Conduction electron spin-lattice relaxation time in the MgB2 superconductor
F. Simon,∗ F. Murányi,† T. Fehér, A. Jánossy
Budapest University of Technology and Economics, Institute of Physics
and Condensed Matter Research Group of the Hungarian Academy of
Sciences, H-1521 Budapest, PO BOX 91, Hungary
L. Forró
IPMC/SB Swiss Federal Institute of Technology (EPFL), CH-1015 Lausanne-EPFL, Switzerland
C. Petrovic,‡ S.L. Bud’ko, P.C. Canfield
Ames Laboratory, U.S. Department of Energy and Department of Physics
and Astronomy, Iowa State University, Ames, Iowa 50011, USA
(Dated: July 19, 2021)
The spin-lattice relaxation time, T1, of conduction electrons is measured as a function of tem-
perature and magnetic field in MgB2. The method is based on the detection of the z component
of the conduction electron magnetization under electron spin resonance conditions with amplitude
modulated microwave excitation. Measurement of T1 below Tc at 0.32 T allows to disentangle con-
tributions from the two Fermi surfaces of MgB2 as this field restores normal state on the Fermi
surface part with π symmetry only.
INTRODUCTION
The conduction electron spin-lattice relaxation time in
metals, T1, is the characteristic time for the return to
thermal equilibrium of a spin system driven out of equi-
librium by e.g. a microwave field at electron-spin reso-
nance (ESR) or a spin-polarized current. The applica-
bility of metals in “spintronics” devices in which infor-
mation is processed using electron spins [1] depends on a
sufficiently long spin life-time. In pure metals T1 is lim-
ited by the Elliott mechanism [2, 3], i.e. the scattering
of conduction electrons by the random spin-orbit poten-
tial of non-magnetic impurities or phonons. In supercon-
ductors, the Elliott mechanism becomes ineffective and
a long T1 is predicted well below Tc [3]. Here we report
the direct measurement of the spin-lattice relaxation time
of conduction electrons in MgB2 in the superconducting
state. The motivation to study the magnetic field and
temperature dependence of T1 is two-fold: i) to test the
predicted lengthening of T1 to temperatures well below
Tc, ii) to measure the contributions to T1 from different
Fermi surface sheets and to compare with the correspond-
ing momentum life-times, τ .
The lengthening of T1 has been observed in a re-
stricted temperature range below Tc in the fulleride
superconductor, K3C60 by measuring the conduction
electron-spin resonance (CESR) line-width, ∆H [4]. This
method assumes 1/T1 = 1/T2 = γe∆H , where γe/2π =
28.0 GHz/T is the electron gyromagnetic ratio, and 1/T2
is the spin-spin or transversal relaxation rate. It is
limited to cases where the homogeneous broadening of
the CESR line due to a finite spin lifetime outweighs
∆Hinhom, the line broadening from inhomogeneities of
the magnetic field. In a superconducting powder sam-
ple, the CESR line is inhomogeneously broadened below
the irreversibility line due to the distribution of vorte-
ces, which is temperature and magnetic field dependent.
This prevents the measurement of T1 from the line-width
and calls for a method to directly measure T1. Electron
spin echo techniques, which usually enable the measure-
ment of T1, are not available for the required nanosec-
ond time resolution range. The magnetic resonance tech-
nique, termed longitudinally detected (LOD) ESR [5, 6]
used in this work allows to measure T1’s as short as a
few ns. The method is based on the measurement of the
electron spin magnetization along the magnetic field, Mz,
using modulated microwave excitation. Mz recovers to
its equilibrium value with the T1 time-constant, thus the
method allows the direct measurement of T1 independent
of magnetic field inhomogeneities.
MgB2 has a high superconducting transition tempera-
ture of Tc = 39 K [7] and its unusual physical properties
[8, 9, 10, 11] are attributed [12, 13] to its disconnected,
weakly interacting Fermi surface (FS) parts. The Fermi
surface sheets derived from B-B bonds with π and σ char-
acters (π and σ FS) have smaller and higher electron-
phonon couplings and superconductor gaps, respectively,
and contribute roughly equally to the density of states
(DOS). The strange band structure leads to unique ther-
modynamic properties: magnetic fields of about 0.3-0.4
T restore the π FS well below Tc for all field orienta-
tions in polycrystalline samples but the material remains
superconducting to much higher fields. This is character-
ized by a small and nearly isotropic upper critical field,
c2 ∼ 0.3 − 0.4 T [10, 14] and a strongly anisotropic
one, Hσ
c2 = 2 − 16 T, [10, 15, 16] related to the π and σ
Fermi surface sheets, respectively. Our measurements at
low fields and low temperatures determine T1 from the π
FS alone, while high field and high temperature experi-
ments measure T1 averaged over the whole FS. We find
http://arxiv.org/abs/0704.0466v1
that spin relaxation in high purity MgB2 is temperature
independent in the high field normal state between 3 K
and 50 K, indicating that it arises from non magnetic im-
purities. Spin relaxation times for electrons on the π and
σ Fermi surface sheets are widely different but are not
proportional to the corresponding momentum relaxation
times.
EXPERIMENTAL
The same MgB2 samples were used as in a previous
study [17]. Thorough grinding, particle size selection
and mixing with SnO2, an ESR silent oxide, produced
a fine powder with well separated small metallic par-
ticles. The nearly symmetric appearance of the CESR
signal [18] proves that penetration of microwaves is ho-
mogeneous and that the particles are smaller than the
microwave penetration depth of ∼ 1µm. SQUID mag-
netometry showed that grinding and particle selection
do not affect the superconducting properties. The parti-
cles are not single crystals but rather aggregates of small
sized single crystals. Continuous wave (cw) and longitu-
dinally detected ESR experiments were performed in a
home-built apparatus [6] at 9.1 and 35.4 GHz microwave
frequencies, corresponding to 0.32 and 1.27 T resonance
magnetic fields. The 9.1 GHz apparatus is based on a
loop-gap resonator with a low quality factor (Q ∼ 200)
and the 35.4 GHz instrument does not employ a mi-
crowave cavity at all. The cw-ESR was detected using an
audio frequency magnetic field modulation. Line-widths
are determined by Lorentzian fits to the cw-ESR data.
For the LOD-ESR, the microwaves are amplitude modu-
lated with f = Ω/2π of typically 10 MHz and the result-
ing varying Mz component of the sample magnetization
is detected with a coil which is parallel to the external
magnetic field and is part of a resonant circuit that is
tuned to f and is matched to 50 Ohms. cw-ESR at 420
GHz (centered at 15.0 T) was performed at EPFL using a
quasi-optical microwave bridge with no resonant cavities.
