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Title

Observation of terahertz magnon of Kaplan-Kittel exchange resonancein yttrium-iron garnet by Raman spectroscopy

Authors

Wei-Hung Hsu, Ka Shen, Yasuhiro Fujii, Akitoshi Koreeda, Takuya
SatohCitation

Physical Review B, Vol. 102, Issue 17, pp. 174432-1-7

Pub. date

2020, 11

Copyright(c) 2020 American Physical Society

DOI

http://dx.doi.org/10.1103/PhysRevB.102.174432

Powered by T2R2 (Tokyo Institute Research Repository)

PHYSICAL REVIEW B 102, 174432 (2020)

Observation of terahertz magnon of Kaplan-Kittel exchange resonancein yttrium-iron garnet by Raman spectroscopy
Wei-Hung Hsu ,1,* Ka Shen,2 Yasuhiro Fujii ,3 Akitoshi Koreeda,3 and Takuya Satoh

1,4,†

1

2

Department of Physics, Kyushu University, Fukuoka 819-0395, Japan
Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, Beijing 100875, China
3
Department of Physical Sciences, Ritsumeikan University, Kusatsu 525-8577, Japan
4
Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan(Received 7 May 2020; revised 4 November 2020; accepted 5 November 2020; published 20 November 2020)
Backscattering Raman spectroscopic investigations were performed on an yttrium-iron garnet single crystal
using linearly and circularly polarized light. A terahertz (THz) magnon of the Kaplan-Kittel (KK) exchange
resonance was discovered, which had been regarded as unobservable via optical methods. The KK exchange
resonance had a 7.8-THz frequency at 80 K, and the polarization selection rule led to an antisymmetric Raman
tensor of the A2 mode. Moreover, the assignment of all the Raman-active phonon modes, 3A1g , 8Eg , and 14T2g ,was proposed. This study will stimulate further investigation of the coupling of THz magnons and phonons andpave the way toward THz optomagnonics.
DOI: 10.1103/PhysRevB.102.174432
I. INTRODUCTION

Yttrium-iron garnet (Y3 Fe5 O12 , YIG), which possesses
superior properties such as long-lived spin waves with very
low damping (α ≈ 10−5 ) and a high Curie temperature(T c = 560 K), has been regarded as an important magnetic
material since its discovery over 60 years ago [1–4]. YIG
is a ferrimagnetic insulator that is one of the best candidates for applications in spintronics [5], magnonics [6,7],
and spin caloritronics [8]. For developing magnetic devices
with ultrafast responses, studies on terahertz (THz) magnon
mechanisms are inevitable [9–14]. However, previous studies
on YIG mostly focused on applications of acoustic magnons
with frequencies in the gigahertz (GHz) range [15–19]. Existing studies provide limited information about the propertiesof THz optical magnons in YIG [4]. Recent investigationsof full magnon band structures have been based on several
theoretical methods [20–24]. In experimental studies, the THz
magnons in YIG have been identified only using inelastic neutron scattering spectroscopy [25–28]. Hence, to understandthe behaviors of the THz magnons in YIG in further detail,it is necessary to investigate other optical properties such as
the selection rule using Raman spectroscopy.
YIG contains 20 Fe atoms in the primitive bcc unit cell.
The magnetic moment is carried by Fe3+ spins in the 12
tetrahedral d sites (majority) and 8 octahedral a sites (minority) with an antiparallel state in the unit cell, forming a
two-sublattice ferrimagnet [3]. Among the 20 magnon modes,
there are two types of modes with opposite precession directions. Twelve modes belong to the counterclockwise (along
the direction of the applied magnetic field) mode, which hasa relatively low frequency and inconspicuous temperature-*†dependent frequency shift. The other eight modes belong to
the clockwise mode, which has a relatively higher frequency
and significant temperature-dependent frequency shift [20].
On the other hand, in a ferrimagnet with two sublattices, there
are two uniform modes at wave number k = 0, namely, an
acoustic mode (Kittel mode) with counterclockwise rotation
and an optical mode with clockwise rotation [29–31]. The
optical mode is identical to the Kaplan-Kittel (KK) exchange
resonance [32] between a-site and d-site Fe sublattices, which
has been considered unobservable via optical [infrared (IR)or Raman] methods [29] and has not been observed to date[33,34].
Therefore, in this study, we investigated the magnetic
and crystallographic properties of a single-crystal YIG using
polarized-backscattering Raman spectroscopy. Considering
the temperature dependence of Raman shift from 80 to 140 K,
we clearly identified a THz magnon of the KK exchange resonance. With regard to the crystallographic properties, although
the phonon modes of YIG have been studied using Raman
spectroscopy for decades, not all the phonon modes havebeen observed [34–37]. Here, using linearly and circularly
polarized light, we completed the phonon-mode assignmentof YIG.
II. MATERIALS AND METHODS

