows we define
(105)J(τ) =/integraldisplay
(1−χA(τ))|ξ(τ)|2dy+/ba∇dblQb(τ)/ba∇dbl2
L2−/ba∇dblQ/ba∇dbl2
L2+2(ξ(τ),Qb(τ))
−b(τ)f(τ) +/integraldisplayb(τ)
0f(v)dv,
wherefis defined in (95) and Qis the unique positive solution to (8).
Lemma 4.8. There exist a1=a1(ν)>0such that for any a < a1and anyt∈[0,T1), there exist
c>0such that
(106)d
dτJ(τ)/lessorsimilarb(τ)sc+Γ2a
b(τ)/ba∇dbl∇ξ(τ)/ba∇dbl2
L2
−b(τ)/parenleftbigg
Γb(τ)+/integraldisplay
|∇˜ξ(τ)|2+|˜ξ(τ)|2e−|y|dy/parenrightbigg
.
Proof.By multiplying inequality (94) by b, we obtain
(107) C2b/parenleftbigg/integraldisplay
|∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg
≤Cb/parenleftbiggd
dτf+sc/parenrightbigg
+b/integraldisplay
{A≤|y|≤2A}|ξ|2dy.
The last term on the right-hand side of (107) may be estimated by (9 9), so that
(108) b/integraldisplay
{A(τ)≤|y|≤2A(τ)}|ξ|2dy/lessorsimilarλ−2scd
dτ/parenleftbigg
λ2sc/integraldisplay
χA|ξ|2dy/parenrightbigg
+Γ3
2−10ν
b+Γ2a
b/ba∇dbl∇ξ/ba∇dbl2
L2.
In order to find a satisfactory bound on the first term on the right -hand side of the inequality above,
we exploit the fact that the total mass is non-increasing. By writing
d
dτ/ba∇dblψ/ba∇dbl2
L2=d
dτ/parenleftbigg
λ2sc/integraldisplay
(χA+(1−χA))|ξ|2+|Qb|2dy+2λ2sc(ξ,Qb)/parenrightbigg
≤0,
we have that
(109)λ−2scd
dτ/parenleftbigg
λ2sc/integraldisplay
χA|ξ|2dy/parenrightbigg
≤ −d
dτ/parenleftbigg/integraldisplay
(1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg
−2sc˙λ
λ/parenleftbigg/integraldisplay
(1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg
.
From the definition of χAgiven in (98) and by Hardy’s inequality, we have
/integraldisplay
(1−χA)|ξ|2≤/integraldisplay
|y|≤3A|y|2
|y|2|ξ|2dy/lessorsimilarA2/ba∇dbl∇ξ/ba∇dbl2
L2.
Similarly, we can obtain a comparable bound in dimensions one and two, s ee [34, Appendix C]. In
particular, for any dimension we conclude that
(110)/integraldisplay
(1−χA)|ξ|2/lessorsimilarA2/integraldisplay
|∇ξ|2+|ξ|e−|y|dy.BLOW-UP OF THE DAMPED NLS EQUATION 27
Thus, by using estimates (80), (74) and (56) we can bound the sec ond term on the right-hand side
of (109) by/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesc˙λ
λ/parenleftbigg/integraldisplay
(1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesc/parenleftBigg˙λ
λ+b/parenrightBigg/parenleftbigg/integraldisplay
(1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+/vextendsingle/vextendsingle/vextendsingle/vextendsinglescb/parenleftbigg/integraldisplay
(1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle
/lessorsimilarsc/parenleftbig
Γ1−20ν
b+b/parenrightbig/parenleftbigg
(1+A2)/integraldisplay
|∇ξ|2+|ξ|e−|y|dy+/ba∇dblQb/ba∇dbl2
L2/parenrightbigg
.
Now we want that
(111) A2/integraldisplay
|∇ξ|2+|ξ|e−|y|dy≤A2Γ1−20ν
b= Γ−2a+1−20ν
b/lessorsimilar1,
where we used the definition of Ain (86) and (56). That is amust satisfy the following inequality
a≤1−20ν
2.
This implies that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesc˙λ
λ/parenleftbigg/integraldisplay
(1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarscb.
By combining (108) with (109), we obtain
b/integraldisplay
{A≤|y|≤2A}|ξ|2dy/lessorsimilar−d
dτ/parenleftbigg/integraldisplay
(1−χA)|ξ|2+|Qb|2dy+2(ξ,Qb)/parenrightbigg
+scb+Γ3
2−10ν
b+Γ2a
b/ba∇dbl∇ξ/ba∇dbl2
L2.
By plugging the above inequality into (107), we infer
(112)b/parenleftbigg/integraldisplay
|∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg
/lessorsimilarbd
dτf
−d
dτ/parenleftbigg/integraldisplay
(1−χA)|ξ|2dy+/ba∇dblQb/ba∇dbl2
L2+2(ξ,Qb)/parenrightbigg
+scb+Γ3
2−10ν
b+Γ2a
b/ba∇dbl∇ξ/ba∇dbl2
L2.
Finally, let us consider f=f(τ) as defined in (95). By the monotonicity property of b, see (85) for
instance, we denote - by some abuse of notation - f=f(b(τ)). In this way, we may write
bd
dτf=d
dτ(bf)−d
dτ/integraldisplayb
0f(v)dv.
Let us now recall the definition of J(τ) given in (105), we have
J(τ) =/integraldisplay
(1−χA(τ))|ξ(τ)|2dy+/ba∇dblQb(τ)/ba∇dbl2
L2−/ba∇dblQ/ba∇dbl2
L2+2(ξ(τ),Qb(τ))
−b(τ)f(τ)+/integraldisplayb(τ)
0f(v)dv.28 P. ANTONELLI AND B. SHAKAROV
By using the previous identities and (52), we see that (112) implies th e following estimate
(113) b/parenleftbigg/integraldisplay
|∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg
/lessorsimilar−d
dτJ+bsc+Γ2a
b/ba∇dbl∇ξ/ba∇dbl2
L2,
which readily gives (106). /square
Let us now discuss how the functional Jis related to the control parameter b. First, we define
K(τ) as
(114) K(τ) =/ba∇dblQb(τ)/ba∇dbl2
L2−/ba∇dblQ/ba∇dbl2
L2−b(τ)f(τ) +/integraldisplayb(τ)
0f(v)dv
whereQis the ground state profile to (1), see (8). In this way, we obtain th at
J(τ)−K(τ) =/integraldisplay
(1−χA(τ))|ξ(τ)|2dy+2(ξ(τ),Qb(τ)).
Thus, by using (110) and (73) we get
(115) J(τ)−K(τ)/lessorsimilar(1+A2)/integraldisplay
|ξ|2e−|y|+|∇ξ|2dy+Γ1−ν
b.
We stress that to provide a satisfactory upper bound on the differ enceJ−K, it is not sufficient to
consider the weaker bound (74) in the inequality above. Indeed, in t he proof of the bootstrap in the
next section, we will first show that the control (56) is not satisfa ctory to obtain the better bound
(61).
To obtain a lower bound we observe that, by using (77), we have
J−K=/integraldisplay
(1−χA)|ξ|2dy−2λ2−2scE[ψ]+2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2
L2
+2(iscβ∂bQb+ibΛQb+Ψb,ξ)−/integraldisplay
R(2)(ξ)dy
whereR(2)(ξ) is defined in (76). We rewrite the equation above as
(116)J−K=−2λ2−2scE[ψ]+2E[Qb]+2(iscβ∂bQb+Ψb,ξ)
+(L(1)ξ,ξ)−/integraldisplay
χA|ξ|2dy−1
σ1+1/integraldisplay
R(3)(ξ)dy
where
(117) L(1)ξ=−∆ξ+ξ−2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb−|Qb|2σ1ξ
and
R(3)(ξ) =|Qb+ξ|2σ1+2−|Qb|2σ1+2−(2σ1+2)|Qb|2σ1Re(Qb¯ξ)
−(2σ1+2)|Qb|2σ1−2/parenleftbig
|Qb|2|ξ|2+2σ1Re(Qb¯ξ)2/parenrightbig
.
We recall the coercivity property
(L(1)ξ,ξ)−/integraldisplay
χA|ξ|2dy≥C/integraldisplay
|ξ|2e−|y|+|∇ξ|2dy−Γ3
2−10ν
b
which was proved in [34, Appendix D]. Then in (116) by further using (54), (32), (70) and (71) to
bound the rests as /vextendsingle/vextendsingle/vextendsingle/vextendsingle2(iscβ∂bQb+Ψb,ξ)−1
σ1+1/integraldisplay
R(3)(ξ)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓbBLOW-UP OF THE DAMPED NLS EQUATION 29
we obtain
(118) J−K/greaterorsimilar/integraldisplay
|ξ|2e−|y|+|∇ξ|2dy−Γb−sc.
By combining (118) and (115), we obtain that
(119)/integraldisplay
|ξ|2e−|y|+|∇ξ|2dy−Γb−sc/lessorsimilarJ−K/lessorsimilar(1+A2)/integraldisplay
|ξ|2e−|y|+|∇ξ|2dy+Γ1−ν
b.
Now we want to prove that K(τ) is a small perturbation of b2(τ). By slightly abusing notations
again, we write K(τ) =K(b(τ)). The following lemma can be found in [36, Section 4].
Lemma 4.9. There exists b(1)>0such that for any 0<b<b 1<b(1), we have
(120) K(b)−K(b1)/lessorsimilarsc
and
(121) b2−sc/lessorsimilarK(b)/lessorsimilarb2+sc.
The proofof this lemma follows from the properties of Qbstated in (32), the definition of fin (95)
the decomposition (90), and the smallness of the outgoing radiation (89). Observe that by collecting
together (119) and (121), we have that J(τ) is close to b2(τ) up to smaller order corrections and
up to the term A2/integraltext
|ξ|2e−|y|+|∇ξ|2dy. Consequently, if aandbare small enough, we obtain the
following inequality
(122)/vextendsingle/vextendsingleJ(τ)−db2(τ)/vextendsingle/vextendsingle/lessorsimilarΓ1−20ν
b(τ)+A2/integraldisplay
|ξ|2e−|y|+|∇ξ|2dy≤νb2(τ).
5.Proof of the Bootstrap
In this section, we are going to prove Proposition 3.3. We proceed in t he following order.
(1) Using the monotonicity properties (85) and (106), we refine th e control over the remainder
ξas in (61).
(2) Inequalities (85) and (106) also imply the dynamical trapping of b(57). In particular, bis
almost constant and close to the value b∗(sc)>0 defined in (9).
(3) From (80), we obtain the equation for the scaling parameter
˙λ
λ∼ −b+Γb.
The previous points yield (58) and a precise law for λ.
(4) By finding suitable bounds on the time derivatives of the energy a nd the momentum, we
will prove (59).
(5) Finally, we will deduce the ˙Hs-norm control of ξin (60).
We shall stress that the first three points of our scheme are cons equences of the local virial law
(85) and the monotonicity formula (106) and their proofs are very similar to those in [36]. On the
other hand, in the undamped case, the fourth point comes natura lly from the conservation of the
energy and the momentum. In our case, we will show that the growt h of the time-dependent energy
and momentum (see equations (18) and (17)) can be controlled unt il the blow-up time by our choice
ofη. Finally, with respect to the proof in [36], the fifth point requires an a dditional change to treat
the newdissipative term. Forthe convenienceofthe reader, we will restatethe bootstrapproposition
below.30 P. ANTONELLI AND B. SHAKAROV
Proposition 5.1. There exists s∗
c>0,s∗>s∗
c,ν∗>0anda∗(ν∗)>a∗(ν∗)>0, such that for any
sc<s∗
c,sc<s<s∗,ν <ν∗anda∗<a<a∗and for any t∈[0,T1),the following inequalities are
true:
Γ1+ν4
b(t)≤sc≤Γ1−ν4
b(t), (123)
0≤λ(t)≤Γ20
b(t), (124)
λ2−2sc|E[ψ(t)]|+λ1−2sc|P[ψ(t)]| ≤Γ3−10ν
b(t), (125)
/integraldisplay
||∇|sξ(t)|2≤Γ1−45ν
b(t), (126)
/integraldisplay
|∇ξ(t)|2+|ξ(t)|2e−|y|dy≤Γ1−10ν
b(t). (127)
Proof.We chooses∗
cto be the minimum of all the conditions for scfound in previous sections. Then
we fixsc<s∗
cand we choose ssuch that
sc<s<min/parenleftbiggdσ1
2σ1+2,dσ2
2σ2+2,1
2/parenrightbigg
such that (103), (78) and (79) are true. Now we choose an initial c onditionψ0∈ O, where the set O
is defined in 3.1. Consequently, there exists a time interval [0 ,T1] where the estimates in Proposition
3.2 are true. In particular, by choosing scsmall enough we obtain that b(t) is small for any t∈[0,T1]
and also that Γ b(t)<1.
We start by proving inequality (127). We proceed by contradiction. First, suppose that there exists
τ0∈[0,τ−1(T1)) such that
/integraldisplay
|∇ξ(τ0)|2+|ξ(τ0)|2e−|y|dy>Γ1−10ν
b(τ0).
Notice that the initial condition satisfies (45), and in particular, sinc e Γb<1, for anyt∈[0,T1), see
(52), we have/integraldisplay
|∇ξ(0)|2+|ξ(0)|2e−|y|dy<Γ1−7ν
b(0),
Then by continuity of ξ,band Γb, there exists a time interval [ τ1,τ2]⊂[0,τ0) such that
(128)/integraldisplay
|∇ξ(τ1)|2+|ξ(τ1)|2e−|y|dy= Γ1−7ν
b(τ1),
/integraldisplay
|∇ξ(τ2)|2+|ξ(τ2)|2e−|y|dy= Γ1−10ν
b(τ2),
and for any τ∈[τ1,τ2]
(129)/integraldisplay
|∇ξ(τ)|2+|ξ(τ)|2e−|y|dy≥Γ1−7ν
b(τ).
The virial estimate (85), controls (52), (53) and (54) imply that fo r anyτ∈[τ1,τ2], we have
˙b/greaterorsimilarsc+/integraldisplay
|∇ξ(τ)|2+|ξ(τ)|2e−|y|dy−Γ1−ν6
b/greaterorsimilarΓ1+ν2
b+Γ1−7ν
b−Γ1−ν6
b>0,
forνsmall enough and hence
(130) b(τ2)≥b(τ1).
We notice that the definition of Γ b(25) implies also that
(131) Γ b(τ2)≥Γb(τ1).BLOW-UP OF THE DAMPED NLS EQUATION 31
On the other hand, from the smallness of ˜ζb(89), we obtain the inequality
(132)/integraldisplay
|∇˜ξ|2+|˜ξ|2e−|y|dy≥1
2/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy−Γ1−ν
b
which combined with the second monotonicity estimate (106) and (52 ) yields
d
dτJ/lessorsimilarbsc+Γ2a
b/ba∇dbl∇ξ/ba∇dbl2
L2−bΓb−b/integraldisplay
|∇˜ξ(τ)|2+|˜ξ(τ)|2e−|y|dy
/lessorsimilarb/parenleftbigg
sc−Γb−1
2/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy+Γ1−ν
b+Γ2a
b
b/ba∇dbl∇ξ/ba∇dbl2
L2/parenrightbigg
/lessorsimilarb/parenleftBig
Γ1−ν2
b−Γb−Γ1−7ν
b+Γ1−ν
b/parenrightBig
≤0,
forνsmall enough where we have chosen a>0 so that
Γ2a
b
b≤1
4.
Thus we obtain that
(133) J(τ2)≤J(τ1).
Next, using (119) and (133) we get
K(τ2)+/integraldisplay
|ξ(τ2)|2e−|y|+|∇ξ(τ2)|2dy−Γb(τ2)−sc/lessorsimilarJ(τ2)≤J(τ1)
/lessorsimilarK(τ1)+(1+A2)/integraldisplay
|ξ(τ1)|2e−|y|+|∇ξ(τ1)|2dy+Γ1−ν
b(τ1).
Equivalently, by using (128) we get
/integraldisplay
|ξ(τ2)|2e−|y|+|∇ξ(τ2)|2dy= Γ1−10ν
b(τ2)
/lessorsimilarK(τ1)−K(τ2)+sc+Γb(τ2)+(1+A2)Γ1−7ν
b(τ1)+Γ1−ν
b(τ1).
On the right-hand side of the equation above, we use (120) to get
|K(τ1)−K(τ2)|/lessorsimilarsc.
Thus using the inequality sc≤Γ1−ν2
b(τ1)(52) and the definition of A(86), we obtain
Γ1−10ν
b(τ2)/lessorsimilarΓ1−ν2
b(τ1)+Γb(τ2)+Γ1−7ν−2a
b(τ1)+Γ1−ν
b(τ1).
Now we suppose that a≤ν
2. This implies that there exists C >0 such that
Γ1−10ν
b(τ2)≤CΓ1−8ν
b(τ1).
By exploiting (131), and for bsmall enough, this implies that
Γ1−10ν
b(τ2)≤CΓ1−8ν
b(τ1)≤CΓ1−8ν
b(τ2)=CΓν
b(τ2)Γ1−9ν
b(τ2)≤Γ1−9ν
b(τ2),
which is a contradiction because Γ b(τ2)<1.
As our next step, we will prove (123). Assume by contradiction tha t there exists τ0∈[0,τ−1(T1)]
such thatsc>Γ1−ν4
b(τ0).By continuity, this implies that there exists τ1<τ0such that
sc= Γ1−ν5
b(τ1)
and for any τ∈[τ1,τ0],
sc>Γ1−ν5
b(τ).32 P. ANTONELLI AND B. SHAKAROV
In particular, we have that
d
dτΓb(τ1)<0
which is equivalent to ˙b(τ1)<0 from the definition of Γ bin (25). On the other hand, the local virial
inequality (85) implies that
˙b(τ1)/greaterorsimilarsc−Γ1−ν6
b(s6)/greaterorsimilarΓ1−ν5
b(s6)−Γ1−ν6
b(s6)>0,
which is a contradiction.
Similarly, suppose by contradiction that there exists τ0∈[0,τ−1(T1)) such that sc<Γ1+ν4
b(τ0).Then,
by using (122), we get that
sc≤Γ1+ν5
/radicalBig
J(τ0)
d
Now letτ1∈[0,τ0] be the largest time such that
sc= Γ1+ν6
/radicalBig
J(τ1)
d
Thus, by definition of τ1and Γb(25) we obtain that
d
dτJ(τ1)≥0,
while from the monotonicity formula (106) and (132) we obtain
d
dτJ(τ1)/lessorsimilarb(τ1)/parenleftBigg
sc−Γb(τ1)−1
2/integraldisplay
|∇ξ(τ1)|2+|ξ(τ1)|2e−|y|dy+Γ1−ν
b(τ1)+Γ2a
b(τ1)
b(τ1)/ba∇dbl∇ξ(τ1)/ba∇dbl2
L2/parenrightBigg
/lessorsimilarb(τ1)/parenleftBigg
Γ1+ν6
/radicalBig
J(τ1)
d−Γ1+ν2
/radicalBig
J(τ1)
d/parenrightBigg
≤0,
which is a contradiction.
The next step will be to proveinequality (124). From the control (1 23), it followsthat the parameter
bis dynamically trapped around b0, that is, for any t∈[0,T1), we have the following upper and
lower bounds
1−ν4
1+ν10b0≤b(t)≤1+ν4
1−ν10b0.
Thus, we notice that the control on the parameter (80), and ineq uality (61) imply that for any
t∈[0,T1)
0<1−2ν4
1+ν10b0≤ −˙λ
λ=−1
2d
dtλ2≤1+2ν4
1−ν10b0,
where we used that
d
dτλ=˙λ=λ2d
dtλ
from (64). Equivalently, we obtain the law of the parameter λ
(134) λ2
0−2/parenleftbigg1+2ν4
1−ν10/parenrightbigg
b0t≤λ2(t)≤λ2
0−2/parenleftbigg1−2ν4
1+ν10/parenrightbigg
b0t.
in particular, λ(t) is a non-increasing function of time. This estimate and the dynamica l trapping
of the parameter b(123) imply inequality (124).BLOW-UP OF THE DAMPED NLS EQUATION 33
The next step is to obtain the bound on the total energy Eand momentum P(125). We start by
deriving a suitable bound for the energy functional. We use (17) to o btain that
E[ψ(t)] =E[ψ0]+η/integraldisplayt
0/integraldisplay
|ψ|2(σ1+σ2+1)−|ψ|2σ2|∇ψ|2−2σ2|ψ|2σ2−2Re/parenleftbig¯ψ∇ψ/parenrightbig2dxdv
≤E[ψ0]+η/integraldisplayt
0λ2sc−2σ2
σ1−2/integraldisplay
|Qb+ξ|2(σ1+σ2+1)dydv.
We notice that if d≤3 andσ2≤σ1<1
(d−2)+or ifd≥4, andσ2< σ∗, thenH1(Rd)֒→
L2(σ1+σ2+1)(Rd). Thus, for scsmall enough we can use the Jensen inequality, (61), (55) and inter -
polation
/ba∇dblξ/ba∇dbl2(σ1+σ2+1)
L2(σ1+σ2+1)/lessorsimilar/ba∇dblξ/ba∇dbl2θ(σ1+σ2+1)
˙H1 /ba∇dblξ/ba∇dbl2(1−θ)(σ1+σ2+1)
˙Hs /lessorsimilar1
for someθ(σ1,σ2,s)∈(0,1), to obtain
/ba∇dblQb+ξ/ba∇dbl2(σ1+σ2+1)
L2(σ1+σ2+1)≤C/parenleftBig
/ba∇dblQb/ba∇dbl2(σ1+σ2+1)
L2(σ1+σ2+1)+/ba∇dblξ/ba∇dbl2(σ1+σ2+1)
L2(σ1+σ2+1)/parenrightBig
≤C
whereC=C(σ1,σ2,b∗)>0. If follows that
λ2−2scE[ψ(t)]≤λ2−2scE[ψ0]+Cηλ2−2sc/integraldisplayt
0λ2sc−2σ2
σ1−2dv. (135)
Now we observe that, from (134), there exists a constant 0 < c=c(ν,b0)≪1 such that, for any
t∈[0,T1),
λ2
0−2(1+c)t≤λ2(t)≤λ2
0−2(1−c)t.
Integrating λin time, it is straightforward to see that for any α∈R, we have
(136)/integraldisplayt
0λ(τ)αdv/lessorsimilarλ(t)α+2.
Thus, from (135), it follows that
(137) λ2−2scE[ψ(t)]≤λ2−2scE[ψ0]+Cηλ2−2σ2
σ1.
From (58) and (43) we have
λ2−2sc|E[ψ0]| ≤Γ50
b.
Moreover, we bound the last term using the smallness of η(48), (57) and (57)
Cηλ2−2σ2
σ1≤Cs3
c≤Γ3−9ν
b
forνsmall enough. This implies the bootstrapped control on the energy
λ2−2sc|E[ψ(t)]| ≤Γ3−10ν
b.
We use the same procedure to bound the momentum functional P. From (18), we have
P[ψ(t)] =P[ψ0]+2η/integraldisplayt
0/integraldisplay
|ψ|2σ2Im(¯ψ∇ψ)dxdτ
=P[ψ0]+2η/integraldisplayt
0λ−2σ2
σ1+2sc−1/ba∇dblQb+ξ/ba∇dbl2σ2+1
L4σ2+2/ba∇dbl∇(Qb+ξ)/ba∇dblL2dv.
Again, ifd≤3,scis small enough and σ2≤σ1or ifd≥4, andσ2<σ∗, then we use use the Jensen
inequality, (61), (55), interpolation and (136) to prove that ther e existsC=C(σ2,b∗)>0 such that
(138) λ1−2scP[ψ(t)]≤λ1−2scP[ψ0]+Cηλ2−2σ2
σ1.34 P. ANTONELLI AND B. SHAKAROV
Consequently, (43), (48) and (57) imply that
λ1−2scP[ψ]≤Γ3−10ν
b,
forνsmall enough. This concludes the proof of (59).
The last step is to obtain the ˙Hs-norm control (126). We define
ˆQ(t,x) =λ−1
σ1Qb/parenleftbiggx−x(t)
λ/parenrightbigg
eiγ,
ˆξ(t,x) =λ−1
σ1ξ/parenleftbigg
t,x−x(t)
λ/parenrightbigg
eiγ,
we decompose the solution as
ψ=ˆQ+ˆξ.
From (1), we see that the function ˆξsatisfies the equation
(139) i ˆξt+∆ˆξ=−E(ˆQ)−N1(ˆξ)−iηN2(ˆξ),
whereEdoes not depend on ˆξ
E(ˆQ) = i∂tˆQ+∆ˆQ+|ˆQ|2σ1ˆQ+iη|ˆQ|2σ2˜Q
and
N1(ˆξ) =|ˆQ+ˆξ|2σ1(ˆQ+ˆξ)−|ˆQ|2σ1ˆQ,
N2(ˆξ) =|ˆQ+ˆξ|2σ2(ˆQ+ˆξ)−|ˆQ|2σ2ˆQ.
Using the equation satisfied by Qb(29), we obtain that
E(ˆQ) =1
λ2+1
σ1eiγ/parenleftBigg
−Ψb+i(˙b−βsc)∂bQb−i/parenleftBigg˙λ
λ+b/parenrightBigg
ΛQb−i˙x
λ·∇Qb
−(˙γ−1)Qb+ iηλ2−2σ2
σ1|Qb|2σ2Qb/parenrightBig
.
We notice that inequality (126) is equivalent to
(140)/integraldisplay
||∇|sˆξ|2dx≤λ2(sc−s)Γ1−45ν
b
since /integraldisplay
||∇|sˆξ|2dx=λ2(sc−s)/integraldisplay
||∇|sξ|2dy,
and thus we will now prove (140). We define for j= 1,2 where
(141) rj=d(2σj+2)
d+2sσj, γj=4(σj+1)
σj(d−2s),2
γj=d
2−d
rj.
We notice that ( γj,rj) are Strichartz admissible pairs (see Definition 2.1). Now we write equ ation
(139) in the integral form
ˆξ(t) =ei∆tˆξ(0)+/integraldisplayt
0ei(t−τ)∆/parenleftBig
E(Qb)+N1(ˆξ)+iηN2(ˆξ)/parenrightBig
dτ.
and we use the Strichartz estimates in Theorem 2.2 to obtain
(142)/ba∇dbl|∇|sˆξ/ba∇dblL∞([0,T1],L2)/lessorsimilar/ba∇dbl|∇|sˆξ0/ba∇dblL2+/ba∇dbl|∇|sE/ba∇dblL1([0,T1],L2)
+/ba∇dbl|∇|sN1(ˆξ)/ba∇dblLγ1[0,T1],Lr1+η/ba∇dbl|∇|sN2(ˆξ)/ba∇dblLγ2([0,t],Lr2).BLOW-UP OF THE DAMPED NLS EQUATION 35
From the initial bound (45), we have
λ2(s−sc)/integraldisplay
||∇|sˆξ(0)|2dx=/integraldisplay
||∇|sξ0|2dy≤Γ1−ν
b0,
and thus the bootstrapped controls on b, see (57) and λ, see (58) imply
(143) /ba∇dbl|∇|sˆξ0/ba∇dbl2
L2≤Γ1−ν
b0λ2(sc−s).
We claim that the remaining terms in (142) are bounded as
(144) /ba∇dbl|∇|sE/ba∇dbl2
L1([0,T1],L2)≤Γ1−15ν
bλ2(sc−s),
and
(145) /ba∇dbl|∇|sN1(ˆξ)/ba∇dblLγ1([0,T1],Lr1)+η/ba∇dbl|∇|sN2(ˆξ)/ba∇dblLγ2([0,T1],Lr2)≤Γ1
2(1−41ν)
bλsc−s.
Notice that (144) and the bound on N1(ˆξ) in (145) has been already proven in [36, Section 4], up
to the term coming from the damping in E. In particular, from the estimates on the parameters
(81), (80), the bound on Ψ b(31), and the bootstrap bounds for b,λandξ, see (57), (58) and (61)
respectively, and from the smallness condition on ηin (48) there holds for any t∈[0,T1)
/ba∇dbl|∇|sE(ˆQ)/ba∇dblL2/lessorsimilar1
λ2+s−sc/parenleftbigg/integraldisplay
|∇ξ|2dy+/integraldisplay
|ξ|2e−|y|dy+Γ1−11ν
b+ηλ2−2σ2/σ1/parenrightbigg1
2
/lessorsimilarΓ1
2(1−12ν)
b
λ2+s−sc.(146)
We use inequality (136) to integrate (146) in time which yields
/integraldisplayt
0/ba∇dbl|∇|sE(ˆQ)/ba∇dblL2dτ/lessorsimilar/integraldisplayt
0Γ1
2(1−12ν)
b0
λ2+(s−sc)dτ/lessorsimilarΓ1
2(1−12ν)
b0
λs−sc.
Lastly, we will show inequality (145). The main ingredient is the inequalit y
(147) /ba∇dbl|∇|sNj(ˆξ)/ba∇dblLr′
j/lessorsimilarλ(2σj+1)(sc−s+2/γj)/ba∇dbl|∇|s+2/γjξ/ba∇dblL2
for anyj= 1,2 which has been proven in [36, Appendix] and will be shown in Appendix C. Notice
that since
s+2/γj=s+(d−2sc)σj
2σj+2,
for anyj= 1,2, then if we choose sto be close enough to sc, we haves<s+2/γj<1. In particular,
we can interpolate and use (61) and (55) to obtain
/ba∇dbl|∇|s+2/γjξ/ba∇dbl2
L2/lessorsimilar/ba∇dbl|∇|sξ/ba∇dbl2θj
L2/ba∇dbl∇ξ/ba∇dbl2(1−θj)
L2≤Γ1−(10+40θj)ν
b
where
θj=1−s−2/γj
1−s.
Forj= 1, we choose sc≪1 small enough and sclose enough to scto have
/ba∇dbl|∇|s+2/γjξ/ba∇dblL2/lessorsimilarΓ1
2(1−40ν)
b0.
It follows that
/ba∇dbl|∇|sN1(ˆξ)/ba∇dblLγ1[0,T1],Lr1/lessorsimilarΓ1
2(1−40ν)
b0/parenleftbigg/integraldisplay
λγ′
1(2σ1+1)(sc−s+2/γj)/parenrightbigg1
γ′
1.
Since
γ′
1(2σ1+1)(s+2/γ1) = 2+γ′
1(s−sc),36 P. ANTONELLI AND B. SHAKAROV
we can use (136) to integrate in time and obtain the first inequality in ( 145)
/ba∇dbl|∇|sN1(ˆξ)/ba∇dblLγ1[0,T1],Lr1/lessorsimilarΓ1
2(1−41ν)
b0
λ(s−sc).
Moreover, we make the same computations for j= 2 and use the control on η(48) and (57) to
conclude that
(148) η/ba∇dbl|∇|sN2(ˆξ)/ba∇dblLγ2([0,T1],Lr2)≤ηΓ1−(10+40θ2)ν
bλsc−s≤Γ2−50ν
bλsc−s.
/square
Thus Proposition 3.3 has been proven for any t∈[0,T1). Consequently, we can extend by
continuity the time interval of the self-similar regime, that is the time interval where the bounds in
Proposition 3.2 are true. Recursively, we extend the self-similar reg ime to the whole time interval
[0,Tmax) whereTmax(ψ0) is the maximal time of existence of the solution stemming from ψ0. Now
we show that ψexperiences a collapse in finite time.
Corollary 5.2. There exists a time Tmax<∞such that
lim
t→Tmaxλ(t) = 0.
Moreover, there exists x∞∈Rdsuch that
lim
t→Tmaxx(t)→x∞,
wherex(t)is defined in (50).
Proof.Letψ0∈ OwhereOis defined in 3.1 and let Tmax≤ ∞be the maximal time of existence
of the corresponding solution to (1). Then Proposition 3.3 implies tha t for anyt∈[0,Tmax), the
bounds on the scaling parameter in (134) are true, namely, there e xists 0< c=c(ν,b0)≪1 such
that
(149) λ2
0−2(1+c)t≤λ2(t)≤λ2
0−2(1−c)t.
Consequently, there exists a time 0 <T=T(λ0,ν,b0)<∞such that lim t→Tλ(t) = 0. Furthermore,
by using decomposition (49) we also obtain that
lim
t→T/ba∇dbl∇ψ(t)/ba∇dbl2
L2= lim
t→Tλ2(sc−1)(t)/ba∇dbl∇(Qb(t)+ξ(t))/ba∇dbl2
L2=∞.
Indeed, in the limit above, the exponent 2( sc−1) is negative because sc<1, and for any t∈[0,T],
Qb(t)∈H1(Rd),/vextenddouble/vextenddouble∇Qb(t)/vextenddouble/vextenddouble
L2≥C >0,
while (123) and (127) imply that
/vextenddouble/vextenddouble∇ξ(t)/vextenddouble/vextenddouble2
L2≤Γ1−10ν
b/lessorsimilars1−10ν
1+ν4
c.
in particular, the maximal time of existence is given by T <∞. Finally, we prove the convergence
to a blow-up point. By exploiting (81) and (149), we get
/vextendsingle/vextendsinglex(T)−x(t)/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT
tdx
dsds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplayT
t1
λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle1
λd
dτx/vextendsingle/vextendsingle/vextendsingle/vextendsingleds
/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg/integraldisplay
|∇ξ(t)|2+|ξ(t)|2e−|y|dy/parenrightbigg1
2
+Γ1−20ν
b(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
L∞([0,T])/integraldisplayT
t/parenleftbig
λ2
0−2(1+c)s/parenrightbig−1
2ds.BLOW-UP OF THE DAMPED NLS EQUATION 37
We use (57) and (61) to obtain
/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg/integraldisplay
|∇ξ(t)|2+|ξ(t)|2e−|y|dy/parenrightbigg1
2
+Γ1−20ν
b(t)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble
L∞([0,T])/lessorsimilar1.
Moreover, we compute the integral as follows
/integraldisplayT
t/parenleftbig
λ2
0−2(1+c)s/parenrightbig−1
2ds=−2
2+c(λ2
0−2(1+c)s)1
2|T
t.
This implies that
lim
t→T/vextendsingle/vextendsinglex(T)−x(t)/vextendsingle/vextendsingle/lessorsimilarlim
t→T(λ2
0−2(1+c)s)1
2|T
t= 0.
/square
6.The case σ2<σ1
In this section, we discuss the case where the exponent of the dam ping term is strictly smaller
than that of the power-type nonlinearity.
First, we observe that the condition which we have chosen for the d amping parameter η≤s3
cis not
strictly necessary. In fact, this condition was used to prove that the damping term is of smaller order
with respect to other terms while the solution is in the self-similar regim e. In particular, it was used
for instance in (104), (84), (146), (148), (137) and (138), whe re the damping term contributions can
be always bounded by
ηλ2−2σ2
σ1(t)/parenleftBigg
C(Qb(t))+c/parenleftbigg/integraldisplay
|∇ξ(t)|2+|ξ(t)|2e−|y|dy/parenrightbigg1
2/parenrightBigg
,
whereC(Qb)>0 is a constant depending only on Qb. We know that in the self-similar regime, from
(56), we have that
/parenleftbigg/integraldisplay
|∇ξ(t)|2+|ξ(t)|2e−|y|dy/parenrightbigg1
2
/lessorsimilarΓ1
2−10ν
b(t)/lessorsimilarC(Qb(t))
forb(t) small enough. Thus, in order to prove that the damping term is sma ller than the leading
order terms in the equation referred to above, which are of the siz e Γb, we shall have for example
that
ηλ2−2σ2
σ1(t)≤Γ2
b(t)
for anyt∈[0,T1). This is equivalent to asking that the scaling parameter satisfies th e inequality
λ(t)<η−1Γσ1
σ1−σ2
b(t).
Notice that the right-hand side of this inequality degenerates for σ2→σ1, and consequently, for
σ2=σ1, the supposition on the smallness of ηbecomes necessary to carry on with the bootstrap.
On the other hand, for σ2<σ1, we can use this condition to proceed with the bootstrap instead of
that in (53). We observe that in this case, the set of initial condition s would depend on η,σ1and
σ2. For instance, it would be necessary to choose the initial condition s atisfying, say
λ0<η−1Γ10σ1
σ1−σ2
b(t)
instead of the weaker bound (42). In particular, the bigger ηis and the closer σ2is toσ1, the more
focused the initial condition needs to be to experience a collapse in fin ite time. This implies the
following.38 P. ANTONELLI AND B. SHAKAROV
Theorem 6.1. There exists s∗
c>0such that for any 0< sc< s∗
c, anyσ∗< σ2< σ1whend≤3
orσ∗< σ2< σ∗whend≥4and anyη >0there exists a set Oη,σ1,σ2⊂H1(Rd)such that if
ψ0∈ Oη,σ1,σ2then the corresponding solution ψ∈C([0,Tmax),H1(Rd))to(1)develops a singularity
in finite time.
We emphasize againthat ourargumentworksonlywhen σ1>σ2. Whenσ1=σ2, it is crucialthat
σ1>2
dandηto be small enough. On the other hand, it is already known that when σ1=σ2=2
d,
solutions are global in time. This means that a solution escapes the se lf-similar regime no matter
how smallηandλ0are. Heuristically, this is explained by the fact that, in the self-similar regime,
whenσ1=2
d, the control parameter converges to zero in time, that is b(t)→0 fort→Tmax(see
[34] for example). Consequently, for σ2=σ1=2
d, there would exist a time T=T(η,b0,λ0)>0
such that for some t>T, the critical inequality η/lessorsimilarΓb(t)is not true. Thus the damping term would
become of the leading order inside the self-similar dynamics and interr upt it.
As our final remark, we notice that if σ2> σ1, then heuristically the self-similar regime would be
interrupted. Indeed the convergence of λto zero implies that contributions of damping terms will
eventually be of the leading order in the dynamics because
λ2−2σ2
σ1(t)→ ∞.
7.Appendix
7.1.Appendix A. In this appendix we will report the proof of [36, Proposition 3 .3] that contains
computations used in Lemma 4.5. We state the result below for the re ader’s convenience.
Lemma 7.1. Suppose that η= 0. Then there exists s(1)
c>0such that for any sc< s(1)
cand for
anyt∈[0,T1), there exists C >0such that
(150) ˙b≥C/parenleftbigg
sc+/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy−Γ1−ν6
b/parenrightbigg
.
We now present some preliminary computations. We start with the fo llowing.
Lemma 7.2. LetQbbe a solution to (29). Then
(151) (i ∂bQb,ΛQb) =−1
4/ba∇dblxQ/ba∇dbl2
L2(1+δ1(sc,b))+sc(iQb,∂bQb),
whereδ1(sc,b)→0assc,b→0andQis the unique positive solution to (8).
Proof.We recall the third property in (32)
(152) (iQb,x·∇Qb) = (iQb,ΛQb) =−b
2/ba∇dblxQ/ba∇dbl2
L2(1+δ1(sc,b))
whereδ1(sc,b)→0 assc,b→0. By taking the derivative of (152) with respect to bwe obtain
d
db(iQb,ΛQb) = (i∂bQb,ΛQb)+(iQb,Λ∂bQb) =−1
2/ba∇dblxQ/ba∇dbl2
L2(1+δ1(sc,b)).
Let us also recall identity (14),
(iQb,Λ∂bQb) =−2sc(iQb,∂bQb)+(i∂bQb,ΛQb).
By plugging it into the previous formula, we obtain
(i∂bQb,ΛQb) =−1
4/ba∇dblxQ/ba∇dbl2
L2(1+δ1(sc,b))+sc(iQb,∂bQb).
/squareBLOW-UP OF THE DAMPED NLS EQUATION 39
Next, we recall the definition of the linearized operator around Qb, see (66)
(153) Lξ= ∆ξ−ξ+ibΛξ+2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb+|Qb|2σ1ξ,
and the nonlinear remainder term in the equation (63) for the pertu rbation, namely
R(ξ) =|Qb+ξ|2σ1(Qb+ξ)−|Qb|2σ1Qb−2σ1Re(ξ¯Qb)|Qb|2σ1−2Qb−|Qb|2σ1ξ.
Lemma 7.3. We have that
(154)(Lξ+R(ξ),ΛQb) =−2scb(iξ,ΛQb)−βsc(ξ,iΛ∂bQb)−(ξ,ΛΨb)
−2λ2−2scE[ψ]+2E[Qb]+H(ξ,ξ)+E(ξ,ξ)
+1
σ1+1/integraldisplay
R(3)(ξ)dy+(R3(ξ),ΛQb).
whereHis the quadratic form defined in (37),E(ξ,ξ)satisfies the following bound
|E(ξ,ξ)| ≤δ2(sc)/integraldisplay
|ξ|2e−|y|+|∇ξ|2dy,
withδ2(sc)→0assc→0and the remainder terms R(3)(ξ),R3(ξ), are given by
(155)R(3)(ξ) =|Qb+ξ|2σ1+2−|Qb|2σ1+2−(2σ1+2)|Qb|2σ1Re(Qb¯ξ)
−(2σ1+2)|Qb|2σ1−2/parenleftbig
|Qb|2|ξ|2+2σ1Re(Qb¯ξ)2/parenrightbig
,
and
(156) R3(ξ) =R(ξ)−2σ1|Qb|2σ1−2/parenleftbigg
Re(Qb¯ξ)ξ+(2σ1−1)|Qb|−2Re(Qb¯ξ)2Qb+Qb|ξ|2/parenrightbigg
.
Proof.We observe that by using properties (13), (14) we have
(∆ξ,ΛQb) = 2(ξ,∆Qb)+(ξ,Λ∆Qb)
and
(ibΛξ,ΛQb) =−2scb(iξ,ΛQb)+(ξ,ibΛ(ΛQb)).
Consequently, by using the equation (36) satisfied by Λ Qband the definition (153), we obtain that
(Lξ,ΛQb) = 2(ξ,∆Qb)−2scb(iξ,ΛQb)
+(ξ,Λ∆Qb−ΛQb+ibΛ(ΛQb)+|Qb|2σ1ΛQb+2σ1Re(¯QbΛQb)|Qb|2σ1−2Qb)
= 2(ξ,∆Qb)−2scb(iξ,ΛQb)+2σ1(ξ,Re(¯Qb,1
σ1Qb)|Qb|2σ1−2Qb)
+(ξ,Λ∆Qb−ΛQb+ibΛ(ΛQb)+|Qb|2σ1ΛQb+2σ1Re(¯Qby·∇Qb)|Qb|2σ1−2Qb)
= 2(ξ,∆Qb+|Qb|2σ1Qb)−2scb(iξ,ΛQb)−βsc(ξ,iΛ∂bQb)−(ξ,ΛΨb).
Now for the first term on the right-hand side of the equation above we notice that (75) implies
2(∆Qb+|Qb|2σ1Qb,ξ) =−2λ2−2scE[ψ]+2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2
L2−/integraldisplay
R(2)(ξ)dy,
where
R(2)(ξ) =1
σ1+1/parenleftbig
|Qb+ξ|2σ1+2−|Qb|2σ1+2−(2σ1+2)|Qb|2σ1Re(Qb¯ξ)/parenrightbig
.
Thus we arrive to the preliminary equation
(Lξ+R(ξ),ΛQb) =−2scb(iξ,ΛQb)−βsc(ξ,iΛ∂bQb)−(ξ,ΛΨb)
−2λ2−2scE[ψ]+2E[Qb]+/ba∇dbl∇ξ/ba∇dbl2
L2−/integraldisplay
R(2)(ξ)dy+(R(ξ),ΛQb).40 P. ANTONELLI AND B. SHAKAROV
It remains to extract the quadratic terms in −/integraltext
R(2)(ξ)dy+(R(ξ),ΛQb). We write
/integraldisplay
R(2)(ξ)dy= 2/integraldisplay
|Qb|2σ1−2/parenleftbig
|Qb|2|ξ|2+2σ1Re(Qb¯ξ)2/parenrightbig
+1
σ1+1/integraldisplay
R(3)(ξ)dy
whereR(3)(ξ) is the rest defined in (155). Then we also write that
(R(ξ),ΛQb) = (R3(ξ),ΛQb)
+2σ1/parenleftbigg
|Qb|2σ1−2/parenleftbigg
Re(Qb¯ξ)ξ+(2σ1−1)|Qb|−2Re(Qb¯ξ)2Qb+Qb|ξ|2/parenrightbigg
,
1
σ1Qb+y·∇Qb/parenrightbigg
whereR3(ξ) is defined in (156). In the equation above, we notice that
2σ1/parenleftbigg
|Qb|2σ1−2/parenleftbigg
Re(Qb¯ξ)ξ+(2σ1−1)|Qb|−2Re(Qb¯ξ)2Qb+Qb|ξ|2/parenrightbigg
,1
σ1Qb/parenrightbigg
= 2/integraldisplay
|Qb|2σ1−2/parenleftbig
|Qb|2|ξ|2+2σ1Re(Qb¯ξ)2/parenrightbig
dy
that is we can write that
−/integraldisplay
R(2)(ξ)dy+(R(ξ),ΛQb) =−1
σ1+1/integraldisplay
R(3)(ξ)dy+(R3(ξ),ΛQb)
+2σ1/parenleftbigg
|Qb|2σ1−2/parenleftbigg
Re(Qb¯ξ)ξ+(2σ1−1)|Qb|−2Re(Qb¯ξ)2Qb
+Qb|ξ|2/parenrightbigg
,y·∇Qb/parenrightbigg
.
Consequently, by replacing Qbwith the ground state Qgenerating an error which we denote by
E(ξ,ξ), one obtains that
/ba∇dbl∇ξ/ba∇dbl2
L2−/integraldisplay
R(2)(ξ)dy+(R(ξ),ΛQb) =H(ξ,ξ)+E(ξ,ξ)
−1
σ1+1/integraldisplay
R(3)(ξ)dy+(R3(ξ),ΛQb)
whereH(ξ,ξ) is the quadratic form defined in (37). For scsmall enough, we also obtain that E(ξ,ξ)
can be bounded by
(157) |E(ξ,ξ)| ≤δ2(sc)/integraldisplay
|ξ|2e−|y|+|∇ξ|2dy,
whereδ2(sc)→0 assc→0. /square
For details of the proof above, we refer to [36, Proposition 3 .3]. We are now ready to prove the
local virial property 4.5 when η= 0.BLOW-UP OF THE DAMPED NLS EQUATION 41
Proof of Lemma 7.1. By taking the scalar product of equation (63) with Λ Qb, we obtain
(158)0 = (i∂τξ,ΛQb)+˙b(i∂bQb,ΛQb)+(2E[Qb]−(iβsc∂bQb+Ψb,ΛQb))
+(Lξ+R(ξ),ΛQb)−2E[Qb]
−/parenleftBigg
i/parenleftBigg˙λ
λ+b/parenrightBigg
Λ(Qb+ξ)+i˙x
λ·∇Qb+(˙γ−1)Qb,ΛQb/parenrightBigg
−/parenleftBigg
i/parenleftBigg˙λ
λ+b/parenrightBigg
Λξ+(˙γ−1)ξ+i˙x
λ·∇ξ,ΛQb/parenrightBigg
.
We will study the contributions of the terms in (158) separately. Fo r the first term, we observe that
(159) (i ∂τξ,ΛQb) =d
dτ(iξ,ΛQb)−(iξ,∂τΛQb) =d
dτ(iξ,ΛQb)−˙b(iξ,Λ∂bQb).
For the second term, we use (151) to get
˙b(i∂bQb,ΛQb) =−˙b1
4/ba∇dblxQ/ba∇dbl2
L2(1+δ1(sc,b))+sc(iQb,∂bQb).
Moreover, from (70) and (56) , we bound the second term on the r ight-hand side of (159) as
|(iξ,Λ∂bQb)|/lessorsimilar/parenleftbigg/integraldisplay
|ξ|2e−|y|+|∇ξ|2dy/parenrightbigg1
2
≤Γ1
2−10ν
b,
which implies that there exists c>0 such that
˙b((i∂bQb,ΛQb)−(iξ,Λ∂bQb)) =˙bsc(∂bQb,iQb)−˙b/parenleftbigg/ba∇dblyQ/ba∇dbl2
L2
4(1+δ1(sc,b))−cΓ1
2−10ν
b/parenrightbigg
.
For the third term in (158), we use the Pohozaev-type estimate (3 5) to obtain
2E[Qb]−(iβsc∂bQb+Ψb,ΛQb) =sc(2E[Qb]+/ba∇dblQb/ba∇dbl2
L2).
For the fourth term, we use (154) to obtain that
(Lξ+R(ξ),ΛQb)−2E[Qb] =−2scb(iξ,ΛQb)−βsc(ξ,iΛ∂bQb)−(ξ,ΛΨb)
−2λ2−2scE[ψ]+H(ξ,ξ)+E(ξ,ξ)
−1
σ1+1/integraldisplay
R(3)(ξ)dy+(R3(ξ),ΛQb)
whereR(3)is defined in (155). For the fifth term in (158), by straightforward computations we get
−/parenleftBigg
i/parenleftBigg˙λ
λ+b/parenrightBigg
ΛQb+i˙x
λ·∇Qb+(˙γ−1)Qb,ΛQb/parenrightBigg
=sc(˙γ−1)/ba∇dblQb/ba∇dbl2
L2.
By combining the previous computations, we have
(160)−˙b((i∂bQb,ΛQb)−(iξ,Λ∂bQb)) =d
dτ(iξ,ΛQb)
+sc/parenleftbig
−2b(iξ,ΛQb)−β(ξ,iΛ∂bQb)+2E[Qb]+/ba∇dblQb/ba∇dbl2
L2+(˙γ−1)/ba∇dblQb/ba∇dbl2
L2/parenrightbig
−(ξ,ΛΨb)−2λ2−2scE[ψ]+H(ξ,ξ)+E(ξ,ξ)+1
σ1+1/integraldisplay
R(3)(ξ)dy
−/parenleftBigg
i/parenleftBigg˙λ
λ+b/parenrightBigg
Λξ+(˙γ−1)ξ+i˙x
λ·∇ξ,ΛQb/parenrightBigg
,42 P. ANTONELLI AND B. SHAKAROV
or equivalently
(161)˙b/parenleftbigg/ba∇dblyQ/ba∇dbl2
L2
4(1+δ1(sc,b))−cΓ1
2−10ν
b/parenrightbigg
≥d
dτ(iξ,ΛQb)
+sc/parenleftBig
˙b(∂bQb,iQb)−2b(iξ,ΛQb)−β(ξ,iΛ∂bQb)+2E[Qb]+/ba∇dblQb/ba∇dbl2
L2+(˙γ−1)/ba∇dblQb/ba∇dbl2
L2/parenrightBig
−(ξ,ΛΨb)−2λ2−2scE[ψ]+H(ξ,ξ)+E(ξ,ξ)+1
σ1+1/integraldisplay
R(3)(ξ)dy
−/parenleftBigg
i/parenleftBigg˙λ
λ+b/parenrightBigg
Λξ+(˙γ−1)ξ+i˙x
λ·∇ξ,ΛQb/parenrightBigg
.
We will now study term by term the right-hand side of (161). The firs t term vanishes for any
t∈[0,T1) because of one orthogonality condition in (51). Furthermore, by choosingscsmall
enough, we notice that
/parenleftbigg/ba∇dblyQ/ba∇dbl2
L2
4(1+δ1(sc,b))−cΓ1
2−10ν
b/parenrightbigg
≥/ba∇dblyQ/ba∇dbl2
L2
8.
For the second term, we observe that by using estimates (52), (5 6) and (80), we obtain that
sc/parenleftBig
˙b(∂bQb,iQb)−2b(iξ,ΛQb)−β(ξ,iΛ∂bQb)+2E[Qb]+/ba∇dblQb/ba∇dbl2
L2+(˙γ−1)/ba∇dblQb/ba∇dbl2
L2/parenrightBig
≥sc/ba∇dblQb/ba∇dbl2
L2
2,
forbsmall enough. Furthermore, we use (70) to bound the term
|(ξ,ΛΨb)|/lessorsimilarΓ1−ν
b,
and the control on the energy (54) to bound
/vextendsingle/vextendsingle2λ2−2scE[ψ]/vextendsingle/vextendsingle/lessorsimilarΓ2
b.
All the rests are bounded using (157) and (69)
/vextendsingle/vextendsingle/vextendsingle/vextendsingleE(ξ,ξ)−1
σ1+1/integraldisplay
R(3)(ξ)dy+(R3(ξ),ΛQb)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤δ4(sc)/integraldisplay
|ξ|2e−|y|+|∇ξ|2dy
for someδ4(sc)>0 withδ4(sc)→0 assc→0. For the last term on the right-hand side of (160),
we notice that using (13) and two of the orthogonality conditions in ( 51), we have
(iΛξ,ΛQb) =−(iξ,Λ(ΛQb))−2sc(iξ,ΛQb) = 0.
Moreover, by using (81), Cauchy-Schwartz inequality and (56), w e obtain
/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg
i˙x
λ·∇ξ,ΛQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/ba∇dbl∇ξ/ba∇dblL2/parenleftBigg
δ2(sc)/parenleftbigg/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1
2
+/integraldisplay
|∇ξ|2dy+Γ1−11ν
b/parenrightBigg
/lessorsimilarδ2(sc)Γ1−20ν
b+Γ3
2−31ν
b.
In the same way, we observe that
−((˙γ−1)ξ,ΛQb) =−/parenleftbigg/parenleftbigg
(˙γ−1)−1
/ba∇dblΛQb/ba∇dbl2
L2(ξ,LΛ(ΛQb)/parenrightbigg
ξ,ΛQb/parenrightbigg
−/parenleftbigg/parenleftbigg1
/ba∇dblΛQb/ba∇dbl2
L2(ξ,LΛ(ΛQb)/parenrightbigg
ξ,ΛQb/parenrightbiggBLOW-UP OF THE DAMPED NLS EQUATION 43
and we use again (81) to obtain that
/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/parenleftbigg/parenleftbigg
(˙γ−1)−1
/ba∇dblΛQb/ba∇dbl2
L2(ξ,LΛ(ΛQb)/parenrightbigg
ξ,ΛQb/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓb.
Now we use the coercivity property in Proposition 2.12 to obtain that there exists c>0 such that
H(ξ,ξ)−((˙γ−1)ξ,ΛQb)≥c/integraldisplay
|ξ|2e−|y|+|∇ξ|2dy
−c/parenleftbig
(ξ,Q)2+(ξ,|y|2Q)2+(ξ,yQ)2
+(ξ,iDQ)2+(ξ,iD(DQ))2+(ξ,i∇Q)2/parenrightbig
,
whereDξis defined in (11). By using property (12), (56) and the closeness o fQbwithQ, we have
that there exists a constant δ5(sc)>0 such that
(ξ,Q)2+(ξ,|y|2Q)2+(ξ,yQ)2+(ξ,iDQ)2+(ξ,iD(DQ))2+(ξ,i∇Q)2
≥(ξ,Qb)2+(ξ,|y|2Qb)2+(ξ,yQb)2+(ξ,iΛQb)2+(ξ,iΛ(ΛQb))2+(ξ,i∇Qb)2
−δ5(sc)(sc+Γb)
whereδ5(sc)→0 assc→0. Finally, we use the orthogonality conditions (51), inequalities (74) and
(79) to obtain
/vextendsingle/vextendsingle(ξ,Qb)2+(ξ,|y|2Qb)2+(ξ,yQb)2+(ξ,iΛQb)2+(ξ,iΛ(ΛQb))2+(ξ,i∇Qb)2/vextendsingle/vextendsingle/lessorsimilarΓ3
2−20ν
b.
Thus (160) implies (150). /square
7.2.Appendix B. In this appendix we report the computations needed in Lemma 4.6, ba sed on
[36, Section 3 .3] and [34, Lemma 6] and stated in Lemma 7.4 below. We recall that the remainder
term˜ξin (90) satisfies
i∂τ˜ξ+˜L˜ξ+i(˙b−βsc)∂b˜Qb−i/parenleftBigg˙λ
λ+b/parenrightBigg
Λ(˜Qb+˜ξ)−i˙x
λ·∇(˜Qb+˜ξ)
−(˙γ−1)(˜Qb+˜ξ)+iηλ2−2σ2
σ1|˜Qb+˜ξ|2σ2(˜Qb+˜ξ)+˜R(˜ξ)−˜Φb+F= 0,(162)
where
˜L˜ξ= ∆˜ξ−˜ξ+ibΛ˜ξ+2σ1Re(˜ξ˜Qb)|˜Qb|2σ1−2˜Qb+|˜Qb|2σ1˜ξ,
˜R(˜ξ) =|˜Qb+˜ξ|2σ1(˜Qb+˜ξ)−|˜Qb|2σ1˜Qb−2σ1Re(˜ξ˜Qb)|˜Qb|2σ1−2˜Qb−|˜Qb|2σ1˜ξ.
(163) F= (∆φA)ζb+2∇φA·∇ζb+iby·∇φAζb,
and
(164) ˜Φb=−Φb−iscb˜ζb+iβ∂b˜ζb+|Qb+˜ζb|2σ1(Qb+˜ζb)−|Qb|2σ1Qb.
Here Φ bis defined in (30), ζbis the outgoing radiation of Lemma 2.8, ˜ζb=φAζbis the localized
outgoing radiation where φAis a smooth cut-off defined in the beginning of Subsection 4.2. By
construction, ˜ζbsatisfies the following inequality
(165)/vextenddouble/vextenddouble/vextenddouble(1+|y|)10(|˜ζb|+|∇˜ζb|2/vextenddouble/vextenddouble/vextenddouble
L2+/vextenddouble/vextenddouble/vextenddouble(1+|y|)10(|∂b˜ζb|+|∇∂b˜ζb|2/vextenddouble/vextenddouble/vextenddouble2
L2≤Γ1−cρ
b,
wherecρ≪1 is defined in Lemma 2.8. By exploiting equation (162) and reproducing the steps in
Lemma 7.1, we obtain the following.44 P. ANTONELLI AND B. SHAKAROV
Lemma 7.4. Letη= 0. Then for any t∈[0,T1), there exists C >0such that
d
dτf(τ)≥C/parenleftbigg/integraldisplay
|∇˜ξ|2+|˜ξ|2e−|y|dy+Γb/parenrightbigg
−1
C/parenleftBigg
sc+/integraldisplay2A
A|ξ|2dy/parenrightBigg
, (166)
where
(167) f=−1
2Im/parenleftbigg/integraldisplay
y·∇˜Qb˜Qbdy/parenrightbigg
−/parenleftBig
˜ζb,iΛ˜Qb/parenrightBig
+(ξ,iΛ˜ζ).
Proof.By taking the scalar product of equation (162) with Λ ˜Qb, we obtain that
(168)0 = (i∂τ˜ξ,Λ˜Qb)+˙b(i∂b˜Qb,Λ˜Qb)+/parenleftBig
2E[˜Qb]−/parenleftBig
iβsc∂b˜Qb+˜Φb−F,Λ˜Qb/parenrightBig/parenrightBig
+(˜L˜ξ+˜R(˜ξ),Λ˜Qb)−2E[˜Qb]
−/parenleftBigg
i/parenleftBigg˙λ
λ+b/parenrightBigg
Λ(˜Qb+˜ξ)+i˙x
λ·∇˜Qb+(˙γ−1)˜Qb,Λ˜Qb/parenrightBigg
−/parenleftBigg
i/parenleftBigg˙λ
λ+b/parenrightBigg
Λ˜ξ+(˙γ−1)˜ξ+i˙x
λ·∇˜ξ,Λ˜Qb/parenrightBigg
.
Now we repeat the steps in Lemma 7.1, and arrive to the preliminary es timate
(169)−˙b/parenleftBig
(i∂b˜Qb,Λ˜Qb)−(i˜ξ,Λ∂b˜Qb)/parenrightBig
−d
dτ(i˜ξ,Λ˜Qb)
=sc/parenleftBig
−2b(i˜ξ,Λ˜Qb)−β(˜ξ,iΛ∂b˜Qb)+2E[˜Qb]+/ba∇dbl˜Qb/ba∇dbl2
L2+(˙γ−1)/ba∇dbl˜Qb/ba∇dbl2
L2/parenrightBig
−(˜ξ,Λ(˜Φb−F))−2λ2−2scE[ψ]+H(˜ξ,˜ξ)+E(˜ξ,˜ξ)+1
σ1+1/integraldisplay
R(3)(˜ξ)dy
−/parenleftBigg
i/parenleftBigg˙λ
λ+b/parenrightBigg
Λ˜ξ+(˙γ−1)˜ξ+i˙x
λ·∇˜ξ,Λ˜Qb/parenrightBigg
.
The most important difference between (169) and (160) is the leadin g order term ( ˜ξ,Λ(˜Φb−F))
instead of ( ˜ξ,ΛΨb). First, by integration by parts and by using (13), we have that
(i∂b˜Qb,Λ˜Qb) =∂b(i˜Qb,Λ˜Qb)−(i˜Qb,Λ∂b˜Qb)
=∂b(i˜Qb,y·∇˜Qb)+(iΛ˜Qb,∂b˜Qb)+2sc(i˜Qb,∂b˜Qb),
which equivalently implies that
−˙b(i∂b˜Qb,Λ˜Qb) =−1
2d
dτ(i˜Qb,y·∇˜Qb)−˙bsc(i˜Qb,∂b˜Qb).
Thus we can write the left-hand side of (169) as
−˙b/parenleftBig
(i∂b˜Qb,Λ˜Qb)−(i˜ξ,Λ∂b˜Qb)/parenrightBig
−d
dτ(i˜ξ,Λ˜Qb) =d
dτf(τ)+˙b(i˜ξ,Λ∂b˜Qb)−˙bsc(i˜Qb,∂b˜Qb)
where
f(τ) =−1
2(i˜Qb,y·∇˜Qb)−(i˜ξ,Λ˜Qb).BLOW-UP OF THE DAMPED NLS EQUATION 45
We notice that by using the estimate on |˙b|(80), the smallness of ˜ζb(165), (69) and the controls
(56), (52) we obtain
−˙b(i˜ξ,Λ∂b˜Qb) =−˙b(i(ξ−˜ζb),Λ∂b(Qb+˜ζb))
/greaterorsimilarΓ1−20ν
b/parenleftBigg/parenleftbigg/integraldisplay
|∇ξ|2+|ξ|2e−|y|dy/parenrightbigg1
2
+Γ1−cρ
b/parenrightBigg
/greaterorsimilarΓ3
2−31ν
b.
Next, all the terms on the right-hand side of (169) are bounded in t he same way as in Lemma 7.1
except for the scalar product ( ˜ξ,Λ(˜Φb−F)). In this way, we obtain that there exists a constant
C >0 such that
(170)d
dτf≥C/integraldisplay
|˜ξ|2e−|y|+|∇˜ξ|2dy−C(sc+Γ3
2−50ν
b)−(˜ξ,Λ(˜Φb−F)).
and Φ bis defined in (29) and Fin (88). We notice that from the definition of ˜Φbin (92) and the
bound on Φ bin (31) and the control on the outgoing radiation (165) we get
/vextenddouble/vextenddouble/vextenddouble(1+|y|10)(|˜Φb|+|∇˜Φb|)/vextenddouble/vextenddouble/vextenddouble2
L2≤Γ1+ν4
b+sc,
which implies that/vextendsingle/vextendsingle/vextendsingle(˜ξ,Λ˜Φb)/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓ1+ν4
b.
Finally, it remains to study the last term ( ˜ξ,ΛF) in (170). We observe that
(˜ξ,ΛF) = (ξ−˜ζb,ΛF) = (ξ,ΛF)−(˜ζb,ΛF).
For the second term on the right-hand side of the equation above, we notice that
−(˜ζb,ΛF) =−(˜ζb,DF)+sc(˜ζb,F)
and from (165)/vextendsingle/vextendsingle/vextendsinglesc(˜ζb,F)/vextendsingle/vextendsingle/vextendsingle≤scΓ1−cρ≤Γ3
2,
while using (25)
−(˜ζb,DF)≥C1Γb
for someC1>0. Finally, we use the Young inequality and (165) to obtain
|(ξ,DF)|/lessorsimilar/parenleftBigg/integraldisplay2A
A|ξ|2dy/parenrightBigg1
2/parenleftBigg/integraldisplay2A
A|F|2/parenrightBigg1
2
≤1
C2/integraldisplay2A
A|ξ|2dy+C1
2Γb.
for someC2>0. Collecting everything we see that from (170) we obtain
d
dτf≥C/integraldisplay
|˜ξ|2e−|y|+|∇˜ξ|2dy−C(sc+Γ1+ν2
b)−(˜ξ,Λ(˜Φb−F))
≥C/integraldisplay
|˜ξ|2e−|y|+|∇˜ξ|2dy+C1Γb−C1
2Γb−C3/parenleftBigg
sc+/integraldisplay2A
A|ξ|2dy/parenrightBigg
for someC3>0. This is equivalent to (166) /square46 P. ANTONELLI AND B. SHAKAROV
7.3.Appendix C. In this appendix, we will report the proof inequality (147) which has a lready
been done in the appendix of [36].
Proof.We give a proof only for the term N1(ξ), since that for N2(ξ) is equivalent. We define the
functionF:C→Cas
F(z) =|z|2σ1z.
From the definition of r1(141) it follows that
/ba∇dbl|∇|sN1(ˆξ)/ba∇dblLr′/lessorsimilar1
λ(2σ1+1)(˜s−sc)/ba∇dbl|∇|s(F(Qb+ξ)−F(Qb))/ba∇dblLr′
where
(171) ˜ s=s+d
2−d
r.
We claim the following estimate:
/ba∇dbl|∇|s(F(Qb+ξ)−F(Qb))/ba∇dblLr′/lessorsimilar/ba∇dbl|∇|˜sξ/ba∇dblL2.
Indeed observe that
(172) F(Qb+ξ)−F(Qb) =/parenleftbigg/integraldisplay1
0∂zF(Qb+τξ)dτ/parenrightbigg
ξ+/parenleftbigg/integraldisplay1
0∂¯zF(Qb+τξ)dτ/parenrightbigg
¯ξ.
Both terms on the right-hand side of (147) are treated in the same way. We define q∈Rsuch that
(173)1
q=1
r′−1
r.
For any function h, by the definition of r(141), ˜s(171) and by Sobolev embedding we have
(174) /ba∇dblh/ba∇dblL2σ1q/lessorsimilar/ba∇dbl|∇|sh/ba∇dblLr/lessorsimilar/ba∇dbl|∇|˜sh/ba∇dblL2.
By the fractional Leibniz rule (see for example [22]), it follows that
/vextenddouble/vextenddouble/vextenddouble/vextenddouble|∇|s/parenleftbigg
ξ/integraldisplay1
0∂zF(Qb+τξ)dτ/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lr′/lessorsimilar/ba∇dbl|∇|sξ/ba∇dblL2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay1
0∂zF(Qb+τξ)dτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lq(175)
+/ba∇dblξ/ba∇dblL2σ1q/vextenddouble/vextenddouble/vextenddouble/vextenddouble|∇|s/integraldisplay1
0∂zF(Qb+τξ)dτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lu,
where
(176)1
u=1
r′−1
2σ1q.
Then, from (174), we have
/vextenddouble/vextenddouble/vextenddouble/vextenddouble|∇|s/parenleftbigg
ξ/integraldisplay1
0∂zF(Qb+τξ)dτ/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble
Lr′/lessorsimilar/ba∇dbl|∇|˜sξ/ba∇dblL2/parenleftbigg/integraldisplay1
0/ba∇dbl∂zF(Qb+τξ)/ba∇dblLqdτ (177)
+/integraldisplay1
0/ba∇dbl|∇|s∂zF(Qb+τξ)/ba∇dblLudτ/parenrightbigg
,
Now it remains to prove that
(178)/parenleftbigg/integraldisplay1
0/ba∇dbl∂zF(Qb+τξ)/ba∇dblLqdτ+/integraldisplay1
0/ba∇dbl|∇|s∂zF(Qb+τξ)/ba∇dblLudτ/parenrightbigg
/lessorsimilar1.
By homogeneity
∀τ∈[0,1],|∂zF(Qb+τξ)|/lessorsimilar|Qb|2σ1+|ξ|2σ1,BLOW-UP OF THE DAMPED NLS EQUATION 47
and so
/integraldisplay1
0/ba∇dbl∂zF(Qb+τξ)/ba∇dblLqdτ/lessorsimilar/integraldisplay1
0(|Qb|2σ1+|ξ|2σ1
L2σ1q)dτ/lessorsimilar/integraldisplay1
0(|1+|ξ|2σ1
L2)dτ/lessorsimilar1.
Moreover, recall [22] the equivalent definition of the homogeneous Besov norm:
∀0<˜s<1,/ba∇dblu/ba∇dbl2
˙B˜s
q,2∼/integraldisplay∞
0/parenleftbig
R−˜ssup
|y|≤R|u(.−y)−u(.)|Lq/parenrightbig21
RdR.
Recall also that /ba∇dbl|∇|˜sψ/ba∇dblLq/lessorsimilar/ba∇dblψ/ba∇dbl˙B˜s
q,2.
Observe that, for 1 ≤d≤3, 2σ1>2 and it follows by homogeneity,
|∂zF(u)−∂zF(v)|/lessorsimilar|u−v|(|u|2σ1−1+|v|2σ1−1).
We define
hτ=Qb+τξ,0≤τ≤1.
We estimate from H¨ older and (174)
|∂zF(hτ)(.−y)−∂zF(hτ)(.)/ba∇dblLu/lessorsimilar/vextenddouble/vextenddouble((hτ)(.−y)−(hτ)(.)|)(|hτ(.−y)|2σ1−1+|hτ(.)|2σ1+1)/vextenddouble/vextenddouble
Lu
/lessorsimilar/ba∇dblhτ(.−y)−hτ(.)/ba∇dblLr/ba∇dblhτ/ba∇dbl2σ1−1
L2σ1q
/lessorsimilar/ba∇dblhτ(.−y)−hτ(.)/ba∇dblLr/ba∇dbl|∇|˜shτ/ba∇dbl2σ1−1
L2,
and so we have that
/ba∇dbl|∇|s∂zF(hτ)/ba∇dblLu/lessorsimilar/integraldisplay∞
0/parenleftbig
R−˜ssup
|y|≤R|∂zF(hτ)(.−y)−∂zF(hτ)(.)|Lu/parenrightbig21
RdR
/lessorsimilar/ba∇dbl|∇|˜shτ/ba∇dbl2σ1−1
L2/ba∇dbl|∇|shτ/ba∇dblLr/lessorsimilar/ba∇dbl|∇|˜shτ/ba∇dbl2σ1
L2,
and this concludes the proof of (147). /square
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Gran Sasso Science Institute, viale Francesco Crispi, 7, 67 100 L’Aquila, Italy
Email address :paolo.antonelli@gssi.it
Gran Sasso Science Institute, viale Francesco Crispi, 7, 67 100 L’Aquila, Italy
Email address :boris.shakarov@gssi.it |
0805.0893v1.Comparison_Between_Damping_Coefficients_of_Measured_Perforated_Micromechanical_Test_Structures_and_Compact_Models.pdf | 9-11 April 2008
©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 Comparison Between Damping Coefficients of
Measured Perforated Structures and Compact Models
T. Veijola1, G. De Pasquale2, and A. Somá2
1 Department of Radio Science and Engineering, Helsinky University of Technology
P.O. Box 3000, 02015 TKK, Finland.
2 Department of Mechanics, Polytechnic of Torino
Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
Abstract - Measured damping coefficients of six different
perforated micromechanical test structures are compared with damping coefficients given by published compact models. The motion of the perforated plates is almost translational, the
surface shape is rectangular, and the perforation is uniform
validating the assumptions made for compact models. In the structures, the perforation ratio varies from 24% - 59%. The study of the structure shows that the compressibility and inertia do not contribute to the damping at the frequencies used (130kHz - 220kHz). The damping coefficients given by all four compact models underestimate the measured damping coefficient by approximately 20%. The reasons for this
underestimation are discussed by studying the various flow
components in the models.
I. I NTRODUCTION
Perforations are used in micromechanical squeeze-film
dampers for several reasons. The main purpose is to reduce
the damping and spring forces of oscillating structures due to the gas flow in small air gaps. Generally, the modeling problem is quite complicated, since the damping force acting on the moving structure depends on the 3D fluid flow in the perforations, in the air gap, and also around the structure. Compact models have been published in the literature, but their verification is generally questionable. Verification methods used are FEM solutions of the Navier-Stokes equations of the fluid volume and measurements [1].