RESULTS
Relaxation in the normal state
The low temperature behavior of the spin-lattice re-
laxation time in MgB2 in the normal state can be mea-
sured using cw-ESR from the homogeneous line-width,
∆Hhom, using 1/T1 = γe∆Hhom at high fields, H > Hc2
that suppresses superconductivity. The maximum upper
critical field is Hc2,max ∼ 16 T for particles with field
in the (a, b) crystallographic plane in the polycrystalline
sample at zero temperature [19]. We did not observe any
effects of superconductivity on the CESR, at 15 T it is
suppressed in the full sample above a temperature of a
0 10 20 30 40 50 60
T (K)
FIG. 1: CESR line-width of MgB2 as a function of tempera-
ture for the 15 T CESR measurement (�). Open circles (©)
show the homogeneous line-width (∆Hhom) after correcting
for the field dependent broadening as explained in the text.
Representative error bars are shown at the lowest tempera-
ture.
0 2 4 6 8 10 12 14 16
Magnetic field (T)
FIG. 2: ESR line-width of MgB2 as a function of magnetic
field measured at 40 K (�). Solid curve is a linear fit to the
data with parameters given in the text.
few K. Fig. 1 shows that the temperature dependence of
the CESR line-width at 15 T is small below 40 K.
The CESR line-width is magnetic field dependent as
shown in Fig. 2 at 40 K: it is linear as function
of magnetic field with ∆H = ∆H0 + b ∗ H , where
∆H0 = 0.90(1) mT is the residual line-width and b =
0.057(1) mT/T. The residual homogeneous line-width
corresponds to T1 = 6.3 ns at 40 K. The linear relation
can be used to correct the 15 T CESR line-width data
to obtain the homogeneous contribution, ∆HHom(T ) =
∆H(15 T, T )− 15T · b as the magnetic field dependence
is expected to be temperature independent. We show
the homogeneous line-width in Fig. 1. We find that it is
0 10 20 30 40 50 60
32 34 36 38
T (K)
T (K)
FIG. 3: Inhomogeneous CESR line broadening in MgB2 below
Tc at 0.32 T. Full and open symbols show the CESR line-width
for up and down magnetic field sweeps, respectively. Inset
shows the data near Tc. Note the line narrowing between Tc
and Tirr and the field sweep direction dependent line-widths
below the irreversibility temperature.
temperature independent within experimental precision
between 3 and 50 K. This means that the spin-lattice re-
laxation time flattens to a residual value that is given by
non-magnetic impurities.
Relaxation in the superconducting state
In type II superconductors, CESR arises from thermal
excitations and from normal state vortex cores. The in-
homogeneity of the magnetic field in the vortex lattice
or glass states does not broaden the CESR line. The lo-
cal magnetic field inhomogeneity is averaged since within
the spin life-time itinerant electrons travel long distances
compared to the inter-vortex distance [4]. This is in con-
trast to the NMR case where the line-shape is affected:
the nuclei are fixed to the crystal and nuclei inside and
outside the vortex cores experience different local fields
[20]. In other words, a superconducting a single crystal
sample would display a narrow conduction electron ESR
line if there were no irreversible effects. However, the
CESR line is inhomogeneously broadened below the irre-
versibility line for a superconducting powder sample: the
vortex distribution depends on a number of factors such
as the thermal and magnetic field history, grain size and,
for an anisotropic superconductor such as MgB2, on the
crystal orientation with respect to the magnetic field also.
The resulting inhomogeneous broadening of the CESR
line gives 1/γ∆Hinhom = T
2 ≪ T1,2, and T1 cannot be
measured from the line-width. In Fig. 3 we show this
effect: above Tc MgB2 has a relaxationally broadened
line-width of ∆H = 0.9 mT. Between Tc and the irre-
versibility temperature at the given field, Tirr, the CESR
remains homogeneous and narrows with the lengthening
of T1. However, below Tirr it broadens abruptly and the
line-width depends on the direction of the magnetic field
sweep: for up sweep it is broader than for down sweeps
due to the irreversibility of vortex insertion and removal.
To enable a direct measurement of the T1 spin lattice
relaxation time, one has to resort to time resolved experi-
ments. Conventional spin-echo ESR methods are limited
to T1’s larger than a few 100 ns. To measure T1’s of
a few nanoseconds, the so-called longitudinally detected
ESR was invented in the 1960’s by Hervé and Pescia [21]
and improved by several groups [22, 23]. The method
is based on the deep amplitude modulation of the mi-
crowave excitation with an angular frequency, Ω ∼ 1/T1.
When the sample is irradiated with the amplitude mod-
ulated microwaves at ESR resonance, the component of
the magnetization along the static magnetic field, Mz,
decays from the equilibrium value, M0, with a time con-
stant T1. Mz relaxes back to M0 with a T1 relaxation
time when the microwaves are turned off. The oscillat-
ing Mz is detected using a coil which is part of a resonant
rf circuit. The phase sensitive detection of the oscillat-
ing voltage using lock-in detection allows the measure-
ment of T1 using ΩT1 = v/u [5, 21], where u and v are
the amplitudes of the in- and out-of-phase components
of the oscillating magnetization after corrections for in-
strument related phase shifts. The principal limitation
of the LOD-ESR technique is its 3-4 orders of magnitude
lower sensitivity compared to conventional cw-ESR. The
LOD-ESR method and the experimental apparatus are
detailed in Refs. [5, 6].
To prove that the LOD-ESR signal of the itinerant
electrons is detected in the superconducting phase, we
compare in Fig. 4 the LOD-ESR signal with that mea-
sured with conventional continuous-wave CESR (referred
to as CESR in the following) of MgB2 in the normal and
superconducting states. The CESR signal is the deriva-
tive of the absorption due to magnetic field modulation
used for lock-in detection. This signal was previously
identified as the ESR of conduction electrons in MgB2 in
the superconducting and normal states [15, 17, 24] and
its characteristics have been discussed in detail [15, 17].
Above Tc at 40 K, the CESR line is relaxationally broad-
ened. Below Tc, it is inhomogeneously broadened and is
diamagnetically shifted, i.e. to higher resonance fields.
The irreversible effects also contribute to a non-linear
baseline known as the non-resonant microwave absorp-
tion [25]. The intensity of the CESR signal decreases
below Tc as we discussed previously [17], due to the van-
ishing of normal state electrons.
The LOD-ESR signal shows the same characteristics
as the CESR below Tc: it is broadened, shifted to higher
fields and its intensity decreases. The values for the
temperature dependent diamagnetic shifts and broad-
ening and the relative intensity change agree for the
two kinds of measurements within experimental precision
(not shown). This unambiguously proves that the LOD-
0.30 0.32 0.340.30 0.32 0.34
ESR LOD-ESR
Magnetic field (T)
15 Kb)
40 Ka)
FIG. 4: ESR (a-b) and LOD-ESR (c-d) spectra of MgB2 at
9.1 GHz (0.32 T). a) and c) at 40 K in the normal state,
and b) and d) in the superconducting state at 15 K. Solid
and dashed curves are the in- and out-of-phase LOD signals,
respectively and are offset for clarity. Vertical solid lines in-
dicate the resonance field above Tc. Note the diamagnetic
shift and broadening for for both kinds of spectra below Tc.