The a-site and d-site magnetizations of the Fe sublattices
in YIG are defined as Ma and Md , respectively. Under an effective magnetic field, the magnetization during precessionalmotion in response to torques is described by the LandauLifshitz-Gilbert (LLG) equation. The LLG equations for Ma
and Md neglecting damping can be written as follows:dMa= −γ Ma × Ha ,dtdMd
= −γ Md × Hd ,dtjacky81418@gmail.com
satoh@phys.titech.ac.jp2469-9950/2020/102(17)/174432(7)174432-1(1)©2020 American Physical Society

HSU, SHEN, FUJII, KOREEDA, AND SATOH
(a)

(b)

(c)

PHYSICAL REVIEW B 102, 174432 (2020)(d)

M

785 Diode Laser
CL

HWP PBSBPF BPF
M
M
HWP

x

PF

Monochromator

FIG. 1. Schematic of the KK exchange resonance mode. (a) Sublattice magnetizations Ma and Md under effective magnetic fields
Ha and Hd in the ground state, respectively. (b) Ferromagnetic andantiferromagnetic vectors M and L under effective magnetic fields
H and h in the ground state, respectively. (c) In the KK exchange
resonance, Ma and Md precess in the clockwise direction. (d) L also
precesses in the same direction, while M is stationary.

where γ is the gyromagnetic ratio of iron that is invariant inboth a- and d-site sublattices. The effective magnetic fields Ha
and Hd acting on Ma and Md can be expressed as Ha = −λMd
and Hd = −λMa , respectively, as shown in Fig. 1(a). The
Weiss constant λ is positive. Assuming Ma = (−Ma , may , maz )and Md = (Md , mdy , mdz ), the precessional motions with timedependence exp(iωt ) can be derived from Eq. (1). The
frequency of the KK exchange resonance is obtained asfollows [32]:ω = λγ |Ma − Md |.

(2)

Here, the effective fields arising from magnetic anisotropy,
demagnetization, and magnetic dipoles are omitted because
their contributions to the resonance frequency are negligibly
small.
Because the magnetizations in the a- and d-site sublattices orientate in opposite directions with strength ratio|Md | : |Ma | = 3 : 2, the ferromagnetic M = Ma + Md andantiferromagnetic vectors L = Md − Ma coexist in YIG [seeFig. 1(b)]. M and L would also exhibit precessional motion
under the effective fields H and h, expressed as follows:
H = Hd + Ha = −λM,
h = Hd − Ha = λL.

(3)

The LLG equations for the vectors M and L are as follows:dMγ= − (M × H + L × h),
dt
2
dL
γ
= − (M × h + L × H).
(4)dt
2
Assuming M = (M, my , mz ) and L = (5M, ly , lz ) according to
the ratio of Md and Ma in YIG, the time-dependent motionsof M and L in the y-z plane are expressed as follows:
dmy
dmz
=
= 0,
dt
dt

Sample

z

y

(5)

dly
= −λγ M(5mz − lz ),dtdlz= −λγ M(ly − 5my ).
(6)dt
Equation (5) shows that the ferromagnetic vector M is notin precessional motion [see Fig. 1(d)]; this implies that the

L
CCD

L

PH L

EF

A

BS

HWP
(QWP)

