# Sublattice-selective inverse Faraday effect in ferrimagnetic rare-earth iron
garnet

Toshiki Hiraoka  Department of Physics, Tokyo Institute of Technology, Meguro,
Tokyo 152-8551, Japan  Ryo Kainuma  Department of Physics, Tokyo Institute of
Technology, Meguro, Tokyo 152-8551, Japan  Keita Matsumoto  Department of
Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan  Kihiro
T. Yamada  Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo
152-8551, Japan  Takuya Satoh<EMAIL_ADDRESS>Department of Physics,
Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan  Quantum Research
Center for Chirality, Institute for Molecular Science, Okazaki, Aichi
444-8585, Japan

###### Abstract

We performed time-resolved pump–probe measurements using rare-earth iron
garnet Gd3/2Yb1/2BiFe5O12 as a two-sublattice ferrimagnet. We measured the
initial phases of the magnetic resonance modes below and above the
magnetization compensation temperature to clarify the sublattice selectivity
of the inverse Faraday effect in ferrimagnets. A comparison of the time
evolution of magnetization estimated using the equations of motion revealed
that the inverse Faraday effect occurring in ferrimagnetic materials has
sublattice selectivity. This is in striking contrast to antiferromagnets, in
which the inverse Faraday effect acts on each sublattice identically. The
initial phase analysis can be applied to other ferrimagnets with compensation
temperatures.

The ultrafast control of magnetic materials using light pulses has attracted
considerable interest over the years. In magnetic insulators, the inverse
Faraday effect (IFE) and Faraday effect (FE) have been applied to excite and
detect magnetization dynamics, respectively.[1] The IFE, which is the reverse
of the magneto-optical FE, was theoretically proposed by Pitaevskii[2] and
Pershan [3] and demonstrated by van der Ziel et al.[4] Ultrafast magnetization
control has been demonstrated in weak ferromagnets, such as DyFeO3[1] and
FeBO3,[5] and pure antiferromagnets, such as NiO,[6] via the IFE using
femtosecond light pulses. Here, an effective magnetic field pulse
$\mathbf{H}_{\mathrm{IFE}}$ was induced by a circularly polarized light pulse.
The direction of $\mathbf{H}_{\mathrm{IFE}}$ is determined by the helicity of
the circularly polarized light. The impulsive $\mathbf{H}_{\mathrm{IFE}}$ acts
on the sublattice magnetization of transition metal ions (e.g., Fe3+ ions in
DyFeO3 and FeBO3 and Ni2+ ions in NiO), followed by precession as free
induction decay. In this theory, the magnitude and direction of
$\mathbf{H}_{\mathrm{IFE}}$ acting on each sublattice are identical when
$\mathbf{H}_{\mathrm{IFE}}$ is perpendicular to the sublattice
magnetizations.[7]

Ferrimagnets consist of magnetic sublattices aligned in opposite directions
with different magnetic ions of different magnitudes, such that the vector sum
is not zero. Thus, ferrimagnets exhibit a net magnetization in the ground
state. Strong antiferromagnetic exchange interactions occur between the two
magnetic sublattices. Accordingly, the magnetic resonance in ferrimagnets has
two modes: the ferromagnetic resonance (FMR) mode (GHz range), which is
similar to that in ferromagnets, and the exchange resonance mode (sub-THz
range), which is unique to ferrimagnets.[8, 9, 10] Furthermore, when the
temperature dependence of the sublattice magnetization differs, the net
magnetization can vanish at the magnetization compensation temperature
$T_{\text{M}}$. Magnetization dynamics in ferrimagnets have been actively
studied in the field of ferrimagnetic spintronics, which combines
controllability, similar to ferromagnetism, with ultrafastness, similar to
antiferromagnetism.[11, 12, 13]

Figure 1: (Color online) Experimental geometry. The GdYb-BIG single crystal
was placed in an electromagnet, which produces an in-plane magnetic field $\pm
H_{0}$.

Ultrafast IFE using circularly polarized light pulses has been reported in
ferrimagnets.[14, 15, 16, 17, 18, 19, 20] However, the magnetic field and
temperature dependence of the initial phase were not explicitly discussed;
moreover, whether the IFE generated $\mathbf{H}_{\mathrm{IFE}}$ in the
opposite or same direction for each sublattice has not been studied. We
attempt to clarify this issue by measuring the initial phase of the coherently
excited resonance modes using a two-sublattice rare-earth (RE) iron garnet.