In this paper, responses of four compact models are
compared with measured responses of test structures. Six different perforated plates (figures 1 and 2) with different topologies have been measured at their first out-of-plane resonant frequencies, and the damping coefficients have been calculated from the quality factors (Q values) and effective masses. The measurement setup, the testing procedure and specimens characteristics are presented in [2]; here it is observed that dynamic parameters of the microsystem characterizing the fluidic and structural coupling can be extracted from the experimental frequency response function (FRF). The dynamic performance of microstructures are
discussed based on the analytical solutions to perforated
parallel-plate problems in [3], [4] and [5]. Since the perforation is uniform, the motion is almost translational, and also since the shape of the surface is rectangular, analytic damping models are applicable.
First, the oscillating flow is analyzed using several
characteristic numbers, the applicable modeling method is then chosen, the damping coefficients are calculated and compared with the measured ones. Finally, the results are discussed.
II. T EST STRUCTURES AND MEASUREMENTS
Figures 1 and 2 show the structures of the test specimen.
The height of the plate hc = 15µm, the air gap height h =
1.6µm. Table I shows the other dimensions of the measured devices. In the table, q is the perforation ratio in percent, in
this case q = MNs
02/(LW), M and N are the number of holes in
the length and width directions, respectively.
Fig. 1. Geometrical shape and dimensions of the vibrating structures.
The measurements are made using the interferometric
microscope Fogale ZoomSurf 3D (figure 3), with 20x objective magnification factor, 0.1nm of vertical resolution and 0.6µm of lateral resolution.
The Frequency Shift technique is used, consisting of the
excitation of the structure by an alternate voltage, the frequency of which is progressively increased by discrete steps. For each level of actuation frequency, the corresponding amplitude of vibration is stored and the experimental FRF is plotted for the detection of the resonance
peak (figure 4). The first detection is made across a wide 9-11 April 2008
©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 frequency range (0-500kHz) in order to roughly locate the
resonance; five successive identical detections are then performed across a more precise and narrow range. These are statistically treated to extract the values of resonance reported in Table II.
TABLE I
DIMENSIONS OF MEASURED TOPOLOGIES
type L
[µm] W
[µm] M×N L:W s0
[µm] s1
[µm] q %
A 372.4 66.4 36x6 6:1 5.0 5.2 24
B 363.9 63.9 36x6 6:1 6.1 3.9 37
C 373.8 64.8 36x6 6:1 7.3 3.0 50
D 369.5 64.5 36x6 6:1 7.9 2.3 59
E 363.8 123.8 36x12 3:1 6.2 3.8 38
F 363.8 243.8 36x24 3:2 6.2 3.8 38
Fig. 2. Microscope image of specimen F.
The measurement technique described uses a red
monochromatic light source for the interferometric fringes detection. The vibration amplitude is detected optically in correspondence of a selectable region ( detection window ) of
the specimen, located at the center of the suspended plate. The output value of the oscillation amplitude is averaged between the values captured by each pixel of the CCD
camera inside the active window.
Fig. 3. Interferometric microscope Fogale ZoomSurf 3D (a) and the 20x
Nikon objective (b).
020406080100120140
180 190 200 210 220 230 240
Frequency (kHz)Amplitude (nm)
Fig. 4. Displacement vs. frequency diagram of specimen C. The quality factor is extracted from the experimental
curve, that was previously interpolated by a 6-order polynomial; the damping coefficient is finally calculated from the quality factor, resonant frequency and the effective mass by means of the method of the half power bandwidth .
These are shown in Table II. The effective mass is calculated from FEM eigenmode analysis. The mass ratio α is the ratio
between the modal mass and total mass.
TABLE II
MEASURED DAMPING COEFFICIENTS AND RESONANT FREQUENCIES OF SIX
DIFFERENT TEST STRUCTURES
type cm measured
[10-6Ns/m] f0 measured
[kHz] mass ratio α
A 47.38 201.637 0.918
B 19.46 204.329 0.893
C 9.863 211.011 0.885
D 7.609 222.282 0.856
E 38.22 173.904 0.946
F 67.44 138.564 0.974
It is ensured that the amplitude of the oscillation is small
compared with the gap height, and the static bias voltage caused by the excitation signal and DC bias voltage does not deflect the plate changing the air gap height.
III. M ODELING OF THE DAMPING COEFFICIENT
A. Analysis of the fluid flow
The actual mass is supported with thin beams in such a
way that the movement of the mass is approximately translational. This justifies starting with an analysis where the velocity of the plate surface is constant.
Flow patterns
In perforated dampers in perpendicular motion, two
different flow patterns can be distinguished. The first is the “closed borders” pattern, where the fluid flows only through the holes. The second pattern, the “closed holes” pattern
considers only the flow from the damper borders. In practical
dampers both patterns exist simultaneously. The perforation ratios q (the area of the surface without holes divided by the
area of the holes) are here considerably high, ranging from 24% - 59%. Also, the holes are relatively wide compared to the air gap height. It is then evident that the first “closed borders” flow pattern is strong here. Here compact models that consider both flow patterns are selected; the contribution the flow patterns in the measured cases will be discussed later in this paper.
Rarefied gas
Air at standard atmospheric conditions is used in the
measurements. The pressure P
A = 101kPa, the density ρ =
1.155 kg/m3, the viscosity coefficient µ = 18.5 ⋅10-6 Ns/m2,
and the mean free path λ = 65nm. The air gap height h =
1.6µm, which makes the Knudsen number of the air gap flow
Kch = λ/h to be 0.04, and the smallest hole diameter is 5µm
making the Knudsen number of the perforation Ktb = λ/s0 to
be 0.013. The estimated contributions to the damping coefficient are 1/(1 + 6 K
ch) and 1/(1 + 7.567 Ktb), that is -24% 9-11 April 2008
©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 and -9.8%, respectively. The assumption of a slightly rarefied
gas is justified, and the slip velocity model is sufficient.
Compressibility
Next, the contribution of compressibility is analyzed. This
can be made by studying the squeeze number. For a rigid surface, the squeeze number is
2212
hPW
Aω ησ= (1)
W is here the smallest, “dominating” characteristic
dimension. For example, for case A without holes
ω σ6108.3−⋅ = (2)
At 200kHz σ = 4.8. This means that without perforation,
compressibility should be considered (when σ ∼ 20, the
viscous and spring forces are equal). When the surface is
perforated, the situation changes completely. The characteristic dimension can now be estimated to be the space between the holes. In case A this is 5.2µm. The squeeze
number now becomes σ = 0.03 at 200kHz. This is an
overestimate of the real situation, since accounting for the
damping in the holes will make the effective squeeze number considerably smaller.
According to the squeeze number analysis, the spring
forces are negligible compared with the damping forces, and
noncompressible gas can be assumed without loss of any
accuracy. The damping coefficient can be considered constant up to several MHz.
Also, it is expected that the spring force due to the gas is
much smaller than the force due to the effective spring of the mechanical structure. The frequency shift due to gas compressibility is expected to be very small. The spring force in the system is only due the effective spring of the mass-spring system. The spring coefficient can also be considered constant at least up to several MHz.
Gas Inertia
One can suspect that the inertia of the gas may contribute
to the damping coefficient. The place where the inertia is the largest is the “widest” flow channel, that is the perforation holes.
The contribution of inertia is characterized by the
Reynolds number R
e, specified for a circular channel as [6]:
µωρ2
erR= (3)
The “worst” case, where the inertial is the largest, is case
D where the hole diameter is the largest. If the square channel is approximated with a circular channel having a radius of r = 4µm, that’s half of the hole side, gives R
e =
0.998·10-6ω, and at 200 kHz R e = 1.255. The real and
imaginary parts of the impedance are equal when Re ~ 6. The
additional imaginary part does not directly influence the damping coefficient, but the change is due to the frequency dependent real part. At R
e ~ 6, the change in the real part in
only 3.2% [6].
This Reynolds number study shows that the inertia needs
not to be considered even in the accurate analysis. B. Compact models
A model for the noncompressible perforation cell is
sufficient, as indicated in the study above. Four models that consider both “closed holes” and “closed borders” flow patterns are selected to be compared. The first one, M1, is a model by Bao [3] for a rectangular damper that is much longer than wide. In this model the air gap regime flow
resistance, the flow resistance of a circular perforation, and
its constant elongation are included. Continuum flow conditions are assumed. The 2
nd model M2 has been also
presented in [3], but now an arbitrary rectangular surface is assumed. Next, the model M3 in [4] is used. The air gap flow resistance model, the circular perforation flow channel model, and four different elongations of the flow channels, that vary depending on the ratios of the cell dimensions, are included in the model. Slip velocity conditions are used for the air gap and the perforations. The 4th model, M4, is made especially for square holes [5]. It includes similar
components as the previous model and accounts also for the
rarefied gas in the slip flow regime.
Model M4 for square perforations accepts directly the
dimensions given in Table I (size of the perforation cell s
x =
s0 + s1). To apply the other models, an effective radius of the
circular perforation r0 and the cell rx need to be specified
first. Matching the areas of the actual cell sx2, and the
equivalent circular cell πr02 gives
πx
xsr= (4)
The radius r0 is determined by requiring the acoustic
impedances of square and rectangular channels to match. For relatively small Knudsen numbers this leads to approximately [3], [4]
00
0 547.02096.1ssr ≈ = (5)
The Appendix shows all equations needed in computing
the damping coefficients using models M1…M4.
IV. S IMULATION RESULTS AND DISCUSSION
The results of the comparison are shown in Table III. ∆i is
the relative error of the simulated damping coefficient cs of
model M i compared to the measured damping coefficients.
TABLE III
RELATIVE ERRORS OF THE COMPACT MODELS
type ∆1 [%]
M1 ∆2 [%]
M2 ∆3 [%]
M3 ∆4 [%]
M4
A -23.53 -25.74 -33.51 -33.27
B -16.36 -18.06 -21.02 -21.96
C -5.21 -6.59 -4.11 -6.65
D -14.66 -15.72 -12.46 -15.29
E -17.27 -18.94 -19.03 -20.14
F -4.77 -6.70 -5.19 -6.52
The results of models M3 and M4 are quite close to each
other, the largest error between them is only 2.8%-points, showing that the effective radius approach is sufficient. Continuum conditions were assumed for M1 and M2, giving approximately 10% larger values than with slip velocity conditions. If these models are corrected to account for the 9-11 April 2008
©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 rarefied gas, the errors become approximately 10% worse,
than those shown in Table III.
The drag on the sidewalls is expected to increase the
damping coefficient since the structure is relatively high. The models do not account for this drag force. The moving supporting beams will have also an additional contribution to the damping. The length of the supporting beams in all cases is about L
b = 122µm, and their widths are about Wb = 4µm. A
rough approximation for the contribution of the supporting beams is
) 61()3.1 (
34
ch33
b b
bK hh WLc
++=µ
(6)
This gives a damping coefficient of cb = 0.16 ⋅10-6 Ns/m.
This approximation shows that the damping due to the beams
is very small.
The responses of models M1 and M2 are quite close. One
could expect that the error of M1 would be larger in case E and especially in case F, since the length to width ratio is quite small in these cases. The explanation for this can be found by studying the contribution of the different flow patterns. This can be easily done using the perforation cell model that assumes the “closed borders” flow pattern, where the pressure distribution is independent of the shape of the
damper. The flow resistances R
P for a perforation cell in [4]
and [5] are derived using this assumption. The damping coefficient becomes simply c
P = NMR P (MN is the number of
holes). Table IV shows the errors of the damping coefficients c
P compared with the measured values using the models for
circular cells, M5, and rectangular cells, M6, as presented in [4] and [5], respectively.
TABLE IV
RELATIVE ERROR OF “PERFORATION CELL” COMPACT MODELS
type ∆5 [%]
M5 ∆6 [%]
M6
A -17.25 -16.92
B -7.81 -9.00
C 7.38 4.36
D -3.55 -6.83
E -11.45 -12.73
F 0.37 -1.08
The results in Table IV show that the “closed borders”
flow pattern is the dominant one. The contribution of the “closed holes” flow is only 6% - 16% of the damping coefficient. This explains why models M1 and M2 differ only slightly in this case: the contribution of the shape-dependent damping is quite small.
TABLE V
RELATIVE CONTRIBUTIONS OF THE FLOW RESISTANCES OF MODEL M5
type Rs [%] Ris [%] R ib [%] Ric [%] Rc [%] Re [%]
A 8.15 9.78 0.78 5.63 68.01 7.65
B 7.62 12.94 1.87 5.13 64.05 8.40
C 6.48 15.30 3.50 4.55 61.19 8.98
D 4.51 14.49 5.03 4.06 62.59 9.31
E 7.51 13.18 2.00 5.07 63.80 8.45
F 7.51 13.18 2.00 5.07 63.80 8.45
To study further the sources of damping, Table V shows
the contributions of the flow resistance components in M5 [4]. The flow resistance of the perforations R
S is the most significant source of damping; its contribution is
approximately 65%. The 2nd important contribution comes
from the intermediate region resistances RIS, RIB, and RIC: 15
- 20%. Next important is the elongation at the perforation outlet R
E, about 8%.
V. C ONCLUSIONS
Measured damping coefficients have been compared to
those obtained with four different compact models for
perforated dampers. After analyzing the oscillating flow with several characteristic numbers, sufficient models were selected. Only translational motion was assumed. The results of all models were quite close to each other, a systematic underestimate of the damping coefficient was about -20%. The reasons for this were discussed and the contribution of various flow components were presented. For a more accurate analysis, the realistic modes of the plates should be considered. It is expected that the “closed holes” flow pattern will become relatively stronger in this case.
The comparison showed also how a model for circular
perforations can be used to model square holes.
REFERENCES
[1] E.S. Kim, Y.H. Cho and M.U. Kim, “Effect of Holes and Edges on the
Squeeze Film Damping of Perforated Micromechanical Structures”
Proceedings of IEEE Micro Electro Mechanical Systems Conference ,
pp. 296-301, 1999.
[2] A. Somà and G. De Pasquale, “Identification of Test Structures for
Reduced Order Modeling of the Squeeze Film Damping in MEMS”,
Proc. DTIP Symposium on Design, Test, Integration and Packaging of
MEMS & MOEMS, pp. 230-239, 2007.
[3] M. Bao, H. Yang, Y. Sun and P.J. French, “Modified Reynolds’
equation and analytical analysis of perforated structures”, J. Micromech.
Microeng. , vol. 13, pp. 795-800, 2003.
[4] T. Veijola, “Analytic Damping Model for an MEM Perforation Cell”,
Microfluidics and Nanofluidics, vol. 2, pp. 249-260, 2006.
[5] T. Veijola, “Analytic Damping Model for a Square Perforation Cell”,
Proc. of the 9th International Conference on Modeling and Simulation
of Microsystems , pp. 554-557, 2006.
[6] C. J. Morris and F. K. Forster, “Oscillatory Flow in Microchannels”,
Experiments in Fluids , vol. 36, pp. 924-937, 2004.
9-11 April 2008
©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 APPENDIX
This Appendix contains equations for four compact
models M1…M4. The dimensions and symbols in Fig. 1 are used: the length and width of the perforated plate are L and
W. The side lengths of the square holes and the square
perforation cells are s
0 and sX = s 0 + s 1, respectively.
A. Model M1 equations
The equations for a narrow hole plate (L>>W) are given
in [3]. Note, in the following equations a = W/2 and b = L/2.
The equivalent radii for the circular cell and hole are given in Eqs. (4) and (5). The damping coefficient c is
−
+ =la
al
hhKr
rhaL c
CCtanh 1
16)( 318234
0
2
02β
βµ
where
()
()
()()
831631323 ln4 4
0
C eff03
C4
02
02eff34 2
rh HrrhhKrrHhlK
C
πβββηββηβ β β β
+ ==+==− − − =
B. Model M2 equations
The equations for an arbitrary shaped rectangular plate are
also included in [3]. Also, in the following equations a = W/2
and b = L/2. The equivalent radii for the circular cell and hole
are given in Eqs. (4) and (5). The damping coefficient c is
() ()
332 2
hb acµγ−=
where
()
()
()
()[]∑∞
=++− − =
,...5,3,123
2 22
232
3 2
2/ 12/ 1tanh24/2 sinh/1 sinh6 3
nn nn
παακπα
πκαααα α γ
al
ba= = α κ ,
Above, l is the same as used in M1 equations.
C. Model M3 equations
A model for a circular perforation cell is derived in [4],
and the damping coefficient of a rectangular perforated plate
is given in the paper. Note, in the following equations a = W
and b = L. The equivalent radii for the circular cell and hole are given in Eqs. (4) and (5). The damping coefficient c is
())1(/1 ,1
,...5,3,1 ,... 5,3,1 , eff eff ,CR ba Gc
mn nm nm∑∑∞
=∞
= +=
Where the effective surface dimensions are
()
() hK b bhK a a
ch effch eff
3.313.13.313.1
+ +=+ +=
and
()
422P
,ch3622
22
22
,
64ab 768,
πµπ
nmMNRRQh nm
bn
amba G
nmnm
=
+ =
The flow resistance of a single perforation cell is
()
()
∆+ = +∆ =∆ =∆−=
− +− =+ + + + + =
0E
tbC
E CC0 ICB0 IBS 2
022
02
X
IS4
X4
0
2
X2
0
0X
3
ch4
X
SE C IC 4
04
X
IB IS S P
88868 283ln21 12
rQhR Rr Rr RhrrrRrr
rr
rr
hQrRR R R
rrR R R R
πµπµπµπµπµ
where the elongations are
++
− =∆++ −
=∆
hh
hrfKK
rrKrr
rr
C 0
B
chtb
2
X2
0
Bch2
X2
0
X0
S
,1732.01812.0133.15.2186.0 32.0 56.0
()
()
− +×+ ⋅=∆
− − +=∆
hrf
rr
rrKrr
rrK
0
E 4
X4
0
2
X2
0tb
E2
X2
0
X0
tb C
754.0 2.0116216.013 944.025.0 41.0 66.0 61
π
where the functions are
()()
()()ch5.3
E334
B
5.17117811 4311.71 ,
Kxxfyyxyxf
++=++=
The flow rate coefficients and Knudsen numbers for the
air gap and the holes are 9-11 April 2008
©EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5 0tb tb tbch ch ch
, 41, 61
rK K QhK K Q
λλ
= +== +=
D. Model M4 equations
A model for a rectangular perforation cell has been given
in [5]. Note, in the following equations a = W and b = L. The
damping coefficient c is given by (C1), where RP for a
rectangular hole is
()
()
∆+ = +∆ ==∆−=
− +− =+ + + + + =
0E
sqC
E CC0 ICIBS 2
022
02
X
IS4
X4
E0
2
X2
E0
E0X
3
ch4
X
SE C IC 4
04
X
IB IS S P
454.28454.28038 283ln21 12
sQhR Rs RRhss sRrr
rr
rr
hQrRR R R
ssR R R R
µµµπµ
where the elongations are
()
() ()
+ − + =∆=∆− + =∆
83.2
0 4
sq EC2
S
019.01 1 41242.0302.08.3 5.61122.0
hsK ξξ ξ
where
X0
ss=ξ
The equation for ∆E includes a misprint in [5]. The
corrected equation is shown above.
The flow rate coefficients and Knudsen numbers for the
square hole are
0sq sq sq , 567.71sK K Qλ= +=
The effective radius is
4 20
E0008.0 02108.0158076.0
ξ ξ+ +=sr
|
2209.00558v1.Growth_parameters_of_Bi0_1Y2_9Fe5O12_thin_films_for_high_frequency_applications.pdf | 1
Growth parameters of Bi 0.1Y2.9Fe5O12 thin films for high frequency
applications
Ganesh Gurjar1,4, Vinay Sharma2, S. Patnaik1,*, Bijoy K. Kuanr3
1School of Physical S ciences, Jawaharlal Neh ru University, New Delhi, INDIA 110067
2Department of Physics, Morgan State University, Baltimore, MD, USA 21251
3Special C entre for Nanosciences, Jawaharlal Nehru University, New Delhi , INDIA 110067
4Shaheed Rajguru College of Applied Sciences for Women, University of Delhi, INDIA 110096
Abst ract
The growth and characterization of Bismuth (Bi) substituted YIG ( Bi-YIG, Bi0.1Y2.9Fe5O12) thin
films are reported. Pulsed laser deposited (PLD) films with thicknesses ranging from 20 to 150 nm
were grown o n Gadolinium Gallium Garnet substrates . Two substrate orientations of (100) and
(111) were considered . The enhanced distribution of Bi3+ ions at dodecahedral site along (111) is
observed to lead to an increment in lattice constant from 12.379 Å in (1 00) to 12.415 Å in (1 11)
orient ed films. Atomic force microscopy images show ed decreasing roughness with increasing
film thickness. Compared to (100) grown films, (111) orient ed films showed an increase in
ferromagnetic resonance linewid th and consequent increase in Gilbert dampin g. The lowest
Gilbert damping values are found to be (1.06±0. 12) × 10-4 for (100) and (2.30±0. 36) × 10-4 for (111)
oriented films with thickness of ≈150 nm . The observed value s of extrinsic linewidth, effective
magnetization , and anisotropic field are related to thickness of the films and substrate orientation.
In addition, the in-plane angular variation establishe d four-fold symmetry for the (100) deposited
films unlike the case of (111) deposited films. This study prescribes growth condition s for PLD
grow n single-crystalline Bi -YIG films towards desired high frequency and magneto -optic al device
applications.
Keyword s: Bi-Yttrium iron oxide; Thin film; Lattice mismatch; Pulsed Laser Deposition;
Ferromagnetic resonance; Gilbert damping; Inhomogeneous br oadening .
Corresponding authors: spatnaik@mail.jnu.ac.in 2
1.1 Introduction
One of the most important magnetic materials for studying high frequency magnetization
dynamics is the Yttrium Iron Garnet (YIG, Y 3Fe5O12). Thin film form of YIG have attracted a
huge attention in the field of spintronic devices due to its large spin -wave propagation length , high
Curie temperature T c ≈ 560 K [1], lowest Gilbert damping and strong magneto -crystalline
anisotropy [2-7]. Due to these merits of YIG, it finds several ap plications such as in magneto -
optical (MO) devices, spin-caloritronics [8,9] , and microwave resonators and filters [10-14].
The crystal structure of YIG is body centered cubic under Ia3̅d space group . In Wyckoff
notation, t he yttrium (Y) ions are located at the dodecahedral 24c sites, whereas the Fe ions are
located at two distinct sites ; octahedral 16a and tetrahedral 24d . The oxygen ions are located in
the 96h sites [7]. The ferrimagnetism of YIG is induced via a super -exchange interaction at the ‘d’
and ‘a’ site between the non-equivalent Fe3+ ions. It has already been observed that substituting
Bi/Ce for Y in YIG improves magneto -optical responsiv ity [13,15 -21]. In addition, Bi substitution
in YIG (Bi -YIG) is known to generate growth -induced anisotropy, therefore, perpendicular
magnetic anisotropy (PMA) can be achieved in Bi doped YIG, which is beneficial in applications
like magnetic memory and logic devices [7,22,23] . Due to its u sage in magnon -spintronics and
related disciplines such as caloritronics, the study of fundamental characteristics of Bi -YIG
materials is of major current interest due to their high uniaxial anisotropy and F araday rotation
[17, 24-27]. Variations in the concentration of Bi3+ in YIG, as well as substrate orientation and
film thickness, can improve strain tuned structural properties and magneto -optic characteristics .
As a result, selecting the appropriate substrate orientation and film thickness is important for
identifying the growth of Bi-YIG thin films. 3
The structural and magnetic characteristics of Bi -YIG [Bi 0.1Y2.9Fe5O12] thin film have been
studied in the current study. Gadolinium Gallium Garnet (GGG) substrates with orientations of
(100) and (111) were used to grow thin films . The Bi-YIG films of four different thickness (≈20
nm, 50 nm, 100 nm and 150 nm ) were deposited in -situ by pulsed laser deposit ion (PLD) method
[19,2 8] over single -crystalline GGG substrates . Along with structural characterization of PLD
grown films , magnetic properties were ascertained by using vibrating sample magnetometer
(VSM) in conjunction with ferromagnetic resonance (FMR) techniques. FMR is a highly effective
tool for studying magnetization dynamics. The FMR response not only provides information about
the magnetization dynamic s of the material such as Gilbert damping and anisotropic field, but also
about the static magnetic properties such as saturation magnetization and anisotropy field.
1.2 Experiment
Polycrystalline YIG an d Bi -YIG targets were synthesized via the solid -state reaction
method. Briefly, yttrium oxide (Y 2O3) and iron oxide (Fe 2O3) powders from Sigma -Aldrich were
grounded for ≈14 hours before calcination at 1100 ℃. The calcined powders were pressed into
pellets of one inch and sintered at 1300 ℃. Using these polycrystalline YIG and Bi -YIG targets,
thin films of four thicknesses ( ≈20 nm, 50 nm, 100 nm, and 150 nm) were synthesized in -situ on
(100) - and (111) -oriented GGG substrates using the PLD method. The samples are labelled in the
text as 20 nm (100), 20 nm (111) , 50 nm (100) , 50 nm (111), 100 nm (100) , 100 nm (111) , 150 nm
(100), and 150 nm (111) . Before deposition, GGG substrates were cleaned in an ultrasonic bath
with acetone and isopropanol for 30 minute s. The deposition chamber was cleaned and evacuated
to 5.3×10-7 mbar. For PLD growth, a 248 nm KrF excimer laser (Laser fluence (2.3 J cm-2) with
10 Hz pulse rate was used to ablate the material from the target . Oxygen pressure, target -to-4
substrate distance, and substrate temperature were maintained at 0.15 mbar, 5.0 cm, and 825 oC,
respectively. Growth rate of deposited films were 6 nm/min . The as -grown films were annealed
in-situ for 2 hours at 825 oC in the presence of oxygen (0.15 mbar). The structural characterization
of thin films were ascertained using X -ray diffraction (XRD) with Cu-Kα radiation (1.5406 Å). We
have performed the XRD me asurement at room temperature in -2 geometry and incidence angle
are 20 degrees. The film's surface morphology and thickness were estimated using atomic force
microscopy (AFM) (WITec GmbH , Germany ). The magnetic properties were studied using a
vibrating sample magnetomet ry (VSM) in Cryogenic 14 Tesla Physical Property Measurement
System (PPMS). FMR measurements were done on a coplanar waveguide (CPW) in a flip -chip
arrangement with a dc magnetic field applied perpendicular to the high -frequency magnetic field
(hRF). A Keysight Vector Network Analyzer was used for this purpose. The CPW was rotated in
the film plane from 0º to 360º for in -plane () measurement s and from 0º to 18 0º for out of plane
(θ) measurement.
In this study, the thickness of Bi -YIG was determined by employing methods such as laser
lithography and AFM. We have calibrated the thickness of thin films with PLD laser shots.
Photoresist by spin coating is applied to a silicon substrate, and then straight-line patterns were
drawn on the photoresist coated substrates using laser photolithography. The PLD technique was
used to deposit thin films of the required material onto a pattern -drawn substrate. It is then
necessary to wet etch the PLD grown thin fi lm in order to remove the photoresist coating. Then,
AFM tip is scanned over the line pattern region in order to estimate the thickness of the grown
samples from the AFM profile image.
5
1.3 Results and Discussion
1.3.1 Structural properties
Figure 1 (a)-(d) show the XRD pattern of (100)- and (111)-oriented Bi-YIG grown thin
films with thickness ≈20-150 nm (Insets depict the zoomed image of XRD patterns) . XRD data
indicate single -crystalline growth of Bi -YIG thin films . Figures 1 (e) and 1 (f) show the l attice
constant and lattice mismatch (with respect to substrate) determined from XRD data, respectively .
The cubic lattice constant 𝒂 is calculated using the formula ,
𝒂=𝜆√ℎ2+𝑘2+𝑙2
2sin𝜃 (1)
where the wavelength of Cu -Kα radiation is represented by 𝜆, diffraction angle by 𝜃, and the Miller
indices of the corresponding XRD peak by [h, k, l] . Further, the l attice mismatch parameter (𝛥𝑎
𝑎)
is calculated using the equation ,
𝛥𝑎
𝑎=(𝑎𝑓𝑖𝑙𝑚 − 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 )
𝑎𝑓𝑖𝑙𝑚 100 (2)
Here lattice constant of film and substrate are represented by 𝑎𝑓𝑖𝑙𝑚 and 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 , respectively .
The reported lattice constant values are consistent with prior findings [15,17,21]. Lattice constant
slightly increases with the increase in thickness of the film in the case of (111) as compared to
(100) . Since the distribution of Bi3+ in the dodecahedral sit e is dependent on the substrate
orientation [7,23,2 9], the (111) oriented films show an increase in the lattice constant . In Bi -YIG
films, this slight increase in the lattice constant (in the 111 direction) leads to a
compar atively larger lattice mismatch as seen in Fig. 1 (f). For 50 nm (111) Bi -YIG film, we
achieved a lattice mismatch of ~0.47 , which is close to what has been reported earlier [30,31]. 6
Smaller value of lattice mismatch can reduc e the damping constant of the film [31]. We want to
underline the importance of lattice plane dependen t growth in conjunction with film thickness in
indicating structural and magnetic property changes.
1.3.2 Surface morphology
Figure 2 (a) -(h) shows room temperature AFM images with root mean square (RMS)
roughness. Roughness is essential from an application standpoint because the roughness directly
impacts the inhomogeneous linewidth broadening which leads to increase in the Gil bert damping.
We have observed RMS roughness around 0.5 nm or less for all grown Bi -YIG films which are
comparable to previous reported YIG films [32,33]. We have observed that RMS roughness
decreases with increase in thickness of the film. With (100) and (111) orientations, there is no
discernible difference in roughness. Furthermore, roughness would be more affected by changes
in growth factors and by substrate orientation [7,33,34].
1.3.3 Static magnetization study
The room temperature ( ≈296 K ) VSM magnetization measurements were carried out with
applied magnetic field parallel to the film plane (in-plane) . The paramagnetic contribution s from
the GGG substrate were carefully subtracted. F igure 3 (a)-(h) show s the magnetization plot s of Bi-
YIG thin films of thickness ≈20-150 nm . Inset of Fig. 3 (i) shows the measured saturation
magnetization ( µ0MS) data of as-grown (100) and (111) -oriented Bi -YIG films which are
consistent with the previous reports [6,17,22,3 5,36]. Figure 3 ( i) shows plot of µ0Ms × t Vs. t,
where ‘t’ is film thickness . This is done t o determine thickness of dead layer via linear 7
extrapolati on plot to the x -axis. The obtaine d magnetic dead -layer for (100) and (111) -oriented
GGG substrates are 2.88 nm and 5.41 nm , which are comparable to previous reports [37-39]. The
saturation magnetization of Bi -YIG films increases as the thickness of the films increases . The
increase in saturation magnetization with increase in thickness can be understood by the following
ways . Firstly, ferromagnetic thin films are generally deposited with a thin magnetically dead layer
over the interface with the substrate. This magnetic dead layer effect is larger in thinner films that
leads to the decrease in net magnetization with the decrease in thickness [40,41]. Figure 3 (i) shows
the effect of magnetic dead laye r region near to the substrate. Secondly, t hicker films exhibit the
bulk effect of YIG which, in turn, results in increas ed magnetization.
1.3.4 Ferromagnetic r esonance study
Figure 4 (a) -(d) shows the FMR absorption spectra of (100) and (111) -oriented films that
are labeled with open circle ( Ο) and open triangle ( Δ) respectively . FMR experiment s were carried
out at room temperature. In -plane dc magnetic field was a pplied parallel to film surface . To find
the effective magnetization and Gilbert damping, the FMR linewidth (∆H) and resonance magnetic
field (H r) are calculated using a Lorentzian fit of the FMR absorption spectra measured at 𝑓 = 1
GHz to 12 GHz. Effective magnetization field ( 0𝑀𝑒𝑓𝑓) were obtained from the fitting of Kittel's
in-plane equation (Eq. 3) [42].
𝑓=𝛾
2𝜋0√(𝐻𝑟)(𝐻𝑟+𝑀𝑒𝑓𝑓) (3),
Here, 0𝑀𝑒𝑓𝑓=0(𝑀𝑠−𝐻𝑎𝑛𝑖), anisotropy field 𝐻𝑎𝑛𝑖=2𝐾1
0𝑀𝑠, and 𝛾 being the gyromagnetic
ratio. Further, the dependence of FMR linewidth on microwave frequency shows a linear variation
(Eq. 4) [42] from which the Gilbert damping parameter (α) and FMR linewidth broad ening ( 𝛥𝐻 0)
were obtained:
µ0𝛥𝐻=µ0𝛥𝐻 0+4𝜋𝛼
𝛾𝑓 (4) 8
where, 𝛥𝐻 0 is the inhomogeneous broadening linewidth and α is the Gilbert damping. Figures 4
(e) and 4 (f) show Kittel and linewidth fitted graphs, respectively . Figure 5 (a) -(d) shows the
derived parameters acquired from the FMR study. The estimated Gilbert damping is con sistent
with data reported for sp in-wave propagation [3,22] . The value of α decreases as the thickness of
the film increases ( Fig. 5 (c)). Howe ver, in the instance of Bi -YIG with (111) orientation, there is
a substantial increase. This might be attributed qualitatively to the presence of Bi3+ ions, which
cause strong spin-orbit coupling [43-45] as well as electron scattering inside the lattice when the
lattice mismatch (or strain) increases [46]. Our earlier study [7] revealed a clear distribution of
Bi3+ ions along (11 1) planes, as well as slightly larger lattice mismatch in Bi -YIG (111). These
results explain the larger values of Gilbe rt damping, 0𝑀𝑒𝑓𝑓, and ΔH 0 values in Bi -YIG (111) (Fig.
5). The change in 0𝑀𝑒𝑓𝑓 is due to u niaxial in -plane magne tic anisotropy and it is observed from
magnetization measurements using 0𝑀𝑒𝑓𝑓=0(𝑀𝑠−𝐻𝑎𝑛𝑖) [36,47,48]. The enhanced
anisotropy field in the lower thic kness of Bi -YIG ( Fig. 5 (d) ) signifies the effect of dead magnetic
layer at the interface. The lattice mismatch between films and GGG substrates induces uniaxial in -
plane magnetic anisotropy [36,47]. ΔH 0 has a magnitude that is similar to previously published
values for the same substrate orientation [7,47]. In conclusion, Bi -YIG with (100) orientation
produces the lowest Gilbert damping facto r and inhomogeneous broadening linewidth . These are
the required optimal parameters for spintronics based devices.
Figure 6 (a) shows the variation of resonance field with polar angle ( ) for the grown 20
nm-150 nm films , H is the angle measured between applied magnetic field and surface of film
(shown in inset of Fig. 4 (a)). The FMR linewidth (ΔH) were extracted fr om fitting of FMR spectra
with L orentzian absorption functions. From Fig. 6 (a), we observe change in H r value for 50 nm
Bi-YIG film as 0.22 T and 0.27 T for (100) and (111) orientation respectively. Similarly, 0.21 T
and 0.31 T change is observed in (100) and (111) orientation respectively for 100 nm Bi -YIG film .
We see that H r increases slightly in case o f (111) oriented film by changing the di rection of H from
0º to 90º with regard to sample surface (inset of Fig. 4 (a)). The change in H r decreases with
increase in film thickness in cas e of (100) while it is reversed in case of (111 ). Figure 6 (b) shows 9
the variation of FMR linewidth with polar angle for 150 nm Bi -YIG film . Maximum FMR
linewidth is observed at 90º and it is slightly more as compared with (100) orientation. The
enhanced variation of FMR linewidth in (111) oriented samples is generated due to the higher
contribution of two -magnon scatte ring in perpendicular geometry [49]. This can be understood
due to the higher anisotropy field in (111) oriented samples ( Fig. 5 (d)).
Figure 6 (c) & (e) shows the azimuthal angle ( ) variation of H r. Frequency of 5 GHz is
used in the measurement . From variation data (by changing the direction of H from 0 º to 360
with regard to sample surface (inset of Fig. 4 (a)). We can see clearly in-plane anisotropy of four-
fold in Bi-YIG (100) (Fig. 6 (c)) unlike in Bi-YIG (111) (Fig. 6 (e)). According to crystalline
surface symmetry there would be six -fold in -plane anisotropy in case of (111) orientation but we
have not observe d it, based on previous reports, it can be superseded by a mis cut-induced uniaxial
anisotropy [33,50]. This reinforces our grown films' single -crystalline nature . The observed change
in H r (H=0 to 45) is 6 .6 mT in 50 nm (100 ), 0.17 mT for 50 nm (111) , 6.2 mT in 100 nm (100) ,
0.17 mT for 100 nm (111) ) and 5.1 mT in 150 nm (100 ). As a result, during in -plane rotation, the
higher FMR field change observed along the (100) orientation. The dependent FMR field data
shown in figure 6 (c) were fitted using the following Kittel relation [50]
𝑓=𝛾
2𝜋0√([𝐻𝑟cos(𝐻−𝑀)+𝐻𝑐cos4(𝑀−𝐶)+𝐻𝑢cos2(𝑀−𝑢)])×
(𝐻𝑟cos(𝐻−𝑀)+𝑀𝑒𝑓𝑓+1
4𝐻𝑐(3+cos4(𝑀−𝐶))+𝐻𝑢𝑐𝑜𝑠2(𝑀−𝑢)) (5)
With respect to the [100] direction of the GGG substrate, in -plane directions of the magnetic field,
magnetization, uniaxial, and cubic anisotropies are given by H, M, u and c, respectively.
𝐻𝑢=2𝐾𝑢
µ0𝑀𝑠 and 𝐻𝑐=2𝐾𝑐
µ0𝑀𝑠 correspond to the uniaxial and cubic anisotropy fields, respectively,
with 𝐾𝑢 and 𝐾𝑐 being the uniaxial and cubic magnetic anisotropy constants, respectively. 10
Figure 6 (d) shows t he obtained uniaxial anisotropy field, cubic anisotropy field and saturation
magnetization field for (100) orientation. The obtained saturation magnetization field follows the
same pattern as we have obtained from the VSM measurements. The cubic anisotropy field
increases and then saturates with the thickness of the film. A large drop in the uniaxial anisotropy
field is observed with the thickness of the grown films. We have not got the in -plane angular
variation data for the 20 nm thick Bi -YIG sample and m ay be due to the low thickness of the Bi -
YIG, it is not detected by our FMR setup.
1.4 Conclusion
In conclusion, we compare the properties of high-quality Bi -YIG thin films of four distinct
thicknesses (20 nm, 50 nm, 100 nm, and 150 nm) grown on GGG substrates with orientations of
(100) and (111). Pulsed laser deposition was used to synthesize the se films. AFM and XRD
characterizations reveal th at the deposited thin films have smooth surfaces and are phase pure.
According to FMR data, t he Gilbert damping value decreases with increase in film thickness . This
is explained i n the context of a dead m agnetic layer . The (100) orientation has a lower va lue of
Gilbert damping, indicating that it is the preferable substrate for doped YIG thin films for high
frequency application . Bi-YIG on (111) orientation , on the other hand, exhibits anisotropic
dominance, which is necessary for magneto -optic devices. Th e spin -orbit coupled Bi3+ ions are
responsible for the enhanced Gilbert damping in (111). We have also correlated ∆H 0, anisotropic
field, and effective magnetization to the variations in film thickness and substrate ori entation . In
(100) oriented films, there is unambiguous observation of four-fold in -plane anisotropy. In
particular, Bi-YIG grown on (111) GGG substrates yields best result for optim al magnetization
dynamics. This is linked to an enhanced magnetic anisotropy. Therefore, proper substrate 11
orientation and thickness are found to be important parameters for growth of Bi-YIG thin film
towards high frequency applications.
Acknowledgments
This work is supported by the MHRD -IMPRINT grant, DST (SERB, AMT , and PURSE -
II) gran t of Govt. of India. Ganesh Gurjar acknowledges CSIR, New Delhi for financial support .
We acknowledge AIRF, JNU for access of PPMS facility.
12
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List of f igure caption s
Figure 1: (a)-(d) X -ray diffraction (XRD) patterns of 20 nm -150 nm Bi -substituted YIG films in
(100) and (111) orientations. Insets in (a) -(d) depict the zoomed image of XRD patterns. Variation
of lattice constant (e) and (f) lattice mismatch with thickness are shown .
Figure 2: (a)-(h) A tomic force microscopy images of 20 nm -150 nm Bi -YIG film in (100) and
(111) orientations are shown .
Figure 3: (a)-(h) Static magnetization graph of 2 0 nm -150 nm Bi-substituted YIG (Bi-YIG) films
in (100) and (111) orientations. ( i) Graph to determine the magnetic dead -layer thickness of Bi -
YIG films on (100) and (111) -oriented GGG substrates is depicted (inset shows the variation of
saturation magnetiz ation value with the film thickness).
Figure 4: (a)-(d) Ferromagnetic resonance ( FMR ) absorption spectra of 20 nm -150 nm Bi-
substituted YIG films with (100) and (111) orientations. Inset in (a) shows the geometry of an
applied field angle measured from the sample surface. (e) shows frequency -dependent FMR
magnetic field data fitted with Kittel Eq. 3 . (f) shows frequency -dependent FMR linewidth data
fitted with Eq. 4 .
Figure 5: Variation s of (a) extrinsic linewidth, (b) effective magnetization, (c) Gilbert damping,
and (d) magnetic anisotropy with thickness for (100) and (111) oriented Bi-substituted YIG films
are depicted . 21
Figure 6: (a) Angular variation of Ferromagnetic resonance (FMR) magnetic field for 20 nm -150
nm Bi -substituted YIG (Bi -YIG) film with (100) and (111) orientations is shown. (b) Angular
variation of FMR linewidth of 150 nm thick Bi -YIG film with (100) and (111) orientation is
shown. Variations of FMR magnetic field as a function of azimuthal angle ( ) for (c) 50 nm, 100
nm and 150 nm Bi -YIG film with (100) orientation is depicted (d) obtained uniaxial anisotropy
field, cubic anisotropy field and saturation magnetization field for (100) orientation. (e)
dependent FMR fi eld data for 50 nm and 100 nm Bi -YIG film with (111) orientation is depicted.
22
Figure 1
23
Figure 2
24
Figure 3
25
Figure 4
26
Figure 5
27
Figure 6
|
1212.1772v3.A_note_on_the_lifespan_of_solutions_to_the_semilinear_damped_wave_equation.pdf | arXiv:1212.1772v3 [math.AP] 14 Mar 2013A NOTE ON THE LIFESPAN OF SOLUTIONS TO THE
SEMILINEAR DAMPED WAVE EQUATION
MASAHIRO IKEDA AND YUTA WAKASUGI
Abstract. This paper concerns estimates of the lifespan of solutions t o the
semilinear damped wave equation /squareu+Φ(t,x)ut=|u|pin (t,x)∈[0,∞)×Rn,
where the coefficient of the damping term is Φ( t,x) =/angbracketleftx/angbracketright−α(1 +t)−βwith
α∈[0,1), β∈(−1,1) andαβ= 0. Our novelty is to prove an upper bound
of the lifespan of solutions in subcritical cases 1 < p <2/(n−α).
1.Introduction
We consider the semilinear damped wave equation
(1.1) utt−∆u+Φ(t,x)ut=|u|p,(t,x)∈[0,∞)×Rn,
with the initial condition
(1.2) ( u,ut)(0,x) =ε(u0,u1)(x), x∈Rn,
whereu=u(t,x) is a real-valued unknown function of ( t,x), 1< p, (u0,u1)∈
H1(Rn)×L2(Rn) andεis a positive small parameter. The coefficient of the
damping term is given by
Φ(t,x) =/an}b∇acketle{tx/an}b∇acket∇i}ht−α(1+t)−β
withα∈[0,1), β∈(−1,1) andαβ= 0. Here /an}b∇acketle{tx/an}b∇acket∇i}htdenotes/radicalbig
1+|x|2.
Our aim is to obtain an upper bound of the lifespan of solutions to (1.1) .
We recall some previous results for (1.1). There are many results a bout global
existenceofsolutionsfor(1.1) andmanyauthorshavetried todet ermine the critical
exponent(see [3, 4, 6,8, 10,11, 12, 13, 16,18, 20, 23, 24]and t he referencestherein).
Here “critical” means that if pc<p, all small data solutions of (1.1) are global; if
1<p≤pc, the local solution cannot be extended globally even for small data.
In the constant coefficient case α=β= 0, Todorova and Yordanov [18] and
Zhang [23] determined the critical exponent of (1.1) with compactly supported
data as
pc=pF= 1+2
n.
Thisisalsothe criticalexponentofthe correspondingheatequatio n−∆v+vt=|v|p
and called the Fujita exponent (see [2]).
On the other hand, there are few results about upper estimates o f the lifespan
for (1.1). When n= 1,2, Li and Zhou [10] obtained the sharp upper bound:
(1.3) Tε≤/braceleftbiggexp(Cε−2/n),ifp= 1+2/n,
Cε−1/κ,if 1<p<1+2/n,
2000Mathematics Subject Classification. 35L71.
Key words and phrases. semilinear damped wave equation; lifespan, upper bound.
12 M. IKEDA AND Y. WAKASUGI
whereC=C(n,p,u0,u1)>0andκ= 1/(p−1)−n/2forthe data u0,u1∈C∞
0(Rn)
satisfying/integraltext
(u0+u1)dx>0. Nishihara [14] extended this result to n= 3 by using
the explicit formula of the solution to the linear part of (1.1) with initial data
(0,u1):
u(t,x) =e−t/2W(t)u1+J0(t)u1.
HereW(t)u1is the solution of the wave equation /squareu= 0 with initial data (0 ,u1)
andJ0(t)u1behaves like a solution of the heat equation −∆v+vt= 0. However,
both the methods of [10] and [14] do not work in higher dimensional ca sesn≥4,
because they used the positivity of W(t), which is valid only in the case n≤3.
In this paper we shall extend both of the results to n≥4 in subcritical cases
1<p<1+2/n.
Next, we recall some results of variable coefficient in cases α/ne}ationslash= 0 orβ/ne}ationslash= 0.
There are many results on asymptotic behavior of solutions in conne ction with the
diffusion phenomenon, Here the diffusion phenomenon means that so lution of the
damped wave equation behaves like a solution for the corresponding heat equation
ast→+∞. For more details about the diffusion phenomenon, see, for example
[19, 21, 22].
For the case α∈[0,1),β= 0, Ikehata, Todorova and Yordanov [8] determined
the critical exponent for (1.1) as pc= 1+ 2/(n−α), which also agrees with that
of the corresponding heat equation −∆v+/an}b∇acketle{tx/an}b∇acket∇i}ht−αvt=|v|p. Here we emphasize that
in this case there are no results about upper estimates for the lifes pan. It will be
given in this paper.
Next, for the case β∈(−1,1),α= 0, Nishihara [15] and Lin, Nishihara and
Zhai [11] proved pc= 1 + 2/n, which is also same as that of the heat equation
−∆v+(1+t)−βvt=|v|p. On the other hand, upper estimates of the lifespan have
not been well studied. Recently, Nishihara [15] obtained a similar resu lt of [10, 14]:
letn≥1,β≥0and(u0,u1)satisfy/integraltext
Rnui(x)dx≥0(i= 0,1),/integraltext
Rn(u0+u1)(x)dx>
0. Then there exists a constant C >0 such that
Tε≤/braceleftbigg
eCε−(1+β)/n,ifp= 1+(1+β)/n,
Cε−1/ˆκ,if 1+2β/n≤p<1+(1+β)/n,
where ˆκ= (1 +β)/(p−1)−n. We note that the rate ˆ κis not optimal, because
it is not same as that of the corresponding heat equation. Moreove r, there are no
results for 1+(1+ β)/n<p≤1+2/n. We note that the proof by Todorova and
Yordanov [18] also gives the same upper bound in the case β= 0,1<p<1+1/n.
In this paper we will improve the above result for all 1 <p<1+2/nand give the
sharp upper estimate.
Finally, we mention that our method is not applicable to αβ/ne}ationslash= 0. On the
other hand, the second author [20] proved a small data global exis tence result for
(1.1) withα,β≥0, α+β≤1, whenp>1+2/(n−α). This also agrees with the
criticalexponent of the correspondingheat equation −∆v+/an}b∇acketle{tx/an}b∇acket∇i}ht−α(1+t)−βvt=|v|p.
Therefore,it isexpectedthat when1 <p≤1+2/(n−α),thereisablow-upsolution
for (1.1) in this case.
2.Main Result
First,wedefinethesolutionof(1.1). Wesaythat u∈X(T) :=C([0,T);H1(Rn))∩
C1([0,T);L2(Rn)) is a solution of (1.1) with initial data (1.2) on the interval [0 ,T)LIFESPAN OF SOLUTIONS TO THE SEMILINEAR DAMPED WAVE EQUATIO N 3
if the identity/integraldisplay
[0,T)×Rnu(t,x)(∂2
tψ(t,x)−∆ψ(t,x)−∂t(Φ(t,x)ψ(t,x)))dxdt
=ε/integraldisplay
Rn{(Φ(0,x)u0(x)+u1(x))ψ(0,x)−u0(x)∂tψ(0,x)}dx (2.1)
+/integraldisplay
[0,T)×Rn|u(t,x)|pψ(t,x)dxdt
holds for any ψ∈C∞
0([0,T)×Rn). We also define the lifespan for the local solution
of (1.1)-(1.2) by
Tε:= sup{T∈(0,∞];there exists a unique solution u∈X(T) of (1.1)-(1.2) }.
We first describe the local existence result.
Proposition 2.1. Letα≥0,β∈R,1<p≤n/(n−2) (n≥3),1<p<∞(n=
1,2),ε >0and(u0,u1)∈H1(Rn)×L2(Rn). ThenTε>0, that is, there exists
a unique solution u∈X(Tε)to(1.1)-(1.2). Moreover, if Tε<+∞, then it follows
that
lim
t→Tε−0/ba∇dbl(u,ut)(t,·)/ba∇dblH1×L2= +∞.
For the proof, see, for example [7]. Next, we give an alomost optimal lower
estimate of Tε.
Proposition 2.2. Let(u0,u1)∈H1(Rn)×L2(Rn)be compactly supported and δ
any positive number. We assume that α∈[0,1),β∈(−1,1),αβ≥0andα+β <1.
Then there exists a constant C=C(δ,n,p,α,β,u 0,u1)>0such that for any ε>0,
we have
Cε−1/κ+δ≤Tε,
where
κ=2(1+β)
2−α/parenleftbigg1
p−1−n−α
2/parenrightbigg
.
The proof of this proposition follows from the a priori estimate for t he energy
of solutions. For the proof, see [11, 8, 20]. We note that the above proposition is
valid even for the case αβ/ne}ationslash= 0.
Next, we state our main result, which gives an upper bound of Tε.
Theorem 2.3. Letα∈[0,1),β∈(−1,1),αβ= 0and let1<p <1+2/(n−α).
We assume that the initial data (u0,u1)∈H1(Rn)×L2(Rn)satisfy
(2.2) /an}b∇acketle{tx/an}b∇acket∇i}ht−αBu0+u1∈L1(Rn)and/integraldisplay
Rn(/an}b∇acketle{tx/an}b∇acket∇i}ht−αBu0(x)+u1(x))dx>0,
where
B=/parenleftbigg/integraldisplay∞
0e−/integraltextt
0(1+s)−βdsdt/parenrightbigg−1
.
Then there exists C >0depending only on n,p,α,β and(u0,u1)such thatTεis
estimated as
Tε≤C
ε−1/κif1+α/(n−α)<p<1+2/(n−α),
ε−(p−1)(log(ε−1))p−1ifα>0, p= 1+α/(n−α),
ε−(p−1)ifα>0,1<p<1+α/(n−α)4 M. IKEDA AND Y. WAKASUGI
for anyε∈(0,1], where
κ=2(1+β)
2−α/parenleftbigg1
p−1−n−α
2/parenrightbigg
.
Remark 2.1. The results of Theorem 2.3 and Proposition 2.2 can be express ed by
the following table:
α= 0 β= 0
pc 1+2
n1+2
n−α
Tε/lessorsimilar ε−1/κ
ε−1/κ,/parenleftbigg
1+α
n−α<p<1+2
n−α/parenrightbigg
ε−(p−1)(log(ε−1))p−1,/parenleftbigg
p= 1+α
n−α/parenrightbigg
ε−(p−1),/parenleftbigg
1<p<1+α
n−α/parenrightbigg
Tε/greaterorsimilarε−1/κ+δε−1/κ+δ
κ(1+β)/parenleftbigg1
p−1−n
2/parenrightbigg2
2−α/parenleftbigg1
p−1−n−α
2/parenrightbigg
Remark 2.2. It is expected that the rate κin Theorems 2.3 is sharp except for the
caseα>0,1<p≤1+α/(n−α)from Proposition 2.2.
Remark 2.3. The explicit form of Φ =/an}b∇acketle{tx/an}b∇acket∇i}ht−α(1+t)−βis not necessary. Indeed,
we can treat more general coefficients, for example, Φ(t,x) =a(x)satisfyinga∈
C(Rn)and0≤a(x)/lessorsimilar/an}b∇acketle{tx/an}b∇acket∇i}ht−α, orΦ(t,x) =b(t)satisfyingb∈C1([0,∞))and
b(t)∼(1+t)−β.
Remark 2.4. The same conclusion of Theorem 2.3 is valid for the correspon ding
heat equation −∆v+Φ(t,x)vt=|v|pin the same manner as our proof.
Our proof is based on a test function method. Zhang [23] also used a similar way
to determine the critical exponent for the case α=β= 0. By using his method,
many blow-up results were obtained for variable coefficient cases (s ee [1, 8, 11]).
However, the method of [23] was based on a contradiction argumen t and so upper
estimates of the lifespan cannot be obtained. To avoid the contrad iction argument,
we use an idea by Kuiper [9]. He obtained an upper bound of the lifespan for some
parabolic equations (see also [5, 17]). We note that to treat the time -dependent
damping case, we also use a transformation of equation by Lin, Nishih ara and Zhai
[11] (see also [1]).
At the end of this section, we explain some notation and terminology u sed
throughout this paper. We put
/ba∇dblf/ba∇dblLp(Rn):=/parenleftbigg/integraldisplay
Rn|f(x)|pdx/parenrightbigg1/p
.
We denote the usual Sobolev space by H1(Rn). For an interval Iand a Banach
spaceX, we define Cr(I;X) as the Banach space whose element is an r-timesLIFESPAN OF SOLUTIONS TO THE SEMILINEAR DAMPED WAVE EQUATIO N 5
continuously differentiable mapping from ItoXwith respect to the topology in
X. The letter Cindicates the generic constant, which may change from line to line.
We also use the symbols /lessorsimilarand∼. The relation f/lessorsimilargmeansf≤Cgwith some
constantC >0 andf∼gmeansf/lessorsimilargandg/lessorsimilarf.
3.Proof of Theorem 2.3
We first note that if Tε≤C, whereCis a positive constant depending only on
n,p,α,β,u 0,u1, then it is obvious that Tε≤Cε−1/κfor anyκ >0 andε∈(0,1].
Therefore, once a constant C=C(n,p,α,β,u 0,u1) is given, we may assume that
Tε>C.
In the case β/ne}ationslash= 0, (1.1) is not divergence form and so we cannot apply the
test function method. Therefore, we need to transform the equ ation (1.1) into
divergence form. The following idea was introduced by Lin, Nishihara a nd Zhai
[11]. Letg(t) be the solution of the ordinary differential equation
/braceleftBigg
−g′(t)+(1+t)−βg(t) = 1,
g(0) =B−1.
The solution g(t) is explicitly given by
g(t) =e/integraltextt
0(1+s)−βds/parenleftbigg
B−1−/integraldisplayt
0e−/integraltextτ
0(1+s)−βdsdτ/parenrightbigg
.
By the de l’Hˆ opital theorem, we have
lim
t→∞(1+t)−βg(t) = 1
and sog(t)∼(1+t)β. We note that B= 1 andg(t)≡1 ifβ= 0. By the definition
ofg(t), we also have |g′(t)|/lessorsimilar|(1+t)−βg(t)−1|/lessorsimilar1. Multiplying the equation (1.1)
byg(t), we obtain the divergence form
(3.1) ( gu)tt−∆(gu)−((g′−1)/an}b∇acketle{tx/an}b∇acket∇i}ht−αu)t=g|u|p,
here we note that αβ= 0. Therefore, we can apply the test function method to
(3.1).
We introduce the following test functions:
φ(x) :=/braceleftBigg
exp(−1/(1−|x|2)) (|x|<1),
0 ( |x| ≥1),
η(t) :=
exp(−1/(1−t2))
exp(−1/(t2−1/4))+exp( −1/(1−t2))(1/2<t<1),
1 (0≤t≤1/2),
0 (t≥1).
It is obvious that φ∈C∞
0(Rn),η∈C∞
0([0,∞)) and there exists a constant C >0
such that for all |x|<1 we have
|∇φ(x)|2
φ(x)≤C.
Using this estimate, we can prove that there exists a constant C >0 such that the
estimate
(3.2) |∆φ(x)| ≤Cφ(x)1/p6 M. IKEDA AND Y. WAKASUGI
is true for all |x|<1. Indeed, putting ϕ:=φ1/qwithq=p/(p−1), we have
|∆φ(x)|=|∆(ϕ(x)q)|/lessorsimilar|∆ϕ(x)|ϕ(x)q+|∇ϕ(x)|2ϕ(x)q−2/lessorsimilarϕ(x)q−1=φ(x)1/p.
In the same way, we can also prove that
(3.3) |η′(t)| ≤Cη(t)1/p,|η′′(t)| ≤Cη(t)1/p
fort∈[0,1).
Letube a solution on [0 ,Tε) andτ∈(τ0,Tε),R≥R0parameters, where τ0∈
[1,Tε),R0>0 are defined later. We define
ψτ,R(t,x) :=ητ(t)φR(x) :=η(t/τ)φ(x/R)
and
Iτ,R:=/integraldisplay
[0,τ)×BRg(t)|u(t,x)|pψτ,R(t,x)dxdt,
JR:=ε/integraldisplay
BR(/an}b∇acketle{tx/an}b∇acket∇i}ht−αBu0(x)+u1(x))φR(x)dx,
whereBR={|x|< R}. Sinceψτ,R∈C∞
0([0,Tε)×Rn) anduis a solution on
[0,Tε), we have
Iτ,R+JR=/integraldisplay
[0,τ)×BRg(t)u∂2
tψτ,Rdxdt−/integraldisplay
[0,τ)×BRg(t)u∆ψτ,Rdxdt
+/integraldisplay
[0,τ)×BR(g′(t)−1)/an}b∇acketle{tx/an}b∇acket∇i}ht−αu∂tψτ,Rdxdt
=:K1+K2+K3.
Here we have used the property ∂tψ(0,x) = 0 and substituted the test function
g(t)ψ(t,x) into the definition of solution (2.1). We note that for the correspo nding
heat equation, we have the same decomposition without the term K1and so we
can obtain the same conclusion (see Remark 2.4). We first estimate K1. By the
H¨ older inequality and (3.3), we have
K1≤τ−2/integraldisplay
[0,τ)×BRg(t)|u||η′′(t/τ)|φR(x)dxdt (3.4)
≤Cτ−2/integraldisplay
[τ/2,τ)×BRg(t)|u|η(t)1/pφR(x)dxdt
≤τ−2I1/p
τ,R/parenleftBigg/integraldisplayτ
τ/2g(t)dt·/integraldisplay
BRφR(x)dx/parenrightBigg1/q
≤Cτ−2+1/q(1+τ)β/qRn/qI1/p
τ,R.LIFESPAN OF SOLUTIONS TO THE SEMILINEAR DAMPED WAVE EQUATIO N 7
Using (3.2) and a similar calculation, we obtain
K2≤R−2/integraldisplay
[0,τ)×BRg(t)|u||∆φ(x/R)|η(t/τ)dxdt (3.5)
≤CR−2/integraldisplay
[0,τ)×BRg(t)|u||φ(x/R)|1/pη(t/τ)dxdt
≤CR−2I1/p
τ,R/parenleftbigg/integraldisplayτ
0g(t)η(t/τ)dt·/integraldisplay
BR1dx/parenrightbigg1/q
≤C(1+τ)(1+β)/qR−2+n/qI1/p
τ,R.
ForK3, using (3.3) and |g′(t)−1|/lessorsimilarC, we have
K3≤τ−1/integraldisplay
[0,τ)×BR/an}b∇acketle{tx/an}b∇acket∇i}ht−α|u||η′(t/τ)|φR(x)dxdt (3.6)
≤τ−1I1/p
τ,R/parenleftBigg/integraldisplayτ
τ/2g(t)−q/pdt·/integraldisplay
BR/an}b∇acketle{tx/an}b∇acket∇i}ht−αqφR(x)dx/parenrightBigg1/q
≤Cτ−1+1/q(1+τ)−β/pFp,α(R)I1/p
τ,R,
where
Fp,α(R) =
R−α+n/q(αq<n),
(log(1+R))1/q(αq=n),
1 ( αq>n).
Thus, putting
D(τ,R) :=τ−(1+β)/p(τ−1+βRq/n+τ1+βR−2+q/n+Fp,α(R))
and combining this with the estimates (3.4)-(3.6), we have
(3.7) JR≤CD(τ,R)I1/p
τ,R−Iτ,R.
Now we use a fact that the inequality
acb−c≤(1−b)bb/(1−b)a1/(1−b)
holds for all a >0,0< b <1,c≥0. We can immediately prove it by considering
the maximal value of the function f(c) =acb−c. From this and (3.7), we obtain
(3.8) JR≤CD(τ,R)q.
On the other hand, by the assumption on the data and the Lebesgu e dominated
convergence theorem, there exist C >0 andR0such thatJR≥Cεholds for all
R>R 0. Combining this with (3.8), we have
(3.9) ε≤CD(τ,R)q
for allτ∈(τ0,Tε) andR>R 0. Now we difine
τ0:= max{1,R(2−α)/(1+β)
0 },
and we substitute
(3.10) R=/braceleftBigg
τ(1+β)/(2−α)(αq<n),
τ (αq≥n)8 M. IKEDA AND Y. WAKASUGI
into (3.9). Here we note that R>R 0ifRis given by (3.10). As was mentioned at
the beginning of this section, we may assume that τ0<Tε. Finally, we have
ε≤C
τ−κ(αq<n),
τ−1/(p−1)log(1+τ) (αq=n),
τ−1/(p−1)(αq>n),
with
κ=2(1+β)
2−α/parenleftbigg1
p−1−n−α
2/parenrightbigg
.
We can rewrite this relation as
τ≤C
ε−1/κif 1+α/(n−α)<p<1+2/(n−α),
ε−(p−1)(log(ε−1))p−1ifα>0, p= 1+α/(n−α),
ε−(p−1)ifα>0,1<p<1+α/(n−α).
Here we note that κ >0 if and only if 1 < p <1+2/(n−α) and that αq=nis
equivalent to p= 1+α/(n−α). Sinceτis arbitrary in ( τ0,Tε), we can obtain the
conclusion of the theorem.
Acknowledgement
The authorsaredeeply gratefultoProfessorTatsuoNishitani. H e gaveus helpful
advice. Theywouldalsoliketoexpresstheirdeepgratitudetoanano nymousreferee
for many useful suggestions and comments.
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E-mail address :m-ikeda@cr.math.sci.osaka-u.ac.jp
E-mail address :y-wakasugi@cr.math.sci.osaka-u.ac.jp
Department of Mathematics, Graduate School of Science, Osa ka University, Toy-
onaka, Osaka, 560-0043, Japan |
2401.00714v1.Calculation_of_Gilbert_damping_and_magnetic_moment_of_inertia_using_torque_torque_correlation_model_within_ab_initio_Wannier_framework.pdf | Calculation of Gilbert damping and magnetic moment of inertia
using torque-torque correlation model within ab initio Wannier
framework
Robin Bajaj1, Seung-Cheol Lee2, H. R. Krishnamurthy1,
Satadeep Bhattacharjee2,∗and Manish Jain1†
1Centre for Condensed Matter Theory,
Department of Physics,
Indian Institute of Science,
Bangalore 560012, India
2Indo-Korea Science and Technology Center,
Bangalore 560065, India
(Dated: January 2, 2024)
1arXiv:2401.00714v1 [cond-mat.mtrl-sci] 1 Jan 2024Abstract
Magnetization dynamics in magnetic materials are well described by the modified semiclassical
Landau-Lifshitz-Gilbert (LLG) equation, which includes the magnetic damping ˆαand the magnetic
moment of inertia ˆItensors as key parameters. Both parameters are material-specific and physi-
cally represent the time scales of damping of precession and nutation in magnetization dynamics.
ˆαandˆIcan be calculated quantum mechanically within the framework of the torque-torque corre-
lation model. The quantities required for the calculation are torque matrix elements, the real and
imaginary parts of the Green’s function and its derivatives. Here, we calculate these parameters
for the elemental magnets such as Fe, Co and Ni in an ab initio framework using density functional
theory and Wannier functions. We also propose a method to calculate the torque matrix elements
within the Wannier framework. We demonstrate the effectiveness of the method by comparing it
with the experiments and the previous ab initio and empirical studies and show its potential to
improve our understanding of spin dynamics and to facilitate the design of spintronic devices.
I. INTRODUCTION
In recent years, the study of spin dynamics[1–5] in magnetic materials has garnered
significant attention due to its potential applications in spintronic devices and magnetic
storage technologies[6–9]. Understanding the behaviour of magnetic moments and their
interactions with external perturbations is crucial for the development of efficient and reliable
spin-based devices. Among the various parameters characterizing this dynamics, Gilbert
damping[10] and magnetic moment of inertia play pivotal roles. The fundamental semi-
classical equation describing the magnetisation dynamics using these two crucial parameters
is the Landau-Lifshitz-Gilbert (LLG) equation[11, 12], given by:
∂M
∂t=M×
−γH+ˆα
M∂M
∂t+ˆI
M∂2M
∂t2!
(1)
where Mis the magnetisation, His the effective magnetic field including both external
and internal fields, ˆαandˆIare the Gilbert damping and moment of inertia tensors with
the tensor components defined as αµνand Iµν, respectively, and γis the gyromagnetic
∗s.bhattacharjee@ikst.res.in
†mjain@iisc.ac.in
2ratio. Gilbert damping, ˆαis a fundamental parameter that describes the dissipation of
energy during the precession of magnetic moments in response to the external magnetic
field. Accurate determination of Gilbert damping is essential for predicting the dynamic
behaviour of magnetic materials and optimizing their performance in spintronic devices. On
the other hand, the magnetic moment of inertia, ˆIrepresents the resistance of a magnetic
moment to changes in its orientation. It governs the response time of magnetic moments to
external stimuli and influences their ability to store and transfer information. The moment
of inertia[13] is the magnetic equivalent of the inertia in classical mechanics[14, 15] and acts
as the magnetic inertial mass in the LLG equation.
Experimental investigations of Gilbert damping[16–23] and moment of inertia involve
various techniques, such as ferromagnetic resonance (FMR) spectroscopy[24, 25], spin-torque
ferromagnetic resonance (ST-FMR), and time-resolved magneto-optical Kerr effect (TR-
MOKE)[26, 27]. Interpreting the results obtained from these techniques in terms of the
LLG equation provide insights into the dynamical behaviour of magnetic materials and can
be used to extract the damping and moment of inertia parameters. In order to explain
the experimental observations in terms of more macroscopic theoretical description, various
studies[28–34] based on linear response theory and Kambersky theory have been carried out.
Linear response theory-based studies of Gilbert damping and moment of inertia involve
perturbing the system and calculating the response of the magnetization to the pertur-
bation. By analyzing the response, one can extract the damping parameter. Ab initio
calculations based on linear response theory[33] can provide valuable insights into the mi-
croscopic mechanisms responsible for the damping process. While formal expression for the
moment of inertia in terms of Green’s functions have been derived within the Linear response
framework[11], to the best of our knowledge, there hasn’t been any first principle electronic
structure-based calculation for the moment of inertia within this formalism.
Kambersky’s theory[35–37] describes the damping phenomena using a breathing Fermi
surface [38] and torque-torque correlation model [39], wherein the spin-orbit coupling acts
as the perturbation and describes the change in the non-equilibrium population of electronic
states with the change in the magnetic moment direction. Gilmore et al.[32, 34] have reported
the damping for ferromagnets like Fe, Ni and Co using Kambersky’s theory in the projector
augmented wave method[40].
The damping and magnetic inertia have been derived within the torque-torque correlation
3model by expanding the effective dissipation field in the first and second-time derivatives
of magnetisation[29–31]. In this work, the damping and inertia were calculated using the
combination of first-principles fully relativistic multiple scattering Korringa–Kohn–Rostoker
(KKR) method and the tight-binding model for parameterisation[41]. However, there hasn’t
been any full ab intio implementation using density functional theory (DFT) and Wannier
functions to study these magnetic parameters.
The expressions for the damping and inertia involves integration over crystal momentum
kin the first Brillouin zone. Accurate evaluation of the integrals involved required a dense k-
point mesh of the order of 106−108points for obtaining converged values. Calculating these
quantities using full ab initio DFT is hence time-consuming. To overcome this problem,
here we propose an alternative. To begin with, the first principles calculations are done on
a coarse kmesh instead of dense kmesh. We then utilize the maximally localised Wannier
functions (MLWFs)[42] for obtaining the interpolated integrands required for the denser k
meshes. In this method, the gauge freedom of Bloch wavefunctions is utilised to transform
them into a basis of smooth, highly localised Wannier wavefunctions. The required real
space quantities like the Hamiltonian and the torque-matrix elements are calculated in the
Wannier basis using Fourier transforms. The integrands of integrals can then be interpolated
on the fine kmesh by an inverse Fourier transform of the maximally localised quantities,
thereby enabling the accurate calculations of the damping and inertia.
The rest of this paper is organized as follows: In Sec. II, we introduce the expressions for
the damping and the inertia. We describe the formalism to calculate the two key quantities:
Green’s function and torque matrix elements, using the Wannier interpolation. In Sec. III,
we describe the computational details and workflow. In Sec. IV, we discuss the results for
ferromagnets like Fe, Co and Ni, and discuss the agreement with the experimental values
and the previous studies. In Sec. V, we conclude with a short summary and the future
prospects for the methods we have developed.
II. THEORETICAL FORMALISM
First, we describe the expressions for Gilbert damping and moment of inertia within the
torque-torque correlation model. Then, we provide a brief description of the MLWFs and
the corresponding Wannier formalism for the calculation of torque matrix elements and the
4Green’s function.
A. Gilbert damping and Moment of inertia within torque-torque correlation
model
If we consider the case when there is no external magnetic field, the electronic structure
of the system can be described by the Hamiltonian,
H=H0+Hexc+Hso=Hsp+Hso (2)
The paramagnetic band structure is described by H0andHexcdescribes the effective lo-
cal electron-electron interaction, treated within a spin-polarised (sp) local Kohn-Sham
exchange-correlation (exc) functional approach, which gives rise to the ferromagnetism. Hso
is the spin-orbit Hamiltonian. As we are dealing with ferromagnetic materials only, we can
club the first two terms as Hsp=H0+Hexc. During magnetization dynamics, (when the
magnetization precesses), only the spin-orbit energy of a Bloch state |ψnk⟩is affected, where
nis the band index of the state. The magnetization precesses around an effective field
Heff=Hint+Hdamp+HI, where Hintis the internal field due to the magnetic anisotropy
and exchange energies, Hdampis the damping field, and HIis the inertial field, respectively.
From Eqn. (1), we can see that the damping field Hdamp =α
Mγ∂M
∂t, while HI=I
Mγ∂2M
∂t2.