Also note the rotated phase of the in-phase and out-of phase
channels upon cooling.
ESR signal originates from the conduction electrons.
The change of the relaxation time T1 is visible in the
LOD-ESR spectra in Fig. 4 as a change in the relative
intensities of the in- and out-of-phase signals. At 40 K
v/u = 0.47 and at 15 K v/u = 0.95, which together with
Ω/2π = 11.4 MHz gives 6.3 and 13.3 ns relaxation times,
respectively. In Fig. 5, we show the T1 data inferred from
the LOD-ESR spectra at 0.32 and 1.27 T as a function
of the reduced temperature T/Tc.
DISCUSSION
The observed lengthening of T1 below Tc ( Fig. 5) is
expected from theory for non-magnetic scattering cen-
ters and low magnetic fields where the susceptibility is
dominated by excitations over the superconducting gap.
On the other hand, the field independence between 0.32
and 1.27 T of T1 below Tc is surprising. The lengthen-
ing of T1 below Tc in zero magnetic field for an isotropic,
type I superconductor was calculated in the framework
of weak-coupled BCS theory by Yafet [3]. He concluded
that T1 lengthens as a result of the freezing of normal
state excitations. However, no theory exists for a type II
superconductor in finite fields with Hc2 anistotropy such
as MgB2, thus here the T1 data are analyzed phenomeno-
0.0 0.5 1.0 1.5 2.0
FIG. 5: Spin-lattice relaxation time as a function of the re-
duced temperature in MgB2 at 0.32 (�) and 1.27 T (©) mag-
netic fields. Representative error bars are shown for some of
the data. Dashed curve shows T1 corresponding to ∆HHom
in the 15 T measurement such as in Fig. 1 with the reduced
temperature normalized to 39 K.
logically in the framework of the two-band/gap model of
MgB2.
In the following, we deduce the residual (low tem-
perature), impurity related spin scattering contributions
of the σ and π Fermi surface sheets. The DOS is
distributed almost equally on the FS sheets of MgB2:
Nπ/(Nπ + Nσ) = 0.56 [13], where Nπ and Nσ are the
DOS of the two types of FS sheets. A magnetic field
of ∼ 0.3 − 0.4 T closes the gap on the π FS sheets but
leaves the gap on the σ sheet almost intact. [10, 17].
This suggests that well below Tc, our experiment at 0.32
T measures exclusively the relaxation of electrons on the
fully closed π FS sheets. Since T1 at 0.32 T increases
slowly with temperature between 10 and 20 K, we ex-
trapolate T1π ≈ T1(10 K, 0.32 T) = 20(2) ns for the π
In order to separate the contribution of the σ FS to
the relaxation rate in the normal state, 1/T1n, we assume
that inter-band relaxation is negligible and 1/T1n is equal
to the average of the spin-lattice relaxation rates on the
two FS’s weighted by the corresponding DOS:
Nπ/T1π +Nσ/T1σ
Nπ +Nσ
Here T1σ is the spin-lattice relaxation time on the σ FS.
The 15 T measurement shows that 1/T1n changes little
with temperature between 3 K and 40 K. Thus we find
T1σ = 3.4(5) ns for the contribution of the σ FS sheets
using T1n = T1(Tc) = 6.3 ns, T1π = 20(2) ns and Eq. 1.
For normal metals with a simple Fermi surface, the so-
called Elliott relation [2, 3, 26, 27] holds, which states
that for a given type of disorder (e.g. phonons or dislo-
cations) T1 is proportional to the momentum relaxation
time, τ . The proportionality constant depends on the
spin orbit splitting of the conduction electron bands and
has been estimated in a number of metals from the shift
of the CESR from the free electron value. Metals with
complicated Fermi surfaces i.e. with great variations of
the electron-phonon coupling on the different FS parts
are known to deviate from the Elliott relation [28] and
calculation of T1 requires to take into account the de-
tails of the band-structure [29, 30, 31]. Examples include
polyvalent elemental metals such as Mg or Al. Clearly, a
calculation of T1 is required for MgB2, which takes into
account its band structure peculiarities. Comparison of
spin scattering and momentum scattering times of the
two types of Fermi surfaces is instructive. The relative
values of τ for the two FS parts, τπ and τσ, and for in-
terband scattering were estimated by Mazin et al. [32].
A very small impurity interband scattering and τπ < τσ,
i.e. a larger π intraband scattering relative to σ intra-
band scattering was required to explain the rather small
depression of Tc in materials with widely different con-
ductivities. De Haas-van Alphen [33] and magnetore-
sistance [34] measurements of high purity samples yield
τπ < τσ also. Such a behaviour could rise fromMg vacan-
cies, which perturb more electrons of the π band relative
to the σ band. However, our spin scattering data do not
support this. In contrast to momentum scattering, spin
scattering is stronger on the σ FS: T1π : T1σ = 6 : 1
in high purity samples and low temperatures. The rela-
tive values of T1 and τ for the two FS do not necessarily
follow the same trend, spin relaxation times at low tem-
peratures depend on spin orbit relaxation on impurities
while momentum relaxation is due to potential scatter-
ing. However, a defect center such as a Mg vacancy with a
strong modification of the electron-phonon coupling and
an atomic number strongly differing from that of the reg-
ular atoms constituting the compound would greatly af-
fect T1 compared to τ . In the two gap model Mg defects
are expected to shorten T1π more than T1σ and thus are
unlikely to be the dominant scatterers.
A final note concerns the validity of the above analysis
of T1’s in the framework of the two-band/gapmodel. The
field independence of the lowest temperature T1 for 0.32
and 1.27 T is unexpected within this model. The spin
susceptibility increases strongly between these fields and
more normal states are restored at 1.27 T than expected
from the closing of the gap on the π FS sheets alone [17].
Based on this, one would expect to observe additional
spin scattering from the restored σ FS parts, which is
clearly not the case. This also indicates that a theoret-
ical study, which takes into account the peculiarities of
MgB2 is required to explain the anomalous spin-lattice
relaxation times.
In conclusion, we presented the measurement of the
spin-lattice relaxation time, T1, of conduction electrons
as a function of temperature and magnetic field in the
MgB2 superconductor. We use a novel method based on
the detection of the z component of the conduction elec-
tron magnetization during electron spin resonance con-
ditions with amplitude modulated microwave excitation.
Lengthening of T1 below Tc is observed irrespective of the
significant CESR line broadening due to irreversible dia-
magnetism in the polycrystalline sample. The field inde-
pendence of T1 for 0.32 T and 1.27 T allows to measure
the separate contributions to T1 from the two distinct
types of the Fermi surface.
