PHYSICAL REVIEW B 95, 174407 (2017)

Ultrafast optical excitation of coherent magnons in antiferromagnetic NiO
Christian Tzschaschel,1,* Kensuke Otani,2 Ryugo Iida,2 Tsutomu Shimura,2 Hiroaki Ueda,3Stefan Günther,1 Manfred Fiebig,1 and Takuya Satoh1,2,4
1

Department of Materials, ETH Zurich, 8093 Zurich, Switzerland
Institute of Industrial Science, The University of Tokyo, Tokyo 153-5805, Japan3
Department of Chemistry, Kyoto University, Kyoto 606-8502, Japan
4
Department of Physics, Kyushu University, Fukuoka 819-0395, Japan(Received 30 January 2017; revised manuscript received 3 April 2017; published 5 May 2017)
2

In experiment and theory, we resolve the mechanism of ultrafast optical magnon excitation in antiferromagnetic
NiO. We employ time-resolved optical two-color pump-probe measurements to study the coherent nonthermal
spin dynamics. Optical pumping and probing with linearly and circularly polarized light along the optic axisof the NiO crystal scrutinizes the mechanism behind the ultrafast magnon excitation. A phenomenologicalsymmetry-based theory links these experimental results to expressions for the optically induced magnetization
via the inverse Faraday effect and the inverse Cotton-Mouton effect. We obtain striking agreement between
experiment and theory that, furthermore, allows us to extract information about the spin domain distribution. We
also find that in NiO the energy transfer into the magnon mode via the inverse Cotton-Mouton effect is about
three orders of magnitude more efficient than via the inverse Faraday effect.
DOI: 10.1103/PhysRevB.95.174407
I. INTRODUCTION

Antiferromagnetism is rapidly gaining importance as a crucial ingredient of spintronics applications [1,2]. Because of theabsence of a net magnetization in the ground state, it is robust
against externally applied fields and the formation of domains
is not obstructed by magnetic stray fields. Accordingly, thetechnologies envisaged are mainly based on the application ofspin currents instead of magnetic fields [3–9]. In addition, theintimate coupling of the sublattice magnetizations in antiferromagnets in combination with a strong exchange interaction
between neighboring spins implies magnetization-dynamicaltimescales, which are typically orders of magnitude faster thanthose of ferro- or ferrimagnetic materials [10]. Naturally, ultrashort laser pulses come to mind when accessing the dynamical
properties of the antiferromagnetic order. In contrast to thermal
approaches, which are based on local heating of the electronic
and magnetic systems [11], nonthermal excitations would provide a quasi-instantaneous access to the antiferromagnetic spin
system via spin-orbit coupling. Thus they can fully exploit thefaster timescales inherent to antiferromagnets. The two mostprominent nonthermal magneto-optical effects are the inverse
Faraday effect (IFE) [12] and the inverse Cotton-Mouton
effect (ICME) [13]. In a qualitative picture, they represent
impulsive stimulated Raman scattering processes, where theIFE is described by an antisymmetric tensor and the ICME by asymmetric tensor [14–16]. Consequently, the magneto-optical
coupling effectively exerts a torque onto the spin system.
The IFE and ICME have been applied to a variety of material systems [17–24], but a clean discrimination in experimentand theory between the two effects for a pure antiferromagnet
is still due. A particularly obvious candidate for such ananalysis is antiferromagnetic NiO because of its high ordering
temperature, its simple crystallographic structure, and its wellresearched physical properties [25–34]. In addition, it may be*christian.tzschaschel@mat.ethz.ch2469-9950/2017/95(17)/174407(11)an excellent candidate for a clear and insightful experimental
and theoretical discrimination between IFE and ICME because
it has been speculated that in NiO the symmetric part issignificantly larger than the antisymmetric part of the Raman
scattering tensor [31]. Consequently, the ICME would be more
pronounced than the IFE, even though the ICME is a secondorder effect in the magnetic order parameter. Unfortunately,
the pronounced magnetic birefringence of NiO [28] leads toan inseparable mixture of the polarization-dependent Raman
contributions. Hence the spin oscillations observed in NiO
are to date generally induced by such mixture of IFE andICME. Consequently, the mechanism behind the nonthermal
excitation of coherent magnons in NiO has not been identified,
let alone quantified [35–40].
In this paper, we present a comprehensive experimental
and theoretical analysis of IFE and ICME in antiferromagnetic
NiO. We separate the two effects in a nonthermal polarizationdependent two-color pump-probe measurement. The birefringence resulting from the optical anisotropy is avoided byapplying our measurements to a specific single-domain state.
The combination with a symmetry-based phenomenologicaltheory that we develop for quantifying IFE and ICME allows
us to distinguish between the two effects and clarify the driving
force exciting the magnon oscillations in NiO. Moreover, we
compare the magnon generation efficiencies of the two effects.
The paper is organized as follows: the crystallographic
and magnetic lattices of NiO are reviewed in Sec. I A witha special focus on the domain structure. We describe themagneto-optical properties in Sec. II A. Subsequently, basedon that description, we develop a theory for the inverse
magneto-optical effects in NiO in Sec. II B. In Sec. III A, theoptical pump-probe setup is described, and the results of thetheory sections are converted into experimental configurations
that enable IFE and ICME to be measured and distinguished.
Sections III B and III C present the experimental results obtained by linear and circular pump polarizations, respectively.
They are discussed in detail in Sec. IV, where we showthat magnon excitation via the ICME in NiO is significantly