Equating these damping and inertial fields to the effective field corresponding to the change
in band energies as magnetization processes, we obtain the mathematical description of the
Gilbert damping and inertia. It was proposed by Kambersky [39] that the change of the
band energies∂εnk
∂θµ(θ=θˆndefines the vector for the rotation) can be related to torque
operator or matrix depending on how the Hamiltonian is being viewed Γµ= [σµ,Hso], where
σµare the Pauli matrices. Eventually, within the so-called torque-torque correlation model,
the Gilbert damping tensor can be expressed as follows:
αµν=g
msπZ Z
−d f(ϵ)
dϵ
Tr[Γµ(IG)(Γν)†(IG)]d3k
(2π)3dϵ (3)
Here trace Tris over the band indices, and msis the magnetic moment. Recently, Thonig
et al [30] have extended such an approach to the case of moment of inertia also, where they
5deduced the moment of inertia tensor components to be,
Iνµ=gℏ
msπZ Z
f(ϵ)Tr[Γν(IG)(Γµ)†∂2
∂ϵ2(RG)
+Γν∂2
∂ϵ2(RG)(Γµ)†(IG)]d3k
(2π)3dϵ (4)
Here f(ϵ) is the Fermi function, ( RG) and ( IG) are the real and imaginary parts of Green’s
function G=(ϵ+ιη− H)−1with ηas a broadening parameter, mis the magnetization in
units of the Bohr magneton, Γµ= [σµ,Hso] is the µthcomponent of torque matrix element
operator or matrix, µ=x, y, z .αis a dimensionless parameter, and I has units of time,
usually of the order of femtoseconds.
To obtain the Gilbert damping and moment of inertia tensors from the above two k-
integrals calculated as sums of discrete k-meshes, we need a large number of k-points, such
as around 106, and more than 107, for the converged values of αand I respectively. The
reason for the large k-point sampling in the first Brillouin zone (BZ) is because Green’s
function becomes more sharply peaked at its poles at the smaller broadening, ηwhich need
to be used. For I, the number of k-points needed is more than what is needed for αbecause
∂2RG/∂ϵ2has cubic powers in RGand/or IGas:
∂2RG
∂ϵ2= 2
(RG)3−RG(IG)2−IGRGIG−(IG)2RG
(5)
We note that to carry out energy integration in αit is sufficient to consider a limited
number of energy points within a narrow range ( ∼kBT) around the Fermi level. This
is mainly due to the exponential decay of the derivative of the Fermi function away from
the Fermi level. However, the integral for I involves the Fermi function itself and not its
derivative. Consequently, while the Gilbert damping is associated with the Fermi surface, the
moment of inertia is associated with the entire Fermi sea. Therefore, in order to adequately
capture both aspects, it is necessary to include energy points between the bottom of the
valence band and the Fermi level.
6B. Wannier Interpolation
1.Maximally Localised Wannier Functions (MLWFs)
The real-space Wannier functions are written as the Fourier transform of Bloch wave-
functions,
|wnR⟩=v0
(2π)3Z
BZdke−ιk.R|ψnk⟩ (6)
where |ψnk⟩are the Bloch wavefunctions obtained by the diagonalisation of the Hamiltonian
at each kpoint using plane-wave DFT calculations. v0is the volume of the unit cell in the
real space.
In general, the Wannier functions obtained by Eqn. (6) are not localised. Usually, the
Fourier transforms of smooth functions result in localised functions. But there exists a phase
arbitrariness of eιϕnkin the Bloch functions because of independent diagonalisation at each
k, which messes up the localisation of the Wannier functions in real space.
To mitigate this problem, we use the Marzari-Vanderbilt (MV) localisation procedure[42–
44] to construct the MLWFs, which are given by,
|wnR⟩=1
NX
qNqX
m=1e−ιq.RUq
mn|ψmq⟩ (7)
where Uq
mnis a (Nq×N) dimensional matrix chosen by disentanglement and Wannierisation
procedure. Nare the number of target Wannier functions, and Nqare the original Bloch
states at each qon the coarse mesh, from which Nsmooth Bloch states on the fine k-mesh
are extracted requiring Nq>Nfor all q,Nis the number of uniformly distributed qpoints
in the BZ. The interpolated wavefunctions on a dense k-mesh, therefore, are given via inverse
Fourier transform as:
|ψnk⟩=X
Reιk.R|wnR⟩ (8)
Throughout the manuscript, we use qandkfor coarse and fine meshes in the BZ, respec-
tively.
7FIG. 1: The figure shows the schematic of the localisation of the Wannier functions on a
Rgrid. The matrix elements of the quantities like Hamiltonian on the Rgrid are
exponentially decaying. Therefore, most elements on the Rgrid are zero (shown in blue).
We can hence do the summation till a cutoff Rcut(shown in red) to interpolate the
quantities on a fine kgrid.
2.Torque Matrix elements
As described in the expressions of αµνand Iνµin Eqns. (3) and (4), the µthcomponent of
the torque matrix is given by the commutator of µthcomponent of Pauli matrices and spin-
orbit coupling matrix i.e.Γµ= [σµ,Hso]. Physically, we define the spin-orbit coupling (SOC)
and spin-orbit torque (SOT) as the dot and cross products of orbital angular momentum and
spin angular momentum operator, respectively, such that Hso=ξℓ.where ξis the coupling
amplitude. Using this definition of Hso, one can show easily that −ι[σ,Hso] = 2ξℓ×which
represents the torque.
There have been several studies on how to calculate the spin-orbit coupling using ab
initio numerical approach. Shubhayan et al. [45] describe the method to obtain SOC matrix
elements in the Wannier basis calculated without SO interaction, using an approximation of
weak SOC in the organic semiconductors considered in their work. Their method involves
DFT in the atomic orbital basis, wherein the SOC in the Bloch basis can be related to
the SOC in the atomic basis. Then, by the basis transformation, they get the SOC in
the Wannier basis calculated in the absence of SO interaction. Farzad et al. [46] calculate
the SOC by extracting the coupling amplitude from the Hamiltonian in the Wannier basis,
treating the Wannier functions as atomic-like orbitals.
8We present a different approach wherein we can do the DFT calculation in any basis (plane
wave or atomic orbital). Unlike the previous approaches, we perform two DFT calculations
and two Wannierisations: one is with spin-orbit interaction and finite magnetisation (SO)
and the other is spin-polarised without spin-orbit coupling (SP). The spin-orbit Hamiltonian,
Hsocan then be obtained by subtracting the spin-polarized Hamiltonian, Hspfrom the full
Hamiltonian, HasHso=H−H sp. This, however, can only be done if both the Hamiltonians,
HandHsp, are written in the same basis. We choose to use the corresponding Wannier
functions as a basis. It should however be noted that, when one Wannierises the SO and
SP wavefunctions, one will get two different Wannier bases. As a result, we can not directly
subtract the HandHspin these close but different Wannier bases. In order to do the
subtraction, we find the transformation between two Wannier bases , i.e. express one set of
Wannier functions in terms of the other. Subsequently, we can express the matrix elements
ofHandHspin the same basis and hence calculate Hso. In the equations below, the Wannier
functions, the Bloch wavefunctions and the operators defined in the corresponding bases in
SP and SO calculations are represented with and without the tilde ( ∼) symbol, respectively.
TheNSO Wannier functions are given by:
|wnR⟩=1
NX
qNqX
m=1e−ιq.RUq
mn|ψmq⟩ (9)
where Uq
mnis a (Nq× N) dimensional matrix. The wavefunctions and Wannier functions
from the SO calculation for a particular qandRare a mixture of up and down spin states
and are represented as spinors:
|ψnq⟩=
|ψ↑
nq⟩
|ψ↓
nq⟩
|wnR⟩=
|w↑
nR⟩
|w↓
nR⟩
(10)
The ˜NsSP Wannier functions are given by:
|˜ws
nR⟩=1
NX
q˜Ns
qX
m=1e−ιq.R˜Uqs
mn|˜ψs
mq⟩ (11)
where s=↑,↓.˜Uqs
mnis a ( ˜Ns
qטNs) dimensional matrix. Since the spinor Hamiltonian
doesn’t have off-diagonal terms corresponding to opposite spins in the absence of SOC, the
wavefunctions will be |˜ψs
nq⟩=|˜ψnq⟩⊗|s⟩. The combined expression for ˜Uqfor˜N↑+˜N↓=˜N
9FIG. 2: This figure shows the implementation flow chart of the theoretical formalism
described in Sec. II
SP Wannier functions is:
˜Uq=
˜Uq↑0
0˜Uq↓
(12)
where ˜Uqis˜NqטNdimensional matrix with ˜Nq=˜N↑
q+˜N↓
q. Dropping the sindex for
SP kets results in the following expression for ˜NSP Wannier functions:
|˜wnR⟩=1
NX
q˜NqX
m=1e−ιq.R˜Uq
mn|˜ψmq⟩ (13)
10We now define the matrix of the transformation between SO and SP Wannier bases as:
TRR′
mn=⟨˜wmR|wnR′⟩
=1
N2X
qq′˜Nq,Nq′X
p,l=1eι(q.R−q′.R′)˜Uq†
mp⟨˜ψpq|ψlq′⟩Uq′
ln
=1
N2X
qq′eι(q.R−q′.R′)[˜Uq†Vqq′Uq′]mn (14)
HereVqq′
pl=⟨˜ψpq|ψlq′⟩. Eqn. (14) is the most general expression to get the transformation
matrix. We can reduce this quantity to a much simpler one using the orthogonality of
wavefunctions of different q. Eqn. (14) hence becomes,
TRR′
mn=1
N2X
qeιq.(R−R′)[˜Uq†(NVq)Uq]mn
=1
NX
qeιq.(R−R′)[˜Uq†VqUq]mn (15)
where Vq
pl=⟨˜ψpq|ψlq⟩. Using this transformation, we write SP Hamiltonian in SO Wannier
bases as:
(Hsp)RR′
mn=⟨wmR|Hsp|wnR′⟩
=X
plR′′R′′′⟨wmR|˜wpR′′⟩
⟨˜wpR′′|Hsp|˜wlR′′′⟩⟨˜wlR′′′|wnR′⟩
=X
plR′′R′′′(T†)RR′′
mp(˜Hsp)R′′R′′′
pl TR′′′R′
ln (16)
Since Wannier functions are maximally localised and generally atomic-like, the major con-
tribution to the overlap TRR′
mnis for R=R′. Therefore, we can write TRR
mn=T0
mn. The
reason is that it depends on relative R−R′, we can just consider overlaps at R=0. Eqn.
16 becomes,
(Hsp)RR′
mn=X
pl(T†)0
mp(˜Hsp)RR′
plT0
ln (17)
Therefore, we write the Hsoin Wannier basis as,
(Hso)RR′
mn=HRR′
mn−(Hsp)RR′
mn (18)
11The torque matrix elements in SO Wannier bases are given by,
(Γµ)RR′
mn= (σµHso)RR′
mn−(Hsoσµ)RR′
mn (19)
Consider ( σµHso)RR′
mnand insert the completeness relation of the Wannier functions, and
also neglecting SO matrix elements between the Wannier functions at different sites because
of their being atomic-like.
(σµHso)RR′
mn=P
pR′′(σµ)RR′′
mp(Hso)R′′R′
pn
= (σµ)RR′
mp(Hso)0
pn (20)
(σµ)RR′
mpis calculated by the Fourier transform of the spin operator written in the Bloch
basis, just like the Hamiltonian.
(σµ)RR′
mp=1
NX
qe−ιq.(R′−R)
Uq†(σµ)qUq
mp(21)
We interpolate the SOT matrix elements on a fine k-mesh as follows:
(Γµ)k
mn=X
R′−Reιk.(R′−R)(σµ)RR′
mn (22)
This yields the torque matrix elements in the Wannier basis. In the subsequent expres-
sions, WandHsubscripts represent the Wannier and Hamiltonian basis, respectively. In
order to rotate to the Hamiltonian gauge, which diagonalises the Hamiltonian interpolated
on the fine kmesh using its matrix elements in the Wannier basis.
(HW)k
mn=X
R′−Reιk.(R′−R)HRR′
mn (23)
(HH)k
mn=
(Uk)†(HW)kUk
mn(24)
Here Uk(not to be confused with Uq) are the matrices with columns as the eigenvectors of
(HW)k, and ( HH)k
mn=ϵmkδmn. We use these matrices to rotate the SOT matrix elements
in Eqn. (22) to the Hamiltonian basis as:
(Γµ
H)k
mn=
(Uk)†(Γµ
W)kUk
mn(25)
123.Green’s functions
The Green’s function at an arbitrary kandϵon a fine k-mesh in the Hamiltonian basis
is given by:
Gk
H(ϵ+ιη) = (ϵ+ιη−(HH)k)−1(26)
where ηis a broadening factor and is caused by electron-phonon coupling and is generally
of the order 5 −10 meV. Gk
H(ϵ+ιη) is aN × N dimensional matrix.
Therefore, we can calculate RG,IGand∂2RG/∂ϵ2as defined in Eqn. (5) and hence, α
and I.
III. COMPUTATIONAL DETAILS
Plane-wave pseudopotential calculations were carried out for the bulk ferromagnetic tran-
sition metals bcc Fe, hcp Co and fcc Ni using Quantum Espresso package[47, 48]. The
conventional unit cell lattice constants ( a) used for bcc Fe and fcc Ni were 5.424 and 6.670
bohrs, respectively and for hcp Co, a=4.738 bohr and c/a=1.623 were used. The non-
collinear spin-orbit and spin-polarised calculations were performed using fully relativistic
norm-conserving pseudopotentials. The kinetic energy cutoff was set to 80 Ry. Exchange-
correlation effects were treated within the PBE-GGA approximation. The self-consistent
calculations were carried out on 16 ×16×16 Monkhorst-Pack Grid using Fermi smearing of
0.02 Ry. Non-self-consistent calculations were carried out using the calculated charge den-
sities on Γ-centered 10 ×10×10 coarse k-point grid. For bcc Fe and fcc Ni, 64 bands were
calculated and hcp Co 96 bands were calculated (because there are two atoms per unit cell
for Co). We define a set of 18 trial orbitals sp3d2,dxy,dxz, and dyzfor Fe, 18 orbitals
per atom s,panddfor Co and Ni, to generate 18 disentangled spinor maximally-localized
Wannier functions per atom using Wannier90 package [43].
From the Wannier90 calculations, we get the checkpoint file .chk, which contains all the
information about disentanglement and gauge matrices. We use .spnand.eigfiles generated
bypw2wannier90 to get the spin operator and the Hamiltonian in the Wannier basis. We
evaluate the SOT matrix elements in the Wannier Basis.
We get αby simply summing up on a fine- kgrid with appropriate weights for the k-
integration, and we use the trapezoidal rule in the range [-8 δ,8δ] for energy integration
1310−610−410−2100
10−610−410−210010−610−410−210010
10
10
10
10
101
0
1−2
−2
−310−410−310−210−1100101
10−410−310−210−1100101
Fe Ni Co(a) (b) (c)
FIG. 3: (a), (b) and (c) shows the αvsηfor Fe, Ni and Co, respectively. Damping
constants calculated using the Wannier implementation are shown in blue. Damping
calculated using the tight binding method based on Lorentzian broadening and Green’s
function by Thonig et al[29] are shown in brown and green, respectively. Comparison with
damping constants calculated by Gilmore et al[32] using local spin density approximation
(LSDA) are shown in red. The dotted lines are guides to the eye.
around the Fermi level where δis the width of the derivative of Fermi function ∼kBT. We
consider 34 energy points in this energy range. We perform the calculation for T= 300K.
For the calculation of I, we use a very fine grid of 400 ×400×400k-points. For η >0.1,
we use 320 energy points between VBM and Fermi energy. For 0 .01< η < 0.1, we use
3200-6400 energy points for the energy integration.
TABLE I
Material η(meV) -I (fs) α(×10−3) τ(ps)
Fe 6 0.210 3.14 0.42
8 0.114 2.77 0.26
10 0.069 2.51 0.17
Ni 10 2.655 34.2 0.48
Co 10 0.061 1.9 0.21
14FIG. 4: Schematic explaining the dependence of intraband and interband contribution in
αwith η.
IV. RESULTS AND DISCUSSION
A. Damping constant
In this section, we report the damping constants calculated for the bulk iron, cobalt and
nickel. The magnetic moments are oriented in the z-direction. For reference direction z, the
damping tensor is diagonal resulting in the effective damping constant α=αxx+αyy.
In Fig. 3, we report the damping constants calculated by the Wannier implementation as
a function of broadening, ηknown to be caused by electron-phonon scattering and scattering
with impurities. We consider the values of ηranging from 10−6to 2 eV to understand the
role of intraband and interband transitions as reported in the previous studies[29, 32]. We
note that the experimental range is for the broadening is expected to be much smaller with
η∼5−10 meV. The results are found to be in very good agreement with the ones calculated
using local spin density approximation (LSDA)[32] and tight binding paramterisation[29].
The expression for Gilbert damping[3] is written in terms of the imaginary part of Green’s
functions. Using the spectral representation of Green’s function, Ank(ω), we can rewrite Eqn.
15(3) as:
αµν=gπ
msX
nmZ
Tµ
nm(k)T∗ν
nm(k)Snmdk (27)
where Snm=R
η(ϵ)Ank(ϵ)Amk(ϵ)dϵis the spectral overlap. Although we are working in the
basis where the Hamiltonian is diagonal, the non-zero off-diagonal elements in the torque
matrix lead to both intraband ( m=n) and interband ( m̸=n) contributions. For the
sake of simple physical understanding, we consider the contribution of the spectral overlaps
at the Fermi level for both intraband and interband transitions in Fig. 4. But in the
numerical calculation temperature broadening has also been considered. For the smaller
η, the contribution of intraband transitions decreases almost linearly with the increase in
ηbecause the overlaps become less peaked. Above a certain η, the interband transitions
become dominant and the contribution due to the overlap of two spectral functions at
different band indices m and n becomes more pronounced at the Fermi level. Above the
minimum, the interband contribution increases till η∼1 eV. Because of the finite Wannier
orbitals basis, we have the accurate description of energy bands only within the approximate
range of ( ϵF−10, ϵF+ 5) eV for the ferromagnets in consideration. The decreasing trend
after η∼1 eV is, therefore, an artifact.
10−2
0.000.050.100.150.20
10−1100-0.001 0.000 0.001 0.002
10−1100
FIG. 5: Plot showing moment of inertia, −I versus broadening, η. The moment of inertia
in the range 0 .03−3.0 eV is shown as an inset. The values using the Wannier
implementation and the tight binding method[30] are shown in blue and green, respectively.
160.020.040.060.08
Co(a)
10−210−11000.0010−1100−0.0075−0.0050−0.0050−0.0025 0.000
10−210−1100−
0.00.51.0 1.01.5
Ni(b)
0.000.010.02
100101
FIG. 6: (a) and (b) show negative of the moment of inertia, −I versus broadening, ηfor
Co and Ni, respectively. The values using the Wannier implementation and the tight
binding method are shown in magenta and cyan, respectively. The moment of inertia is
shown as an inset in the range 0 .03−2.0 eV and 0 .03−3.0 eV for Co and Ni, respectively.
B. Moment of inertia
In Fig. 5, we report the values for the moment of inertia calculated for bulk Fe, Co
and Ni. Analogous to the damping, the inertia tensor is diagonal, resulting in the effective
moment of inertia I = Ixx+ Iyy.
The behaviour for I vs ηis similar to that of the damping, with smaller and larger ηtrends
arising because of intraband and interband contributions, respectively. The overlap term
in the moment of inertia is between the ∂2RG/∂ϵ2andIGunlike just IGin the damping.
In Ref. [30], the moment of inertia is defined in terms of torque matrix elements and the
overlap matrix as:
Iµν=−gℏ
msX
nmZ
Tµ
nm(k)T∗ν
nm(k)Vnmdk (28)
where Vnmis an overlap function, given byR
f(ϵ)(Ank(ϵ)Bmk(ϵ) +Bnk(ϵ)Amk(ϵ))dϵand
Bmk(ϵ) is given by 2( ϵ−ϵmk)((ϵ−ϵmk)2−3η2)/((ϵ−ϵmk)2+η2)3. There are other notable
features different from the damping. In the limit η→0, the overlap Vmnreduces to 2 /(ϵmk−
ϵnk)3. For intraband transitions ( m=n), this leads to I → −∞ . In the limit η→ ∞ ,
Vmn≈1/η5which leads to I →0. The behaviour at these two limits is evident from Fig. 5.
The large τ(small η) behaviour is consistent with the expression I = −α.τ/2πderived by
17104
−10−310−2−10−1100
10−2−10−110−1100101102
FIG. 7: The damping, magnetic moment of inertia and relaxation rate are shown as a
function of broadening, ηin blue, green and red, respectively. The grey-shaded region
shows the observed experimental relaxation rate, τ, ranging from 0 .12 to 0 .47 ps. The
corresponding range of ηis shown in purple and is 6 −12 meV. This agrees with the
experimental broadening in the range of 5 −10 meV, arising from electron-phonon
coupling. The numbers are tabulated in Table I
F¨ahnle et al. [49]. Here τis the Bloch relaxation lifetime. The behaviour of τas a function
ofηusing the above expression in the low ηlimit is shown in Fig. 7. Apart from these
limits, the sign change has been observed in a certain range of ηfor Fe and Co. This change
in sign can be explained by the Eqn. (5). In the regime of intraband contribution, at a
certain η, the negative and positive terms integrated over ϵandkbecome the same, leading
to zero inertia. Above that η, the contribution due to the negative terms decreases until the
interband contribution plays a major role leading to maxima in I (minima in −I). Interband
contribution leads to the sign change from + to - and eventually zero at larger η.
The expression I = −α.τ/2πderived from the Kambersky model is valid for η <10 meV,
which indicates that damping and moment of inertia have opposite signs. By analyzing the
rate of change of magnetic energy, Ref. [11] shows that Gilbert damping and the moment
of inertia have opposite signs when magnetization dynamics are sufficiently slow (compared
toτ).
Experimentally, the stiffening of FMR frequency is caused by negative inertia. The
softening caused by positive inertia is not observed experimentally. This is because the
experimentally realized broadening, ηcaused by electron-phonon scattering and scattering
with impurities, is of the order of 5 −10 meV. The values of Bloch relaxation lifetime, τ
18measured at the room temperature with the FMR in the high-frequency regime for Ni 79Fe21
and Co films of different thickness, range from 0 .12−0.47 ps. The theoretically calculated
values for Fe,Ni and Co using the Wannier implementation for the ηranging from 5 −10
meV are reported in Table. I and lies roughly in the above-mentioned experimental range
for the ferromagnetic films.
V. CONCLUSIONS
In summary, this paper presents a numerical method to obtain the Gilbert damping
and moment of inertia based on the torque-torque correlation model within an ab initio
Wannier framework. We have also described a technique to calculate the spin-orbit coupling
matrix elements via the transformation between the spin-orbit and spin-polarised basis. The
damping and inertia calculated using this method for the transition metals like Fe, Co and
Ni are in good agreement with the previous studies based on tight binding method[29, 30]
and local spin density approximation[32]. We have calculated the Bloch relaxation time
for the approximate physical range of broadening caused by electron-phonon coupling and
lattice defects. The Bloch relaxation time is in good agreement with experimentally reported
values using FMR[27]. The calculated damping and moment of inertia can be used to study
the magnetisation dynamics in the sub-ps regime. In future studies, we plan to use the
Wannier implementation to study the contribution of spin pumping terms, arising from
the spin currents at the interface of ferromagnetic-normal metal bilayer systems due to
the spin-orbit coupling and inversion symmetry breaking to the damping. We also plan to
study the magnetic damping and anisotropy in experimentally reported 2D ferromagnetic
materials[50] like CrGeTe 3,CrTe ,Cr3Te4etc. The increasing interest in investigating the
magnetic properties in 2D ferromagnets is due to magnetic anisotropy, which stabilises the
long-range ferromagnetic order in such materials. Moreover, the reduction in dimensionality
from bulk to 2D leads to intriguingly distinct magnetic properties compared to the bulk.
VI. ACKNOWLEDGEMENTS
This work has been supported by a financial grant through the Indo-Korea Science and
Technology Center (IKST). We thank the Supercomputer Education and Research Centre
19(SERC) at the Indian Institute of Science (IISc) and the Korea Institute of Science and
Technology (KIST) for providing the computational facilities.
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1608.04071v1.Mechanical_energy_and_mean_equivalent_viscous_damping_for_SDOF_fractional_oscillators.pdf | arXiv:1608.04071v1 [physics.class-ph] 14 Aug 2016Mechanical energy and mean equivalent viscous
damping for SDOF fractional oscillators
Jian Yuan1,∗, Bao Shi, Mingjiu Gai, Shujie Yang
Institute of System Science and Mathematics, Naval Aeronau tical and Astronautical
University, Yantai 264001, P.R.China
Abstract
Thispaperaddressesthetotalmechanical energyofasingledegr eeoffreedom
fractional oscillator. Based on the energy storage and dissipation properties
of the Caputo fractional derivatives, the expression for total m echanical en-
ergy in the single degree of freedom fractional oscillator is firstly pr esented.
The energy regeneration due to the external exciting force and t he energy
loss due to the fractional damping force during the vibratory motio n are an-
alyzed. Furthermore, based on the mean energy dissipation of the fractional
damping element in steady-state vibration, a new concept of mean e quiva-
lent viscous damping is suggested and the value of the damping coeffic ient is
evaluated.
Keywords: Fractional oscillators, linear viscoelasticity, fractional
constitutive relations, mechanical energy, mean equivalent viscou s damping
1. Introduction
Viscoelasticmaterialsanddampingtreatmenttechniqueshavebeen widely
applied in structural vibration control engineering, such as aeros pace indus-
try, military industry, mechanical engineering, civil and architectu ral engi-
neering[1]. Describingtheconstitutive relationsforviscoelastic mat erialsisa
top priority to seek for the dynamics of the viscoelastically damped s tructure
and to design vibration control systems.
∗Corresponding author. Tel.: +8613589862375.
Email address: yuanjianscar@gmail.com (Jian Yuan)
Preprint submitted to Elsevier September 19, 2018Recently, theconstitutiverelationsemploying fractionalderivativ es which
relatestressandstraininmaterials, alsotermedasfractionalvisc oelasticcon-
stitutive relations, have witnessed rapid development. They may be viewed
asanaturalgeneralizationoftheconventional constitutiverelat ionsinvolving
integer order derivatives or integrals, and have been proven to be a power-
ful tool of describing the mechanical properties of the materials. Over the
conventional integer order constitutive models, the fractional o nes have vast
superiority. The first attractive feature is that they are capable of fitting ex-
perimental results perfectly and describing mechanical propertie s accurately
in both the frequency and time domain with only three to five empirical pa-
rameters[2]. Thesecondisthattheyarenotonlyconsistent withth ephysical
principles involved [3] and the molecule theory [4], but also represent t he fad-
ing memory effect [2] and high energy dissipation capacity [5]. Finally, fr om
mathematical perspectives the fractional constitutive equation s and the re-
sulting fractional differential equations of vibratory motion are co mpact and
analytic [6].
Nowadays many types of fractional order constitutive relations h ave been
established via a large number of experiments. The most frequently used
models include the fractional Kelvin-Voigt model with three paramet ers [2]:
σ(t) =b0ε(t)+b1Dαε(t), the fractional Zener model with four parameters
[3]:σ(t)+aDασ(t) =b0ε(t)+b1Dαε(t), and the fractional Pritz model with
five parameters [7]: σ(t)+aDασ(t) =b0ε+b1Dα1ε(t)+b2Dα2ε(t).
Fractionaloscillators, orfractionallydampedstructures, aresy stemswhere
the viscoelastic damping forces in governing equations of motion are de-
scribed by constitutive relations involving fractional order derivat ives [8].
The differential equations of motion for the fractional oscillators a re frac-
tional differential equations. Researches on fractional oscillator s are mainly
concentratedontheoreticalandnumericalanalysisofthevibrat ionresponses.
Investigations on dynamical responses of SDOF linear and nonlinear frac-
tional oscillators, MDOF fractional oscillators and infinite-DOF frac tional
oscillators have been reviewed in [8]. Asymptotically steady state beh avior
of fractional oscillators have been studied in [9, 10]. Based on the fu nc-
tional analytic approach, the criteria for the existence and the be havior of
solutions have been obtained in [11-13], and particularly in which the im-
pulsive response function for the linear SDOF fractional oscillator is derived.
The asymptotically steady state response of fractional oscillator s with more
than one fractional derivatives have been analyzed in [14]. Consider ing the
memory effect and prehistory of fractional oscillators, the histor y effect or
2initialization problems for fractionally damped vibration equations has been
proposed by Fukunaga, M. [15-17] and Hartley, T.T., and Lorenzo, C.F. [18,
19].
Stability synthesis for nonlinear fractional differential equations h ave re-
ceived extensive attention in the last five years. Mittag-Leffler sta bility theo-
rems [20, 21]andtheindirect Lyapunov approach[22] based onthe frequency
distributed model are two main techniques to analyze the stability of non-
linear systems, though there is controversy between the above t wo theories
due to state space description and initial conditions for fractional systems
[23]. In spite of the increasing interest in stability of fractional differ ential
equations, there’s little results on the stability of fractionally dampe d sys-
tems. For the reasons that Lyapunov functions are required to c orrespond to
physical energy and that there exist fractional derivatives in the differential
equations of motion for fractionally damped systems, it is a primary t ask to
define the energies stored in fractional operators.
Fractional energy storage and dissipation properties of Riemann- Liouville
fractional integrals is defined [24, 25] utilizing the infinite state appr oach.
Based on the fractional energies, Lyapunov functions are propo sed and sta-
bility conditions of fractional systems involving implicit fractional der iva-
tives are derived respectively by the dissipation function [24, 25] an d the
energy balance approach [26, 27]. The energy storage properties of frac-
tional integrator and differentiator in fractional circuit systems h ave been
investigated in [28-30]. Particularly in [29], the fractional energy for mula-
tion by the infinite-state approach has been validated and the conv entional
pseudo-energy formulations based on pseudo state variables has been inval-
idated. Moreover, energy aspects of fractional damping forces described by
the fractional derivative of displacement in mechanical elements ha ve been
considered in [31, 32], in which the effect on the energy input and ener gy
return, as well as the history or initialization effect on energy respo nse has
been presented.
On the basis of the recently established fractional energy definitio ns for
fractional operators, our main objective in this paper is to deal wit h the
total mechanical energy of a single degree of freedom fractional oscillator.
To this end, we firstly present the mechanical model and the differe ntial
equation of motion for the fractional oscillator. Then based on the energy
storage and dissipation in fractional operators, we provide the ex pression of
total mechanical energy in the single degree of freedom fractiona l oscillator.
Furthermore, we analyze the energy regeneration due to the ext ernal exciting
3force and the energy loss due to the fractional damping force in th e vibration
processes. Finally, based on the mean energy dissipation of the fra ctional
dampingelementinsteady-statevibration, weproposeanewconce ptofmean
equivalent viscous damping and determine the expression of the dam ping
coefficient.
The rest of the paper is organized as follows: Section 2 retrospect some
basic definitions and lemmas about fractional calculus. Section 3 intr oduces
the mechanical model and establishes the differential equation of m otion for
the single degree of freedom fractional oscillator. Section 4 provid es the
expression of total mechanical energy for the SDOF fractional o scillator and
analyzes the energy regeneration and dissipation in the vibration pr ocesses.
Section 5 suggests a new concept of mean equivalent viscous dampin g and
evaluatesthevalueofthedamping coefficient. Finally, thepaperisco ncluded
in section 6 with perspectives.
2. Preliminaries
Definition 1. The Riemann-Liouville fractional integral for the function
f(t) is defined as
aIα
tf(t) =1
Γ(α)/integraldisplayt
a(t−τ)α−1f(τ)dτ, (1)
whereα∈R+is an non-integer order of the factional integral, the subscripts
aandtare lower and upper terminals respectively.
Definition 2. The Caputo definition of fractional derivatives is
aDα
tf(t) =1
Γ(n−α)/integraldisplayt
af(n)(τ)dτ
(t−τ)α−n+1,n−1< α < n. (2)
Lemma 1. The frequency distributed model for the fractional integra tor [33-
35]Theinput ofthe Riemann-Liouvilleintegralis denotedb yv(t)andoutput x(t),
thenaIα
tv(t)is equivalent to
/braceleftBigg
∂z(ω,t)
∂t=−ωz(ω,t)+v(t),
x(t) =aIα
tv(t) =/integraltext+∞
0µα(ω)z(ω,t)dω,(3)
withµα(ω) =sin(απ)
πω−α.
4System (3) is the frequency distributed model for fractiona l integrator,
which is also named as the diffusive representation.
Lemma 2. The following relation holds[26]
/integraldisplay∞
0ωµα(ω)
ω2+Ω2dω=sinαπ
2Ωαsinαπ
2. (4)
3. Differential equation of motion for the fractional oscill ator
This section will establish the differential equation of motion for a sin-
gle degree of freedom fractional oscillator, which consists of a mas s and a
spring with one end fixed and the other side attached to the mass, d epicted
in Fig.1. The spring is a solid rod made of some viscoelastic material with
the cross-sectional area Aand length L, and provides stiffness and damping
for the oscillator.
m
Figure 1: Mechanical model for the SDOF fractional oscillator.
In accordance with Newton’s second law, the dynamical equation fo r the
SDOF fractional oscillator is
m¨x(t)+fd(t) =f(t),fd(t) =Aσ(t), (5)
wherefd(t) is the forceprovided by the viscoelastic rodandcan be separated
into two parts: the resilience and the damping force. f(t) is the vibration
exciting force acted on the mass.
The kinematic relation is
ε(t) =x(t)
L. (6)
5As for the constitutive equation of viscoelastic material, the followin g frac-
tional Kelvin-Voigt model (7) with three parameters will be adopted
σ(t) =b0ε(t)+b1Dα
tε(t), (7)
whereα∈(0,1) is the order of fractional derivative, b0andb1are positive
constant coefficients.
The above three relations (5) (6) and (7) form the following differen tial equa-
tion of motion for the single degree of freedom fractional oscillator
m¨x(t)+cDαx(t)+kx(t) =f(t), (8)
wherec=Ab1
L,k=Ab0
L.
For the reason that the Caputo derivative is fully compatible with the clas-
sical theory of viscoelasticity on the basis of integral and different ial consti-
tutive equations [36], the adoption of the Caputo derivative appear s to be
the most suitable choice in the fractional oscillators. For the simplific ation
of the notation, the Caputo fractional-order derivativeC
0Dα
tis denoted as Dα
in this paper.
Comparing the forms of differential equations for the fractional o scillator (8)
with the following classical ones
m¨x(t)+c˙x(t)+kx(t) =f(t), (9)
onecansee thatthefractionalone(8)isthegeneralizationofthe classical one
(9)byreplacing thefirst orderderivative ˙ xwiththefractionalorderderivative
Dαx. However, the generalization induces the following essential differe nces
between them.
•In view of the formalization of the mechanical model, the classical os -
cillator is composed of a mass, a spring and a dashpot, where kis the
stiffness coefficient of the spring offering restoring force kxandcis
the damping coefficient of the dashpot offering the damping force c˙x.
The fractional oscillator is formed by a mass and an viscoelastic rod.
The rod offers not only resilience but also damping force. In fraction al
differential equation(8), the coefficient candkare determined by both
the constitute equation (7) for the viscoelastic material and the g eo-
metrical parameters for the rod, which can be interpreted respe ctively
as the fractional damping coefficient and the stiffness coefficient. A s a
6result, the physical meaning of candkin the fractional oscillator (8)
andtheclassical one(9) aredifferent. Thefractional damping for cecan
be viewed as a parallel of a spring component kx(t) and a springpot
component cDαx(t) which is termed in [37] and illustrated in Fig.2.
The hysteresis loop of the fractional damping force is dipicted in Fig.3 .
mkx
cD xD
Figure 2: Abstract mechanical model for the SDOF fractional osc illator.
−2−1.5 −1−0.5 00.5 11.5−4−3−2−101234
x(t)Dalpha x(t)
Figure 3: Hysteresis loop of the fractional damping force.
•Fractional operators are characterized by non-locality and memo ry
properties, so fractional oscillators (8) also exhibit memory effect and
the vibration response is influenced by prehistory. While the classica l
one (9) has no memory effect and the vibration response is irrelevan t
with prehistory.
•In the aspect of mechanical energy, the fractional term Dαxin (8) not
only stores potential energy but also consumes energy due to the fact
7that fractional operators exhibit energy storage and dissipation simul-
taneously [24]. As a result, the total mechanical energy in fraction al
oscillator consists of three parts: the kinetic energy1
2m˙x2stored in the
mass, the potential energy corresponding to the spring element1
2kx2,
and the potential energy e(t) stored in the fractional derivative. How-
ever, in [38] the potential energy e(t) stored in the fractional term
Dαxhas been neglected and the expression1
2m˙x2+1
2kx2for the total
mechanical energy is incomplete.
4. The total mechanical energy
Given the above considerations, we present the total mechanical energy
of the SDOF fractional oscillator (8)in this section. The fractional system is
assumed to be at rest before exposed to the external excitation . We firstly
analyze the energy stored in the Caputo derivative, based on which the ex-
pression for total mechanical energy is derived. Then we obtain th e energy
regeneration due to external excitation and the energy dissipatio n due to the
fractional viscoelastic damping.
Bydefinitions(1)and(2),theCaputoderivativeiscomposedofone Riemann-
Liouville fractional order integral and one integer order derivative ,
Dαx(t) =I1−α˙x(t).
In view of Lemma 1, the frequency distributed model for the Caput o deriva-
tive is /braceleftBigg
∂z(ω,t)
∂t=−ωz(ω,t)+ ˙x,
Dαx(t) =/integraltext∞
0µ1−α(ω)z(ω,t)dω.(10)
In terms of the fractional potential energy expression for the f ractional inte-
gral operator in [24], the stored energy in the Caputo derivative is
e(t) =1
2/integraldisplay∞
0µ1−α(ω)z2(ω,t)dω. (11)
The total mechanical energy of the SDOF fractional oscillator is th e sum of
the kinetic energy of the mass1
2m˙x2, the potential energy corresponding to
the spring element1
2kx2, and the potential energy stored in the fractional
derivative ce(t)
E(t) =1
2m˙x2+1
2kx2+c
2/integraldisplay∞
0µ1−α(ω)z2(ω,t)dω. (12)
8To analyze the energy consumption in the fractional viscoelastic os cillator,
taking the first order time derivative of E(t),one derives
dE(t)
dt=m˙x¨x+kx˙x+c/integraldisplay∞
0µ1−α(ω)z(ω,t)∂z(ω,t)
∂tdω.(13)
Substituting the first equation in the frequency distributed model (10) into
the third term of the above equation (13), one derives
dE(t)
dt=m˙x¨x+kx˙x+c/integraldisplay∞
0µ1−α(ω)z(ω,t)[−ωz(ω,t)+ ˙x]dω
=m˙x¨x+kx˙x+c˙x/integraldisplay∞
0µ1−α(ω)z(ω,t)dω
−c/integraldisplay∞
0ωµ1−α(ω)z2(ω,t)dω. (14)
Substituting the second equation in the frequency distributed mod el (10)
into the second term of the above equation (14), one derives
dE(t)
dt=m˙x¨x+kx˙x+c˙xDαx−c/integraldisplay∞
0ωµ1−α(ω)z2(ω,t)dω
= ˙x[m¨x+cDαx+kx]−c/integraldisplay∞
0ωµ1−α(ω)z2(ω,t)dω. (15)
Substituting the differential equation of motion (8) for the fractio nal oscilla-
tor into the first term of the above equation (15), one derives
dE(t)
dt=f(t) ˙x(t)−c/integraldisplay∞
0ωµ1−α(ω)z2(ω,t)dω. (16)
From Eq. (16) it is clear that the energy regeneration in the fractio nal
oscillator due to the work done by the external excitation in unit time is
P(t) =f(t) ˙x(t). (17)
On the other hand, the energy consumption or the Joule losses due to the
fractional viscoelastic damping is
J(t) =c/integraldisplay∞
0ωµ1−α(ω)z2(ω,t)dω. (18)
9The mechanical energy changes in the vibration process can be obs erved
through the following numerical simulations. Parameters in the frac tional
oscillator (8) are taken respectively as m= 1,c= 0.4,k= 2,α= 0.56, the
external force are assumed to be f(t) = 30cos6 t. Fig.4 shows the fractional
potential energy ce(t); Fig.5 shows comparison between the fractional energy
ce(t)and the total mechanical energy E(t); Fig.6 illustrates the mechanical
energy consumption J(t).
0 5 10 15 20 25 3000.050.10.150.20.250.30.350.4
Time(sec)Fractional Energy Ec(t)
Figure 4: Fractional energy of the SDOF fractional oscillator.
0 5 10 15 20 25 30051015202530354045
Time(sec)Energy
Figure 5: Comparison between the fractional energy and the tota l mechanical energy.
100 5 10 15 20 25 3000.511.522.533.54
Time(sec)Energy Consumption J (t)
Figure 6: Mechanical energy consumption in the SDOF fractional os cillator.
Remark 1. IfthefollowingmodifiedfractionalKelvin-Voigtconstituteequa-
tion (19)which is proposed in [39] is taken to describe the viscoelastic stress-
strain relation
σ(t) =b0ε(t)+b1Dα1ε(t)+b2Dα2ε(t), (19)
withα1,α2∈(0,1), the differential equation of motion for the SDOF frac-
tional oscillator is
m¨x(t)+c1Dα1x(t)+c2Dα2x(t)+kx(t) =f(t), (20)
wherec1=Ab1
L,c2=Ab2
L,k=Ab0
L.
In view of the following equivalences (21) and (22) between the Capu to
derivatives and the frequency distributed models
Dα1x(t) =I1−α1˙x(t)⇔/braceleftBigg
∂z1(ω,t)
∂t=−ωz1(ω,t)+ ˙x(t)
Dα1x(t) =/integraltext∞
0µ1−α1(ω)z1(ω,t)dω(21)
and
Dα2x(t) =I1−α2˙x(t)⇔/braceleftBigg
∂z2(ω,t)
∂t=−ωz21(ω,t)+ ˙x(t)
Dα2x(t) =/integraltext∞
0µ1−α2(ω)z2(ω,t)dω(22)
11the total mechanical energy of the fractional oscillator (20) is ex pressed as
E(t) =1
2m˙x2+1
2kx2+c1
2/integraldisplay∞
0µ1−α1(ω)z2
1(ω,t)dω
+c2
2/integraldisplay∞
0µ1−α2(ω)z2
2(ω,t)dω. (23)
In the above expression (23) for the total mechanical energy,
P1(t) =c1
2/integraldisplay∞
0µ1−α1(ω)z2
1(ω,t)dω
represents the potential energy stored in Dα1x(t), whereas
P2(t) =c2
2/integraldisplay∞
0µ1−α2(ω)z2
2(ω,t)dω
represents thepotential energystored in Dα2x(t). Taking the first order time
derivative of E(t) in Eq.(23), one derives
˙E(t) =f(t) ˙x(t)−c1/integraldisplay∞
0ωµ1−α1(ω)z2
1(ω,t)dω
−c2/integraldisplay∞
0ωµ1−α2(ω)z2
2(ω,t)dω.
It is clear that the energy dissipation due to the fractional viscoela stic damp-
ingc1Dα1xis
Jα1(t) =c1/integraldisplay∞
0ωµ1−α1(ω)z2
1(ω,t)dω, (24)
and the energy dissipation due to the fractional viscoelastic dampin gc2Dα2x
is
Jα2(t) =c2/integraldisplay∞
0ωµ1−α2(ω)z2
2(ω,t)dω. (25)
5. The mean equivalent viscous damping
The resulting differential equations of motion for structures incor porating
fractional viscoelastic constitutive relations to dampen vibratory motion are
fractional differential equations, which are strange and intricate ly to tackled
withfor engineers. In engineering, complex descriptions fordampin g areusu-
allyapproximately represented by equivalent viscous damping to simp lify the
12theoretical analysis. Inspired by this idea, we suggest a new conce pt of mean
equivalent viscous damping based on the expression of fractional e nergy (18).
Using thismethod, fractionaldifferential equations aretransfor med into clas-
sical ordinary differential equations by replacing the fractional da mping with
the mean equivalent viscous damping. The principle for the equivalenc y is
that the mean energy dissipation due to the desired equivalent damp ing and
the fractional viscoelastic damping are identical.
To begin with, some comparisons of the energy dissipation between t he frac-
tional oscillator (8) and the classical one (9) are made in the following .In
view of the concept of work and energy in classical physics, the wor k done
by any type of damping force is expressed as
W(t) =/integraldisplayt
0fc(τ)dx(τ), (26)
wherefc(t) is some type of damping force, x(t) is the displacement of the
mass.
In the classical oscillators, the viscous damping force is
fc1(t) =c˙x(t).
The work done by the viscous damping force is
W1(t) =/integraldisplayt
0c˙x(τ)dx(τ) =/integraldisplayt
0c˙x2(τ)dτ. (27)
It is well known that the energy consumption in unit time is
J1(t) =c˙x2(t), (28)
which is equal to the rate of the work done by the viscous damping fo rce
J1(t) =dW1(t)
dt.
Obviously, the entire work done by the viscous damping force is conv erted
to heat energy.
However, the case in the fractional oscillators is different. As a mat ter fact,
the fractional damping force is
fc2(t) =cDαx(t).
13The work done by the fractional damping force is
W2(t) =/integraldisplayt
0cDαx(τ)dx(τ).
Due to the property of energy storage and dissipation in fractiona l deriva-
tives, the entire work done by the fractional damping force W2is converted
to two types of energy: one of which is the heat energy
J(t) =c/integraldisplay∞
0ωµ1−α(ω)z2(ω,t)dω,
and the other is the fractional potential energy
P(t) =c
2/integraldisplay∞
0µ1−α(ω)z2(ω,t)dω.
However, in [40] the equivalent viscous damping coefficient was obtain ed by
the equivalency
/contintegraldisplay
cDαx(τ)dx(τ) =/contintegraldisplay
ceq˙x(τ)dx(τ)
Bythisequivalencythepropertiesoffractionalderivativehavebe enneglected
and the work done by the fractional damping force is considered to be con-
vertedintotheheatentirely. Asaresult, theaboveequivalencyisp roblematic
and the value of the derived equivalent viscous damping coefficient is la rger
than the actual value.
In terms of the energy consumption (18), (24) and (25) due to th e fractional
damping force, we suggest a new the concept of mean equivalent vis cous
damping and evaluate the expression of the damping coefficient.
Assuming the steady-state response of the fractional oscillator (8) is
x(t) =XejΩt,
whereXis the amplitude and Ω is the vibration frequency.
Step1. We firstly need to calculate the mean energy consumption due to
the fractional viscoelastic damping element, i.e.
Jα(t) =c/integraldisplay∞
0ωµ1−α(ω)z(ω,t)2dω. (29)
14To this end, we evaluate the mean square of z(ω,t), i.e.z(ω,t)2.
In terms of the first equation in the diffusive representation of Cap uto deriva-
tive (10)
˙z(ω,t) =−ωz(ω,t)+ ˙x(t),
we get
z(ω,t) =˙x(t)
ω+jΩ=jΩxejΩt
√
ω2+Ω2ejθ,
whereθ= arctanΩ
ω.
Furthermore we get
z(ω,t)2=1
2z(ω,t)z(ω,t)∗=1
2Ω2x2
ω2+Ω2, (30)
wherez(ω,t)∗is the complex conjugate of z(ω,t).
Substituting Eq. (30) into Eq.(29), one derives
Jα(t) =c/integraldisplay∞
0ωµ1−α(ω)z(ω,t)2dω
=c
2Ω2X2/integraldisplay∞
0ωµ1−α(ω)
ω2+Ω2dω. (31)
Applying the relation (4) in Lemma 2 ,one derives
/integraldisplay∞
0ωµ1−α(ω)
ω2+Ω2dω=sin(1−α)π
2Ωαsin/parenleftbig1−α
2/parenrightbig
π. (32)
Substituting Eq.(32) into Eq. (31) one derives
Jα(t) =c
4Ω1+αX2sin(1−α)π
sin/parenleftbig1−α
2/parenrightbig
π. (33)
Step2. Now we calculate the mean energy loss due to the viscous damping
force in the classical oscillator. From the relation(28), we have
J(t) =cmeq˙x2(t),
wherecmeqis denoted as the mean equivalent viscous damping coefficient for
the fractional viscoelastic damping.
15Then the mean of the energy loss is derived as
J(t) =cmeq˙x2=1
2cmeq˙x˙x∗=1
2cmeqΩ2X2. (34)
Step3. Letting Jα(t) =J(t) and from the relations (33) and (34) one
derives
c
4Ω1+αX21
2sin(1−α)π
sin/parenleftbig1−α
2/parenrightbig
π=1
2cmeqΩ2X2.
Consequently, we obtain the mean equivalent viscous damping coeffic ient for
the fractional viscoelastic damping
cmeq=c
2Ωα−1sin(1−α)π
sin/parenleftbig1−α
2/parenrightbig
π. (35)
It is clear from (35) that the mean equivalent viscous damping coeffic ient for
the fractional viscoelastic damping is a function of the vibration fre quency
Ω and the order αof the fractional derivative. To this point, the fractional
differential equations for the SDOF fractional oscillator (8) is appr oximately
simplified to the following classical ordinary differential equation
m¨x(t)+cmeq˙x(t)+kx(t) =f(t). (36)
With the aid of numerical simulations, we compare the vibration respo nses
of the approximate integer-order oscillator (36) with the fraction al one (8).
The coefficients are respectively taken as m= 1,c= 0.4,k= 2,α= 0.56,
the external force is taken asthe form f=FcosΩt, whereF= 30, Ω = 6. In
terms of Eq.(35), we derive the mean equivalent viscous damping coe fficient
cmeq= 0.14.
160510152025303540−2−1.5−1−0.500.511.5
Time(sec)x(t)
Figure 7: The mean equilvalent damping coefficient for the SDOF fract ional oscillator of
Kelvin-Voigt type.
Remark 2. By the above procedure, we can furthermore evaluate the mean
equivalent viscous damping coefficient for the SDOF fractional oscilla tor (20)
containing two fractional viscoelastic damping elements. Letting Jα1(t) +
Jα2(t) =J(t) and from the relations (24) (25) and (34), we get
c1
4Ωα1+1X2sin(1−α1)π
sin/parenleftbig1−α1
2/parenrightbig
π+c2
4Ωα2+1X2sin(1−α2)π
sin/parenleftbig1−α2
2/parenrightbig
π
=1
2c(α1,α2,Ω)Ω2X2.(37)
From (37) we obtain the mean equivalent viscous damping coefficient
cmeq=c1
2Ωα1−1sin(1−α1)π
sin/parenleftbig1−α1
2/parenrightbig
π+c2
2Ωα2−1sin(1−α2)π
sin/parenleftbig1−α2
2/parenrightbig
π.(38)
With the aid of numerical simulations, we compare the vibration respo nses
of the approximate integer-order oscillator (36) with the fraction al one(20).
The coefficients are respectively taken as m= 1,c1= 0.4,c2= 0.2,k= 2,
17α1= 0.56,α2= 0.2, the external force is taken as the form f=FcosΩt,
whereF= 30, Ω = 6. In terms of Eq.(38), we derive the mean equivalent
viscous damping coefficient cmeq= 0.56.
0510152025303540−2−1.5−1−0.500.511.5
Time(sec)x(t)
Figure 8: The mean equilvalent damping coefficient for the SDOF fract ional oscillator of
modified Kelvin-Voigt type.
6. Discussion
The total mechanical energy in single degree of freedom fractiona l oscil-
lators has been dealt with in this paper. Based on the energy storag e and
dissipation properties of the Caputo fractional derivative, the to tal mechani-
cal energy is expressed asthe sum ofthe kinetic energy of themas s1
2m˙x2, the
potential energy corresponding to the spring element1
2kx2, and the potential
energy stored in the fractional derivative e(t) =1
2/integraltext∞
0µ1−α(ω)z2(ω,t)dω.
The energy regeneration and loss in vibratory motion have been ana lyzed by
means of the total mechanical energy. Furthermore, based on t he mean en-
ergy dissipation of the fractional damping element in steady-state vibration,
a new concept of mean equivalent viscous damping has been suggest ed and
the expression of the damping coefficient has been evaluated.
18By virtue of the total mechanical energy in SDOF fractional oscillat ors,
it becomes possible to formulate Lyapunov functions for stability an alysis
and control design for fractionally damped systems as well as othe r types
of fractional dynamic systems. As for the future perspectives, our research
efforts will be focused on fractional control design for fractiona lly damped
oscillators and structures.
Acknowledgements
The author Yuan Jian expresses his thanks to Prof. Dong Kehai fr om
Naval Aeronautical and Astronautical University, and Prof. Jian g Jianping
from national University of Defense technology. All the authors a cknowledge
the valuable suggestions from the peer reviewers. This work was su pported
by the Natural Science Foundation of the Province Shandong of Ch ina ti-
tled Controls for fractional systems with applications to hyperson ic vehicles
(Grant Nos. ZR2014AM006).
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23 |
2307.05981v1.Parabolic_elliptic_Keller_Segel_s_system.pdf | arXiv:2307.05981v1 [math.AP] 12 Jul 2023PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM
VALENTIN LEMARIÉ
Abstract. We study on the whole space Rdthe compressible Euler system with damping coupled to the
Poisson equation when the damping coefficient tends towards i nfinity. We first prove a result of global
existence for the Euler-Poisson system in the case where the damping is large enough, then, in a second step,
we rigorously justify the passage to the limit to the parabol ic-elliptic Keller-Segel after performing a diffusive
rescaling, and get an explicit convergence rate. The overal l study is carried out in ‘critical’ Besov spaces, in
the spirit of the recent survey [16] by R. Danchin devoted to p artially dissipative systems.
1.Introduction
In this article, we will focus on two systems : Euler-Poisson and parabolic-elliptic Keller-Segel system.
Let us first present these systems and motivate their study.
The Euler-Poisson system with damping, set on the whole spac eRd(whered≥2) reads:
∂tρε+div(ρεvε) = 0,
∂t(ρεvε)+div(ρεvε⊗vε)+∇(P(ρε))+ε−1ρεvε=−ρε∇Vε,
−∆Vε=ρε−ρ(1.1)
whereε >0,ρε=ρε(t,x)∈R+represents the density of the gas (with ρ >0a constant state), vε=
vε(t,x)∈Rdthe velocity, the pressure P(z) =Azγwithγ >1andA >0, as well as Vε=Vε(t,x)the
potential.
This is the classical Euler compressible system with dampin g, to which we added a coupling with the
Poisson equation.
Without this coupling, System (1.1) reduces to the compress ible Euler system with the damping coeffi-
cientε−1: /braceleftbigg
∂tρε+div(ρεvε) = 0,
∂t(ρεvε)+div(ρεvε⊗vε)+∇(P(ρε))+ε−1ρεvε= 0.
This system was recently studied by Crin-Barat and Danchin in [11] where they established a result of
existence and uniqueness of global solutions for sufficientl y small data, and obtained optimal time decay
estimates for these solutions. In parallel, they studied th e singular limit of the system when the damping
coefficient tends towards infinity. They then obtained the equ ation of porous media. In this article, we will
draw freely on this study and the method used to obtain a prior i estimates.
The system that we will study is usually used to describe the t ransport of charge carriers (electrons
and ions) in semiconductor devices or plasmas. The system co nsists of conservation laws for mass density
and current density for carriers, with a Poisson equation fo r electrostatic potential. This system is also of
interest in other fields like e.g. in chemotaxis: then, ρεrepresents cell density and Vε,the concentration of
chemoattractant secreted by cells. For recent results on th e Euler-Poisson system: well-posedness, existence
of global solutions, study of long-time behavior or other si ngular limits, the reader can refer to [1], [2], [3],
[9], [18],[27], [31], [39] and [41].
As in [16], in order to investigate the asymptotics of soluti ons of (1.1) when εgoes to0,we perform the
following so-called diffusive change of variable:
(˜̺ε,˜vε)(τ,x) = (̺ε,ε−1vε)(ε−1τ,x) (1.2)
so that we have
∂t˜̺ε+div(˜̺ε˜vε) = 0
ε2(∂t˜vε+ ˜vε·∇˜vε)+∇(P(˜̺ε))
˜̺ε+ ˜vε+∇(−∆)−1(˜̺ε−̺) = 0.(1.3)
12 VALENTIN LEMARIÉ
We then define the damped mode:
˜Wε:=∇(P(˜̺ε))
˜̺ε+ ˜vε+∇(−∆)−1(˜̺ε−̺). (1.4)
As the first equation of (1.3) can be rewritten as
∂t˜̺ε−∆(P(˜̺ε))−div/parenleftbig
˜̺ε∇(−∆)−1(˜̺ε−̺)/parenrightbig
= div(˜̺ε˜Wε),
we expect the limit density Nto satisfy the following parabolic-elliptic Keller-Segel system :
/braceleftbigg
∂tN−∆(P(N)) = div( N∇V)
−∆V=N−̺(1.5)
supplemented with the initial data lim
ε→0˜̺ε
0.
Our second aim is to justify the passage to the limit when ε→0of the Euler-Poisson system towards the
parabolic-elliptic Keller-Segel system.
Recall that (1.5) is a model for describing the evolution of d ensityN=N(t,x)∈R+of a biological
population under the influence of a chemical agent with conce ntration V=V(t,x)∈Rd. Chemotaxis are an
important means of cell communication. How cells are arrang ed and organized is determined by communi-
cation by chemical signals. Studying such a biological proc ess is important because it has repercussions in
many branches of medicine such as cancer [8], [32], embryoni c development [30] or vascular networks [7], [20].
The previous system is famous in biology and comes from E.F Ke ller and L.A Segel in [26]. This basic model
was used to describe the collective movement of bacteria pos sibly leading to cell aggregation by chemotactic
effect. We refer to the articles [25] and [21] for more details and information about the different Keller-Segel
models studied since the 1970s.
Our aim here is to demonstrate that (1.5) may be obtained from the Euler-Poisson system with damping
when the parameter εtends to 0.This question has been addressed in [28] on the torus case and Sobolev
spaces in a situation where the potential satisfies a less sin gular equation : the author justifies the passage
to the limit for regular periodic solutions. A lot of article s justify another limit: the passage from the
parabolic-parabolic Keller-Segel system to the parabolic -elliptic Keller-Segel system (see e.g. the paper [29]
by P-G. Lemarié-Rieusset for the case of Morrey spaces).
In the same spirit as this article, T. Crin-Barat, Q. He and L. S hou in [13] justified the high relaxation
asymptotics for the (less singular) parabolic-parabolic K eller-Segel system (the potential satisfies the equation
−∆V+bV=aNwitha,b >0) : this other system comes from the system (HPC) (hyperbolic -parabolic-
chemotaxis) which is a damped isentropic compressible Eule r system with a potential satisfying an elliptical
equation. In comparison with what is done here, T. Crin-Barat et alused a parabolic approach to justify
their passage to the limit. Here, we have to handle the more si ngular case where the limit system is parabolic-
elliptic.
2.Main results and sketch of the proof
In this section, we will first present and motivate the functi onal spaces used. Secondly we will state
the results and the sketch of the proofs about the well-posed ness behavior of Euler-Poisson system and the
justification of the passage to the limit to parabolic-ellip tic Keller-Segel system.
2.1.Functional spaces.
Before describing the main results of this article, we introd uce the different notations and definitions used
throughout this document. We will designate by C >0an independent constant of εand time, and f/lessorsimilarg
will mean f≤Cg. For any Banach space Xand all functions f,g∈X, we denote /ba∇dbl(f,g)/ba∇dblX:=/ba∇dblf/ba∇dblX+/ba∇dblg/ba∇dblX.
We designate by L2(R+;X)the set of measurable functions f: [0,+∞[→Xsuch that t/mapsto→ /ba∇dblf(t)/ba∇dblXbelongs
toL2(R+)and write /ba∇dbl·/ba∇dblL2(R+;X):=/ba∇dbl·/ba∇dblL2(X).
In this article we will use a decomposition in Fourier space, called the homogeneous Littlewood-Paley
decomposition . To this end, we introduce a regular non-negative function ϕonRdwith support in thePARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 3
annulus{ξ∈Rd,3/4≤ |ξ| ≤8/3}and satisfying
/summationdisplay
j∈Zϕ(2−jξ) = 1, ξ/ne}ationslash= 0.
For allj∈Z, the dyadic homogeneous blocks ˙∆jand the low frequency truncation operator ˙Sjare defined
by
˙∆j:=F−1(φ(2−j·)Fu),˙Sju:=F−1(χ(2−j·)Fu),
whereFandF−1designate respectively the Fourier transform and its inver se. From now on, we will use
the following shorter notation :
uj:=˙∆ju.
LetS′
hthe set of tempered distributions uonRdsuch that lim
j→−∞/ba∇dbl˙Sju/ba∇dblL∞= 0. We have then :
u=/summationdisplay
j∈Zuj∈ S′,˙Sju=/summationdisplay
j′≤j−1uj′,∀u∈ S′
h.
Homogeneous Besov spaces ˙Bs
p,rfor allp,r∈[1,+∞]ands∈Rare defined by:
˙Bs
p,r:=/braceleftBig
u∈ S′
h/vextendsingle/vextendsingle/vextendsingle/ba∇dblu/ba∇dbl˙Bsp,r:=/ba∇dbl{2js/ba∇dbluj/ba∇dblLp}j∈Z/ba∇dbllr<∞/bracerightBig
·
In this article, we will only consider Besov spaces of indices p= 2andr= 1.
As we will need to restrict our Besov norms in specific regions o f low and high frequencies, we introduce
the following notations :
/ba∇dblu/ba∇dblh
˙Bs
2,1:=/summationdisplay
j≥−12js/ba∇dbluj/ba∇dblL2,/ba∇dblu/ba∇dbll
˙Bs
2,1:=/summationdisplay
j≤−12js/ba∇dbluj/ba∇dblL2,/ba∇dblu/ba∇dbll−,ε
˙Bs
2,1:=/summationdisplay
j≤−1
2j≤ε2js/ba∇dbluj/ba∇dblL2,
/ba∇dblu/ba∇dbll+,ε
˙Bs
2,1:=/summationdisplay
j≤−1
2j≥ε2js/ba∇dbluj/ba∇dblL2.
We put in the appendix several results about Besov spaces: the reader may refer to Chapter 2 of [4] for more
information on this topic.
2.2.Main results, sketch of the proofs and article organization .
The starting point is that, formally, (1.1) rewrites :
/braceleftbigg∂tρε+div(ρεvε) = 0,
∂t(ρεvε)+div(ρεvε⊗vε)+∇(P(ρε))+ε−1ρεvε=−ρε∇(−∆)−1(ρε−ρ).(2.1)
By the variable change
(ρ,v) = (ρε,vε)(εt,εx), (2.2)
we get the following system: :
/braceleftbigg∂tρ+div(ρv) = 0,
∂t(ρv)+div(ρv⊗v)+∇(P(ρ))+ρv=−ε2ρ∇(−∆)−1(ρ−ρ).(2.3)
For the study of this system, the key is to obtain suitable glo bal-in-time a priori estimates. Then, very
classical arguments lead to existence and uniqueness of glo bal solutions (see Theorem 2.1 below).
Obtaining estimates will be strongly inspired by the work do ne by Crin-Barat et al in [11] where we
consider the classic compressible Euler system. As the syst em we are studying is very close to a partially
dissipative system, we will follow [16] so as to obtain a prio ri estimates : the standard energy method is not
enough to conclude because we do not obtain all the informati on through this (mainly at the low frequencies)
on the dissipated part. Hence, we must better use the couplin g, exhibiting a combination of unknowns (the
"purely" damped mode) that will allow us to recover the whole dissipation.
For the estimates, therefore, we follow the strategy propos ed by Danchin in [16]: first (second part of
the third section), we analyze how to obtain the estimates fo r the linear system in Besov space ˙Bs
2,1where4 VALENTIN LEMARIÉ
s∈Ris any at the moment. For high frequencies, we follow step by s tep the approach of [16], by putting
the negligible term containing ε2∇∆−1̺, in order to obtain exponential decay. For low frequencies, the task
is slightly more complicated because the latter term is no mo re negligible : we lose the symmetry condition
and we can not apply Danchin’s method. But by looking at the sys tem differently (essentially by changing
vto another variable dependent on ε), we obtain a symmetrical system for which we find the associa ted
estimates (go back to the initial unknowns to get the estimat es). We then see the appearance of two regimes
within the low frequencies that will be named respectively v ery low frequencies (frequencies below ε) and
medium frequencies (frequencies of magnitude between εand1). To recover the full dissipative properties
of the system, we introduce the damped mode W:=−∂tv.Then, from the estimate satisfied by W,we will
improve the estimate for the low frequency part of v.
Let us next explain how to choose the indices of regularity fo r the solution. Since our system is very
similar to Euler’s without coupling, we make the same choice as in [11] : s=d
2for the medium frequencies
ands=d
2+1for the high frequencies. We taked
2−1for the very low frequencies of the density in view of
the estimate obtained for the linear system.
For proving a priori estimates for (2.3), we have to take into account non-linear terms now. To do this, for
high frequencies, we again follow the method described by [1 6] with a precise analysis of the system using
commutators.
Concerning the low frequencies, we need some information on the damped mode W=−∂tv, before
studying the estimates of the density and velocity. By combin ing the obtained inequalities,we deduce the
desired a priori estimates.
Before stating our main results, we provide the reader with th e following diagram so as to clarify the
notations of the theorem:
1•
ε−1•
|ξ|l l+,1, ε−1h, ε−1
l−, ε−1
Our first result states the global well-posedness of the Eule r Poisson system with high relaxation. We
point out an explicit dependence of the estimates with respe ct to the damping parameter, that we believe
to be optimal:
Theorem 2.1. Letε >0. There exists a positive constante αsuch that for all ε≤1/2and initial data
Zε
0= (̺ε
0−̺,vε
0)∈/parenleftbigg
˙Bd
2−1
2,1∩˙Bd
2+1
2,1/parenrightbigg
×/parenleftbigg
˙Bd
2
2,1∩˙Bd
2+1
2,1/parenrightbigg
satisfying :
Zε
0:=/ba∇dbl̺ε
0−̺/ba∇dbll
˙Bd
2−1
2,1+/ba∇dbl̺ε
0−̺/ba∇dbll+,1, ε−1
˙Bd
2
2,1+/ba∇dblvε
0/ba∇dbll−, ε−1
˙Bd
2
2,1+ε/ba∇dbl(̺ε
0−̺,vε
0)/ba∇dblh, ε−1
˙Bd
2+1
2,1≤α,
System (1.1)supplemented with the initial data (̺ε
0−¯ρ,vε
0)admits a unique global-in-time solution Zε=
(̺ε−¯ρ,vε)in the set
E:=/braceleftbigg
(̺ε−̺,vε)/vextendsingle/vextendsingle/vextendsingle/vextendsingle(̺ε−̺)l∈ Cb(R+:˙Bd
2−1
2,1), ε(̺ε−̺)l∈L1(R+;˙Bd
2−1
2,1),
(̺ε−̺)l+,1, ε−1∈ Cb(R+:˙Bd
2
2,1), ε(̺ε−̺)l+,1, ε−1∈L1(R+;˙Bd
2+1
2,1),
(vε)l−, ε−1∈ Cb(R+;˙Bd
2
2,1),(vε)l∈L1(R+;˙Bd
2
2,1),(vε)l+,1, ε−1∈L1(R+;˙Bd
2+1
2,1),
(̺ε−̺,vε)h,ε−1∈ Cb(R+;˙Bd
2−1
2,1)∩L1(R+;˙Bd
2+1
2,1), wε∈ Cb(R+;˙Bd
2
2,1)∩L1(R+;˙Bd
2
2,1)/bracerightbigg
where we have denoted wε:=ε∇(P(̺ε))
̺ε+vε+ε∇(−∆)−1(̺ε−̺).PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 5
Moreover, we have the following inequality :
Zε(t)≤CZε
0
where
Zε(t):=/ba∇dbl̺ε−̺/ba∇dbll
L∞/parenleftbig
˙Bd
2−1
2,1/parenrightbig+/ba∇dbl̺ε−̺/ba∇dbll+,1, ε−1
L∞/parenleftbig
˙Bd
2
2,1/parenrightbig+/ba∇dblvε/ba∇dbll−, ε−1
L∞/parenleftbig
˙Bd
2
2,1/parenrightbig+ε/ba∇dbl(̺ε−̺,v)/ba∇dblh, ε−1
L∞/parenleftbig
˙Bd
2+1
2,1/parenrightbig
+/ba∇dblε(̺ε−̺)/ba∇dbll
L1/parenleftbig
˙Bd
2−1
2,1/parenrightbig+/ba∇dblvε/ba∇dbll
L1/parenleftbig
˙Bd
2
2,1/parenrightbig+ε/ba∇dbl̺ε−̺/ba∇dbll+,1, ε−1
L1/parenleftbig
˙Bd
2+2
2,1/parenrightbig+/ba∇dblvε/ba∇dbll+,1, ε−1
L1/parenleftbig
˙Bd
2+1
2,1/parenrightbig
+/ba∇dbl(̺ε−̺,v)/ba∇dblh,ε−1
L1/parenleftbig
˙Bd
2+1
2,1/parenrightbig+/ba∇dblwε/ba∇dbl
L∞/parenleftbig
˙Bd
2
2,1/parenrightbig+ε−1/ba∇dblwε/ba∇dbl
L1/parenleftbig
˙Bd
2
2,1/parenrightbig.
Remark 2.1. In summary, for large enough damping and small enough initia l data, we get a global solution
to the Euler-Poisson system. The demonstration will allow u s to understand the choice of regularity indices
for the different frequency groups. In addition, the control of the damped mode wεin the above theorem which
enable us to obtain on the one hand the a priori estimate of the theorem and on the other hand the following
theorem on the singular limit of (1.1)when the damping coefficient ε−1tends to infinity.
By the change of variable (1.2) and the existence theorem on Eu ler-Poisson, we then have ˜Wε=O(ε)
inL1(R+;˙Bd
2
2,1)where˜Wεis defined by (1.4) and we need to look at the solutions of (1.3) in the following
functional space :
˜E:=/braceleftbigg
(̺ε−̺,vε)/vextendsingle/vextendsingle/vextendsingle/vextendsingle(̺ε−̺)l∈ Cb(R+:˙Bd
2−1
2,1), ε(̺ε−̺)l∈L1(R+;˙Bd
2−1
2,1),
(̺ε−̺)l+,1, ε−1∈ Cb(R+:˙Bd
2
2,1), ε(̺ε−̺)l∈L1(R+;˙Bd
2+1
2,1),(̺ε−̺)l+,1, ε−1∈L1(R+;˙Bd
2+1
2,1),
ε(vε)l−, ε−1∈ Cb(R+;˙Bd
2
2,1),(vε)l∈L1(R+;˙Bd
2
2,1),(vε)l+,1, ε−1∈L1(R+;˙Bd
2+1
2,1),
(̺ε−̺,vε)h,ε−1∈ Cb(R+;˙Bd
2−1
2,1)∩L1(R+;˙Bd
2+1
2,1), wε∈ Cb(R+;˙Bd
2
2,1)∩L1(R+;˙Bd
2
2,1)/bracerightbigg
·
By studying the system satisfied by the difference between the s olution of Euler-Poisson and that of Keller-
Segel parabolic-elliptic and thanks to the previous theore m, we manage to justify that the solutions of the
Euler-Poisson system that has been scaled back will converg e to the solutions of the Keller-Segel system
towards the following theorem :
Theorem 2.2. We consider (1.3)forε >0small enough. Then, there exists a positive constant α(inde-
pendent of ε) such that for all initial data N0∈˙Bd
2−1
2,1∩˙Bd
2
2,1for(1.5)and˜Zε
0∈˜Efor(1.3)satisfying
/ba∇dblN0/ba∇dbl˙Bd
2
2,1∩˙Bd
2−1
2,1≤α, (2.4)
˜Zε
0:=/ba∇dbl˜̺ε
0−̺/ba∇dbll
˙Bd
2−1
2,1+/ba∇dbl˜̺ε
0−̺/ba∇dbll+,1, ε−1
˙Bd
2
2,1+ε/ba∇dbl˜vε
0/ba∇dbll, ε−1
˙Bd
2
2,1+ε/ba∇dbl/parenleftbig
˜̺ε
0−̺,ε˜vε
0/parenrightbig
/ba∇dblh, ε−1
˙Bd
2+1
2,1≤α,
the system (1.5)admits an unique solution Nin the space
Cb/parenleftbig
R+;˙Bd
2−2
2,1∩˙Bd
2
2,1/parenrightbig
∩L1/parenleftbig
R+;˙Bd
2−2
2,1∩˙Bd
2
2,1/parenrightbig
,
satisfying for all t≥0,
/ba∇dblN(t)−̺/ba∇dbl˙Bd
2−1
2,1∩˙Bd
2
2,1+/integraldisplayt
0/ba∇dblN−̺/ba∇dbl˙Bd
2+2
2,1∩˙Bd
2+1
2,1dτ≤C/ba∇dblN0−̺/ba∇dbl˙Bd
2−1
2,1∩˙Bd
2
2,1, (2.5)
and the system (1.3)has an unique global-in-time solution ˜Zεin˜Esuch that
˜Z(t)≤C˜Z06 VALENTIN LEMARIÉ
where
(2.6)˜Zε(t):=/ba∇dbl˜̺ε−̺/ba∇dbll
L∞/parenleftbig
˙Bd
2−1
2,1/parenrightbig+/ba∇dbl˜̺ε−̺/ba∇dbll+,1, ε−1
L∞/parenleftbig
˙Bd
2
2,1/parenrightbig+ε/ba∇dbl˜vε/ba∇dbll, ε−1
L∞/parenleftbig
˙Bd
2
2,1/parenrightbig+ε/ba∇dbl(˜̺ε−̺,ε˜vε)/ba∇dblh, ε−1
L∞/parenleftbig
˙Bd
2+1
2,1/parenrightbig
+/ba∇dbl˜̺ε−̺/ba∇dbll
L1/parenleftbig
˙Bd
2−1
2,1/parenrightbig+/ba∇dbl˜vε/ba∇dbll
L1/parenleftbig
˙Bd
2
2,1/parenrightbig+/ba∇dbl˜̺ε−̺/ba∇dbll+,1, ε−1
L1/parenleftbig
˙Bd
2+2
2,1/parenrightbig+/ba∇dbl˜vε/ba∇dbll+,1, ε−1
L1/parenleftbig
˙Bd
2+1
2,1/parenrightbig
+ε−1/ba∇dbl(˜̺ε−̺,ε˜vε)/ba∇dblh,ε−1
L1/parenleftbig
˙Bd
2+1
2,1/parenrightbig+ε/ba∇dbl˜Wε/ba∇dbl
L∞/parenleftbig
˙Bd
2
2,1/parenrightbig+ε−1/ba∇dbl˜Wε/ba∇dbl
L1/parenleftbig
˙Bd
2
2,1/parenrightbig
where˜Wεhas been defined in (1.4).
