PHYSICAL REVIEW B 84, 064402 (2011)

Spectral dependence of photoinduced spin precession in DyFeO3
Ryugo Iida,1 Takuya Satoh,1,2,* Tsutomu Shimura,1 Kazuo Kuroda,1 B. A. Ivanov,1,3
Yusuke Tokunaga,4,5 and Yoshinori Tokura4,5,6
1

Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan
2
PRESTO, Japan Science and Technology Agency, Tokyo 102-0075, Japan
3
Institute of Magnetism, Ukrainian Academy of Science, Vernadskii Avenue 36B, 03142 Kiev, Ukraine
4
Multiferroic Project, ERATO, Japan Science and Technology Agency, Wako, Saitama 351-0198, Japan
5
Cross-Correlated Materials Research Group (CMRG) and Correlated Electron Research Group (CERG),RIKEN Advanced Science Institute (ASI), Wako, Saitama 351-0198, Japan
6
Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan(Received 1 March 2011; revised manuscript received 16 May 2011; published 2 August 2011)
Spin precession was nonthermally induced by an ultrashort laser pulse in orthoferrite DyFeO3 with a pumpprobe technique. Both circularly and linearly polarized pulses led to spin precessions; these phenomena areinterpreted as the inverse Faraday effect and the inverse Cotton-Mouton effect, respectively. For both cases, thesame mode of spin precession was excited; the precession frequencies and polarization were the same, but thephases of oscillations were different. We have shown theoretically and experimentally that the analysis of phases
can distinguish between these two mechanisms. We have demonstrated experimentally that in the visible region,
the inverse Faraday effect was dominant, whereas the inverse Cotton-Mouton effect became relatively prominent
in the near-infrared region.
DOI: 10.1103/PhysRevB.84.064402

PACS number(s): 78.20.Ls, 75.50.Ee, 75.40.Gb, 78.47.J−

I. INTRODUCTION

Magnetization switching triggered by femtosecond laser
pulses has been studied in recent years. Ultrafast demagnetization in ferromagnetic metals and semiconductors has also
been reported.1,2 These phenomena show thermal magnetic
switching with light pulses on picosecond time scales.3
However, heat-assisted spin reorientation is relatively slow
because of the thermal diffusion time.
A light pulse with a certain polarization nonthermally modifies the electron spin state.4,5 Recently, it has been reported
that spin precession is induced by a circularly polarized pulse
in antiferromagnetic (AFM) DyFeO3 with weak ferromagnetic(FM) moment.6 The phase of spin precession changes by 180◦
on reversal of the pump helicity. The interpretation of thisphenomenon is that an effective magnetic field pulse parallelto the pump wave vector is induced by the circularly polarized
light pulse, giving rise to the precession. The magnetic field
generation effect is referred to as the inverse Faraday effect(IFE). The same effect has also been observed even in pure
AFM NiO with no net magnetic moment in the groundstate.7 The resonance frequencies of AFM materials reachthe terahertz range, which is several orders of magnitude
higher than that of FM materials. For that reason, AFM
materials attract much attention in the context of ultrafast spin
control.7–15 Spin precession is also observed with a linearlypolarized pump pulse, in particular, a pulse polarized in a
direction nonparallel to the crystal axes. This phenomenon
is called the inverse Cotton-Mouton effect (ICME).16,17 A
detailed review of these phenomena can be found in Ref. 18.
The ultrafast IFE and ICME are interpreted as impulsive
stimulated Raman scattering (ISRS).19–21 An electron in the
ground state is excited by the pump pulse into a virtual state,
which changes the orbital momentum of the electron. The
nonzero orbital momentum flips the electron spin with spinorbit coupling in the virtual state. The excited electron radiates
1098-0121/2011/84(6)/064402(11)a photon and transits to the final state. The energy gap between
the final and ground states corresponds to the spin precession
energy.
ISRS is a modulation of the dielectric permittivity by thepump pulse and should be dependent on the properties ofthe pulse, such as its polarization, wavelength, and fluence.
Therefore, examining the dependence of the photoinduced spin
precession on these properties will help us to understand the
ISRS mechanism. In particular, it is not obvious how the pump
photon energy influences spin precession. An action spectrumof photoinduced spin precession should indicate the relation
between the optical excited state and the spin precession via
ISRS.
In the majority of previous publications, the excitation ofspin oscillations by ultrashort laser pulses was associated with
IFE and ICME separately. In the present work, we report spin
precession induced via ISRS as functions of the pump pulse
polarization and wavelength. We found that both effects, IFE
and ICME, are working in the same way, exciting the samemode of spin precession. The phases of the spin precession via
IFE and ICME differ by 90◦ , allowing the two effects to be
distinguished. We found an essential dependence of the phaseon the pump wavelength and demonstrated that the IFE andICME are dominating effects in different spectral regions, inthe visible region and in the near-infrared region, respectively.
Thus, the analysis of the phase difference of the spin precession
reveals the mechanism of ISRS.
II. PHYSICAL PROPERTIES
A. Crystallographic and magnetic properties

DyFeO3 is a rare-earth orthoferrite and crystallizes in an
16
orthorhombic structure D2h(P bnm).22 Spins of the Dy3+ ions
are not ordered above 4 K. Four Fe3+ ions occupy positions(1/2, 0, 0), (1/2, 0, 1/2), (0, 1/2, 1/2), and (0, 1/2, 0) in the064402-1©2011 American Physical Society

RYUGO IIDA et al.