OL

FIG. 2. Homemade backscattering Raman-spectroscopy system.
The incident light from the 785-nm diode laser (LD785-SEV300,
Thorlabs) passes through a collimating lens (CL), half-wave plate(HWP), polarized beam splitter (PBS), bandpass filters (BPF), polarized filter (PF), beam splitter (BS), quarter-wave plate (QWP), andobjective lens (OL). The diameter of the focused laser spot equals
25 μm. After sample irradiation by the incident polarized light at
40 mW, the scattered light from the sample is polarized using a
Glan-Taylor prism (denoted by A). Subsequently, the inelasticallyscattered light is separated from its elastically scattered counterpart
using an edge filter (EF), combination of lenses (L), and pinhole(PH). The scattered light comprising different wavelengths is spatially separated by the grating (1200 g/mm grating blazed at 500 nm)
in the monochromator (Acton SpectraPro 2500i, Princeton Instruments) followed by subsequent detection by CCD image sensors(S11501-1007S, Hamamatsu Photonics). The frequency resolutionof this Raman system equals 1.0 cm−1 . The heating caused by the
focused laser spot results in a 1–2 cm−1 redshift in the Raman
spectrum.

projecting vectors of the y-z plane, namely, Md and Ma , are
not only antiparallel but also of the same magnitude [seeFig. 1(c)]. On the other hand, the antiferromagnetic vector L precesses clockwise, and the precessional frequency
obtained using Eq. (6) is the same as that obtained using
Eq. (2), namely, ω = λγ M. This frequency is equivalent to(12Sd − 8Sa )|Jad | = 10|Jad |, where Sa = Sd = 5/2 and Jad isthe exchange constant between the magnetizations at the aand d-site sublattices [23]. Thus, the KK exchange resonanceis identical to the clockwise uniform mode at k = 0 [29,30](α1 mode in Ref. [23]).
In the proposed 180◦ backscattering Raman-spectroscopy
system (Fig. 2), the diode-laser radiation with 785-nm wavelength was linearly and circularly polarized using polarizers
as well as half- and quarter-wave plates. In accordance withthe selection of polarization of the incident and scattered light,
two linearly polarized configurations [parallel () and crossed(⊥)] and four circularly polarized configurations (RL, LR,
RR, and LL) were defined as shown in Fig. 3. The helicityof the circularly polarized light was defined as the direction ofpolarization rotation in the sample plane, irrespective of the
propagation direction.
The sample was mounted under vacuum maintained inside
an optical cryostat (VPF-100, JANIS) capable of controlling
temperatures over a wide range from the boiling point ofliquid nitrogen to 500 K. The temperature sensor was attachedto the sample holder made of copper.
Before we searched for the KK exchange resonance mode,
we investigated the phonon modes using angle-resolved polarized Raman spectroscopy [38] to distinguish the magnon and

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PHYSICAL REVIEW B 102, 174432 (2020)

Parallel

RL

RR

CrossedLR

LL

linearly polarized light propagating along the z direction.
On the other hand, we have the Jones vectors |H = (1, 0)
for horizontal polarization,|V = (0, 1) for vertical polariza√
2
forright-handed
circular polarization,
tion, |R = (1, −i)/√and |L = (1, i)/ 2 for left-handed circular polarization [40].
Therefore, the Raman intensity ratios [I : I⊥ : IRL : IRR ] of
A1g , Eg , and T2g are calculated to be [1 : 0 : 0 : 1], [1 : 1 :2 : 0], and [3 : 2 : 4 : 1], respectively [41]. Further, ILR = IRL ,and ILL = IRR . In addition, an in-plane magnetic field of 2.5
kOe was applied using permanent magnets to saturate oursample magnetically. This ensured that Faraday rotation of theincident and scattered light in the sample did not influence theRaman selection rules.

yxIII. RESULTS AND DISCUSSION
FIG. 3. Schematics of linearly and circularly polarized configurations for selection rules. The parallel- and crossed-polarized lightrays indicate that the polarizations of the incident and selected
scattered light are parallel and perpendicular, respectively. R andL represent the right- and left-handed helicities, respectively, of thecircularly polarized light.

phonon modes. A YIG single crystal grown by the floatingzone technique was oriented along the [111] direction andpolished to a thickness of 160 μm. YIG belongs to the cubic
centrosymmetric space group Ia3̄d and crystallographic point
group m3̄m [2]. The phonon Raman tensors are described in
the Appendix, where the bases of the Raman tensors [39] aretransformed to x̂  [112̄], ŷ  [1̄10], and ẑ  [111].
For the crystallographic point group m3̄m, all three
Raman-active phonon modes, namely, A1g , Eg , and T2g , arenot expected to have polarization azimuth dependence on