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PHYSICAL REVIEW B 95, 174407 (2017)

corresponding to a reduced crystallographic point symmetry
2/m, which is also the point symmetry of the magnetic lattice[32,42]. This distortion, as well as the resulting linear birefringence, are approximately two orders of magnitude smaller
than that associated with the T domains [29] so that they havenegligible influence on the polarization of the propagating
pump and probe light. For the symmetry-based polarization
analysis, however, the full magnetic 2/m symmetry needs
to be considered, as we shall see later. Antiferromagnetic
ordering along the [112̄] direction breaks the threefold
rotational symmetry; for the resulting 2/m symmetry thetwofold axis is perpendicular to both the rhombohedral
distortion and the easy axis of the spins, i.e., along [11̄0].
With the two sublattice magnetizations M1 (t) and M2 (t),we define the ferromagnetic vector M(t) = M1 (t) + M2 (t) andthe antiferromagnetic vector L(t) = M1 (t) − M2 (t). To study
dynamics, it is convenient to split both quantities into a timeindependent ground state and describe the excitation by a timedependent contribution:

FIG. 1. (a) Crystallographic and magnetic structure of NiO in
the defined coordinate system. (b) Graphical representation of thespin dynamics for the in-plane mode and (c) out-of-plane mode. (d)Schematic of the experimental geometry. θ denotes the azimuth angleof the pump polarization relative to the easy axis of the spins, whereasψ parameterizes the setting of the Wollaston prism. The probe pulse
is always circularly polarized.

more efficient than via the IFE. In Sec. V, conclusions arepresented.
A. NiO structure

NiO is a type-II antiferromagnet with a Néel temperature
TN of 523 K [27]. In the paramagnetic phase, the crystal
has the NaCl-type structure (point group m3̄m). Below TN ,spins are coupled ferromagnetically within the {111} planes
with neighboring planes being coupled antiferromagnetically[Fig. 1(a)] [26]. Furthermore, in the antiferromagnetic phase,
there is a rhombohedral distortion along the 111 direction
arising from exchange striction. This distortion corresponds
to a reduction of the crystallographic point symmetry to
3̄m and induces a significant uniaxial optical anisotropy ofn = 0.003 [28]. The optic axis forms along the direction of
the distortion. Because the four independent 111 directions([111], [111̄], [11̄1], [1̄11]) are energetically degenerate in theparamagnetic phase, the rhombohedral distortion can occur
along any of those directions leading to four twin-domainstates commonly referred to as T -domain states (T0 −T3 ).
The four T -domain states can be distinguished by their linear
birefringence [41].
Within each T -domain state, spins point in one of three
independent 112̄ directions that are perpendicular to thedirection of the rhombohedral distortion [30]. This creates theformation of three spin domain states, commonly referred to as
S-domain states, S1 −S3 , leading to a total of twelve possible
orientation domain states in NiO [34]. The formation of the
S domains leads to another small magnetostrictive distortion,

M(t) = M0 + m(t) = m(t),(1a)

L(t) = L0 + l(t).

(1b)

The dynamic contribution may be a superposition of the two
eigenmodes of the two sublattice antiferromagnetic system,
both of which are optically excitable in NiO [35]. For thein-plane mode (IPM), also termed Bg mode, the modulation of
the antiferromagnetic vector l(t) is along the [11̄0] direction,i.e., it occurs within the sheets of ferromagnetically coupledspins. The oscillating magnetization m(t), in contrast, isalong the [111] out-of-plane direction. The frequency of thismode is IPM /2π  0.14 THz at 77 K [33,35]. The opposite
behavior occurs for the out-of-plane mode (OPM), also termed
Ag mode. The antiferromagnetic vector is modulated along
the [111] direction, whereas the magnetization oscillates
along [11̄0]. The eigenfrequency of the out-of-plane modeis OPM /2π  1.07 THz at 77 K [27,31,35–40,43,44].
In contrast to previous publications [35], we specificallyconsider a T0 domain on a (111)-cut NiO sample, where therhombohedral distortion is along the surface normal. Therefore, the optic axis coincides with the propagation direction oflight at normal incidence and optical anisotropy, especially linear birefringence, can be avoided. For this situation, we define areference system: we choose the x axis to be along the surface
normal, i.e., the [111] direction, the z axis to be along themagnetic easy axis, i.e., the [112̄] direction, and the y axis perpendicular to both to form a right-handed coordinate system,
i.e., along [11̄0]. The orientation is shown in Fig. 1 together
with a schematic representation of the spin motion for the inplane mode [Fig. 1(b)] and the out-of-plane mode [Fig. 1(c)].
Using this notation, Eqs. (1a) and (1b) can be expressed as174407-2⎛⎞
mx (t)
M(t) = ⎝my (t)⎠,
0
⎛ ⎞ ⎛
⎞
lx (t)
0
L(t) = ⎝ 0 ⎠ + ⎝ly (t)⎠.
Lz
0

(2a)

(2b)

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PHYSICAL REVIEW B 95, 174407 (2017)

Here, mx and ly are contributions purely from the in-planemode, whereas my and lx originate from the out-of-planemode.

We briefly review the phenomenological theory of the Faraday effect as well as the Cotton-Mouton effect, both of which
are used to detect magnon oscillations in NiO. Furthermore,a phenomenological theory of the inverse magneto-opticaleffects, i.e., the IFE and the ICME, is presented, whichenables the different magnon excitation mechanisms to be
distinguished. These discussions are accompanied by a special
consideration of the point-group symmetry of NiO.
Light-matter interaction is typically described by an interaction Hamiltonian which, in cgs units, reads [45]ij (M,L)Ei (t)Ej∗ (t).
16πij = ij(0) + ikijMk Mk + ikijL k Lk
(4)

As |M|  |L|, the term quadratic in M can be neglected.
For symmetry reasons, only even orders in L can givenonvanishing contributions to the dielectric tensor. This leads
to the simplified equation
ij = ij(0) + ikijMk Mk + gijLLkl Lk Ll .