PHYSICAL REVIEW B 84, 064402 (2011)

unit cell. In the exchange approximation, the arrangement of
their magnetic moments, M 1 , M 2 , M 3 , M 4 , corresponds toone of the four patterns Gi : M1i = −M2i = M3i = −M4i , Fi :M1i = M2i = M3i = M4i , Ai : M1i = −M2i = −M3i = M4i ,and Ci : M1i = M2i = −M3i = −M4i (i = x, y, z). DyFeO3
crystal has the spin arrangement 4 (Gx Ay Fz ) and belongs
to the magnetic point group m m m above the Morin point
and below the Néel temperature, at 37 K < T < TN = 645
K.23–27 Because of the superexchange interaction, the spinsare almost completely arranged antiferromagnetically along
the x axis. Due to the Dzyaloshinskii-Moriya interaction, allspins tilt by about 0.5◦ toward the z axis.28,29 Usually theconditions M 1 ∼ M 3 and M 2 ∼ M 4 are valid and a simpler
model with just two different sublattice magnetic moments,
M 1 and M 2 , with |M 1 | = |M 2 | = M0 , can be employed.26,30
In what follows, this two-sublattice model will be used. We
denote the FM vector by M = M 1 + M 2 and the AFM vector
by L = M 1 − M 2 . These vectors are subject to constraints(M · L) = 0,

M +L
2

2

= 4M02 .

are described as follows:
M(t) = Mz ẑ + m(t),L(t) = Lx x̂ + l(t),where ẑ and x̂ are unit vectors parallel to the z axis andx axis, respectively, and the variables m(t) and l(t) correspond to two eigenfrequency modes, as shown in Figs. 1(b)and 1(c). The components mx , my , and lz oscillate at thequasi-ferromagnetic resonance (F mode) with the angularfrequency ωF . On the other hand, lx , ly , and mz oscillate at
the quasi-antiferromagnetic resonance (AF mode) with theangular frequency ωAF .30 Those resonance frequencies are
given by33–35
ωF = 2γ M0 A(q3 − q1 ),ωAF = 2γ M0 d 2 − Aq1 ,d M(t)= −γ {[M(t) × H eff ] + [L(t) × heff ]},dt
d L(t)= −γ {[M(t) × heff ] + [L(t) × H eff ]},
dt

(2)
(3)

A 2 p1 2 p3 2 q1 2
M + Mx + Mz + Lx
2
2
2
2
q3 2
+ Lz − d · (M × L).
2

B. Optical and magneto-optical properties

DyFeO3 has the d-d transitions 6 A1g → 4 Eg , 4 A1g centered
at a wavelength of 500 nm, 6 A1g → 4 T2g at 700 nm,
and 6 A1g → 4 T1g at 1000 nm.36,37 This crystal is opticallybiaxial, so the components of the dielectric permittivity tensor000
are εxx= εyy= εzz. The birefringence ρ stems from thedifference in the refractive indices, such as nxy = nx − ny(ρ = 2π n/λ). On the other hand, the magnetization Mz leads
to the Faraday rotation for the light propagating along thez axis. The Faraday effect is much smaller than the effectof birefringence. In DyFeO3 , the Faraday rotation φ andthe birefringence per unit length are φ = 1.6 × 103 deg/cm
and ρ = 1.2 × 105 deg/cm, respectively, for light with awavelength of 800 nm propagating along the z axis.38–40(4)The last term describes the Dzyaloshinskii-Moriya interaction
and d is parallel to the y axis. Equations (2) and (3), linearized
above the ground state determined by the energy (4), yield
eigenmodes of oscillations of the vectors M(t) and L(t).
These spin precession modes for the 4 ground state withthe equilibrium values of Mz = 0 and Lx = 0 [see Fig. 1(a)]y(8)

where M0 = |M 1 | = |M 2 |, and the anisotropy constants p1
and p3 are omitted, because their contribution to frequencies
is negligible. (It is worth noting here that the exchangerelativistic constant of the Dzyaloshinskii-Moriya interaction
has the same order of magnitude as a square root
√ of the productof exchange and relativistic constants like A|q1 | and thus
should be kept in the above expressions.) The orbits of thespin precession and the temporal response of M and L are
different in the two modes.

where γ = gμB /h̄ (> 0) is the gyromagnetic constant, μB
is the modulus of the Bohr magneton, g is the gyromagnetic
ratio, g ≈ 2 for orthoferrites, and H eff and heff are the effective
magnetic fields. Using the magnetic energy of an orthoferrite,
the effective fields are denoted as H eff = −∂H/∂ M and heff =−∂H/∂ L, where the Hamiltonian is given by26,33

(a)

(7)

(1)

The dynamics of M(t) and L(t) is described by LandauLifshitz equations26,31,32

H=

(5)
(6)

y

(b)

y

(c)

4

x

L

M

M2
z

M1

1
4

M1

M2

2
3

2

3

z 4

L

M
1

2
3

1 x
4

3

1
2,4

2

M2

4

M

3
1
2,4 M1

3
4

1

z

1

L
3 x
2

FIG. 1. (Color online) (a) The static magnetic structure of DyFeO3 , with the four Fe3+ spins regarded as satisfying M1(b) Quasi-ferromagnetic and (c) quasi-antiferromagnetic spin precession.
064402-2

3
1
2

M3 and M2

M4 .

SPECTRAL DEPENDENCE OF PHOTOINDUCED SPIN . . .

PHYSICAL REVIEW B 84, 064402 (2011)

TABLE I. The dielectric permittivity tensor εij as a function of magnetic component. Modulation of the dielectric permittivity in F modeand AF mode is shown in the third and fourth columns, respectively. No external magnetic field is present. εija = −εjai and εijs = εjs i .
TensorElement

Static(m(t) = 0, l(t) = 0)

F Mode(mx ,my ,lz = 0)

AF Mode(lx ,ly ,mz = 0)s
εxxaxxzz Mz2 + bxxxx L2x + cxxzx Mz Lx

0

s
εyy

ayyzz Mz2 + byyxx L2x + cyyzx Mz Lx

0

s
εzz

azzzz Mz2 + bzzxx L2x + czzzx Mz Lx

0

s
εxy
s
εzx

0
0

s
εyz
a
εxy
a
εzx
a
εyz

0
ifxyz Mz + igxyx Lx
000(2azxxz Mz + czxxx Lx )mx+ (2bzxxz Lx + czxzz Mz )lz(2ayzyz Mz + cyzyx Lx )my0
ifzxy myifyzx mx + igyzz lz(2axxzz Mz + cxxzx Lx )mz+ (2bxxxx Lx + cxxzx Mz )lx(2ayyzz Mz + cyyzx Lx )mz+ (2byyxx Lx + cyyzx Mz )lx(2azzzz Mz + czzzx Lx )mz+ (2bzzxx Lx + czzzx Mz )lx(2bxyxy Lx + cxyzy Mz )ly0