Moreover, if, /ba∇dblN0−˜̺ε
0/ba∇dbl˙Bd
2−1
2,1≤Cε, then we have
/ba∇dblN−˜̺ε/ba∇dbl
L∞/parenleftbig
R+;˙Bd
2−1
2,1/parenrightbig+/ba∇dblN−˜̺ε/ba∇dblh
L1/parenleftbig
R+;˙Bd
2+1
2,1/parenrightbig+/ba∇dblN−˜̺ε/ba∇dbll
L1/parenleftbig
R+;˙Bd
2
2,1/parenrightbig≤Cε.
Remark 2.2. This theorem ensures that the solution densities of the Eule r-Poisson system converge (in the
space highlighted in the theorem) towards the only solution of the Keller-Segel system. For the velocity limit,
we can take the limit in (1.4).
3.Study of the Euler-Poisson system with damping
This section is devoted to the proof of theorem 2.1.
3.1.Study of the linearized system.
Linearizing (2.3) around (̺,0)yields the following system :
/braceleftbigg∂t̺+divv= 0
∂tv+P′(ρ)∇̺+v=−ε2∇(−∆)−1̺.
Performing the change of unknown ˜ρ:=̺(t,P′(ρ)x)reduces the study to the case P′(ρ) = 1 (after
changing εintoε′:=ε/radicalbig
P′(ρ)). Hence we focus on the following linear system :
/braceleftbigg∂t̺+divv= 0,
∂tv+∇̺+v=−ε2∇(−∆)−1̺.(3.1)
By the Fourier transform, we get :
d
dt/parenleftbigg
/hatwide̺
/hatwidev/parenrightbigg
+/parenleftBigg0 iξ
i/parenleftBig
1+ε2|ξ|−2/parenrightBig
ξtId/parenrightBigg/parenleftbigg
/hatwide̺
/hatwidev/parenrightbigg
= 0.
The eigenvalues of the matrix of this system are:
•1is with multiplicity d−1,
•
λ±(ξ) =1
2(1±/radicalbig
1−4(|ξ|2+ε2))if|ξ|2+ε2<1
4
λ±(ξ) =1
2(1±i/radicalbig
4(|ξ|2+ε2)−1)elseare the two remaining eigenvalues.
For the high frequencies, we thus have :
Re/parenleftbig
λ±(ξ)/parenrightbig
= 1.
As for partially dissipative hyperbolic systems, we expect exponential decay for high frequencies.
For the low frequencies, in the case ε <1/2,two regimes have to be considered: a very low frequency
regime (i.e for |ξ| ≤/radicalbig
1/4−ε2) and another for the medium frequencies (for |ξ| ≥/radicalbig
1/4−ε2).
In what follows, we prove a priori estimates for a smooth enou gh solution (̺,v)of (3.1).PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 7
3.1.1. High frequency behavior: case |ξ| ≥1/2.
Proposition 3.1. LetZ= (̺,v)and/ba∇dblZ/ba∇dblh
˙Bs
2,1:=/summationtext
j≥−12js/ba∇dbl˙∆jZ/ba∇dblL2. Then we have :
/ba∇dblZ(t)/ba∇dblh
˙Bs
2,1+/integraldisplayt
0/ba∇dblZ/ba∇dblh
˙Bs
2,1dτ/lessorsimilar/ba∇dblZ0/ba∇dblh
˙Bs
2,1.
Proof. Since the classical energy method does not provide enough in formation, we consider, as in [16], the
evolution equation of ∇̺·v, namely
∂t(∇̺·v) =∇∂t̺·v+∇̺·∂tv=−∇divv·v−∇̺·/parenleftbig
∇̺+v+ε2∇(−∆)−1̺/parenrightbig
·
We then integrate on Rdand by integration by parts :/integraldisplay
Rd∂t(∇̺·v)dx+/integraldisplay
Rd∇̺·vdx=/ba∇dbldivv/ba∇dbl2
L2−/ba∇dbl∇̺/ba∇dbl2
L2−ε2/ba∇dbl̺/ba∇dbl2
L2.
By taking the gradient in (3.1) and taking the scalar product w ith∇̺(respectively ∇v), we get :
1
2d
dt/ba∇dbl∇̺/ba∇dbl2
L2−/integraldisplay
Rd∇∇̺·∇vdx= 0
1
2d
dt/ba∇dbl∇v/ba∇dbl2
L2+/integraldisplay
Rd∇∇̺·∇vdx+/ba∇dbl∇v/ba∇dblL2=−ε2/integraldisplayt
0∇∇(−∆)−1̺·∇vdx.
These identities are also true for (̺j,vj)(where̺j=˙∆j̺andvj=˙∆jv) since the studied system is linear
with constant coefficients. To study high frequencies, we wil l further assume that j∈N.
We then set Lj:=1
2/ba∇dbl∇̺j/ba∇dbl2
L2+1
2/ba∇dbl∇vj/ba∇dbl2
L2+1
4/integraldisplay
Rd∇̺j·vjdxwhich verifies :
3
8/parenleftbig
/ba∇dbl∇̺j/ba∇dbl2
L2+/ba∇dbl∇vj/ba∇dbl2
L2/parenrightbig
≤ Lj≤5
8/parenleftbig
/ba∇dbl∇̺j/ba∇dbl2
L2+/ba∇dbl∇vj/ba∇dbl2
L2/parenrightbig
because for j≥0,/ba∇dblvj/ba∇dblL2≤1
2/ba∇dbl∇vj/ba∇dblL2by Bernstein’s inequality.
Then we have :
d
dtLj+/ba∇dbl∇vj/ba∇dbl2
L2+1
4/integraldisplay
Rd∇̺j·vjdx−1
4/ba∇dbldivv/ba∇dbl2
L2+1
4/ba∇dbl∇̺j/ba∇dbl2
L2≤ε2/ba∇dbl̺j/ba∇dblL2/ba∇dbl∇vj/ba∇dblL2+ε2
4/ba∇dbl̺j/ba∇dbl2
L2
≤2ε2
4/ba∇dbl∇̺j/ba∇dbl2
L2+2ε2
4/ba∇dbl∇vj/ba∇dbl2
L2.
By the inequalities of Cauchy-Schwarz and Bernstein, we have :
/ba∇dbl∇vj/ba∇dbl2
L2+1
4/integraldisplay
Rd∇̺j·vjdx−1
4/ba∇dbldivv/ba∇dbl2
L2+1
4/ba∇dbl∇̺j/ba∇dbl2
L2−2ε2
4/ba∇dbl∇̺j/ba∇dbl2
L2+2ε2
4/ba∇dbl∇vj/ba∇dbl2
L2
≥ −1
4/ba∇dbl∇̺j/ba∇dblL2/ba∇dblvj/ba∇dblL2+3−2ε2
4/ba∇dbl∇vj/ba∇dbl2
L2+1−2ε2
4/ba∇dbl∇̺j/ba∇dbl2
L2
≥4−4ε2
8/ba∇dbl∇vj/ba∇dbl2
L2+1−4ε2
8/ba∇dbl∇̺j/ba∇dbl2
L2
≥1−4ε2
5Lj.
By the inequality on Lj, we thus get :
d
dtLj+1−4ε2
5Lj≤0.
Forε≤1
4, so we get :
/ba∇dbl∇(̺j,vj)(t)/ba∇dblL2+3
20/integraldisplayt
0/ba∇dbl∇(̺j,vj)/ba∇dblL2dτ≤ /ba∇dbl∇(̺j,vj)(0)/ba∇dblL2.
We multiply by 2j(s−1)and we sum up on j∈N, we get then the inequality announced in the proposition
(the case j=−1that presents no particular difficulty can be studied separat ely). /square8 VALENTIN LEMARIÉ
3.1.2. Low frequency behavior. We have to proceed differently since the term −ε2∇(−∆)−1is of order −1.
The goal will be here to understand his role.
Let us use the Helmholtz decomposition to highlight the two b ehaviors corresponding respectively to the
solenoid and the irrotational part of v:
v=Pv+Qv wherePandQverifyP= Id+∇(−∆)−1divandQ=−∇(−∆)−1div.
First, let us look at the equation verified by Pv. By applying Pto the second equation of the system
(3.1) and using that P∇= 0, we get :
∂tPv+Pv= 0.
By applying ˙∆jand taking the scalar product with Pvj, we have:
d
dt/ba∇dblPvj/ba∇dbl2
L2+/ba∇dblPvj/ba∇dbl2
L2= 0.
Then we have by (A.1):
/ba∇dblPvj(t)/ba∇dblL2+/integraldisplayt
0/ba∇dblPvj/ba∇dblL2dτ≤ /ba∇dblPv0,j/ba∇dblL2. (3.2)
Now let us look at the system satisfied by the divergence of vand̺. We then set u= divv. By taking the
divergence in the second equation of (3.1), we get the follow ing2×2system :
/braceleftbigg
∂t̺+u= 0,
∂tu+∆̺+u=ε2̺.
In Fourier variables, it becomes :
/braceleftbigg
∂t/hatwide̺+/hatwideu= 0,
∂t/hatwideu−(|ξ|2+ε2)/hatwide̺+/hatwideu= 0.
Setting/hatwidew:=1/radicalbig
|ξ|2+ε2/hatwideuyields :
∂t˜Z+A(D)˜Z+B(D)˜Z= 0 (3.3)
where ˜Z=/parenleftbigg
̺
w/parenrightbigg
, A(ξ) =/parenleftbigg
0/radicalbig
|ξ|2+ε2
−/radicalbig
|ξ|2+ε20/parenrightbigg
, B(ξ) =/parenleftbigg
0 0
0 1/parenrightbigg
·
Let us build by hand a Lyapunov functional, allowing us to rec over the dissipative properties of the system
on˜Z. By taking the scalar product with ˜Zin (3.3) and looking at the time derivative of Re(/hatwide̺·/hatwidew), we have:
1
2d
dt|/hatwide̺|2+/radicalbig
|ξ|2+ε2Re(/hatwide̺·/hatwidew) = 0,
1
2d
dt|/hatwidew|2−/radicalbig
|ξ|2+ε2Re(/hatwide̺·/hatwidew)+|/hatwidew|2= 0,
d
dtRe(/hatwide̺·/hatwidew) =−/radicalbig
|ξ|2+ε2|/hatwidew|2+/radicalbig
|ξ|2+ε2|/hatwide̺|2−Re(/hatwide̺·/hatwidew).
With these equations, we can easily deduce the Lyapunov func tional and the equation it verifies:
d
dt/parenleftBig
|/hatwide̺|2+|/hatwidew|2−/radicalbig
|ξ|2+ε2Re(/hatwide̺·/hatwidew)/parenrightBig
+(2−(|ξ|2+ε2))|/hatwidew|2+(|ξ|2+ε2)|/hatwide̺|2−/radicalbig
|ξ|2+ε2Re(/hatwide̺·/hatwidew) = 0.PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 9
Yet
/radicalbig
|ξ|2+ε2Re(/hatwide̺·/hatwidew)≤|ξ|2+ε2
2|/hatwide̺|2+|/hatwidew|2
2·
Thus for |ξ|2+ε2≤1, we have:
d
dt/parenleftBig
|/hatwide̺|2+|/hatwidew|2−/radicalbig
|ξ|2+ε2Re(/hatwide̺·/hatwidew)/parenrightBig
+|ξ|2+ε2
2|/hatwidew|2+|ξ|2+ε2
2|/hatwide̺|2≤0.
We have then:
|(/hatwide̺,/hatwidew)(t,ξ)| ≤2e−1
8(|ξ|2+ε2)t|(/hatwide̺0,/hatwidew0)(ξ)|. (3.4)
So we have after spectral localization of the system by means of˙∆jwithj≤ −1and forεsmall enough:
/ba∇dbl(̺j,wj)(t)/ba∇dblL2+1
8(22j+ε2)/integraldisplayt
0/ba∇dbl(̺j,wj)/ba∇dblL2dτ≤2/ba∇dbl(̺0,j,w0,j)/ba∇dblL2. (3.5)
We have a priori estimate on ˜Zj. A similar estimate has yet to be obtained for (̺j,uj).
By definition of wand by multiplying (3.5) by/radicalbig
22j+ε2, we have as an estimate :
/ba∇dbl(/radicalbig
22j+ε2̺j,uj)(t)/ba∇dblL2+/parenleftbig
22j+ε2/parenrightbig/integraldisplayt
0/ba∇dbl(/radicalbig
22j+ε2̺j,uj)/ba∇dblL2dτ/lessorsimilar/ba∇dbl(/radicalbig
22j+ε2̺0,j,u0,j)/ba∇dblL2.
These estimates reveal two distinct regimes within the low f requencies : |ξ| ≤ε("very low frequencies")
and|ξ| ≥ε("medium frequencies"). For very low frequencies, we have 22j+ε2≃ε2, and thus
/ba∇dbl(ε˜̺j,uj)(t)/ba∇dblL2+ε2/integraldisplayt
0/ba∇dbl(ε˜̺j,uj)/ba∇dblL2dτ/lessorsimilar/ba∇dbl(ε˜̺0,j,u0,j)/ba∇dblL2withu= divv,
and for the medium frequencies, since 22j+ε2≃22j,
/ba∇dbl(˜̺j,vj)(t)/ba∇dblL2+22j/integraldisplayt
0/ba∇dbl(˜̺j,vj)/ba∇dblL2dτ/lessorsimilar/ba∇dbl(˜̺0,j,v0,j)/ba∇dblL2.
Consequently, if we denote :
/ba∇dblZ/ba∇dbll−,ε
˙Bs
2,1:=/summationdisplay
j≤−1
2j≤ε2js/ba∇dblZj/ba∇dblL2et/ba∇dblZ/ba∇dbll+,ε
˙Bs
2,1:=/summationdisplay
j≤−1
2j≥ε2js/ba∇dblZj/ba∇dblL2. (3.6)
then, we obtain:
/ba∇dbl(ε˜̺,div(v))(t)/ba∇dbll−,ε
˙Bs
2,1+ε2/integraldisplayt
0/ba∇dbl(ε˜̺,div(v))/ba∇dbll−,ε
˙Bs
2,1dτ/lessorsimilar/ba∇dbl(ε˜̺0,div(v0))/ba∇dbll−,ε
˙Bs
2,1
/ba∇dbl(˜̺,v)(t)/ba∇dbll+,ε
˙Bs
2,1+/integraldisplayt
0/ba∇dbl(˜̺,v)/ba∇dbll+,ε
˙Bs+2
2,1dτ/lessorsimilar/ba∇dbl(˜̺0,v0)/ba∇dbll+,ε
˙Bs
2,1,
/ba∇dblPv(t)/ba∇dbll
˙Bs
2,1+/integraldisplayt
0/ba∇dblPv/ba∇dbll
˙Bs
2,1dτ/lessorsimilar/ba∇dblPv0/ba∇dbll
˙Bs
2,1(incompressible part) .
3.1.3. Damped mode and improvement of estimates for v.
Like in [16], we consider the damped mode :
˜W:=−∂tv=∇̺+ε2∇(−∆)−1̺+v.
We have :
∂t˜W+˜W=−(∇+ε2∇(−∆)−1)divv.10 VALENTIN LEMARIÉ
By applying the localization operator ˙∆j, taking the scalar product with ˜Wj, multiplying by 2js, summing
up onjcorresponding to very low and medium frequencies and applyi ng the lemma A.1, we get :
/ba∇dbl˜W(t)/ba∇dbll−,ε
˙Bs
2,1+/integraldisplayt
0/ba∇dbl˜W/ba∇dbll−,ε
˙Bs
2,1dτ/lessorsimilar/ba∇dbl˜W0/ba∇dbll−,ε
˙Bs
2,1+/integraldisplayt
0ε2/ba∇dblv/ba∇dbll−,ε
˙Bs
2,1dτ
/ba∇dbl˜W(t)/ba∇dbll+,ε
˙Bs
2,1+/integraldisplayt
0/ba∇dbl˜W/ba∇dbll+,ε
˙Bs
2,1dτ/lessorsimilar/ba∇dbl˜W0/ba∇dbll+,ε
˙Bs
2,1+/integraldisplayt
0/ba∇dblv/ba∇dbll+,ε
˙Bs+2
2,1dτ.
In particular, we have :
/ba∇dbl˜W0/ba∇dbll−,ε
˙Bs
2,1/lessorsimilarε2/ba∇dbl̺0/ba∇dbll−,ε
˙Bs−1
2,1+/ba∇dblv0/ba∇dbll−,ε
˙Bs
2,1
/ba∇dbl˜W0/ba∇dbll+,ε
˙Bs
2,1/lessorsimilar/ba∇dbl̺0/ba∇dbll+,ε
˙Bs+1
2,1+/ba∇dblv0/ba∇dbll+,ε
˙Bs
2,1.
By using the fact v=˜W−∇̺−ε2∇(−∆)−1̺and the estimates on the low frequencies obtained previousl y,
we have :
/ba∇dblv(t)/ba∇dbll−,ε
˙Bs
2,1≤ /ba∇dbl˜W(t)/ba∇dbll−,ε
˙Bs
2,1+ε2/ba∇dbl̺(t)/ba∇dbll−,ε
˙Bs−1
2,1/lessorsimilarε2/ba∇dbl̺0/ba∇dbll−,ε
˙Bs−1
2,1+/ba∇dblv0/ba∇dbll−,ε
˙Bs
2,1+/integraldisplayt
0ε2/ba∇dblv/ba∇dbll−,ε
˙Bs
2,1dτ
/integraldisplayt
0ε/ba∇dblv/ba∇dbll−,ε
˙Bs
2,1dτ≤/integraldisplayt
0ε/ba∇dbl˜W/ba∇dbll−,ε
˙Bs
2,1dτ+/integraldisplayt
0ε3/ba∇dbl̺/ba∇dbll−,ε
˙Bs−1
2,1dτ/lessorsimilarε/ba∇dbl̺0/ba∇dbll−,ε
˙Bs−1
2,1+/ba∇dblv0/ba∇dbll−,ε
˙Bs
2,1+/integraldisplayt
0ε2/ba∇dblv/ba∇dbll−,ε
˙Bs
2,1dτ
/ba∇dblv(t)/ba∇dbll+,ε
˙Bs
2,1/lessorsimilar/ba∇dbl˜W(t)/ba∇dbll+,ε
˙Bs
2,1+/ba∇dbl̺(t)/ba∇dbll+,ε
˙Bs+1
2,1/lessorsimilar/ba∇dbl̺0/ba∇dbll+,ε
˙Bs+1
2,1+/ba∇dblv0/ba∇dbll+,ε
˙Bs
2,1+/integraldisplayt
0/ba∇dblv/ba∇dbll+,ε
˙Bs+2
2,1dτ
/integraldisplayt
0/ba∇dblv/ba∇dbll+,ε
˙Bs+1
2,1dτ≤/integraldisplayt
0/ba∇dbl˜W/ba∇dbll+,ε
˙Bs+1
2,1dτ+/integraldisplayt
0/ba∇dbl̺/ba∇dbll+,ε
˙Bs+2
2,1dτ/lessorsimilar/ba∇dbl̺0/ba∇dbll+,ε
˙Bs
2,1+/ba∇dblv0/ba∇dbll+,ε
˙Bs
2,1+/integraldisplayt
0/ba∇dblv/ba∇dbll+,ε
˙Bs+2
2,1dτ
By summing up these inequalities and noticing that some terms in the right-hand side are negligible compared
to those of the left-hand side, we get :
/ba∇dblv(t)/ba∇dbll−,ε
˙Bs
2,1+/integraldisplayt
0ε/ba∇dblv/ba∇dbll−,ε
˙Bs
2,1dτ/lessorsimilarε/ba∇dbl̺0/ba∇dbll−,ε
˙Bs−1
2,1+/ba∇dblv0/ba∇dbll−,ε
˙Bs
2,1
/ba∇dblv(t)/ba∇dbll+,ε
˙Bs
2,1+/integraldisplayt
0/ba∇dblv/ba∇dbll+,ε
˙Bs+1
2,1dτ/lessorsimilar/ba∇dbl̺0/ba∇dbll+,ε
˙Bs
2,1+/ba∇dblv0/ba∇dbll+,ε
˙Bs
2,1
3.2.A priori estimates for the non-linear system.
Let us now prove similar estimates for the non-linear system . To do this, we use Makino symmetrization,
which consists in setting
c:=(γA)1
2
˜γ̺˜γwith˜γ=γ−1
2· (3.7)
After this change of unknown, we obtain:
∂tc+v·∇c+ ˜γcdiv(v) = 0
∂tv+v·∇v+ ˜γc∇c+v=−ε2∇(−∆)−1/parenleftBigg/parenleftbigg
˜γ
(γA)1
2/parenrightbigg1
˜γ
c1
˜γ−̺/parenrightBigg
.(3.8)
We setf(x):=/parenleftBigg
˜γ
(γA)1
2/parenrightBigg1
˜γ
x1
˜γ.So we have by Taylor’s formula with integral rest :
/parenleftBigg
˜γ
(γA)1
2/parenrightBigg1
˜γ/parenleftBig
c1
˜γ−c1
˜γ/parenrightBig
=f(˜c+c)−f(c) = ˜cf′(c)+/integraldisplayc
c(c−y)f′′(y)dy.
Let us set F(˜c) =/integraldisplayc
c(c−y)f′′(y)dyandG(˜c) = ˜cf′(c) +/integraldisplayc
c(c−y)f′′(y)dywhich vanishes at 0 where
c:=(γA)1
2
˜γ̺˜γand˜c=c−c.PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 11
Then we get:
/braceleftbigg∂t˜c+v·∇˜c+ ˜γcdiv(v)+ ˜γ˜cdiv(v) = 0,
∂tv+v·∇v+ ˜γc∇c+ ˜γ˜c∇c+v=−ε2∇(−∆)−1G(˜c).(3.9)
After changing v(t,x)andc(t,x)intov(t,˜γcx)andc(t,˜γcx), respectively, we can look at the following
system (keeping the previous notations) :
∂t˜c+1
˜γcv·∇˜c+div(v)+˜c
cdiv(v) = 0,
∂tv+1
˜γcv·∇v+∇c+˜c
c∇c+v=−ε2∇(−∆)−1G(˜c).(3.10)
To simplify the presentation, suppose from now on that ˜γc= 1: the general case works the same way.
We’re going to assume throughout this section that:
/ba∇dbl(˜c,v)/ba∇dbl˙Bd
2
2,1≪1. (3.11)
In view of the linear analysis, we will start the following st udy with the choice of index:
•d
2−1for very low frequencies,
•d
2for medium frequencies,
•d
2+1for high frequencies.
This choice of index is strongly inspired by the results of [1 5] where to study the relaxation limit, a similar
choice is taken.
Let us pose then :
˜L(t):=/ba∇dbl(ε˜c,divv)(t)/ba∇dbll−,ε
˙Bd
2−1
2,1+/ba∇dbl(˜c,v)(t)/ba∇dbll+,ε
˙Bd
2
2,1+/ba∇dblPv(t)/ba∇dbll
˙Bd
2
2,1+/ba∇dbl(˜c,v)(t)/ba∇dblh
˙Bd
2+1
2,1
and
˜H(t):=ε2/ba∇dbl(ε˜c,divv)(t)/ba∇dbll−,ε
˙Bd
2−1
2,1+/ba∇dbl(˜c,v)(t)/ba∇dbll+,ε
˙Bd
2+2
2,1+/ba∇dblPv(t)/ba∇dbll
˙Bd
2
2,1+/ba∇dbl(˜c,v)(t)/ba∇dblh
˙Bd
2+1
2,1.
Lemma 3.2. We have the following inequalities :
/ba∇dbl(˜c,v)(t)/ba∇dbll
˙Bd
2
2,1+/ba∇dbl(˜c,v)(t)/ba∇dblh
˙Bd
2+1
2,1/lessorsimilar˜L(t),
/ba∇dbl(˜c,v)(t)/ba∇dbll
˙Bd
2+2
2,1+/ba∇dbl(˜c,v)(t)/ba∇dblh
˙Bd
2+1
2,1/lessorsimilarε2/ba∇dbl(˜c,v)(t)/ba∇dbll−,ε
˙Bd
2
2,1+/ba∇dbl(˜c,v)(t)/ba∇dbll+,ε
˙Bd
2+2
2,1+/ba∇dbl(˜c,v)(t)/ba∇dblh
˙Bd
2+1
2,1/lessorsimilar˜H(t),
ε2/ba∇dbl(˜c,v)(t)/ba∇dbl2
˙Bd
2
2,1+/ba∇dbl(˜c,v)(t)/ba∇dbl2
˙Bd
2+1
2,1/lessorsimilar˜L(t)˜H(t).
Proof. The first two inequalities are easily deduced from the definit ion (3.6).
Let us set Z= (˜c,v). We have by definition and the second inequality:
ε2/ba∇dblZ/ba∇dbl2
˙Bd
2
2,1=/parenleftBigg
ε2/ba∇dblZ/ba∇dbll−,ε
˙Bd
2
2,1+ε2/ba∇dblZ/ba∇dbll+,ε
˙Bd
2
2,1+ε2/ba∇dblZ/ba∇dblh
˙Bd
2
2,1/parenrightBigg
/ba∇dblZ/ba∇dbl˙Bd
2
2,1
/lessorsimilar/parenleftBigg
ε2/ba∇dblZ/ba∇dbll−,ε
˙Bd
2
2,1+/ba∇dblZ/ba∇dbll+,ε
˙Bd
2+2
2,1+ε2/ba∇dblZ/ba∇dblh
˙Bd
2+1
2,1/parenrightBigg
/ba∇dblZ/ba∇dbl˙Bd
2
2,1
/lessorsimilar˜L˜H.12 VALENTIN LEMARIÉ
We have also :
/ba∇dblZ/ba∇dbl2
˙Bd
2+1
2,1/lessorsimilar/parenleftBigg
/ba∇dblZ/ba∇dbll−,ε
˙Bd
2+1
2,1/parenrightBigg2
+/parenleftBigg
/ba∇dblZ/ba∇dbll+,ε
˙Bd
2+1
2,1/parenrightBigg2
+/parenleftBigg
/ba∇dblZ/ba∇dblh
˙Bd
2+1
2,1/parenrightBigg2
/lessorsimilarε2/ba∇dblZ/ba∇dbll−,ε
˙Bd
2
2,1+/ba∇dblZ/ba∇dbll+,ε
˙Bd
2
2,1/ba∇dblZ/ba∇dbll+,ε
˙Bd
2+2
2,1+˜L˜H
/lessorsimilar˜L˜H.
/square
3.2.1. High Frequency Estimates.
We rely on the high-frequency analysis carried out in [16].
Proposition 3.3. For high frequencies, we have the estimate:
/ba∇dblZ(t)/ba∇dblh
˙Bd
2+1
2,1+/integraldisplayt
0/ba∇dblZ/ba∇dblh
˙Bd
2+1
2,1dτ≤ /ba∇dblZ0/ba∇dblh
˙Bd
2+1
2,1+C/integraldisplayt
0˜L(τ)˜H(τ)dτ.
Proof. Let us denote Lj:=1
2/ba∇dblZj/ba∇dbl2
L2+2−2j
4/integraldisplay
Rd∇˜cj·vjdx.
The following lemma will enable us to handle the first term of Lj:
Lemma 3.4. We have the following inequality with Z=/parenleftbigg
˜c
v/parenrightbigg
andε′=/radicalbig
f(c)ε:
1
2d
dt/ba∇dblZj/ba∇dbl2
L2+/ba∇dblvj/ba∇dbl2
L2+/integraldisplayt
0ε′2∇(−∆)−1˜cj·vjdx≤Caj2−j(d
2+1)/parenleftbig
/ba∇dbl∇Z/ba∇dbl˙Bd
2
2,1/ba∇dblZ/ba∇dbl˙Bd
2+1
2,1+ε2/ba∇dbl˜c/ba∇dbl2
˙Bd
2
2,1/parenrightbig
/ba∇dblZj/ba∇dblL2
with(aj)verifying
/summationdisplay
j∈Zaj= 1. (3.12)
Proof. Let us apply ˙∆jto the system (3.10), then we get:
∂tcj+v·∇˜cj+c
cdiv(vj)+/integraldisplayt
0ε′2∇(−∆)−1˜cj·vjdx= [v·∇,˙∆j]˜c+[˜c
c,˙∆j]div(v),
∂tvj+v·∇vj+c
c∇cj+vj+ε′2∇(−∆)−1˜cj=−ε2˙∆j∇(−∆)−1F(˜c)+[v·∇,˙∆j]v+[˜c
c,˙∆j]∇˜c.
By performing the scalar product with Zj:=/parenleftbigg
˜cj
vj/parenrightbigg
inL2(Rd;Rn), we get:
1
2d
dt/ba∇dblZj/ba∇dbl2
L2+/ba∇dblvj/ba∇dbl2
L2=−/integraldisplay
Rd/parenleftBig
v·∇˜cj+c
cdiv(vj)/parenrightBig
˜cj+/integraldisplay
Rd[v·∇,˙∆j]˜c˜cjdx
+/integraldisplay
Rd[˜c
c,˙∆j]div(v)˜cjdx−/integraldisplay
Rd/parenleftBig
v·∇vj+c
c∇cj/parenrightBig
vjdx
+/integraldisplay
Rd/bracketleftBig
v·∇,˙∆j/bracketrightBig
vvjdx+/integraldisplay
Rd[˜c
c,˙∆j]∇˜c·vjdx
−ε2/integraldisplay
Rd˙∆j∇(−∆)−1F(˜c)·vjdx.
In order to bound the right-hand side, we use the following fa cts:
• /ba∇dblε2∇(−∆)−1F(˜c)/ba∇dbl˙Bd
2+1
2,1≤ε2/ba∇dblF(˜c)/ba∇dbl˙Bd
2
2,1≤ε2C(/ba∇dbl˜c/ba∇dblL∞)/ba∇dbl˜c/ba∇dbl2
˙Bd
2
2,1/lessorsimilarε2/ba∇dbl˜c/ba∇dbl2
˙Bd
2
2,1.PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 13
•The following commutator estimates with s′=d
2+1:
/ba∇dbl[v·∇,˙∆j]Z/ba∇dblL2≤Caj2−js′/ba∇dbl∇Z/ba∇dbl˙Bd
2
2,1/ba∇dblZ/ba∇dbl˙Bs′
2,1where/summationtext
j∈Zaj= 1,
/ba∇dbl[˜c
c,˙∆j]div(v)/ba∇dblL2≤Caj2−js′/ba∇dbl∇Z/ba∇dbl˙Bd
2
2,1/ba∇dblZ/ba∇dbl˙Bs′
2,1,
/ba∇dbl[˜c
c,˙∆j]∇˜c/ba∇dblL2≤Caj2−js′/ba∇dbl∇Z/ba∇dbl˙Bd
2
2,1/ba∇dblZ/ba∇dbl˙Bs′
2,1.
•The integration by parts:
/integraldisplay
Rdv·∇˜cj˜cjdx=−/integraldisplay
Rd1
2div(v)|˜cj|2dx.
•/integraldisplay
Rdv·∇˜vj˜vjdx=−/integraldisplay
Rd1
2div(v)|˜vj|2dx,
•/integraldisplay
Rdc
c(div(vj)˜cj+∇˜cjvj)dx=−1
c/integraldisplay
Rdvj∇(c˜cj)dx+1
c/integraldisplay
Rdc∇˜cjvjdx=−/integraldisplay
Rdvj˜cj∇c
cdx,
hence using the injection ˙Bd
2
2,1(Rd)֒→ Cb(Rd)
/integraldisplay
Rdv·∇˜cj˜cjdx+/integraldisplay
Rdv·∇˜vj˜vjdx+/integraldisplay
Rdc
c(div(vj)˜cj+∇˜cjvj)dx
≤Caj2−js′/ba∇dbl∇Z/ba∇dbl˙Bd
2
2,1/ba∇dblZ/ba∇dbl˙Bs′
2,1/ba∇dblZj/ba∇dblL2.
Whence the result by putting together all the above inequali ties. /square
For the other term of Lj, look at (3.10) as:
∂t˜c+div(v) =−v·∇c−˜c
cdiv(v)
∂tv+∇˜c+ε′2∇(−∆)−1˜c+v=−v·∇v−˜c
c∇˜c−ε2∇(−∆)−1F(˜c)
whereε′=/radicalbig
f(c)ε. (3.13)
Let us denote S1:=−v·∇c−˜c
cdiv(v)andS2:=−v·∇v−˜c
c∇˜c−ε2∇(−∆)−1F(˜c)as well as S:= (S1,S2).
Analogously to linear analysis, we obtain:
d
dt/parenleftbigg/integraldisplay
Rd∇˜cj·vjdx/parenrightbigg
+/integraldisplay
Rd∇˜cj·vjdx−/ba∇dbldivvj/ba∇dbl2
L2+/ba∇dbl∇˜cj/ba∇dbl2
L2+ε′2/ba∇dbl˜cj/ba∇dblL2=/integraldisplay
RdRe(˙∆jS1·v)dx
+/integraldisplay
RdRe(∇cj·˙∆jS2)dx.
By Cauchy-Schwarz and Bernstein inequalities, we see that Lj:=1
2/ba∇dblZj/ba∇dbl2
L2+2−2j
4/integraldisplay
Rd∇˜cj·vjdxverifies :
(3.14)3
8/ba∇dblZj/ba∇dbl2
L2≤ Lj≤1
2/ba∇dblZj/ba∇dbl2
L2+2−2j
8/ba∇dbl∇˜cj/ba∇dbl2
L2+1
8/ba∇dblvj/ba∇dbl2
L2≤5
8/ba∇dblZj/ba∇dbl2
L2.
We then summarize the previous inequality with the inequali ty of the lemma 3.4:
d
dtLj+/ba∇dblvj/ba∇dbl2
L2+/integraldisplayt
0ε′2∇(−∆)−1˜cj·vjdx−2−2j
4/ba∇dbldivvj/ba∇dbl2
L2+2−2j
4/ba∇dbl∇cj/ba∇dbl2
L2+ε′22−2j
4/ba∇dbl˜cj/ba∇dbl2
L2+2−2j
4/integraldisplay
Rd∇cj·vjdx
≤Caj2−js′/parenleftBigg
/ba∇dbl∇Z/ba∇dbl˙Bd
2
2,1/ba∇dblZ/ba∇dbl˙Bs′
2,1+ε2/ba∇dbl˜c/ba∇dbl2
˙Bd
2
2,1/parenrightBigg
/ba∇dblZj/ba∇dblL2+2−2j
4(/ba∇dbl˙∆jS1/ba∇dblL2/ba∇dblvj/ba∇dblL2+/ba∇dbl˙∆jS2/ba∇dblL2/ba∇dbl˜cj/ba∇dblL2).14 VALENTIN LEMARIÉ
But owing to the Cauchy-Schwarz and Bernstein inequalities, w e have
/integraldisplayt
0ε′2∇(−∆)−1˜cj·vjdx+2−2j
4/integraldisplay
Rd∇˜cj·vjdx≥ −1−2ε′2
8/ba∇dbl˜cj/ba∇dbl2
L2−1−2ε′2
8/ba∇dblvj/ba∇dbl2
L2
which allows us to obtain, thanks to (3.14),
/ba∇dblvj/ba∇dblL2+/integraldisplayt
0ε′2∇(−∆)−1˜cj·vjdx−2−2j
4/ba∇dbldivvj/ba∇dbl2
L2+1
4/ba∇dblcj/ba∇dbl2
L2+ε′22−2j
4/ba∇dbl˜cj/ba∇dbl2
L2+2−2j
4/integraldisplay
Rd∇cj·vjdx
≥5−2ε′2
8/ba∇dblvj/ba∇dbl2
L2+1−2ε′2
8/ba∇dbl˜cj/ba∇dbl2
L2
≥1−2ε′2
5Lj.
We then get for ε′≤1/2:
d
dtLj+Lj/lessorsimilar/parenleftBigg
2−j/ba∇dbl˙∆jS/ba∇dblL2+aj2−js′(/ba∇dblZ/ba∇dbl2
˙Bs′
2,1+ε2/ba∇dbl˜c/ba∇dbl2
˙Bd
2
2,1)/parenrightBigg
/ba∇dblZj/ba∇dblL2. (3.15)
By decomposing Z=Zl+Zhwhere/hatwiderZl=/hatwideZ1|ξ|≤1and/hatwiderZh=/hatwideZ1|ξ|≥1as well as using the lemmas 3.2 and
A.3, we have :
/ba∇dblZ∇Z/ba∇dblh
˙Bd
2
2,1≤ /ba∇dblZ∇Zl/ba∇dbl˙Bd
2+1
2,1+/ba∇dblZ∇Zh/ba∇dbl˙Bd
2
2,1/lessorsimilar/ba∇dblZ/ba∇dbl2
˙Bd
2+1
2,1+/ba∇dblZ/ba∇dbl˙Bd
2
2,1/ba∇dblZ/ba∇dbll
˙Bd
2+2
2,1+/ba∇dblZ/ba∇dbl˙Bd
2
2,1/ba∇dblZ/ba∇dblh
˙Bd
2+1
2,1/lessorsimilar˜L˜H
/ba∇dbl−ε2∇(−∆)−1F(˜c)/ba∇dblh
˙Bd
2
2,1/lessorsimilarε2/ba∇dblF(˜c)/ba∇dblh
˙Bd
2−1
2,1≤ε2/ba∇dblF(˜c)/ba∇dbl˙Bd
2
2,1/lessorsimilarε2/ba∇dblZ/ba∇dbl2
˙Bd
2
2,1/lessorsimilar˜L˜H.
Consequently, we have :
2js′/radicalBig
Lj(t)+c2js′/integraldisplayt
0/radicalbig
Ljdτ≤2js′/radicalBig
Lj(0)+C/integraldisplayt
0aj˜L(τ)˜H(τ)dτ,withs′=d
2+1
We deduce, by summing on j∈N, the estimate for the high frequencies of the proposition. /square
3.2.2. Damped mode. As explained in [16], at the low frequency level, we have a los s of information on the
part undergoing dissipation (here v). To overcome this loss, Danchin highlighted "the damped mo de" to
recover the missing information: we then pose W:=−∂tv.
However here we cannot proceed as in linear analysis: some no n-linear terms do not allow us to conclude.
So we are going to proceed differently here: we will first study the damped mode, which will allow us to
have very precise estimates. In a second step, we will study t he equation on cusing this damped mode and
finally we will get the information on vthat we need using the estimates on Wandc.
From now on, let us take the constants (other than ε) appearing in our system equal to 1to simplify the
presentation: they play no role in what will follow.
Let us first look at this damped mode at the high frequencies.
Lemma 3.5. We have the following estimate for the damped mode:
/ba∇dblW(t)/ba∇dblh
˙Bd
2
2,1+/integraldisplayt
0/ba∇dblW/ba∇dblh
˙Bd
2
2,1dτ/lessorsimilar˜L(t)+/integraldisplayt
0˜H(τ)dτ+/integraldisplayt
0˜L(τ)˜H(τ)dτ.
Proof. By definition of W, we have :
W=v+∇c+ε2∇(−∆)1˜c+v·v+˜c∇˜c+ε2∇(−∆)−1F(˜c).
We then have (by using the fact that ˙Bd
2
2,1is a multiplicative algebra) :
/ba∇dblW(t)/ba∇dblh
˙Bd
2
2,1/lessorsimilar/ba∇dblv(t)/ba∇dblh
˙Bd
2
2,1+/ba∇dbl˜c(t)/ba∇dblh
˙Bd
2+1
2,1+ε2/ba∇dbl˜c(t)/ba∇dblh
˙Bd
2−1
2,1+/ba∇dblv(t)/ba∇dblh
˙Bd
2
2,1/ba∇dblv(t)/ba∇dblh
˙Bd
2+1
2,1+/ba∇dbl˜c(t)/ba∇dblh
˙Bd
2
2,1/ba∇dbl˜c(t)/ba∇dblh
˙Bd
2+1
2,1
+ε2/ba∇dbl˜c(t)/ba∇dblh
˙Bd
2
2,1/ba∇dbl˜c(t)/ba∇dblh
˙Bd
2−1
2,1.PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 15
By the lemma 3.2, we have : /ba∇dblW(t)/ba∇dblh
˙Bd
2
2,1/lessorsimilar˜L(t)and/integraldisplayt
0/ba∇dblW(τ)/ba∇dblh
˙Bd
2
2,1dτ/lessorsimilar/integraldisplayt
0˜H(τ)dτ+/integraldisplayt
0˜L(τ)˜H(τ)dτ.
/square
As∂tv+W= 0, thus we have :
∂tW+W=∂tv+W+∂tv·∇v+v·∇(∂tv)+∇(∂tc)+(∂tc)∇c+˜c∇(∂tc)+ε2∇(−∆)−1∂tc
+ε2∇(−∆)−1F′(˜c)∂tc
=−W·∇v−v·∇W−∇(v·∇˜c)−∇divv−∇(˜cdivv)−v·∇˜c∇˜c−divv∇˜c−˜cdivv∇˜c
−˜c∇(v·∇˜c)−˜c(∇divv)−˜c∇(divv∇˜c)−ε2∇(−∆)−1(v·∇˜c)−ε2∇(−∆)−1(divv)
−ε2∇(−∆)−1(˜cdivv).
Lemma 3.6. The following estimate holds true:
(3.16)/ba∇dblW(t)/ba∇dbll
˙Bd
2
2,1+/integraldisplayt
0/ba∇dblW/ba∇dbll
˙Bd
2
2,1dτ/lessorsimilar/ba∇dblW0/ba∇dbll
˙B2,1+/integraldisplayt
0˜L(τ)˜H(τ)dτ+/integraldisplayt
0/ba∇dblv/ba∇dbll
˙Bd
2+2
2,1dτ+ε2/integraldisplayt
0/ba∇dbldivv/ba∇dbll
˙Bd
2−1
2,1dτ.
Proof. By applying ˙∆jto the previous equation verified by W, taking the scalar product with Wj, multiplying
by2jd
2, summing up on j∈Z−and using the lemma A.1, we get owing to the product laws of the lemma
A.3 (each non-linear term in the right-hand side appears in t he same order as the following estimate):
/ba∇dblW(t)/ba∇dbll
˙Bd
2
2,1+/integraldisplayt
0/ba∇dblW/ba∇dbll
˙Bd
2
2,1dτ/lessorsimilar/ba∇dblW0/ba∇dbll
˙Bd
2
2,1+/integraldisplayt
0/ba∇dblW/ba∇dbl˙Bd
2
2,1/ba∇dblv/ba∇dbl˙Bd
2+1
2,1dτ+/integraldisplayt
0/ba∇dblv·W/ba∇dbll
˙Bd
2−1
2,1dτ
+/integraldisplayt
0/parenleftBigg
/ba∇dbl∇v·∇˜c/ba∇dbl˙Bd
2
2,1+/ba∇dblv·∇∇c/ba∇dbll
˙Bd
2
2,1/parenrightBigg
dτ+/integraldisplayt
0/ba∇dblv/ba∇dbll
˙Bd
2+2
2,1dτ
+/integraldisplayt
0/parenleftBigg
/ba∇dbl∇˜cdivv/ba∇dbl˙Bd
2
2,1+/ba∇dbl˜c∇divv/ba∇dbl˙Bd
2
2,1/parenrightBigg
dτ+/integraldisplayt
0/ba∇dblv/ba∇dbl˙Bd
2
2,1/ba∇dbl˜c/ba∇dbl2
˙Bd
2+1
2,1dτ
+/integraldisplayt
0/ba∇dblv/ba∇dbl˙Bd
2+1
2,1/ba∇dbl˜c/ba∇dbl˙Bd
2+1
2,1dτ+/integraldisplayt
0/ba∇dbl˜c/ba∇dbl˙Bd
2+1
2,1/ba∇dblv/ba∇dbl˙Bd
2+1
2,1/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1dτ
+/integraldisplayt
0/ba∇dbl˜c∇(v·∇˜c)/ba∇dbll
˙Bd
2
2,1dτ+/integraldisplayt
0/ba∇dbl˜c(∇divv)/ba∇dbll
˙Bd
2
2,1dτ+/integraldisplayt
0/ba∇dbl˜c∇(divv∇˜c)/ba∇dbll
˙Bd
2
2,1dτ
+ε2/integraldisplayt
0/ba∇dblv/ba∇dbl˙Bd
2
2,1/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1dτ+/integraldisplayt
0ε2/ba∇dbldivv/ba∇dbll
˙Bd
2−1
2,1dτ+ε2/integraldisplayt
0/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1/ba∇dblv/ba∇dbl˙Bd
2
2,1dτ.
Let us estimate one by one the terms appearing in the right-ha nd side:
•/integraldisplayt
0/ba∇dblW/ba∇dbll
˙Bd
2
2,1/ba∇dblv/ba∇dbl˙Bd
2+1
2,1dτcan be absorbed by the left-hand side as we know that /ba∇dblZ/ba∇dbl
L∞(˙Bd
2
2,1)is small;
•/integraldisplayt
0/ba∇dblW/ba∇dblh
˙Bd
2
2,1/ba∇dblv/ba∇dbl˙Bd
2+1
2,1dτ/lessorsimilar/integraldisplayt
0˜L(τ)˜H(τ)dτby the lemma 3.5;
•/integraldisplayt
0/ba∇dblv·W/ba∇dbll
˙Bd
2−1
2,1dτ/lessorsimilar/integraldisplayt
0/ba∇dblv/ba∇dbl˙Bd
2
2,1(/ba∇dblW/ba∇dbll
˙Bd
2
2,1+/ba∇dblW/ba∇dblh
˙Bd
2
2,1)dτhas the low frequency part absorbed by the
left-hand side and the other part below/integraldisplayt
0˜L(τ)˜H(τ)dτby the lemma 3.5;
•By using the lemma 3.2 and the fact that /ba∇dblZ/ba∇dbl
L∞(˙Bd
2
2,1)is small, we have :
/integraldisplayt
0/parenleftbigg
/ba∇dbl∇v·∇c/ba∇dbl˙Bd
2
2,1+/ba∇dbl∇˜c·divv/ba∇dbl˙Bd
2
2,1+/ba∇dblv/ba∇dbl˙Bd
2
2,1/ba∇dbl˜c/ba∇dbl2
˙Bd
2+1
2,1+/ba∇dblv/ba∇dbl˙Bd
2+1
2,1/ba∇dblc/ba∇dbl˙Bd
2+1
2,1+/ba∇dbl˜c/ba∇dbl˙Bd
2+1
2,1/ba∇dblv/ba∇dbl˙Bd
2+1
2,1/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1/parenrightbigg
dτ
/lessorsimilar/integraldisplayt
0/ba∇dblZ/ba∇dbl2
˙Bd
2+1
2,1dτ/lessorsimilar/integraldisplayt
0˜L(τ)˜H(τ)dτ;16 VALENTIN LEMARIÉ
•/integraldisplayt
0ε2/ba∇dblv/ba∇dbl˙Bd
2
2,1/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1dτ+ε2/integraldisplayt
0/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1/ba∇dblv/ba∇dbl˙Bd
2
2,1/lessorsimilar/integraldisplayt
0˜L(τ)˜H(τ)dτby using the lemma 3.2;
• /ba∇dblv·∇∇c/ba∇dbll
˙Bd
2
2,1/lessorsimilar/ba∇dblv·∇∇cl/ba∇dbl˙Bd
2
2,1+/ba∇dblv·∇∇ch/ba∇dbl˙Bd
2
2,1/lessorsimilar/ba∇dblv/ba∇dbl˙Bd
2
2,1/ba∇dbl˜c/ba∇dbll
˙Bd
2+2
2,1+/ba∇dblv·∇∇ch/ba∇dbll
˙Bd
2−1
2,1
/lessorsimilar˜L˜H+/ba∇dblv/ba∇dbl˙Bd
2
2,1/ba∇dbl˜c/ba∇dblh
˙Bd
2+1
2,1/lessorsimilar˜L˜Hby the lemmas 3.2 and A.3;
•By proceeding in the same way as previously, we discover that :
/ba∇dbl˜c∇divv/ba∇dbll
˙Bd
2
2,1/lessorsimilar/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1/parenleftBigg
/ba∇dblv/ba∇dbll
˙Bd
2+2
2,1+/ba∇dblv/ba∇dblh
˙Bd
2+1
2,1/parenrightBigg
/lessorsimilar˜L˜H;
• /ba∇dbl˜c∇(v·∇˜c)/ba∇dbll
˙Bd
2
2,1≤ /ba∇dbl˜cdivv∇˜c/ba∇dbl˙Bd
2
2,1+/ba∇dbl˜cv·∇∇˜c/ba∇dbl˙Bd
2
2,1.
For the first term, the lemma 3.2 is used and for the second, we d o the same as for the previous
two points. We find that this term is less than ˜L˜H.
So we have inequality (3.16). /square
3.2.3. Study of the low frequencies of ˜c.
We have :
∂t˜c+divv=−˜cdivv−v·∇˜c.
However , divv= divW−∆c+ε2˜c+ε2F(˜c)−div(˜c∇˜c)−div(v·∇v).
So by rewriting the equation of ˜c, we get
∂t˜c−∆˜c+ε2c=−divW−ε2F(˜c)+div(˜c∇˜c)+div(v·∇v)−˜cdivv−v·∇˜c
After spectral localization by means of ˙∆j, taking the scalar product with ˙∆j˜c, multiplying by ε2j(d
2−1)
(respectively 2jd
2) and, finally, summing up on 2j≤ε(respectively ε≤2j≤1), we get
Lemma 3.7.
The following estimates are satisfied by ˜c:
/ba∇dblε˜c(t)/ba∇dbll−,ε
˙Bd
2−1
2,1+ε2/integraldisplayt
0/ba∇dblε˜c(t)/ba∇dbll−,ε
˙Bd
2−1
2,1dτ/lessorsimilar/ba∇dblε˜c0/ba∇dbll−,ε
˙Bd
2−1
2,1+ε/integraldisplayt
0/ba∇dblW/ba∇dbll−,ε
˙Bd
2
2,1dτ+/integraldisplayt
0C(τ)˜C(τ)dτ
+ε/integraldisplayt
0/ba∇dblv·∇v/ba∇dbll−,ε
˙Bd
2
2,1+ε/integraldisplayt
0/ba∇dbl˜cdivv/ba∇dbl˙Bd
2−1
2,1dτ+ε/integraldisplayt
0/ba∇dblv·∇˜c/ba∇dbl˙Bd
2−1
2,1dτ,
/ba∇dbl˜c(t)/ba∇dbll+,ε
˙Bd
2
2,1+/integraldisplayt
0/ba∇dbl˜c/ba∇dbll+,ε
˙Bd
2+2
2,1dτ/lessorsimilar/ba∇dbl˜c0/ba∇dbll+,ε
˙Bd
2
2,1+/integraldisplayt
0/ba∇dblW/ba∇dbll+,ε
˙Bd
2+1
2,1dτ+/integraldisplayt
0C(τ)˜C(τ)dτ+/integraldisplayt
0/ba∇dbldiv(v·∇v)/ba∇dbll+,ε
˙Bd
2
2,1dτ
+/integraldisplayt
0/ba∇dbl˜cdivv/ba∇dbl˙Bd
2
2,1dτ+/integraldisplayt
0/ba∇dblv·∇˜c/ba∇dbl˙Bd
2
2,1dτ
where, in view of the linear estimates, we set C(t):=/ba∇dblε˜c(t)/ba∇dbll−,ε
˙Bd
2−1
2,1+/ba∇dbl˜c(t)/ba∇dbll+,ε
˙Bd
2
2,1+/ba∇dbl˜c(t)/ba∇dblh
˙Bd
2+1
2,1and˜C(t):=
ε2/ba∇dblε˜c(t)/ba∇dbll−,ε
˙Bd
2−1
2,1+/ba∇dbl˜c(t)/ba∇dbll+,ε
˙Bd
2+2
2,1+/ba∇dbl˜c(t)/ba∇dblh
˙Bd
2+1
2,1.
3.2.4. Study of the low frequencies of v.
It is now possible to deduce optimal bounds for vfrom the ones we have just derived for Wand˜c. We need
to decompose vby using the damped mode as follows :
v=W−∇c+ε2∇(−∆)−1˜c+v·∇v+˜c∇c+ε2∇(−∆)−1F(˜c) (3.17)
Lemma 3.8. Based on the following estimates, we will set:
V(t):=/ba∇dblv(t)/ba∇dbll
˙Bd
2
2,1+/ba∇dblv(t)/ba∇dblh
˙Bd
2+1
2,1and˜V(t):=ε/ba∇dblv(t)/ba∇dbll−,ε
˙Bd
2
2,1+/ba∇dblv(t)/ba∇dbll+,ε
˙Bd
2+1
2,1+/ba∇dblv(t)/ba∇dblh
˙Bd
2+1
2,1.PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 17
We then obtain the following estimates:
/ba∇dblv(t)/ba∇dbll−,ε
˙Bd
2
2,1+ε/integraldisplayt
0/ba∇dblv/ba∇dbll−,ε
˙Bd
2
2,1dτ/lessorsimilar/ba∇dblW(t)/ba∇dbll−,ε
˙Bd
2
2,1+(ε2+/ba∇dblZ(t)/ba∇dbl˙Bd
2+1
2,1+εC(t))C(t)+V(t)/ba∇dblZ(t)/ba∇dbl˙Bd
2+1
2,1
+/integraldisplayt
0ε/ba∇dblW/ba∇dbll−,ε
˙Bd
2
2,1dτ+/integraldisplayt
0˜C(τ)dτ+/integraldisplayt
0C(τ)˜C(τ)dτ+ε/integraldisplayt
0V(τ)˜V(τ)dτ.
/ba∇dblv(t)/ba∇dbll+,ε
˙Bd
2
2,1+/integraldisplayt
0/ba∇dblv(t)/ba∇dbll+,ε
˙Bd
2+1
2,1dτ/lessorsimilar/ba∇dblW(t)/ba∇dbll+,ε
˙Bd
2
2,1+/ba∇dbl˜c(t)/ba∇dbll+,ε
˙Bd
2+1
2,1+(V(t)+C(t))/ba∇dblZ(t)/ba∇dbl˙Bd
2+1
2,1+ε(C(t))2
+/integraldisplayt
0/ba∇dblW/ba∇dbll+,ε
˙Bd
2+1
2,1dτ+/integraldisplayt
0˜C(τ)dτ+/integraldisplayt
0V(τ)˜V(τ)dτ+/integraldisplayt
0C(τ)˜C(τ)dτ.
Proof. By lemmas A.3, A.5 and (3.17), we obtain :
/ba∇dblv(t)/ba∇dbll−,ε
˙Bd
2
2,1/lessorsimilar/ba∇dblW/ba∇dbll−,ε
˙Bd
2
2,1+ε2/ba∇dbl˜c/ba∇dbll−,ε
˙Bd
2−1
2,1+/ba∇dblv/ba∇dbl˙Bd
2
2,1/ba∇dblv/ba∇dbl˙Bd
2+1
2,1+/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1/ba∇dbl˜c/ba∇dbl˙Bd
2+1
2,1+ε2/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1/ba∇dbl˜c/ba∇dbl˙Bd
2−1
2,1.
We deduce :
/ba∇dblv(t)/ba∇dbll−,ε
˙Bd
2
2,1/lessorsimilar/ba∇dblW(t)/ba∇dbll−,ε
˙Bd
2
2,1+(ε+/ba∇dblZ(t)/ba∇dbl˙Bd
2+1
2,1+εC(t))C(t) +V(t)/ba∇dblv(t)/ba∇dbl˙Bd
2+1
2,1
ε/integraldisplayt
0/ba∇dblv/ba∇dbll−,ε
˙Bd
2
2,1dτ/lessorsimilar/integraldisplayt
0ε/ba∇dblW/ba∇dbll−,ε
˙Bd
2
2,1dτ+/integraldisplayt
0˜C(τ)dτ+/integraldisplayt
0C(τ)˜C(τ)dτ+ε/integraldisplayt
0V(τ)˜V(τ)dτ.
Similarly, we get :
/ba∇dblv(t)/ba∇dbll+,ε
˙Bd
2
2,1/lessorsimilar/ba∇dblW(t)/ba∇dbll+,ε
˙Bd
2
2,1+/ba∇dbl˜c(t)/ba∇dbll+,ε
˙Bd
2+1
2,1+(V(t)+C(t))/ba∇dblZ(t)/ba∇dbl˙Bd
2+1
2,1+ε(C(t))2.
By lemmas A.3 and A.5 and inequality /ba∇dblZ/ba∇dbll+,ε
˙Bd
2+1
2,1≤ /ba∇dblZ/ba∇dbll+,ε
˙Bd
2
2,1, we have also :
/ba∇dblv/ba∇dbll+,ε
˙Bd
2+1
2,1/lessorsimilar/ba∇dblW/ba∇dbll+,ε
˙Bd
2+1
2,1+/ba∇dbl˜c/ba∇dbll+,ε
˙Bd
2+2
2,1+/ba∇dblv/ba∇dbl˙Bd
2
2,1/ba∇dblv/ba∇dbl˙Bd
2+1
2,1+/ba∇dbl˜c·∇c/ba∇dbll+,ε
˙Bd
2+1
2,1+ε2/ba∇dbl˜c/ba∇dbl2
˙Bd
2
2,1.
Like the proof of lemma 3.2, we have that ε2/ba∇dbl˜c/ba∇dbl2
˙Bd
2
2,1/lessorsimilarC˜C.
Moreover,
/ba∇dbl˜c·∇c/ba∇dbll+,ε
˙Bd
2+1
2,1/lessorsimilar/ba∇dbl˜c·∇cl/ba∇dbll
˙Bd
2+1
2,1+/ba∇dbl˜c·∇ch/ba∇dbll
˙Bd
2+1
2,1
/lessorsimilar/ba∇dbl˜c/ba∇dbl˙Bd
2+1
2,1/ba∇dbl˜c/ba∇dbll
˙Bd
2+1
2,1+/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1/ba∇dbl˜c/ba∇dbll
˙Bd
2+2
2,1+/ba∇dbl˜c·∇ch/ba∇dbll
˙Bd
2
2,1
/lessorsimilarC˜C+/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1/ba∇dbl˜c/ba∇dbl˙Bd
2+1
2,1/lessorsimilarC˜C.
We deduce :
/integraldisplayt
0/ba∇dblv/ba∇dbll+,ε
˙Bd
2
2,1dτ/lessorsimilar/integraldisplayt
0/ba∇dblW/ba∇dbll+,ε
˙Bd
2+1
2,1dτ+/integraldisplayt
0˜C(τ)dτ+/integraldisplayt
0V(τ)˜V(τ)dτ+/integraldisplayt
0C(τ)˜C(τ)dτ.
/square
3.2.5. Final a priori estimates.
Let us denote
(3.18) L(t):=C(t)+V(t)+/ba∇dblW(t)/ba∇dbll
˙Bd
2
2,1andH(t):=˜C(t)+˜V(t)+/ba∇dblW(t)/ba∇dbll
˙Bd
2
2,1.
We notice that : ˜L(t)≤ L(t)and˜H(t)≤ H(t).18 VALENTIN LEMARIÉ
Proposition 3.9. We have the following estimate :
L(t)+/integraldisplayt
0H(τ)dτ/lessorsimilarL(0)+/integraldisplayt
0L(τ)H(τ)dτ.
If we take L(0)sufficiently small, thus we obtain the final a priori estimate :
L(t)+/integraldisplayt
0H(τ)dτ/lessorsimilarL(0).
Proof. First, we note that by summing up the previous inequalities ( lemmas 3.6, 3.7, 3.8), terms in the
right-hand side can be absorbed by those of the left-hand sid e. Indeed :
•In (3.16), we have that the term/integraldisplayt
0/ba∇dblv/ba∇dbll
˙Bd
2+2
2,1dτ+ε2/integraldisplayt
0/ba∇dbldivv/ba∇dbl˙Bd
2−1
2,1dτis negligible compared to
/integraldisplayt
0˜V(τ)dτ(so also to/integraldisplayt
0H(τ)dτ).
•In the estimates of the lemma 3.7, we have that/integraldisplayt
0ε/ba∇dblW/ba∇dbll−,ε
˙Bd
2
2,1dτ,/integraldisplayt
0/ba∇dblW/ba∇dbll+,ε
˙Bd
2+1
2,1dτare negligible
compared to/integraldisplayt
0/ba∇dblW/ba∇dbll
˙Bd
2
2,1dτ. By using that /ba∇dblZ/ba∇dbl˙Bd
2
2,1is small and the lemma A.3, we also have that
termsε/integraldisplayt
0/ba∇dblv·∇v/ba∇dbll−,ε
˙Bd
2
2,1dτand/integraldisplayt
0/ba∇dbldiv(c·∇v)/ba∇dbll+,ε
˙Bd
2
2,1dτare negligible compared to/integraldisplayt
0˜V(τ)dτ.
•(ε2+εC(t))C(t)and/ba∇dbl˜c(t)/ba∇dbll+,ε
˙Bd
2+1
2,1are negligible compared to L(t).
Using the definitions of the various introduced norms and L,Hand the lemma A.3, we have :
•/integraldisplayt
0˜L˜Hdτ/lessorsimilar/integraldisplayt
0LHdτ,
•/integraldisplayt
0C(τ)˜C(τ)dτ+/integraldisplayt
0V(τ)˜V(τ)dτ/lessorsimilar/integraldisplayt
0L(τ)H(τ)dτ,
•/integraldisplayt
0ε/ba∇dbl˜cdivv/ba∇dbll−,ε
˙Bd
2−1
2,1dτ/lessorsimilar/integraldisplayt
0/ba∇dblε˜c/ba∇dbl˙Bd
2−1
2,1/ba∇dblv/ba∇dbl˙Bd
2+1
2,1dτ/lessorsimilar/integraldisplayt
0L(τ)H(τ)dτ,
•/integraldisplayt
0/ba∇dbl˜cdivv/ba∇dbll−,ε
˙Bd
2
2,1dτ/lessorsimilar/integraldisplayt
0/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1/ba∇dblv/ba∇dbl˙Bd
2+1
2,1dτ/lessorsimilar/integraldisplayt
0L(τ)H(τ)dτ,
•ε/ba∇dblv·∇c/ba∇dbll−,ε
˙Bd
2−1
2,1dτ/lessorsimilar/integraldisplayt
0/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1/ba∇dblεv/ba∇dbl˙Bd
2
2,1dτ/lessorsimilar/integraldisplayt
0L(τ)H(τ)dτ.
Now, if we sum up the previous inequalities by using what we ju st did before and by removing "negligible
terms compared to the right term", we get:
L(t)+/integraldisplayt
0H(τ)dτ/lessorsimilarL(0)+/integraldisplayt
0L(τ)H(τ)dτ+/integraldisplayt
0/ba∇dblv·∇c/ba∇dbll+,ε
˙Bd
2
2,1dτ.
To handle the last term, let us use the fact that vandware interrelated as follows:
v=W−∇c+ε2∇(−∆)−1˜c+v·∇v+˜c∇c+ε2∇(−∆)−1F(˜c).
By the lemma A.3, we have also :
/ba∇dblv·∇c/ba∇dbl˙Bd
2
2,1/lessorsimilar/ba∇dblW/ba∇dbl˙Bd
2
2,1/ba∇dbl˜c/ba∇dbl˙Bd
2+1
2,1+/ba∇dblε˜c/ba∇dbl˙Bd
2−1
2,1/ba∇dbl˜c/ba∇dbl˙Bd
2+1
2,1+/ba∇dbl˜c/ba∇dbl2
˙Bd
2+1
2,1+/ba∇dblv/ba∇dbl˙Bd
2
2,1/ba∇dblv/ba∇dbl˙Bd
2+1
2,1/ba∇dbl˜c/ba∇dbl˙Bd
2+1
2,1
+/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1/ba∇dbl˜c/ba∇dbl2
˙Bd
2+1
2,1+/ba∇dbl˜c/ba∇dbl˙Bd
2
2,1/ba∇dblε˜c/ba∇dbl˙Bd
2−1
2,1/ba∇dblε˜c/ba∇dbl˙Bd
2+1
2,1.
We then have by lemma 3.2 and definition of L,H:
/ba∇dblv·∇c/ba∇dbl˙Bd
2
2,1/lessorsimilar˜L˜H+LH/lessorsimilarLH,PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 19
hence the result. We have the second inequality of the propos ition by using lemma A.2. /square
3.3.A global well-posedness theorem. Here is the theorem that we will prove in the rest of this secti on:
Theorem 3.10.
We assume ε′≤1/2withε′defined in (3.13). Then, there exists a positive constant αsuch that for all
Zε
0= (˜c0,v0)∈˙Bd
2
2,1∩˙Bd
2+1
2,1satisfying
Zε
0:=/ba∇dblε˜c0/ba∇dbll−,ε′
˙Bd
2−1
2,1+/ba∇dbl˜c0/ba∇dbll+,ε′
˙Bd
2
2,1+/ba∇dblv0/ba∇dbll
˙Bd
2
2,1+/ba∇dbl(˜c0,v0)/ba∇dblh
˙Bd
2+1
2,1≤α,
the system (3.8)with the initial data (c0,v0)admits a unique global-in-time solution Z= (˜c,v)in the set
E:=/braceleftbigg
(˜c,v)/vextendsingle/vextendsingle/vextendsingle/vextendsingleε˜cl−,ε′∈ Cb(R+:˙Bd
2−1
2,1), ε3˜cl−,ε′∈L1(R+;˙Bd
2−1
2,1),˜cl+,ε′∈ Cb(R+:˙Bd
2
2,1),
˜cl+,ε′∈L1(R+;˙Bd
2+2
2,1), vl∈ Cb(R+;˙Bd
2
2,1),εvl−,ε′∈L1(R+;˙Bd
2
2,1),vl+,ε′∈L1(R+;˙Bd
2+1
2,1),
(˜c,v)h∈ Cb(R+;˙Bd
2+1
2,1)∩L1(R+,˙Bd
2+1
2,1), Wl∈ Cb(R+;˙Bd
2
2,1)∩L1(R+;˙Bd
2
2,1)/bracerightbigg
where we denote W:=−∂tv.
Moreover, we have the following inequality :
Z(t)≤CZε′
0
where
Z(t):=/ba∇dblε˜c/ba∇dbll−,ε′
L∞/parenleftbig
˙Bd
2−1
2,1/parenrightbig+/ba∇dbl˜c/ba∇dbll+,ε′
L∞/parenleftbig
˙Bd
2
2,1/parenrightbig+/ba∇dblv/ba∇dbll
L∞/parenleftbig
˙Bd
2
2,1/parenrightbig+/ba∇dbl(˜c,v)/ba∇dblh
L∞/parenleftbig
˙Bd
2+1
2,1/parenrightbig+ε2/ba∇dblε˜c/ba∇dbll−,ε′
L1/parenleftbig
˙Bd
2−1
2,1/parenrightbig+ε/ba∇dblv/ba∇dbll−,ε′
L1/parenleftbig
˙Bd
2
2,1/parenrightbig
+/ba∇dbl˜c/ba∇dbll+,ε′
L1/parenleftbig
˙Bd
2+2
2,1/parenrightbig+/ba∇dblv/ba∇dbll+,ε′
L1/parenleftbig
˙Bd
2+1
2,1/parenrightbig+/ba∇dbl(˜c,v)/ba∇dblh
L1/parenleftbig
˙Bd
2+1
2,1/parenrightbig+/ba∇dblW/ba∇dbl
L∞/parenleftbig
˙Bd
2
2,1/parenrightbig+/ba∇dblW/ba∇dbl
L1/parenleftbig
˙Bd
2
2,1/parenrightbig.
The first step is to approximate (3.9).
(1)Approximate systems
Let us take Jnthe spectral truncation operator on {ξ∈Rd, n−1≤ |ξ| ≤n}.
We consider the following system :
d
dt/parenleftbigg
˜c
v/parenrightbigg
+/parenleftbigg
Jn(Jn(v)·(∇Jn(c)))+ ˜γJn(Jn(c)div(Jn(v)))
Jn(Jn(v)·∇(Jn(v)))+ ˜γJn(Jn(c)∇(Jn(c)))+Jn(v)/parenrightbigg
=/parenleftbigg
0
−ε2∇(−∆)−1Jn(G(Jn(˜c)))/parenrightbigg
.
•By the Cauchy-Lipschitz theorem, we have (using the spectral truncation operator) that this system
admits a maximal solution (cn,vn)∈ C1([0,Tn[:L2)with initial data ( Jnc0,Jnv0)for alln∈N.
•We have Jncn=cnandJn(vn) =vn(by using the uniqueness in the previous system) and thus:
/braceleftbigg∂tcn+Jn(vn·∇cn)+ ˜γJn(cndiv(vn)) = 0,
∂tvn+Jn(vn·∇vn)+ ˜γJn(cn∇cn)+vn=−ε2∇(−∆)−1Jn(G(˜cn)).
•From the lemma 3.9, we deduce (the nindex corresponding to the sequence (cn,vn)):
Ln(t)+/integraldisplayt
0Hn(τ)dτ/lessorsimilarLn(0)≤ L(0).
In particular (by argument of extension of the maximal solut ion), we have that Tn= +∞.
(2)Convergence of the sequence
The previous estimates guarantee that (˜cn,vn)n∈Nis a bounded sequence of E,whereEis the functional
space described in the theorem.20 VALENTIN LEMARIÉ
In particular, (˜cn,vn)n∈Nis bounded in L∞(R+;˙Bd
2
2,1)∩L1(R+;˙Bd
2+2
2,1)and inL∞(R+;˙Bd
2+1
2,1)∩L1(R+;˙Bd
2+1
2,1)
at low and respectively high frequencies level, so bounded ( by interpolation) in L2/parenleftbigg
˙Bd
2+1
2,1/parenrightbigg
.
We know that ˙Bd
2
2,1is included continuously in L∞, hence˙Bd
2+1
2,1is locally compact in L2.
We can therefore apply the Ascoli theorem and after diagonal extraction, we gather that, up to subse-
quence,(cn,vn)n∈Nconverges to some (c,v)inC([0,T[;S′(Rd)).
By classical arguments of weak compactness, one can conclude as in e.g [4] that (c,v)belongs to Eand
that(c,v)is a solution of the initial system.
3.4.Proof of uniqueness.