(5)

Note that because of the ac character of Mk () and Lk () themagneto-optical coupling constants kijMk and gijLLkl will dependon the frequency . For simplicity, we will henceforth omit
the superscripts M and LL as well as the argument .
Considering the complex dielectric tensor ij (M,L) and theOnsager principle, the absence of absorption leads toij (M,L) = j∗i (M,L) = ij∗ (−M,L) = ij∗ (M,−L).

(7)αy = gyyzz Lz Lz + gyyzx Lz lx ,(8a)αz = gzzzz Lz Lz + gzzzx Lz lx ,(8b)

β = gyzzy Lz ly ,(8c)

ξ = kyzx mx .

(8d)

All other possible contributions to gij kl and kij k vanish incompliance with the 2/m symmetry of the antiferromagnetic
order [42,48]. As the static magnetic linear birefringenceexpressed by Eqs. (8a) and (8b) was not resolved, we assume
gyyzz ≈ gzzzz and redefine:(3)Here, ij (M,L) is the dielectric tensor, which is in general
a complex function of M and L, and Ei is the electric field
amplitude with Ei (t) = [Ei (t)eiωt ] [18,46]. We assume lightpropagating in x direction. Thus E = (0,Ey (t),Ez (t)).
Expanding the dielectric tensor into a power series in M
and L, we obtain with magneto-optical coupling constants kij kand gij kl [47,48]:

LL
ML+ gijMMkl Mk Ml + gij kl Lk Ll + gij kl Mk Ll .


β + iξ
.
 (0) + αzWith Eq. (5) we identify

II. PHENOMENOLOGICAL THEORY
OF MAGNETO-OPTICAL AND INVERSEMAGNETO-OPTICAL EFFECTS IN NiO

Hint = −remaining tensor components:  (0) +αyyy yz=zy zz
β − iξ

(6)

Here, ij∗ denotes the complex conjugate of ij . Equation (6)indicates that the diagonal components ii are purely real,
whereas the off-diagonal components are in general complex.
[ij ] is a symmetric tensor, whereas [ij ] is antisymmetric.
Consequently, the nonzero coefficients in Eq. (5) are realvalued and satisfy kij k = −kj ik and gij kl = gj ikl = gij lk =gj ilk . As the birefringence caused by the magnetostriction is
(0)
(0)
= zz
≡
neglected in our symmetry analysis [29], we set yy
(0)
(0)
(0)
 and yz = zy ≡ 0. Considering an electromagnetic wave
propagating in x direction, we neglect all x components ofthe dielectric tensor and assume the following ansatz for the

αy =  + αy ,
 (0) + (9a)+αz =  + αz ,

(9b)



(0)

with
αy = gyyzx Lz lx ,(10a)αz = gzzzx Lz lx .

(10b)

A. Magneto-optical effects

We now discuss the eigenvalues and eigenpolarizations of
Eq. (7) in the simplified case, where only one of the quantities
α, β or ξ is nonzero. The square roots of these eigenvalues are
the refractive indices corresponding to the eigenpolarizations.
We show that ξ leads to the Faraday effect, i.e., circularbirefringence, whereas α and β induce a linear birefringence
thus leading to the Cotton-Mouton effect.
1. ξ  = 0 and α = β = 0

The refractive indices N± and corresponding eigenpolarizations E± are
N± =

 ±ξ



N±
E0
E± = √ exp iω t −
x
c
2

(11a)


(ŷ ∓ i ẑ).

(11b)Here, ŷ and ẑ correspond to unit vectors along the y andz directions, ω is the angular frequency of the light, andc is the speed of light. Thus the eigenpolarizations E±
describe circularly polarized waves (σ ± ), which are subject todifferent refractive indices N± . Typically, with   |kyzx√mx |,
the circular birefringence N = N+ − N− ≈ kyzx mx /  islinear in mx and results in a rotation of the plane of polarizationof linearly polarized light by
φF = −

ωdkyzx mx∝ mx ,√c (12)where d is the sample thickness. Therefore the magnetization
component mx can be studied by analyzing the Faraday
rotation of linearly polarized light.

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PHYSICAL REVIEW B 95, 174407 (2017)

2. β  = 0 and α = ξ = 0

The eigenvalues N
izations E±45◦ are±45◦

N±45◦ =

 ± β,and the corresponding eigenpolar-



N±45◦
E0
x
E±45◦ = √ exp iω t −
c
2

(13a)



(ŷ ± ẑ). (13b)The eigenpolarizations are linearly polarized with angle ± 45◦
relative to the y direction. Over a propagation distance d, thislinear birefringence induces a phase difference ofωdgyzzy Lz ly∝ ly .
(14)
φ45◦ ≈√c 
Incident circularly polarized light thus becomes ellipticallypolarized with principal axes along the y and z directions.

since the tensor components are dependent on the frequency ofboth the laser and the magnetization, the actual values for the
direct and inverse effects can still be different, as suggested inRef. [50]. However, this has no influence on the results of theensuing investigation as we focus our discussions exclusively
on the coupling parameters for the inverse effects.
We define the effective magnetic fields Heff and heff for m
and l as the partial derivative of the interaction Hamiltonian
with respect to m and l:
Heff = −

(15a)

(15b)