C. Interaction of the light pulse and the medium

The interaction of the magnetic medium and transmitting
light is described by the dielectric permittivity tensor εij .41,42
By virtue of the Onsager principle, if absorption is negligible,
εij can be divided into antisymmetric and symmetric parts,(εija = −εjai ) and (εijs = εjs i ), with real and imaginary components, respectively. For a transparent medium, the tensor
components can be written in the following general form:
εij = εij(0) + ifij k Mk + igij k Lk
+ aij kl Mk Ml + bij kl Lk Ll + cij kl Mk Ll ,εij(0)(9)whereis a magnetization-independent term having asymmetric part only. By taking into account the symmetry oforthoferrite, the terms in the first line (except the εij(0) ) represent
the antisymmetric part of εij , and the terms in the second line
describe the spin-dependent symmetric part of the permittivity
tensor. The symmetry of the fourth-rank tensors aij kl , bij kl , andcij kl is determined by the magnetic and crystal point groups,
and fij k and gij k are the third-rank tensors, antisymmetric overthe first pair of indices, e.g., fij k = −fj ik . Tensors cij kl andgij k originate from the Dzyaloshinskii-Moriya interaction. The
Hamiltonian of the interaction between the light pulse and the
medium is42
εij
Ei (t)Ej∗ (t),Hint =(10)16π
where Ei (t) is the time-dependent amplitude of the light in
the pulse. A circularly polarized pulse propagating along the√ (1, ±
z axis can be described in the form (Ex (t),Ey (t)) = E(t)2i), where the ± indicate the opposite senses of helicity. A
linearly polarized pulse with the polarization inclined at anangle θ with respect to the x axis can be described in theform (Ex (t),Ey (t)) = E(t)(cos θ, sin θ ). Then a straightforward
calculation gives the Hamiltonian of the interaction with themedium of the form
E(t)E ∗ (t)  sσ±saεxx + εyy,(11)=
∓ 2iεxy
Hint
32π
∗

E(t)E (t)  slinss
εxx cos2 θ + εyy
Hint=
sin2 θ + εxy
sin 2θ
16π(12)0ifxyz mz + igxyx lx
00

for circularly and linearly polarized pulses, respectively.
Nonzero components of the tensors εijs and εija are listed inTable I.43,44
When a pulse is incident on a medium, the interaction
between the pulse and the medium is given by Eqs. (10)–(12).
The incident pump pulse generates effective pulsed fields
H eff = −∂Hint /∂ M and heff = −∂Hint /∂ L. Both effective
fields are proportional to the intensity of the light, E(t)E ∗ (t).
If the pulse duration  is much shorter than the period ofspin oscillations, 
1/ωF ,1/ωAF , the real pulse shape canbe replaced by the Dirac delta function, E(t)E ∗ (t) → I0 δ(t),where I0 = E(t)E ∗ (t)dt is the integrated pulse intensity. The
light-induced effective fields H eff and heff can be regarded as
being proportional to the delta function δ(t) as well. For a lightpulse propagating along the z axis, H eff and heff generated bya circularly polarized pulse are±

I0 δ(t)[2(axxzz + ayyzz )Mz
32π+ (cxxzx + cyyzx )Lx ± 2fxyz ]ẑ,(13)

I0 δ(t)[2(bxxxx + byyxx )Lx
32π+ (cxxzx + cyyzx )Mz ± 2gxyx ]x̂,(14)

H eff,σ = −±heff,σ = −respectively. The phenomenon of generating these effective
magnetic fields is known as IFE.
For a magnetic field pulse of a short duration, the actionof the light-induced effective fields within the delta-function
approximation can be described as an instantaneous deviationof the FM and AFM vectors, M = M(t = +0) − M(t =−0) and L = L(t = +0) − L(t = −0), from their equilibrium positions, M(t = −0) = Mz ẑ and L(t = −0) = Lx x̂,respectively. M and L can be found by integration of
Eqs. (2) and (3) from t = −0 to t = +0. After vanishing of
the pulsed effective field, the spins precess around the effective
fields corresponding to their equilibrium directions following
the Landau-Lifshitz equations, based on the Hamiltonian(4). Thus the action of the pulse can be regarded as acreation of some (nonequilibrium) initial conditions for theLandau-Lifshitz equations. The deviation of the FM and AFM

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RYUGO IIDA et al.

PHYSICAL REVIEW B 84, 064402 (2011)

vectors induced by the circularly polarized pulse is describedby±M σ = 0,±

Amplified
Ti:Sapphire laserdelay line
P λ/2

(15)

γ I0
[2(bxxxx + byyxx − axxzz − ayyzz )Mz Lx
32π+ (cxxzx + cyyzx )Mz2 − (cxxzx + cyyzx )L2x

L σ = −∓ 2(fxyz Lx − gxyx Mz )]ŷ.