LetZ1= (c1,v1)andZ2= (c2,v2)be two solutions. We denote δc:=c1−c2, δv:=v1−v2andδZ:=Z1−Z2.
In particular, we have :
/braceleftBigg
∂tδc+v2·∇δc+ ˜γc2div(δv) =−δv∇c1−˜γδcdiv(v1)
∂tδv+v2·∇δv+ ˜γc2∇δc+ε′2∇(−∆)−1δc+δv=−δv·∇v1−˜γδc∇c1−ε2∇(−∆)−1(F(˜c1)−F(˜c2)).
Lemma 3.11. We have the inequality :
/ba∇dblδZ(t)/ba∇dblh
˙Bd
2
2,1+/ba∇dblδv(t)/ba∇dbll
˙Bd
2
2,1+/ba∇dblδc(t)/ba∇dbll+,ε
˙Bd
2
2,1+ε/ba∇dblδc(t)/ba∇dbll−,ε
˙Bd
2−1
2,1/lessorsimilar/integraldisplayt
0(L1+L2+H1+H2+1)(τ)/parenleftbigg
/ba∇dblδZ(t)/ba∇dblh
˙Bd
2
2,1
+/ba∇dblδv(t)/ba∇dbll
˙Bd
2
2,1+/ba∇dblδc(t)/ba∇dbll+,ε
˙Bd
2
2,1+ε/ba∇dblδc(t)/ba∇dbll−,ε
˙Bd
2−1
2,1/parenrightbigg
dτ
whereLi,Hifori∈ {1,2}correspond to LandHin(3.18) forZi.
Once this lemma is proven, it is easy to conclude the uniquene ss by Grönwall’s lemma.
Proof.
(1) Estimate for high frequencies :
Let us start by proving an estimate for high frequencies. By ap plying the localization operation
˙∆j, we get :
∂tδcj+v2·∇δcj+ ˜γc2div(δvj) =/bracketleftBig
v2·∇,˙∆j/bracketrightBig
δc+/bracketleftBig
˜γc2,˙∆j/bracketrightBig
div(δc)−˙∆j(δv·∇c1+ ˜γδcdiv(v1))
∂tδvj+v2·∇δvj+ ˜γc2∇δcj+ε′2∇(−∆)−1δcj+δvj=/bracketleftBig
v2·∇,˙∆j/bracketrightBig
δv+/bracketleftBig
˜γc2∇,˙∆j/bracketrightBig
δc
−˙∆j/parenleftbig
δv·∇v1+ ˜γδc∇c1+ε2∇(−∆)−1(F(˜c1)−F(˜c2))/parenrightbig
On the one hand, owing to commutator estimates (see e.g [4]), we have:
/ba∇dbl/bracketleftBig
v2·∇,˙∆j/bracketrightBig
δc+/bracketleftBig
˜γc2,˙∆j/bracketrightBig
div(δc) +/bracketleftBig
v2·∇,˙∆j/bracketrightBig
δv+/bracketleftBig
˜γc2∇,˙∆j/bracketrightBig
δc/ba∇dblh
˙Bd
2
2,1/lessorsimilar/ba∇dblZ2/ba∇dbl˙Bd
2+1
2,1/ba∇dblδZ/ba∇dblh
˙Bd
2
2,1
/lessorsimilarH2(τ)/ba∇dblδZ/ba∇dblh
˙Bd
2
2,1.
On the other hand, we have by integration by parts:
• −/integraldisplay
Rdv2·∇δcjδcjdx=1
2/integraldisplay
Rddiv(v2)|δcj|2dx,
• −/integraldisplay
Rdv2·∇δvj·δvjdx=1
2/integraldisplay
Rddiv(v2)|δvj|2dx,
• −/integraldisplay
Rd˜γ˜c2div(δvj)δcjdx−/integraldisplay
Rd˜γ˜c2∇δcj·δvjdx=−/integraldisplay
Rd˜γ˜c2div(δcjδvj)dx=/integraldisplay
Rd˜γ∇c2·(δcjδvj)dx.PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 21
Then we have for all j≤ −1:
−/integraldisplay
Rdv2·∇δcjδcjdx−/integraldisplay
Rdv2·∇δvj·δvjdx−/integraldisplay
Rd˜γ˜c2div(δvj)δcjdx−/integraldisplay
Rd˜γ˜c2∇δcj·δvjdx
=1
2/integraldisplay
Rddiv(v2)|δZj|2dx+/integraldisplay
Rd˜γ∇c2·(δcjδvj)dx
/lessorsimilaraj2−jd
2/ba∇dblZ2/ba∇dbl˙Bd
2+1
2,1/ba∇dblδZ/ba∇dbl˙Bd
2
2,1/ba∇dblδZ/ba∇dbl˙Bd
2
2,1
/lessorsimilaraj2−jd
2H2(t)/ba∇dblδZ/ba∇dbl˙Bd
2
2,1/ba∇dblδZ/ba∇dbl˙Bd
2
2,1.
We have also :
/ba∇dbl−˙∆j(δv·∇c1+ ˜γδcdiv(v1))−˙∆j/parenleftbig
δv·∇v1+ ˜γδc∇c1+ε2∇(−∆)−1(F(˜c1)−F(˜c2))/parenrightbig
/ba∇dblL2
/lessorsimilaraj2−jd
2/ba∇dblδZ/ba∇dbl˙Bd
2
2,1/ba∇dblZ1/ba∇dbl˙Bd
2+1
2,1
/lessorsimilaraj2−jd
2/ba∇dblδZ/ba∇dbl˙Bd
2
2,1H1.
We then have (taking the scalar product with δZjin the previous system) the following estimate:
/ba∇dblδZ/ba∇dblh
˙Bd
2
2,1+/integraldisplayt
0/ba∇dblδZ/ba∇dblh
˙Bd
2
2,1dτ/lessorsimilar/integraldisplayt
0(H1+H2)(τ)/ba∇dblδZ/ba∇dblh
˙Bd
2
2,1dτ.
(2) Estimates for low frequencies :
As for the study of the Euler-Poisson system, we will look at δvin˙Bd
2
2,1, the very low frequencies
ofεδcin˙Bd
2−1
2,1and the medium ones of δcin˙Bd
2
2,1.
For the above system, we obtain the following estimates:
/ba∇dblεδc(t)/ba∇dbll−,ε
˙Bd
2−1
2,1/lessorsimilar/integraldisplayt
0ε/bracketleftBigg
/ba∇dblδv/ba∇dbll−,ε
˙Bd
2
2,1+/ba∇dbl(Z1,Z2)/ba∇dbl˙Bd
2
2,1/ba∇dblδZ/ba∇dbl˙Bd
2
2,1/bracketrightBigg
dτ
/ba∇dblδc(t)/ba∇dbll+,ε
˙Bd
2
2,1/lessorsimilar/integraldisplayt
0/bracketleftBigg
/ba∇dblδv/ba∇dbll+,ε
˙Bd
2+1
2,1+/ba∇dblZ2/ba∇dbl˙Bd
2
2,1/parenleftBigg
/ba∇dblδZ/ba∇dbll
˙Bd
2+1
2,1+/ba∇dblδZ/ba∇dblh
˙Bd
2
2,1/parenrightBigg
+/ba∇dblδZ/ba∇dbl˙Bd
2
2,1/ba∇dblZ1/ba∇dbl˙Bd
2+1
2,1/bracketrightBigg
dτ
/ba∇dblδv/ba∇dbll
˙Bd
2+1
2,1+/integraldisplayt
0/ba∇dblδv/ba∇dbll
˙Bd
2+1
2,1dτ/lessorsimilar/integraldisplayt
0/bracketleftbigg
ε2/ba∇dblδc/ba∇dbll−,ε
˙Bd
2−1
2,1+/ba∇dblδc/ba∇dbll+,ε
˙Bd
2+1
2,1+/ba∇dblZ2/ba∇dbl˙Bd
2
2,1/parenleftBigg
/ba∇dblδZ/ba∇dbll
˙Bd
2+1
2,1+/ba∇dblδZ/ba∇dblh
˙Bd
2
2,1/parenrightBigg
+/ba∇dblδZ/ba∇dbl˙Bd
2
2,1/ba∇dblZ1/ba∇dbl˙Bd
2+1
2,1+ε2/parenleftbig
/ba∇dblδc/ba∇dbl˙Bd
2
2,1/ba∇dbl(c1,c2)/ba∇dbl˙Bd
2−1
2,1
+/ba∇dblδc/ba∇dbl˙Bd
2−1
2,1/ba∇dbl(c1,c2)/ba∇dbl˙Bd
2
2,1/parenrightbig/bracketrightbigg
dτ.
(3) Final estimate : We now put together all the estimates and observe that some t erms in the right-hand
side are negligible compared to the left-hand side (thanks i n particular to our lemma about 3.9 a
priori estimates) and we obtain the final result.
/square
We deduce by change of variable (2.2) and lemma A.4 the statem ent of the theorem 2.1.
4.Keller-Segel parabolic/elliptical system
The goal of this section is to justify the convergence of the d ensity solution of the first equation of (1.3)
to the unique solution of (1.5) when εtends to 0.
We deduce from the theorem 2.1 the following theorem which wi ll allow us to study the singular limit of
the Euler-Poisson system:22 VALENTIN LEMARIÉ
Theorem 4.1. Let beε >0. Letε′be defined as (3.13). There exists a positive constant αsuch that for all
ε′≤1
2and data Zε
0= (̺ε
0−̺,vε
0)/parenleftbigg
˙Bd
2−1
2,1∩˙Bd
2+1
2,1/parenrightbigg
×/parenleftbigg
˙Bd
2
2,1∩˙Bd
2+1
2,1/parenrightbigg
satisfying :
Zε
0:=/ba∇dbl̺ε
0−̺/ba∇dbll−, ε′ε−1
˙Bd
2−1
2,1+/ba∇dbl̺ε
0−̺/ba∇dbll+, ε′ε−1, ε
˙Bd
2
2,1+ε/ba∇dblvε
0/ba∇dbll, ε−1
˙Bd
2
2,1+ε/ba∇dbl(̺ε
0−̺,εvε
0)/ba∇dblh, ε−1
˙Bd
2+1
2,1≤α,
the system (1.1)with the initial data (̺ε
0,vε
0)admits a unique global-in-time solution Zε= (̺ε−¯ρ,vε)in the
set
˜E:=/braceleftbigg
(̺ε−̺,vε)/vextendsingle/vextendsingle/vextendsingle/vextendsingle(̺ε−̺)l−, ε′ε−1∈ Cb(R+:˙Bd
2−1
2,1), ε(̺ε−̺)l−, ε′ε−1∈L1(R+;˙Bd
2−1
2,1),
(̺ε−̺)l+, ε′ε−1, ε−1∈ Cb(R+:˙Bd
2
2,1), ε(̺ε−̺)l+, ε′ε−1∈L1(R+;˙Bd
2+1
2,1),(̺ε−̺)l+, ε′ε−1, ε−1∈L1(R+;˙Bd
2+1
2,1),
ε(vε)l, ε−1∈ Cb(R+;˙Bd
2
2,1),(vε)l−, ε′ε−1∈L1(R+;˙Bd
2
2,1),(vε)l+, ε′ε−1∈L1(R+;˙Bd
2+1
2,1),
(̺ε−̺,vε)h,ε−1∈ Cb(R+;˙Bd
2−1
2,1)∩L1(R+;˙Bd
2+1
2,1), wε∈ Cb(R+;˙Bd
2
2,1)∩L1(R+;˙Bd
2
2,1)/bracerightbigg
where we have denoted wε:=ε∇(P(̺ε))
̺ε+vε+ε∇(−∆)−1(̺ε−̺).
Moreover, we have the following inequality :
Zε(t)≤CZε
0
where
Zε(t):=/ba∇dbl̺ε−̺/ba∇dbll−, ε′ε−1
L∞/parenleftbigg
˙Bd
2−1
2,1/parenrightbigg+/ba∇dbl̺ε−̺/ba∇dbll+, ε′ε−1, ε−1
L∞/parenleftbigg
˙Bd
2
2,1/parenrightbigg+ε/ba∇dblvε/ba∇dbll, ε−1
L∞/parenleftbigg
˙Bd
2
2,1/parenrightbigg+ε/ba∇dbl(̺ε−̺,εvε)/ba∇dblh, ε−1
L∞/parenleftbigg
˙Bd
2+1
2,1/parenrightbigg
+/ba∇dbl̺ε−̺/ba∇dbll−, ε′ε−1
L1/parenleftbigg
˙Bd
2−1
2,1/parenrightbigg+/ba∇dblvε/ba∇dbll−, ε′ε−1
L1/parenleftbigg
˙Bd
2
2,1/parenrightbigg+/ba∇dbl̺ε−̺/ba∇dbll+, ε
L1/parenleftbigg
˙Bd
2+2
2,1/parenrightbigg+/ba∇dblvε/ba∇dbll+, ε
L1/parenleftbigg
˙Bd
2+1
2,1/parenrightbigg
+ε−1/ba∇dbl(̺ε−̺,εvε)/ba∇dblh,ε−1
L1/parenleftbigg
˙Bd
2+1
2,1/parenrightbigg+/ba∇dblwε/ba∇dbl
L∞/parenleftbigg
˙Bd
2
2,1/parenrightbigg+ε−2/ba∇dblwε/ba∇dbl
L1/parenleftbigg
˙Bd
2
2,1/parenrightbigg.
We notice with theorem 4.1 ensures that ˜Wε=O(ε)inL1(R+;˙Bd
2
2,1). As the first equation of (1.3) can
be rewritten as
∂t˜̺ε−∆(P(˜̺ε))−div/parenleftbig
˜̺ε∇(−∆)−1(˜̺ε−̺)/parenrightbig
= div(˜̺ε˜Wε),
we suspect that the density will tend to satisfy the paraboli c-elliptic Keller-Segel system (1.5) supplemented
with the initial data lim
ε→0˜̺ε
0.
Let us now rigorously prove the theorem 2.2 :
Proof. Let us justify quickly that for all N0satisfying (2.4), there exists a unique global-in-time sol utionN
of (1.5) in Cb/parenleftbigg
R+;˙Bd
2−1
2,1∩˙Bd
2
2,1/parenrightbigg
∩L1/parenleftbigg
R+;˙Bd
2+2
2,1∩˙Bd
2−1
2,1/parenrightbigg
satisfying (2.5).
In terms of ˜N:=N−̺,the equation (1.5) is rewritten :
∂t˜N−∆(P(N))−div/parenleftBig
N∇(−∆)−1˜N/parenrightBig
= 0.
By the Taylor-Lagrange formula, we notice that :
P(N)−P(̺) =˜NP′(̺)+g(˜N)
wheregis a smooth function vanishing at 0(and also its first derivative).
(1) Low frequency analysis
We can rewrite this system as :
∂t˜N+̺˜N= ∆/parenleftbig˜NP′(̺)+g(˜N)/parenrightbig
+div/parenleftBig
˜N∇(−∆)−1˜N/parenrightBig
·PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 23
We then obtain the following estimates:
/ba∇dbl˜N(t)/ba∇dbll
˙Bd
2−1
2,1+/integraldisplayt
0/ba∇dbl˜N/ba∇dbll
˙Bd
2−1
2,1dτ/lessorsimilar/ba∇dbl˜N0/ba∇dbll
˙Bd
2−1
2,1+/integraldisplayt
0/ba∇dblg(˜N)/ba∇dbl˙Bd
2+1
2,1dτ+/integraldisplayt
0/ba∇dbl˜N/ba∇dbll
˙Bd
2+1
2,1dτ+/integraldisplayt
0/ba∇dbl˜N/ba∇dbl˙Bd
2
2,1/ba∇dbl˜N/ba∇dbl˙Bd
2−1
2,1dτ,
We note that the term/integraldisplayt
0/ba∇dbl˜N/ba∇dbll
˙Bd
2+1
2,1dτis negligible compared to/integraldisplayt
0/ba∇dbl˜N/ba∇dbll
˙Bd
2−1
2,1dτ.
We deduce that :
/ba∇dbl˜N(t)/ba∇dbll
˙Bd
2−1
2,1+/integraldisplayt
0/ba∇dbl˜N/ba∇dbll
˙Bd
2−1
2,1dτ/lessorsimilar/ba∇dbl˜N0/ba∇dbll
˙Bd
2−1
2,1+/integraldisplayt
0/ba∇dbl˜N/ba∇dbl˙Bd
2−1
2,1∩˙Bd
2
2,1/ba∇dbl˜N/ba∇dbl˙Bd
2+2
2,1∩˙Bd
2−1
2,1dτ.
(2) High frequency analysis
We can rewrite the system as :
∂t˜N−P′(̺)∆˜N= ∆(g(˜N))−div(N∇(−∆)−1˜N).
Then, we get :
/ba∇dbl˜N(t)/ba∇dblh
˙Bd
2
2,1+/integraldisplayt
0/ba∇dbl˜N/ba∇dblh
˙Bd
2+2
2,1dτ/lessorsimilar/ba∇dbl˜N0/ba∇dblh
˙Bd
2
2,1+/integraldisplayt
0/ba∇dbl˜N/ba∇dbl˙Bd
2+2
2,1/ba∇dbl˜N/ba∇dbl˙Bd
2
2,1dτ+/integraldisplayt
0/ba∇dbl˜N/ba∇dblh
˙Bd
2
2,1dτ
+/integraldisplayt
0/parenleftBigg
/ba∇dbl˜N/ba∇dbl˙Bd
2+1
2,1/ba∇dbl˜N/ba∇dbl˙Bd
2−1
2,1+/ba∇dbl˜N/ba∇dbl2
˙Bd
2
2,1/parenrightBigg
dτ.
Thus, we have :
/ba∇dbl˜N(t)/ba∇dblh
˙Bd
2
2,1+/integraldisplayt
0/ba∇dbl˜N/ba∇dblh
˙Bd
2+2
2,1dτ/lessorsimilar/ba∇dbl˜N0/ba∇dblh
˙Bd
2
2,1+/integraldisplayt
0/parenleftBigg
/ba∇dbl˜N/ba∇dbl˙Bd
2+2
2,1/ba∇dbl˜N/ba∇dbl˙Bd
2
2,1+/ba∇dbl˜N/ba∇dbl˙Bd
2−1
2,1/ba∇dbl˜N/ba∇dbl˙Bd
2+1
2,1/parenrightBigg
dτ.
(3) A priori estimate
By gathering the previous information, we get:
/ba∇dbl˜N(t)/ba∇dbl˙Bd
2−1
2,1∩˙Bd
2
2,1+/integraldisplayt
0/ba∇dbl˜N/ba∇dbl˙Bd
2+2
2,1∩˙Bd
2−1
2,1dτ/lessorsimilar/ba∇dbl˜N0/ba∇dbl˙Bd
2−1
2,1∩˙Bd
2
2,1+/integraldisplayt
0/ba∇dbl˜N/ba∇dbl˙Bd
2−1
2,1∩˙Bd
2
2,1/ba∇dbl˜N/ba∇dbl˙Bd
2+2
2,1∩˙Bd
2−1
2,1dτ.
Then, we have:
/ba∇dbl˜N(t)/ba∇dbl˙Bd
2−1
2,1∩˙Bd
2
2,1+/integraldisplayt
0/ba∇dbl˜N/ba∇dbl˙Bd
2+2
2,1∩˙Bd
2−1
2,1dτ/lessorsimilar/ba∇dbl˜N0/ba∇dbl˙Bd
2−1
2,1∩˙Bd
2
2,1.
Hence (2.5).
By taking advantage of the Picard fixed point theorem in the fun ctional framework given by the
inequalities above, we obtain an unique global-in-time sol ution˜Nof (1.5) in Cb/parenleftBig
R+;˙Bd
2−1
2,1∩˙Bd
2
2,1/parenrightBig
∩
L1/parenleftBig
R+;˙Bd
2+2
2,1∩˙Bd
2−1
2,1/parenrightBig
satisfying (2.5).
In order to prove the last part of this theorem, let us observe that(N,˜̺ε)satisfies :
/braceleftBigg
∂t˜̺ε−∆(P(˜̺ε))−div/parenleftbig
˜̺ε∇(−∆)−1(˜̺ε−̺/parenrightbig
= div/parenleftBig
˜̺ε˜Wε/parenrightBig
,
∂tN−∆(P(N))−div/parenleftbig
N∇(−∆)−1(N−̺)/parenrightbig
= 0.
Let us denote δN=N−˜̺ε. We obtain :
∂tδN+∆(P(˜̺ε)−∆(P(N))+̺ δN−div/parenleftbig
δN∇(−∆)−1(N−̺)/parenrightbig
−div/parenleftbig
(˜̺ε−̺)∇(−∆)−1δN/parenrightbig
=−div/parenleftBig
˜̺ε˜Wε/parenrightBig
·
Let us look estimate δNat the level of regularity ˙Bd
2−1
2,1. We have :
/integraldisplayt
0/ba∇dbldiv(˜̺ε˜Wε)/ba∇dbl˙Bd
2−1
2,1dτ/lessorsimilar/integraldisplayt
0/ba∇dbl˜̺ε˜Wε/ba∇dbl˙Bd
2
2,1dτ/lessorsimilar/ba∇dbl˜̺ε/ba∇dbl
L∞
t/parenleftbigg
˙Bd
2
2,1/parenrightbigg/integraldisplayt
0/ba∇dbl˜Wε/ba∇dbl˙Bd
2
2,1dτ/lessorsimilarαε,24 VALENTIN LEMARIÉ
/integraldisplayt
0/ba∇dbldiv/parenleftbig
δN∇(−∆)−1(N−̺/parenrightbig
/ba∇dbl˙Bd
2−1
2,1dτ/lessorsimilar/integraldisplayt
0/ba∇dblδN∇(−∆)−1(N−̺)/ba∇dbl˙Bd
2
2,1dτ
/lessorsimilar/integraldisplayt
0/ba∇dblδN/ba∇dbl˙Bd
2
2,1/ba∇dbl(N−̺)/ba∇dbl˙Bd
2−1
2,1dτ
/lessorsimilarα/integraldisplayt
0/ba∇dblδN/ba∇dbl˙Bd
2
2,1dτ,
/integraldisplayt
0/ba∇dbldiv/parenleftbig
(˜̺ε−̺)∇(−∆)−1δN/parenrightbig
/ba∇dbl˙Bd
2−1
2,1dτ/lessorsimilar/integraldisplayt
0/ba∇dbl(˜̺ε−̺)∇(−∆)−1δN/ba∇dbl˙Bd
2
2,1dτ
/lessorsimilar/ba∇dbl˜̺ε−̺/ba∇dbl
L∞(˙Bd
2
2,1)/integraldisplayt
0/ba∇dblδN/ba∇dbl˙Bd
2−1
2,1dτ
/lessorsimilarα/integraldisplayt
0/ba∇dblδN/ba∇dbl˙Bd
2−1
2,1dτ.
In order to study ∆(P(˜̺ε)−P(N)), we use the identity :
∆(P(φ)) = ∆φP′(φ)+|∇φ|2P′′(φ).
Hence, we have :
∆(P(˜̺ε)−P(N)) =P′(̺)∆(δN)+(P′(N)−P′(̺))∆(δN) +∆˜̺ε/parenleftbig
P′(N)−P′(˜̺ε)/parenrightbig
+/parenleftbig
|∇N|2−|∇˜̺ε|2/parenrightbig
P′′(N)+|∇˜̺ε|2/parenleftbig
P′′(N)−P′′(˜̺ε)/parenrightbig
·
Let bound the r.h.s in L1(R+;˙Bd
2−1
2,1)(in particular, the estimate of Theorem 4.1 will be used but a lso the
different lemmas in the appendix) :
/integraldisplayt
0/ba∇dbl∆(δN)/parenleftbig
P′(N)−P′(̺)/parenrightbig
/ba∇dbl˙Bd
2−1
2,1dτ/lessorsimilar/integraldisplayt
0/ba∇dblδN/ba∇dbl˙Bd
2+1
2,1/ba∇dblN−̺/ba∇dbl˙Bd
2
2,1dτ/lessorsimilarα/integraldisplayt
0/ba∇dblδN/ba∇dbl˙Bd
2+1
2,1dτ,
/integraldisplayt
0/ba∇dbl∆˜̺ε/parenleftbig
P′(N)−P′(˜̺ε)/parenrightbig
/ba∇dbl˙Bd
2−1
2,1dτ/lessorsimilar/integraldisplayt
0/ba∇dbl˜̺ε−̺/ba∇dbl˙Bd
2+1
2,1/ba∇dblδN/ba∇dbl˙Bd
2
2,1/ba∇dbl(N,˜̺ε)/ba∇dbl˙Bd
2
2,1dτ
/lessorsimilar(α2+α)/ba∇dblδN/ba∇dbl
L∞(˙Bd
2
2,1),
/integraldisplayt
0/ba∇dbl|∇˜̺ε|2/parenleftbig
P′′(N)−P′′(̺)/parenrightbig
/ba∇dbl˙Bd
2−1
2,1dτ/lessorsimilar/integraldisplayt
0/ba∇dbl˜̺ε/ba∇dbl˙Bd
2+1
2,1/ba∇dbl˜̺ε/ba∇dbl˙Bd
2
2,1/ba∇dblδN/ba∇dbl˙Bd
2
2,1/ba∇dbl(˜N,˜̺ε)/ba∇dbl˙Bd
2
2,1dτ
/lessorsimilar(α3+α2)/integraldisplayt
0/ba∇dblδN/ba∇dbl˙Bd
2
2,1dτ,
/integraldisplayt
0/ba∇dbl/parenleftbig
|∇N|2−|∇˜̺ε|2/parenrightbig
P′′(N)/ba∇dbl˙Bd
2−1
2,1dτ/lessorsimilar/integraldisplayt
0/ba∇dbl∇N−∇˜̺ε/ba∇dbl˙Bd
2
2,1/ba∇dbl(∇N,∇˜̺ε)/ba∇dbl˙Bd
2−1
2,1/ba∇dblN/ba∇dbl˙Bd
2
2,1dτ
/lessorsimilar(α2+α)/integraldisplayt
0/ba∇dblδN/ba∇dbl˙Bd
2+1
2,1dτ.
In particular, δNsatisfies :
∂tδN−P′(̺)∆δN+̺δN=−div/parenleftBig
˜̺ε˜Wε/parenrightBig
+div/parenleftbig
δN∇(−∆)−1(N−̺/parenrightbig
+div/parenleftbig
(˜̺ε−̺)∇(−∆)−1δN/parenrightbig
−∆(δN)(P′(N)−P′(̺))−∆˜̺ε/parenleftbig
P′(N)−P′(˜̺ε)/parenrightbig
−|∇˜̺ε|2/parenleftbig
P′′(N)−P′′(̺)/parenrightbig
−/parenleftbig
|∇N|2−|∇˜̺ε|2/parenrightbig
P′′(N).PARABOLIC-ELLIPTIC KELLER-SEGEL’S SYSTEM 25
By using all previous inequalities, we get :
/ba∇dblδN/ba∇dbl
L∞
t(R+;˙Bd
2−1
2,1)∩L1(R+;˙Bd
2−1
2,1)∩L1(R+;˙Bd
2+1
2,1)/lessorsimilar/ba∇dblδN(0)/ba∇dbl˙Bd
2−2
2,1+αε
+(α+α2+α3)/ba∇dblδN/ba∇dbl
L∞
t(R+;˙Bd
2−1
2,1)∩L1(R+;˙Bd
2−1
2,1)∩L1(R+;˙Bd
2+1
2,1).
Therefore, for αsmall enough, we get :
/ba∇dblδN/ba∇dbl
L∞(R+;˙Bd
2−1
2,1)∩L1(R+;˙Bd
2−1
2,1∩˙Bd
2+1
2,1)/lessorsimilar/ba∇dblδN(0)/ba∇dbl˙Bd
2−1
2,1+αε
which completes the proof of the theorem. /square
Appendix A.
Here we recall classic lemmas involving differential inequa lities and some basic properties on Besov spaces
and product estimates have been be used repeatedly in the art icle.
Lemma A.1. LetX: [0,T]→R+be a continuous function such that X2is differentiable. Assume that
there exist a constant c≥0and a measurable function A: [0,T]→R+such that
1
2d
dtX2+cX2≤AXa.e. on[0,T].
Then, for all t∈[0,T], we have:
X(t)+c/integraldisplayt
0X(τ)dτ≤X0+/integraldisplayt
0A(τ)dτ.
This classical lemma can be found for instance in [16] :
Lemma A.2. LetT >0. LetL: [0,T]→RandH: [0,T]→Rtwo continous positive functions on [0,T]
such that
L(t)+c/integraldisplayt
0H(τ)dτ≤ L(0)+C/integraldisplayt
0L(τ)H(τ)dτ,
andL(0)≤α <<1then for all t∈[0,T], we have :
L(t)+c
2/integraldisplayt
0H(τ)dτ≤ L(0).
Proof. Letα∈]0,c
2C[. We set T0= sup
T1∈[0,T]/braceleftBigg
sup
t∈[0,T1]L(t)≤α/bracerightBigg
. Thissupexists because the previous set is
non empty ( 0belongs to this set) and since Lis continuous, T0>0. In time t=T0, we get :
L(T0)+c/integraldisplayT0
0H(τ)dτ≤ L(0)+C/integraldisplayT0
0H(τ)L(τ)dτ≤ L(0)+αC/integraldisplayT0
0H(τ)dτ.
Hence we have :
L(T0)+c
2/integraldisplayT0
0H(τ)dτ≤ L(0).
AsL(t)≤ L(T0)for allt∈[0,T0]and/integraldisplayt
0H(τ)dτ≤/integraldisplayT0
0H(τ)dτ, we obtain with the previous inequality :
L(t)≤α∀t∈[0,T0].
With the continuity of L, we must have T0=T, whence the result. /square
The following lemmas are classic results on Besov spaces ( see e.g. [4]).26 VALENTIN LEMARIÉ
Lemma A.3. For alld≥2, the pointwise product extends in a continuous application of˙Bd
2−1
2,1(Rd)×˙Bd
2
2,1(Rd)
to˙Bd
2−1
2,1(Rd)and˙Bd
2
2,1is a multiplicative algebra for all d≥1.
For alld≥1, we have for (u,v)∈˙Bd
2
2,1∩˙Bd
2+1
2,1thatuv∈˙Bd
2+1
2,1and the following inequality :
/ba∇dbluv/ba∇dbl˙Bd
2+1
2,1/lessorsimilar/ba∇dblu/ba∇dbl˙Bd
2
2,1/ba∇dblv/ba∇dbl˙Bd
2+1
2,1+/ba∇dblu/ba∇dbl˙Bd
2+1
2,1/ba∇dblv/ba∇dbl˙Bd
2
2,1.
The following lemma comes from the proof in [4] of the followi ng well-known property on Besov spaces :
/ba∇dblz(α·)/ba∇dbl˙Bs
2,1≃αs−d
2/ba∇dblz/ba∇dbl˙Bs
2,1for allα >0.
Lemma A.4. Lets∈Randz∈˙Bs
2,1.We have that :
/ba∇dblz(α·)/ba∇dbll−,ε
˙Bs
2,1≃αs−d
2/ba∇dblz/ba∇dbll−,εα−1
˙Bs
2,1;/ba∇dblz(α·)/ba∇dbll+,ε
˙Bs
2,1≃αs−d
2/ba∇dblz/ba∇dbll+,εα−1, α−1
˙Bs
2,1;/ba∇dblz(α·)/ba∇dblh
˙Bs
2,1≃αs−d
2/ba∇dblz/ba∇dblh, εα−1
˙Bs
2,1
where we denote /ba∇dbl·/ba∇dbll+, α, β
˙Bs
2,1:=/summationdisplay
j∈Z
α≤2j≤β
42js/ba∇dbl·/ba∇dblL2and/ba∇dbl·/ba∇dblh,α
˙Bs
2,1:=/summationdisplay
j≥−2
2j≥α2js/ba∇dbl·/ba∇dblL2.
Lemma A.5. LetF:Rd→Rpa smooth function with F(0) = 0 . Then for all (p,r)∈[1,∞]2,s >0and
u∈˙Bs
p,r∩L∞, we have F(u)∈˙Bs
p,r∩L∞and
/ba∇dblF(u)/ba∇dbl˙Bsp,r≤C/ba∇dblu/ba∇dbl˙Bsp,r,
withCdepending only on /ba∇dblu/ba∇dblL∞,F′(and higher order derivatives), s,pandd.
Lemma A.6. Letg∈C∞(R)such that g′(0) = 0 . Then, for all u,v∈˙Bs
2,1∩L∞withs >0,
/ba∇dblg(v)−g(u)/ba∇dbl˙Bs
2,1≤C/parenleftBig
/ba∇dblv−u/ba∇dblL∞/ba∇dbl(u,v)/ba∇dbl˙Bs
2,1+/ba∇dblv−u/ba∇dbl˙Bs
2,1/ba∇dbl(u,v)/ba∇dblL∞/parenrightBig
·
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1511.04802v1.Determination_of_intrinsic_damping_of_perpendicularly_magnetized_ultrathin_films_from_time_resolved_precessional_magnetization_measurements.pdf | 1
Determination of intrinsic damping of perpendicularly magnetized
ultrathin films from time resolved precessional magnetization
measurements
Amir Capua1,*, See -hun Yang1, Timothy Phung1, Stuart S. P. Parkin1,2
1 IBM Research Division, Almaden Research Center, 650 Harry Rd., San Jose, California
95120, USA
2 Max Planck Institute for Microstructure Physics, Halle (Saale), D -06120, Germany
*e-mail: acapua@us.ibm.com
PACS number(s) : 75.78. -n
Abstract:
Magnetization dynamics are strongly influenced by damping, namely the loss of spin
angular momentum from the magnetic system to the lattice. An “effective” damping
constant αeff is often determined experimentally from the spectral linewidth of the free
induction decay of the magnetization after the system is excited to its non -equilibrium state .
Such an αeff, however, reflects both intrinsic damping as well as inhomogeneous
broadening that arises , for example, from spatial variations of the anisotropy field. In this
paper we compare measurements of the m agnetization dynamics in ultrathin non -epitaxial
films having perpendicular magnetic anisotropy using two different techniques, time-
resolved magneto optical Kerr effect (TRMOKE ) and hybrid optical -electrical
ferromagnetic resonance (OFMR) . By using a n external magnetic field that is applied at
very small angles to the film plane in the TRMOKE studies , we develop an explicit closed -
form analytical expression for the TRMOKE spectral linewidth and show how this can be
used to reliably extract the intrinsic Gilbert damping constant. The damping constant
determined in this way is in exc ellent agreement with that determined from the OFMR
method on the same samples. Our studies indicate that the asymptotic high -field approach
that is often used in the TRMOKE method to distinguish the intrinsic damping from the 2
effective damping may result in significant error , because such high external magnetic
fields are required to make this approach valid that they are out of reach . The error becomes
larger the lower is the intrinsic damping constant, and thus may account for the
anomalously high damping constants that are often reported in TRMOKE studies . In
conventional ferromagnetic resonance ( FMR ) studies , inhomogeneous contributions can
be readily distinguished from intrinsic damping contributions from the magnetic field
dependence of the FMR linewidth. Using the analogous approach, w e show how reliable
values of the intrinsic damping can be extracted from TRMOKE in two distinct magnetic
systems with significant perpendicular magnetic anisotropy: ultrathin CoFeB layers and
Co/Ni/Co trilayers.
3
I. Introduction
Spintronic nano -devices have been identified in recent years as one of the most
promisin g emerging technologies for future low power microelectronic circuits1, 2. In the
heart of the dynamical spin -state transi tion stands the energy loss parameter of the Gilbert
damping . Its accurate dete rmination is of paramount importance as it determines the
performance of key building blocks required for spin manipulation such as t he switching
current threshold of the spin transfer torq ue magnetic tunnel junction (MTJ) used in
magnetic random access memory (MRAM) as w ell as the skyrmion velocities and the
domain wall motion current threshold . Up-scaling for high logic and data capacities while
obtaining stability with high retention energies require in addition that large magnetic
anisotropies be ind uced. T hese cannot be achieved simply by engineering the geometrical
asymmetries in the nanometer -scale range , but rather require harnessing the induced spin-
orbit interaction a t the interface of the ferromagnet ic film to obtain perpendicular magnetic
anisotropy (PMA)2. Hence an increasing effort is invested in the quest for perpendicular
magnetized materials having large anisotropies with low Gilbert damping3-11.
Two distinct families of experimental methods are typically used for measurement
of Gilbert damping , namely, time-resolved pump -probe and continuous microwave
stimulated ferromagnetic resonance ( FMR ), either of which can be implemented using
optical and/or electrical methods . While in some cases good agreement between these
distinct techniques have been reported12, 13, there is often significant disagreement between
the methods14, 15. 4
When the time resolved pump -probe method is implemented using the magneto
optical Kerr effect ( TRMOKE), a clear advantage over the FMR method is gained in the
ability to operate at very high fields and frequencies16, 17. On the other hand , the FMR
method a llows operation over a wide r range of geometrical configurations . The
fundamental geometrical restriction of the TRMOKE comes from the fact that the
magnetization precession s are initiated from the perturbation of the effective anisotropy
field by the pump pulse , by momentarily increasing the lattice temperature18, 19. In cases
where the torque exerted by the effective anisotropy field is n egligible , the pump pulse
cannot sufficiently perturb the magnetization . Such a case occurs for example whenever
the magnetization lays in the plane of the sample in uniaxial thin films having
perpendicular magneti c anisotropy . Similar limitations exist if the magnetic field i s applied
perpendicular to the film . Hence in TRMOKE experiments , the external field is usually
applied at angles typically not smaller than about
10
from either the film plane or its
normal. This fact has however the consequence that the steady state magnetization
orientation , determined by the bal ancing condition for the torques , cannot be described
using a n explicit -form algebraic expression , but rather a numerical approach should be
taken5. Alternatively , the dynamics can be described using an effective damping from
which the intrinsic damping , or at least an upper bound o f its value , is estimated at the high
magnetic field limit with the limit being undetermined . These approaches are hence less
intuitive while the latter does not indicate directly on the energy losses but rather on the
combination of the energy loss rate , coherence time of the spin ensemble and geometry of
the measurement . 5
In this paper , we present an approach where the TRMOKE system is operated while
applying the magnetic field at very sm all angles with respect to the sample plane. This
enables us to use explicit closed -form analytical expressions derived for a perfectly in -
plane external magnetic field as an approximate solution. Hence , extraction of the intrinsic
Gilbert damping using an analytical model becomes possible without the need to drive the
system to the high magnetic field limit providing at the same time an intuitive
understanding of the measured responses. The validity of t he method is verified using a
highly sensitive hybrid optical -electrical FMR system (OFMR) capable of operating with
a perfectly in -plane magnetic field where the analytical expressions hold. In particular, we
bring to test the high -field asymptotic approa ch used for evaluation of the intrinsic damping
from the effective damping and show that in order for it to truly indicate the intrinsic
damping, extremely high fields need to be applied. Our analysis reveals the resonance
frequency dispersion relation as well as the inhomogeneous broadening to be the source of
this requirement which becomes more difficult to fulfill the smaller the intrinsic damping
is. The presented method is applied on two distinct families of technologically relevant
perpendicularly mag netized systems; CoFeB4, 6 and Co/Ni/Co20-23. Interestingly, the results
indicate that the Ta seed layer thickness used in CoFeB films strongly affects the intrinsic
damping , while t he static characteristics of the films remain intact . In the Co/Ni/Co trilayer
system which has in contrast a large effective anisotropy field, unexpected ly large spectral
linewidth s are measured when the external magnetic field is comparable to the effective
anisotropy field, which cannot be explained by the conventional model of no n-interacting
spins describing the inhomogeneous broadening . This suggest s that under the low stiffness 6
conditions associated with such bias fi elds, cooperative exchange interactions, as two
magnon scattering, become relevant8, 24.
II. EXPERIMENT
The experiments present ed were carried out on three PMA samples: two samples
consist ed of Co36Fe44B20 which differed by the thickness of the underlayer and a third
sample consisting of Co/Ni/Co trilayer . The CoFeB samples were characterized by low
effective anisotropy (Hkeff) values as well as by small distribution of its value in contrast to
the Co /Ni/Co trilayer system . We define here Hkeff as 2Ku/Ms-4πMs where Ku is the
anisotropy energy constant and Ms being the saturation magnetization.
The structure s of the two CoFeB samples were 50Ta|11CoFeB |11MgO |30Ta, and
100Ta|11CoFeB |11MgO |30Ta (units are in Å) and had similar Ms value s of 1200 emu/cc
and Hkeff of 1400 Oe and 1350 Oe respectively. The t hird system studied was
100AlO x|20TaN |15Pt|8Pt 75Bi25|3Co|7Ni|1.5Co |50TaN with Ms of 600 emu/cc and Hkeff
value of about 4200 Oe . All samples were grown on oxidized Si substrates using DC
magnetron sputtering and exhibited sharp perpendicular switching characteristics . The
samples consisting of CoFeB were annealed for 30 min at
275
C in contrast to the
Co/Ni/Co which was measured as deposited. Since the resultant film has a polycrystalline
texture , the in -plane anisotropy is averaged out and the films are regarded as uniaxial
crystals with the symmetry axis being perpendicular to the film plane. 7
The t wo configurations of the experimental setup were driven by a Ti:Sapphire laser
emitting 70 fs pulses at 800 nm having energy of 6 nJ. In the first configuration a standard
polar pump -probe TRMOKE was implemented with the probe pulse being a ttenuated by
15 dB compared to the pump pulse. Both beams were focused on the sample to an estimated
spot size of 10.5 m defined by the full width at half maximum (FWHM) . In the hybrid
optical -electrical OFMR system , the Ti:Sapphire laser served to pro be the magnetization
state via the magneto -optical Kerr effect after being attenuated to pulse energies of about
200 pJ and was phase -locked with a microwave oscillator in a similar configuration to the
one reported in Ref. [ 25]. For this measurement , the film was patterned into a 20 m x 20
m square island with a Au wire deposited in proximity to it, which was driven by the
microwave signal. Prior to reaching the sample, the probing laser beam traversed the
optical delay line that enabled mapping of the time axis and in particular the out of plane -
mz component of the magnetization as in the polar TRMOKE experiment . With this
configuration the OFMR realizes a conventional FMR system where the magnetization
state is read in the time-domain using the magneto optical Kerr effect and hence its high
sensitivity . The OFMR system therefore enables operation even when the external field is
applied fully in the sample plane.
III. RESULTS AND DISCUSSION
A. TRMOKE measurements on 50 Å-Ta CoFeB film
The first experiments we present were performed on the 50 Å-Ta CoFeB system
which is similar to the one studied in Ref. [4]. The TRMOKE measurement was carried 8
out at two angles of applied magnetic field,
H, of
4
and
1
measured from the surface
plane as indicated in Fig. 1. We de fine here in addition the comple mentary angle measured
from the surface normal,
2HH . Having i ts origin in the effective anisotropy, the
torque generated by the optical pump is proportional to
cos( )sins keffMH with θ being
the angle of the magnetization relative to the normal of the sample plane. Hence, f or
1H
, the angle θ becomes close to
/2 , and the resultant torque generated by the optical pump
is not strong enough to initiate reasonable precessions . For the same reason, the maximum
field measureable for the
1H
case is significantly lower than for the
4H
case. This
is clearly demonstrated in the m easured MOKE signals for the two
H angles in Fig. 2 (a).
While for
4H
the precessional motion is clearly seen even at a bias field of 12 kOe,
with
1H
the precessions are hardly observable already at a bias field of 5.5 kOe.
Additionally, it is also possible that the lower signal to noise ratio observed for
1H
may
be due to a breakdown into domains with the almost in -plane applied magnetic field26.
After reduction of the background signal, the measured data can be fitted to a decaying
sinusoidal response from which t he frequency and decay time can be extracted in the usual
manner 6 (Fig. 2(b)) . The measured precession frequency as a function of the applied
external field , H0, is plotted in Fig. 3(a). Significant differences near Hkeff are observed for
merely a change of three degrees in the angle of the applied magnetic field . In particular,
the trace for
1H
exhibits a minimum point at approximately Hkeff in contrast to the
monotonic behavior of the
4H
case. The theoretical dependence of the resonance 9
frequency on the magnetic bias field expressed in normalized units,
/keffH , with
being the resonance angular frequency and
the gyromagnetic ratio, is presented in Fig.
3(b) for several representative angles of the applied field. The resonance frequency at the
vicinity of Hkeff is very sensitive to slight changes in the angle of the applied field as
observed also in the experiment . Actually the derivative of the resonance frequency with
respect to the applied field at the vicinity of Hkeff is even more sensitive where it diverges
for
90
but reaches a value of zero for the slightest angle divergence. A discrepancy
between the measurement and the theoretical solution exists however. At field values much
higher than Hkeff the precession frequenc y should be identical for all angles (Fig. 3(b)) but
in practice the resonance frequency measured for
H of
4
is consistently higher by nearly
2 GHz than at
1
. The t heory also predicts that for the case of
4
, the resonance frequencies
should exhibit a minimum point as well which is not observed in the measurement . The
origin of the difference is not clear and may be related to the inhomogeneities in the local
fields or to the higher orders of the interface induced anisotropy which were neglected in
the theoretical calculation .
In Fig. 3(c), we plot the effective Lorentzian resonance linewidth in the frequency
domain ,
eff , defined by
2/eff eff with
eff being the measured decay time extracted
from the measured responses. Decompos ing the measured linewidth t o an intrinsic
contribution that represent s the energy loss es upon precession and an extrinsic contribution
which represent s the inhomogeneities in the local fields and is not related to energy loss of 10
the spin system , we express the linewidth as :
int eff IH .
int is given by the
Smit -Suhl formula27, 28 and equals
2/ with
denoting the intrinsic spin precession decay
time where as
IH represents the dispersion in the resonance frequencies due to the
inhomogeneities. If the variations in the resonance frequency are assumed to be primarily
caused by variations in the local effective anisotrop y field
keffH ,
IH may be given by :
/IH keff keff d dH H
. For the case of
/2H or
0H ,
eff has a closed
mathematical form. In PMA films with bias field applied in the sample plane , the
expression for
eff becomes :
0
002
00
0
0022
002 for H
2
2 for Heff keff keff keff
keff
keff keff
eff keff keff
keffkeffHH H H H
H H H
HH HH H HHH HH
, (1)
with
denot ing the Gilbert damping . The first term s in Eq. (1) stem from the intrinsic
damping , while the second term s stem from the inhomogeneous broadening . Eq. (1) shows
that while the contribution of the intrinsic part to the total spectral linewidth is finite, as the
external field approaches Hkeff either from higher or lower field values, the inhomogeneous
contribution diverges. Equation (1) further shows that for H0 >> Hkeff , the slope of
eff
becomes
2 with a constant offset given by
/2keffH . Although Eq. (1) is valid only
for
/2H , it is still instructi ve to apply it on the measured linewidth for the
4H
case. 11
The theoretical intrinsic linewidth for
/2H , inhomogeneous contribution and the sum
of the two a fter fitting
and
keffH in the range H 0 > 5000 Oe are plotted in Fig. 3(c). The
resul tant fitting values were 0.023 ±0.002 for the Gilbert damping and 175 Oe for
keffH . At
external fields comparable to Hkeff the theoretical expression derived for the
inhomogeneous broadening for a perfectly in -plane field does not describe properly the
experiment . In the theoretical analysis , at fields comparable to Hkeff, the derivative
0/d dH
diverges and therefore also the derivative
/keff d dH as understood from Fig. 3(b). In the
experiment however ,
/2H and the actual derivative
/keff d dH approache s zero.
Hence any variation in Hkeff result s in minor variation of the frequency . This mean s that the
contribution of the inhomogeneous broadening to the total linewidth is suppressed near
Hkeff in the experiment as opposed to being expanded in the theoretical calculation which
was carried out for
/2H . The result is an overestimate d theoretical linewidth near
Hkeff. After reduction of the inhomogeneous broadening , the extracted intrinsic measured
linewidth is presented in Fig. 3(c) as well showing the deviation from the theor etical
intrinsic contribution as the field approaches Hkeff.
To further investigate the e ffect of tilt ing the magnetic field , we study the TRMOKE
responses for the
1H
case. The measured linewidth for this case is presented in Fig.
3(d). In contrast to the
4H
case, the measured linewidth now increases at fields near
Hkeff as expected theoretically . Furthermore, the measured linewidth for the
1H
case is 12
well describe d by Eq. (1) even in the vicinity of Hkeff as well as for bias fields smaller than
Hkeff. The fitting result s in the same damping value of 0.023 ±0.0015 as with the
4H
case, and a variation in
keffH of 155 Oe, which is 20 Oe smaller than the value fitted for
the
4H
case.
We next turn to examine the G ilbert damping. In the absence of the demagnetization
and crystalline anisotrop y fields, the expression for the intrinsic Gilbert damping is given
by:
1 . (2)
Once the anisotropy and the demagnetization field s are included , the expression for the
intrinsic Gilbert damping becomes :
0
0
0
0
001 for
21 for
2keff
keff
keff keffdHHHd
dHHHd H H H H
, (3)
and is valid only for
2H and for crystals having uniaxial symmetry. At oth er angles
a numerical method5 should be used to relate the precession decay time to the Gilbert
damping. Eq. (3) is merely the intrinsic contribution in Eq. (1) written in the form
resembling Eq. (2) . At high fields both Eq s. (2) and (3) converge to the same result since 13
1 0dH
d. As seen in Fig. 3(b), at bias field s comparable to Hkeff the additional derivative
term of Eq. (3) becomes very significant . When substituting the measured decay time,
eff
, for
, Eq. (2) gives what is often interpreted as the “effective ” damping , αeff, from which
the intrinsic damping is measured by evaluating it at high fields when the damping becomes
asymptotically field independent. Additionally, t he asymptotic limit should be reached
with respect to the inhomogeneous contribution of Eq. (1). In Fig. 3(e), we plot the effective
damping using
eff and Eq. (2) . We further show the intrinsic damping value after
extracting the intrinsic linewidth and using Eq. (3). Examining first the effective damping
values, we see that for the two angles , the values are distinctively different at low fields
but converge at approximately 41 00 Oe (Beyond 5500 Oe the data for the
1H
case
could not be measured). In fact , the behavior of the effective damping seems to be related
to the dependence of the resonance frequency on H0 (Fig. 3(a)) in which for the
1H
case reaches an extremum while the
4H
case exhibits a monotonic behavior . Since Eq.
(2) lacks the derivative term
0/dH d , near Hkeff the effective damping is related to the
Gilbert damping by the relation:
01
effd
dH for H0 > Hkeff. Furthermore, since
does not
depend on the magnetic field to the first order, the dependence of the effective damping,
eff , on
the bias field stems from the derivative term
0 d dH which becomes larger and eventually
diverges to infinity when the magnetic field reaches Hkeff as can be inferred from Fig. 3(b)
for the case of
0H
for which Eq. (3) was derived . Hence the increase in
eff at bias 14
fields near Hkeff. The same considerations apply also for H0 < Hkeff. As the angle
H increases ,
this analysis becomes valid only for bias fields which are large enough or small enough
relative to Hkeff. When examined separately, each effective damping trace may give the
impression that at the higher fields it has become bias field independent and reached its
asymptotic value from which two very distinct Gilbert damping values of ~0.027 and
~0.039 are extracted at field values of 12 kOe and 5.5 kOe for the
4H
and
1H
measurements , respectively . These values are also rather different from the intrinsic
damping value of 0.023 extracted using the analytical model . In contrast to the effective
damping , the intrinsic damping obtained from the analytical model reveal s a constant and
continuous behavior which is field and angle independent. The presumably negative values
measure d for the
4H
case stem of course from the fac t that the expressions in Eqs. (1)
and (3) are derived for the
2H case. The error in using the effective damping in
conjunction with the asymptotic approximation compared to using the analytical model is
therefore 17% and 70% for the
4H
and
1H
measurements respectively.
It is important in addition to understand th e conse quence of using Eq. (2) rathe r than
Eq. (3) . In Fig. 3(f) we present the error in the damping value after accounting for the
inhomogeneous broadening using Eq. (2) instead of the complete expression of Eq. (3) . As
expected , the error increases as the applied field approaches Hkeff. For the measurement
taken with
4H
the error is significantly smaller due to the smaller value of the
derivative
0/d dH . 15
As mentioned previously, i n order to evaluate the intrinsic damping from the total
measured linewidth , the asymptotic limit should be reached with respect to the
inhomogeneous broadening as well (Eq. (1) ). In Fig s. 3(c) and 3(d) we see that this is not
the case where the contribution of the inhomogeneous linewidth is still large compared to
the intrinsic l inewidth . Examining Figs. 3(d) and 3(f) for the case of
1H
, we see that
the overall error of 70% resulting in the asymptotic evaluation stems from both the
contribution of inhomogeneous broadening as well as from the use of Eq. ( 2) rather than
Eq. (3) while for
4H
(Figs. 3(c) and 3(f)) the error of 17% is solely due to contribution
of the inhomogeneous broadening which was not as negligible as conceived when applying
the asymptotic approximation .
B. Comparison of TRMOKE and OFMR measurements in 100 Å-Ta CoFeB
film
We next turn to study the magnetization dynamics using the OFMR system where
the precession s are driven with the microwave signal . Hence, the external magnetic field
can be applied perfectly in the sample plane. The 100 Å-Ta CoFeB sample was used for
this experiment. Before patterning the film for the OFMR measurement, a TRMOKE
measurement was carried out at
4H
which exhibited a similar behavior to that observed
with the sample having 50 Å Ta as a seeding layer . The dependence of the resonance
frequency on the magnetic field as well as the measured linewidth and its different
contributions are presented in Figs. 4(a) and 4(b). Before reduction of the inhomogeneous 16
broadening the asymptotic effective damping was measured to be ~0.0168 while after
extraction of the intrinsic damping a value of 0.0109 ±0.0015 was measured marking a
difference of 54% (Fig. 4(c)). The fitted
keffH was 205 Oe. Fig. 4(b) shows that the origin
of the error stems from significan t contribution of the inhomogeneous broadening
compared to the intrinsi c contribution which plays a mor e significant role when the
damping is low. By us ing the criteria for the minimum field that results in
10IH eff
to estimat e the point where the asymptotic approximation would be valid , we arrive to a
value of at least 4.6 T which is rather impractic al. The threshold of this minimal f ield is
highly dependent on the damping so that for a lower damping an even higher field would
be required.
An example of a measured trace using the OFMR system at a low microwave
frequency of 2.5 GHz is presented in Fig. 4(d). The square root of the magn etization
amplitude (out of plane mz component) while preserving its sign is plotted to show detail .
The high sensitivity of the OFMR system enable s operation at very low frequencies and
bias fields. For every frequency and DC magnetic field value , several cycles of the
magnetization precession were recorded by scanning the optical delay line. The magnetic
field was then swept to fully capture the resonance . The trace should be examined
separately in two sections, be low Hkeff and above Hkeff (marked in the figure by black dashed
line). For frequencies of up to
keffH two resonances are crossed as indicated by the guiding
red dashed line which represents the out -of-phase component of the magnetization, namely
the imaginary part of the magnetic susceptibility . Hence the cross section along this line 17
gives the field dependent absorption spectrum from which the resonance frequency and
linewi dth can be identified. This spectrum is show n in Fig. 4( e) together with the fitted
lorentzian lineshapes for bias fields below and above Hkeff. The resultant resonance
frequencies of all measurements are plotted in addition in Fig. 4(a).
The resonance linewidth s extracted for bias fields larger than Hkeff, are presented in
Fig. 4(f). Here the effective magnetic field linewidth , ΔHeff, that includes the contribution
of the inhomogeneous broadening derived from the same principles that led to Eq. (1) with
/2H
is given by:
02
2
0
00
2 2
0211 for 2
4
2
with keff
eff keff keff
keff
keff keff
eff keff
keff
keffHH H H H
H
HH HHHH H H
HH
0 for keff HH (5).
The second terms in Eq. (5) denote the contribution of the inhomogeneous broadening ,
IHH
, and are frequency dependent as opposed to the case where the field is applied out
of the sample plane9. The dispersion in the effective anisotropy ,
keffH , and the intrinsic
Gilbert damping were found by fitting the linewidth in the seemingly linear range at
frequencies larger th an 7.5 GHz . The contribution s of the intrinsic and inhomogeneous
parts and the ir sum are presented as well in Fig. 4(f). 18
It is apparent that the measured linewidth at the lower frequencies is much broader
than the theoretical one. The reason for that lies in the fact that in practice the bias field is
not applied perfectly in the sample plane as well as in the fact that there migh t be locally
different orientation s of the polycrystalline grains due to the natural imperfections of the
interfaces that further result in angle distribution of
H . Since the measured field linewidth
is a projection of the spectral linewidth into the magnetic field domain, the relation between
the frequency and the field intrinsic linewidth s is given by:
1
int int
0dHdH
. The
intrinsic linewidth ,
int , in the frequency domain near Hkeff is finite , as easily seen from
Eq. (1) while the derivative term near Hkeff is zero for even the slightest angle misalignment
as already seen. H ence the field-domain linewidth diverg es to infinity as observed
experimentally. The inhomogeneous broadening component does not diverge in that
manner but is rather suppressed . To show that the excessive linewidth at low field s is
indeed related to the derivative of
0/d dH we empirically multiply the total theoretical
linewidth by the factor
0 / ( )d d H which turns out to fit the data surprisingly well (Fig.
4(f)). This is merely a phenomenological qualitative description, and a rigorous description
should still be derived.
The fitted linewidth of Fig. 4(f) results in the intrinsic damping value of
0.011 ±0.0005 and is identical to the value obtained by the TRMOKE method . Often
concerns regarding the differences between the TRMOKE and FMR measurements such
as spin wave emission away from the pump laser spot in the TRMOKE29, increase of 19
damping due to thermal heating by the pump pulse as well as differences in the nature of
the inhomogeneous broadening are raised. Such effects do not seem to be significant here .
Additionally , it is worth noting that s ince the linewidth seems to reach a linear dependence
with respect to the field at high fields , it may be naively fitted using a constant frequency -
independent inhomogeneous broadening factor . In that case an underestimated value of
~0.0096 would have been obtained . The origin of this misinterpretation is seen clearly by
examining the inhomogeneous broadening contribu tion in Fig. 4(f) that show s it as well to
exhibit a seemingly linear dependence at the high fields. Regarding the inhomogeneous
broadening , the anisotropy field dispersion,
effKH , obtained with the TRMOKE was 205
Oe while the value obtained from the OFMR system was 169 Oe . Although these values
are of the same order of magnitude , the difference is rather significant. It is possible that
the discrepancy is related to the differences in the measurement techniques. For instance,
the fact that both the pump and probe beams have the same spot size may cause an uneven
excitation across the probed region in the case of the TRMOKE measurement while in the
case of the OFMR measurement the amplitude of the microwave field decays at increasin g
distances away from the microwire. These effects may be reflect ed in the measurements as
inhomogeneous broadening. Nevertheless , the measured intrinsic damping values are
similar.
Finally , we compare the effective damping of the OFMR and the TRMOKE
measu rements without correcting for the inhomogeneous broadening in Fig. 4(g). The 20
figure shows a deviation in the low field values which is by now understood to be unrelated
to the energy losses of the system .
Furthermore, we observe that the thickness of the Ta underlayer affects the
damping. The comparison of the 50 Å -Ta CoFeB and the 100 Å -Ta CoFeB samples shows
that the increase by merely 50 Å of Ta, reduced significantly the damping while leaving
the anisot ropy field unaffected.
C. TRMOKE and OFMR measurements in Co /Ni/Co film
In the last set of measurements we study the Co /Ni/Co film which has distinctively
different static properties compared to the CoFeB samples . The sample was studied using
the TRMOKE setup at two
H angles of
1
and
4
and using the OFMR system at
0H
. The resultant resonance frequency traces are depicted in Fig. 5(a). The spectral linewidth
measure d for
4H
using the TRMOKE setup is presented in Fig . 5(b). A linear fit at the
quasi linear high field range results in a large damping value of 0.081 ±0.015 and in a very
large
effKH of 630 Oe . The large damping is attributed to the efficient spin pumping into
PtBi30 layer having large spin -orbit coupling . When the angle of the applied magnetic f ield
is reduced to
1H
a clearer picture of the contribution of the inhomogeneous broadening
to the total linewidth is obtained (Fig. 5(c)) revealing that it cannot explain solely the
measured spectral linewidth s. While the theoretical model predicts that the increase in
bandwidth spans a relatively narrow field range around Hkeff, the measurement shows an
increase over a much larger range around Hkeff. The linewidth broadening originating from 21
the anisotropy dispersion was theoretically calculated under the assumption of a small
perturbation of the resonance frequency. A large
keffH value was measured however from
the TRMOKE measurement taken at
4H
. Calculating numerically the exact variation
of the resonance frequency improved slightly the fit but definitely did not resolve the
discrepancy (not presented) . From this fact we understand that there should be an additional
source contributing to the line broadening at least near Hkeff. A possible explanation may
be related to the low stiffness27 associated with the
0 keff HH conditions . Under such
conditions weaker torques which are usually neglected may become relevant24, 31. These
torques could possibly originate from dipolar or exchange coupling resulting in two
magnon scattering processes or even in a breakdown into magnetic domains as described
by Grolier et al.26. From the limited data range at this angle, the damping could not be
measured.
The OFMR system enabl ed a wider range of fields and frequencies than the ones
measured with the TRMOKE for
1H
(Fig. 5(a)). Fig. 5(d) presents t he measured
OFMR linewidth . The quasi -linear regime of the linewidth seems to be reached at
frequencies of 12 GHz corresponding to bias field values which are larger than 7500 Oe .
The resultant intrinsic damping after fitting to this range was 0.09±0.005 with a
keffH of
660 Oe which differ by approximately 10% from the values obtained from the TRMOKE
measurement . The effective measured damping is plotted in Fig. 5(e). The asymptotic
damping value , though not fully reached for this high damping sample , would be about 0.1. 22
This represents an error of about 10% which is smaller compared to the errors of 17% and
54% encountered in the CoFeB samples because of the larger damping of the Co /Ni/Co
sample.
D. Considerations of two -magnon scattering
In general, two -magnon spin wave scattering by impurities may exist in our
measurements at all field ranges32, 33, not only near Hkeff as suggested in the discussion of
the previous section32, 33. The resultant additional linewidth broadening would then be
regarded as an extr insic contribution to the damping34-36. While in isotropic films which
exhibit low crystalline anisotropy or in films having in -plane crystalline anisotropy, two -
magnon scattering is maximized when the external field is applied in the film plane, in
PMA films this is not necessarily the case and the highest efficiency of two -magnon
scattering may be obtained at some oblique angle35.
In films where two -magnon scatt ering is significant , the measured linewidth should
exhibit an additional nonlinear dependence on the external field which cannot be accounted
for by the present model . In such case, a s trong dependence on the external field would be
observed for fields below Hkeff due to the variation in the orientation of the magnet ization
with the external magnetic field. At higher fields the dependence on the external field is
expected to be moderate35.
While at bias field values below Hkeff our data is relatively limited, at external
magnetic fields that are larger than Hkeff, the observed linewidth seems to be described well 23
by our model resulting in a field independent Gilbert damping coefficient . This seems to
support our model that the scattering of spin waves does not have a prominent effect. It is
possible however that a moderate dependence on the bias field, especially at high field
values, may have been “linearized” and classified as intrinsic damping.
IV. CONCLUSION
In conclusion, in this paper we studied the time domain magnetization dynamics in
non-epitaxial thin films having perpendicular magnetic anisotropy using the TRMOKE and
OFMR system s. The analytical model used to interpret the magnetization dynamics from
the TRMOKE responses indicated that the asymptotic high -field approach often used to
distinguish the intrinsic damping from the effective damping may result in significant error
that increases the lower the damping is . Two sources for the error were identified while
validity of th e asymptotic approach was shown to require very high magnetic fields.
Additionally, the effective damping was shown to be highly affected by the derivative of
the resonance frequency with respect to the magnetic field
0/d dH . The analytical
approach developed here was verified by use of the OFMR measurement showing excellent
agreement whenever the intrinsic damping was compared and ruled out the possibility of
thermal heating by the laser or emission of spin waves away from the probed area.
24
As to the systems studied, a large impact of the seed layer on the intrinsic damping
with minor effect on the static characteristics of the CoFeB system was observed and may
greatly aid in engineering the proper materials for the MTJ. Interestingly, the use of the
analytical model enabled identification of an additional exchange torque when low stiffness
conditions prevailed. While effort still remains to understand th e limits on the angle of the
applied magnetic field to which the analytical solution is valid , the approach presented is
believed to accelerate the discovery of novel materials for new applications . 25
Acknowledgments:
A.C. thanks the Viterbi foundation and the Feder Family foundation for supporting this
research.
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27
Figure 1
FIG 1. Illustration of the angles
H ,
H and
. M and H0 vectors denote the magnetization
and external magnetic field , respectively.
28
Figure 2
FIG. 2. Measured TRMOKE responses at
H angles of
4
and
1
. (a) TRMOKE signal at
low and high external magnetic field values. Traces are shifted for clarity. (b) Measured
magnetization responses after reduction of background signal (open circles)
29
superimposed with the fitted decaying sine wave (solid lines). Traces are shifted and
normalized to have the same peak amplitude. Data presented for low and high external
magnetic field values.
30
Figure 3
FIG. 3. TRMOKE measurements at
4H
and
1H
. (a) Measured resonance
frequency versus magnetic field. (b) Theoretical dependence of resonance frequency on
magnetic field presented in normalized units . (c) & (d) Measured linewidth (blue) , fitted
theoretical con tributions to l inewidth (green, cyan, magenta) and extracted intrinsic
linewidth from measurement (red) for
4H
and
1H
, respectively. (e) Intrinsic and
effective damping. (f) Error in damping value when using Eq. (2) instead of Eq. (3 ).
31
Figure 4
FIG. 4. TRMOKE and OFMR measurements at
4H
and
0H
, respectively. (a)
Measured resonance frequency versus magnetic field. (b) Measured linewidth (blue), fitted
theoretical contributions to linewidth (green, cyan, magenta) and extracted intrinsic
32
linewidth from measurement (red) using the TRMOKE with
4H
. (c) Intrinsic and
effective damping using TRMOKE . (d) Representative OFMR trace at 2.5 GHz. The
function sign( mz)(mz)1/2 is plotted. (e) Field dependent absorption spectrum (blue)
extracted from the cross section along the red dashed lined of (d) together with fitted
lorentzian lineshapes (red). (f) Measured linewidth (blue), fitted theoretical contributions
to linewidth (green, cyan, black) and empirical fit that describes the angle misalignment
(magenta) using the OFMR with
0H
. (g) Effective damping using the OFMR and
TRMOKE . 33
Figure 5
FIG. 5. TRMOKE at
4H
and
1H
and OFMR measurement at
0H
for Co/Ni/Co
sample . (a) Measured resonance frequency versus magnetic field. (b ) Measured linewidth
(blue), fitted theoretical contributions to linewidth (green, cyan, magenta) and extracted
intrinsic linewidth from measurement (red) using the TRMOKE with
4H
. (c)
Measured linewidth (blue), fitted theoretical c ontributions to linewidth (green, cyan,
magenta) using the TRMOKE with
1H
. (d) Measured linewidth (blue), fitted
theoretical contributions to linewidth (green, cyan, black) using the OFMR with
0H
.
34
(e) Effecti ve (blue) and intrinsic (black ) damping using the TRMOKE at
4H
and
effective damping measured with the OFMR at
0H
(red). |
1604.04688v1.A_broadband_Ferromagnetic_Resonance_dipper_probe_for_magnetic_damping_measurements_from_4_2_K_to_300_K.pdf | arXiv:1604.04688v1 [cond-mat.mtrl-sci] 16 Apr 2016A broadband Ferromagnetic Resonance dipper probe for magne tic damping
measurements from 4.2 K to 300 K
Shikun Hea)and Christos Panagopoulosb)
Division of Physics and Applied Physics, School of Physical
and Mathematical Sciences, Nanyang Technological Univers ity,
Singapore 637371
Adipper probefor broadband FerromagneticResonance (FMR)op erating from4.2K
to room temperature is described. The apparatus is based on a 2-p ort transmitted
microwave signal measurement with a grounded coplanar waveguide . The waveguide
generates a microwave field and records the sample response. A 3- stagedipper design
is adopted for fast and stable temperature control. The tempera ture variation due
to FMR is in the milli-Kelvin range at liquid helium temperature. We also desig ned
a novel FMR probe head with a spring-loaded sample holder. Improve d signal-to-
noise ratio and stability compared to a common FMR head are achieved . Using a
superconducting vector magnet we demonstrate Gilbert damping m easurements on
two thin film samples using a vector network analyzer with frequency up to 26GHz:
1) A Permalloy film of 5 nm thickness and 2) a CoFeB film of 1.5nm thicknes s. Ex-
periments were performed with the applied magnetic field parallel and perpendicular
to the film plane.
a)Electronic mail: skhe@ntu.edu.sg
b)Electronic mail: christos@ntu.edu.sg
1I. INTRODUCTION
In recent years, the switching of a nanomagnet by spin transfer t orque (STT) using a spin
polarized current has been realized and intensively studied.1–3This provides avenues to new
types of magnetic memory and devices, reviving the interest on mag netization dynamics in
ultrathinfilms.4,5Highfrequencytechniques playanimportantroleinthisresearchdir ection.
Amongthem, FerromagneticResonance(FMR)isapowerfultool. M ostFMRmeasurements
have been performed using commercially available systems, such as e lectron paramagnetic
resonance (EPR) or electron spin resonance (ESR).6These techniques take advantage of
the high Q-factor of a microwave cavity, where the field modulation a pproach allows for the
utilization of a lock-in amplifier.7The high signal-to-noise ratio enables the measurement of
evensub-nanometerthickmagneticfilms. However, theoperating frequencyofametalcavity
is defined by its geometry and thus is fixed. To determine the damping of magnetization
precession, whichisinprincipleanisotropic, several cavitiesarereq uiredtostudytherelation
between the linewidth and microwave frequency at a given magnetiza tion direction.8–10The
apparent disadvantage isthat changing cavities canbetedious and prolongthe measurement
time.
Recently, an alternative FMR spectrometer has attracted consid erable attention.11–17
The technique is based on a state of the art vector network analyz er (VNA) and a coplanar
waveguide (CPW). Both VNA and CPW can operate in a wide frequency range hence this
technique is also referred to as broadband FMR or VNA-FMR. The br oadband FMR tech-
nique offers several advantages. First, it is rather straightforw ard to measure FMR over a
wide frequency range. Second, one may fix the applied magnetic field and acquire spectra
with sweeping frequency in a matter of minutes.17Furthermore, a CPW fabricated on a chip
using standard photolithography enables FMR measurements on pa tterned films as well as
on a single device.18In brief, it is a versatile tool suitable for the characterization of ma g-
netic anisotropy, investigation of magnetization dynamics and the s tudy of high frequency
response of materials requiring a fixed field essential to avoid any ph ase changes caused by
sweeping the applied field.
Although homebuilt VNA-FMRs are designed mainly for room temperat ure measure-
ments, a setup with variable temperature capability is of great inter est both for fundamental
studies and applications. Denysenkov et al. designed a probe with va riable sample temper-
ature, namely, 4-420K,19however, the spectrometer only operates in reflection mode. In
a more recent effort, Harward et al. developed a system operating at frequencies up to
70GHz.12However the lower bound temperature of the apparatus is limited to 27K. Here
we present a 2-port broadband FMR apparatus based on a superc onducting magnet. A
3-stage dipper probe has been developed which allows us to work in th e temperature range
4.2- 300K. Taking advantage of a superconducting vector magnet , measurements can be
performed with the magnetic moment saturated either parallel or p erpendicular to the film
plane. Wealsodesigned aspring-loadedsample holderforfastandre liablesample mounting,
quicktemperatureresponseandimprovedstability. Thissetupallow sforswiftchangesofthe
FMR probe heads and requires little effort for the measurement of d evices. To demonstrate
the capability of this FMR apparatus we measured the temperature dependence of magne-
2FIG. 1. View of the FMR dipper probe. Top panel: The schematic of the entire design with a
straight type FMR head. All RF connectors are 2.4mm. The vacu um cap mounted on the 4K
stage, using In seal, and the radiation shield mounted on the second stage are not shown for clarity.
Bottom panel: photograph of the components inside the vacuu m cap.
tization dynamics of thin film samples of Permalloy (Py) and CoFeB in diffe rent applied
magnetic field configurations.