The bismuth-doped RE iron garnet Gd3/2Yb1/2BiFe5O12 (GdYb-BIG) is a
ferrimagnet with a Curie temperature of 573 K and $T_{\text{M}}=96$ K, which
was obtained from the temperature dependence of magnetization and the sign
change of Faraday rotation.[21] The exchange interaction between Fe ions in
the tetragonal and octahedral sites is much stronger than that between Fe and
RE ions. Therefore, in the sub-THz frequency range, one can regard the two
magnetic sublattices to be composed of Fe magnetization
$\mathbf{M}^{\text{Fe}}$ and RE magnetization $\mathbf{M}^{\text{RE}}$ (Ref.
[20]).

We used a GdYb-BIG single crystal with a (111) plane orientation and a
thickness of 140 $\mu$m, which was grown by the liquid-phase epitaxy method.
As shown in Fig. 1, we performed time-resolved pump–probe measurements in
transmission geometry. The polarization of pump pulse (wavelength 1300 nm,
pulse energy 4 $\mu$J, repetition rate 500 Hz, pulse width 60 fs, spot
diameter 280 $\mu$m) was circular ($\sigma^{\pm}$) to excite the magnetic
resonance modes via the IFE. The Faraday rotation angle of the linearly
polarized probe pulse (wavelength 800 nm, pulse energy less than 10 nJ,
repetition rate 1 kHz, pulse width 60 fs, spot diameter 80 $\mu$m) is mainly
sensitive to the out-of-plane component $M_{z}^{\text{Fe}}$ over the entire
temperature range.[22, 21] We applied an external in-plane magnetic field
$H_{0}=2$ kOe in the positive and negative directions, which was sufficient to
align $\mathbf{M}^{\text{Fe}}$ and $\mathbf{M}^{\text{RE}}$ in the plane,
resulting in a single-domain structure below 75 K and above 130 K. Thus, we
measured the dependences of the magnetization dynamics on the helicity
($\sigma^{\pm}$) of the circular pump polarization, the direction of the
external magnetic field ($\pm H_{0}$), and the temperature ($T\gtrless
T_{\text{M}}$). The direction of $\mathbf{H}_{\mathrm{IFE}}$ can be controlled
by $\sigma^{\pm}$; the directions of $\mathbf{M}^{\text{Fe}}$ and
$\mathbf{M}^{\text{RE}}$ can be controlled by $\pm H_{0}$; and the relative
magnitudes of $\mathbf{M}^{\text{Fe}}$ and $\mathbf{M}^{\text{RE}}$ can be
controlled by $T$.

Figure 2: (Color online) Magnetization dynamics at 300 K. HF mode at external
field: $+H_{0}$ (a) and $-H_{0}$ (b). LF mode at external field: $+H_{0}$ (c)
and $-H_{0}$ (d). Pump helicity: $\sigma^{+}$ (red dots) and $\sigma^{-}$
(blue dots). The solid lines are the damped sinusoidal functions with
$\alpha_{\mathrm{HF}}\sim 0.03$ and $\alpha_{\mathrm{LF}}\sim 0.05$ for the HF
and LF modes, respectively. Figure 3: (Color online) Magnetization dynamics at
60 K. HF mode at external field: $+H_{0}$ (a) and $-H_{0}$ (b). LF mode at
external field: $+H_{0}$ (c) and $-H_{0}$ (d). Pump helicity: $\sigma^{+}$
(red dots) and $\sigma^{-}$ (blue dots). The solid lines are the damped
sinusoidal functions with $\alpha_{\mathrm{HF}}\sim 0.01$ and
$\alpha_{\mathrm{LF}}\sim 0.35$ for the HF and LF modes, respectively.