OPA

(16)

H eff,lin = −(17)

I0 δ(t)[{(2bxxxx Lx + cxxzx Mz ) cos2 θ16π+ (2byyxx Lx + cyyzx Mz ) sin2 θ }x̂

heff,lin = −+ (2bxyxy Lx + cxyzy Mz ) sin 2θ · ŷ].

detectorpump

Here, M is not affected by the effective field directly, whereas
ly of L(t) takes nonzero deviations. The resonance mode withly = 0 is AF mode. In Fig. 1(c), two spins M 1 and M 2 , as wellas their sum and difference M and L, move toward positions 2
or 4, and the spins precess around their ground-state directions.
Because the spins have only an ly variable component whenthe effective magnetic field disappears, spin precession starts
at position 2 or 4 [see Fig. 1(c)].
Similarly, effective magnetic fields induced by a linearlypolarized pulse are
I0 δ(t)[(2axxzz Mz + cxxzx Lx ) cos2 θ16π+ (2ayyzz Mz + cyyzx Lx ) sin2 θ ]ẑ,sample WP
probe

(18)

These effective magnetic fields are induced via ICME. The
deviations of the FM and AFM vectors created by the effective
field areγ I0 
M lin =2bxyxy L2x + cxyzy Mz Lx sin 2θ · ẑ, (19)16π
γ I0 
2bxxxx Mz Lx + cxxzx Mz2
L lin =16π
− 2axxzz Mz Lx − cxxzx L2x cos2 θ+ 2byyxx Mz Lx + cyyzx Mz2− 2ayyzz Mz Lx − cyyzx L2x sin2 θ ŷ
− 2bxyxy Mz Lx + cxyzy Mz2 sin 2θ · x̂ . (20)Here, the components mz of M and lx , ly of L are affected by
the effective field. This precession mode is also an AF mode,
but the initial direction of the spin deviation differs from thatfor the circularly polarized pulse case.
A pulse propagating along the x or y axis should trigger
the spin precession with both F and AF modes. The amplitude
and the phase of the precession depends on the polarization of
the pulse.

λ/2 GTP

λ/4

(BBO)

FIG. 2. (Color online) The experimental setup geometry. BBO
was used for frequency doubling of the pump pulse, if necessary.
WP: Wollaston prism, GTP: Glan-Taylor prism, P: polarizer, λ/2:
half-wave plate, λ/4: quarter-wave plate.

of a few millimeters were mechanically polished. The sample
thickness was 140 μm, except for a thickness-dependentmeasurement. The sample was placed in a cryostat at
77 K with no external magnetic field. Optical pulses witha central wavelength of 790 nm, a duration of 150 fs, anda repetition rate of 1 kHz were emitted from an amplified
Ti:sapphire laser. The beam was separated into two beams
by a beam splitter. One was employed as the probe beam,and the other was injected into an optical parametric amplifier(OPA), which converted the incident beam to signal and idler
beams, in the wavelength ranges 1140–1580 nm and 1580–
2570 nm, respectively. Furthermore, the signal and idler beams
were frequency-doubled with a β-BaB2 O4 (BBO) crystal if
necessary. Then unwanted beams were cut by color filters.
The ranges of the pump wavelength were 600–750 nm (second
harmonic of the signal pulse), 850–1100 nm (second harmonicof the idler pulse), and 1140–1500 nm (the signal pulse).
Figure 3 illustrates the circular and linear polarizations
employed for the pump and probe pulses. Circularly polarized
pulses are denoted as σ ± . Linearly polarized pulses, denoted
L1, L2, L3, L4, L5, and L6, were tilted at −π/4,π/4,0,π/2,
and ∓α from the x axis, respectively, where tan α = 2. The
fluence of the pump pulse was varied from 15 to 130 mJ/cm2 ,depending on the wavelength. The pump pulses were focusedon the sample to spot sizes of 50–100 μm. The probe pulses
were linearly polarized and had a pulse fluence of 1 mJ/cm2 .
The probe pulse was vertically incident on the surface of thesample, whereas the pump pulse was incident at the angle of 7◦ .
The transmitted probe pulse was divided into two orthogonallypolarized pulses by a Wollaston prism, and each pulse was
detected with a Si photodiode. The ratio of the signals fromthe detectors allowed us to determine the angle of the probe
polarization.
yσIII. EXPERIMENTAL RESULTS

+

σ

−

L5
L1

L4

L6
L2
L3

x

A. Experimental setup

We studied photoinduced spin precession in DyFeO3 using
a pump-probe magneto-optical technique, as shown in Fig. 2.
DyFeO3 single crystals were grown by the floating-zone
method, and the orientation of the faces were determined byback-reflection x-ray Laue photographs.27 Faces with a width

FIG. 3. (Color online) Pulse polarizations. Circularly polarized
pulses are denoted σ ± . Linearly polarized pulses, denoted L1, L2,
L3, L4, L5, and L6, were tilted at −π/4,π/4,0,π/2, and ∓α fromthe x axis, respectively, where tan α = 2.

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SPECTRAL DEPENDENCE OF PHOTOINDUCED SPIN . . .

PHYSICAL REVIEW B 84, 064402 (2011)

σ+
y

x
z

L4

FIG. 4. (Color online) Polarizations of the propagating pulses in
the medium with birefringence.
B. Dependence of the polarization rotationon the pump pulse polarization

Figure 4 illustrates the polarization of the propagating
pulses in the medium with birefringence. For the sake ofsimplicity, we will discuss the picture of the light propagation
without taking the Faraday effect into account. Pulses withcircular polarization or linear polarization nonparallel tothe crystal axis are transformed, whereas pulses with linear
polarization parallel to the crystal axis, corresponding to thenormal modes of light in the media, are not. Here, it is worth
noting that in the former case, even though a pure circularly
or linearly polarized pulse is incident, ICME and IFE are also
induced, respectively, because the birefringence transforms
light polarization in the medium. For the pulses with generallinear polarization or pulses with circular polarization, thereal and imaginary parts of Ex Ey , which are responsible
for the terms in Eqs. (13)–(20) including sin 2θ , and fxyz
and gxyx , respectively, will oscillate in space along thepulse propagation direction, while they remain uniform only

Modulation of the dielectric permittivity leads to oscillationof the probe polarization in the sample. The origin of theprobe polarization change can be attributed to the CottonMouton effect and the Faraday effect. The Cotton-Moutons, whereas
effect is magnetic linear birefringence based on εxy
the Faraday effect is magnetic circular birefringence basedaon εxy.
In order to identify the effect giving rise to the polarization
rotation as observed in Fig. 5, we set σ ± for the pumppolarization and L5 and L6 for the probe polarization. For
L5 and L6, the Faraday effect leads to rotation of the probe
polarization in the same direction for both probe polarizations.
On the other hand, the Cotton-Mouton effect leads to the