ACKNOWLEDGEMENTS
The authors are grateful to Richárd Gaál for the de-
velopment of the ESR instrument at the EPFL. F.S.
and F.M. acknowledge the Zoltán Magyary postdoc-
toral programme, the Bolyai fellowship of the Hungarian
Academy of Sciences and the Alexander von Humboldt
Foundation for support, respectively. Work supported
by the Hungarian State Grants (OTKA) No. TS049881,
F61733, PF63954 and NK60984 and by the Swiss NSF
and its NCCR ”MaNEP”. Ames Laboratory is operated
for the U.S. Department of Energy by Iowa State Uni-
versity under Contract No. W-7405-Eng-82.
∗ Corresponding author: simon@esr.phy.bme.hu
† Present address: Leibniz Institute for Solid State and
Materials Research Dresden, PF 270116 D-01171 Dres-
den, Germany
‡ Present address: Condensed Matter Physics and Ma-
terials Science Department, Brookhaven National Labo-
ratory, Upton, New York 11973-5000, USA
[1] I. Žutić, J. Fabian, and S. D. Sarma, Rev. Mod. Phys.
76, 323 (2004).
[2] R. J. Elliott, Physical Review 96, 266279 (1954).
[3] Y. Yafet, Phys. Lett. A 98, 287 (1983).
[4] N. M. Nemes, J. E. Fischer, G. Baumgartner, L. Forró,
T. Fehér, G. Oszlányi, F. Simon, and A. Jánossy, Phys.
Rev. B 61, 7118 (2000).
[5] F. Murányi, F. Simon, F. Fülöp, and A. Jánossy, J.
Magn. Res. 167, 221 (2004).
[6] F. Simon and F. Murányi, J. Magn. Res. 173, 288 (2005).
[7] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani,
and J. Akimitsu, Nature 410, 63 (2001).
[8] F. Bouquet, R. A. Fisher, N. E. Phillips, D. G. Hinks,
and J. D. Jorgensen, Physical Review Letters 87, 047001
(2001).
[9] P. Szabó, P. Samuely, J. Kacmarćık, T. Klein, J. Marcus,
D. Fruchart, S. Miraglia, C. Marcenat, and A. G. M.
Jansen, Phys. Rev. Lett. 87, 137005 (2001).
[10] F. Bouquet, Y. Wang, I. Sheikin, T. Plackowski,
A. Junod, S. Lee, and S. Tajima, Physical Review Letters
89, 257001 (2002).
[11] S. Tsuda, T. Yokoya, Y. Takano, H. Kito, A. Matsushita,
F. Yin, J. Itoh, H. Harima, and S. Shin, Physical Review
Letters 91, 127001 (2003).
[12] J. Kortus, I. I. Mazin, K. D. Belashchenko, V. P.
Antropov, and L. L. Boyer, Phys. Rev. Lett. 86, 4656
(2001).
[13] H. J. Choi, D. Roundy, H. Sun, M. L. Cohen, and S. G.
Louie, Nature 418, 758 (2002).
[14] M. R. Eskildsen, M. Kugler, S. Tanaka, J. Jun, S. M.
Kazakov, J. Karpinski, and R. Fischer, Phys. Rev. Lett.
89, 187003 (2002).
[15] F. Simon, A. Jánossy, T. Fehér, F. Murányi, S. Garaj,
L. Forró, C. Petrovic, S. L. Bud’ko, G. Lapertot, V. Ko-
gan, et al., Phys. Rev. Lett. 87, 047002 (2001).
[16] M. Angst, R. Puzniak, A. Wisniewski, J. Jun, S. M.
Kazakov, J. Karpinski, J. Roos, and H. Keller, Phys.
Rev. Lett. 88, 167004 (2002).
[17] F. Simon, A. Jánossy, T. Fehér, F. Murányi, S. Garaj,
L. Forró, C. Petrovic, S. L. Bud’ko, R. A. Ribeiro, and
P. C. Canfield, Phys. Rev. B 72, 012511 (2005).
[18] F. J. Dyson, Physical Review 98, 349 (1955).
[19] D. K. Finnemore, J. E. Ostenson, S. L. Bud’ko, G. Laper-
tot, and P. C. Canfield, Phys. Rev. Lett. 86, 2420 (2001).
[20] P. Pincus, Phys. Lett. 13, 21 (1964).
[21] J. Hervé and J. Pescia, C. R. Acad. Sci.(Paris) 251, 665
(1960).
[22] V. A. Atsarkin, V. V. Demidov, and G. A. Vasneva, Phys-
ical Review B 52, 1290 (1995).
[23] J. Granwehr, J. Forrer, and A. Schweiger, J. Magn. Res.
151, 78 (2001).
[24] R. R. Urbano, P. G. Pagliuso, C. Rettori, Y. Kopelevich,
N. O. Moreno, and J. L. Sarrao, Phys. Rev. Lett. 89,
087602 (2002).
[25] K. W. Blazey, K. A. Muller, J. G. Bednorz, W. Berlinger,
G. Amoretti, E. Buluggiu, A. Vera, and F. C. Matacotta,
Phys. Rev. B 36, 7241 (1987).
[26] Y. Yafet, in Solid State Physics, edited by F. Seitz and
D. Turnbull (1963), vol. 14, p. 1.
[27] F. Beuneu and P. Monod, Phys. Rev. B 18, 2422 (1977).
[28] P. Monod and F. Beuneu, Phys. Rev. B 19, 911 (1979).
[29] R. H. Silsbee and F. Beuneu, Phys. Rev. B 27, 2682
(1983).
[30] J. Fabian and S. Das Sarma, Phys. Rev. Lett. 81, 5624
(1998).
[31] J. Fabian and S. Das Sarma, Phys. Rev. Lett. 83, 1211
(1999).
[32] I. I. Mazin, O. K. Andersen, O. Jepsen, O. V. Dol-
gov, J. Kortus, A. A. Golubov, A. B. Kuzmenko, and
D. van der Marel, Phys. Rev. Lett. 89, 107002 (2002).
[33] E. A. Yelland, J. C. Cooper, A. Carrington, N. E. Hussey,
P. J. Meeson, S. Lee, A. Yamammoto, and S. Tajima,
Phys. Rev. Lett. 88, 217002 (2002).
[34] I. Pallecchi, V. Ferrando, E. G. D’Agliano, D. Marre,
M. Monni, M. Putti, C. Tarantini, F. Gatti, H. U. Ae-
bersold, E. Lehmann, et al., Phys. Rev. B 72, 184512
(2005).

  The spin-lattice relaxation time, $T_{1}$, of conduction electrons is
measured as a function of temperature and magnetic field in MgB$_2$. The method
is based on the detection of the $z$ component of the conduction electron
magnetization under electron spin resonance conditions with amplitude modulated
microwave excitation. Measurement of $T_{1}$ below $T_c$ at 0.32 T allows to
disentangle contributions from the two Fermi surfaces of MgB$_{2}$ as this
field restores normal state on the Fermi surface part with $\pi$ symmetry only.