Hence the eigenpolarizations are linearly polarized along they and z directions with different refractive indices Ny,z . Overa propagation distance d, this linear birefringence induces aphase difference ofωd(gyyzx − gzzzx )Lz lx∝ lx .
(16)√2c Consequently, circularly polarized light becomes elliptically polarized with principal axes aligned at ± 45◦ . The
magnetically induced linear birefringence observed in cases
2 and 3 are also known collectively as the Cotton-Mouton
effect.
To summarize, because each component of the dielectric
tensor has a specific dynamical modification, the polarizationof light propagating through the material is altered in a highly
selective way. This selectivity enables the different physicalmechanisms that are responsible for a certain modulationof the magnetization to be distinguished experimentally. In
particular, only the in-plane magnon mode causes oscillations
in mx and ly and can thus be observed via the Faraday effect(case 1) or the Cotton-Mouton effect (case 2). The out-of-planemagnon mode causes oscillations of lx and my and is therefore
only observable via the Cotton-Mouton effect (case 3).
B. Inverse magneto-optical effects

(17)

(18a)
(18b)

where γ is the gyromagnetic ratio. Anisotropy terms leading
to the elliptical precession of m and l, and damping terms are
subsumed into Rm,l . Combining these with Eq. (17) and theinitial conditions M(t = 0) = 0 and L(t = 0) = (0,0,Lz ), we
obtain∂mγ 2=
L [gyzzy {Ey (t)Ez∗ (t) + Ez (t)Ey∗ (t)}x̂∂t
16π z− {gyyxz Ey (t)Ey∗ (t) + gzzxz Ez (t)Ez∗ (t)}ŷ] + Rm ,φyz ≈The Hamiltonian in Eq. (3) with the dielectric tensor definedin Eq. (7) can also be used to describe the inverse magnetooptical effects. In accordance with the previous notion, kij k andgij kl then mediate the IFE and the ICME, respectively, withdirect and inverse effects parameterized by the same coupling
tensors. This assumption should be valid in the region of opticaltransparency [49] to which our investigation is restricted. Yet,

∂Hint
.
∂l

∂m
= −γ {M × Heff + L × heff } + Rm ,∂t∂l= −γ {M × heff + L × Heff } + Rl ,∂t

The refractive indices Ny,z and eigenpolarizations Ey,z are + αy,z ,


Ny
x ŷ,
Ey = E0 exp iω t −cNz
Ez = E0 exp iω t −x ẑ.
cheff = −When an ultrashort light pulse irradiates a sample, theseeffective magnetic fields become the driving force of thenonthermal magnetization dynamics.
The Landau-Lifshitz-Gilbert equations for m and l are [45]3. α  = 0 and β = ξ = 0

Ny,z =

∂Hint
,
∂m

(19a)
iγ
∂l
=−
Lz kyzx {Ey (t)Ez∗ (t) − Ez (t)Ey∗ (t)}ŷ + Rl .
∂t
16π
(19b)
If the magnetization dynamics are induced by an ultrafast
laser pulse of pulse duration τ , which is short compared to thespin oscillation period, i.e., E(t)E ∗ (t) ≈ EE ∗ τ δ(t) = I0 /cδ(t),the terms Rm and Rl can be neglected and Eqs. (19a) and (19b)
can be integrated around t = 0:γτ 2
L [gyzzy {Ey Ez∗ + Ez Ey∗ }x̂
16π z− {gyyxz Ey Ey∗ + gzzxz Ez Ez∗ }ŷ],γτl = −iLz kyzx {Ey Ez∗ − Ey∗ Ez }ŷ.
16π

m =

(20a)
(20b)

These optically induced changes occur instantaneously during
the excitation.
1. Excitation by linearly polarized lightWith linearly polarized light specified by (Ey (t),Ez (t)) =E(t)(sin θ, cos θ ), where θ denotes the angle between the
direction of polarization and the z axis (cf. Fig. 1), Eqs. (20a)174407-4

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PHYSICAL REVIEW B 95, 174407 (2017)

and (20b) lead tomlin =γ
L2 I0 [gyzzy sin(2θ )x̂ − (g1 − g2 cos(2θ ))ŷ],
16π c z(21a)

llin = 0.

(21b)Here, g1 = (gyyxz + gzzxz )/2 and g2 = (gyyxz − gzzxz )/2. After the quasi-instantaneous generation of mx and my , thespins start to precess around their easy axis orientation witha strong ellipticity that reflects the pronounced magnetic
anisotropy perpendicular to this axis. The short axis of theellipse is along m, whereas the long axis is along l [41,51].
The precession can be separated into in-plane and out-ofplane contributions, where for the in-plane mode mx and ly
oscillate with a π/2 phase difference at frequency IPM andfor the out-of-plane mode my and lx oscillate at frequency
OPM . The magnetization dynamics lead toγ
L2 I0 gyzzy sin 2θ cos IPM t,
16π c zγ
AIPM L2z I0 gyzzy sin 2θ sin IPM t,lylin (t) = −16π cγL2 I0 (g1 − g2 cos 2θ ) cos OPM t,
mliny (t) = −16π c zmlin
x (t) =

(22a)
(22b)

(22c)
γ
AOPM L2z I0 (g1 − g2 cos 2θ ) sin OPM t.
lxlin (t) = −
16π c
(22d)
Here, we introduced the anisotropy factors AIPM and AOPM ,which account for the magnetic anisotropy and parametrize theellipticity of the spin precession [18]. The coupling between
the light field and the magnetization is purely described byparameters based on the tensor gij kl , and therefore basedon magnetic linear birefringence. Therefore both modes are
excited by the ICME.
2. Excitation by circularly polarized lightWith circularly
polarized light, σ ± = (Ey (t),Ez (t)) =√E(t)(1, ∓ i)/ 2, analogous considerations as for linearlypolarized light lead toγ
L2 I0 g1 ŷ,
16π c zγ±lσ = ∓
Lz I0 kyzx ŷ.
16π c±mσ = −

(23a)
(23b)