σ−

0.2

Polarization RotationΔθ (deg)

σ+

0.2

0.1
0.0

0.1
0.0

-0.1

C. The influence of magnetization on the probe polarization

(b)

0.3

Polarization RotationΔθ (deg)(a)for pulses linearly polarized parallel to the crystalline axis.
Therefore, the effective magnetic field and spin precession
generated by IFE and ICME will be different at different
positions in the sample.
Figure 5 shows the polarization rotation of the probe
pulse as a function of the delay time between the pumpand probe pulses. The pump wavelength was 1050 nm, andthe polarizations were σ ± , L1, L2, L3, and L4. The probe
polarization was L4. When the pump polarizations were
σ ± , L1, and L2, oscillation of the polarization rotation wasobserved. The frequency of the oscillation was 210 GHz at thetemperature T = 77 K, in agreement with previous infrared
and Raman experiments.31,35,46 In Figs. 5(c) and 5(d), the pumppulses with polarizations L3 and L4 did not induce oscillationof the probe polarization. Polarizations σ ± , L1, and L2 had
Ex Ey components, but L3 and L4 did not, as shown in Fig. 4.

-0.1

0

5

10

15

20

25

30

-0.2

35

L1
L2
0

5

Delay (ps)

15

20

25

30

35

(d)
0.2

Polarization RotationΔθ (deg)

0.2

Polarization RotationΔθ (deg)(c)

10

Delay (ps)

L3

0.1
0.0

0.0

-0.1
-0.2

L4

0.1

-0.1

0

5

10

15

20

25

30

35

Delay (ps)

-0.2

0

5

10

15

20

25

30

35

Delay (ps)FIG. 5. (Color online) Oscillation of the probe polarization θ(t) as a function of the time delay between the pump and probe pulses. Six
types of probe polarizations were used: (a) circular polarization σ ± , and linear polarizations tilted at (b) ∓45◦ , (c) 0◦ , and (d) 90◦ with respect
to the x axis.
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PHYSICAL REVIEW B 84, 064402 (2011)

L5
L6

0.0
-0.1
-0.2
-0.3

270

precession (deg)Initial phase of the spin

0.1

circular polarization
linear polarization

180

(a)
90
0
-90

-180

0

5

10

15

20

Delay (ps)

25

30

35
Normalized amplitude (a.u.)

Polarization Rotation Δθ (deg)RYUGO IIDA et al.

FIG. 6. (Color online) Time-resolved pump-induced probe polarization θ(t). The pump polarization was circular, and the graphshows the difference of the probe polarizations for the pumps withcircular polarizations σ + and σ − . The pump wavelengths were
750 nm and the probe polarizations were L5 and L6.

rotation of the major axis of the probe polarization in the
opposite direction. Therefore, the dominance of the rotationof the probe polarization can be distinguished. Figure 6
shows that the polarization rotations of two probe pulses withpolarizations L5 and L6 oscillated in the same direction. This
indicates that the contribution of the Faraday effect is dominant
and that of the Cotton-Mouton effect is negligible for the probe
wavelength of 800 nm. In addition, this is consistent with thefact that IFE is dominant for the pump wavelength of 800 nm(see below).
D. Dependence of the polarization rotationon the pump wavelength

The oscillation of the probe polarization originates fromspin precession. Therefore, the phase of the oscillation indicates the direction of an effective magnetic field induced by thepump beam. The dependence of the effective magnetic field
and reorientation of magnetization on the pump wavelength
gives information about the interaction of the light pulse andthe magnetic medium.
An experiment was performed with four types of pumppolarizations, σ ± , L1, and L2. The differences between theoscillations for σ + and σ − and between those for L1 and L2
were measured. Figure 7(a) shows the initial phase ξ of theoscillation of the probe polarization versus pump wavelength.
The oscillation is described by θ (t) = A sin(ωt + ξ ) at t > 0,
where A is the amplitude, ω is the angular frequency, and ξ isthe initial phase. The initial phase was close to 0◦ (or ±180◦ ),when the pump wavelength was 800 nm. This is consistent with
Ref. 45. When the pump wavelength was between 1000 nm and1100 nm, the initial phase was closer to ±90◦ . When the pumpwavelength was above 1200 nm, the initial phase was between0◦ and 90◦ . By comparing two samples with thicknesses of
140 μm and 170 μm, it was confirmed that the sample
thickness does not affect the phase shift (data not shown).
Figure 7(b) represents the amplitude A of the oscillation
as a function of the pump wavelength. The amplitude A is
proportional to the pump fluence, thus justifying normalizationof the amplitude by the fluence. Because of the transformationof the pulse polarization in the birefringent crystal, theresulting effective field is different at different positions along

0.08

(b)
0.06
0.04
0.02
0.00
600

800 1000 1200 1400
Pump wavelength (nm)1600

FIG. 7. (Color online) (a) Initial phase and (b) amplitude of theoscillation of the polarization as a function of the pump wavelength.
The amplitude is normalized by the pump fluence.

pump pulse propagation. Therefore, the amplitude A wasnot simply proportional to the magnitude of the generated
magnetic field. However, when the pump wavelength was from
700 nm (6 A1 → 4 T2 ) to 1000 nm (6 A1 → 4 T1 ), the amplitude
was larger than that of the other region in Fig. 7(a). This result
suggests that the photoinduced spin precession is related to theelectron transition.
E. Pump-probe measurement in (100) and(010) oriented crystals

To determine all dielectric permittivities, we performed
pump-probe measurements in (100) and (010) oriented crystals. The pump wavelength was 750 nm, and the crystal
thickness in both cases was 100 μm. However, in contrast
to the previous experiments,6 oscillation of the polarization ofneither F nor AF modes was observed in either propagation
direction.
F. The dependence of polarization rotation on temperature