Figures 4(a) and 4(b) show the Raman spectra ranging
from 100 to 775 cm−1 at 80 K and their magnifications,
respectively. In previous studies, with only linearly polarized
configurations, some ambiguous peaks could not be distinguished [34,35]. However, in the present study, using bothlinearly and circularly polarized configurations, all the phononmodes, namely, 3A1g , 8Eg , and 14T2g , in YIG were successfully assigned and are listed in Table I.
By considering the THz magnon frequency range reported in [21–24,26,27], we focused on the spectra in the
240–270-cm−1 range, as depicted in Figs. 5(a) and 5(b) for
the RR- and LL-polarized configurations, respectively. For
the RR-polarized configuration, we observed a tiny signal toshift significantly from 80 to 140 K and blend into the tailof the T2g phonon signal above 140 K. Additionally, a nearly
identical signal was observed for the LL-polarized configuration. Because of the significant temperature dependence ofFIG. 4. (a) Full Raman spectra with various polarization configurations at 80 K. For linear polarization, dark and light green spectra
represent parallel- and crossed-polarized configurations, respectively. For circular polarization, black, red, blue, and purple spectra represent
the RL-, LR-, RR-, and LL-polarized configurations, respectively. (b) Magnification of (a).
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PHYSICAL REVIEW B 102, 174432 (2020)

TABLE I. Assignment of phonon signals in YIG (111) Raman
spectra at 80 K. All shifts are in cm−1 .
Ref. [34]Mode

Ref. [35]ModeThis study

Mode

130
175

T2g
T2g

131
174

T2g
T2g

193
237
274

T2g
T2g
Eg

194
238
274

T2g
T2g
Eg + T2g

132
174
178
194
239
272
274
290

T2g
T2g
T2g
T2g
T2g
Eg + T2g
T2g
Eg

315

Eg

319
324

Eg
T2g

323
340
345
377
396
416

T2g
A1g
Eg
T2g
T2g
Eg + T2g

446

T2g

505
507
581
587
620
675

Eg
A1g
T2g
T2g
Eg
Eg

347
380

Eg + A1g
T2g

346
378

Eg
T2g

420
449

Eg + T2g
T2g + A1g

416
419
445
456

Eg
T2g
T2g
Eg

507

A1g

504

A1g

593

T2g

592
624

T2g
Eg

698

A1g + Eg

685
692
704
711
736

740

A1g

Eg + T2g
Eg + T2g
A1g
T2g
A1g orEg + T2g

737
739

A1g
Eg

the frequency shift, the tiny signal was assigned to the THz
magnon. The wave number k of the detected magnon in the180◦ backscattering geometry was 4π n/λ (n is the refractive
index, λ is the incident light wavelength). Therefore, a0 k ≈0.05 rad  1 (the lattice constant of YIG a0 = 1.24 nm [2]),meaning that the THz magnon of the KK exchange resonancewith k ≈ 0 was excited at 7.8 THz, the frequency corresponding to 260 cm−1 at 80 K.
Furthermore, the temperature-dependent frequency shiftsof the THz magnon in our RR and LL spectra were identified
and compared to the results obtained from neutron scattering
measurements [25,28] and simulations [23,24], as shown in
Fig. 6(a). Our results show the same tendency as reported
in other studies. However, there is a frequency deviation
between our results and the others. The exchange constant
Jad was determined in previous studies based on the resultsof magnetization, specific heat measurement, inelastic neutron scattering, or first-principles calculations. However, weconsider that Raman spectroscopy provides a more precise
frequency resolution than other techniques. By extrapolating
the magnon frequency of 7.8 THz at 80 K down to 4 K
based on the temperature dependence of magnetization [42],we obtained a frequency of 8.0 THz at 4 K. This yields
Jad = −38 K according to 10|Jad | = 8.0 THz. Furthermore,

FIG. 5. YIG (111) Raman spectra in the 240–270-cm−1 range
with (a) RR- and (b) LL-polarized configurations in the 80–140 K
temperature range. The red, green, and purple curves fit the magnon
and two T2g phonons, respectively. The blue curve denotes the sumof the red, green, and purple curves. Fitting was performed usinga Voigt function that represents the convolution of the Gaussianand Lorentzian functions with full widths at the half maxima of
1.0 and 2.5 cm−1 , respectively. Unlike other phonon signals, themagnon peaks indicated by arrows represent significant temperaturedependent frequency shifts.