Conduction electron spin-lattice relaxation time in the MgB2 superconductor
F. Simon,∗ F. Murányi,† T. Fehér, A. Jánossy
Budapest University of Technology and Economics, Institute of Physics
and Condensed Matter Research Group of the Hungarian Academy of
Sciences, H-1521 Budapest, PO BOX 91, Hungary
L. Forró
IPMC/SB Swiss Federal Institute of Technology (EPFL), CH-1015 Lausanne-EPFL, Switzerland
C. Petrovic,‡ S.L. Bud’ko, P.C. Canfield
Ames Laboratory, U.S. Department of Energy and Department of Physics
and Astronomy, Iowa State University, Ames, Iowa 50011, USA
(Dated: July 19, 2021)
The spin-lattice relaxation time, T1, of conduction electrons is measured as a function of tem-
perature and magnetic field in MgB2. The method is based on the detection of the z component
of the conduction electron magnetization under electron spin resonance conditions with amplitude
modulated microwave excitation. Measurement of T1 below Tc at 0.32 T allows to disentangle con-
tributions from the two Fermi surfaces of MgB2 as this field restores normal state on the Fermi
surface part with π symmetry only.
INTRODUCTION
The conduction electron spin-lattice relaxation time in
metals, T1, is the characteristic time for the return to
thermal equilibrium of a spin system driven out of equi-
librium by e.g. a microwave field at electron-spin reso-
nance (ESR) or a spin-polarized current. The applica-
bility of metals in “spintronics” devices in which infor-
mation is processed using electron spins [1] depends on a
sufficiently long spin life-time. In pure metals T1 is lim-
ited by the Elliott mechanism [2, 3], i.e. the scattering
of conduction electrons by the random spin-orbit poten-
tial of non-magnetic impurities or phonons. In supercon-
ductors, the Elliott mechanism becomes ineffective and
a long T1 is predicted well below Tc [3]. Here we report
the direct measurement of the spin-lattice relaxation time
of conduction electrons in MgB2 in the superconducting
state. The motivation to study the magnetic field and
temperature dependence of T1 is two-fold: i) to test the
predicted lengthening of T1 to temperatures well below
Tc, ii) to measure the contributions to T1 from different
Fermi surface sheets and to compare with the correspond-
ing momentum life-times, τ .
The lengthening of T1 has been observed in a re-
stricted temperature range below Tc in the fulleride
superconductor, K3C60 by measuring the conduction
electron-spin resonance (CESR) line-width, ∆H [4]. This
method assumes 1/T1 = 1/T2 = γe∆H , where γe/2π =
28.0 GHz/T is the electron gyromagnetic ratio, and 1/T2
is the spin-spin or transversal relaxation rate. It is
limited to cases where the homogeneous broadening of
the CESR line due to a finite spin lifetime outweighs
∆Hinhom, the line broadening from inhomogeneities of
the magnetic field. In a superconducting powder sam-
ple, the CESR line is inhomogeneously broadened below
the irreversibility line due to the distribution of vorte-
ces, which is temperature and magnetic field dependent.
This prevents the measurement of T1 from the line-width
and calls for a method to directly measure T1. Electron
spin echo techniques, which usually enable the measure-
ment of T1, are not available for the required nanosec-
ond time resolution range. The magnetic resonance tech-
nique, termed longitudinally detected (LOD) ESR [5, 6]
used in this work allows to measure T1’s as short as a
few ns. The method is based on the measurement of the
electron spin magnetization along the magnetic field, Mz,
using modulated microwave excitation. Mz recovers to
its equilibrium value with the T1 time-constant, thus the
method allows the direct measurement of T1 independent
of magnetic field inhomogeneities.
MgB2 has a high superconducting transition tempera-
ture of Tc = 39 K [7] and its unusual physical properties
[8, 9, 10, 11] are attributed [12, 13] to its disconnected,
weakly interacting Fermi surface (FS) parts. The Fermi
surface sheets derived from B-B bonds with π and σ char-
acters (π and σ FS) have smaller and higher electron-
phonon couplings and superconductor gaps, respectively,
and contribute roughly equally to the density of states
(DOS). The strange band structure leads to unique ther-
modynamic properties: magnetic fields of about 0.3-0.4
T restore the π FS well below Tc for all field orienta-
tions in polycrystalline samples but the material remains
superconducting to much higher fields. This is character-
ized by a small and nearly isotropic upper critical field,
c2 ∼ 0.3 − 0.4 T [10, 14] and a strongly anisotropic
one, Hσ
c2 = 2 − 16 T, [10, 15, 16] related to the π and σ
Fermi surface sheets, respectively. Our measurements at
low fields and low temperatures determine T1 from the π
FS alone, while high field and high temperature experi-
ments measure T1 averaged over the whole FS. We find
http://arxiv.org/abs/0704.0466v1
that spin relaxation in high purity MgB2 is temperature
independent in the high field normal state between 3 K
and 50 K, indicating that it arises from non magnetic im-
purities. Spin relaxation times for electrons on the π and
σ Fermi surface sheets are widely different but are not
proportional to the corresponding momentum relaxation
times.
EXPERIMENTAL
The same MgB2 samples were used as in a previous
study [17]. Thorough grinding, particle size selection
and mixing with SnO2, an ESR silent oxide, produced
a fine powder with well separated small metallic par-
ticles. The nearly symmetric appearance of the CESR
signal [18] proves that penetration of microwaves is ho-
mogeneous and that the particles are smaller than the
microwave penetration depth of ∼ 1µm. SQUID mag-
netometry showed that grinding and particle selection
do not affect the superconducting properties. The parti-
cles are not single crystals but rather aggregates of small
sized single crystals. Continuous wave (cw) and longitu-
dinally detected ESR experiments were performed in a
home-built apparatus [6] at 9.1 and 35.4 GHz microwave
frequencies, corresponding to 0.32 and 1.27 T resonance
magnetic fields. The 9.1 GHz apparatus is based on a
loop-gap resonator with a low quality factor (Q ∼ 200)
and the 35.4 GHz instrument does not employ a mi-
crowave cavity at all. The cw-ESR was detected using an
audio frequency magnetic field modulation. Line-widths
are determined by Lorentzian fits to the cw-ESR data.
For the LOD-ESR, the microwaves are amplitude modu-
lated with f = Ω/2π of typically 10 MHz and the result-
ing varying Mz component of the sample magnetization
is detected with a coil which is parallel to the external
magnetic field and is part of a resonant circuit that is
tuned to f and is matched to 50 Ohms. cw-ESR at 420
GHz (centered at 15.0 T) was performed at EPFL using a
quasi-optical microwave bridge with no resonant cavities.