It is well known that magnon frequencies in orthoferrites
strongly depend on the temperature.6,31,35,46 We measured
the temperature dependence of the spin precession properties
in DyFeO3 . The frequencies of the oscillations for pumpwavelengths of 750 nm and 1200 nm are shown in Fig. 8(a),in comparison with previously reported spin precession.31
Our data show excellent agreement with Refs. 6 and 31,
regardless of the pump wavelength. The frequency decreases
with approaching the Morin point Tr = 37 K because ofmagnon softening associated with the spin reorientation.31,35,46
The temperature dependence of the initial phase ξ of thespin precession for pump wavelength of 750 nm is shownin Fig. 8(b). The initial phase was close to 0◦ or 180◦ with a

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PHYSICAL REVIEW B 84, 064402 (2011)

gxyx lx ,ifxyz (Mz + mz ) + igxyx (Lx + lx ). Because fxyz mzathe phase of mz corresponds mostly to that of εxy
and that of
the oscillation of the polarization. In the AF-mode column ofTable I, the tensor elements proportional to lx are negligible.
These findings simplify the dielectric permittivity tensor.
s
The remaining elements in AF mode are εxy= (bxyxy Lx +acxyzy Mz )ly and εxy = ifxyz mz . Here we suppose that a pulse
is incident on a (001) oriented crystal. We can simplify theeffective magnetic field and the dynamics of the magnetizationinduced by the circular polarization:

500

Frequency (GHz)

450

(a)

400
350
300
750 nm1200 nm

250
200

I0 δ(t)fxyz
ẑ,
16π
±
heff,σ = 0,±
M σ = 0,
γ I0 fxyz Lx±
L σ = ±ŷ.
16π±

H eff,σ = ∓Initial phase of the spin
precession (deg)

(b)

180

90

0

(21)
(22)
(23)
(24)

In the case of the linear polarization one in turn obtains-90
60 80 100 120 140 160 180 200 220 240

H eff,lin = 0,Temperature (K)I0 δ(t)(2bxyxy Lx + cxyzy Mz ) sin 2θ · ŷ,
heff,lin = −16π
γ I0(2bxyxy L2x + cxyzy Mz Lx ) sin 2θ · ẑ,M lin =16π
γ I0(2bxyxy Mz Lx + cxyzy Mz2 ) sin 2θ · x̂.
L lin = −16π

FIG. 8. (a) Temperature dependence of the induced spin precession frequency. Pump wavelengths were 750 nm and 1200 nm.
The solid line shows the magnon frequency taken from Ref. 31. (b)Temperature dependence of the initial spin precession phase for thepump wavelength of 750 nm.

jump at T = 150 K. It is worth noting that at this temperature
the frequencies of F mode and AF mode become equal, that
is, A(q3 − q1 ) = d 2 − Aq1 . Furthermore, the energies of two
domain walls with the spin rotation in (010) and (001) planes
become equal at this point, which leads to the reconstruction of
domain walls.47 However, we were not able to find the relation
between the properties described above and the initial phaseshift.
IV. DISCUSSION
A. Landau-Lifshitz equations

According to the results of the previous section, the numberof essential dielectric permittivity components can be reduced.
First, we found that pump-probe measurement in (100)sssaand (010) oriented crystals revealed that εzz
, εxz
, εyz
, εxz
,
a
and εyzwere negligible. In addition, pump pulses with L3 andL4 polarizations did not trigger spin precession in the (001)oriented crystal in Fig. 5. Polarizations L3 and L4 had only
electric field components Ex and Ey , respectively. Therefore,
the terms containing cos2 θ and sin2 θ in Eqs. (17), (18), and(20) were also negligible.
Moreover, it has been reported that fxyz Mz and gxyx Lx
are of the same order of magnitude for orthoferrites.48 In
contrast to that, for the AF mode the ratio of mz and lx is|mz / lx | = |Lx /Mz | 100. Thus, fxyz mz
gxyx lx , and one
can ignore the term gxyx lx . In addition, Fig. 6 indicates thatthe observed oscillation of the polarization was dominatedaby the imaginary part of the dielectric permittivity εxy
=

(25)
(26)
(27)
(28)

The second terms are much smaller than the first ones in
Eqs. (26), (27), and (28), respectively. As a result, IFEas
and ICME are induced by the contributions of εxy
and εxy,
respectively.
Equation (24) indicates that the circular polarization causes
the AFM component ly and rotation torque of the AF mode.
On the other hand, the FM component does not change in
Eq. (23). As a result, IFE leads to oscillations proportional tosin ωAF t. On the other hand, Eqs. (27) and (28) indicate thatlinear polarization causes components mz and lx . As shownin Fig. 1(c), mz and lx have the same phase, so ICME leads
to oscillations proportional to cos ωAF t. Therefore, the initial
phases of mz excited by IFE and ICME differ by 90◦ .
Because the phase of mz is nearly equal to that of theoscillation of the polarization, we can estimate the phase ofspin precession from the result in Fig. 7. Since the polarizationof the pump pulse is transformed by birefringence, the effective
magnetic field and spin precession differ at different positionsin the medium, as shown in Fig. 4. However, if one ofmagnetization dynamics via IFE and ICME is dominant
and the other is negligible, the time dependence of mz andthe oscillation of the probe polarization are proportional tosin ωAF t or cos ωAF t, respectively.

B. Sigma model

The nonlinear sigma model is a convenient tool for thedescription of linear and especially nonlinear spin dynamics

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RYUGO IIDA et al.