This induces oscillations of m and l according to1γ
Lz I0 kyzx sin IPM t,
16π c AIPMγ±
Lz I0 kyzx cos IPM t,lyσ (t) = ∓
16π c
γ
±
L2 I0 g1 cos OPM t
mσy (t) = −16π c zγ±
AOPM L2z I0 g1 sin OPM t.
lxσ (t) = −16π c±mσx (t) = ∓

(24a)
(24b)
(24c)

Thus, the in-plane mode is linearly dependent on Lz ,obtains a 180◦ phase change upon changing the pump helicity,and couples via kyzx , which is related to magnetic circularbirefringence. Accordingly, it is excited by the IFE, which
creates an effective magnetic field that exerts a torque on thespin system and contributes the term l. Meanwhile, eventhough induced by circularly polarized light, the out-of-planemode is excited via the ICME.
III. EXPERIMENTAL RESULTS
A. Optical setup

We study the magnon dynamics in NiO by performing
impulsive stimulated Raman scattering experiments in thetime domain, which was realized by a pump-probe setup.
We optically excite the sample using a 0.98-eV 90-fs laser
pulse and probe the transient optical properties of the material
with a 1.55-eV 50-fs pulse [35]. The absorption coefficient of
NiO for the pump pulse is approximately 20 cm−1 at 77 K
[25]. By pumping and probing the sample in the highly
transparent regime, we are able to excite and measure theentire volume of our 260-μm-thick NiO slice. Furthermore, weavoid heating effects, which allows us to study the nonthermal
magnetization dynamics. The polarization of both pulses
can be tuned such that any linear or circular polarization
can be realized for the pump and for the probe pulse. The
transmitted part of the probe pulse is split into orthogonal
contributions by a Wollaston prism and measured as intensities
I1 and I2 on a balanced pair of photodiodes. The theory
presented in the previous section allows to predict the resulting
imbalanceη =I1 − I2
I1 + I2−
pumponI1 − I2
I1 + I2(25)
pumpoff

between the photodiodes as a function of the orientation of theWollaston prism, which is parameterized by the angle ψ [cf.
Fig. 1(c)], as well as by the pump and probe polarizations.
We focus on the Cotton-Mouton effect by probing withcircularly polarized light and measuring the ellipticity of thetransmitted light. This enables both in-plane and out-of-planemodes to be observed. The sample is kept at 77 K for all
measurements.
Eliminating ly by combining Eqs. (14) and (22b), we findfor the in-plane mode excited by linearly polarized light thefollowing dependence of the ellipticity on the pump and probe
conditions:
lin
pu
pr
ηIPM
= CAIPM L3z I0 gyzzygyzzy
sin 2θ cos 2ψ sin IPM t. (26)√Here, we defined C = −γ ωd/(16π c2  ). Furthermore, themagneto-optical coupling constants are in general frequency
dependent and can therefore be different for the pump and thepuprobe pulse. This is taken into account by introducing gyzzyprand gyzzy .
Analogously, combining Eqs. (14) and (24b) yields thefollowing dependence for the observation of the in-planemode, when excited by circularly polarized light:(24d)
174407-5

±

σ
pu pr
ηIPM
= ±CL2z I0 kyzx
gyzzy cos 2ψ cos IPM t.

(27)

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PHYSICAL REVIEW B 95, 174407 (2017)

Similar considerations based on Eqs. (16) and (22d) as well as(24d) yield for the out-of-plane mode:
lin
ηOPM
= CAOPM L3z I0
pu

pu

pr

(28)

pu pr

(29)

× g1 − g2 cos 2θ g2 sin 2ψ sin OPM t,±σ= CAOPM L3z I0 g1 g2 sin 2ψ sin OPM t.
ηOPMThus the present model clearly predicts the measurable signalof the magnon dynamics as a function of pump and probe
polarizations. In reverse, it allows the determination of themechanisms leading to magnon excitation. Experimentally
verifying the predictions, which are ultimately summarized in
Eqs. (26) to (29), is the core part of the following section. We
shall first consider excitations using linearly polarized pumppulses and subsequently circularly polarized light.
B. Excitation by linearly polarized light

To verify the predictions regarding linearly polarized pumppulses, i.e., Eqs. (26) and (28), we performed time-resolved
measurements for three different settings: (i) the detection
angle ψ is fixed at 0◦ and the pump polarization angle θ isvaried; (ii) the pump polarization angle θ is fixed at 45◦ andthe detection angle ψ is varied; (iii) the detection angle ψ is
fixed at 45◦ and the pump polarization angle θ is varied.
Figure 2(a) shows time-resolved ellipticity measurements
for setting (i). Here and in all following time-dependentmeasurements the pump pulse triggers an instantaneous
step-like change of η. This step occurs independent ofpolarization which suggests a thermal origin [22] rather
than displacive mechanisms like photoinduced magnetic
anisotropy effects, which are mostly restricted to systems
with dopants or impurities [52,53]. Figure 2(a) reveals asingle oscillation with a periodicity of approximately 8 ps.
The solid lines are fits according to the equation ηIPM =η0 − A exp (−t/τ ) sin (t + B). [Note that A corresponds topupr
CAIPM L3z I0 gyzzy gyzzy sin 2θ cos 2ψ in Eq. (26).] Fitting yields
an oscillation frequency /2π of 0.13 THz, which is inagreement with the expected value of 0.14 THz for the in-planemode [35]. The slight deviation may be temperature related.
The initial phase B turns out to be close to zero, confirming
the sinelike time-dependence of Eq. (26). The red curve in
Fig. 2(b) shows the behavior of the signed amplitude A. It
resembles the predicted sin 2θ function, but a fit proportional
to sin 2(θ − ζ ) reveals a small shift ζ = −6.9◦ ± 0.7◦ , andthus a deviation from the predicted behavior. As we shall see
in Sec. IV A, this phase shift originates from the S-domain
substructure of our single T domain. Distinct from the red
curve, the blue curve in Fig. 2(b) shows the signed amplitude
A of the magnon oscillation for setting (ii). It confirms the
expected cos 2ψ dependence of the in-plane mode amplitude
in both Eqs. (26) and (27). To verify the linear dependence
on the pump intensity, the pump fluence was reduced from
80 mJ cm−2 for setting (i) to 40 mJ cm−2 for setting (ii). The
observed maximum amplitudes of the two curves in Fig. 2(b)differ by a factor of about 2, confirming the predicted behavior.
Figure 3(a) shows time-resolved measurements of themagnetically induced linear birefringence for setting (iii).
According to our model, this allows for the most efficient