RESULTS
Relaxation in the normal state
The low temperature behavior of the spin-lattice re-
laxation time in MgB2 in the normal state can be mea-
sured using cw-ESR from the homogeneous line-width,
∆Hhom, using 1/T1 = γe∆Hhom at high fields, H > Hc2
that suppresses superconductivity. The maximum upper
critical field is Hc2,max ∼ 16 T for particles with field
in the (a, b) crystallographic plane in the polycrystalline
sample at zero temperature [19]. We did not observe any
effects of superconductivity on the CESR, at 15 T it is
suppressed in the full sample above a temperature of a
0 10 20 30 40 50 60
T (K)
FIG. 1: CESR line-width of MgB2 as a function of tempera-
ture for the 15 T CESR measurement (�). Open circles (©)
show the homogeneous line-width (∆Hhom) after correcting
for the field dependent broadening as explained in the text.
Representative error bars are shown at the lowest tempera-
ture.
0 2 4 6 8 10 12 14 16
Magnetic field (T)
FIG. 2: ESR line-width of MgB2 as a function of magnetic
field measured at 40 K (�). Solid curve is a linear fit to the
data with parameters given in the text.
few K. Fig. 1 shows that the temperature dependence of
the CESR line-width at 15 T is small below 40 K.
The CESR line-width is magnetic field dependent as
shown in Fig. 2 at 40 K: it is linear as function
of magnetic field with ∆H = ∆H0 + b ∗ H , where
∆H0 = 0.90(1) mT is the residual line-width and b =
0.057(1) mT/T. The residual homogeneous line-width
corresponds to T1 = 6.3 ns at 40 K. The linear relation
can be used to correct the 15 T CESR line-width data
to obtain the homogeneous contribution, ∆HHom(T ) =
∆H(15 T, T )− 15T · b as the magnetic field dependence
is expected to be temperature independent. We show
the homogeneous line-width in Fig. 1. We find that it is
0 10 20 30 40 50 60
32 34 36 38
T (K)
T (K)
FIG. 3: Inhomogeneous CESR line broadening in MgB2 below
Tc at 0.32 T. Full and open symbols show the CESR line-width
for up and down magnetic field sweeps, respectively. Inset
shows the data near Tc. Note the line narrowing between Tc
and Tirr and the field sweep direction dependent line-widths
below the irreversibility temperature.
temperature independent within experimental precision
between 3 and 50 K. This means that the spin-lattice re-
laxation time flattens to a residual value that is given by
non-magnetic impurities.
Relaxation in the superconducting state
In type II superconductors, CESR arises from thermal
excitations and from normal state vortex cores. The in-
homogeneity of the magnetic field in the vortex lattice
or glass states does not broaden the CESR line. The lo-
cal magnetic field inhomogeneity is averaged since within
the spin life-time itinerant electrons travel long distances
compared to the inter-vortex distance [4]. This is in con-
trast to the NMR case where the line-shape is affected:
the nuclei are fixed to the crystal and nuclei inside and
outside the vortex cores experience different local fields
[20]. In other words, a superconducting a single crystal
sample would display a narrow conduction electron ESR
line if there were no irreversible effects. However, the
CESR line is inhomogeneously broadened below the irre-
versibility line for a superconducting powder sample: the
vortex distribution depends on a number of factors such
as the thermal and magnetic field history, grain size and,
for an anisotropic superconductor such as MgB2, on the
crystal orientation with respect to the magnetic field also.
The resulting inhomogeneous broadening of the CESR
line gives 1/γ∆Hinhom = T
2 ≪ T1,2, and T1 cannot be
measured from the line-width. In Fig. 3 we show this
effect: above Tc MgB2 has a relaxationally broadened
line-width of ∆H = 0.9 mT. Between Tc and the irre-
versibility temperature at the given field, Tirr, the CESR
remains homogeneous and narrows with the lengthening
of T1. However, below Tirr it broadens abruptly and the
line-width depends on the direction of the magnetic field
sweep: for up sweep it is broader than for down sweeps
due to the irreversibility of vortex insertion and removal.
To enable a direct measurement of the T1 spin lattice
relaxation time, one has to resort to time resolved experi-
ments. Conventional spin-echo ESR methods are limited
to T1’s larger than a few 100 ns. To measure T1’s of
a few nanoseconds, the so-called longitudinally detected
ESR was invented in the 1960’s by Hervé and Pescia [21]
and improved by several groups [22, 23]. The method
is based on the deep amplitude modulation of the mi-
crowave excitation with an angular frequency, Ω ∼ 1/T1.
When the sample is irradiated with the amplitude mod-
ulated microwaves at ESR resonance, the component of
the magnetization along the static magnetic field, Mz,
decays from the equilibrium value, M0, with a time con-
stant T1. Mz relaxes back to M0 with a T1 relaxation
time when the microwaves are turned off. The oscillat-
ing Mz is detected using a coil which is part of a resonant
rf circuit. The phase sensitive detection of the oscillat-
ing voltage using lock-in detection allows the measure-
ment of T1 using ΩT1 = v/u [5, 21], where u and v are
the amplitudes of the in- and out-of-phase components
of the oscillating magnetization after corrections for in-
strument related phase shifts. The principal limitation
of the LOD-ESR technique is its 3-4 orders of magnitude
lower sensitivity compared to conventional cw-ESR. The
LOD-ESR method and the experimental apparatus are
detailed in Refs. [5, 6].
To prove that the LOD-ESR signal of the itinerant
electrons is detected in the superconducting phase, we
compare in Fig. 4 the LOD-ESR signal with that mea-
sured with conventional continuous-wave CESR (referred
to as CESR in the following) of MgB2 in the normal and
superconducting states. The CESR signal is the deriva-
tive of the absorption due to magnetic field modulation
used for lock-in detection. This signal was previously
identified as the ESR of conduction electrons in MgB2 in
the superconducting and normal states [15, 17, 24] and
its characteristics have been discussed in detail [15, 17].
Above Tc at 40 K, the CESR line is relaxationally broad-
ened. Below Tc, it is inhomogeneously broadened and is
diamagnetically shifted, i.e. to higher resonance fields.
The irreversible effects also contribute to a non-linear
baseline known as the non-resonant microwave absorp-
tion [25]. The intensity of the CESR signal decreases
below Tc as we discussed previously [17], due to the van-
ishing of normal state electrons.