PHYSICAL REVIEW B 84, 064402 (2011)of antiferromagnets; see Ref. 26 for details. It is based on the
dynamical equation for the vector L only that is of the second
order in time derivatives, whereas the vector M is a slave
variable so it can be expressed through the vector L and its
time derivative. Recently, two alternative scenarios of laserinduced excitations of spin oscillations in antiferromagnets
have been discussed within the framework of this model. The
so-called inertial mechanism has been proposed for canted
antiferromagnets and has been realized experimentally forholmium orthoferrite.11 Within the sigma-model approach,the inertial mechanism is associated with an action of thelaser-induced pulse of the magnetic field on the vector L as apulse of force on the massive particle. In this mechanism, thelaser pulse creates an initial value of the time derivative d L/dt
that in principle can lead to quite large deviations of the vector
L after the action of the pulse. In the alternative mechanism,
the time derivative of the effective magnetic field plays a roleof the driving force, leading to an initial deviation of the vector
L from its equilibrium direction.7,49 For this field-derivative
mechanism, the amplitudes of spin deviations are expected tobe smaller than for inertial mechanism, but can be realized
for any antiferromagnet, even a purely compensated one. The
latter mechanism has been observed experimentally in AFM
nickel oxide, where the Dzyaloshinskii-Moriya interaction isforbidden by symmetry.7 It is interesting to understand whichmechanism is responsible for the spin oscillations observed in
the present work.
The Lagrangian density of the sigma model can be written
as follows:26
L=

∂L 2
1
1
H·
+
2
2
8γ AM0 ∂t
4γ AM02
1+ [H · (L × d)] − Wa (L) ,
A

∂L
×L
∂t
(29)

where H is the effective magnetic field and Wa (L) isthe effective anisotropy energy that includes L-dependent
terms from the Hamiltonian (4) and a contribution from the
Dzyaloshinskii-Moriya interaction, see Eq. (31) below. The
slave variable, magnetic moment M, can be easily expressed
via vector L and its time derivative,M=

HL2 − L(L · H)
1
L×d
+
+
A
AL2
γ AL2

∂L
×L
∂t

,
(30)

where L = |L|. Within the sigma-model approximation, thelength of the vector L should be treated as a constant, L2x +L2y + L2z = const (2M0 )2 . Thus, in the linear approximation
Lx 2M0 − (ly2 + lz2 )/4M0 and the two components ly and lz
can be considered as independent variables. It is in line with ourexperimental observation that the component lx is completely
negligible. The effective anisotropy energy can be taken in theform
Wa (L) =1
1(q3 − q1 )lz2 +22d2− q1 ly2 ,A(31)

where the additive constant is omitted. Free oscillations of
the two components at H = 0 correspond to two independent

magnon modes (F and AF modes), described by the following
equations:d 2 lz+ ωF2 lz = 0,
dt 2

dlz
d
1
m = x̂ lz + ŷ,A
2γ AM0 dt

d 2 ly2+ ωAF
ly = 0,
dt 2

dly
1
.
m = −ẑ2γ AM0 dt(32)

Now let us discuss the excitations of the modes by lightpulses. The interaction of the spin system with the light is
described by the Hamiltonian (10), which for the specific caseof circularly or linearly polarized light reads as (11) or (12),respectively. Within the sigma-model approach, for different
polarizations the interaction terms enter different parts of the
Lagrangian (29): The circularly polarized light contributes±
to the effective field H = H eff,σ , whereas the effect of thelinearly polarized light is described by the time-dependent
contributionδWa (L,t) =1
Ei (t)Ej∗ (t)bij kl Lk Ll
16π

(33)

to the effective anisotropy energy Wa (L). Among all thesecontributions to the Lagrangian, we need to find terms linear
on ly and lz , which produce the “driving force,” i.e., lead to anonzero right-hand side in the equations of motion (32).
The light-induced effective field is directed along z axis,
and it is easy to see that the term (H · (d × L)) gives no“driving force” contributions for both modes. The gyroscopic
term with dl/dt provides such a term for y component of±
the vector l, proportional to Hzeff,σ Lx (dly /dt), but not for
its z component. Thus, for the state of interest (L = Lx x̂ inthe ground state), the IFE can excite the AF mode only. In
the discussion presented above, the only part proportional tobxyxy Lx ly gives an essential contribution to δWa (L,t). Using
these relations, one can find that all the terms do not affect theequation for lz (F mode), whereas the equation for ly describing
the AF mode acquires nonzero right-hand side and reads as±

dHzeff,σd 2 ly2+ A(2γ M0 )2 hlin
+
ω
l
=
−2γ
M
y
0
AF
y ,
dt 2
dt

(34)

where hliny = −∂δWa (L,t)/∂ly is the effective field. To find thesolution of the equation, we will use, as it has been done in theSec. II C, the delta function substitution, E(t)E ∗ (t) → I0 δ(t).
Using the concrete forms for the effective fields, see Eqs. (21)and (26), the right-hand side of the Eq. (34) can be present as
d 2 ly± dδ(t)2+  lin δ(t) ,+ ωAFly =  σ
dt 2
dt

(35)

where the notations
γ M0 fxyz σ ±I0 ,8π
γ 2 AM03 bxyxy sin 2θ lin
I0 lin = −π±σ = ±(36)(37)

are introduced to shorten the formulas. Because the analysisof the second-order equations (34) or (35) is a bit more
complicated than for first-order Landau-Lifshits equations, let
us discuss it in more details. Note that at the short time intervalof the action of the pulse t ∼ t
1/ωAF the behavior of

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PHYSICAL REVIEW B 84, 064402 (2011)

the solution can be described by Eq. (35) where only the
first term and the right-hand side are retained. Integrationof (35) over t from some value t0 < 0, where ly = 0 anddly /dt = 0, till small but positive t, 0 < t
1/ωAF , gives
t
lin
σ±
dly /dt =  (t) +  δ(t). Here (t) ≡ −∞ δ(t  )dt  isthe Heaviside step function, (t) = 0 at t < 0 and (t) = 1
at t > 0. Thus, the initial value of dly /dt is determined by the
amplitude  lin only, dly /dt =  lin at t = +0. Then one more
integration of the above expression for dly /dt over t from±t = −0 till t = +0 gives us ly (t = +0) =  σ . Combining
this results we arrive at the following initial conditions:±

γ M0 fxyz I0σ(ly )t=0 = ±,8π
γ 2 AM03 bxyxy I0lin sin 2θ
dly=−,
dt t=0
π

(38)
(39)