FIG. 2. (a) Observation of the in-plane mode from measurementsof the Cotton-Mouton effect in setting (i), i.e., ψ = 0◦ , θ varied.
Curves are vertically displaced for clarity. (b) Signed amplitude A of
optically induced magnon oscillation in setting (i) (red) and setting(ii) (blue). A(θ) ∝ sin 2(θ + ζ ). A(ψ) ∝ cos 2ψ. The difference inthe modulation amplitude reflects the difference in pump power as
mentioned in the text.

observation of the out-of-plane mode. Measurements were
performed on the same spot as for Fig. 2(a). A high-frequency
modulation of the underlying in-plane mode is clearlyvisible. The solid curves are fits according to η = η0 +A0 exp (−t/τ0 ) − A exp (−t/τ ) sin (t+B) − A exp (−t/τ )sin ( t + B ). The exponential terms (∼τ0 ,τ,τ ) and thephase shifts (∼B,B ) are phenomenological additions
parameterizing the magnetic damping and the aforementioned
S-domain substructure of our single T domain, respectively.
The fit reveals /2π = 0.13 THz and  /2π = 1.07 THz
confirming the origin of the observed oscillations as a magnon
excitation. A magnified representation of the region aroundt = 0 is given in Fig. 3(b). The sinelike behavior of theout-of-plane mode is in agreement with Eq. (28). Figure 3(c)shows the dependence of the signed oscillation amplitudeof the out-of-plane mode A on the pump polarization angle
θ . As expected from Eq. (28), we observe a cos 2(θ − ζ )dependence with an isotropic, i.e., polarization-independent
background. This contribution is allowed in materials withmagnetic point groups permitting gyyxz = −gzzxz such as NiO
with its magnetic symmetry 2/m [48]. This phenomenologicaldescription captures all features of the measurements. Thus,

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FIG. 4. Magnon oscillations induced by circularly polarized light.
Curves have been displaced by ±0.1 mrad for clarity. (a) Only thein-plane mode is observed; the signal displays a 180◦ phase shift
following a change in the pump helicity, indicating excitation via the
IFE. (b) The out-of-plane mode has no pump-helicity dependence,indicating excitation via the ICME.

FIG. 3. (a) Observation of the out-of-plane mode by measurementof the Cotton-Mouton effect in setting (iii), i.e., ψ = 45◦ , θ varied.
Curves are vertically displaced for clarity. (b) Magnification of region
between −1 and +4 ps. The out-of-plane mode is sinelike. Data points
around 0 are out of scale. (c) Dependence of the signed amplitude
A of the out-of-plane mode on the linear pump polarization angle θ.
The solid red line is a fit using A = X1 − X2 cos 2(θ − ζ ).

in NiO, with its collinear spin structure, the excitation
mechanism relies on a magneto-optical coupling that issymmetry-allowed in the magnetic point group. Note thatin principle, the noncollinearity of the spins in the domain
walls can lead to an additional isotropic contribution via
the inverse magneto-refractive effect [54,55]. However, with150 nm the width of a NiO domain wall is about an order ofmagnitude smaller than the lateral extension of the domain[56], so that any isotropic contribution from the walls would
be of the order of a few percent only. This value is expected todiminish further by compensation effects among the variety
of wall types that are possible between the several domainstates within a T0 domain of NiO. We therefore neglect the
influence of the inverse magneto-refractive effect and restrict
our description to the simplest possible model required to
explain our results.
The red line in Fig. 3(c) plots the fitting function X1 −X2 cos 2(θ − ζ ) with X1 = (9 ± 1) × 10−5 , X2 = −(1.6 ±0.1) × 10−4 , and ζ = 3.7◦ ± 2.2◦ . The phase shift of 3.7◦
and the presence of the in-plane mode are again caused by
the admixture of additional S domains to the anticipatedsingle-domain state, which are discussed in detail in Sec. IV A.
Summarizing, we are able to observe both magnon
modes of NiO by studying the magnetically induced linear

birefringence, which can be efficiently probed by circularly
polarized light. Furthermore, based on the striking agreement
between measurement and theory, we can identify the ICME
as the driving mechanism for the optical magnon excitation bylinearly polarized light in NiO.
C. Excitation by circularly polarized light