II. APPARATUS
A. Cryostat and superconducting magnet system
Our customized cryogenic system was developed by Janis Research Company Inc. and
includes a superconducting vector magnet manufactured by Cryo magnetics Inc. A vertical
field up to 9T is generated by a superconducting solenoid. The field ho mogeneity is ±0.1%
over a 10mm region. A horizontal split pair superconducting magnet provides a field up to
4T with uniformity ±0.5% over a 10mm region. The vector magnet is controlled by a
Model 4G-Dual power supply. Although the power supply gives field r eadings according to
the initial calibration, to avoid the influence of remnant field we employ an additional Hall
sensor. The cryostat has a 50mm vertical bore to accommodate v ariable temperature
inserts and dipper probes. Our dipper probe described below is confi gured for this
cryostat, however, the principle can be applied also to other comme rcially available
superconducting magnets and cryostats.
3B. Dipper probe
Fig. 1shows a schematic of our dipper probe assembly and a photograph o f the com-
ponents inside the vacuum cap. The dipper probe is 1.2m long and is mou nted to the
cryogenic system via a KF50 flange. The sliding seal allows a slow insert ion of the dipper
probe directly into the liquid helium space. Supporting arms (not show n) lock the probe
and minimize vibration, with the sample aligned to the field center. The c onnector box on
top has vacuum tight Lemo and Amphenol connectors for 18 DC sign al feedthrough. Two
2.4mm RF connection ports allow for frequencies up to 50GHz . A vacu um pump port can
be shut by a Swagelok valve. We adopted a three stage design as sho wn in the photograph
ofFig. 1. The 4K stage and the vacuum cap immersed in the He bath provide co oling power
for the probe. The intermediate second stage acts as an isolator o f heat flow and as thermal
sink for the RF cables, providing improved temperature control. Fu rthermore, it allows one
to change probe heads conveniently as we discuss later. A separat e temperature sensor on
the second stage is used for monitoring purpose. The third stage, namely, the FMR probe
head with the spring loaded sample holder, is attached to the lower en d of the intermediate
stage using stainless steel rods.
Apairof0.086”stainlesssteelSemi-RigidRFcablesrunfromthetopo ftheconnectorbox
to the non-magnetic bulkhead connector (KEYCOM Corp.) mounted on the second stage.
BeCu non-magnetic Semi-Rigid cables (GGB Industries, Inc.) are use d for the connection
between the second stage and the probe head. The cables are car efully bent to minimize
losses. The rods connecting the stages are locked by set screws. Loosening the set screw
allows the rod length to be adjusted to match the length of the RF ca bles. Reflection
coefficient (S 11) and transmission coefficient (S 21) can be recorded simultaneously with this
2-port design. The leads for the temperature sensors, heater, Hall sensor and for optional
transport measurements are wrapped around Cu heat-sinks at t he 4K stage before being
soldered to the connection pins.
C. Probe head with spring-loaded sample holder
The key part of the dipper probe, namely, the FMR probe head is sch ematically shown in
Fig. 2. Theassemblyisplacedinaradiationshieldtubewithaninnerdiametero f32mm. To
maximize thermal conduction between parts, homebuilt component s are machined from Au
plated Cu. The 1” long customized grounded coplanar waveguide (GC PW) has a nominal
impedanceof50Ohm. Thestraight-lineshapeGCPWwasmadeonduro idR/circlecopyrtR6010(Rogers)
board, with a thickness of 254 µm and dielectric constant 10.2. The width of the center
conductor is 117 µm and the gap between the latter and the ground planes is 76 µm. For
the connection, first the GCPW is soldered to the probe head, and s ubsequently the center
pin of the flange connector (Southwest Microwave) is soldered to t he center conductor of the
GCPW. The response of the dipper with the straight-line shape GCPW installed is shown
inFig. 3. The relatively large insertion loss (-16.9dB at 26GHZ) is due to a tota l cable
length of more than 3m and multiple connectors. The high frequency current flowing in the
CPW generates a magnetic field of the same frequency. This RF field d rives the precession
4FIG. 2. Schematic of the spring-loaded FMR probe head with st raight shape grounded coplanar
waveguide (GCPW). 1 Au plated Cu housing; 2 straight shape GC PW; 3 flange connector; 4 strain
gauge thin film heater; 5 CernoxTMtemperature sensor; 6 Hall-sensor housing; 7 housing for 4- pin
Dip socket or pingo pin; 8 sample; 9 sample holder; 10 Cu sprin g; 11 spring housing; 12 sample
holder handle nut.
of the magnetic material placed on top of the signal line, and gives ris e to a change in the
system’s impedance, which in turn alters the transmitted and reflec ted signals.
A spring-loaded sample holder depicted in Fig. 2by items 9 to 12 is designed to mount
the sample. The procedure for loading a sample is as follows: 1) Pull up the handle nut
and apply a thin layer of grease (Apiezon N type) to the sample holder ; 2) Place the sample
at the center of the sample holder; 3) Mount the spring-loaded sam ple holder to the FMR
head; 4) Release the handle nut gradually so that the spring pushes the sample towards
the waveguide. The mounting-hole of the spring-housing is slightly lar ger than the outer
diameterofthespring. Thisallowsthesampleholdertomatchthesur faceoftheGCPWself-
adaptively. With the spring-loaded FMR head design, the sample moun ting is simple and
leaves no residue from the commonly used tapes. It maximizes the sig nal by minimizing the
gap between waveguide and sample, and enhances the stability. Fur thermore, it is suitable
for variable temperature measurements due to the enhanced the rmal coupling between the
sample, cold head and sensors ( items 9 to 12 in Fig. 2.).
The temperature sensor is mounted at the backside of the probe h ead. Due to limited
space, the heater consists of three parallel connected strain ga uges with a resistance of 120
Ohm. TheHallsensor canbemountedaccordingtotherequiredmea surement configuration.
The position of the Hall sensor shown in Fig. 2is an example for measurements in the
presence of a magnetic field applied parallel to the sample surface.
D. Probe-head using end-launch connector
Although the probe head with straight-line CPW works well in our expe riment, the
necessary replacement of CPW due to unavoidable performance fa tigue over time, or for
testing new CPW designs can be time consuming. In response, end-la unch connectors
(ELC) utilizing a clamping mechanism allow for a smooth transition from R F cables to
CPW. Soldering the launch pin to the center conductor of CPW is optio nal and reduces
the effort for modifications. In Fig. 4, we show our design of a FMR probe-head using ELC
50 5 10 15 20 25-30-20-100S (dB)
f (GHz) S11
S21
FIG. 3. The reflection (S 11) and transmission (S 21) coefficients of the dipper probe with the
straight-line shape GCPW mounted. The measurement was perf ormed at room temperature.
form Southwest Microwave, Inc. and a homebuilt U-shape GCPW. Sim ilar to the design of
Fig. 2, a Au plated Cu housing is used to mount the GCPW, ELC and the tempe rature and
Hall sensors. There are two locations for sample mounting. In posit ion A, the vertical field
is used for measurements with the magnetic field applied parallel to th e surface of the thin
film sample whereas the horizontal field is used for measurements wit h field perpendicular
to the sample surface. On the other hand, measurements for bot h configurations can be
accomplishedonlybyusingthehorizontalfieldifthesampleisplacedinp ositionB.Asshown
inFig. 4(b) and (c), to change between configurations simply requires rot ating the dipper
probeby90degrees. Nevertheless, weprefertoplacethesample inpositionAfortheparallel
configurationsincethesolenoidfieldismoreuniform. However, weno tethatthesamedesign
with the sample placed in position B is suitable also for an electromagnet . Furthermore,
adding a rotary stage to the probe enables angular dependent FMR measurements.
III. EXPERIMENTAL TEST
In this section, we present data to assess the performance of th e FMR probe head and
discuss two sets of magnetic damping measurements, demonstrat ing the capabilities and
performance of the appratus.
A. Spring-loaded sample holder
We tested our setup using a Keysight PNA N5222A vector network a nalyzer with maxi-
mum frequency 26.5GHz. The output power of the VNA is always 0dBm in our test. Note
that with 2.4mm connectors and customized GCPW, our design can in p rinciple operate
up to 50GHz. The performance of the spring-loaded sample holder is first studied at room
temperature with a 2nm thick Co 40Fe40B20film. For direct comparison, the FMR spectra
are recorded with two sample loading methods: One with a spring-load ed sample holder
6FIG. 4. FMR probe-head with u-shape GCPW and end-launch conn ector. (a) Photograph of the
probe-head using end-launch connector and U-shape GCPW. Se nsors are mounted at the backside
and at the bottom of the Cu housing. Simplified sketch of the co nfiguration for measuring with an
external field generated by the split coils (b) parallel and ( c) perpendicular to the sample plane.
Rotating the dipper probe in the horizontal plane changes fr om one configuration to the other.
(Fig. 2) and the other using the common method12which only requires Kapton tape. The
magnetic field is applied parallel to the plane of the thin film sample. Six se ts of data were
obtained by reloading the sample for each measurement. In Fig. 5, we show the amplitude
of the power transmission coefficient from Port 1 to Port 2 (S 21) at a frequency of 10GHz
and a temperature of 300K. The open circles represent a spectru m for a spring-loaded sam-
ple mounting whereas the open squares is the spectrum showing larg est signal for the six
flip-sample loadings. The averaged spectra for all six spectra are s hown by solid line and
dotted line, for spring and flip-sample loading, respectively. Two obs ervations are evident:
First, the best signal we obtained using the flip sample method is appr oximately 20 percent
lower compared to the spring-loaded method. Thus the spring-load ed method gives a better
signal to noise ratio and sensitivity. Second, for the spring-loaded method, the difference
between the averaged spectrum and single spectra is negligibly small. On the other hand the
variation between measurements for the flip-sample method can be as large as 20 percent.
Hence the spring-loaded method has better stability and is reprodu cible.
B. Temperature response
As detailed in the previous section, the probe head is made of Au plate d Cu blocks with
highinternalthermalconductionandgoodthermalcontact. Con sequently, theresponsetime
of the temperature control will be small as the characteristic the rmal relaxation time of a
system is C/k, whereCis the heat capacity and kis the overall thermal conduction. Also,
the temperature difference between sample and sensor is minimized e ven with the heater
turned on. Shown in Fig. 6are the FMR spectra and temperature variation for a CoFeB
film of 3nm thickness measured at 4.4K. The external field was swept at a rate of about -
10Oe/s. Forfields close to which FMRpeaks areobserved, we detec ted a temperaturerise of
7FIG. 5. Comparison between S 21signals obtained using spring-loaded sample holder mounti ng and
flip-sample mounting at 300K. The sample has a stack of MgO(3n m)—CoFeB(2nm)—MgO(3nm)
deposited on silicon substrate. (Numbers in parenthesis of the sample composition represent the
thickness of the respective layer.) The frequency is 10 GHz a nd the FMR center field is at 1520
Oe.
a few mK. In fact, the field values corresponding to maximum temper atures are about 20Oe
lower than the fields satisfying FMR condition, showing that the char acteristic relaxation
time between the sample and cold head is approximately 2 seconds. Th e temperature rise
of the probe head due to FMR indicates that the magnetic system ab sorbs energy from
the microwave and dissipates into the thermal bath. Specifically, at the field satisfying
the FMR condition, the damping torque is balanced by the torque gen erated by the RF
field. However, the dissipation power of such process is propotiona l to the thickness of
the magnetic film hence is very small. The successful detection of a t emperature rise adds
credence to the high thermal conduction within the probe head and relative low thermal
conduction between different stages. This demonstration shows t hat the probe head is
capable of measuring samples with phase transitions in a narrow temp erature range, such
as a superconducting/ferromagnetic bilayer system.20
C. Magnetic damping measurements
Although the FMR probe can be used to determine the energy anisot ropy of magnetic
materials, our primary purpose is to study magnetic damping parame ter. In the following,
two examples of such measurements are briefly described. Shown in Fig. 7is FMR response
of a Py film of 5nm thickness deposited on a silicon substrate, measur ed at 4.4K. The
sweeping external magnetic field is parallel to the sample surface. R eal and Imaginary parts
of the spectra obtained at selected frequencies are plotted with o pen circles in Fig. 7(a)
and (b), respectively. In FMR measurements, the change in the tr ansmittance, S21, is a
direct measure of the field-dependent susceptibility of the magnet ic layer. According to the
8FIG. 6. Sample temperature variation due to FMR at selected f requencies. (upper panel) Ampli-
tude of S 21and (lower panel) temperature variation of MgO(3nm)—CoFeB (3nm)—MgO(3nm) at
4.4K measured with external field parallel to the film plane.
LandauLifshitzGilbert formalism, the dynamic susceptibility of the ma gnetic material in the
configuration where the field is applied parallel to the plane of the thin film be described
as:21
χIP=4πMs(H0+Huni+4πMeff+i∆H/2)
(H0+Huni)(H0+Huni+4πMeff)−H2
f+i(∆H/2)·[2(H0+Huni)+4πMeff](1)
Here, 4πMsis the saturation magnetization, Huniis the in-plane uniaxial anisotropy,
4πMeffis the effective magnetization, Hf= 2πf/γ, and ∆His the linewidth of the spectrum
– the last term is of key importance to determine the damping parame ter. As shown by
solid lines in Fig. 7(a) and (b), the spectra can be fitted very well by adding a backgr ound,
a drift proportional to time, and a phase factor.11,22The field linewidth as a function of
frequency – ∆ H(f) is plotted in Fig. 7(c). The data points fall on a straight line. The
damping parameter αGL= 0.012±0.001 is therefore determined by the slope through9,23:
∆H=4π
γαGLf+∆H0 (2)
The error bar here is calculated from the confidence interval of th e fit.
We have also tested the setup with a magnetic field applied perpendicu lar to the sample
plane. The results for a MgO(3nm)—Co 40Fe40B20(1.5nm)—MgO(3nm) stack deposited on
silicon substrate are shown in Fig. 8. Comparing the spectra obtained at different tempera-
tures and fixed frequency, two observations are evident. First, the FMR peak position shifts
to higher field as the temperature is lowered due to changes in the eff ective magnetization.
9FIG. 7. FMR data of a Py thin film of thickness 5nm measured at 4. 4K with magnetic field
applied parallel to the sample plane. (a) Real and (b) Imagin ary parts of transmitted signal S 21
at selected frequencies. The data are normalized and the rel ative strength between the spectra at
different frequencies are kept. (c) FMR linewidth as a functio n of frequency. The damping was
calculated to be 0.012 ±0.001, using a linear fit.
Second, the FMR linewidth increases with decreasing temperature. Although the interfacial
anisotropy can be determined by fitting the FMR peak positions to th e Kittel formula,24
here, we are more interested in the damping parameter as a functio n of temperature. The
dynamic susceptibility in this configuration is25:
χOP=4πMs(H−4πMeff−i∆H/2)
(H−4πMeff)2−H2
f+i∆H·(H−4πMeff)(3)
Following the same procedure as for Py, the real and imaginary part of the spectra are
fitted simultaneously to obtain the linewidth. In Fig. 8(b), we plot the linewidth as a
function of frequency at the two boundaries of our measured tem peratures. Although the
measured linewidth at lower temperature is larger, the slope of the t wo curves is in good
agreement. The additional linewidth at 6K is primarily due to zero freq uency broadening,
which quantifies the magnitude of dispersion of the effective magnet ization. The results
are summarized in Fig. 8(c). Gilbert damping is essentially independent of temperature
although there is a minimum at 40 K. The room temperature value obta ined is in agreement
with the value for a thicker CoFeB.21,26On the other hand, the inhomogeneous broaden-
ing increases with lowering temperatue. The value at 6K is more than d ouble compared
to room temperatue. Notably, neglecting the zero-frequency off set ∆H(0), arising due to
inhomogeneity, would give rise to an enhanced effective damping comp ared to the intrinsic
contribution. Cavity based, angulardependent FMRmayalso disting uish theGilbertdamp-
ing from inhomogeneity effects. A shortcoming however, is the need to take into account
the possible contribution of two magnon scattering, which causes in creased complications
in the analysis of the data.27,28On the other hand, broadband FMR using a dipper probe
with the applied magnetic field in the perpendicular configuration, rule s out two magnon
scattering making this technique relatively straightforward to imple ment.29
Thedipperprobediscussedhereisnotlimitedtomeasurementsofth edampingcoefficient.
The broadband design is also useful for time-domain measurements .30Furthermore, a spin
10FIG. 8. Temperature dependent FMR measurement for a CoFeB th in film of thickness 1.5nm
with the magnetic field applied perpendicular to the film plan e. (a) Transmitted FMR signal at
20GHz obtained at different temperatures. (b) FMR linewidth a s a function of frequency at 6K
and 280K. (c) Damping constant and inhomogeneous broadenin g as a function of temperature.
The solid lines are the guides for the eye.
transfer torque ferromagnetic resonance31measurement on a single device can be performed
with variable temperature using bias Tee and a separate sample holde r.
IV. CONCLUSION
We have developed a variable temperature FMR to measure the magn etic damping pa-
rameter in ultra thin films. The 3-stage dipper and FMR head with a spr ing-loaded sample
holder design have a temperature stability of milli Kelvin during the FMR measurements.
This apparatus demonstrates improved signal stability compared t o traditional flip-sample
mounting. The results for Py and CoFeB thin films show that the FMR d ipper can measure
the damping parameter of ultra thin films with: Field parallel and perpe ndicular to the film
plane in the temperature range 4.2-300K and frequency up to at lea st 26 GHz.
ACKNOWLEDGMENTS
TheauthorsaregratefultoSzeTerLimatDataStorageInstitut eforpreparingtheCoFeB
samples. We acknowledge Singapore Ministry of Education (MOE), Ac ademic Research
Fund Tier 2 (Reference No: MOE2014-T2-1-050) and National Res earch Foundation (NRF)
of Singapore, NRF-Investigatorship (Reference No: NRF-NRFI2 015-04) for the funding of
this research.
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13 |
2010.01044v4.Multilevel_quasi_Monte_Carlo_for_random_elliptic_eigenvalue_problems_I__Regularity_and_error_analysis.pdf | arXiv:2010.01044v4 [math.NA] 6 Oct 2022Multilevel quasi-Monte Carlo for random elliptic eigenval ue
problems I: Regularity and error analysis
Alexander D. Gilbert1Robert Scheichl2
October 7, 2022
Abstract
Stochastic PDE eigenvalue problems are useful models for quantify ing the uncer-
tainty in several applications from the physical sciences and engine ering, e.g., struc-
tural vibration analysis, the criticality of a nuclear reactor or phot onic crystal struc-
tures. In this paper we present a multilevel quasi-Monte Carlo (MLQ MC) method for
approximatingtheexpectationofthe minimaleigenvalueofanelliptic e igenvalueprob-
lem with coefficients that are given as a series expansion of countably -many stochastic
parameters. The MLQMC algorithm is based on a hierarchy of discret isations of the
spatial domain and truncations of the dimension of the stochastic p arameter domain.
To approximate the expectations, randomly shifted lattice rules ar e employed. This
paper is primarily dedicated to giving a rigorous analysis of the error o f this algo-
rithm. A key step in the error analysis requires bounds on the mixed d erivatives of
the eigenfunction with respect to both the stochastic and spatial variables simulta-
neously. Under stronger smoothness assumptions on the parame tric dependence, our
analysis also extends to multilevel higher-order quasi-Monte Carlo r ules. An accom-
panying paper [Gilbert and Scheichl, 2022], focusses on practical ex tensions of the
MLQMC algorithm to improve efficiency, and presents numerical resu lts.
1 Introduction
Consider the following elliptic eigenvalue problem (EVP)
−∇·/parenleftbig
a(x,y)∇u(x,y)/parenrightbig
+b(x,y)u(x,y) =λ(y)c(x,y)u(x,y),forx∈D,
u(x,y) = 0 for x∈∂D,(1.1)
where the differential operator ∇is with respect to the physical variable x, which belongs
to a bounded, convex domain D⊂Rd(d= 1,2,3), and where the stochastic parameter
y= (yj)j∈N∈Ω:= [−1
2,1
2]N,
is an infinite-dimensional vector of independently and iden tically distributed (i.i.d.) uni-
form random variables on [ −1
2,1
2].
The dependence of the coefficients on the stochastic paramete rs carries through to the
eigenvalues λ(y), and corresponding eigenfunctions u(y):=u(·,y), and as such, in this
1School of Mathematics and Statistics, University of New Sou th Wales, Sydney NSW 2052, Australia.
alexander.gilbert@unsw.edu.au
2Institute for Applied Mathematics & Interdisciplinary Cen tre for Scientific Computing, Universit¨ at
Heidelberg, 69120 Heidelberg, Germany and Department of Ma thematical Sciences, University of Bath,
Bath BA2 7AY UK.
r.scheichl@uni-heidelberg.de
1paper we are interested in computing statistics of the eigen values and of linear functionals
of the corresponding eigenfunction. In particular, we woul d like to compute the expecta-
tion, with respect to the countable product of uniform densi ties, of the smallest eigenvalue
λ, which is an infinite-dimensional integral defined as
Ey[λ] =/integraldisplay
Ωλ(y) dy:= lim
s→∞/integraldisplay
[−1
2,1
2]sλ(y1,y2,...,ys,0,0...) dy1dy2···dys.
The multilevel Monte Carlo (MLMC) method [22, 30] is a varian ce reduction scheme
that has been successfully applied to many stochastic simul ation problems. When applied
to stochastic PDE problems (see, e.g., [6, 8]), the MLMC meth od is based on a hierarchy
ofL+ 1 increasingly fine finite element meshes {Tℓ}L
ℓ=0(corresponding to a decreasing
sequenceof meshwidths h0> h1>···> hL>0), andanincreasing sequenceoftruncation
dimensions s0< s1<···< sL<∞. Letting the dimension-truncated FE approximation
on level ℓbe denoted by λℓ:=λhℓ,sℓ, by linearity, we can write the expectation on the
finest level as
Ey[λL] =Ey[λ0]+L/summationdisplay
ℓ=1Ey[λℓ−λℓ−1]. (1.2)
Each expectation Ey[λℓ−λℓ−1] is then approximated by an independent Monte Carlo
method. Defining uℓ:=uhℓ,sℓwe can write a similar telescoping sum for Ey[G(uL)] for
any linear functional G(u).
Quasi-Monte Carlo (QMC) methods are equal-weight quadratu re rules where the sam-
ples are deterministically chosen to be well-distributed, see [13]. Multilevel quasi-Monte
Carlo (MLQMC) methods, whereby a QMC quadrature rule to appr oximate the expec-
tation on each level, were first developed in [23] for path sim ulation with applications in
option pricing and then later applied to stochastic PDE prob lems (e.g., [34, 33]). For
certain problems, MLQMC methods can be shown to converge fas ter than their Monte
Carlo counterpart, and for most problems the gains from usin g multilevel and QMC are
complementary.
In this paper we present a rigorous analysis of the error of a M LQMC algorithm for
approximating the expectation of the smallest eigenvalue o f (1.1) in the case where the
coefficients are given by a Karhunen–Lo` eve type series expan sion. The main result proved
in this paper is that under some common assumptions on the sum mability of the terms in
thecoefficientexpansion, theroot-mean-squareerror(RMSE )ofaMLQMCapproximation
ofEy[λ], which on each level ℓ= 0,1,...,Luses a randomly shifted lattice rule with Nℓ
points, a FE discretisation with meshwidth hℓ>0 and a fixed truncation dimension, is
bounded by
RMSE/lessorsimilarh2
L+L/summationdisplay
ℓ=0N−1+δ
ℓh2
ℓ,forδ >0, (1.3)
with a similar result for the eigenfunction (see Theorems 3. 1, 3.2 and Remark 5.1). This
error boundis clearly better thanthe correspondingresult fora MLMC method, which has
N−1+δ
ℓreplaced by N−1/2
ℓ, and in terms of the overall complexity to achieve a RMSE less
than some tolerance ε >0 the total cost compared to a single level QMC approximation is
reducedbyafactor of ε−1inspatial dimensions d≥2(seeCorollary3.1). Underequivalent
assumptions, the convergence rates in (1.3) coincide with t he rates in the corresponding
error bound for source problems from [34, 33]. Although it is not unexpected that we
are able to obtain the same convergence rates as for source pr oblems, the analysis here
is completely new and because of the nonlinear nature of eige nvalue problems presents
several added difficulties not encountered previously in the analysis of source problems.
2Indeed, the key intermediate step is an in-depth analysis of the mixed regularity of the
eigenfunction, simultaneously in both the spatial and stoc hastic variables. The result,
presented in Theorem 4.1, is a collection of explicit bounds on the mixed derivatives of the
eigenfunction, where the derivatives are second order with respect to the spatial variable
xand arbitrarily high order with respect to the stochastic va riabley. The proof of these
bounds forms a substantial proportion of this paper, and req uires a delicate multistage
induction argument along with a considerable amount of tech nical analysis (see Section 4
and the Appendix). These bounds significantly extend the pre vious regularity results
for stochastic EVPs from [1], which didn’t give any bounds on the derivatives, and [19],
which gave bounds that were first-order with respect to xand higher order with respect to
y. Furthermore, many other multilevel methods require simil ar mixed regularity bounds
for their analysis, e.g., multilevel stochastic collocati on [42]. Hence, the bounds are of
independent interest and open the door for further research into methods for uncertainty
quantification for stochastic EVPs. In particular, we show h ow the mixed regularity
bounds can be immediately applied to extend the analysis als o to multilevel quasi-Monte
Carlo methods for EVPs based on higher-order interlaced polynomial lattice rules [9, 24],
following the papers [11, 12] for source problems (see Secti on 5.4).
The focus of this paper is the theoretical analysis of our MLQ MC algorithm for EVPs.
As such numerical results and practical details on how to effic iently implement the algo-
rithm will be given in a separate paper [21].
EVPs provide a useful way to model problems from a diverse ran ge of applications,
such as structural vibration analysis [43], the nuclear cri ticality problem [15, 31, 44] and
photonic crystal structures [14, 18, 32, 36]. More recently , interest in stochastic EVPs has
been driven by a desire to quantify the uncertainty in applic ations such as nuclear physics
[2, 3, 45, 46], structural analysis [40] and aerospace engin eering [39]. The most widely
used numerical methods for stochastic EVPs are Monte Carlo m ethods [40]. More recently
stochastic collocation methods [1] and stochastic Galerki n/polynomial chaos methods [17,
45, 46] have been developed. In particular, to deal with the h igh-dimensionality of the
parameter space, sparse and low-rank methods have been cons idered, see [1, 16, 26, 28,
29]. Additionally, the present authors (along with colleag ues) have applied quasi-Monte
Carlo methods to (1.1) and proved some key properties of the m inimal eigenvalue and its
corresponding eigenfunction, see [19, 20].
Although we consider the smallest eigenvalue, the MLQMC met hod and analysis in
this paper can easily be extended to any simple eigenvalue th at is well-separated from the
rest of the spectrum for all parameters y. If the quantity of interest depends on a cluster
of eigenvalues, or on the corresponding subspace of eigenfu nctions, then, in principle, the
method in this paper could be used in conjunction with a subsp ace-based eigensolver.
Again, one important point for the theory would be that the ei genvalue cluster is well-
separated from the rest of the spectrum, uniformly in y.
The structure of the paper is as follows. In Section 2 we give a brief summary of the
required mathematical material. Then in Section 3 we presen t the MLQMC algorithm
along with a cost analysis. Section 4 proves the key regulari ty bounds, which are then
required for the error analysis in Section 5. Finally, in the appendix we give the proof of
the two key lemmas from Section 5.
2 Mathematical background
In this section we briefly summarise the relevant material on variational EVPs, finite
element methods and quasi-Monte Carlo methods. For further details we refer the reader
to the references indicated throughout, or [19].
3As a start, we make the following assumptions on the coefficien ts, which will ensure
that the problem (1.1) is well-posed and admits fast converg ence rates of our MLQMC
algorithm. In particular, we assume that all coefficients are bounded from above and
below, independently of xandy.
Assumption A1.
1.aandbare of the form
a(x,y) =a0(x)+∞/summationdisplay
j=1yjaj(x)andb(x,y) =b0(x)+∞/summationdisplay
j=1yjbj(x),(2.1)
whereaj, bj∈L∞(D), for allj≥0, andc∈L∞(D)depend onxbut noty.
2. There exists amin>0such that a(x,y)≥amin,b(x,y)≥0andc(x)≥amin, for all
x∈D,y∈Ω.
3. There exist p,q∈(0,1)such that
∞/summationdisplay
j=1max/parenleftbig
/⌊a∇d⌊laj/⌊a∇d⌊lL∞,/⌊a∇d⌊lbj/⌊a∇d⌊lL∞/parenrightbigp<∞and∞/summationdisplay
j=1/⌊a∇d⌊l∇aj/⌊a∇d⌊lq
L∞(D)<∞.
For convenience, we define amax<∞so that
max/braceleftbig
/⌊a∇d⌊la(y)/⌊a∇d⌊lL∞,/⌊a∇d⌊l∇a(y)/⌊a∇d⌊lL∞,/⌊a∇d⌊lb(y)/⌊a∇d⌊lL∞,/⌊a∇d⌊lc/⌊a∇d⌊lL∞/bracerightbig
≤amaxfor ally∈Ω.(2.2)
2.1 Variational eigenvalue problems
To introduce the variational form of the PDE (1.1), we let V:=H1
0(D), the first order
Sobolev space of functions with vanishing trace, and equip Vwith the norm /⌊a∇d⌊lv/⌊a∇d⌊lV:=
/⌊a∇d⌊l∇v/⌊a∇d⌊lL2(D). The space Vtogether with its dual, which we denote by V∗, satisfy the well-
known chain of compact embeddings V⊂⊂L2(D)⊂⊂V∗, where the pivot space L2(D)
is identified with its own dual.
Forv,w∈V, define the inner products A(y;·,·),M(·,·) :V×V→Rby
A(y;w,v):=/integraldisplay
Da(x,y)∇w(x)·∇v(x)dx+/integraldisplay
Db(x,y)w(x)v(x)dx,
M(w,v):=/integraldisplay
Dc(x)w(x)v(x)dx,
and let their respective induced norms be given by /⌊a∇d⌊lv/⌊a∇d⌊lA(y):=/radicalbig
A(y;v,v) and/⌊a∇d⌊lv/⌊a∇d⌊lM:=/radicalbig
M(v,v). Further, let M(·,·) also denote the duality paring on V×V∗.
In the usual way, multiplying (1.1) by v∈Vand performing integration by parts with
respect tox, we arrive at the following variational EVP, which is equiva lent to (1.1). Find
λ(y)∈R,u(y)∈Vsuch that
A(y;u(y),v) =λ(y)M(u(y),v) for all v∈V, (2.3)
/⌊a∇d⌊lu(y)/⌊a∇d⌊lM= 1.
The classical theory for symmetric EVPs (see, e.g., [5]) ens ures that the variational
EVP (2.3) has countably many strictly positive eigenvalues , which, counting multiplicities,
we label in ascending order as
0< λ1(y)≤λ2(y)≤ ···.
4The corresponding eigenfunctions,
u1(y), u2(y), ...,
can be chosen to form a basis of Vthat is orthonormal with respect to the inner product
M(·,·), and, by (2.3), also orthogonal with respect to A(y;·,·).
Proposition 2.1. The smallest eigenvalue is simple for all y∈Ω. Furthermore, there
existsρ >0, independent of y, such that
λ2(y)−λ1(y)≥ρfor ally∈Ω. (2.4)
Proof.The Krein–Rutmann Theorem and [19, Proposition 2.4].
Henceforth, we will let the smallest eigenvalue and its corr esponding eigenfunction be
simply denoted by λ=λ1andu=u1.
It is often useful to compare the eigenvalues λkto the eigenvalues of the negative
Laplacian on D, also with homogeneous Dirichlet boundary conditions and w ith respect
to the standard L2inner product. These are denoted by
0< χ1< χ2≤χ3≤ ···, (2.5)
and will often simply be referred to as Laplacian eigenvalue s or eigenvalues of the Lapla-
cian, without explicitly stating the domain or boundary con ditions.
The following form of the Poincar´ e inequality will also be u seful throughout this paper
/⌊a∇d⌊lv/⌊a∇d⌊lL2≤χ−1/2
1/⌊a∇d⌊lv/⌊a∇d⌊lV,forv∈V. (2.6)
It follows by the min-max representation for the Laplacian e igenvalue χ1.
The upper and lower bounds on the coefficients (2.2), along wit h the Poincar´ e inequal-
ity (2.6), ensure that the A(y)- andM-norms are equivalent to the V- andL2-norms,
respectively, with
√amin/⌊a∇d⌊lv/⌊a∇d⌊lV≤/⌊a∇d⌊lv/⌊a∇d⌊lA(y)≤/radicalBigg
amax/parenleftbigg
1+1
χ1/parenrightbigg
/⌊a∇d⌊lv/⌊a∇d⌊lV, (2.7)
√amin/⌊a∇d⌊lv/⌊a∇d⌊lL2≤/⌊a∇d⌊lv/⌊a∇d⌊lM≤√amax/⌊a∇d⌊lv/⌊a∇d⌊lL2. (2.8)
Finally, as is to be expected, for our finite element error ana lysis we require second-
order smoothness with respect to the spatial variables, whi ch we characterise by the space
Z=H2(D)∩V, equipped with the norm
/⌊a∇d⌊lv/⌊a∇d⌊lZ:=/parenleftbig
/⌊a∇d⌊lv/⌊a∇d⌊l2
L2+/⌊a∇d⌊l∆v/⌊a∇d⌊l2
L2/parenrightbig1/2.
In particular, the eigenfunctions belong to Z, see [19, Proposition 2.1].
2.2 Stochastic dimension truncation
The first type of approximation we make is to truncate the infin ite dimensional stochastic
domain to finitely many dimensions, which, for a truncation d imension s∈N, we do by
simply setting yj= 0 for all j > s. The result is that the coefficients aandbnow only
depend on sterms. We define the following notation: ys= (y1,y2,...,ys),
as(x,y):=a0(x)+s/summationdisplay
j=1yjaj(x), bs(x,y):=b0(x)+s/summationdisplay
j=1yjbj(x),
5and
As(y;w,v):=/integraldisplay
Das(x,y)∇w(x)·∇v(x)dx+/integraldisplay
Dbs(x,y)w(x)v(x)dx.
So that the truncated approximations, denoted by ( λs(y),us(y)), satisfy
As(y;us(y),v) =λs(y)M(us(y),v) for all v∈V. (2.9)
2.3 Finite element methods for EVPs
To begin with, we first describe the finite element (FE) spaces used to discretise the
EVP (2.3). Let {Vh}h>0be a family of conforming FE spaces of dimension Mh, where
eachVhcorresponds to a shape regular triangulation ThofDand the index parameter
h= max{diam(τ) :τ∈Th}is called the meshwidth. Since we have only assumed that the
domainDis convex and a∈W1,∞(D), throughoutthis paperweonly consider continuous,
piecewise linear FE spaces. However, under stricter condit ions on the smoothness of the
domain and the coefficients, one could easily extend our algor ithm to higher-order FE
methods. Furthermore, we assume that the number of FE degree s of freedom is of the
order of h−d, so that Mh/equalorsimilarh−d. This condition is satisfied by quasi-uniform meshes and
also allows for local refinement.
Forh >0, eachy∈Ω yields a FE (or discrete) EVP, which is formulated as: Find
λh(y)∈R,uh(y)∈Vhsuch that
A(y;uh(y),vh) =λh(y)M(uh(y),vh) for all vh∈Vh, (2.10)
/⌊a∇d⌊luh(y)/⌊a∇d⌊lM= 1.
The discrete EVP (2.10) has Mheigenvalues
0< λ1,h(y)≤λ2,h(y)≤ ··· ≤ λMh,h(y),
and corresponding eigenfunctions
u1,h(y), u2,h(y), ..., u Mh,h(y),
which are known to converge to the first Mheigenvalues and eigenfunctions of (2.3) as
h→0, see, e.g., [5] or [19] for the stochastic case.
From [19, Theorem 2.6] we have the following bounds on the FE e rror for the minimal
eigenpair, which we restate here because they will be used ex tensively in our error analysis
in Section 5.
Theorem 2.2. Leth >0be sufficiently small and suppose that Assumption A1 holds.
Then, for all y∈Ω,λhsatisfies
|λ(y)−λh(y)| ≤Cλh2, (2.11)
the corresponding eigenfunction uhcan be chosen such that
/⌊a∇d⌊lu(y)−uh(y)/⌊a∇d⌊lV≤Cuh, (2.12)
and forG ∈H−1+t(D)witht∈[0,1]
/vextendsingle/vextendsingleG(u(y))−G(uh(y))/vextendsingle/vextendsingle≤CGh1+t, (2.13)
where0< Cλ, Cu, CGare positive constants independent of yandh.
6We have already seen that the minimal eigenvalue of the conti nuous problem (2.3) is
simple for all y, and that the spectral gap is bounded independently of y. It turns out
that the spectral gap of the FE eigenproblem (2.10) is also bo unded independently of y
andh, provided that the FE eigenvalues are sufficiently accurate. Specifically, if
h≤h:=/radicalbiggρ
2Cλ, (2.14)
then
λ2,h(y)−λ1,h(y)≥λ2(y)−λ1(y)−/parenleftbig
λ1,h(y)−λ1(y)/parenrightbig
≥ρ−Cλh2≥ρ
2,(2.15)
where we have used the FE error estimate (2.11) and that λ1,h(y) converges from above.
In fact, it is well known that for conforming methods all of th e FE eigenvalues con-
verge from above, so that λk,h(y)≥λk(y). Then, as in [19], we can use the eigenvalues
of the Laplacian (or rather their FE approximations) to boun d the FE eigenvalues and
eigenfunctions independently of y. Hence, for k= 1,2,...,M hand for ally∈Ω, there
existλkanduk, which are independent of both yandh, such that
λk:=amin
amaxχk≤λk(y)≤λk,h(y)≤amax
amin(χk,h+1)≤λk, (2.16)
max/braceleftbig
/⌊a∇d⌊luk(y)/⌊a∇d⌊lV,/⌊a∇d⌊luk,h(y)/⌊a∇d⌊lV/bracerightbig
≤/radicalbig
amax(χk,h+1)
amin≤uk,(2.17)
whereχk,his the FE approximation of the kth Laplacian eigenvalue χk. In addition
to converging from above, for the Laplacian eigenvalues it i s known that χk≤χk,h≤
χk+Ckh2, for some constant that is independent of h(see [7, Theorem 10.4]). As such,
forhsufficiently small there exists an upper bound on χk,hthat is independent of h, which
in turn allows us to choose the final upper bounds λkandukso that they are independent
of bothyandh.
To conclude this section we introduce some notation and some properties of Vhthat
will be useful later on. First, the spaces Vhsatisfy the best approximation property :
inf
vh∈Vh/⌊a∇d⌊lw−vh/⌊a∇d⌊lV/lessorsimilarh/⌊a∇d⌊lw/⌊a∇d⌊lZ,for allw∈Z. (2.18)
Then, for h >0, letPh(y) :V→Vhdenote the A(y)-orthogonal projection of Vonto
Vh, which satisfies
A(y;w−Ph(y)w,vh) = 0,for allw∈V, vh∈Vh, (2.19)
and hence also
/⌊a∇d⌊lw−Ph(y)w/⌊a∇d⌊lA(y)= inf
vh∈Vh/⌊a∇d⌊lw−vh/⌊a∇d⌊lA(y).
2.4 Quasi-Monte Carlo integration
Quasi-Monte Carlo (QMC) methods are a class of equal-weight quadrature rules that can
be used to efficiently approximate an integral over the s-dimensional (translated) unit
cube
Isf:=/integraldisplay
[−1
2,1
2]sf(y)dy.
There are several different flavours of QMC rules, however in th is paper we focus on
randomly shifted rank-1 lattice rules . In Section 5.4 we will also briefly discuss how to
7extend our method to higher-order interlaced polynomial lattice rules , see [9, 24]. For
further details on different QMC methods see, e.g., [13].
A randomly shifted rank-1 lattice rule approximation to IsfusingNpoints is
Qs,N(∆)f:=1
NN−1/summationdisplay
k=0f(tk−1
2), (2.20)
where for a generating vector z∈Nsand a uniformly distributed random shift ∆∈[0,1)s,
the pointstkare given by
tk=tk(∆) =/braceleftbiggkz
N+∆/bracerightbigg
fork= 0,1,...,N−1.
Here{·}denotes taking the fractional part of each component of a vec tor and1
2:=
(1
2,1
2,...1
2).
The standard spaces for analysing randomly shifted lattice s rules are the so-called
weighted Sobolev spaces that were introduced in [41]. Here the term “w eighted” is used
to indicate that the space depends on a collection of positiv e numbers called “weights”
that model the importance of different subsets of variables an d enter the space through its
norm. To be more explicit, given a collection of weights γ:={γu>0 :u⊆ {1,2,...,s}},
letWs,γbe thes-dimensional weighted Sobolev space of functions with squa re-integrable
mixed first derivatives, equipped with the (unanchored) nor m
/⌊a∇d⌊lf/⌊a∇d⌊l2
Ws,γ=/summationdisplay
u⊆{1:s}1
γu/integraldisplay
[−1
2,1
2]|u|/parenleftbigg/integraldisplay
[−1
2,1
2]s−|u|∂|u|
∂yuf(y) dy−u/parenrightbigg2
dyu.(2.21)
Hereyu:= (yj)j∈uandy−u:= (yj)j∈{1:s}\u. Note also that we have used here setnotation
to denote the mixed first derivatives, as this is the conventi on in the QMC literature.
However, when we later give results for higher-order mixed d erivatives we will switch to
multi-index notation.
A generating vector that leads to a good randomly shifted lat tice rule in practice can
be constructed using the component-by-component (CBC) algorithm, or the more efficient
fast CBC construction [37, 38]. In particular, it can be shown (see, e .g., [13, Theorem
5.10]) that the root-mean-square (RMS) error of a randomly s hifted lattice rule using a
generating vector constructed by the CBC algorithm satisfie s
/radicalBig
E∆/bracketleftbig
|Isf−Qs,Nf|2/bracketrightbig
≤/parenleftBigg
1
ϕ(N)/summationdisplay
∅/\e}atio\slash=u⊆{1:s}γξ
u/parenleftbigg2ζ(2ξ)
(2π2)ξ/parenrightbigg|u|/parenrightBigg1/2ξ
/⌊a∇d⌊lf/⌊a∇d⌊lWs,γfor allξ∈(1
2,1].(2.22)
Hereϕis the Euler totient function, ζis the Riemann zeta function and E∆denotes the
expectation with respect to the random shift ∆. ForNprime one has ϕ(N) =N−1 or
forNa power of 2 one has ϕ(N) =N/2, and so in both cases taking ξclose to 1 /2 in
(2.22) results in the RMS error converging close to O(N−1).
In practice, it is beneficial to perform several independent QMC approximations corre-
spondingto a small number of independent random shifts, and then take the final approxi-
mation to betheaverage over thedifferent shifts. In particul ar, let∆(1),∆(2),...,∆(R)be
Rindependent uniform random shifts, and let the average over the QMC approximations
with random shift ∆(r)be denoted by
/hatwideQs,N,Rf:=1
RR/summationdisplay
r=1Qs,N(∆(r))f.
8Then, the sample variance,
/hatwideV[/hatwideQs,N,R]:=1
R(R−1)R/summationdisplay
r=1/bracketleftbig/hatwideQs,N,Rf−Qs,N(∆(r))f/bracketrightbig2, (2.23)
can be used as an estimate of the mean-square error of /hatwideQs,N,Rf.
3 MLQMC for random EVPs
Applying a QMC rule to each term in the telescoping sum (1.2), using a different number
Nℓof samples on each level, a simple MLQMC approximation of Ey[λ] is given by
QML
L(∆)λ:=L/summationdisplay
ℓ=0Qℓ(∆ℓ)/parenleftbig
λℓ−λℓ−1/parenrightbig
. (3.1)
Here, we define Qℓ(∆ℓ):=Qsℓ,Nℓ(∆ℓ) (see (2.20)) and we treat the L+ 1 independent
random shifts, ∆ℓ∈[0,1)sℓ, as a single vector of dimension/summationtextL
ℓ=0sℓ, denoted by ∆=
(∆0,∆1,...,∆L). Recall also that λℓ=λhℓ,sℓforℓ= 0,1,...,L, and for simplicity
denoteλ−1= 0. By using a different random shift for each level, the approx imations
across different levels will be statistically independent. F or a linear functional G ∈V∗, the
MLQMC approximation to Ey[G(u)] is defined in a similar fashion.
As for single level QMC rules, it is beneficial to use multiple random shifts, so that we
can estimate the variance on each level. Letting ∆(1),∆(2),...,∆(R)beRindependent
random shifts of dimension/summationtextL
ℓ=0sℓ, the shift-averaged MLQMC approximation is
/hatwideQML
L,Rλ:=L/summationdisplay
ℓ=01
RR/summationdisplay
r=1Qℓ(∆(r)
ℓ)/parenleftbig
λℓ−λℓ−1/parenrightbig
. (3.2)
Ifin practice theparameters arenot specified beforehand, t henwe set hℓ/equalorsimilar2−ℓ,sℓ/equalorsimilar2ℓ
and use the adaptive algorithm from [23] to choose the number of QMC points Nℓ.
The mean-square error (with respect to the random shift(s) ∆) of the MLQMC esti-
mator can be written as the sum of the bias and the total varian ce as follows
E∆/bracketleftbig
|Ey[λ]−/hatwideQL(∆)λ|2/bracketrightbig
=|Ey[λ−λL]|2+L/summationdisplay
ℓ=0V∆[Qℓ(λℓ−λℓ−1)].(3.3)
In the equation above, we have simplified the first term (corre sponding to the bias) by the
telescoping property, and the variance on each level is defin ed by
V∆[Qℓ(λℓ−λℓ−1)]:=E∆/bracketleftbig
|Ey[λℓ−λℓ−1]−Qℓ(∆ℓ)(λℓ−λℓ−1)/vextendsingle/vextendsingle2/bracketrightbig
,
where the cross-terms have vanished because randomly shift ed QMC rules are unbiased.
By the linearity of G ∈V∗, the error for the eigenfunction approximation can be decom -
posed in the same way.
Assuming that the total bias and the variance on each level de cay at some given
rates, then the decomposition of the mean-square error (3.3 ) gives the following abstract
complexity theorems (oneeach, fortheeigenvalue andforfu nctionals oftheeigenfunction).
As is usual with the analysis of multilevel algorithms, the d ifficult part is to verify the
assumptions on the decay of the variance and to determine the corresponding parameters.
This analysis will be performed in Section 5.
9Theorem 3.1 (Eigenvalues) .Suppose that E∆[Qℓ(λℓ−λℓ−1)] =Ey[λℓ−λℓ−1], and that
there exist positive constants αλ,α′,βλ,β′,ηsuch that
M1.|Ey[λ−λL]|/lessorsimilarhαλ
L+s−α′
L, and
M2.V∆[Qℓ(λℓ−λℓ−1)]/lessorsimilarR−1N−η
ℓ/parenleftBig
hβλ
ℓ−1+s−β′
ℓ−1/parenrightBig
, for allℓ= 0,1,2,...,L.
Then
E∆/bracketleftBig/vextendsingle/vextendsingleEy[λ]−/hatwideQML
L,R(λ)/vextendsingle/vextendsingle2/bracketrightBig
/lessorsimilarhαλ
L+sα′
L+1
RL/summationdisplay
ℓ=01
Nη
ℓ/parenleftBig
hβλ
ℓ−1+s−β′
ℓ−1/parenrightBig
.
Theorem 3.2 (Functionals) .ForG ∈V∗, suppose E∆[G(uℓ−uℓ−1)] =Ey[G(uℓ−uℓ−1)],
and that there exist positive constants αG,α′,βG,β′,ηsuch that
M1.|Ey[G(u−uL)]|/lessorsimilarhαG
L+s−α′
L, and
M2.V∆[Qℓ(G(uℓ−uℓ−1))]/lessorsimilarR−1N−η
ℓ/parenleftBig
hβG
ℓ−1+s−β′
ℓ−1/parenrightBig
, for allℓ= 0,1,2,...,L.
Then
E∆/bracketleftBig/vextendsingle/vextendsingleEy[G(u)]−/hatwideQML
L,R(G(u))/vextendsingle/vextendsingle2/bracketrightBig
/lessorsimilarhαG
L+sα′
L+1
RL/summationdisplay
ℓ=01
Nη
ℓ/parenleftBig
hβG
ℓ−1+s−β′
ℓ−1/parenrightBig
.
Remark 3.1. In the case of a single truncation dimension, sℓ=sLfor allℓ= 1,2,...,L,
the terms s−β′
ℓ−1can be dropped from the theorems above.
In Section 5, we verify that if Assumption A1 on the coefficient s holds, then Assump-
tions M1 and M2 above are satisfied, and we give explicit value s of the rates.
To better illustrate the power of our MLQMC algorithm, we giv e here the following
complexity bound for the special case of geometrically deca ying meshwidths and a fixed
truncation dimension. We only give the eigenvalue result, b ut an analogous result holds
also for linear functionals G ∈L2(D). For less smooth functionals, G ∈H−1+t(D) for
t∈[0,1], similar results hold but with slightly adjusted rates.
Corollary 3.1. Let0< ε≤e−1and suppose that Assumption A1 holds with p,q≤2/3.
Also, let hℓ/equalorsimilar2−ℓwithh0sufficiently small and let sℓ=sL/equalorsimilarh2p/(2−p)
L. Finally, suppose
that each Qℓis anNℓ-point lattice rule corresponding to a CBC-constructed gener ating
vector. If there exists 0< γ < d+1such that the cost on each level ℓ∈Nsatisfies
M3.cost/parenleftbig
Qℓ(λℓ−λℓ−1)/parenrightbig
/lessorsimilarRNℓ/parenleftbig
sℓh−d
ℓ+h−γ
ℓ/parenrightbig
,
then,LandNℓ= 2nℓ, fornℓ∈N, can be chosen such that
E∆/bracketleftBig/vextendsingle/vextendsingleEy[λ]−/hatwideQML
L,R(λ)/vextendsingle/vextendsingle2/bracketrightBig
/lessorsimilarε2
and forδ >0
cost/parenleftbig/hatwideQML
L,R(λ)/parenrightbig
/lessorsimilar
ε−1−p/(2−p)−δifd= 1,
ε−1−p/(2−p)−δlog2(ε−1)3/2+δifd= 2,
ε−d/2−p/(2−p)ifd >2.
Proof.In Section 5 (cf., (5.1) and Theorem 5.3) we verify that Assum ptions M1, M2 from
Theorem 3.1 hold with αλ= 2,α′= 2/p−1,βλ= 2αλ= 4 and η= 2−δ. The remainder
of the proof follows by a standard minimisation argument as i n, e.g., [33, Cor. 2].
Remark 3.2. In [21] we verify that the cost does indeed satisfy Assumptio n M3 with
γ≈d, which is the same order cost as the source problem.
104 Stochastic regularity
In order for a randomly shifted lattice rule approximation t o achieve the error bound
(2.22), we require that the integrand belongs to Ws,γ, which in turn requires bounds on
the mixed first derivatives. For the eigenproblem (1.1), thi s means that we need to study
the regularity of eigenvalues (and eigenfunctions) with re spect to the stochastic parameter
y. In order to bound the variance on each level of our MLQMC esti mator, it is necessary
to also study the FE error in Ws,γ(cf. (5.4)), whereas the single level analysis in [19]
only required the expected FE error. This analysis of the FE e rror in a stronger norm
requires mixed regularity of the solution with respect to bo thxandysimultaneously,
which has not been shown previously. The theorem below prese nts the required bounds
foru, along with the bounds from [19] with respect to yonly, which are included here for
completeness. Analyticity of simple eigenvalues and eigen functions with respect to ywas
shown in [1], however, explicit bounds on the derivatives we re not given there and they
also did not consider the mixed xandyregularity required for the ML analysis.
Although the analysis of randomly shifted lattice rules req uires only the mixed first
derivatives (cf., (2.21)), we also give results for arbitra ry higher-order mixed derivatives.
We do this because the proof technique is the same, and also si nce these bounds may be
useful for the analysis of higher-order methods, e.g., high er-order QMC (see Section 5.4)
or sparse grid rules (see, e.g., [25, 47]). As such, to simpli fy notation we will write mixed
higher-order derivatives using multi-index notation inst ead of the set notation used in
Section 2.4. For a multi-index ν= (νj)j∈Nwithνj∈N∪ {0}and only finitely-many
nonzero components, let ∂ν
ydenote the mixed partial differential operator where the orde r
of derivative with respect to the variable yjisνj. Define|ν|:=/summationtext
j≥1νjand denote the set
of all admissible multi-indices by F:={ν∈NN:|ν|<∞}. All operations and relations
between multi-indices will be performed componentwise, e. g., forν,m∈ Faddition is
given byν+m= (νj+mj)j∈N, andν≤mif and only if νj≤mjfor allj∈N. Similarly,
forν,m∈ Fand a sequence β∈ℓ∞define the following shorthand for products
/parenleftbiggν
m/parenrightbigg
:=∞/productdisplay
j=1/parenleftbiggνj
mj/parenrightbigg
andβν:=∞/productdisplay
j=1βνj
j.
Note that since ν,m∈ Fhave finite support these products have finitely-many terms.
Theorem 4.1. Letν∈ Fbe a multi-index, let ǫ∈(0,1), and suppose that Assumption A1
holds. Also, define the sequences β= (βj)j∈Nandβ= (βj)j∈Nby
βj:=Cβmax/parenleftbig
/⌊a∇d⌊laj/⌊a∇d⌊lL∞,/⌊a∇d⌊lbj/⌊a∇d⌊lL∞/parenrightbig
, (4.1)
βj:=Cβmax/parenleftbig
/⌊a∇d⌊laj/⌊a∇d⌊lL∞,/⌊a∇d⌊lbj/⌊a∇d⌊lL∞,/⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞/parenrightbig
, (4.2)
whereCβ≥1, given explicitly below in (4.6), is independent of ybut depends on ǫ.
Then, for all y∈Ω, the derivative of the minimal eigenvalue with respect to yis
bounded by
|∂ν
yλ(y)| ≤λ|ν|!1+ǫβν, (4.3)
and the derivative of the corresponding eigenfunction sati sfies both
/⌊a∇d⌊l∂ν
yu(y)/⌊a∇d⌊lV≤u|ν|!1+ǫβν, (4.4)
/⌊a∇d⌊l∂ν
yu(y)/⌊a∇d⌊lZ≤C|ν|!1+ǫβν, (4.5)
whereλ,uare as in (2.16),(2.17), respectively, and Cin(4.5)is independent of ybut
depends on ǫ.
Moreover, for h >0sufficiently small, the bounds (4.3)and(4.4)are also satisfied by
λh(y)anduh(y), respectively.
11Proof.To facilitate the proof with a single constant for both seque ncesβandβwe define
Cβ:=2λ2
ρaminλ
a2maxλ/parenleftbigg3λ
λCǫ+1/parenrightbigg
, (4.6)
whereCǫfrom [19, Lemma 3.3] is given by
Cǫ:=21−ǫ
1−2−ǫ/parenleftbigge2
√
2π/parenrightbiggǫ
,
which is independent of yandh. Then clearly it follows that Cβis independent of yand
h. Later we will use that 1 /amin≤Cβ/2 and 1/(aminχ1/2)≤Cβ/2, which both follow
from the lower bounds Cǫ≥1 for all ǫ∈(0,1) andλ/λ≥a2
max/a2
min(1+1/χ).
Theproofforthebounds(4.3) and(4.4)isgiven in[19, Theor em3.4]. If hissufficiently
small such that the FE eigenvalues resolve the spectral gap ( i.e., (2.15) holds) then the
bounds also hold for λh(y) anduh(y) because Vh⊂V, cf. [19, Rem. 3.2 and 3.5].
For the bound (4.5), we first prove a recursive bound on /⌊a∇d⌊l∂νu(y)/⌊a∇d⌊lZand then use
an induction result from [12] to prove the final bound. Consid er the strong form of the
eigenproblem (1.1) for the pair ( λ(y),u(y)), which, omitting the xandydependence, is
given by
−∇·(a∇u)+bu=cλu.
Theνth derivative with respect to ycommutes with the spatial derivatives ∇. Thus,
using the Leibniz general product rule we have
−∇·(a∇∂ν
yu)+b∂ν
yu+∞/summationdisplay
j=1νj/parenleftbig
−∇·(aj∇∂ν−ej
yu)−bj∂ν−ej
yu/parenrightbig
=c/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
∂m
yλ ∂ν−m
yu,
whereejis the multi-index that is 1 in the jth entry and zero elsewhere. Then we can
use the identity ∇·(φψ) =φ∇·ψ+∇φ·ψto simplify this to
a∆∂ν
yu=−∇a·∇∂ν
yu+b∂ν
yu−c/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
∂m
yλ ∂ν−m
yu
+∞/summationdisplay
j=1νj/parenleftbig
−aj∆∂ν−ej
yu−∇aj·∇∂ν−e
yu+bj∂ν−ej
yu/parenrightbig
.
Sincea≥amin>0;a,aj∈W1,∞andb,bj∈L∞for allj∈N; and∂m
yu∈Vfor
allm∈ F, it follows by induction on |ν|that ∆∂ν
yu∈L2. This allows us to take the
L2-norm of both sides, which, after using the triangle inequal ity and the bounds in (2.2),
gives the following recursive bound for ∆ ∂ν
yu
/⌊a∇d⌊l∆∂ν
yu/⌊a∇d⌊lL2≤/⌊a∇d⌊l∇a/⌊a∇d⌊lL∞
amin/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lV+/⌊a∇d⌊lb/⌊a∇d⌊lL∞
amin/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lL2
+/⌊a∇d⌊lc/⌊a∇d⌊lL∞
amin/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
|∂m
yλ|/⌊a∇d⌊l∂ν−m
yu/⌊a∇d⌊lL2+∞/summationdisplay
j=1νj/⌊a∇d⌊laj/⌊a∇d⌊lL∞
amin/⌊a∇d⌊l∆∂ν−ej
yu/⌊a∇d⌊lL2
+1
amin∞/summationdisplay
j=1νj/parenleftbig
/⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞/⌊a∇d⌊l∂ν−ej
yu/⌊a∇d⌊lV+/⌊a∇d⌊lbj/⌊a∇d⌊lL∞/⌊a∇d⌊l∂ν−ej
yu/⌊a∇d⌊lL2/parenrightbig
.
12Adding/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lL2to both sides and then using the definition of βj, we can write this
bound in terms of the Z-norm as
/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lZ≤ /⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lL2+/⌊a∇d⌊l∆∂ν
yu/⌊a∇d⌊lL2≤∞/summationdisplay
j=1νjβj/⌊a∇d⌊l∂ν−ej
yu/⌊a∇d⌊lZ+Bν, (4.7)
where we used that 1 /amin≤Cβ, and then defined
Bν:=/⌊a∇d⌊l∇a/⌊a∇d⌊lL∞
amin/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lV+/⌊a∇d⌊lc/⌊a∇d⌊lL∞
amin/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
|∂m
yλ|/⌊a∇d⌊l∂ν−m
yu/⌊a∇d⌊lL2
/parenleftbigg/⌊a∇d⌊lb/⌊a∇d⌊lL∞
amin+1/parenrightbigg
/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lL2+1
amin∞/summationdisplay
j=1νj/parenleftbig
/⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞/⌊a∇d⌊l∂ν−ej
yu/⌊a∇d⌊lV+/⌊a∇d⌊lbj/⌊a∇d⌊lL∞/⌊a∇d⌊l∂ν−ej
yu/⌊a∇d⌊lL2/parenrightbig
.
Now, the sum on the right of (4.7) only involves lower-order v ersions of the object
we are interested in bounding (namely, /⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lZ), whereas the terms in Bνonly involve
derivatives that can be bounded using one of (4.3) or (4.4).
We bound the remaining L2-norms in Bνby the Poincar´ e inequality (2.6) to give
Bν≤/parenleftbigg/⌊a∇d⌊l∇a/⌊a∇d⌊lL∞
amin+/⌊a∇d⌊lb/⌊a∇d⌊lL∞+amin
amin√χ1/parenrightbigg
/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lV+/⌊a∇d⌊lc/⌊a∇d⌊lL∞
amin√χ1/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
|∂m
yλ|/⌊a∇d⌊l∂ν−m
yu/⌊a∇d⌊lV
+1
amin∞/summationdisplay
j=1νj/parenleftbigg
/⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞+/⌊a∇d⌊lbj/⌊a∇d⌊lL∞
√χ1/parenrightbigg
/⌊a∇d⌊l∂ν−ej
yu/⌊a∇d⌊lV
≤amax
amin/parenleftbigg
1+2√χ1/parenrightbigg/parenleftBigg
/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lV+/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
|∂m
yλ|/⌊a∇d⌊l∂ν−m
yu/⌊a∇d⌊lV/parenrightBigg
+1
amin∞/summationdisplay
j=1νj/parenleftbigg
/⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞+/⌊a∇d⌊lbj/⌊a∇d⌊lL∞
√χ1/parenrightbigg
/⌊a∇d⌊l∂ν−ej
yu/⌊a∇d⌊lV,
where in the last inequality we have boundedthe L∞-norms on the second line using (2.2),
and then simplified. Then, substituting in the bounds (4.3) a nd (4.4) gives
Bν≤amax
amin/parenleftbigg
1+2√χ1/parenrightbigg/parenleftBigg
u|ν|!1+ǫβν+/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
λ|m|!1+ǫβm·u|ν−m|!1+ǫβν−m/parenrightBigg
+1
amin∞/summationdisplay
j=1νj/parenleftbigg
/⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞+/⌊a∇d⌊lbj/⌊a∇d⌊lL∞√χ1/parenrightbigg
u(|ν|−1)!1+ǫβν−ej
=uamax
amin/parenleftbigg
1+2√χ1/parenrightbigg
βν/parenleftBigg
|ν|!1+ǫ+λ/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
|m|!1+ǫ|ν−m|!1+ǫ/parenrightBigg
+u
amin(|ν|−1)!1+ǫ∞/summationdisplay
j=1νj/parenleftbigg
/⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞+/⌊a∇d⌊lbj/⌊a∇d⌊lL∞
√χ1/parenrightbigg
βν−ej.
13Using the fact that (1+ χ−1/2)/amin≤Cβand also that clearly βj≤βj, we have
Bν≤uamax
amin/parenleftbigg
1+2√χ1/parenrightbigg
βν/parenleftBigg
|ν|!1+ǫ+λ/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
|m|!1+ǫ|ν−m|!1+ǫ/parenrightBigg
+u
amin(|ν|−1)!1+ǫ∞/summationdisplay
j=1νj/parenleftbig
1+χ−1/2/parenrightbig
max/parenleftbig
/⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞,/⌊a∇d⌊lbj/⌊a∇d⌊lL∞/parenrightbig
βν−ej
≤uamax
amin/parenleftbigg
1+2√χ1/parenrightbigg
βν/parenleftBigg
|ν|!1+ǫ+λ/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
|m|!1+ǫ|ν−m|!1+ǫ/parenrightBigg
+u|ν|!1+ǫβν.
The sum that remains can be bounded using the same strategy as in the proof of [19,
Lemma 3.4], as follows
/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
|m|!1+ǫ|ν−m|!1+ǫ=2|ν|!1+ǫ+|ν|−1/summationdisplay
k=1k!1+ǫ(|ν|−k)!1+ǫ/summationdisplay
m≤ν,|m|=k/parenleftbiggν
m/parenrightbigg
=2|ν|!1+ǫ+|ν|−1/summationdisplay
k=1k!1+ǫ(|ν|−k)!1+ǫ/parenleftbigg|ν|
k/parenrightbigg
=|ν|!1+ǫ/parenleftbigg
2+|ν|−1/summationdisplay
k=1/parenleftbigg|ν|
k/parenrightbigg−ǫ/parenrightbigg
≤|ν|!1+ǫ/parenleftBigg
2+21−ǫ
1−2−ǫ/parenleftbigge2
√
2π/parenrightbiggǫ
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
Cǫ/parenrightBigg
, (4.8)
where for the inequality on the last line we have used [19, Lem ma 3.3].
Hence,Bνis bounded above by
Bν≤CB|ν|!1+ǫβν,
where
CB:=u/bracketleftbiggamax
amin/parenleftbigg
1+2√χ1/parenrightbigg/parenleftbig
1+λ(2+Cǫ)/parenrightbig
+1/bracketrightbigg
<∞
is clearly independent of yandν.
Now we can bound the recursive formula (4.7) using the bound a bove on Bν, which
gives
/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lZ≤∞/summationdisplay
j=1νjβj/⌊a∇d⌊l∂ν−ej
yu/⌊a∇d⌊lZ+CB|ν|!1+ǫβν.
Finally, by [12, Lemma 4] we can bound this above by
/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lZ≤/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
|m|!βmCB|ν−m|!1+ǫβν−m
=CBβν/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
|m|!|ν−m|!1+ǫ≤CB(2+Cǫ)|ν|!1+ǫβν,
where to obtain the final result we have again used (4.8).
145 Error analysis
We now provide a rigorous analysis of the error for (3.1), whi ch we do by verifying the
assumptions from Theorems 3.1 and 3.2.
Recall that we use the shorthand λℓ:=λhℓ,sℓfor the dimension-truncated FE approx-
imation of the minimal eigenvalue on level ℓ, whereas λsdenotes the minimal eigenvalue
of the dimension-truncated version of the continuous EVP (2 .9). The bias (the first term)
in (3.3) can be bounded by the triangle inequality to give
|Ey[λ−λhL,sL]| ≤ |Ey[λ−λsL]|+|Ey[λsL−λhL,sL]|,
and similarly for the eigenfunction. Now, both terms on the r ight can be bounded above
using the results from the single level algorithm. Explicit ly, forG ∈H−1+t(D) with
t∈[0,1] using Theorem 4.1 from [19] and then Theorem 2.2 gives the b ounds
|Ey[λ−λL]|/lessorsimilars−2/p+1
L+h2
L, (5.1)
|Ey[G(u−uL)]|/lessorsimilars−2/p+1
L+h1+t
L, (5.2)
with constants independent of sLandhL. That is, we have verified Assumptions M1 from
both Theorems 3.1 and 3.2 with αλ= 2,αG= 1+tandα′= 2/p−1.
For the variance terms on each level in (3.3) (alternatively to verify Assumption M2),
we must study the QMC error of the differences λℓ−λℓ−1. Sinceλℓ−λℓ−1∈ Wsℓ,γfor all
ℓ= 0,1,2,...,Land each QMC rule Qℓuses CBC-constructed generating vector zℓ, by
(2.22) we have the upper bound
V∆[Qℓ(λℓ−λℓ−1)]≤C2
ξ,ℓ
ϕ(Nℓ)1/ξ/⌊a∇d⌊lλℓ−λℓ−1/⌊a∇d⌊l2
Wsℓ,γfor allξ∈(1/2,1],(5.3)
whereCξ,ℓis the constant from (2.22) with s=sℓ. Thus, in Assumption M2 we can take
η= 1/ξ∈[1,2) and for the other parameters we must study the norm of the di fference
on each level.
By the triangle inequality, we can separate truncation and F E components of the error
/⌊a∇d⌊lλℓ−λℓ−1/⌊a∇d⌊lWsℓ,γ
≤ /⌊a∇d⌊lλsℓ−λsℓ−1/⌊a∇d⌊lWsℓ,γ+/⌊a∇d⌊lλsℓ−λhℓ,sℓ/⌊a∇d⌊lWsℓ,γ+/⌊a∇d⌊lλsℓ−1−λhℓ−1,sℓ−1/⌊a∇d⌊lWsℓ−1,γ.(5.4)
In contrast to the single level setting [19], here we need to s tudy the truncation and FE
errors in the weighted QMC norm (2.21) instead of simply the e xpected truncation and
FE errors. Each term will be handled separately in the subsec tions that follow.
Thekey ingredient intheerroranalysisaretheboundsofthe derivatives oftheminimal
eigenvalue and its eigenfunction that were given in Section 4.
5.1 Estimating the FE error
As a first step towards bounding the FE errors in the Ws,γ-norm, we bound their deriva-
tives with respect to y, which are given below in Theorem 5.1. The bulk of the work
to bound the FE error in Ws,γis dedicated to proving these regularity bounds. As in
Theorem 4.1 we also present bounds on higher-order mixed der ivatives instead of simply
the mixed first derivatives required in the Ws,γnorm.
The strategy for proving these bounds is similar to the proof [19, Lemma 3.4], except
in the current multilevel setting we need to bound the deriva tives of the FE errors of
the eigenvalue and eigenfunction, in addition to the deriva tives of the eigenvalue and
15eigenfunction themselves. First, we differentiate variatio nal equations involving the errors
to obtain a recursive formula for each of the eigenvalue and e igenfunction errors, and then
prove the bounds by induction on the cardinality of |ν|. Once we have proved the bound
for the eigenfunction in (5.11), the result for any function alG(u(y)) in (5.12) follows by
a duality argument. Throughout the proofs in this section we will omit the xandy
dependence. Note also that throughout we must explicitly tr ack the constants to ensure
that they are independent of yandh, but also to make sure that in both of the inductive
steps the constants are not growing, since this could interf ere with the summability of /hatwideβ.
Also, the results in this section are all shown for hsufficiently small, wherehere sufficiently
smallmeans that the FE eigenvalues resolve the spectral gap. Expl icitly, we assume that
h≤h(see (2.14) and (2.15)) for some h >0 that is independent of y. This ensures that
the condition that his sufficiently small (i.e., h≤h) is also independent of y.
In the following key lemma, we bound the derivative of the diffe rence between the
eigenfunctionanditsprojection Phu(y)ontoVh, whichisnotequaltotheFEeigenfunction
uh(y), but is easier to handle. The proof relies on the new mixed re gularity estimate (4.5).
Lemma 5.1. Letν∈ Fbe a multi-index, let h >0be sufficiently small and suppose that
Assumption A1 holds. Then
/⌊a∇d⌊l∂ν
yu−Ph∂ν
yu/⌊a∇d⌊lV≤CPh|ν|!1+ǫβν, (5.5)
whereβis as defined in (4.2), andCPis independent of y,handν.
Proof.Using the equivalence of the V-norm and the induced A-norm in (2.7), along with
theA-orthogonality of the projection and the best approximatio n property (2.18), we get
/⌊a∇d⌊l∂ν
yu−Ph∂ν
yu/⌊a∇d⌊lV≤/radicalBigg
amax
amin/parenleftbigg
1+1
χ1/parenrightbigg
inf
vh∈Vh/⌊a∇d⌊l∂ν
yu−vh/⌊a∇d⌊lV
≤/radicalBigg
amax
amin/parenleftbigg
1+1
χ1/parenrightbigg
Ch/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lZ≤CPh|ν|!1+ǫβν,
where for the last inequality we have used the bound (4.5). Th e final constant CPis
independent of y,hand alsoν.
The three recursive formulae presented in the next two lemma s are the key to the
induction proof to bound the derivatives of the FE error. The general strategy is to
differentiate variational equations involving the FE errors . However, the proofs are quite
long and technical, and as such are deferred to the Appendix.
Lemma 5.2. Letν∈ Fbe a multi-index, let h >0be sufficiently small and suppose that
Assumption A1 holds. Then, for all y∈Ω, the following two recursive bounds hold
|∂ν
y(λ−λh)| ≤CI/parenleftBigg
h|ν|!βν+∞/summationdisplay
j=1νjβj/⌊a∇d⌊l∂ν−ej
y(u−uh)/⌊a∇d⌊lV
+/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg
|m|!1+ǫβm/bracketleftBig
/⌊a∇d⌊l∂ν−m
y(u−uh)/⌊a∇d⌊lV+|∂ν−m
y(λ−λh)|/bracketrightBig/parenrightBigg
(5.6)
16and
|∂ν
y(λ−λh)| ≤CII/parenleftBigg/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
/⌊a∇d⌊l∂ν−m
y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊l∂m
y(u−uh)/⌊a∇d⌊lV (5.7)
+∞/summationdisplay
j=1/summationdisplay
m≤ν−ejνj/parenleftbiggν−ej
m/parenrightbigg
βj/⌊a∇d⌊l∂ν−ej−m
y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊l∂m
y(u−uh)/⌊a∇d⌊lV
+/summationdisplay
m≤ν/summationdisplay
k≤m/parenleftbiggν
m/parenrightbigg/parenleftbiggm
k/parenrightbigg
|ν−m|!1+ǫβν−m/⌊a∇d⌊l∂m−k
y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊l∂k
y(u−uh))/⌊a∇d⌊lV/parenrightBigg
,
whereβ,βare defined in (4.1),(4.2), respectively, and CI, CIIare independent of y,h
andν.