Figures 2(a)–2(d) and 3(a)–3(d) show the Faraday rotation of the probe as a
function of delay $t$ at $T=300$ K ($>T_{\text{M}}$) and 60 K
($<T_{\text{M}}$), respectively, in the external magnetic fields $+H_{0}$
[(a): $t\leq 20$ ps, (c): $t\leq 300$ ps] and $-H_{0}$ [(b): $t\leq 20$ ps,
(d): $t\leq 300$ ps], together with fitting with damped sinusoidal functions
$\sin(\omega_{\mathrm{HF}}t)\exp(-\alpha_{\mathrm{HF}}\omega_{\mathrm{HF}}t)$
and
$\sin(\omega_{\mathrm{LF}}t)\exp(-\alpha_{\mathrm{LF}}\omega_{\mathrm{LF}}t)$.
At 300 K, a high-frequency (HF) mode at 403 GHz and a low-frequency (LF) mode
at 5.8 GHz were observed. At 60 K, an HF mode at 222 GHz and an LF mode at
10.7 GHz were observed. The HF and LF modes were attributed to the exchange
resonance and spatially propagating FMR (magnetostatic) modes,
respectively.[17] From Figs. 2(a)–2(d) and 3(a)–3(d), we observe that the
initial phases of the sinusoidal functions [$\sin(\omega t)$ or $-\sin(\omega
t)$] of the LF and HF modes do not depend on the temperature or the direction
of the external field. In contrast, the pump helicity changed the phases of
both modes by 180∘. The results are summarized in Table 1. The peak at
approximately 6–7 ps in Figs. 2(a) and 2(b) is due to the reflection of the
pump pulse from the second face of the sample.[23]

It should be noted that for $T<T_{\text{M}}$, measurements were made at 40 and
50 K in addition to 60 K, with qualitatively the same results. For $T<40$ K,
the analysis is complicated by the contribution of Yb ions. For
$T_{\text{M}}<T$, measurements were performed at 140–300 K and similar results
were obtained. For $75~{}\text{K}<T<130~{}\text{K}$, the applied field was not
sufficient to align the magnetization in-plane, and the two modes could not be
excited.

Table 1: Time evolutions of the HF and LF modes (measurement and Case 1) | Temperature | $T>T_{\mathrm{M}}$
---|---
Pump helicity | $\sigma^{+}$ | $\sigma^{-}$
External field | $+H_{0}$ | $-H_{0}$ | $+H_{0}$ | $-H_{0}$
LF mode | $+\sin\omega_{\mathrm{LF}}t$ | $+\sin\omega_{\mathrm{LF}}t$ | $-\sin\omega_{\mathrm{LF}}t$ | $-\sin\omega_{\mathrm{LF}}t$
HF mode | $+\sin\omega_{\mathrm{HF}}t$ | $+\sin\omega_{\mathrm{HF}}t$ | $-\sin\omega_{\mathrm{HF}}t$ | $-\sin\omega_{\mathrm{HF}}t$
| $T<T_{\mathrm{M}}$
---
$\sigma^{+}$ | $\sigma^{-}$
$+H_{0}$ | $-H_{0}$ | $+H_{0}$ | $-H_{0}$
$+\sin\omega_{\mathrm{LF}}t$ | $+\sin\omega_{\mathrm{LF}}t$ | $-\sin\omega_{\mathrm{LF}}t$ | $-\sin\omega_{\mathrm{LF}}t$
$+\sin\omega_{\mathrm{HF}}t$ | $+\sin\omega_{\mathrm{HF}}t$ | $-\sin\omega_{\mathrm{HF}}t$ | $-\sin\omega_{\mathrm{HF}}t$
Figure 4: Case 1: snapshot of sublattice magnetization deviations by the
action of $\mathbf{H}_{\mathrm{IFE}}$ pulses with the opposite directions on
each magnetic sublattice. The cross and dot circles indicate that the
directions of $\mathbf{H}_{\mathrm{IFE}}$ are from the front to the back and
from the back to the front of the plane, respectively. These deviations can be
decomposed into those of the LF and HF modes. $\mathbf{M}_{\mathrm{1}}$ and
$\mathbf{M}_{\mathrm{2}}$ rotate counterclockwise and clockwise around the
$H_{0}$ axis in LF and HF modes, respectively.  Figure 5: Case 2: snapshot of
sublattice magnetization deviations by the action of
$\mathbf{H}_{\mathrm{IFE}}$ pulses with the same directions on each magnetic
sublattice. The dot circles indicate that the directions of
$\mathbf{H}_{\mathrm{IFE}}$ are from the back to the front of the plane. These
deviations can be decomposed into those of the LF and HF modes. The dominant
motion is in the LF mode (a) or HF mode (b). $\mathbf{M}_{\mathrm{1}}$ and
$\mathbf{M}_{\mathrm{2}}$ rotate counterclockwise and clockwise around the
$H_{0}$ axis in LF and HF modes, respectively.