After confirming our model theory for the generation ofmagnons by linearly polarized light, we now consider magnon
excitations driven by circularly polarized optical pulses.
Similar to the previous section, two cases can be distinguished,
where the detection angle of the Wollaston prism is fixed toeither ψ = 0◦ or ψ = 45◦ . Furthermore, the helicity σ ± of thecircularly polarized pump pulse can be altered. Four individual
measurements are obtained (see Fig. 4).
For ψ = 0◦ [Fig. 4(a)], only the in-plane mode is observedin agreement with the theory. The cosinelike behavior of
the probed birefringence accords also with the prediction.
Moreover, the in-plane mode obtains a 180◦ phase shift when
the pump helicity is changed. This is a distinct signature of the
IFE as the driving mechanism of this oscillation. Qualitatively,
the IFE creates an effective magnetic field pulse in x direction,
which acts as a torque on Lz , effectively rotating L aroundthe x axis. This causes a finite contribution in ly , which canbe consequently probed by the induced birefringence via the
Cotton-Mouton effect.
The out-of-plane mode cannot be probed in this geometry
because of the sin 2ψ dependence [Eqs. (28) and (29)]. To
clarify its excitation mechanism, we also took measurements at
ψ = 45◦ , which allows for the observation of the out-of-planemode. The obtained time-traces [Fig. 4(b)] show the expected
sinelike time dependence. Remarkably, the out-of-plane mode
does not obtain a 180◦ phase shift after a change in the pump174407-7

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PHYSICAL REVIEW B 95, 174407 (2017)

helicity, just as predicted by Eq. (29). Consequently, based on
the excellent agreement between theory and all measurements
presented here, we can identify the Cotton-Mouton effect as
the driving mechanism of the out-of-plane mode, even though
it was excited by circularly polarized light. Note that the weakunderlying signature of the in-plane mode in Fig. 4(b) does
not obtain a 180◦ phase shift, when the pump helicity ischanged. Furthermore, it exhibits a sine-like time-dependence
as opposed to the cosinelike time dependence of the in-planemode in Fig. 4(a). Thus it is not excited by the IFE acting on
the underlying S-domain substructure, but rather by the ICME
[Eq. (26)] due to a slight inevitable ellipticity of the circularly
polarized pump pulse.
FIG. 5. Amplitude of the in-plane mode as a function of pumppolarization angle θ for ψ = −45◦ and 0◦ . The dependence can be
explained by contributions from different S-domain states.

IV. DISCUSSION
A. Influence of S2 and S3 domains

The coordinate system in Fig. 1 was chosen such that theeasy axis of the S1 -domain state lies along the z axis. The
probed single-T -domain area, however, may also include S2and S3 domains. Their easy axes are rotated around the xaxis by 120◦ and 240◦ , respectively. As we will see in thefollowing, measurements on different spots on the sample
indeed yielded varying compositions of S domains. For a moredetailed analysis of our data, we therefore have to expand η
by terms representing the contributions from these domainstates. For pumping with linearly polarized light, probing withcircularly polarized light leads to
lin
pu
pr
= CAIPM L3z I0 gyzzygyzzy
sin IPM tηIPM

Let us now analyze the distribution of the domains probed.
For a single S1 domain, the amplitude for ψ = ±45◦ would
be zero for all pump angles. However, the amplitude of thein-plane-mode as a function of pump polarization for different
detection angles (Fig. 5) immediately reveals the presence of amulti-S-domain composition of the sample. A fit of Eqs. (30)yields
A1 = 0.651 ± 0.004,
A2 = 0.064 ± 0.007,
A3 = 0.285 ± 0.012,
In combination with Eqs. (33) and (34), we find× [A1 cos (2ψ) sin (2θ )+ A2 cos (2ψ − 120◦ ) sin (2θ − 120◦ )◦(35)

δ lin = −7.4◦ ± 0.7◦ ,Alin
eff = 0.744 ± 0.002.

◦+ A3 cos (2ψ − 240 ) sin (2θ − 240 )] (30)

(36)
(37)

for the in-plane mode, where A1,2,3 represent the area fractions
covered by the domain states S1,2,3 of the single T domain.
Thus we impose the boundary condition A1 + A2 + A3 = 1.
Note that for A1 = A2 = A3 = 1/3, the isotropic 3̄m symmetry is recovered as an average across the T domain. In thiscase, Eq. (30) simplifies to

As anticipated, the combination of S1 , S2 , and S3 domains
leads to a phase shift δ lin in the polarization dependence.
A similar analysis of the S-domain composition can beapplied for the excitation of the in-plane mode with circularly
polarized light, that is, via the IFE. In analogy to Eq. (30), we
obtainpu
prgyzzy
sin (2θ − 2ψ) sin IPM t, (31)η = 12 CAIPM L3z I0 gyzzy

σ
pu pr
ηIPM
= CL2z I0 kyzx
gyzzy cos IPM t

indicating that the observed amplitude depends solely on the
difference between pump and detection angle. This behavior
has been observed for instance in FeBO3 [18].
For parameterizing the degree of S-domain mixing within
a NiO T domain, it is convenient to consider the setting ψ = 0
for which Eq. (30) can be rewritten as±

× [A1 cos (2ψ)+ A2 cos (2ψ − 120◦ )+ A3 cos (2ψ − 240◦ )],which can be expressed as
±

σ
pu pr
= CL2z I0 kyzx
gyzzy cos IPM t
ηIPM

lin
pu
pr
= CAIPM L3z I0 gyzzygyzzy
sin IPM t
ηIPM
lin
×Alin
eff sin(2θ − δ )

with

 √

δ

lin

3(A2 − A3 )= arctan
4A1 + A2 + A3

(38)

±

with

δ

(33)

σ±

±

× Aσeff cos(2ψ − δ σ )

(39)


 √
3(A2 − A3 )= arctan
2A1 − A2 − A3

(40)

(32)

and

and
Alin
eff =

4A1 + A2 + A3.
4 cos δ lin(34)
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Aσeff =

2A1 − A2 − A3.
2 cos δ σ ±(41)

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PHYSICAL REVIEW B 95, 174407 (2017)