Lemma 5.3. Letν∈ Fbe a multi-index, let h >0be sufficiently small and suppose that
Assumption A1 holds. Then, for all y∈Ω,
/⌊a∇d⌊l∂ν
y(u−uh)/⌊a∇d⌊lV≤CIII/parenleftBigg
h|ν|!1+ǫβν+∞/summationdisplay
j=1νjβj/⌊a∇d⌊l∂ν−ej
y(u−uh)/⌊a∇d⌊lV
+/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg
|m|!1+ǫβm/bracketleftBig
/⌊a∇d⌊l∂ν−m
y(u−uh)/⌊a∇d⌊lV+|∂ν−m
y(λ−λh)|/bracketrightBig/parenrightBigg
,(5.8)
whereβ,βare defined in (4.1),(4.2), respectively, and CIIIis independent of y,handν.
The astute reader may now ask, why do we need both the bounds (5 .6) and (5.7)
on the derivative of the eigenvalue error? The reason is that the upper bound in (5.6)
depends only on derivatives with order strictly less than ν, whereas the bound in (5.7)
depends on ∂ν
y(u−uh). Hence the inductive step for the eigenfunction (see (5.11) below)
only works with (5.6). On the other hand, (5.6) cannot be used for the inductive step for
theeigenvalue result (see (5.10) below), because it will only result in a bo und of order
O(h). Hence, the second bound (5.7) is required to maintain the o ptimal rate of O(h2)
for the eigenvalue error.
We now have the necessary ingredients to prove thefollowing boundson thederivatives
of the FE error.
Theorem 5.1. Letν∈ Fbe a multi-index, let h >0be sufficiently small and suppose
that Assumption A1 holds. Define the sequence /hatwideβ= (/hatwideβj)j∈Nby
/hatwideβj:=/hatwideCβmax/parenleftbig
/⌊a∇d⌊laj/⌊a∇d⌊lL∞,/⌊a∇d⌊lbj/⌊a∇d⌊lL∞,/⌊a∇d⌊l∇aj/⌊a∇d⌊lL∞/parenrightbig
, (5.9)
where/hatwideCβ, given explicitly below in (5.14), is independent of y,handj. Then
/vextendsingle/vextendsingle∂ν
y/bracketleftbig
λ(y)−λh(y)/bracketrightbig/vextendsingle/vextendsingle≤C1|ν|!1+ǫ/hatwideβνh2, (5.10)
/⌊a∇d⌊l∂ν
y/bracketleftbig
u(y)−uh(y)/bracketrightbig
/⌊a∇d⌊lV≤C2|ν|!1+ǫ/hatwideβνh, (5.11)
and forG ∈H−1+t(D)
/vextendsingle/vextendsingle∂ν
yG(u(y)−uh(y))/vextendsingle/vextendsingle≤C3|ν|!1+ǫ/hatwideβνh1+t, (5.12)
withC1,C2,C3all independent of y,handν.
17Proof.Throughoutwe usethe convention that 0! = 1. Then, dueto thee rror bound(2.11)
for the FE eigenvalue error, the base case of the induction ( ν=0) for the eigenvalue result
(5.10) holds provided C1≥Cλ. Thus, let
C1:= max/braceleftbig
Cλ, CIIC2
u(2+Cǫ)(4+Cǫ)/bracerightbig
. (5.13)
Similarly, defining C2=Cuthe base case of the induction for (5.11) also clearly holds d ue
to (2.12).
For the inductive step, let νbe such that |ν| ≥1 and assume that (5.10) and (5.11)
hold for allmwith|m|<|ν|. Now, since the recursive bound for the eigenvalue (5.7)
still depends on a term of order ν, whereas the recursive bound for the eigenfunction (5.8)
only depends on strictly lower order terms, for our inductiv e step to work we first prove
the result (5.11) for the eigenfunction, before proving the result (5.10) for the eigenvalue.
Substituting the induction assumptions (5.10) and (5.11) f or|m|<|ν|into (5.8) gives
/⌊a∇d⌊l∂ν
y(u−uh)/⌊a∇d⌊lV≤CIII/parenleftBigg
|ν|!1+ǫβνh+∞/summationdisplay
j=1νjβjC2(|ν|−1)!1+ǫ/hatwideβν−ejh
+/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg
|m|!1+ǫβm(C2+C1h)|ν−m|!1+ǫ/hatwideβν−mh/parenrightBigg
≤/hatwideβνhCIII/parenleftBigg/bracketleftbigg/parenleftbiggCβ
/hatwideCβ/parenrightbigg|ν|
+C2Cβ
/hatwideCβ/bracketrightbigg
|ν|!1+ǫ
+/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg
|ν−m|!1+ǫ|m|!1+ǫ(C1h+C2)/parenleftbiggCβ
/hatwideCβ/parenrightbigg|m|/parenrightBigg
where we have rescaled each product by using the definitions o fβ,βand/hatwideβin (4.1), (4.2)
and (5.9). The constants can be simplified by defining
/hatwideCβ:=1
C2Cβmax(1,CIII)/bracketleftbig
1+C2+Cǫ(C1h+C2)/bracketrightbig
, (5.14)
which is independent of y,handν. This guarantees that Cβ//hatwideCβ≤1, and thus since
|m|,|ν−m| ≥1, we have the bound
/vextenddouble/vextenddouble∂ν
y(u−uh)/vextenddouble/vextenddouble
V
≤/hatwideβνhCIIICβ
/hatwideCβ/parenleftBigg
(1+C2)|ν|!1+ǫ+ (C1h+C2)/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg
|ν−m|!1+ǫ|m|!1+ǫ/parenrightBigg
≤CIIICβ
/hatwideCβ/bracketleftbig
1+C2+Cǫ(C1h+C2)/bracketrightbig
|ν|!1+ǫ/hatwideβνh≤C2|ν|!1+ǫ/hatwideβνh,
where we have used [19, Lemma 3.3] to bound the sum from above b yCǫ|ν|!1+ǫ(see also
(4.8)), as well as (5.14) to give the final result.
For the inductive step for the eigenvalue, we substitute the result (5.11), which has
just been shown to hold for all multi-indices of order up to an d including |ν|, into (5.7)
18and then simplify, to give
|∂ν
y(λ−λh)| ≤CII(C2)2/hatwideβνh2/parenleftBigg/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
|ν−m|!1+ǫ|m|!1+ǫ
+∞/summationdisplay
j=1νj/summationdisplay
m≤ν−ej/parenleftbiggν−ej
m/parenrightbigg
|ν−ej−m|!1+ǫ|m|!1+ǫ
+/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
|ν−m|!1+ǫ/summationdisplay
k≤m/parenleftbiggm
k/parenrightbigg
|m−k|!1+ǫ|k|!1+ǫ/parenrightBigg
,
where we have used the fact that βj≤/hatwideβj. The sums can again be bounded using (4.8)
(using it twice for the double sum on the last line), to give
|∂ν
y(λ−λh)| ≤CII(C2)2/hatwideβνh2(2+Cǫ)(4+Cǫ)|ν|!1+ǫ,
which, with C1as defined in (5.13) and C2=Cu, gives our desired result (5.10).
The final result for the derivative of the error of the linear f unctional G(u) (5.12)
follows by considering the same dual problem (A.16) as in [19 ]. But instead, here we let
w=∂ν
y(u−uh) and then use the upper bound (5.11) in the last step.
We can now simply substitute these bounds on the derivatives of the FE error into
(5.4), in order to bound the FE component of the error. For the second and third term in
(5.4), in the case of the eigenvalue, this gives
/⌊a∇d⌊lλs(y)−λh,s(y)/⌊a∇d⌊lWs,γ≤h2/parenleftBigg
C2
1/summationdisplay
u⊆{1:s}|u|!2(1+ǫ)
γu/productdisplay
j∈u/hatwideβ2
j/parenrightBigg1/2
. (5.15)
Similar results hold for u(y) andG(u(y)).
To ensure that the constant on the RHS of (5.15), and the const ants in the bounds
that follow, are independent of the dimension, for ξ∈(1
2,1] to be specified later, we will
choose the weights γby
γj= max/parenleftbig/hatwideβj,βp/q
j/parenrightbig
, γ u=/parenleftBigg
(|u|+3)!2(1+ǫ)/productdisplay
j∈u(2π2)ξ
2ζ(2ξ)γ2
j/parenrightBigg1/(1+ξ)
,(5.16)
wherep,qarethesummability parametersfromAssumptionA1.3, sotha t(γj)j∈N∈ℓq(R).
5.2 Estimating the truncation error
It remains to estimate the first term in (5.4) — the truncation error.
Theorem 5.2. Suppose that Assumption A1 holds and let s,/tildewides∈Nwiths >/tildewides. Addition-
ally, suppose that the weights γare given by (5.16), then
/⌊a∇d⌊lλs−λ/tildewides/⌊a∇d⌊lWs,γ/lessorsimilar/tildewides−1/p+1/q/parenleftBigg/summationdisplay
u⊆{1:/tildewides}(|u|+3)!2(1+ǫ)
γu/productdisplay
j∈uβ2
j/parenrightBigg1/2
,(5.17)
with the constant independent of /tildewidesands.
19Proof.Sinceλsis analytic we can expand it as a Taylor series about 0in the variables
{y/tildewides+1,...,ys}:
λs(ys) =λs(y/tildewides;0)+s/summationdisplay
i=/tildewides+1yi/integraldisplay1
0∂
∂yiλs(y/tildewides;ty{/tildewides+1:s}) dt,
where we use the notation ys= (y1,y2,...,ys), (y/tildewides;0) = (y1,y2,...,y/tildewides,0,...0) and
(y/tildewides;ty{/tildewides+1:s}) = (y1,y2,...,y/tildewides,ty/tildewides+1,ty/tildewides+2,...,ty s).
Sinceλ/tildewides(y/tildewides) =λs(y/tildewides;0) (this is simply different notation for the same object), this
can be rearranged to give
λs(ys)−λ/tildewides(y/tildewides) =s/summationdisplay
i=/tildewides+1yi/integraldisplay1
0∂
∂yi(y/tildewides;ty{/tildewides+1:s}) dt. (5.18)
Letu⊆ {1,2...,/tildewides}, then differentiating (5.18) with respect to yugives
∂|u|
∂yu/parenleftbig
λs(ys)−λ/tildewides(y/tildewides)/parenrightbig
=s/summationdisplay
i=/tildewides+1yi/integraldisplay1
0∂|u|+1
∂yu∪{i}λs(y/tildewides;ty{/tildewides+1:s}) dt.
Taking the absolute value, using the triangle inequality an d the fact that |yj| ≤1/2, we
have the upper bound
/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂|u|
∂yu/parenleftbig
λs(ys)−λ/tildewides(y/tildewides)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1
2s/summationdisplay
i=/tildewides+1/integraldisplay1
0/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂|u|+1
∂yu∪{i}λs(y/tildewides;ty{/tildewides+1:s})/vextendsingle/vextendsingle/vextendsingle/vextendsingledt.
Now, substituting in the upper bound on the derivative of λsfrom [19, Lemma 3.4, equa-
tion (3.6)] gives
/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂|u|
∂yu/parenleftbig
λs(ys)−λ/tildewides(y/tildewides)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤λ
2s/summationdisplay
i=/tildewides+1(|u|+1)!1+ǫβi/productdisplay
j∈uβj
=λ
2/parenleftBiggs/summationdisplay
i=/tildewides+1βi/parenrightBigg
(|u|+1)!1+ǫ/productdisplay
j∈uβj, (5.19)
withβjas in (4.1).
Lettingu⊆ {1,2,...,s}withu∩ {/tildewides+ 1,/tildewides+ 2,...,s} /\e}atio\slash=∅, the derivative λs−λ/tildewidesis
simply/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂|u|
∂yu/parenleftbig
λs(ys)−λ/tildewides(y/tildewides)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂|u|
∂yuλs(ys)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤λ
2|u|!1+ǫ/productdisplay
j∈uβj, (5.20)
where we have again used the upper bound [19, equation (3.6)] .
We now boundthe norm(2.21) of λs−λ/tildewidesinWs,γ. Splitting thesumover u⊆ {1,2,...}
by whether ucontains any of {/tildewides+1,/tildewides+2,...,s}, we can write
/⌊a∇d⌊lλs−λ/tildewides/⌊a∇d⌊l2
Ws,γ=/summationdisplay
u⊆{1:/tildewides}1
γu/integraldisplay
[−1
2,1
2]|u|/parenleftbigg/integraldisplay
[−1
2,1
2]s−|u|∂|u|
∂yu/bracketleftbig
λs(ys)−λ/tildewides(y/tildewides)/bracketrightbig
dy−u/parenrightbigg2
dyu
+/summationdisplay
u⊆{1:s}
u∩{/tildewides+1:s}/\e}atio\slash=∅1
γu/integraldisplay
[−1
2,1
2]|u|/parenleftbigg/integraldisplay
[−1
2,1
2]s−|u|∂|u|
∂yu/bracketleftbig
λs(ys)−λ/tildewides(y/tildewides)/bracketrightbig
dy−u/parenrightbigg2
dyu.
20Substituting in the bounds (5.19) and (5.20) then yields
/⌊a∇d⌊lλs−λ/tildewides/⌊a∇d⌊lWs,γ≤/bracketleftBigg/parenleftBiggs/summationdisplay
i=/tildewides+1βi/parenrightBigg2/summationdisplay
u⊆{1:/tildewides}(|u|+1)!2(1+ǫ)
γu/productdisplay
j∈uβ2
j
+/summationdisplay
u⊆{1:s}
u∩{/tildewides+1:s}/\e}atio\slash=∅|u|!2(1+ǫ)
γu/productdisplay
j∈uβ2
j/bracketrightBigg1/2
. (5.21)
Next, we can bound the sum over iin (5.21) by the whole tail of the sum, which can
then be bounded using [19, eq. (4.7)], to give
s/summationdisplay
i=/tildewides+1βi≤∞/summationdisplay
i=/tildewides+1βi≤min/parenleftbiggp
1−p,1/parenrightbigg
/⌊a∇d⌊lβ/⌊a∇d⌊lℓp/tildewides−(1/p−1). (5.22)
Then for weights given by (5.16), following the proof of [34, Theorem 11] we can bound
/summationdisplay
u⊆{1:s}
u∩{/tildewides+1:s}/\e}atio\slash=∅|u|!2(1+ǫ)
γu/productdisplay
j∈uβ2
j/lessorsimilar/tildewides−2(1/p−1/q)/summationdisplay
u⊆{1:/tildewides}(|u|+3)!2(1+ǫ)
γu/productdisplay
j∈uβ2
j,(5.23)
with a constant that is independent of /tildewidesands.
Since 1/q >1, after substituting the bounds (5.22) and (5.23) into (5.2 1) we obtain
the final result (5.17).
5.3 Final error bound
In the previous two sections we have successfully bounded th e FE and truncation error
in theWs,γnorm, now these bounds can simply be substituted into (5.4) t o bound the
variance on each level.
Theorem 5.3. LetL∈N, let1 =h−1> h0> h1>···> hL>0withh0sufficiently
small, let 1 =s−1=s0≤s1··· ≤ ··· sL, and suppose that Assumption A1 holds with
p < q. Also, let each Qℓbe a lattice rule using Nℓ= 2nℓ,nℓ∈N, points corresponding
to a CBC-constructed generating vector with weights γgiven by (5.16). Then, for all
ℓ= 0,1,2,...L,
V∆[Qℓ(λℓ−λℓ−1)]≤C1N−η
ℓ/parenleftbig
h4
ℓ−1+s−2(1/p−1/q)
ℓ−1/parenrightbig
, (5.24)
and, for G ∈H−1+t(D)with0≤t≤1,
V∆[Qℓ(G(uℓ)−G(uℓ−1))]≤C2N−η
ℓ/parenleftbig
h2(1+t)
ℓ−1+s−2(1/p−1/q)
ℓ−1/parenrightbig
, (5.25)
where, for 0< δ <1,
η=
2−δifq∈(0,2
3],
2
q−1ifq∈(2
3,1).
The second term in (5.24)and(5.25)can be dropped if sℓ=sL, forℓ= 1,2,...L.
Proof.We prove the result for the eigenvalue, since the eigenfunct ion result follows anal-
ogously. For ℓ≥1, substituting the bounds (5.15) and (5.17) into (5.4) give s
/⌊a∇d⌊lλℓ−λℓ−1/⌊a∇d⌊lWs,γ≤C/parenleftBigg/summationdisplay
u⊆{1:sℓ}(|u|+3)!2(1+ǫ)
γu/productdisplay
j∈u/hatwideβ2
j/parenrightBigg1/2/parenleftbig
h2
ℓ−1+s−(1/p−1/q)
ℓ−1/parenrightbig
,
21where we have simplified by using that hℓ< hℓ−1,sℓ−1< sℓ,βj≤/hatwideβj, and also merged all
constants into a generic constant C, which may depend on ǫ.
Substituting the bound above into (5.3), then using that Nℓ= 2nℓand thus ϕ(Nℓ) =
Nℓ/2, the variance on level ℓcan be bounded by
V∆[Qℓ(λℓ−λℓ−1)]≤Cℓ,γ,ξN−1/ξ
ℓ/parenleftbig
s−2(1/p−1/q)
ℓ−1+h4
ℓ−1/parenrightbig
. (5.26)
The constant is given by
Cℓ,γ,ξ:=C221/ξ/parenleftBigg/summationdisplay
u⊆{1:sℓ}(|u|+3)!2(1+ǫ)
γu/productdisplay
j∈u/hatwideβ2
j/parenrightBigg/parenleftBigg/summationdisplay
∅/\e}atio\slash=u⊆{1:sℓ}γξ
u/parenleftbigg2ζ(2ξ)
(2π2)ξ/parenrightbigg|u|/parenrightBigg1/ξ
.
Forℓ= 0 we can similarly substitute (4.3) into the CBC bound (2.22 ), and then since
h−1=s−1= 1 and βj≤/hatwideβjit follows that (5.26) also holds for ℓ= 0 with the constant
C0,γ,ξas above.
All that remains to be shown is that this constant can be bound ed independently of
sℓ−1andsℓ. To this end, substituting the formula (5.16) for γuand using the fact that
/hatwideβj≤γjthen simplifying, we can bound Cℓ,γ,ξabove by
Cℓ,γ,ξ≤C/parenleftBigg/summationdisplay
|u|<∞(|u|+3)!2ξ(1+ǫ)
1+ξ/productdisplay
j∈uγ2ξ
1+ξ
j/parenleftbigg2ζ(2ξ)
(2π2)ξ/parenrightbigg1
1+ξ
/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright
Sξ/parenrightBigg1+ξ
ξ
,
where again Cis a generic constant, which may depend on ǫ.
We now choose the exponents ξandǫso that the sum Sξis finite. For 0 < δ <1, let
ξ=
1
2−δifq∈(0,2
3],
q
2−qifq∈(2
3,1),andǫ=1−ξ
4ξ>0. (5.27)
With this choice of ξwe have (4 ξ−q−3qξ)/(1−ξ)≥qfor anyq∈(0,1), and so
∞/summationdisplay
j=1γ4ξ−q−3qξ
1−ξ
j <∞. (5.28)
Then define the sequence
αj=/parenleftBigg
1+∞/summationdisplay
i=1γq
i/parenrightBigg−1
γq
i,so that∞/summationdisplay
j=1αj<1. (5.29)
Substituting in our choice (5.27) for ǫ, and multiplying and dividing each term by the
product of α(1+3ξ)/(2(1+ξ))
j , we can write
Sξ=/summationdisplay
|u|<∞(|u|+3)!1+3ξ
2(1+ξ)/parenleftBigg/productdisplay
j∈uα1+3ξ
2(1+ξ)
j/parenrightBigg/parenleftBigg/productdisplay
j∈uγ2ξ
1+ξ
jα−1+3ξ
2(1+ξ)
j/parenleftbigg2ζ(2ξ)
(2π2)ξ/parenrightbigg1
1+ξ/parenrightBigg
.
22Applying H¨ older’s inequality with exponents 2(1+ ξ)/(1+3ξ)>1 and 2(1+ ξ)/(1−ξ)>1
gives
Sξ≤/parenleftBigg/summationdisplay
|u|<∞(|u|+3)!/productdisplay
j∈uαj/parenrightBigg1+3ξ
2(1+ξ)/parenleftBigg/summationdisplay
|u|<∞/productdisplay
j∈uγ4ξ
1−ξ
jα−1+3ξ
1−ξ
j/parenleftbigg2ζ(2ξ)
(2π2)ξ/parenrightbigg2
1−ξ/parenrightBigg2−ξ
2(1+ξ)
≤/bracketleftBigg
6/parenleftbigg
1−∞/summationdisplay
j=1αj/parenrightbigg−4/bracketrightBigg1+3ξ
2(1+ξ)
·exp/bracketleftBigg
2−ξ
2(1+ξ)/parenleftbigg2ζ(2ξ)
(2π2)ξ/parenrightbigg2
1−ξ/parenleftbigg
1+∞/summationdisplay
j=1γq
j/parenrightbigg1+3ξ
1−ξ∞/summationdisplay
j=1γ4ξ−q−3qξ
1−ξ
j/bracketrightBigg
,
where we have used [35, Lemma 6.3]. From (5.28) and (5.29) it f ollows that Sξ<∞, and
soCℓ,γ,ξcan be bounded independently of sℓ. Finally, letting η= 1/ξforξas in (5.27)
gives the desired result with a constant independent of sℓ.
Remark 5.1. Hence, we have verified that Assumptions M2 from Theorems 3.1 and 3.2
hold with βλ= 2αλ= 4,βG= 2αG= 2(1+t),β′= 1/p+1/q, andηas given above.
The upper bounds in Theorem 5.1, (5.15) and Theorem 5.2 are th e same as the corre-
sponding bounds from the MLQMC analysis for the source probl em (see [34, Theorems 7,
8, 11]), the only differences are in the values of the constants and in the extra 1+ ǫfactor
in the exponent of |u|!. As such the final variance bounds in Theorem 5.3 also coinci de
with the bounds for the source problem from [34] for all q <1. The only difference is that
our result does not hold for q= 1, whereas the results for the source problem do.
5.4 Extension to higher-order QMC
As mentioned earlier, the bounds on the higher-order deriva tives that we proved in Sec-
tion 4 imply higher order methods can also be used for the quad rature component of our
ML algorithm, which will provide a faster convergence rate i nNℓ. We now provide a brief
discussionofhowtoextendourMLalgorithm, andtheerroran alysis, to higher-order QMC
(HOQMC) rules. From an algorithm point of view, one can simply use HOQ MC points
instead of lattice rules for the quadrature rules Qℓin (3.1). We denote this ML-HOQMC
approximation by QMLHO
L. To extend the error analysis to HOQMC we can again use a
general framework as in Theorems 3.1 and 3.2. We stress that t he difficult part is to verify
the assumptions, and in particular to show the required mixe d higher-order derivative
bounds that we have already proved in Theorem 4.1. The remain der of the analysis then
follows the same steps as in the previous sections with only s light modifications to handle
the higher-order norm as in [12], where ML-HOQMC methods wer e applied to PDE source
problems. As such, we don’t present the full details here but only an outline.
A HOQMC rule is an equal-weight quadrature rule of the form (2 .20) that can achieve
faster than 1 /Nconvergence for sufficiently smooth integrands. A popular cl ass of de-
terministic HOQMC rules are interlaced polynomial lattice rules , see [9, 24] and [11, 12]
for their application to PDE source problems. Loosely speak ing, a polynomial lattice rule
is a QMC rule similar to a lattice rule, except the points are g enerated by a vector of
polynomials instead of integers, the number of points Nis a prime power and the points
are not randomly shifted. Higher order convergence in sdimensions is then achieved by
taking a polynomial lattice rule in a higher dimension, ν·sforν∈N, and cleverly in-
terlacing the digits across the dimensions of each ( νs)-dimensional point to produce an
s-dimensional point. The factor ν∈Nis called the interlacing order and it determines
23the convergence rate. Good interlaced polynomial lattice r ules can also be constructed by
a CBC algorithm. See [24] for the full details.
Following [12], for ν∈Nand 1≤r≤ ∞we introduce the Banach space Wν,r
s,γ, which
is a higher-order analogue of the first-order space Ws,γ, with the norm
/⌊a∇d⌊lf/⌊a∇d⌊lWν,r
s,γ= max
u⊆{1:s}1
γu/parenleftBigg/summationdisplay
v⊆u/summationdisplay
τu\v∈{1:ν}|u\v|
/integraldisplay
[−1
2,1
2]|v|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay
[−1
2,1
2]s−|v|∂(νv,τu\v,0)
y f(y) dy−v/vextendsingle/vextendsingle/vextendsingle/vextendsingler
dyv/parenrightBigg1/r
.(5.30)
Here (νv,τu\v,0)∈ Fis the multi-index with jth entry given by νifj∈v,τjifj∈u\v
and0otherwise. For f∈ Wν,r
s,γ, anorder νinterlaced polynomiallattice ruleusing Npoints
insdimensions can be constructed using a CBC algorithm such tha t the (deterministic)
error converges at a rate N−ηfor 1≤η < ν(see [11, Theorem 3.10]).
Letνp=⌊1/p⌋+ 1 forp <1 as in Assumption A1 and 1 ≤r≤ ∞, then it follows
from (4.3) that λs∈ Wνp,r
s,γfor alls. Hence, the error of a single level QMC approximation
ofEy[λs] using an order νpinterlaced polynomial lattice rule will converge as N−1/p.
Similarly, the ML analysis can be extended to show that a ML-H OQMC method achieves
higher order convergence in Nℓ, where in this case we choose the interlacing factor to be
νq=⌊1/q⌋+ 1 forq <1 as in Assumption A1. Indeed, (5.10) implies that the bound
(5.15) can easily beextended to Wνq,r
s,γand(4.3) implies that (5.17) can also beextended to
Wνq,r
s,γfor alls. In both cases, the sums over uon the right hand sides need to be updated
to account for the form of (5.30), but the exponents of handsremain the same. Hence,
by following the proof of Theorem 5.3 it can be shown that the f ollowing deterministic
analogue of the variance bound (5.3) holds for interlaced po lynomial lattice rules.
Theorem 5.4. Suppose that Assumption A1 holds with p < q <1. Forℓ∈N, letQHO
ℓbe
an interlaced polynomial lattice rule, constructed using a CBC algorithm with Nℓa prime
power number of points and interlacing factor νq=⌊1/q⌋+1. ThenQHO
ℓsatisfies
/vextendsingle/vextendsingleQHO
ℓ(λℓ−λℓ−1)/vextendsingle/vextendsingle/lessorsimilarN−1/q
ℓ/parenleftbig
h2
ℓ+s−1/p+1/q
ℓ/parenrightbig
, (5.31)
where the implied constant is independent of hℓ,sℓandNℓ.
The fact that the implied constant in (5.31) is independent o fsℓcan be shown by
following similar arguments as in [12] using a special form o fγucalledsmoothness-driven,
product and order-dependent (SPOD) weights, as introduced in [11, eq. (3.17)]. Thus
the following deterministic version of Theorem 3.1 holds fo r the error of the ML-HOQMC
approximation.
Theorem 5.5. Suppose that Assumption A1 holds with p < q < 1, letL∈Nand for
ℓ= 0,1,...,LletQHO
ℓbe an interlaced polynomial lattice rule as in Theorem 5.4. The n
the multilevel HOQMC approximation QMLHO
Lwith quadrature rule QHO
ℓon each level
satisfies
/vextendsingle/vextendsingleEy[λ]−QMLHO
L(λ)/vextendsingle/vextendsingle/lessorsimilarh2
L+s−2/p+1
L+L/summationdisplay
ℓ=0N−1/q
ℓ/parenleftbig
h2
ℓ+s−1/p+1/q
ℓ/parenrightbig
,
where the implied constant is independent of hℓ,sℓandNℓfor allℓ= 0,1,...,L.
Similar arguments can also be used to obtain an error bound wi th the same conver-
gence rates for QMLHO
L(G(u)), i.e., for the approximation of the expected value of smoo th
functionals of the eigenfunction.
246 Conclusion
We have presented a MLQMC algorithm for approximating the ex pectation of the eigen-
value of a random elliptic EVP, and then performed a rigorous analysis of the error. The
theoretical results clearly show that for this problem the M LQMC method exhibits better
complexity than both single level MC/QMC and MLMC. In the com panion paper [21],
we will present numerical results that also verify this supe rior performance of MLQMC
in practice. In that paper, we will in addition present novel ideas on how to efficiently
implement the MLQMC algorithm for EVPs.
Other interesting avenues for future research would be to co nsider non-self adjoint
EVPs, e.g., convection-diffusion problems, or to use the mult i-index MC framework from,
e.g., [10, 27] to separate theFE anddimension truncation ap proximations on each level. In
principle, the algorithm studied in this paper can also be ap plied to the lognormal setting
as in, e.g., [33], i.e., whereeach coefficient is theexponent ial of a Gaussian randomfield, by
using QMC rules for integrals on unbounded domains. However , in this case, the difficulty
for both single level and multilevel QMC is that the coefficien ts are no longer uniformly
boundedfromabove and below. As such, it is possiblethat the spectral gap, λ2(y)−λ1(y),
becomes arbitrarily small for certain parameter values. Si nce all aspects of the method
(the stochastic derivative bounds, the FE error, the perfor mance of the eigenvalue solver
etc.) depend inversely on the spectral gap, then both the met hod and the theory fail if the
gap becomes arbitrarily small. Thetechnique for boundingt he spectral gap in [19, 20] fails
in this case because the stochastic parameters belong to an u nbounded domain. On the
other hand, we conjecture that the spectral gap only becomes small with low probability,
and so probabilistic arguments may be able to be used to bound the gap from below. This
is again another example of the differences between stochasti c EVPs and source problems,
and such analysis would make for interesting future work.
Acknowledgements. This work is supported by the Deutsche Forschungsgemeinsch aft
(GermanResearchFoundation)underGermany’sExcellenceS trategyEXC2181/1-390900948
(the Heidelberg STRUCTURES Excellence Cluster).
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A Proofs of recursive bounds on derivatives of the FE error
Here, we give the proofs of the recursive bounds on the deriva tives of the FE error from
Section 5.1 (Lemmas 5.2 and 5.3), which were key to the induct ive steps in the proofs of
the explicit bounds in Theorem 5.1. Throughout we omit the xandydependence.
Proof of Lemma 5.2 (eigenvalue bounds). Letv=vh∈Vhin thevariational eigenproblem
(2.3), andthensubtracttheFEeigenproblem(2.10), withth esamevh,togivethefollowing
variational relationship between the two FE errors
A(u−uh,vh) =λM(u−uh,vh)+(λ−λh)M(uh,vh), (A.1)
which holds for all vh∈Vh.
Differentiating (A.1) using the Leibniz general product rule , gives the following recur-
sive formula for the νth derivatives of the eigenvalue and eigenfunction errors
0 =A(∂ν
y(u−uh),vh)−λM(∂ν
y(u−uh),vh)−(λ−λh)M(∂ν
yuh,vh)
+∞/summationdisplay
j=1νj/parenleftbigg/integraldisplay
Daj∇/bracketleftbig
∂ν−ejy(u−uh)/bracketrightbig
·∇vh+/integraldisplay
Dbj/bracketleftbig
∂ν−ejy(u−uh)/bracketrightbig
vh/parenrightbigg
−/summationdisplay
m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg/bracketleftbig
∂ν−m
yλM(∂m
y(u−uh),vh)+∂ν−m
y(λ−λh)M(∂m
yuh,vh)/bracketrightbig
.
28Adding extra terms and usingthe A-orthogonality of Ph, we can write this in the following
more convenient form
0 =A(Ph∂ν
y(u−uh),vh)−λhM(Ph∂ν
y(u−uh),vh)
−(λ−λh)M(∂ν
yu,vh)−λhM(∂ν
yu−Ph∂ν
yu,vh)
+∞/summationdisplay
j=1νj/parenleftbigg/integraldisplay
Daj∇/bracketleftbig
∂ν−ej
y(u−uh)/bracketrightbig
·∇vh+/integraldisplay
Dbj/bracketleftbig
∂ν−ej
y(u−uh)/bracketrightbig
vh/parenrightbigg
−/summationdisplay
m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg/bracketleftbig
∂ν−m
yλM(∂m
y(u−uh),vh)+∂ν−m
y(λ−λh)M(∂m
yuh,vh)/bracketrightbig
.(A.2)
Lettingvh=uhin (A.2) and separating out the m=0term, we obtain the following
formula for the derivative of the eigenvalue error
∂ν
y(λ−λh) = (λh−λ)M(∂ν
yu,uh)−λhM(∂ν
yu−Ph∂ν
yu,uh)−(∂ν
yλ)M(u−uh,uh)
+∞/summationdisplay
j=1νj/parenleftbigg/integraldisplay
Daj∇/bracketleftbig
∂ν−ej
y(u−uh)/bracketrightbig
·∇uh+/integraldisplay
Dbj/bracketleftbig
∂ν−ej
y(u−uh)/bracketrightbig
uh/parenrightbigg
−/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg/bracketleftbig
∂ν−m
yλM(∂m
y(u−uh),uh)+∂ν−m
y(λ−λh)M(∂m
yuh,uh)/bracketrightbig
,
where we have used the fact that uhis normalised. Also the first two terms in (A.2) cancel
because the bilinear form is symmetric and ( λh,uh) satisfy the FE eigenvalue problem
(2.10) with Ph∂ν
y(u−uh)∈Vhas a test function.
Taking the absolute value, then using the triangle and Cauch y–Schwarz inequalities
gives the upper bound
|∂ν
y(λ−λh)| ≤ |λ−λh|/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lM+λh/⌊a∇d⌊l∂ν
yu−Ph∂ν
yu/⌊a∇d⌊lM+|∂ν
yλ|/⌊a∇d⌊lu−uh/⌊a∇d⌊lM
+∞/summationdisplay
j=1νj/bracketleftbig
/⌊a∇d⌊laj/⌊a∇d⌊lL∞/⌊a∇d⌊l∂ν−ej
y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊luh/⌊a∇d⌊lV+/⌊a∇d⌊lbj/⌊a∇d⌊lL∞/⌊a∇d⌊l∂ν−ej
y(u−uh)/⌊a∇d⌊lL2/⌊a∇d⌊luh/⌊a∇d⌊lL2/bracketrightbig
+/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg/bracketleftbig
|∂ν−m
yλ|/⌊a∇d⌊l∂m
y(u−uh)/⌊a∇d⌊lM+|∂ν−m
y(λ−λh)|/⌊a∇d⌊l∂m
yuh/⌊a∇d⌊lM/bracketrightbig
,
where we have again simplified by using /⌊a∇d⌊luh/⌊a∇d⌊lM= 1. Then, using the equivalence of
norms (2.8) and the Poincar´ e inequality (2.6), we can bound theM- andL2-norms by the
corresponding V-norms, to give
|∂ν
y(λ−λh)| ≤/radicalbiggamax
χ1/bracketleftBig
|λ−λh|/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lV+λ/⌊a∇d⌊l∂ν
yu−Ph∂ν
yu/⌊a∇d⌊lV+|∂ν
yλ|/⌊a∇d⌊lu−uh/⌊a∇d⌊lV/bracketrightBig
+u/parenleftbigg
1+1
χ1/parenrightbigg∞/summationdisplay
j=1νjβj
Cβ/⌊a∇d⌊l∂ν−ej
y(u−uh)/⌊a∇d⌊lV
+/radicalbiggamax
χ1/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg/bracketleftbig
|∂ν−m
yλ|/⌊a∇d⌊l∂m
y(u−uh)/⌊a∇d⌊lV+|∂ν−m
y(λ−λh)|/⌊a∇d⌊l∂m
yuh/⌊a∇d⌊lV/bracketrightbig
,
where we have also used the upper bounds (2.16) and (2.17), an d the definition of βj(4.1).
29Substituting in the upper bounds on the derivatives (4.3) an d (4.4), the bound on the
projection error (5.5), and then the bounds on the FE errors ( 2.11) and (2.12), we have
the upper bound
|∂ν
y(λ−λh)| ≤/radicalbiggamax
χ1/bracketleftbig
Cλhuβν+λCPβν+λCuβν/bracketrightBig
h|ν|!1+ǫ
+u
Cβ/parenleftbigg
1+1
χ1/parenrightbigg∞/summationdisplay
j=1νjβj/⌊a∇d⌊l∂ν−ej
y(u−uh)/⌊a∇d⌊lV+/radicalbiggamax
χ1/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg
·/bracketleftBig
λ|ν−m|!1+ǫβν−m/⌊a∇d⌊l∂m
y(u−uh)/⌊a∇d⌊lV+u|m|!1+ǫβm|∂ν−m
y(λ−λh)|/bracketrightBig
.
Note that we can simplify the sum on the last line using the sym metry of the binomial
coefficient,/parenleftbign
k/parenrightbig
=/parenleftbign
n−k/parenrightbig
, as follows. First, we separate it into two sums
/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg/bracketleftBig
λ|ν−m|!1+ǫβν−m/⌊a∇d⌊l∂m
y(u−uh)/⌊a∇d⌊lV+u|m|!1+ǫβm|∂ν−m
y(λ−λh)|/bracketrightBig
=λ/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg
|ν−m|!1+ǫβν−m/⌊a∇d⌊l∂m
y(u−uh)/⌊a∇d⌊lV
+u/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg
|m|!1+ǫβm|∂ν−m
y(λ−λh)|/bracketrightBig
=(λ+u)/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg
|m|!1+ǫβm/bracketleftBig
/⌊a∇d⌊l∂ν−m
y(u−uh)/⌊a∇d⌊lV+|∂ν−m
y(λ−λh)|/bracketrightBig
,
(A.3)
where to obtain the last equality we have simply relabelled t he indices in the first sum.
Then, since βj≤βjandhis sufficiently small (i.e., h≤hwithhas in (2.14)), the
result (5.6) holds. The constant is given by
CI:= max/braceleftbigg/radicalbiggamax
χ1/bracketleftbig
uhCλ+λ(CP+Cu)/bracketrightbig
,u
Cβ/parenleftbigg
1+1
χ1/parenrightbigg
,/radicalbiggamax
χ1(λ+u)/bracerightbigg
,
which is independent of y,handν.
For the second result (5.7), using [4, Lemma 3.1] the eigenva lue error can also be
written as
λ−λh=−A(u−uh,u−uh)+λM(u−uh,u−uh),
which after taking the νth derivative becomes
∂ν
y(λ−λh) =−/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
A(∂ν−m
y(u−uh),∂m
y(u−uh))
−∞/summationdisplay
j=1/summationdisplay
m≤ν−ejνj/parenleftbiggν−ej
m/parenrightbigg/bracketleftbigg/integraldisplay
Daj∇∂ν−ej−m
y(u−uh)·∇∂m
y(u−uh)
+/integraldisplay
Dbj∂ν−ej−m
y(u−uh)∂m
y(u−uh)/bracketrightbigg
+/summationdisplay
m≤ν/summationdisplay
k≤m/parenleftbiggν
m/parenrightbigg/parenleftbiggm
k/parenrightbigg
∂ν−m
yλM(∂m−k
y(u−uh),∂k
y(u−uh)).
30Taking the absolute value, then usingthe triangle, Cauchy– Schwarz and Poincar´ e (2.6)
inequalities, along with the norm equivalences (2.7), (2.8 ), gives
|∂ν
y(λ−λh)| ≤amax/parenleftbigg
1+1
χ1/parenrightbigg/summationdisplay
m≤ν/parenleftbiggν
m/parenrightbigg
/⌊a∇d⌊l∂ν−m
y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊l∂m
y(u−uh)/⌊a∇d⌊lV
+/parenleftbigg
1+1
χ1/parenrightbigg∞/summationdisplay
j=1/summationdisplay
m≤ν−ejνj/parenleftbiggν−ej
m/parenrightbiggβj
Cβ/⌊a∇d⌊l∂ν−ej−m
y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊l∂m
y(u−uh)/⌊a∇d⌊lV
+amax
χ1/summationdisplay
m≤ν/summationdisplay
k≤m/parenleftbiggν
m/parenrightbigg/parenleftbiggm
k/parenrightbigg
|∂ν−m
yλ|/⌊a∇d⌊l∂m−k
y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊l∂k
y(u−uh))/⌊a∇d⌊lV.
Finally, substituting in the upper bound (4.3) on the deriva tive ofλgives the desired
result (5.7). The constant is given by
CII:=/bracketleftbig
1/Cβ+amax(1+λ)/bracketrightbig/parenleftbigg
1+1
χ1/parenrightbigg
,
which is independent of h,yandν.
Proof of Lemma 5.3 (eigenfunction bound). Wedealwiththeeigenfunctionerrorprojected
ontoVh, as opposed to ∂ν
y(u−uh), because the latter belongs to Vbut not to Vh. As
such, we first separate the error as
/⌊a∇d⌊l∂ν
y(u−uh)/⌊a∇d⌊lV≤ /⌊a∇d⌊lPh∂ν
y(u−uh)/⌊a∇d⌊lV+/⌊a∇d⌊l∂ν
yu−Ph∂ν
yu/⌊a∇d⌊lV
≤ /⌊a∇d⌊lPh∂ν
y(u−uh)/⌊a∇d⌊lV+CPh|ν|!1+ǫβν, (A.4)
where in the second inequality we have used the bound (5.5).
Similar to the proof of [19, Lemma 3.4], the bilinear form tha t acts on Ph∂ν
y(u−uh)
(namely, A −λhM) is only coercive on the orthogonal complement of the eigens pace
corresponding to λh, which we denote by E(λh)⊥. Hence, to obtain the recursive for-
mula for the derivative of the eigenfunction error, we first m ake the following orthogonal
decomposition. The FE eigenfunctions form an orthogonal ba sis forVh, and so we have
Ph∂ν
y(u−uh) =M(Ph∂ν
y(u−uh),uh)uh+ϕh, (A.5)
whereϕh∈E(λh)⊥. Then we can bound the norm by
/⌊a∇d⌊lPh∂ν
y(u−uh)/⌊a∇d⌊lV≤ |M(Ph∂ν
y(u−uh),uh)|/⌊a∇d⌊luh/⌊a∇d⌊lV+/⌊a∇d⌊lϕh/⌊a∇d⌊lV. (A.6)
To bound the first term in this decomposition (A.6), first obse rve that we can write
|M(Ph∂ν
y(u−uh),uh)| ≤|M(∂ν
yu,u)−M(∂ν
yuh,uh)|
+|M(∂ν
yu,u−uh)|+|M(∂ν
yu−Ph∂ν
yu,uh)|.(A.7)
Thefirstterm ontheright in(A.7) can beboundedby differentia ting thenormalisation
equations /⌊a∇d⌊lu/⌊a∇d⌊lM= 1 and /⌊a∇d⌊luh/⌊a∇d⌊lM= 1 (see [19, eq. (3.15)]) to give
M(∂ν
yu,u)−M(∂ν
yuh,uh)
=−1
2/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg/bracketleftbig
M(∂ν−m
yu,∂m
yu)−M(∂ν−m
yuh,∂m
yuh)/bracketrightbig
=−1
2/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg/bracketleftbig
M(∂ν−m
y(u−uh),∂m
yu)+M(∂ν−m
yuh,∂m
y(u−uh))/bracketrightbig
.
31Then, using the triangle inequality, the Cauchy–Schwarz in equality, the equivalence of
norms (2.8) and the Poincar´ e inequality (2.6), gives the up per bound
|M(∂ν
yu,u)−M(∂ν
yuh,uh)|
≤amax
2χ1/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg/bracketleftbig
/⌊a∇d⌊l∂ν−m
y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊l∂m
yu/⌊a∇d⌊lV+/⌊a∇d⌊l∂ν−m
yuh/⌊a∇d⌊lV/⌊a∇d⌊l∂m
y(u−uh)/⌊a∇d⌊lV/bracketrightbig
≤uamax
2χ1/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg/bracketleftbig
|m|!1+ǫβm/⌊a∇d⌊l∂ν−m
y(u−uh)/⌊a∇d⌊lV
+|ν−m|!1+ǫβν−m/⌊a∇d⌊l∂m
y(u−uh)/⌊a∇d⌊lV/bracketrightbig
=uamax
χ1/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg
|m|!1+ǫβm/⌊a∇d⌊l∂ν−m
y(u−uh)/⌊a∇d⌊lV,
where for the second last inequality we have used the upper bo und(4.4) and the analogous
bound for uh. For the equality on the last line, we have simplified the sum u sing the
symmetry of the binomial coefficient as in (A.3).
To bound the second and third terms in (A.7) we use the Cauchy– Schwarz inequality,
the equivalence of norms (2.8), and the Poincar´ e inequalit y (2.6), followed by the bound
on the projection error (5.5) and the bound on the FE error (2. 12), which gives
|M(∂ν
yu,u−uh)|+|M(∂ν
yu−Ph∂ν
yu,uh)|
≤ /⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lM/⌊a∇d⌊lu−uh/⌊a∇d⌊lM+/⌊a∇d⌊l∂ν
yu−Ph∂ν
yu/⌊a∇d⌊lM
≤amax
χ1/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lV/⌊a∇d⌊lu−uh/⌊a∇d⌊lV+/radicalbiggamax
χ1/⌊a∇d⌊l∂ν
yu−Ph∂ν
yu/⌊a∇d⌊lV
≤amax
χ1Cuh|ν|!1+ǫβν+/radicalbiggamax
χ1CPh|ν|!1+ǫβν
≤/parenleftbiggamax
χ1Cu+/radicalbiggamax
χ1CP/parenrightbigg
h|ν|!1+ǫβν, (A.8)
where in the last inequality we have used that βj≤βj.
Substitutingthese twoboundsinto(A.7)then multiplyingb y/⌊a∇d⌊luh/⌊a∇d⌊lVgives thefollowing
upper bound on the first term of the decomposition (A.6)
|M(Ph∂ν
y(u−uh),uh)|/⌊a∇d⌊luh/⌊a∇d⌊lV≤u/parenleftbiggamax
χ1Cu+/radicalbiggamax
χ1CP/parenrightbigg
h|ν|!1+ǫβν
+u2amax
χ1/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν|m|!1+ǫβm/⌊a∇d⌊l∂ν−m
y(u−uh)/⌊a∇d⌊lV.(A.9)
Note that we have also used (2.17).
Next, to bound the norm of ϕh(the second term in the decomposition (A.6)), we let
32vh=ϕhin (A.2) and then rearrange the terms to give
A(Ph∂ν
y(u−uh),ϕh)−λhM(Ph∂ν
y(u−uh),ϕh) = (λ−λh)M(∂ν
yu,ϕh)
+λhM(∂ν
yu−Ph∂ν
yu,ϕh)+∂ν
yλM(u−uh,ϕh)+∂ν
y(λ−λh)M(uh,ϕh)
−∞/summationdisplay
j=1νj/parenleftbigg/integraldisplay
Daj∇/bracketleftbig
∂ν−ej
y(u−uh)/bracketrightbig
·∇ϕh+/integraldisplay
Dbj/bracketleftbig
∂ν−ej
y(u−uh)/bracketrightbig
ϕh/parenrightbigg
(A.10)
+/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg/bracketleftbig
∂ν−m
yλM(∂m
y(u−uh),ϕh)+∂ν−m
y(λ−λh)M(∂m
yuh,ϕh)/bracketrightbig
.
Again using the decomposition (A.5) and the fact that uhsatisfies the eigenproblem
(2.10) with ϕh∈Vhas atest function, the lefthandsideof (A.10) simplifiesto A(ϕh,ϕh)−
λhM(ϕh,ϕh). Since ϕh∈E(λh)⊥, we can use the FE version of the coercivity estimate
[19, Lemma 3.1] (see also Remark 3.2 that follows), to bound t his from below by
A(ϕh,ϕh)−λhM(ϕh,ϕh)≥amin/parenleftbiggλ2,h−λh
λ2,h/parenrightbigg
/⌊a∇d⌊lϕh/⌊a∇d⌊l2
V≥aminρ
2λ2/⌊a∇d⌊lϕh/⌊a∇d⌊l2
V,(A.11)
where in the last inequality we have used the upper bound (2.1 6), along the lower bound
(2.15) on the FE spectral gap, which is applicable for hsufficiently small.
Taking the absolute value, the right hand side of (A.10) can b e bounded using the
triangle inequality, the Cauchy–Schwarz inequality, the e quivalence of norms (2.8), and
the Poincar´ e inequality (2.6), which, combined with the lo wer bound (A.11), gives
aminρ
2λ2/⌊a∇d⌊lϕh/⌊a∇d⌊l2
V≤amax
χ1/parenleftBig
|λ−λh|/⌊a∇d⌊l∂ν
yu/⌊a∇d⌊lV+λ/⌊a∇d⌊l∂ν
yu−Ph∂ν
yu/⌊a∇d⌊lV
+|∂ν
yλ|/⌊a∇d⌊lu−uh/⌊a∇d⌊lV+|∂ν
y(λ−λh)|/⌊a∇d⌊luh/⌊a∇d⌊lV/parenrightBig
/⌊a∇d⌊lϕh/⌊a∇d⌊lV
+∞/summationdisplay
j=1νj/parenleftbigg
/⌊a∇d⌊laj/⌊a∇d⌊lL∞+1
χ1/⌊a∇d⌊lbj/⌊a∇d⌊lL∞/parenrightbigg
/⌊a∇d⌊l∂ν−ej
y(u−uh)/⌊a∇d⌊lV/⌊a∇d⌊lϕh/⌊a∇d⌊lV
+amax
χ1/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg/parenleftbig
|∂ν−m
yλ|/⌊a∇d⌊l∂m
y(u−uh)/⌊a∇d⌊lV+|∂ν−m
y(λ−λh)|/⌊a∇d⌊l∂m
yuh/⌊a∇d⌊lV/parenrightbig
/⌊a∇d⌊lϕh/⌊a∇d⌊lV.
Dividing through by aminρ/(2λ2)/⌊a∇d⌊lϕh/⌊a∇d⌊lV, then using the bounds (2.11), (4.3), (4.4), (5.5),
along with the fact that that βj≤βjfor allj∈Nandh≤h, we have that the norm of
ϕhis bounded by
/⌊a∇d⌊lϕh/⌊a∇d⌊lV≤amax
amin2λ2
ρχ1/bracketleftBig
uhCλ+λCP+λCu/bracketrightBig
h|ν|!1+ǫβν+amax
amin2uλ2
ρχ1|∂ν
y(λ−λh)|(A.12)
+2λ2
aminρCβ/parenleftbigg
1+1
χ1/parenrightbigg∞/summationdisplay
j=1νjβj/⌊a∇d⌊l∂ν−ej
y(u−uh)/⌊a∇d⌊lV
+amax
amin2λ2
ρχ1(λ+u)/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg
|m|!1+ǫβm/bracketleftBig
/⌊a∇d⌊l∂ν−m
y(u−uh)/⌊a∇d⌊lV+/⌊a∇d⌊l∂ν−m
y(λ−λh)|/bracketrightBig
.
Note that to get the sum on the last line we have again simplifie d similarly to (A.3).
33Substituting the bounds (A.6), followed by (A.9) and (A.12) , into the decomposition
(A.4) gives the following recursive bound on the derivative of the eigenfunction error
/⌊a∇d⌊l∂ν
y(u−uh)/⌊a∇d⌊lV
≤/bracketleftbiggamax
amin2λ2
ρχ1/parenleftbig
uhCλ+λCP+λCu/parenrightbig
+u/parenleftbiggamax
χ1Cu+/radicalbiggamax
χ1CP/parenrightbigg
+CP/bracketrightbigg
h|ν|!1+ǫβν
+amax
amin2uλ2
ρχ1|∂ν
y(λ−λh)|+2λ2
aminρCβ/parenleftbigg
1+1
χ1/parenrightbigg∞/summationdisplay
j=1νjβj/⌊a∇d⌊l∂ν−ej
y(u−uh)/⌊a∇d⌊lV
+amax
amin2λ2
ρχ1(λ+u)/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg
|m|!1+ǫβm/bracketleftBig
/⌊a∇d⌊l∂ν−m
y(u−uh)/⌊a∇d⌊lV+/⌊a∇d⌊l∂ν−m
y(λ−λh)|/bracketrightBig
+u2amax
χ1/summationdisplay
0/\e}atio\slash=m≤ν
m/\e}atio\slash=ν/parenleftbiggν
m/parenrightbigg
|m|!1+ǫβm/⌊a∇d⌊l∂ν−m
y(u−uh)/⌊a∇d⌊lV.
Observe that all of the constant terms are independent of y,handν.
To obtain the final result with a right handside that does not d ependon any derivative
of orderν, we now substitute the recursive formula (5.6) for ∂ν
y(λ−λh). After grouping
the similar terms and collecting all of the constants into CIIIwe have the final result.
SinceCIfrom (5.6) and all of the constants above are independent of y,h, andν, the
final constant CIIIis as well.
34 |
2002.08723v2.Stoner_Wohlfarth_switching_of_the_condensate_magnetization_in_a_dipolar_spinor_gas_and_the_metrology_of_excitation_damping.pdf | Stoner-Wohlfarth switching of the condensate magnetization in a dipolar spinor gas
and the metrology of excitation damping
Seong-Ho Shinn,1Daniel Braun,2and Uwe R. Fischer1
1Department of Physics and Astronomy, Seoul National University, 08826 Seoul, Korea
2Eberhard-Karls-Universit at T ubingen, Institut f ur Theoretische Physik, 72076 T ubingen, Germany
(Dated: May 13, 2020)
We consider quasi-one-dimensional dipolar spinor Bose-Einstein condensates in the homogeneous-
local-spin-orientation approximation, that is with unidirectional local magnetization. By analyti-
cally calculating the exact e ective dipole-dipole interaction, we derive a Landau-Lifshitz-Gilbert
equation for the dissipative condensate magnetization dynamics, and show how it leads to the Stoner-
Wohlfarth model of a uni-axial ferro-magnetic particle, where the latter model determines the stable
magnetization patterns and hysteresis curves for switching between them. For an external magnetic
eld pointing along the axial, long direction, we analytically solve the Landau-Lifshitz-Gilbert equa-
tion. The solution explicitly demonstrates that the magnetic dipole-dipole interaction accelerates
the dissipative dynamics of the magnetic moment distribution and the associated dephasing of the
magnetic moment direction. Under suitable conditions, dephasing of the magnetization direction
due to dipole-dipole interactions occurs within time scales up to two orders of magnitude smaller
than the lifetime of currently experimentally realized dipolar spinor condensates, e.g., produced with
the large magnetic-dipole-moment atoms166Er. This enables experimental access to the dissipation
parameter in the Gross-Pitaevski mean- eld equation, for a system currently lacking a complete
quantum kinetic treatment of dissipative processes and, in particular, an experimental check of the
commonly used assumption that is a single scalar independent of spin indices.
I. INTRODUCTION
Ever since a phenomenological theory to describe the
behavior of super
uid helium II near the point has
been developed by Pitaevski [1], the dynamics of Bose-
Einstein condensates (BEC) under dissipation has been
intensely studied, see, e.g., [2{8]. Experimentally, the im-
pact of Bose-Einstein condensation on excitation damp-
ing and its temperature dependence has for example been
demonstrated in [9{12].
Dissipation in the form of condensate loss is de ned
by a dimensionless damping rate entering the left-
hand side of the Gross-Pitaevski equation, replacing the
time derivative as i@t!(i )@t. While a micro-
scopic theory of condensate damping is comparatively
well established in the contact-interaction case, using var-
ious approaches, cf., e.g., [5, 13{15], we emphasize the
absence of a microscopic theory of damping in dipolar
spinor gases. While for scalar dipolar condensates, par-
tial answers as to the degree and origin of condensate-
excitation damping have been found see, e.g., Refs. [16{
19], in spinor or multicomponent gases the interplay of
anisotropic long-range interactions and internal spinor or
multicomponent degrees of freedom leads to a highly in-
tricate and dicult-to-disentangle many-body behavior
of condensate-excitation damping.
In this paper, we propose a method to experimen-
tally access in a dipolar spinor condensate by using
the dynamics of the unidirectional local magnetization in
a quasi-one-dimensional (quasi-1D) dipolar spinor BEC
in the presence of an external magnetic eld. To this
end, we rst derive an equation of motion for the mag-
netization of the BEC that has the form of a Landau-
Lifshitz-Gilbert (LLG) equation [20{22], with an addi-tional term due to the dipole-dipole interaction between
the atoms. The LLG equation is ubiquitous in nano-
magnetism, where it describes the creation and dynam-
ics of magnetization. The static limit of this equation
is, in the limit of homogeneous local spin-orientation, de-
scribed by the well-known Stoner-Wolfarth (SW) model
[23{25] of a small magnetic particle with an easy axis of
magnetization. We then investigate the magnetization
switching after
ipping the sign of the external magnetic
eld, and demonstrate the detailed dependence of the
switching dynamics on the dissipative parameter .
For a quasi-2D spinor BEC with inhomogeneous local
magnetization, Ref. [26] has studied the magnetic domain
wall formation process by deriving a LLG type equa-
tion. Here, we derive the LLG equation in a quasi-1D
spinor BEC with unidirectional local magnetization, in
order to establish a most direct connection to the orig-
inal SW model. In distinction to [27], which studied
the e ective quasi-1D dipole-dipole interaction resulting
from integrating out the two transverse directions within
a simple approximation, we employ below an exact ana-
lytic form of the dipole-dipole interaction. In Section II,
we establish the quasi-1D spinor Gross-Pitaevski (GP)
equation with dissipation, and equations of motion for
the magnetization direction (unit vector) M. Section V
shows how the LLG equation and the SW model result,
and Section VI derives analytical solutions to the equa-
tions of motion for Mwhen the external magnetic eld
points along the long, zaxis. We summarize our results
in section VII.
We defer two longer derivations to Appendices. The
analytical form of the e ective dipole-dipole interaction
energy is deduced in Appendix A, and the quasi-1D GP
mean- eld equation with dissipation is described in detailarXiv:2002.08723v2 [cond-mat.quant-gas] 12 May 20202
in Appendix B. Finally, in Appendix C, we brie
y discuss
to which extent relaxing the usual simplifying assumption
that dissipation even in the spinor case is described by
a single scalar changes the LLG equation, and whether
this a ects the SW model and its predictions.
II. GENERAL DESCRIPTION OF DAMPING IN
BECS
The standard derivation of the quantum kinetics of
Bose-Einstein condensate damping [5] starts from the
microscopic Heisenberg equation of motion for the quan-
tum eld operator ^ (r;t), for a scalar (single compo-
nent) BEC in the s-wave scattering limit. Using their
results, [28] obtained a mean- eld equation to describe
the dissipation of scalar BEC, whose form is
(i )~@
@t=H (1)
where is the (in the large Nlimit) dominant mean- eld
part upon expanding the full bosonic eld operator ^ .
In Ref. [1], Pitaevski obtained a similar but slightly
di erent form of the dissipative mean- eld equation
based on phenomenological considerations, i~@
@t=
(1 i )H , by parametrizing the deviation from exact
continuity for the condensate fraction while minimizing
the energy [1]. The latter deviation is assumed to be
small, which is equivalent to assuming that remains
small. This provides a clear physical interpretation of the
damping mechanism, namely one based on particle loss
from the condensate fraction. The version of Pitaevski
can be written as
(i )~@
@t=
1 + 2
H : (2)
It can thus be simply obtained by rescaling time with a
factor 1 + 2compared to (1). Hence, as long as one
does not predict precisely , the two dissipative equa-
tions (1) and (2) cannot be distinguished experimentally
from the dynamics they induce. From the data of [11],
[4] estimated typical values of '0:03 for a scalar BEC
of23Na atoms (see also [12]), which shows that to distin-
guish between (1) and (2) experimentally the theoretical
predictions of would need to be precise to the order of
10 4.
How eqs.(1) and (2) can be generalized to the dipolar
spinor gases is comparatively little investigated. Using a
symmetry-breaking mean- eld approach by writing the
quantum eld operator as ^ (r;t) as ^ (r;t) = (r;t) +
^ (r;t), with (r;t) =h^ (r;t)iandh^ (r;t)i= 0, [5]
and [28] showed that is derived from the three- eld
correlation function h^ y(r;t)^ (r;t)^ (r;t)iin a ba-
sis whereh^ (r;t)^ (r;t)i= 0. From this microscopicorigin, based on correlation functions, it is clear that in
principle might depend on the spin indices in a spinor
BECs and hence become a tensor (see Appendix C for
a corresponding phenomenological generalization). Nev-
ertheless, it is commonly assumed cf.,e.g., [26, 29], that
does not depend on spin indices, and the scalar value
found speci cally in [4] for a scalar BEC of23Na atoms
is commonly used, while a clear justi cation of this as-
sumption is missing.
Extending the microscopic derivations in [5] and [28] to
the spinor case would be theoretically interesting, but is
beyond the scope of the present paper. Here, we instead
focus on the question whether the standard assumption
that the damping of each spinor component can be de-
scribed by the mean- eld equation [28] leads to exper-
imentally falsi able dynamical signatures. It will turn
out that this assumption introduces an additional strong
dephasing in the spin-degrees of freedom, ampli ed by
the dipolar interaction. Hence, even on time scales on
which the decay of the condensate fraction according to
(1) can be neglected, the relaxation of the magnetization
of the BEC potentially o ers valuable insights whether
the scalar- assumption is justi ed. Indeed, in [30] it
was shown experimentally that on the time scale of the
switching dynamics of the magnetization the number of
particles in the condensates remains approximately con-
stant. One might wonder, then, which dissipative mech-
anism is left. However, as we will show, by assuming the
same GP equation for each component of the spinor as
for scalar bosons, additional dephasing occurs that is in
fact much more rapid than the decay of condensate den-
sity due to dephasing accelerated by the dipole-dipole
interaction.
III. MEAN-FIELD DYNAMICS OF DAMPING
IN DIPOLAR SPINOR BECS
For a spinor BEC, linear and quadratic Zeeman inter-
actions are commonly included in the Hamiltonian. The
quadratic Zeeman interaction is related to a second-order
perturbation term in the total energy that can be induced
by the interaction with an external magnetic eld ( qB)
as well as with the interaction with a microwave eld
(qMW) [31]. Speci cally, by applying a linearly polarized
microwave eld, one can change qMWwithout changing
qB[32, 33]. Hence, we will assume that the quadratic
Zeeman term can be rendered zero by suitably changing
qMW.
Following [26], we thus assert that for a dipolar spinor
BEC without quadratic Zeeman term, the mean- eld
equation can be written as3
(i )~@ (r;t)
@t=
~2
2mr2+Vtr(r) +c0j (r;t)j2 ~fb bdd(r;t)g^f
(r;t)
+SX
k=1c2kX
1;2;;k=x;y;zF1;2;;k(r;t)^f1^f2^fk (r;t): (3)
where (r;t) is a vector quantity whose -th component
in the spinor basis is (r;t) (spin-space indices from
the beginning of the Greek alphabet such as ; ;
;:::
are integers running from StoS). In this expres-
sion, ~^fis the spin- Soperator where the spin ladder
is de ned by ^fzj i= j iandh j i= ; , while
F1;2;;k(r;t):= y(r;t)^f1^f2^fk (r;t) are the
components of the expectation value of ^f1^f2^fk.
The Larmor frequency vector reads b=gFBB=~
(with Land e g-factor gF, Bohr magneton B, and
the external magnetic induction B),~bdd(r;t)e=
cddR
d3r0P
0=x;y;zQ;0(r r0)F0(r0;t):Here,cdd=
0(gFB)2=(4) andeis a unit vector along the
axis [31] (by convention, indices from the middle of the
Greek alphabet such as ;;;;::: =x;y;z denote spa-
tial indices), and Q;0is the spin-space tensor de ned in
Eq. (A2) of Appendix A. Finally, mis the boson mass,
c0the density-density interaction coecient, and c2kthe
interaction coecient parametrizing the spin-spin inter-
actions, where kis an positive integer running from 1
toS[26]. For example, c2is the spin-spin interaction
coecient of a spin-1 gas ( S= 1).
To develop a simple and intuitive physical approach,
we consider a quasi-1D gas for which one can perform
analytical calculations. We set the trap potential as
Vtr(x;y;z ) =1
2m!2
?
x2+y2
+V(z); (4)
so that the long axis of our gas is directed along the z
axis and the gas is strongly con ned perpendicularly.
For a harmonic trap along all directions, i.e. when
V(z) =m!2
zz2=2, we set!? !z. For a box trap along
z, i.e. when V(z) = 0 forjzj LzandV(z) =1
FIG. 1. Schematic of the considered geometry in a quasi-1D
gas (shaded ellipsoid). The length of the red magnetization
arrows, all pointing in the same direction (homogeneous local-
spin-orientation limit), represents jd(z;t)j.forjzj> Lz, our gas will be strongly con ned along z
as long as the quasi-1D condition is satis ed, where we
will discuss below whether the condition is satis ed, in
section VI A.
Single-domain spinor BECs have been already real-
ized, for example, using spin-187Rb [34]. This single-
domain approximation is common in nanomagnetism, see
for example [24], by assuming magnetic particles much
smaller than the typical width of a domain wall. The
local magnetization is related to the expectation value
~F(r;t)~ y(r;t)^f (r;t) of the spatial spin density
operator byd(r;t) =gFBF(r;t). An unidirectional
local magnetization d(z;t) is then given by
dx(z;t) =d(z;t) sin(t) cos (t);
dy(z;t) =d(z;t) sin(t) sin (t); (5)
dz(z;t) =d(z;t) cos(t);
whered(z;t) =d(z;t)eis the-th component of
d(z;t),d(z;t) =jd(z;t)j,(t) is polar angle of d(z;t),
and (t) is azimuthal angle of d(z;t). For an illustra-
tion of the geometry considered, see Fig. 1. For a single
component dipolar BEC, F(r;t) has a xed direction.
To study the relation of the Stoner-Wohlfarth model, in
whichF(r;t) changes its direction, with a dipolar BEC,
a multi-component dipolar BEC should therefore be em-
ployed.
In the quasi-1D approximation, the order parameter
(r;t) is commonly assumed to be of the form
(r;t) =e 2=(2l2
?)
l?p (z;t): (6)
wherel?is the harmonic oscillator length in the x y
plane and=p
x2+y2. Assuming our gas is in the
homogeneous local spin-orientation limit, we may also
apply a single mode approximation in space so that
(z;t) = uni(z;t) (t). The time-dependent spinor
part is
(t) =h je i^fz (t)e i^fy(t)jSi; (7)
for spin-Sparticles [26, 31] and the normalization reads
j(t)j2:=y(t)(t) = 1. Finally, due to the ( i )
factor on the left-hand side of Eq. (3), for the ease of
calculation, we may make the following ansatz for the
(r;t), cf. Ref. [35],
(r;t) =e 2=(2l2
?)
l?p (z;t) (t)e (i+ )!?t=(1+ 2):
(8)4
From our ans atze in Eq. (7) and (8), one concludes that
the expectation value of the (spatial) spin-density oper-
ator is
~Fx(r;t) =~Se 2=l2
?
l2
?j (z;t)j2e 2 !?t=(1+ 2)
sin(t) cos (t);
~Fy(r;t) =~Se 2=l2
?
l2
?j (z;t)j2e 2 !?t=(1+ 2)
sin(t) sin (t);
~Fz(r;t) =~Se 2=l2
?
l2
?j (z;t)j2e 2 !?t=(1+ 2)
cos(t): (9)
The above equations lead to unidirectional local magneti-
zation, which has been assumed in Eqs. (5), in the quasi-
1D limit (after integrating out the strongly con ning xandyaxes). Note however that our ansatz in Eq. (8) is
sucient, but not necessary for the homogeneous-local-
spin-orientation limit, and the homogeneous-local-spin-
orientation ansatz is thus designed to render our ap-
proach as simple as possible.
Because we are not assuming any speci c form of
(z;t) in our ansatz in Eq. (8),we cover every possible
time behavior of j (r;t)j2:= y(t) (t):
j (r;t)j2=e 2=l2
?
l2
?j (z;t)j2e 2 !?t=(1+ 2):(10)
Eq. (10) explicitly shows that Eq. (8) does not imply
an exponentially decaying wavefunction with time since
j (z;t)j2can be any physical function of time t. How-
ever, the ansatz (8) simpli es the resulting equation for
(z;t), Eq.(11) below.
By integrating out the xandydirections, the GP equa-
tion for a quasi-1D spin- SBEC can be written as (see
for a detailed derivation Appendix B)
(i )~@f (z;t) (t)g
@t=
~2
2m@2
@z2+V(z) +c0
2l2
?n(z;t)
(z;t) (t)
+~[ b+SfM(t) 3Mz(t)ezgPdd(z;t)]8
<
:SX
= S
^f
; (z;t) (t)9
=
;
+SX
k=1c2k
2l2
?n(z;t)X
1;2;;k=x;y;zSM1;2;;k(t)8
<
:SX
= S
^f1^f2^fk
; (z;t) (t)9
=
;;
(11)
where we de ned the two functions
M1;2;;k(t):=1
SSX
; = Sy
(t)
^f1^f2^fk
; (t); (12)
Pdd(z;t):=cdd
2~l3
?Z1
1dz0n(z0;t)
Gjz z0j
l?
4
3z z0
l?
; (13)
with the axial density n(z;t):=R
d2j (r;t)j2=
j (z;t)j2e 2 !?t=(1+ 2), whereR
d2 :=R1
1dxR1
1dy. Finally, the function Gappearing
inPddis de ned as
G():=r
2
2+ 1
e2=2Erfcp
2
:(14)
We plotG() as a function of in Fig. 2. Eq.(11) rep-
resents our starting point for analyzing the dynamics of
magnetization. We will now proceed to show how it leads
to the LLG equation and the Stoner-Wolfarth model.IV. EFFECTIVE LANGRANGIAN
DESCRIPTION
To provide a concise phase space picture of the conden-
sate magnetization dynamics, we discuss in this section a
collective coordinate Lagrangian appropriate to our sys-
tem.
LetM(t):=d(z;t)=d(z;t) where the magne-
tizationd(z;t) is de ned in Eq. (5). Explicitly,
the local magnetization direction reads M(t) =
(sin(t) cos (t);sin(t) sin (t);cos(t)). Then, from
Eqs. (9) and (10), F(r;t) =SM(t)j (r;t)j2and one5
1 2 3 4 50.20.40.60.81.01.2
FIG. 2. The function G() de ned in Eq. (14). Note that
G()'2=3+O
5
for 1, soG() is always positive
for0.
obtains (see for a detailed derivation Appendix B)
@M
@t=Mfb+S0
dd(t)Mzezg M@M
@t;(15)
where the renormalized interaction function 0
dd(t) reads
0
dd(t) =3
N(t)Z1
1dzn(z;t)Pdd(z;t);(16)
andN(t):=R
d3rj (r;t)j2=R1
1dz n (z;t). From
Eqs. (A9), (A12), and (13), 0
dd(t) is connected to the
dipole-dipole interaction contribution Vdd(t) by
Vdd(t) =3
2~S2
sin2(t) 2
3Z1
1dzn(z;t)Pdd(z;t)
=~
2S2N(t) 0
dd(t)1
3 cos2(t)
: (17)
We note that in order to obtain the e ective quasi-1D
dipolar interaction (17), we did not use, in distinction
to Ref. [27], any simplifying approximation. A detailed
derivation is provided in Appendix A.
Eq.(15) is the LLG equation with the external mag-
netic eld in z-direction modi ed by the magnetiza-
tion inz-direction due to the dipole-dipole interac-
tion. The corresponding term in units of magnetic eld,
~S0
dd(t)Mzez=(gFB), can be seen as an additional
magnetic eld that is itself proportional to the magne-
tization inz-direction, and which leads to an additional
nonlinearity in the LLG equation.
From Eqs. (13) and (16), to get how 0
dd(t) depends
on timet, one has to calculate the double integral
Z
dzZ
dz0n(z;t)n(z0;t)
Gjz z0j
l?
4
3z z0
l?