To understand the initial phases, two cases can be considered for the
sublattice selectivity of the IFE. The directions of
$\mathbf{H}_{\mathrm{IFE}}^{\text{Fe}}$ acting on $\mathbf{M}^{\text{Fe}}$ and
$\mathbf{H}_{\mathrm{IFE}}^{\text{RE}}$ acting on $\mathbf{M}^{\text{RE}}$ are
opposite (Case 1 in Fig. 4) and the same (Case 2 in Fig. 5), respectively. In
each case, $\mathbf{M}^{\text{Fe}}$ and $\mathbf{M}^{\text{RE}}$ rotate
instantaneously according to the following equations of motion under the
impulsive actions of $\mathbf{H}_{\mathrm{IFE}}^{\text{Fe}}$ and
$\mathbf{H}_{\mathrm{IFE}}^{\text{RE}}$, respectively.

$\displaystyle\frac{d\mathbf{M}^{\text{Fe}}}{dt}$
$\displaystyle=-\gamma\mathbf{M}^{\text{Fe}}\times\mathbf{H}_{\mathrm{IFE}}^{\text{Fe}},$
(1) $\displaystyle\frac{d\mathbf{M}^{\text{RE}}}{dt}$
$\displaystyle=-\gamma\mathbf{M}^{\text{RE}}\times\mathbf{H}_{\mathrm{IFE}}^{\text{RE}}.$
(2)

Here, $\gamma$ is the gyromagnetic ratio, which is equal for Gd and Fe ions.
Even if they were different, the qualitative argument in this paper would
still hold. Equations (1) and (2) are valid only during laser-pulse
excitation, where the interactions between the sublattices, magnetic
anisotropy field, external magnetic field, and damping can be neglected.[24]
After the pulses of $\mathbf{H}_{\mathrm{IFE}}^{\text{Fe}}$ and
$\mathbf{H}_{\mathrm{IFE}}^{\text{RE}}$ disappear, $\mathbf{M}^{\text{Fe}}$
and $\mathbf{M}^{\text{RE}}$ continue to rotate as a superposition of the
counterclockwise LF and clockwise HF modes around the $H_{0}$ axis, which
determine the initial phases of the two modes in Figs. 2(a)–2(d) and
3(a)–3(d). Let the two sublattice magnetizations be $\mathbf{M}_{1}$ and
$\mathbf{M}_{2}$, where
$\left|\mathbf{M}_{1}\right|>\left|\mathbf{M}_{2}\right|$. $\mathbf{M}_{1}$
points toward the external magnetic field. At 300 K,
$\mathbf{M}_{1}=\mathbf{M}^{\text{Fe}}$ and
$\mathbf{M}_{2}=\mathbf{M}^{\text{RE}}$. At 60 K,
$\mathbf{M}_{1}=\mathbf{M}^{\text{RE}}$ and
$\mathbf{M}_{2}=\mathbf{M}^{\text{Fe}}$.

In Case 1, let $a$ and $b$ be the in-plane displacements of $\mathbf{M}_{1}$
and $\mathbf{M}_{2}$ in the LF mode, respectively. The ratio $a/b$ can be
approximated as $\left|\mathbf{M}_{1}\right|/\left|\mathbf{M}_{2}\right|$. In
the HF mode, the in-plane displacements of $\mathbf{M}_{1}$ and
$\mathbf{M}_{2}$ are regarded as identical and are denoted by $c$.[10] The in-
plane displacements of $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$ can be
represented by the superposition of the LF and HF modes if $a\geq c\geq b$
holds true. In special cases where $\mathbf{H}_{\mathrm{IFE}}$ is generated
only in $\mathbf{M}_{1}$ or $\mathbf{M}_{2}$, $a>b=c$ or $a=c>b$ hold true,
respectively. The dependences of the time evolution of $M_{z}^{\text{Fe}}$ on
the pump helicity, the direction of the external magnetic field, and the
temperature were consistent with the experimental results, as presented in
Table 1.