±

where I0σ and I0lin determine independent action of circularly±
and linearly polarized light, respectively, with I0σ and I0lin
being the corresponding integrated pulse intensities. As one
can see from the equation, within the sigma-model approachthe effective magnetic field created by the IFE enters theequation through its time derivative only, whereas the inertial
mechanism is caused solely by ICME. Thus the field-derivative
mechanism of the action of IFE, discussed previously for compensated antiferromagnets,7,49 is responsible for the excitationof spin oscillations in the 4 phase of dysprosium orthoferrite
investigated here. We conclude that it is difficult to realize
the inertial mechanism of the magnetic field pulse action in
the majority of orthoferrites at high temperatures where thesame 4 phase is present. The inertial mechanism has been
observed for a special phase of holmium orthoferrite wherethe vector L is not collinear with the symmetry axis.11 On
the other hand, for the present experiment the ICME leads toinertial mechanism of the spin excitations.
After the action of the pulse, only free spin oscillations
persist in the system. They are described by the solution
lyfree = a cos(ωAF t + ξ ),
d 2 − Aq1
free
mz = asin(ωAF t + ξ ),
A

(40)
(41)

observed phase should take an intermediate value given by
Eq. (42).
It is worth noting that the condition for domination of acertain effect does not translate into a plain comparison of theeffective constant values fxyz and 2M0 bxyxy for IFE and ICME,
respectively. The point is, the ICME contributes throughthe inertial mechanism that is much more effective than thefield-derivative mechanism involved in the action of IFE. In
ourcalculation, this leads to appearance of the large multiplierA/ d 2 − Aq1 = γ Hex /ωAF , where γ Hex ≈ 20 THz, Hex =2AM0 600 T is the exchange field of orthoferrite,22 in thecontribution of ICME; see Eq. (43). Therefore the dominationof IFE, for the same value of the pulse fluence, needs at least
50 times higher value of the corresponding constant, andthe ratio fxyz /2M0 bxyxy is expected to be large enough fororthoferrites. Thus, the above analysis gives us a possibility
to estimate the values of constants responsible for different
inverse magneto-optical effects, IFE and ICME.
C. Comparison between theory and experiment

In the previous discussion, based on the Landau-Lifshitz
equations and the nonlinear sigma model, we came to theconclusion that the time dependence of mz induced via IFE andICME is proportional to sin ωAF t and cos ωAF t, respectively.
The phase of the oscillation is constant and mz is proportional
to either sin ωAF t or cos ωAF t in some region of the pumpwavelength in Fig. 7(a). When the pump pulse is in the visible
region (<800 nm), the probe polarization and mz oscillate assin ωAF t. This property is independent of temperature as shownin Fig. 8(b). On the other hand, when the pump pulse is in thenear-infrared region (1000–1100 nm), the probe polarizationand mz oscillate as cos ωAF t. Thus, we can conclude that thevisible and near-infrared light pulses dominantly induce spin
precession via IFE and ICME, respectively.
A number of reasons can be given for why the dominant
effect varies with pump wavelength. IFE is induced by apulse whose wavelength is near the transition 6 A1 → 4 T2 at
700 nm. On the other hand, ICME is induced by a pulse whose
wavelength is near the transition 6 A1 → 4 T1 at 1000 nm. In
addition, the Faraday rotation angle increases with decreasing
wavelength in DyFeO3 .39,40 This tendency agrees with theresult for the IFE.

where the amplitude a and the phase ξ are determined by the
initial conditions (38) and (39) as follows:
4AM0 bxyxy I0lin sin 2θtan ξ = ∓,±fxyz I0σ d 2 − Aq1

γ M0
a=
4π


2

± 2
fxyz I0σ
2Abxyxy M0 I0lin sin 2θ
.
+4
d 2 − Aq1

(42)

(43)

Finally, we arrive at the previous result: If one of the two
mechanisms, IFE or ICME, is dominating, the the phase of
the mz oscillations takes the values ξ = 0, π or ξ = ±π/2,
respectively. Thus, the observed time dependence of the
Faraday rotation oscillations is proportional to sin ωAF t orcos ωAF t for the dominating role of IFE or ICME, respectively.
If none of the mechanisms is truly dominating, then the

V. CONCLUSIONS

We have studied the dependence of photoinduced spin
precession in DyFeO3 on the wavelength and polarization ofa pump pulse with a pump-probe magneto-optical technique.
The polarization rotation of the probe pulse was dependent
on the pump polarization. Pulses propagating along thez axis with both circular and linear polarizations induced aneffective magnetic field (IFE and ICME) and spin precession.
The dominant component of the dielectric permittivity in both
effects was εxy , and IFE and ICME were induced by its
as
antisymmetric and symmetric parts εxy
and εxy, respectively.
The phase and amplitude of the spin precession weredependent on the pump wavelength in DyFeO3 . A difference inthe pump wavelength changes the dominant effect, giving riseto the spin precession. A visible pulse (wavelength <800 nm)

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RYUGO IIDA et al.

PHYSICAL REVIEW B 84, 064402 (2011)

induced the IFE, and the oscillation of the probe polarization
was proportional to sin ωAF t. On the other hand, a near-infrared
pulse (wavelength of 1000–1100 nm) induced the ICME dominantly, and the oscillation was proportional to cos ωAF t. Whenthe pump wavelength was near the electron transition 6 A1 →4
T2 at 700 nm and 6 A1 → 4 T1 at 1000 nm, the amplitude of
the oscillation was larger than that of the other region.
The ratio of the effective magnetic fields via IFE andICME, fxyz /2M0 bxyxy , is expected to be large enough fororthoferrites. However, the ellipticity of spin precession withAF mode is also very large. Therefore, even though the linearlypolarized light pulse induces a much weaker magnetic field

than the circularly polarized one, ICME can give the sameorder contribution as IFE.

*

21

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ACKNOWLEDGMENTS

This work was supported by KAKENHI (19860020 and
20760008) and JST PRESTO program. B.A.I. was partly
supported by Grant No. 228-11 from the Ukrainian Academyof Sciences and by Grant No. 5210 from STCU. A part of thisresearch is granted by JSPS through FIRST Program initiatedby CSTP. We thank A. K. Kolezhuk for useful discussions andhelp.

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