the in-plane magnetic field of 2.5 kOe shifted the magnon frequency by a few GHz, which is well below the experimental
resolution.
Moreover, we focused on the same frequency range andrepeated the Raman measurements with all linearly and circularly polarized configurations at 80 K; the results are shown in
Fig. 6(b). The Raman intensity ratio of the KK mode was [I :I⊥ : IRL : IRR ] = [0 : 1 : 0 : 1], indicating that the magnon Raman tensor can be expressed in the antisymmetric form as


0
iK
R=
.
−iK
0
This antisymmetric form of the KK mode corresponds to linear (first-order) magnetic excitation, where K is proportional
to mz [43–47]. The similarity of the KK mode to the one in
the GHz acoustic magnon (Kittel mode) excited through lightscattering [17–19] indicates that the KK and Kittel modes
have the same symmetry. In the KK mode, all the atomic
spins from the same sublattice simultaneously rotate with theirspins always parallel and with the same precession amplitude,
similar to the Kittel mode in the ferromagnetic subsystem.
The other sublattice behaves in the same way. Therefore, thedirection of precessional motion of each sublattice is reversed
under mirror operations (σv or σd ), which corresponds to theA2 mode [48]. Note that the antisymmetric form of the Raman
tensor and the corresponding A2 mode cannot be assigned tothe phonon mode under the m3̄m point group.
Symmetric contributions owing to quadratic (second order)
excitation [45,47,49–51] were not observed in the present
study, probably because the linear magneto-optical (Faraday)effect dominated the quadratic magneto-optical (CottonMouton) effect at a wavelength of 785 nm [52].

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PHYSICAL REVIEW B 102, 174432 (2020)

does not precess owing to the equality of γ for Ma and
Md , this mode cannot be observed by IR spectroscopy andmagnetic resonance spectroscopy.
IV. CONCLUSIONSAll the phonon modes, namely, 3A1g , 8Eg , and 14T2g , in
YIG were identified and assigned using linearly and circularly
polarized light through Raman spectroscopy. A THz magnonof the KK exchange resonance in YIG was discovered in
Raman spectra at temperatures from 80 to 140 K, and theexchange constant was Jad = −38 K. The selection rule ofmagneto-optical coupling in YIG was experimentally confirmed, which suggested an antisymmetric magnon Raman
tensor (A2 ) corresponding to the linear magneto-optical effect.
This study will stimulate further investigation of the couplingof THz magnons and phonons for applications involving spin
caloritronics and pave the way toward THz optomagnonics.
ACKNOWLEDGMENTS

FIG. 6. (a) Temperature-dependent frequency shift in the KK
mode. The red and black lines with error bars denote the KK mode
observed in our Raman spectra with RR- and LL-polarized configurations, respectively. The purple line with squares [25] and orange
line with pentagons [28] represent the experimental data for neutron
scattering. The blue line with triangles [24] and green line withhexagons [23] indicate the simulation results obtained using different
models. The KK mode is fitted with normalized magnetization (yellowish green) [42]. (b) The selection rule for the KK mode of YIG
observed with crossed-, RR-, and LL-polarized configurations at 7.8
THz and 80 K.

In addition, unlike the acoustic magnon, the ferromagnetic
vector M of the KK mode does not precess, as shown in
Fig. 1(d). Therefore, the KK mode was considered to be unobservable via optical methods [29]. Nevertheless, the KK mode
was observed in our Raman method as a linear magnetic excitation, which usually results from the precession of M. This
contradiction can be explained as follows. The absorption
coefficient of YIG at 785 nm is 50–100 cm−1 [53–55]. This
is attributed to off-resonant transition with the nearest resonances at 900 nm (6 A1g → 4 T1g ) and 700 nm (6 A1g → 4 T2g )in octahedral Fe3+ ions [54]. The Faraday rotation of 600◦ –800◦ /cm at 785 nm [52,56–59] dominantly originates fromoptical transitions in octahedral Fe3+ ions [59–62]. Therefore,
the Faraday rotation is more sensitive to Ma than Md . The KK
mode resulted from the precessional motion of Ma [63].
The KK exchange resonance between iron and rare-earth
sublattices is identified in rare-earth iron garnets through IR[64–66] and pump-probe [67–69] spectroscopy. However, in
YIG, because the ferromagnetic vector M of the KK mode