The LOD-ESR signal shows the same characteristics
as the CESR below Tc: it is broadened, shifted to higher
fields and its intensity decreases. The values for the
temperature dependent diamagnetic shifts and broad-
ening and the relative intensity change agree for the
two kinds of measurements within experimental precision
(not shown). This unambiguously proves that the LOD-
0.30 0.32 0.340.30 0.32 0.34
ESR LOD-ESR
Magnetic field (T)
15 Kb)
40 Ka)
FIG. 4: ESR (a-b) and LOD-ESR (c-d) spectra of MgB2 at
9.1 GHz (0.32 T). a) and c) at 40 K in the normal state,
and b) and d) in the superconducting state at 15 K. Solid
and dashed curves are the in- and out-of-phase LOD signals,
respectively and are offset for clarity. Vertical solid lines in-
dicate the resonance field above Tc. Note the diamagnetic
shift and broadening for for both kinds of spectra below Tc.
Also note the rotated phase of the in-phase and out-of phase
channels upon cooling.
ESR signal originates from the conduction electrons.
The change of the relaxation time T1 is visible in the
LOD-ESR spectra in Fig. 4 as a change in the relative
intensities of the in- and out-of-phase signals. At 40 K
v/u = 0.47 and at 15 K v/u = 0.95, which together with
Ω/2π = 11.4 MHz gives 6.3 and 13.3 ns relaxation times,
respectively. In Fig. 5, we show the T1 data inferred from
the LOD-ESR spectra at 0.32 and 1.27 T as a function
of the reduced temperature T/Tc.
DISCUSSION
The observed lengthening of T1 below Tc ( Fig. 5) is
expected from theory for non-magnetic scattering cen-
ters and low magnetic fields where the susceptibility is
dominated by excitations over the superconducting gap.
On the other hand, the field independence between 0.32
and 1.27 T of T1 below Tc is surprising. The lengthen-
ing of T1 below Tc in zero magnetic field for an isotropic,
type I superconductor was calculated in the framework
of weak-coupled BCS theory by Yafet [3]. He concluded
that T1 lengthens as a result of the freezing of normal
state excitations. However, no theory exists for a type II
superconductor in finite fields with Hc2 anistotropy such
as MgB2, thus here the T1 data are analyzed phenomeno-
0.0 0.5 1.0 1.5 2.0
FIG. 5: Spin-lattice relaxation time as a function of the re-
duced temperature in MgB2 at 0.32 (�) and 1.27 T (©) mag-
netic fields. Representative error bars are shown for some of
the data. Dashed curve shows T1 corresponding to ∆HHom
in the 15 T measurement such as in Fig. 1 with the reduced
temperature normalized to 39 K.
logically in the framework of the two-band/gap model of
MgB2.
In the following, we deduce the residual (low tem-
perature), impurity related spin scattering contributions
of the σ and π Fermi surface sheets. The DOS is
distributed almost equally on the FS sheets of MgB2:
Nπ/(Nπ + Nσ) = 0.56 [13], where Nπ and Nσ are the
DOS of the two types of FS sheets. A magnetic field
of ∼ 0.3 − 0.4 T closes the gap on the π FS sheets but
leaves the gap on the σ sheet almost intact. [10, 17].
This suggests that well below Tc, our experiment at 0.32
T measures exclusively the relaxation of electrons on the
fully closed π FS sheets. Since T1 at 0.32 T increases
slowly with temperature between 10 and 20 K, we ex-
trapolate T1π ≈ T1(10 K, 0.32 T) = 20(2) ns for the π
In order to separate the contribution of the σ FS to
the relaxation rate in the normal state, 1/T1n, we assume
that inter-band relaxation is negligible and 1/T1n is equal
to the average of the spin-lattice relaxation rates on the
two FS’s weighted by the corresponding DOS:
Nπ/T1π +Nσ/T1σ
Nπ +Nσ
Here T1σ is the spin-lattice relaxation time on the σ FS.
The 15 T measurement shows that 1/T1n changes little
with temperature between 3 K and 40 K. Thus we find
T1σ = 3.4(5) ns for the contribution of the σ FS sheets
using T1n = T1(Tc) = 6.3 ns, T1π = 20(2) ns and Eq. 1.
For normal metals with a simple Fermi surface, the so-
called Elliott relation [2, 3, 26, 27] holds, which states
that for a given type of disorder (e.g. phonons or dislo-
cations) T1 is proportional to the momentum relaxation
time, τ . The proportionality constant depends on the
spin orbit splitting of the conduction electron bands and
has been estimated in a number of metals from the shift
of the CESR from the free electron value. Metals with
complicated Fermi surfaces i.e. with great variations of
the electron-phonon coupling on the different FS parts
are known to deviate from the Elliott relation [28] and
calculation of T1 requires to take into account the de-
tails of the band-structure [29, 30, 31]. Examples include
polyvalent elemental metals such as Mg or Al. Clearly, a
calculation of T1 is required for MgB2, which takes into
account its band structure peculiarities. Comparison of
spin scattering and momentum scattering times of the
two types of Fermi surfaces is instructive. The relative
values of τ for the two FS parts, τπ and τσ, and for in-
terband scattering were estimated by Mazin et al. [32].
A very small impurity interband scattering and τπ < τσ,
i.e. a larger π intraband scattering relative to σ intra-
band scattering was required to explain the rather small
depression of Tc in materials with widely different con-
ductivities. De Haas-van Alphen [33] and magnetore-
sistance [34] measurements of high purity samples yield
τπ < τσ also. Such a behaviour could rise fromMg vacan-
cies, which perturb more electrons of the π band relative
to the σ band. However, our spin scattering data do not
support this. In contrast to momentum scattering, spin
scattering is stronger on the σ FS: T1π : T1σ = 6 : 1
in high purity samples and low temperatures. The rela-
tive values of T1 and τ for the two FS do not necessarily
follow the same trend, spin relaxation times at low tem-
peratures depend on spin orbit relaxation on impurities
while momentum relaxation is due to potential scatter-
ing. However, a defect center such as a Mg vacancy with a
strong modification of the electron-phonon coupling and
an atomic number strongly differing from that of the reg-
ular atoms constituting the compound would greatly af-
fect T1 compared to τ . In the two gap model Mg defects
are expected to shorten T1π more than T1σ and thus are
unlikely to be the dominant scatterers.
A final note concerns the validity of the above analysis
of T1’s in the framework of the two-band/gapmodel. The
field independence of the lowest temperature T1 for 0.32
and 1.27 T is unexpected within this model. The spin
susceptibility increases strongly between these fields and
more normal states are restored at 1.27 T than expected
from the closing of the gap on the π FS sheets alone [17].
Based on this, one would expect to observe additional
spin scattering from the restored σ FS parts, which is
clearly not the case. This also indicates that a theoret-
ical study, which takes into account the peculiarities of
MgB2 is required to explain the anomalous spin-lattice
relaxation times.
In conclusion, we presented the measurement of the
spin-lattice relaxation time, T1, of conduction electrons
as a function of temperature and magnetic field in the
MgB2 superconductor. We use a novel method based on
the detection of the z component of the conduction elec-
tron magnetization during electron spin resonance con-
ditions with amplitude modulated microwave excitation.