:
(18)
To achieve a simple physical picture, we assume that
n(z;t) does not depend on time twithin the time range
we are interested in. Then we may write 0
dd(t) = 0
dd.
The lifetime of a typical dipolar BEC with large atomicmagnetic dipole moments such as164Dy [36],162Dy and
160Dy [37], or166Er [30] is of the order of seconds. Since
taking into account the time dependence of n(z;t) gen-
erally requires a numerical solution of Eq. (11), we here
consider the case where n(z;t) is constant in time tas
in [26], to predominantly extract the e ect of magnetic
dipole-dipole interaction per se .
We also neglect the possible e ect of magnetostriction.
The latter e ect, amounting to a distortion of the aspect
ratio of the condensate in a harmonic trap as a func-
tion of the angle of the external magnetic eld with the
symmetry axis of the trap, was measured in a conden-
sate of Chromium atoms [38] (with a magnetic moment
of 6B). The magnetostriction e ect in that experiment
was of the order of 10%. For alkali atoms with spin-1
the e ect should be a factor 62smaller. In addition, the-
oretical analyses in the Thomas-Fermi limit show that
magnetostriction in harmonic traps becomes particularly
small for very small or very large asymmetries of the trap
[39, 40].
More speci cally, Ref. [41] has shown that magne-
tostriction is due to the force induced by the dipole-dipole
mean- eld potential dd(r;t). In Appendix D, we ap-
ply the approach of [41] to a dipolar spinor BEC. From
Eqs. (16), (17), (A1), and (D5), 0
dd(t) contains dd(z;t)
[the quasi-1D form of dd(r;t) de ned in Eq. (D5)] by
S2
1 3M2
z(t)
N(t)~0
dd(t)
= 3Z1
1dzn(z;t) dd(z;t):(19)
Hence, our LLG-type equation in Eq. (15) e ectively con-
tains the dipole-dipole mean- eld potential which causes
magnetostriction and the form of Eq. (15) itself will not
be changed whether the e ect of magnetostriction is large
or not. Only the value of 0
dd(t) will be changed because
magnetostriction changes the integration domain. Fur-
thermore, we show in Appendix D that for our quasi-1D
system, the e ect of magnetostriction is smaller in a box
trap than in harmonic trap. In fact, for the box trap,
this e ect can be neglected if Lz=l?is suciently large.
Thus, we may neglect the e ect of magnetostriction un-
der suitable limits for both box and harmonic traps.
To get a simple physical idea of the dynamical behavior
of our system, let us, for now, assume that there is no
damping, = 0. When the external magnetic eld is
chosen to lie in the x zplane,B= (Bx;0;Bz), Eq. (15)
becomes
d
dt=bxsin ;
d
dt=bxcotcos bz S0
ddcos: (20)
where we already de ned the Larmor frequency vector
b=gFBB=~below Eq. (3).
By using the Lagrangian formalism introduced in [42],6
the Lagrangian Lof this system then ful lls
L
~=_ cos+bxsincos +bzcos+S
40
ddcos (2);
(21)
where _ =d =dt . The equations of motion are
1
~@L
@= _ sin+bxcoscos bzsin S
20
ddsin (2);
@L
@_= 0;1
~@L
@ = bxsinsin ;1
~@L
@_ = cos:(22)
One easily veri es that Eq. (21) is indeed the Lagrangian
which gives Eqs. (20). Let pbe the conjugate momen-
tum of the coordinate . Sincep= 0 andp =~cos
(~times thezcomponent of M), the Hamiltonian His
given by
H= bxq
~2 p2
cos bzp +~2 2p2
4~S0
dd:(23)
Note that the energy ~E:=H ~S0
dd=4 is conserved.
Hence, if we put p = (p )inand ==2 at some time
t=t0,~E= bz(p )in S0
dd(p )2
in=2~. We can then
express as a function of p as
cos = ~E+bzp +1
2~S0
ddp2
bxq
~2 p2
=
(p )in p bz+S0
dd(p )in+p
2~
bxq
~2 p2
: (24)
The canonical momentum p remains the initial ( p )in
whenbx= 0, implying that does not change when
bx= 0, consistent with Eqs. (20). If jbxjis larger than
jbzS0
ddj, we can have p 6= (p )inwithjcos j1,
which allows for the switching process of the magneti-
zation. Below a threshold value of jbxjthat depends on
bzandS0
dd,p has to remain constant for Eq. (24) to
be satis ed, which corresponds to simple magnetization
precession about the zaxis.
Whenp is a function of time, there are two important
cases:
(a)jbzj S0
dd: cos =bz
bx(p )in p q
~2 p2
;
(b)jbzj S0
dd: cos =S0
dd
2bx(p )2
in p2
~q
~2 p2
:(25)
We plot the corresponding phase diagrams ( vs ) in
Fig. 3.
Let (p )in=~cosin,bx=bsin0, andbz=bcos0.
When case (a) holds jbzj S0
dd, one concludes that
cos0cos+ sin0sincos = cos0cosin, which is
constant. Since db=db(cos0cos+ sin0sincos ),
in case (a) the magnetization dprecesses around the ex-
ternal magnetic eld B, as expected. When (b) holds,
SW switching can occur, to the description of which we
proceed in the following.
0.5 1.0 1.5 2.0
-1.0-0.50.51.0(p )in=~=2 and in==2
0.5 1.0 1.5 2.0
-1.0-0.50.51.0
(p )in= ~=2and in==2
FIG. 3.p =~vs =when = 0 (no dissipation), with initial
values (p )inand in(initial value of ) as shown. (1) Dashed
blue:bz=bx= 0:2 andjbzj S0
dd. (2) Black line: bz=bx=
0:2 andS0
dd=bx= 0:6. (3) Dash-Dotted red: S0
dd=bx= 0:6
andjbzj S0
dd. (4) Dotted orange horizontal line: bx= 0.
V. CONNECTION TO STONER-WOHLFARTH
MODEL
The phenomenological SW model can be directly read
o from the equations in the preceding section. From
Eq. (23), ~H:=H+~S0
dd=4 is given by
~H
~= bxsincos bzcos+S0
dd
2sin2:
(26)
Let (b)crbe the value of bat the stability limit where
@~H=@ = 0 and@2~H=@2= 0. Then one obtains the
critical magnetic elds
(bx)crcos =S0
ddsin3;(bz)cr= S0
ddcos3:
(27)
which satisfy the equation
f(bx)crcos g2=3+ (bz)2=3
cr=fS0
ddg2=3: (28)
We coin the curve in the ( bx;bz)-plane described by
Eq. (28) the switching curve, in accordance with the ter-
minology established in [43]. Because changes in time
[see Eqs. (20) and Fig. 3], the switching curve depends
in general on the timing of the applied external magnetic
elds. We note that, for = 0, Eqs. (26) and (28) are
identical to the SW energy functional
HSW
~= bxsin bzcos+Ksin2 (29)7
and the SW astroid [43], respectively, if we identify K=
S0
dd=2.
The LLG equation in Eq. (15) has stationary solu-
tions withMparallel to the e ective magnetic eld
~fb+S0
dd(t)Mzezg=(gFB). Since we set bto lie
in thexzplane, will go to zero for suciently large
times. Thus Eq. (26) leads to the SW model (29) due
to the damping term in (15) if >0. In Appendix C,
we demonstrate that a more general tensorial damping
coecient introduces additional terms on the right-
hand side of the LLG equation (15), which involve time
derivatives . While these will thus not a ect the SW phe-
nomenology, which results from the steady states as func-
tion of the applied magnetic elds, and which is thus gov-
erned by the vanishing (in the stationary limit) of the rst
term on the right-hand side of the LLG equation, they
a ect the detailed relaxation dynamics of the magneti-
zation and its time scales. These deviations can hence
can be used to probe deviations from assuming a single
scalar .
Before we move on to the next section, we show the
characteristic behavior of 0
ddde ned in Eq. (16), for
a box-trap scenario de ned by n(z;t) =N=(2Lz) for
LzzLzandn(z;t) = 0 otherwise ( Nis number
of particles).
We stress that due to the nite size of the trap along
the \long" zdirection, in variance with the Hohenberg-
Mermin-Wagner theorem holding for in nitely extended
systems in the thermodynamic limit, a quasi-1D BEC can
exist also at nite temperatures [44]. This remains true
up to a ratio of its proper length to the de-Broglie wave-
length [45], beyond which strong phase
uctuations set
in [46]. In fact, these strongly elongated quasi-1D BECs
at nite temperature have been rst realized already long
ago, cf., e.g. [47].
For the box trap, 0
dd= dd(Lz=l?) where
dd() =3Ncdd
2~l3
?1
(Z2
0dv
1 v
2
G(v) 2
3)
:
(30)
From Eq. (14), G(v)'2=v3+O
v 5
forv 1, so that
dd()'Ncdd
2~l3
?1
for=Lz
l? 1: (31)
Hence dd() is a slowly decreasing function of the
cigar's aspect ratio (keeping everything else xed). We
will see below that for the parameters of experiments
such as [30], the e ective magnetic eld due to dipolar
interactions greatly exceeds the externally applied mag-
netic elds (in the range relevant for SW switching to be
observed) [48].
VI. ANALYTICAL RESULTS FOR AXIALLY
DIRECTED EXTERNAL MAGNETIC FIELD
Without dissipation, when bx= 0,p =~cos=~Mz
is rendered constant; see Eq. (20). However, in the pres-ence of dissipation, Mzchanges in time even if bx= 0.
By employing this change, we propose an experimental
method to measure .
For simplicity, we will assume that the number density
is constant in time (also see section IV) and the external
magnetic eld points along the zdirection,B=Bzez.
Let a critical (see for a detailed discussion below) value
of the magnetization be
(Mz)cr:= bz
S0
dd: (32)
Then Eq. (15) can be written as
@M
@t=S0
ddMezfMz (Mz)crg M@M
@t
=Mez(bz+S0
ddMz) M@M
@t:(33)
SinceM@M
@t= 0, by taking the cross product with M
on both sides of Eq. (15), one can derive an expression
forM@M
@t:
@Mz
@t= S0
dd
1 + 2fMz (Mz)crg
M2
z 1
=
1 + 2(bz+S0
ddMz)
M2
z 1
:(34)
SinceMis the scaled magnetization, jMj= 1 with a
condensate. Hence, 1Mz1. Also, according to
the discussion below Eq. (26), the generally positive SW
coecient (with units of frequency) KisS0
dd=2.
From Eq. (34), for time-independent 0
dd, one con-
cludes that there are three time-independent solutions,
Mz= (Mz)crandMz=1. For a box-trapped BEC
and constant number density, 0
dd= ddwhich is always
positive in the quasi-1D limit (cf. Eq. (30) and the discus-
sion following it). For some arbitrary physical quasi-1D
trap potential, in which the number density is not con-
stant in space, from Eqs. (13), (16), and Fig. 2, one can
infer that 0
dd>0, due to the fact that the quasi-1D num-
ber density n(z;t)>0,n(z;t) has its maximum value
nearz= 0 for a symmetric trap centered there, and then
G() also has its maximum value near = 0. Then, if
j(Mz)crj<1,Mz= (Mz)cris an unstable solution and
Mz=1 are stable solutions. When j(Mz)crj<1 and
1< Mz<(Mz)cr,Mzgoes to 1. Likewise, Mzgoes
to 1 when ( Mz)cr< Mz<1. This bifurcation does not
occur ifj(Mz)crj>1. For simplicity, we assume that
j(Mz)crj<1. This is the more interesting case due to
the possibility of a bifurcation of stable solutions leading
to SW switching.
Let (Mz)inbe the value of Mzatt= 0. The analytic8
solution of Eq. (34) satis es
t=1 + 2
S0
dd"
1
f(Mz)crg2 1ln(Mz)in (Mz)cr
Mz (Mz)cr
1
2f1 (Mz)crgln1 Mz
1 (Mz)in
+1
2f1 + (Mz)crgln1 + (Mz)in
1 +Mz
=1 + 2
"
S0
dd
b2z (S0
dd)2lnbz+S0
dd(Mz)in
bz+S0
ddMz
1
2 (bz+S0
dd)ln1 Mz
1 (Mz)in
1
2 (bz S0
dd)ln1 + (Mz)in
1 +Mz
: (35)
The above equation tells us that, if ( Mz)in6= (Mz)crand
(Mz)in6=1,Mzgoes to its stable time-independent
solution (jMzj= 1) at time t=1. Thus, we de ne
acritical switching time tcrto be the time when jMzj=
0:99. Also, note that the form of LLG equation (Eq. (33))
does not change whether BEC is con ned in a quasi-
1D, quasi-2D, or a three-dimensional geometry. This is
because one can nd a connection between 0
ddand the
e ective dipole-dipole-interaction potential Ve , so one
can measure even if the BEC is e ectively con ned in
a space with dimension higher than one, using Eq. (35).
We point out, in particular, that tcris inversely pro-
portional to 0
dd. Hence, for a constant density quasi-
1D BEC con ned between LzzLz, 0
dd=
dd(Lz=l?), and thus tcris also inversely proportional
to the linear number density along z. This follows from
the relation between dd(Lz=l?) and the linear numberdensity along zdisplayed in Eq. (30).
For large dipolar interaction, the asymptotic expres-
sion fortcris, assuming 1
tcr'1
S0
ddln"
5p
2(1 (Mz)2
in)
j(Mz)in (Mz)crj#
(36)
providedS0
dd jbzj () j (Mz)crj 1:
The above tcrdiverges at ( Mz)in= (Mz)cror1, as
expected, since Mz= (Mz)crandMz=1 are time-
independent solutions of the LLG equation. We stress
that Eq. (36) clearly shows that the magnetic dipole-
dipole interaction accelerates the decay of Mz. Hence, by
using a dipolar spinor BEC with large magnetic dipole
moment such as produced from164Dy or166Er one may
observe the relaxation of Mzto the stable state within
the BEC lifetime, enabling the measurement of .
Before we show how the critical switching time tcr
depends on ( Mz)inand , we will qualitatively discuss
when our quasi-1D assumption and homogeneous-local-
spin-orientation assumption are valid. Typically, spin-
spin-interaction couplings are much smaller than their
density-density-interaction counterparts, by two orders
of magnitude. For spin 123Na BEC or spin 187Rb BEC,
c0'100jc2j[31, 34]. Thus we may neglect to a rst ap-
proximation the S2timesc2kterms in Eq. (11) (see the
discussion at the end of Appendix D). We also require
j(Mz)crj<1. Thus, we may additionally neglect the b
term compared to the Pdd(z;t) term since, for b=bzez,
S0
dd>jbjshould be satis ed to make j(Mz)crj<1 (see
Eq. (32)) and 0
ddis related to Pdd(z;t) by Eq. (16).
When = 0, using our ansatz in Eq. (8) and integrating
out thexandydirections, Eq. (D4) can be approximated
by the expression
(t) (z;t) =
~2
2m@2
@z2+V(z) +c0
2l2
?j (z;t)j2+ dd(z;t)
(z;t); (37)
where, from Eqs. (D5), (A1), and (17), the dipole-dipole interaction mean- eld potential reads
dd(z;t) =~S2
1 3M2
z(t)
Pdd(z;t)
=cdd
2l3
?S2
1 3M2
z(t) Z1
1dz0j (z0;t)j2
Gjz0 zj
l?
4
3z0 z
l?
=cdd
2l2
?S2
1 3M2
z(t) Z1
1dzj (z+ zl?;t)j2G(jzj) 4
3j (z;t)j2
: (38)
From Fig. 2, the function G() is positive and decreases
exponentially as increases. Thus, if l?is small enough
such thatj (z+ zl?;t)j2does not change within the
rangejzj5, one may conclude that
dd(z;t)'2
3S2
1 3M2
z(t) cdd
2l2
?j (z;t)j2;(39)due to the propertyR1
0dG () = 1.
A spinor (S= 6) dipolar BEC has been realized using
166Er [30]. For this BEC, c0= 4~2a=m wherea'67aB
(aBis Bohr radius) and 2 S2cdd=3 = 0:4911c0. Due to
jMz(t)j1 from the de nition of M(t), the maximum
value of the chemical potential (t) is achieved when9
Mz(t) = 0, where
(t)'V(z) +
c0+2
3S2cddn(z;t)
2l2
?: (40)
From above Eq. (40), we may regard the 3D number
density as n(z;t)=
2l2
?
. In [30], N= 1:2105,
!?=(2) =p156198 Hz = 175 :75 Hz,!z=(2) =
17:2 Hz,l?= 0:589m, and the measured peak number
density npeakis 6:21020m 3. Using Eq. (37) and (39),
by denoting Lzas the Thomas-Fermi radius along z,
( LzzLz) withV(z) =m!2
zz2=2, one derives
Lz=(
3
c0+ 2S2cdd=3
N
4m!2zl2
?)1=3
; (41)
and the mean number density n= (N=2Lz)=
2l2
?
=
6:7211020m 3as well as chemical potential =(~!?) =
m!2
zL2
z=(2~!?) = 23:22. Note that n'1:1 npeak. Be-
causeis not less than ~!?, the experiment [30] is not
conducted within the quasi-1D limit.
The homogeneous-local-spin-orientation approxima-
tion is valid when the system size is on the order of the
spin healing length sor less, which has been experimen-
tally veri ed in in [34]. Using c0'100jc2j,s'10d
whered=p
~2=(2mc0n) is the density healing length
ands=p
~2=(2mjc2jn) is the spin healing length.
Thus, ifLzis on the order of 10 d, the homogeneous-
local-spin-orientation approximation is justi ed.
Using the S= 6 element166Er, we can provide
numerical values which satisfy both the quasi-1D and
homogeneous-local-spin-orientation limits, as well as
they enable us to explicitly show how tcrdepends on
(Mz)inin a concretely realizable setup. We consider be-
low two cases: (A) box trap along z[49] and (B) harmonic
trap alongz.
A. Box traps
We setV(z) = 0 forjzj< Lzand1other-
wise. Then n(z;t) =N=(2Lz) and we estimate '
c0+ 2S2cdd=3
N=
4l2
?Lz
from Eq. (40). In this
case, 0
dd= dd(Lz=l?) as is calculated in Eq. (30).
FixingBz= 0:03 mG and N= 100, we con-
sider the following two cases: (1) !?=(2) = 2:4
104Hz andLz= 3:125m. Then Lz=l?= 62:03,
=(~!?) = 0:1692, and Lz=d= 29:55. Thus,
the system is in both the quasi-1D and homogeneous-
local-spin-orientation limit. Sdd(Lz=l?) = 4:074
103Hz,~Sdd(Lz=l?)=(gFB) = 0:3969 mG, and cr:=
cos 1(Mz)cris 85:67.
(2)!?=(2) = 1:2104Hz andLz= 6:250m.
ThenLz=l?= 87:72,=(~!?) = 0:0846, and
Lz=d= 29:55. Thus, again the system is
in both the quasi-1D and homogeneous-local-spin-
orientation limits. Sdd(Lz=l?) = 1:028103Hz,
~Sdd(Lz=l?)=(gFB) = 0:1002 mG, and cr= 72:57.
Fig. 4 shows the relation between tcrand (Mz)in.
-1.0 -0.5 0.5 1.00.050.100.15!?=(2) = 2:4104Hz,Lz= 3:125m, andl?= 0:0504m
whereN=
4Lzl2
?
= 10:031020m 3((Mz)cr= 0:0756).
-1.0 -0.5 0.5 1.00.20.40.6
!?=(2) = 1:2104Hz,Lz= 6:250m, andl?= 0:0712m
whereN=
4Lzl2
?
= 2:5081020m 3((Mz)cr= 0:2995).
FIG. 4.tcras a function of ( Mz)inwhenB=Bzezwhere
Bz= 0:03 mG and particle number N= 100. From top to
bottom: Red for = 0 :01, black for = 0 :03, and blue for
= 0:09. Lines are from exact analytic formula in Eq. (35),
and dot-dashed are from asymptotic expression in Eq. (36).
Generally,tcrdecreases as increases. Also, note that tcr
diverges as ( Mz)in!(Mz)cr. For larger mean number den-
sityN=
4Lzl2
?
(top), the asymptotic expression of tcris
essentially indistinguishable from the exact analytic formula
oftcr.
B. Harmonic traps
We setV(z) =m!2
zz2=2. Using the Thomas-Fermi
approximation, from Eq. (40), =m!2
zL2
z=2 whereLz
is given by Eq. (41).
c0+ 2S2cdd=3
n(z;t)=
l2
?
=
m!2
z
L2
z z2
forjzjLzandn(z;t) = 0 forjzj>Lz.
From thisn(z;t), we performed a numerical integration
to calculate 0
ddin Eq. (16). Fixing Bz= 0:03 mG, we
consider the following two cases:
(1)N= 240,!?=(2) = 2000 Hz, and !z=(2) =
50 Hz, for which Lz= 5:703m andLz=l?= 32:68.
We obtain again the quasi-1D and homogeneous-local-
spin-orientation limits since =(~!?) = 0:3337 and
Lz=d= 17:85. Furthermore, S0
dd= 1:644103Hz,
~S0
dd=(gFB) = 1:60210 1mG, andcr= 79:21.
(2)N= 340,!?=(2) = 1000 Hz, and !z=(2) =
25 Hz, where Lz= 8:070m andLz=l?= 32:70. Again,
we have the quasi-1D and with homogeneous-local-spin-
orientation limits ful lled due to =(~!?) = 0:3341 and10
-1.0 -0.5 0.5 1.00.10.20.30.4
N= 240,!?=(2) = 2000 Hz, and !z=(2) = 50 Hz.
Lz= 5:703m andl?= 0:1745m where
N=
4Lzl2
?
= 1:0101020m 3((Mz)cr= 0:1873).
-1.0 -0.5 0.5 1.00.20.40.60.8
N= 340,!?=(2) = 1000 Hz, and !z=(2) = 25 Hz.
Lz= 8:070m andl?= 0:2468m where
N=
4Lzl2
?
= 0:5501020m 3((Mz)cr= 0:3741).
FIG. 5.tcras a function of ( Mz)inwhenB=Bzezwhere
Bz= 0:03 mG, for two particle numbers Nas shown. From
top to bottom: Red for = 0 :01, black for = 0 :03, and
blue for = 0 :09. Lines are from exact analytic formula in
Eq. (35), and dot-dashed are from asymptotic expression in
Eq. (36). Generally, tcrdecreases as increases. Also, note
thattcrdiverges as ( Mz)in!(Mz)cr. For larger mean number
densityN=
4Lzl2
?
(top), the asymptotic expression of tcris
essentially indistinguishable from the exact analytic formula
oftcr.
Lz=d= 17:87. In addition, S0
dd= 8:230102Hz,
~S0
dd=(gFB) = 8:01910 2mG, andcr= 68:03.
Fig. 5 shows for the harmonic traps the relation be-
tweentcrand (Mz)in.
C. Measurability of critical switching time
Figs. 4 and 5 demonstrate that the critical switching
timetcris much smaller than the lifetime of BEC (sev-
eral seconds [30]) and thus, by measuring tcrby varying
(Mz)in, one will be able to obtain the value of , pro-
vided indeed does not depend on spin indices as for
example Refs. [26, 29] have assumed. Conversely, if one
obtains from the measurements a di erent functional re-
lation which does not follow Eq. (35), this implies that
may depend on spin indices.
Note that both gures, Figs. 4 and 5, show that tcrisinversely proportional to the mean number density
N=
4Lzl2
?
. Eq. (36) states that tcris inversely pro-
portional to 0
dd, but except for the box trap case, in
which one can analytically calculate 0
dd= dd(Lz=l?)
in Eq. (30), the dependence of 0
ddand the mean number
densityN=
4Lzl2
?
is not immediately apparent. Thus,
at least for harmonic traps, and in the Thomas-Fermi ap-
proximation, one may use the box trap results of Eq. (30)
for provide an approximate estimate of the behavior of
tcr.
VII. CONCLUSION
For a quasi-1D dipolar spinor condensate with
unidirectional local magnetization (that is in the
homogeneous-local-spin-orientation limit), we provided
an analytical derivation of the Landau-Lifshitz-Gilbert
equation and the Stoner-Wohlfarth model. For an exter-
nal magnetic eld along the long axis, we obtained an
exact solution of the quasi-1D Landau-Lifshitz-Gilbert
equation. Our analytical solution demonstrates that the
magnetic dipole-dipole interaction accelerates the relax-
ation of the magnetization to stable states and hence
strongly facilitates observation of this process within the
lifetime of typical dipolar spinor BECs. Employing this
solution, we hence propose a method to experimentally
access the dissipative parameter(s) .
We expect, in particular, that our proposal provides a
viable tool to verify in experiment whether is indeed
independent of spin indices, as commonly assumed, and
does not have to be replaced by a tensorial quantity for
spinor gases. We hope that this will stimulate further
more detailed investigations of the dissipative mechanism
in dipolar BECs with internal degrees of freedom.
We considered that the magnetization along z,Mz, has
contributions solely from the atoms residing in the con-
densate, an approximation valid at suciently low tem-
peratures. When the magnetization from noncondensed
atoms is not negligible, as considered by Ref. [5] for a
contact interacting scalar BEC, correlation terms mix-
ing the noncondensed part and the mean eld, such asPS
= S
(r;t)h^ (r;t)^ (r;t)iwill appear on the
right-hand side of Eq. (3). Here, ^ (r;t) is the -th
component of quantum eld excitations above the mean-
eld ground state in the spinor basis. Considering the
e ect of these terms is a subject of future studies.
ACKNOWLEDGMENTS
The work of SHS was supported by the National
Research Foundation of Korea (NRF), Grant No.
NRF-2015-033908 (Global PhD Fellowship Program).
SHS also acknowledges the hospitality of the Uni-
versity of T ubingen during his stay in the summer
of 2019. URF has been supported by the NRF11
under Grant No. 2017R1A2A2A05001422 and Grant No. 2020R1A2C2008103.
Appendix A: Derivation of the e ective potential Ve
The dipole-dipole interaction term Vdd(t) in the total energy is given by [31]
Vdd(t) =cdd
2Z
d3rZ
d3r0X
;0=x;y;zF(r;t)Q;0(r r0)F0(r0;t); (A1)
wherecddis dipole-dipole interaction coecient, F(r;t) = y(r;t)^f (r;t), andQ;0(r) is de ned as the tensor
Q;0(r):=r2;0 3rr0
r5(A2)
in spin space, where r=jrjandr=re, withebeing the unit vector along the axis. From now on, we de ne
= (x;y) such that dxdy =d2=d'd where tan'=y=x.
Using the convolution theorem, the dipole-dipole interaction term Vdd(t) can be expressed by
Vdd(t) =cdd
2(2)D=2Z
d3k~n(k;t) ~n( k;t)~Udd(k;t) (A3)
with the Fourier transform
Udd(;t) =1
n(r;t)n(r0;t)X
;0=x;y;zF(r;t)Q;0()F0(r0;t); (A4)
where ~g(k;t) = (2) D=2R
drg(r;t)eikris the Fourier transform of the function g(r;t) inD-dimensional space r
(in our case, D= 3),=r r0, andn(r;t) =j (r;t)j2.
By denoting k= (k;kz), wherek= (kx;ky) withk=q
k2x+k2yand tan'k=ky=kx, with our mean- eld
wavefunction in Eq. (8), one derives
~n(k;t) =1
l2
?1
(2)3=2Z
d2Z1
1dze (=l?)2n(z;t)eikeikzz=1
2~n(kz;t)e k2
l2
?=4; (A5)
wheren(z;t):=j (z;t)j2e 2 !?t=(1+ 2). Note the factor of (2 ) 1appearing, when compared to Eq. (12) in
Ref. [27], which is stemming from our de nition of Fourier transform.
Denoting=jj, by writingefor the unit vector along , we obtain
Udd(;t) = 1
3r
6
5hn
Y2
2(e)e 2i (t)+Y 2
2(e)e2i (t)o
S2sin2(t)
n
Y1
2(e)e i (t) Y 1
2(e)ei (t)o
S2sinf2(t)gi
+1
3r
6
5Y0
2(e)r
2
3S2
3 sin2(t) 2
; (A6)
whereYm
l(e) are the usual spherical harmonics. Its Fourier transform ~Udd(k;t) is
~Udd(k;t) =1
(2)3=24
3S2
1 3
2sin2(t)
3k2
z
k2+k2z 1
+1p
2k2
k2+k2zS2sin2(t) cos
2'k 2 (t)
+r
2
kkz
k2+k2zS2sinf2(t)gcos
'k (t)
:(A7)
By plugging Eq. (A5) and Eq. (A7) into Eq. (A3), we nally obtain Vdd(t) as
Vdd(t) =cdd
2p
2Z1
1dkz~n(kz;t) ~n( kz;t)2S2
l2
?p
2
1 3
2sin2(t)
k2
zl2
?=2
ek2
zl2
?=2E1
k2
zl2
?=2
1
3
;(A8)12
whereE1(x) =R1
xdue u=uis exponential integral.
Note that Eq. (A8) can be also written as
Vdd(t) =cdd
2p
2Z1
1dkz~n(kz;t) ~n( kz;t)~Ve (kz;t) =cdd
2Z1
1dzZ1
1dz0n(z;t)n(z0;t)Ve (z z0;t):(A9)
Due to the fact that ~Ve (kz;t) can be obtained by Eq. (A8), we can get Ve (z;t) by inverse Fourier transform. As a
preliminary step, we rst write down some integrals of E1(x) as follows:
Z1
1dxex2E1
x2
e ikx=Z1
1dxe ikxZ1
x2dte (t x2)
t= ()3=2ek2=4Erfc (jkj=2): (A10)
Di erentiating Eq. (A10) with respect to ktwo times results in
Z1
1dxx2ex2E1
x2
e ikx= ()3=21
2k2
2+ 1
ek2=4Erfc (jkj=2) jkj
2p 2p(k)
: (A11)
Therefore,Ve (z;t) can be calculated as
Ve (z;t) =1p
2Z1
1dkz2S2
l2
?p
2
1 3
2sin2(t)
k2
zl2
?=2
ek2
zl2
?=2E1
k2
zl2
?=2
1
3
e ikzz
=S2
l3
?3
2sin2(t) 1
G(jzj=l?) 4
3(z=l?)
; (A12)
whereG(x) is de ned in Eq. (14), and (x) is the Dirac delta function.
The Fourier transform of Eq. (A12) acquires the form
~Ve (kz;t) =1p
2Z1
1dzV e (z;t)eikzz=r
2
S2
l2
?3
2sin2(t) 1Z1
0dvG (v) cos (kzl?v) 2
3
=r
2
S2
l2
?3
2sin2(t) 1Z1
0dunp
2u2+ 1
eu2Erfc (u) 2uo
cosp
2kzl?u
2
3
:(A13)
From [50], the following integral involving the complementary error function is
Z1
0dueu2Erfc (u) cos (bu) =1
2peb2=4E1
b2=4
: (A14)
By di erentiating Eq. (A14) two times with respect to b, we get
Z1
0duu2eu2Erfc (u) cos (bu) = 1
2p1
2b2
2+ 1
eb2=4E1
b2=4
1 +2
b2
: (A15)
Hence, Eq. (A13) becomes
~Ve (kz;t) =r
2
S2
l2
?3
2sin2(t) 1
1
2
k2
zl2
?+ 1
ek2
zl2
?=2E1
k2
zl2
?=2
1 +1
k2zl2
?
+1
2ek2
zl2
?=2E1
k2
zl2
?=2
+1
k2zl2
? 2
3
=2S2
l2
?p
2
1 3
2sin2(t)
k2
zl2
?=2
ek2
zl2
?=2E1
k2
zl2
?=2
1
3
: (A16)
Comparing Eq. (A8) with Eq. (A16), one veri es that Eq. (A12) is the correct result for the e ective interaction of
the quasi-1D dipolar spinor gas.13
Appendix B: Quasi-1D Gross-Pitaevski equation with dissipation
By introducing an identical damping coecient for each component of the spinor, cf., e.g. Refs. [26, 29] (i.e. as if
each component e ectively behaves as a scalar BEC [28]), and neglecting a possible quadratic Zeeman term, the GP
equation for a spin- SBEC can be written as [26]
(i )~@ (r;t)
@t=
~2
2mr2+Vtr(r) +c0j (r;t)j2
(r;t) ~SX
= Sfb bdd(r;t)g
^f
; (r;t)
+SX
k=1c2kX
1;2;;k=x;y;zF1;2;;k(r;t)SX
= S
^f1^f2^fk
; (r;t);
(B1)
where (r;t) is the -th component of the mean- eld wavefunction (r;t) (the spin-space index is an integer
taking 2S+ 1 values running from SandS),F1;2;;k(r;t):= y(r;t)^f1^f2^fk (r;t),~^fis the spin- S
operator,b=gFBB=~(gFis the Land e g-factor, Bis the Bohr magneton, and Bthe external magnetic eld).
Finally, ~bdd(r;t)e=cddR
d3r0P
0=x;y;zQ;0(r r0)F0(r0;t), whereeis the unit vector along the axis
(=x;y;z ) [31]. Applying the formalism of Ref. [1] to a spinor BEC assuming that does not depend on spin
indices, one just needs to transform t!
1 + 2
tin Eq. (B1) and (8). We then integrate out the xandydirections
in Eq. (B1) to obtain the quasi-1D GP equation.
From Eq. (8) in the main text, we have
Z
d2SX
= Se 2=(2l2
?)
l?p
~bdd(r)
^f
;
(r;t)
=cdd
2l3
?Z1
1dz0n(z0;t)
Gjz z0j
l?
4
3z z0
l?
(z;t)e i+
1+ 2!?tSfM(t) 3Mz(t)ezgSX
= S
^f
; (t);
(B2)
whereR
d2:=R1
1dxR1
1dyandn(z;t):=R
d2j (r;t)j2=j (z;t)j2e 2 !?t=(1+ 2).
For a spin-SBEC, from Eq. (B1), for the trap potential given in Eq. (4) and if we use Eq. (8), by integrating out
thexandydirections, one acquires the expression
(i )~@f (z;t) (t)g
@t=
~2
2m@2
@z2+V(z) +c0
2l2
?n(z;t)
(z;t) (t)
+ [ ~b+~SfM(t) 3Mz(t)ezgPdd(z;t)]8
<
:SX
= S
^f
; (z;t) (t)9
=
;
+SX
k=1c2k
2l2
?n(z;t)X
1;2;;k=x;y;zSM1;2;;k(t)8
<
:SX
= S
^f1^f2^fk
; (z;t) (t)9
=
;;
(B3)
whereM1;2;;k(t) is de ned in Eq. (12) and
Pdd(z;t) =cdd
2~l3
?Z1
1dz0n(z0;t)
Gjz z0j
l?
4
3z z0
l?
=cdd
~S2
3 sin2(t) 2 Z1
1dz0n(z0;t)Ve (z z0;t);
(B4)
withVe de ned in (A12). It is already clear from Eq. (B3) that, besides particle loss from the condensate encoded
in a decayingj (z;t)j, dissipation also leads to a dephasing , i.e. the decay of (t) due to the term @(t)=@t.
From now on, if there is no ambiguity, and for brevity, we drop the arguments such as x;y;z;t from the functions.14
From Eq. (B3), we then get
~@
@t= ~
@
@t +i
1 + 2
~2
2m1
@2
@z2+V+c0
2l2
?n
+ +i
1 + 2f~b S(M 3Mzez)~Pddg8
<
:SX
= S
^f
; 9
=
;
+i
1 + 2SX
k=1c2k
2l2
?nX
1;2;;k=x;y;zSM1;2;;k8
<
:SX
= S
^f1^f2^fk
; 9
=
;; (B5)
Since@jj2
@t= 0 due to the normalization jj2= 1, we then have
0 = 2Re
~
@
@t
1 + 2
~2
2m1
@2
@z2+V+c0
2l2
?n
+i
1 + 2~2
2m1
@2
@z2 1
@2
@z2
+2
1 + 2f~b S(M 3Mzez)~PddgSM 2
1 + 2SX
k=1c2k
2l2
?nX
1;2;;k=x;y;zS2M2
1;2;;k: (B6)
Hence the dynamics of the magnetization direction follows the equation
~S@M
@t= 2Re8
<
:SX
; = Sy
^f
;
~@
@t9
=
;
= 2
1 + 2S2Mf~b S(M 3Mzez)~PddgM+2
1 + 2MSX
k=1c2k
2l2
?nX
1;2;;k=x;y;zS3M2
1;2;;k
+
1 + 2X
=x;y;zf~b S(M 3Mz;z)~PddgSf;+ (2S 1)MMg
1
1 + 2X
;=x;y;zf~b S(M 3Mz;z)~Pddg;;SM
2Re8
<
: +i
1 + 2SX
k=1c2k
2l2
?nX
1;2;;k=x;y;zSM1;2;;kSX
; = Sy
^f^f1^f2^fk
; 9
=
;; (B7)
since the scalar product y
^f ^f +^f ^f
=Sf ; + (2S 1)M M g[26].
By direct comparison, we can identify Eq. (B8) below as being identical to Eq. (B21) in [26], the only di erence
consisting in the de nition of M1;2;;k: We employ a scaled version of M1;2;;k, which is normalized to Sin
[26]. From Eq. (7) in the main text,
X
1;2;;k=x;y;zM1;2;;kSX
; = Sy
^f^f1^f2^fk
; =X
1;2;;k=x;y;zM2
1;2;;kS2M; (B8)
which is real. Therefore, Eq. (B7) can be written in the following form
@M
@t=
1 + 2M[Mfb S(M 3Mzez)Pddg] +1
1 + 2Mfb S(M 3Mzez)Pddg
=1
1 + 2M(b+ 3SPddMzez)
1 + 2M[M(b+ 3SPddMzez)]
=M(b+ 3SPddMzez) M@M
@t; (B9)
sinceM@M
@t= 0 holds.
AsPis a function of zandt, butMis independent of z[Mis the scaled local magnetization and our aim is to
study a dipolar spinor BEC with unidirectional local magnetization (the homogeneous-local-spin-orientation limit)],
by multiplying with n(z;t) both sides of Eq. (B9) and integrating along z, we nally get the LLG equation
@M
@t=M(b+S0
ddMzez) M@M
@t; (B10)
where 0
ddis de ned in Eq. (16). Note here that 0
ddbecomes dd(Lz=l?) de ned in Eq. (30) when n(z;t) =N=(2Lz)
for LzzLzandn(z;t) = 0 otherwise.15
Appendix C: Modi cation of the LLG equation for a spin-space tensor
When depends on spin indices, i.e. is a tensor, Eq. (B3) can be generalized to read
SX
= S(i ; ; )~@f (z;t) (t)g
@t=SX
= SH ; (z;t) (t): (C1)
The spinor part of the wavefunction is normalized to unity, jj2= 1. Hence, we know that@jj2
@t= 0. Therefore, from
Eq. (C1), we derive the expression
SX
; = SRe
i
; @
@t i
; 1
@
@t i1
~
H ;
Re1
@
@t
= 0: (C2)
This then leads us to
@M
@t=2
SSX
; ;
= SRe
i
^f
; ;
@
@t i
^f
; ;
1
@
@t i1
~
^f
; H ;
2Re
M1
@
@t
: (C3)
For scalar , ; ! ; , the equation above becomes Eq. (B7).
From Eqs. (C2) and (C3), one concludes that the stationary solution Mof Eq. (C3) is independent of . In other
words, whether depends on spin indices or not, the SW model (29) is left una ected, also see the discussion in
Section V of the main text.
Appendix D: Description of magnetostriction
For a dipolar spinor BEC without quadratic Zeeman term, when there is no dissipation ( = 0), the mean- eld
equation in Eq. (3) can be written as
(t) (r;t) =8
<
: ~2
2mr2+Vtr(r) +c0SX
= Sj (r;t)j29
=
; (r;t) ~fb bdd(r;t)gSX
= S
^f
; (r;t)
+SX
k=1c2kX
1;2;;k=x;y;zSX
1; 1; = S
^f1^f2^fk
1; 1
^f1^f2^fk
;
1(r;t) 1(r;t) (r;t):
(D1)
where we have substituted i~@ (r;t)
@t= (t) (r;t).
Since we consider the homogeneous-local-spin-orientation limit, we may write (r;t) = uni(r;t) (t). In this
limit, we have
j (r;t)j2:= y(r;t) (r;t) =SX
= S y
(r;t) (r;t) =j uni(r;t)j2; (D2)
sinceSX
= Sj (t)j2= 1 from the de nition of (t) in Eq. (7). Thus j uni(r;t)j2is equal to the number density.
Then Eq. (D1) can be written as
(t) (t) uni(r;t) =
~2
2mr2+Vtr(r) +c0j uni(r;t)j2
(t) uni(r;t)
~fb bdd(r;t)gSX
= S
^f
; (t) uni(r;t)
+SSX
k=1c2kX
1;2;;k=x;y;zSX
= SM1;2;;k(t)
^f1^f2^fk
; (t)j uni(r;t)j2 uni(r;t):
(D3)16
Now, we decompose the chemical potential (t) as(t):=SX
= S (t)j (t)j2. Then one obtains
(t) uni(r;t) ="
~2
2mr2+Vtr(r) +(
c0+S2SX
k=1c2kX
1;2;;k=x;y;zM2
1;2;;k(t))
j uni(r;t)j2#
uni(r;t)
+ [dd(r;t) S~fbM(t)g] uni(r;t);
(D4)
where
dd(r;t):=S2cdd2
4Z
d3r08
<
:X
;0=x;y;zM(t)Q;0(r r0)M0(t)9
=
;j uni(r0;t)j23
5; (D5)
is the dipole-dipole mean- eld potential [41] following from the de nition of bddbelow Eq. (3) in the main text.
Due toMx(t) = sin(t) cos (t),My(t) = sin(t) sin (t), andMz(t) = cos(t), from Eqs. (A4) and (A6), we
have
X
;0=x;y;zM(t)Q;0()M0(t) = 1
3r
6
5hn
Y2
2(e)e 2i (t)+Y 2
2(e)e2i (t)o
sin2(t)
n
Y1
2(e)e i (t) Y 1
2(e)ei (t)o
sinf2(t)gi
+1
3r
6
5Y0
2(e)r
2
3
3 sin2(t) 2
; (D6)
whereYm
l(e) are the usual spherical harmonics.
By using Eq. (A2), an alternative form of Eq. (D6) can be obtained:
X
;0=x;y;zM(t)Q;0()M0(t) =X
;0=x;y;z2;0 30
5M(t)M0(t) =2jM(t)j2 3fM(t)g2
5
=2 3fM(t)g2
5: (D7)
Thus, dd(r;t) can be written as
dd(r;t) =S2cdd"Z
d3r0jr r0j2 3f(r r0)M(t)g2
jr r0j5j uni(r0;t)j2#
=S2cdd"Z
d3r0jr r0j2 3f(r r0)M(t)g2
jr r0j5j uni(r0;t)j2#
= 3
2S2cddsin2(t)Z
d3j uni(+r;t)j21
5h
2 2
z 2fxsin (t) ycos (t)g2i
3S2cddsinf2(t)gZ
d3j uni(+r;t)j2z
5fxcos (t) + ysin (t)g
+1
2S2cdd
1 3 cos2(t) Z
d3j uni(+r;t)j21
5
32
z 2
: (D8)
where r:=r=LwithLbeing some length which scales r(so that ris a dimensionless vector). For example, in
quasi-1D with trap potential being Eq. (4), L=l?. Note that, in the special case where M(t) =Mz(t)ez, the form
of Eq. (D8) becomes identical to Eq.(6) in Ref. [40].
Since we concentrate on quasi-1D gases, with trap potential given by Eq. (4) in the main text, we will explicitly
compute the form of dd(r;t) for the quasi-1D setup. By writing
j uni(r;t)j2=e 2=l2
?
l2
?j (z;t)j2; (D9)17
-20 -10 10 20
-0.4-0.20.20.40.60.81.0
-60 -40 -20 20 40 60
-0.4-0.20.20.40.60.81.0
FIG. 6. Scaled dipole-dipole mean- eld potential dd(z) as a function of zfor a quasi-1D box trap. (Left) Lz=l?= 10.
(Right)Lz=l?= 30.
and integrating out xandydirections, one can get the quasi-1D dipole-dipole-interaction mean- eld potential dd(z;t)
as follows (which is in Eq. (38)):
dd(z;t) =cdd
2l2
?S2
1 3M2
z(t) Z1
1dzj (z+ zl?;t)j2G(jzj) 4
3j (z;t)j2
: (D10)
Now, let us consider box trap in quasi-1D case, i.e. V(z) = 0 forjzjLzandV(z) =1forjzj>LzwhereV(z)
is in Eq. (4). Then we may write
j (z;t)j2=2
64N
2LzforjzjLz,
0 forjzj>Lz,(D11)
sinceV(z) = 0 for LzzLz. Thus, dd(z;t) can be written as
dd(z;t) =2
666664dd(t)(Z(Lz z)=l?
(Lz+z)=l?dzG(jzj) 4
3)
forjzjLz,
dd(t)Z(Lz z)=l?
(Lz+z)=l?dzG(jzj) forjzj>Lz,(D12)
where dd(t):=NcddS2
1 3M2
z(t)
=
2Lzl2
?
. dd(z;t) is discontinuous at z=Lzbecause of the sudden change
of the density at the boundary ( z=Lz) due to box trap potential.
De ning the scaled density-density mean- eld potential dd(z):= dd(z;t)=dd(t), we obtain Fig. 6, for two
di erent axial extensions, Lz=l?= 10 and 30. As Fig. 6 clearly illustrates, in a box-trapped quasi-1D gas, dd(z;t)
becomes approximately constant for jzj<LcandLc!LzforLz=l? 1. Depending on the value of M(t), dd(r;t)
will introduce either a repulsive or an attractive force. This force will however exist only near the boundary for a box
trap, where it can lead to a slight modi cation of the density of atomes. Its relative in
uence decreases with increasing
extension of the trapped gas along the zaxis, and can therefore be consistently neglected in the approximation of
constant particle-density.
However, to assess whether signi cant magnetostriction occurs, one has to consider, in addition to dd, the trap
potentialVtrand the `quasi' density-density interaction mean eld potential 0de ned as
0(r;t):=(
c0+S2SX
k=1c2kX
1;2;;k=x;y;zM2
1;2;;k(t))
j uni(r;t)j2: (D13)
We can coin 0(r;t) a `quasi' density-density interaction mean eld potential because only c0is a density-density
interaction coecient ( c2kare interaction coecients parametrizing the spin-spin interactions for spin- Sgas wherek
is an integer with 1 kS. For example, c2is the spin-spin interaction coecient of a spin-1 gas). In our quasi-1D
case, this 0(r;t) potential is 0(z;t) where
0(z;t):=(
c0
2l2
?+S2SX
k=1c2k
2l2
?X
1;2;;k=x;y;zM2
1;2;;k(t))
j (z;t)j2: (D14)18
In the main text, we assume that c0 S2PS
k=1c2kP
1;2;;k=x;y;zM2
1;2;;k(t). For spin-123Na or87Rb,S= 1
andc0'100jc2j[31, 34], so this is an appropriate assumption (note thatPS
k=1P
1;2;;k=x;y;zM2
1;2;;k(t) = 1).
The values of the c2kare not yet established for166Er. We therefore tacitly assume in the main text, when calculating
concrete numerical examples for166Er, that the above condition also still holds, despite the prefactor S2enhancing
the importance of spin-spin interactions in 0(z;t). When this assumption is not applicable, one is required to take
into account the time dependence of 0(z;t) due toM(t) together with magnetostriction due to dd(z;t), which will
change the system size Lzas a function of t. This will in turn change the integration domain and quasi-1D density
n(z;t) =j (z;t)j2in Eq. (19), and incur also a changed time dependence of 0
dd(t), and the solution of the coupled
system of equations (B10) and (D4) needs to be found self-consistently.
For a harmonic trap, due to the resulting inhomogeneity of j (z;t)j2, dd(z;t) will have more signi cant spatial
dependence than its box trap counterpart shown in Fig. 6. Here, we note that Ref. [40] has already shown, for a spin-
polarized gas, that magnetostriction occurs in a harmonic trap. The e ect of magnetostriction is generally expected
to be larger in a harmonic trap when compared to a box trap with similar geometrical and dynamical parameters for
large relative system size Lz=l? 1, at least under the above condition that the S2c2k=c0are suciently small.
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2304.09366v1.Thickness_dependent_magnetic_properties_in_Pt_CoNi_n_multilayers_with_perpendicular_magnetic_anisotropy.pdf | 1Thickness-dependentmagneticpropertiesinPt/[Co/Ni]n
multilayerswithperpendicularmagneticanisotropy*
ChunjieYan(晏春杰)1,LinaChen(陈丽娜)1,2,#,KaiyuanZhou(周恺元)1,Liupeng
Yang(杨留鹏)1,QingweiFu(付清为)1,WenqiangWang(王文强)1,Wen-Cheng
Yue(岳文诚)3,LikeLiang(梁立克)1,ZuiTao(陶醉)1,JunDu(杜军)1,Yong-Lei
Wang(王永磊)3andRonghuaLiu(刘荣华)1*
1NationalLaboratoryofSolidStateMicrostructures,SchoolofPhysicsand
CollaborativeInnovationCenterofAdvancedMicrostructures,NanjingUniversity,
Nanjing210093,China
2SchoolofScience,NanjingUniversityofPostsandTelecommunications,Nanjing
210023,China.
3SchoolofElectronicsScienceandEngineering,NanjingUniversity,Nanjing210093,
China.
WesystematicallyinvestigatedtheNiandCothickness-dependentperpendicular
magneticanisotropy(PMA)coefficient,magneticdomainstructures,and
magnetizationdynamicsofPt(5nm)/[Co(tConm)/Ni(tNinm)]5/Pt(1nm)multilayersby
combiningthefourstandardmagneticcharacterizationtechniques.The
magnetic-relatedhysteresisloopsobtainedfromthefield-dependentmagnetizationM
andanomalousHallresistivity(AHR)xyfoundthatthetwoserialmultilayerswith
tCo=0.2and0.3nmhavetheoptimumPMAcoefficientKUwellasthehighest
coercivityHCattheNithicknesstNi=0.6nm.Additionally,themagneticdomain
structuresobtainedbyMagneto-opticKerreffect(MOKE)microscopyalso
significantlydependonthethicknessandKUofthefilms.Furthermore,the
thickness-dependentlinewidthofferromagneticresonanceisinverselyproportionalto
KUandHC,indicatingthatinhomogeneousmagneticpropertiesdominatethe
linewidth.However,theintrinsicGilbertdampingconstantdeterminedbyalinear
fittingoffrequency-dependentlinewidthdoesnotdependonNithicknessandKU.OurresultscouldhelppromotethePMA[Co/Ni]multilayerapplicationsinvarious
spintronicandspin-orbitronicdevices.
Keywords:perpendicularmagneticanisotropy,magneticdomain,damping,
multiayers
PACS：75.30.Gw,75.70.Kw,75.40.Gb,68.65.Ac1.INTRODUCTION
Magneticmultilayerswithstrongperpendicularmagneticanisotropy(PMA)and
lowmagneticdampinghaveattractedmuchattentionbecauseoftheirpotential
applicationsinhigh-densitymagneticrandomaccessmemories(MRAM)[1-5]andspin
torquenano-oscillators[6-9].Comparedtothein-planemagnetizedferromagnets,
ferromagneticfilmswithPMAfacilitatetherealizationofnonvolatileMRAMwith
lowerpowerandhigherdensitystoragebecausethelatterhaslowercriticalswitching
currentandhigherthermalstabilitythantheformerasthecontinuousdownscalingof
thecellsizeofdevices[10].Inaddition,PMAcanbeaneffectivemagneticfieldto
achievezeroexternalmagneticfieldworkingspin-torquenano-oscillatorswith
ferromagnetswithstrongPMAandlowdampingasitsfreelayer.[11]Therefore,the
controllabletailoringPMAofmagneticfilmsisanessentialprerequisitefor
developinghigh-performancespintronicdevices.Themagneticmultilayers,e.g.,
[Co/Pd],[Co/Pt],and[Co/Ni],provideanopportunitytotunetheirmagnetic
propertiesbychangingthethicknessratiocontrollablyandthenumberofbilayer
repeatsthankstotheinterface-inducedPMAduetointerfacialspin-orbitcouplingand
interfacialstrainrelevantmagnetoelasticeffects[12-17].AmongthesePMAmultilayers,
thePMA[Co/Ni]multilayeralsoexhibitslowdampingconstant[14],whichgetsmuch
attention,especiallyforthefieldsofcurrent-drivenauto-oscillationofmagnetization
andexcitationandmanipulationofspin-waves[18,19].Furthermore,thePMA[Co/Ni]
multilayeralsousefulforspin-orbittorquedevices[20-22].Therefore,[Co/Ni]
multilayersareconsideredoneofthemostpromisingPMAferromagnetsinvarious
spintronicdevices.Althoughthereareafewstudiesonthemagneticanisotropy,
magnetotransport,andmagneticdampingofPt/[Co/Ni]multilayers[6,14,23],the
systematicallystudiedevolutionofmagnetostaticproperties,includingthetopography
ofmagneticdomainsandmagneticdynamicswiththethicknessratioofCoandNi
layersforthismultilayerfilmstillneedstomakeathoroughinvestigationforfacilitatingitbetterusedinfurtherspintronics.
Here,wesystematicallyinvestigatehowtocontrolthemagneticfilmPMAby
tailoringtheinterfacialeffectbyvaryingthethicknessoftheNilayeranditsimpact
onmagneticdomainstructureanddynamicaldampingintwoserialCo/Nimultilayers
withtCo=0.2and0.3nm.ThehighestPMAcoefficientKU~3×106ergcm-3and
coercivityHC~250OearefoundattheoptimumNithicknesstNi=0.6nmforthe
studiedtwoserials.Thenucleationofthemagneticdomainoccursatonlyafew
nucleationsitesandgraduallyexpandswithmagneticfieldsforthemultilayerswith
theoptimumNithicknesses0.4nm~0.6nm.Finally,theintrinsicGilbertdamping
constantαisnotsensitivetothickness-dependentKUanddomainstructureseven
thoughthelinewidthofferromagneticresonanceisinverselyproportionaltoKUand
HC,whichisdominatedbyinhomogeneousmagneticproperties.
2.EXPERIMENT
TwoserialmultilayersofPt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1)and
Pt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1),namedasPt/[Co(0.2)/Ni(tNi)]andPt/[Co(0.3)/Ni(tNi)]
respectively,weredepositedonSi/SiO2substratesatroomtemperatureby
dc-magnetronsputteringwithArpressure3×10-3torr.Theunitinparenthesesisthe
thicknessinnm.Thebasepressureofthesputteringdepositionchamberisbelow2×
10−8torr.Thedepositionratewasmonitoredbythequartzcrystalmonitorinsituand
calibratedbyspectroscopicellipsometry(SE).Thestaticmagneticpropertieswere
characterizedbythevibratingsamplemagnetometer(VSM),theanomalousHall
resistivity(AHR)measurement，andtheMagneto-opticKerreffect(MOKE)
microscopy,respectively.Thefilms'ferromagneticresonance(FMR)spectra,
obtainedbycombiningcoplanarwaveguide(CPW)andlock-intechniques,werealso
adoptedtocharacterizetheirdynamicmagneticproperties.Allthesemagnetic
characterizationswereperformedatroomtemperature.3.RESULTSANDDISCUSSION
3.1Quasi-staticmagneticproperties
TodirectlyobtainthethicknessdependenceofPMApropertiesintheCo/Nifilms,
wefirstperformedthemagnetichysteresisloopsofsampleswithdifferentthicknesses
usingVSM.Figure1showsthemagnetizationhysteresisloopswiththeout-of-plane
andin-planefieldgeometriesforthetwoserialmultilayersofPt/[Co(0.2)/Ni(tNi)]and
Pt/[Co(0.3)/Ni(tNi)]samples.Thewell-definedsquareM-Hloopsunderout-of-plane
field[Figs.1(a)and(c)]indicatethattwostudiedserialPt/[Co/Ni]multilayersexhibit
aperpendicularmagneticanisotropy.Additionally,thesaturationmagnetizationMSof
themultilayersdecreaseswithincreasingthethicknessoftheNilayertNi,from673
emucm-3to495emucm-3forPt/[Co(0.2)/Ni(tNi)]and723emucm-3to639emucm-3
forPt/[Co(0.3)/Ni(tNi)],whichagreeswiththemuchlowerMS~484emucm-3ofthe
metalnickelcomparedtothatofthecobaltlayerMS~1422emucm-3.Basedonthe
out-of-planeandin-planemagnetizationhysteresisloops,theperpendicular
anisotropyfieldHKwasdeterminedbyusingthedefinedformulaforthePMA[12,24]:
K=2
s0S⊥−0S∥ +4πS.ThecalculatedHK,MSandthe
coercivityHC,obtainedfromtheM-Hloops,weresummarizedbelowinFig.5.
Fig.1.(a)-(b)MagnetizationloopsofthefilmsPt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1)with
out-of-plane(a)andin-plane(b)magneticfield.(c)-(d)Sameas(a)-(b),for
Pt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1).
ThesestaticmagneticpropertiesofthemetalPt/[Co/Ni]multilayerfilmsalsocan
bedeterminedbytheelectrictransportsinmagneticfield,e.g.,anomalousHall
resistivity(AHR)andmagnetoresistiveeffect.Comparedtothestandard
magnetometerabove,theelectrictransportsinmagneticfieldmeasurementsprovide
analternativeapproachand,especially,moreusefulforspintronicnano-devices
becausetheycaneasilyaccessthemagneticpropertiesofthemicroscaleand
nanoscalesamples[25,26].Therefore,wealsoperformtheout-of-planeandin-plane
AHRloopsasafunctionoftheappliedmagneticfieldsforthestudiedtwoserial
multilayers,asshowninFig.2.ThecoercivityHCdeterminedfromtheout-of-plane
AHRloopsarewellconsistentwiththevaluesobtainedbytheM-Hloops,andare
alsosummarizedinthefollowingFig.5.Meanwhile,wecancalculateHKofthe
studiedfilmsfromthein-planeAHRloopsbyusingthefollowingrelation[27]:K=
∥∙tanarcsinxy
xy(0)+4πS,wherexy0istheAHRvalueatzeroin-planefield.
TheevolutionofHKwiththethicknessoftheNilayerisoverallconsistentwiththe
resultsdeterminedbytheVSMmeasurement.Furthermore,theAHRmeasurements
alsoprovideustheadditionalinformation,whichcannotbeeasilyaccessedbyVSM,
aboutthestudiedtwoserialPt/[Co/Ni]multilayers.Forexample,wefindthatthe
in-planeAHRnear-zeromagneticfieldismuchsmallerthantheout-of-planeAHR
forthesampleswithcertainNithickness,indicatingthatthesesamplesformthe
multi-domainstructuresatthelowin-planemagneticfields.Therefore,thevalueof
thedifferencebetweenout-of-planeandin-planeAHRatnear-zerofieldshintsthat
thedifferentNithicknessfilmsmayexhibitdistinctmagneticdomainstructures[28].Fig.2.(a)-(b)AnomalousHallresistivityasafunctionofout-of-plane(a)and
in-planemagneticfieldwith5otiltanglefromthefileplane(b)forthesamples
Pt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1).Theinsetshowsthegeometricrelationshipbetween
magneticfield,magnetizationandeffectiveanisotropicfieldKeff=K–4πS
(c)-(d)Sameas(a)-(b),forPt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1).AllAHRweremeasuredby
usingthefilmspatternedintoa0.3×10mmHallcross.
3.2Magneticdomainstructures
TodirectlyrevealtheevolutionofmagneticdomainstructurewithNithicknessand
thedetailofmagnetizationreversalprocessunderexternalfield,theMOKE
microscopyisalsoperformedforPt/[Co(0.2)/Ni(tNi)]serialsamples.Figure3shows
MOKEloopsandtherepresentativeMOKEimagesduringscanningout-of-plane
magneticfields.Brightanddarkregionsrepresentthedomainswithmagneticmoment
point-upandpoint-down,respectively.Exceptforthemultilayerwiththenickel
thicknesstNi=0.2nm[Fig.3(a)],allfilmsexhibitawell-definedPMAcorresponding
rectangularordumbbell-shapehysteresisloops[Figs.3(b)-(h)].TheMOKEimagesshowthatthesamplewithtNi=0.3nmbeginstonucleatewithnumerousnucleation
pointsapproximatelyuniformdispersiononthewholefilmatthefieldof138Oe,
thengraduallyexpandwithincreasingfield,andsaturatetotheuniformstateat⊥˃
300Oe[Fig.3(b)].Combiningwithfield-dependentmagneticsusceptibilityandAHR
characterizations,HCandHKofPt/[Co(0.2)/Ni(tNi)]firstenhancewithincreasingtNi,
reachthemaximumattNi=0.6nm,andthenreducewithcontinueincreasingtNito0.9
nm.Forthesampleswith0.4nm≤Ni≤0.6nm,theMOKEloopsexhibita
well-definedrectangularshape.Meanwhile,incontrasttotNi=0.3nm,onlyafew
nucleationpointsappearatthecriticalmagneticfield,andthenbegintoexpandthe
magneticdomainwithacontinuouslyincreasingfield.ForincreasingtNitoabove0.7
nm,theMOKEloopsbegintotransferintothedumbbellshapefromtheprior
well-definedrectangularshape.ThecorrespondingMOKEimagesshowthatthefilms
withtNi≥0.7nmformtree-likedomainwallswithmoreirregularbranchingsas
increasingtNiduringthefieldrangenearbelowthesaturationfield[themiddlepicture
ofFig.3].Inotherwords,forthefilmswithtNi≥0.7nm,thelengthofthedomainwall
increaseswithincreasingtNi,whichisconsistentwiththetrendofdependenceofHK
onNithickness.Asweknowthatthetotalenergydensityofthemagneticdomain
wallperunitareawisproportionaltomagneticanisotropyenergy,exchangeenergy
anddemagnetizationenergybasedonthewidelyrecognizedformula[29]:w=2
+
U
2+S2
4µ02
+,whereAistheexchangeconstant,δisthethicknessofthedomain
wall,KUisthePMAcoefficient,MSisthesaturationmagnetization,tisthethickness
oftheentirefilm,andμ0isthepermeabilityofvacuum.Tominimizethetotalenergy
oftheentirefilm,thevolume(orlength)ofthedomainwallneedstoreduceand
increasedemagnetizationenergywhenKUincreaseswithincreasingtNiintherangeof
0.4nm~0.6nm.Fig.3.(a)-(h)Magneto-opticKerrhysteresisloopsandmagneticdomainimagesof
thefilmsPt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1)withlabeledthicknesstNi=0.2nm(a),0.3nm
(b),0.4nm(c),0.5nm(d),0.6nm(e),0.7nm(f),0.8nm(g),0.9nm(h),respectively.
Thecorrespondingmagneticdomainimageswiththesizeof100µm×150µmwereobtainedatthelabeledout-of-planemagneticfields(alsomarkedasthereddots
onloops).
3.3Magnetizationdynamics
TofurtherinvestigatetheNithickness-dependentmagnetizationdynamicsofCo/Ni
multilayers,weperformthebroadbandFMRmeasurementwiththeexternalfield
perpendiculartothefilmplane.AllFMRmeasurementswerecarriedoutwitha
home-madedifferentialFMRmeasurementsystemcombiningLock-intechniqueat
roomtemperature.Acontinuous-waveOerstedfieldwithaselectedradiofrequencyis
generatedviaconnectingcoplanewaveguide(CPW)withanRFgenerator,which
producesamicrowavesignaltoexciteFMRofferromagneticfilm,whichwithfilm
surfacewasadheredontheCPW.TheRFpowerusedintheexperimentsis15dBm.
Toimprovethesignal-to-noiseratio(SNR),alock-indetectiontechniqueisemployed
throughthemodulationofsignals.Themodulationofadirectcurrent(DC)magnetic
fieldHisprovidedbyapairofsecondaryHelmholtzcoilspoweredbyanalternating
current(AC)sourcewith129.9Hz[seeFig.4(a)][16,30].Thedifferentialabsorption
signalismeasuredbysweepingthemagneticfieldwithafixedmicrowavefrequency.
TherepresentativeFMRspectrumofPt(5)/[Co(0.2)/Ni(0.3)]5/Pt(1)obtainedat9GHz
isshownintheinsetofFig.4(b).ThedifferentialFMRspectrumcanbewellfittedby
usingacombinationofsymmetricandantisymmetricLorentzianfunction,asfollows:
=s4∆−res
4−res2+∆22+∆2−4−res2
4−res2+∆22, (1)
whereVSandVArepresentthesymmetricandantisymmetricfactors,Histhe
externalmagneticfield,Hresistheresonancefield,andΔHisthelinewidthofFMR
corresponding3timesofthepeak-to-dipwidthintheFMRspectrum.The
relationshipbetweenthefrequencyfandtheresonancefieldHresofthetwoseriesof
Pt/[Co(0.2)/Ni(tNi)]andPt/[Co(0.3)/Ni(tNi)]samples[Figs.4(b)and(d)]canbewell
fittedbytheKittelequation[31]=γ
2πres+eff, (2)
whereγ
2π=2.8MHzOe−1isthegyromagneticratio,Heffiseffective
demagnetization[32]eff=K−4S.Therefore,themagneticanisotropyfieldHK
alsocanbedirectlydeterminedfromthedispersionrelationoffversusHresbyusinga
parameterMSobtainedbyVSM.Inaddition,wecanobtaintheintrinsicGilbert
dampingαbyfittingtheexperimentaldataoflinewidthΔHversusresonance
frequency[Figs.4(c)and(e)]withtheformula:∆=∆0+4πα
γ,hereΔH0isan
inhomogeneouslinewidthindependentofthefrequency,andthesecondtermisthe
intrinsiclinewidthlinearlyproportionaltothefrequency.Theinhomogeneous
linewidthofsamplesisderivedfromroughness,defectsandinhomogeneousPMA
andmagnetization[33].
Fig.4.(a)DifferentialFMRspectraexperimentalsetup.(b)Dependenceofthe
resonancefieldHresonthefrequencyfwiththeout-of-planefieldforthefilms
Pt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1).SolidlinesindicatetheKittelfittingcurves.Theinsetis
therepresentativeFMRspectrumobtainedat9GHz,whichcanbewellfittedbyEq.(1)(solidredline).(c)Thelinewidthversusfrequency(symbols)forthesamples
Pt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1).Thesolidlineisalinearfitting,whichcanextractthe
correspondingdampingconstantαbasedonEq.(2).(d)-(e)Sameas(b)-(c),forthe
filmsPt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1).
Figure5summarizesthedependenceofthedeterminedmaterialparameters:the
saturationmagnetizationMS,thecoercivityHC,theanisotropyfieldHK,the
inhomogeneouslinewidthΔH0andthemagneticdampingconstantαonNithickness
tNiforthestudiedtwoseriesofPt/[Co(0.2)/Ni(tNi)]andPt/[Co(0.3)/Ni(tNi)]samples.
ThedeterminedHKbythreeindependentmethodsshowsanoverallconsistent
behavior.TheHKbeginstoincreasewithincreasingtNi,andreachesthemaximumat
tNi~0.6nm,whereafterreducesagainwithcontinuingtoincreasetNi.Severalreasons
accountforthisphenomenon.First,themagneticanisotropyofthestudiedmultilayer
ismainlycontributedfromtheinterfacialmagneticanisotropyoftheCo/NiandPt/Co
interfaces[34].Second,theCo/Nimultilayers'interfacequalitydependshighlyonthe
Nilayer'sthickness.Inotherwords,toothinnickellayermaynotgetagoodCo/Ni
interfaceduetoinevitableelementsdiffusionduringsputteringdeposition.However,
theHKwilldropduetoreducingtheratiooftheinterfacialanisotropytothevolume
anisotropyenergyiftheNilayeristoothick.LikeHK,theHCshowsasimilartrend
withvaryingthicknessoftheNilayer.Aswewellknowthatthecoercivitydepends
onPMA,aswellasdefects-inducedpinningeffects.But,inourcase,theresultsshow
thatthecombinationofPMAandmagnetization-relevantdemagnetizationfield
dominatethecoercivity,whichcanbewellexplainedbytheBrownformula[35]:C=
2U
S−S,whereKU=(MS*HK)/2andNarethemagneticanisotropyconstantand
thedemagnetizationfactorofthefilm,respectively.
Figures5(d)and5(i)showtheinhomogeneouslinewidth(H0)ofFMRspectraas
afunctionoftNiforPt/[Co(0.2)/Ni(tNi)]andPt/[Co(0.3)/Ni(tNi)],respectively.Forthin
thicknessNisamples,islandstructuresaremostlikelyformed.Thisresultsina
broadeningoftheresonancelinewidthduetoadistributionofeffectiveinternal
anisotropyanddemagnetizationfields[37].OnecanseethattheminimumlinewidthoftwoserialsamplescorrespondstothemaximumPMAfieldHK,suggestingthe
inhomogeneousmagneticanisotropy-inducedlinearbroadeningistheminimumatthe
optimumPMAcondition[36].Althoughtheintrinsicdampingconstantisalmost
independentoftheNithicknessforthestudiedtwoserials,butthePt/[Co(0.2)/Ni(tNi)]
filmshavealowerdampingconstantα~0.04thanα~0.07ofPt/[Co(0.3)/Ni(tNi)].
Theobviousdifferenceindampingconstantbetweenthetwoserialmultilayer
systemsindicatesthattheformerhasbettermagneticdynamicproperties.
Fig.5.(a)-(e)DependenceofthesaturationmagnetizationMS(a),thecoercivityHC(b),
theanisotropyfieldHK(c),theinhomogeneouslinewidthΔH0(d)andthemagnetic
dampingconstantα(e)ontheNithicknesstNiinthefilms
Pt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1).(f)-(j)Sameas(a)-(e)forthesamples
Pt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1).MS,HC,andHKweredeterminedfromthepreviousmagnetizationloops,AHRloops,MOKEloops,andtheferromagnetresonance
spectra.ThelinewidthwasdeterminedbyfittingtheexperimentalFMRspectrumwith
aLorentzianfunctionbasedonEq.(1).Themagneticdampingconstantwasobtained
byalinearfittingofΔHvs.fcurvesbasedonEq.(2).
4.CONCLUSION
NithicknesseffectonthestaticmagneticpropertiesandmagneticdynamicsofPt(5
nm)/[Co(0.2nmand0.3nm)/Ni(tNinm)]5/Pt(1nm)multilayersdemonstratethatthe
twostudiedserialmultilayersystemsexhibittheoptimumPMAcoefficientKUwellas
thehighestcoercivityHCattheNithicknesstNi=0.6nm.TheMOKEimagesfurther
confirmthatthemaximumKUcorrespondstothemagneticdomainstructurewiththe
shortestlengthofdomainwallthroughminimizingthetotalenergy,whichconsistsof
magneticanisotropyenergy,exchangeenergy,anddemagnetizationenergy.
Furthermore,thefrequency-dependentFMRspectrashowthatthedampingconstant
remainsalmostconstantwiththedifferentNithicknessesforbothserials,butthe
Pt/[Co(0.2)/Ni(tNi)]multilayerserialhasalowerdampingconstantα~0.04than0.07
ofthePt/[Co(0.3)/Ni(tNi)]serial.Accordingtotheobtainedresults,wefindthatthe
optimumPMAcoefficientKU=3.3×106ergcm-3,thehighestcoercivityHC=250
Oe,andaswellasthelowestdampingconstantα=0.04canbeachievedat
Pt(5)/[Co(0.2)/Ni(0.6)]5/Pt(1).Ourresultsofoptimizingmagneticpropertiesofthe
Pt/[Co/Ni]multilayerbytuningtheratioofCo/Nilayersishelpfultofacilitateits
applicationsinvariousspintronicdevices.
Acknowledgement：ProjectissupportedbytheNationalNaturalScienceFoundationof
China(GrantNos.11774150,12074178,12004171,12074189and51971109),theAppliedBasicResearchProgramsofScienceandTechnologyCommissionFoundationofJiangsuProvince,
China(GrantNo.BK20170627),theNationalKeyR&DProgramofChina(GrantNo.
2018YFA0209002),theOpenResearchFundofJiangsuProvincialKeyLaboratoryfor
Nanotechnology,andtheScientificFoundationofNanjingUniversityofPostsand
Telecommunications(NUPTSF)(GrantNo.NY220164).
Correspondingauthors,Email:chenlina@njupt.edu.cn,rhliu@nju.edu.cn
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