In revisiting Fig. 4, the dominance of the in-planemode for ψ = 0 and its small amplitude of approximately
0.05 mrad for ψ = 45◦ are striking. They point to the
pronounced prevalence of the S1 domain state so that, even±
without an explicit fit of Eq. (38), we can conclude that δ σ ≈ 0
σ±and Aeff ≈ 1.0 in the probed area. A refinement of our analysis
by taking S-domain distributions into account as described inthis section enables, in the following, quantitative statements
about the strength of the magneto-optical coupling constantsin NiO to be made.
B. Magneto-optical coupling constants

This section focuses on the quantitative analysis of themagneto-optical coupling tensors kij k and gij kl . During
the analysis of the out-of-plane mode given in Sec. III B, thefitting parameters X1 and X2 were extracted, which are directly
related to the magneto-optical coupling constants gyyxz andgzzxz . The extracted values yield gzzxz /gyyxz ≈ −3.6. Thisis a significant deviation from the isotropic case with 3̄msymmetry, where the ratio would be −1 [42,48]. This is strong
confirmation that, although the deviation from the crystallographic point symmetry 3̄m toward 2/m by magnetostriction
from the S domains is small, the magneto-optical properties
of NiO have to be discussed in the framework of the magnetic
point symmetry 2/m.
Furthermore, by comparing the oscillation amplitude of
the in-plane mode in Figs. 2 and 4, the magnon generation
efficiency via IFE and ICME can be compared. The pumpfluences were 80 mJ cm−2 in both cases. In Fig. 2, themagnon was excited by the ICME with a maximum oscillation
amplitude lyICME of approximately 4.7 mrad. In contrast, forgeneration via the IFE [Fig. 4(a)], the observed oscillation had
an amplitude lyIFE of 0.13 mrad. In both cases, the dynamics
were probed via the contribution of ly to the Cotton-Mouton
effect. The quantitative evaluation of the two excitation paths
is hindered, however, by the multi-S-domain distribution. With
the analysis of the previous section, we can now renormalize
the measured amplitudes for single-S-domain samples.
From Eq. (32) and (39), we see that the ratio of the spin
precession amplitudes is determined bylyICME
lyIFE

pu

Alin Lz gyzzy= AIPM σeff±.
pu
Aeff kyzx(42)

The anisotropy factor AIPM can be derived from the exchange
field [35] HE = 2π × 27.4 THz/γ and the angular frequencyof the mode IPM = 2π × 0.14 THz according to [18]AIPM =

2γ HE≈ 400.
IPM(43)

The second factor in Eq. (42) is geometric and accounts
for the distribution of S domains within the probed area.
σ±With our previously determined values for Alineff and Aeff ,we conclude that the ratio of the induced magnetizationspu
pu
is Lz gyzzy /kyzx ≈ 0.1. Consequently, the ICME induces amagnetization, which is about an order of magnitude smaller
than that of the IFE in NiO. Even though NiO is structurally
different, this value is in line with the values obtained by Raman scattering in rutile structure antiferromagnets [57]. Nevertheless, in NiO, this is overcompensated by the pronounced

magnetic anisotropy so that in total the amplitude ratio of
the induced magnon oscillation on a single S domain equals
AIPM Lz gyzzy /kyzx ≈ 50.
Moreover, we can consider the magnetic anisotropy energy,
which applies to the in-plane mode:a 2 b 2m + l .
(44)
2 x 2 yThis anisotropy leads to an elliptical spin motion. Consequently, mx = 0, when ly is maximized and vice versa.
Therefore the ratio of the energies pumped into the magnetic
system by the ICME and the IFE scales with the square of theratio of the ly amplitudes, which is about 502 , or 2500.
As the IFE and ICME are described by antisymmetricand symmetric tensors kij k and gij kl , respectively, we can
now revisit the apparent contradictions in earlier Raman
scattering experiments [31]. There, it had been argued that thecommonly accepted antisymmetric Raman scattering tensor
is not sufficient to explain their results, but a symmetrictensor would. Moreover, they estimated that the symmetriccontribution would be dominant. This is now confirmed,
explained, and quantified by our measurements.
Haniso =V. CONCLUSIONS

We performed time-resolved pump-probe measurements of
two magnon modes in antiferromagnetic NiO. Measurements
were performed on T0 domains on the (111) surface of thesample. Thus, pump and probe pulses were propagating along
the optic axis of the crystal, which avoids loss of the initial lightpolarization due to birefringence. This allowed us to study thedependence of the amplitude and phase of the induced magnon
oscillations on pump polarization in detail. Comparing themeasurements to an analytical model under consideration of
the full magnetic 2/m point symmetry, we clarified the driving
force of the individual magnon modes. Our model predicts
clear selection rules for the dependence of the optical responseon the probe conditions, which were verified in experiments.
The ICME constitutes the excitation mechanism forboth the in-plane and the out-of-plane magnon modes bylinearly polarized light. Its analysis even provides highly
sensitive quantitative access to the distribution of the elusive
S-domain substructure of the otherwise dominating T -domain
distribution.
When circularly polarized pump pulses are used,
the general behavior of the in-plane mode is qualitatively
different from the out-of-plane mode. Such pulses propagating
along the x axis excite the out-of-plane mode via the ICME;
the IFE becomes the driving mechanism of the in-plane mode.
Comparison of the amplitudes of the magnon oscillations
resulting from ICME and IFE revealed that the energy transfer
into the magnetic system via the ICME is about three ordersof magnitude more efficient than via the IFE. Whereas themagneto-optical coefficients parameterizing the ICME areabout an order of magnitude smaller than those of the IFE,
the dynamics induced by the ICME are significantly more
pronounced due to the strong magnetic anisotropy. This
resolves the long-standing question about the proclaimed
dominance of the second-order ICME over the first-order IFE
derived from Raman scattering experiments.

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ACKNOWLEDGMENTS

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