We would like to thank J. Barker and G. E. W. Bauerfor valuable discussions and a critical proofreading of themanuscript. We also thank P. Maldonado, P. M. Oppeneer,
L.-W. Wang, and K. Xia for sharing with us their data on
phonon dispersion calculations. This study was supportedby the Japan Society for the Promotion of Science (JSPS)
KAKENHI (Grants No. JP15H05454, No. JP17K18765, No.
JP19H01828, No. JP19H05618, No. JP19K21854, and No.
JP26103004) and the JSPS Core-to-Core Program (A. Advanced Research Networks).
APPENDIX: PHONON-MODE RAMAN TENSORSThe Raman tensors R pertaining to the crystallographic
point group m3̄m with x̂  [100], ŷ  [010], and ẑ  [001] canbe expressed as follows [39]:
⎛
⎞
1 0 0
 
R A1g = a⎝0 1 0⎠,
(A1)
0 0 1
⎛
⎞
1 0
0
 1
0 ⎠,
R Eg = b⎝0 1
0 0 −2⎛ √⎞− 3 √00 2
(A2)
R Eg = b⎝ 0
3 0 ⎠,
0
0
0
⎛
⎞
0 0 0
 1
R T2g = d ⎝0 0 1⎠,
0 1 0
⎛
⎞
0 0 1
 2
R T2g = d ⎝0 0 0⎠,
1 0 0
⎛
⎞
0 1 0
 3
(A3)
R T2g = d ⎝1 0 0⎠.
0 0 0
In the above equations, a, b, and d indicate the nonzero Raman
tensor elements.

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HSU, SHEN, FUJII, KOREEDA, AND SATOH

PHYSICAL REVIEW B 102, 174432 (2020)

√
√ ⎞
⎛
−4
−2 3 −√ 2d⎝ √R T 2g =
6 ⎠,
−2√ 3
√0
6
6
4
− 2
√ ⎞
√
⎛
−4
2 3 −√2 √d
R T 22g = ⎝ 2 √30
− 6 ⎠,√
64− 2 − 6
√ ⎞
⎛
2
0
2 2
 
d
0
−6
0 ⎠.
R T 32g = ⎝ √
6
0
4
2 2

Because our YIG sample was cut along the [111] direction,
the original tensors were transformed from the [001] direction
to the [111] direction using the transformation operator T
given by
⎛

√1
⎜ 16
√
T =⎜
⎝ 6
− √26

− √12
√1
2

0

⎞

√1
3
⎟
√1 ⎟.
3⎠
√1
3

(A4)

This equation rotates x̂  [100], ŷ  [010], and ẑ  [001] tox̂  [112̄], ŷ  [1̄10], and ẑ  [111], respectively. Subsequently, the Raman tensors R = T −1 RT can be expressed as
⎛

1 0
 
⎜
R A1g = a⎝0 1
0 0
⎛
−1 0
 1
⎜
1
R E g = b⎝ 0
√
2 0
⎛
0
1
 2
⎜
0
R E g = b⎝1
√
0
2



⎞
0
⎟
0 ⎠,
1
√ ⎞
2
⎟
0 ⎠,

(A7)

The Raman scattering intensities I (êi , ês ) for A1g , Eg , and T2g
can be expressed as
 
2
(A8)
IA1g (êi , ês ) ∝ êi∗ · R A1g · ês ,
 
 
2
2
IEg (êi , ês ) ∝ êi∗ · R Eg1 · ês + êi∗ · R Eg2 · ês ,

(A5)

(A9)

 
 
2
2
IT2g (êi , ês ) ∝ êi∗ · R T2g1 · ês + êi∗ · R T2g2 · ês 2+ êi∗ · R T2g3 · ês .
(A10)

0

⎞
0
√ ⎟
2⎠,


1

(A6)

In the above equations, êi and ês represent Jones vectors
corresponding to the incident- and scattered-light rays, respectively.

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