Lengthening of T1 below Tc is observed irrespective of the
significant CESR line broadening due to irreversible dia-
magnetism in the polycrystalline sample. The field inde-
pendence of T1 for 0.32 T and 1.27 T allows to measure
the separate contributions to T1 from the two distinct
types of the Fermi surface.
ACKNOWLEDGEMENTS
The authors are grateful to Richárd Gaál for the de-
velopment of the ESR instrument at the EPFL. F.S.
and F.M. acknowledge the Zoltán Magyary postdoc-
toral programme, the Bolyai fellowship of the Hungarian
Academy of Sciences and the Alexander von Humboldt
Foundation for support, respectively. Work supported
by the Hungarian State Grants (OTKA) No. TS049881,
F61733, PF63954 and NK60984 and by the Swiss NSF
and its NCCR ”MaNEP”. Ames Laboratory is operated
for the U.S. Department of Energy by Iowa State Uni-
versity under Contract No. W-7405-Eng-82.
∗ Corresponding author: simon@esr.phy.bme.hu
† Present address: Leibniz Institute for Solid State and
Materials Research Dresden, PF 270116 D-01171 Dres-
den, Germany
‡ Present address: Condensed Matter Physics and Ma-
terials Science Department, Brookhaven National Labo-
ratory, Upton, New York 11973-5000, USA
[1] I. Žutić, J. Fabian, and S. D. Sarma, Rev. Mod. Phys.
76, 323 (2004).
[2] R. J. Elliott, Physical Review 96, 266279 (1954).
[3] Y. Yafet, Phys. Lett. A 98, 287 (1983).
[4] N. M. Nemes, J. E. Fischer, G. Baumgartner, L. Forró,
T. Fehér, G. Oszlányi, F. Simon, and A. Jánossy, Phys.
Rev. B 61, 7118 (2000).
[5] F. Murányi, F. Simon, F. Fülöp, and A. Jánossy, J.
Magn. Res. 167, 221 (2004).
[6] F. Simon and F. Murányi, J. Magn. Res. 173, 288 (2005).
[7] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani,
and J. Akimitsu, Nature 410, 63 (2001).
[8] F. Bouquet, R. A. Fisher, N. E. Phillips, D. G. Hinks,
and J. D. Jorgensen, Physical Review Letters 87, 047001
(2001).
[9] P. Szabó, P. Samuely, J. Kacmarćık, T. Klein, J. Marcus,
D. Fruchart, S. Miraglia, C. Marcenat, and A. G. M.
Jansen, Phys. Rev. Lett. 87, 137005 (2001).
[10] F. Bouquet, Y. Wang, I. Sheikin, T. Plackowski,
A. Junod, S. Lee, and S. Tajima, Physical Review Letters
89, 257001 (2002).
[11] S. Tsuda, T. Yokoya, Y. Takano, H. Kito, A. Matsushita,
F. Yin, J. Itoh, H. Harima, and S. Shin, Physical Review
Letters 91, 127001 (2003).
[12] J. Kortus, I. I. Mazin, K. D. Belashchenko, V. P.
Antropov, and L. L. Boyer, Phys. Rev. Lett. 86, 4656
(2001).
[13] H. J. Choi, D. Roundy, H. Sun, M. L. Cohen, and S. G.
Louie, Nature 418, 758 (2002).
[14] M. R. Eskildsen, M. Kugler, S. Tanaka, J. Jun, S. M.
Kazakov, J. Karpinski, and R. Fischer, Phys. Rev. Lett.
89, 187003 (2002).
[15] F. Simon, A. Jánossy, T. Fehér, F. Murányi, S. Garaj,
L. Forró, C. Petrovic, S. L. Bud’ko, G. Lapertot, V. Ko-
gan, et al., Phys. Rev. Lett. 87, 047002 (2001).
[16] M. Angst, R. Puzniak, A. Wisniewski, J. Jun, S. M.
Kazakov, J. Karpinski, J. Roos, and H. Keller, Phys.
Rev. Lett. 88, 167004 (2002).
[17] F. Simon, A. Jánossy, T. Fehér, F. Murányi, S. Garaj,
L. Forró, C. Petrovic, S. L. Bud’ko, R. A. Ribeiro, and
P. C. Canfield, Phys. Rev. B 72, 012511 (2005).
[18] F. J. Dyson, Physical Review 98, 349 (1955).
[19] D. K. Finnemore, J. E. Ostenson, S. L. Bud’ko, G. Laper-
tot, and P. C. Canfield, Phys. Rev. Lett. 86, 2420 (2001).
[20] P. Pincus, Phys. Lett. 13, 21 (1964).
[21] J. Hervé and J. Pescia, C. R. Acad. Sci.(Paris) 251, 665
(1960).
[22] V. A. Atsarkin, V. V. Demidov, and G. A. Vasneva, Phys-
ical Review B 52, 1290 (1995).
[23] J. Granwehr, J. Forrer, and A. Schweiger, J. Magn. Res.
151, 78 (2001).
[24] R. R. Urbano, P. G. Pagliuso, C. Rettori, Y. Kopelevich,
N. O. Moreno, and J. L. Sarrao, Phys. Rev. Lett. 89,
087602 (2002).
[25] K. W. Blazey, K. A. Muller, J. G. Bednorz, W. Berlinger,
G. Amoretti, E. Buluggiu, A. Vera, and F. C. Matacotta,
Phys. Rev. B 36, 7241 (1987).
[26] Y. Yafet, in Solid State Physics, edited by F. Seitz and
D. Turnbull (1963), vol. 14, p. 1.
[27] F. Beuneu and P. Monod, Phys. Rev. B 18, 2422 (1977).
[28] P. Monod and F. Beuneu, Phys. Rev. B 19, 911 (1979).
[29] R. H. Silsbee and F. Beuneu, Phys. Rev. B 27, 2682
(1983).
[30] J. Fabian and S. Das Sarma, Phys. Rev. Lett. 81, 5624
(1998).
[31] J. Fabian and S. Das Sarma, Phys. Rev. Lett. 83, 1211
(1999).
[32] I. I. Mazin, O. K. Andersen, O. Jepsen, O. V. Dol-
gov, J. Kortus, A. A. Golubov, A. B. Kuzmenko, and
D. van der Marel, Phys. Rev. Lett. 89, 107002 (2002).
[33] E. A. Yelland, J. C. Cooper, A. Carrington, N. E. Hussey,
P. J. Meeson, S. Lee, A. Yamammoto, and S. Tajima,
Phys. Rev. Lett. 88, 217002 (2002).
[34] I. Pallecchi, V. Ferrando, E. G. D’Agliano, D. Marre,
M. Monni, M. Putti, C. Tarantini, F. Gatti, H. U. Ae-
bersold, E. Lehmann, et al., Phys. Rev. B 72, 184512
(2005).