In Case 2, the time evolution of $M_{z}^{\text{Fe}}$ differs depending on
whether the dominant motion is in the LF mode (Case 2a) or HF mode (Case 2b),
because the sense of rotation is opposite between the two modes. In Case 2a
[Fig. 5(a)], the $M_{z}^{\text{Fe}}$ for the LF mode can be determined, but
not that for the HF mode. In Case 2b [Fig. 5(b)], the $M_{z}^{\text{Fe}}$ for
the HF mode can be determined, but not that for the LF mode. The time
evolutions of $M_{z}^{\text{Fe}}$ are presented in Tables 2 and 3 for Cases 2a
and 2b, respectively; however, they do not match the experimental results in
Table 1.

Table 2: Time evolutions of the HF and LF modes (Case 2a). The $\pm$ signs of the HF modes are in the same order. | Temperature | $T>T_{\mathrm{M}}$
---|---
Pump helicity | $\sigma^{+}$ | $\sigma^{-}$
External field | $+H_{0}$ | $-H_{0}$ | $+H_{0}$ | $-H_{0}$
LF mode | $+\sin\omega_{\mathrm{LF}}t$ | $+\sin\omega_{\mathrm{LF}}t$ | $-\sin\omega_{\mathrm{LF}}t$ | $-\sin\omega_{\mathrm{LF}}t$
HF mode | $\pm\sin\omega_{\mathrm{HF}}t$ | $\pm\sin\omega_{\mathrm{HF}}t$ | $\mp\sin\omega_{\mathrm{HF}}t$ | $\mp\sin\omega_{\mathrm{HF}}t$
| $T<T_{\mathrm{M}}$
---
$\sigma^{+}$ | $\sigma^{-}$
$+H_{0}$ | $-H_{0}$ | $+H_{0}$ | $-H_{0}$
$-\sin\omega_{\mathrm{LF}}t$ | $-\sin\omega_{\mathrm{LF}}t$ | $+\sin\omega_{\mathrm{LF}}t$ | $+\sin\omega_{\mathrm{LF}}t$
$\mp\sin\omega_{\mathrm{HF}}t$ | $\mp\sin\omega_{\mathrm{HF}}t$ | $\pm\sin\omega_{\mathrm{HF}}t$ | $\pm\sin\omega_{\mathrm{HF}}t$
Table 3: Time evolutions of the HF and LF modes (Case 2b). The $\pm$ signs of the LF modes are in the same order. | Temperature | $T>T_{\mathrm{M}}$
---|---
Pump helicity | $\sigma^{+}$ | $\sigma^{-}$
External field | $+H_{0}$ | $-H_{0}$ | $+H_{0}$ | $-H_{0}$
LF mode | $\pm\sin\omega_{\mathrm{LF}}t$ | $\pm\sin\omega_{\mathrm{LF}}t$ | $\mp\sin\omega_{\mathrm{LF}}t$ | $\mp\sin\omega_{\mathrm{LF}}t$
HF mode | $+\sin\omega_{\mathrm{HF}}t$ | $+\sin\omega_{\mathrm{HF}}t$ | $-\sin\omega_{\mathrm{HF}}t$ | $-\sin\omega_{\mathrm{HF}}t$
| $T<T_{\mathrm{M}}$
---
$\sigma^{+}$ | $\sigma^{-}$
$+H_{0}$ | $-H_{0}$ | $+H_{0}$ | $-H_{0}$
$\mp\sin\omega_{\mathrm{LF}}t$ | $\mp\sin\omega_{\mathrm{LF}}t$ | $\pm\sin\omega_{\mathrm{LF}}t$ | $\pm\sin\omega_{\mathrm{LF}}t$
$-\sin\omega_{\mathrm{HF}}t$ | $-\sin\omega_{\mathrm{HF}}t$ | $+\sin\omega_{\mathrm{HF}}t$ | $+\sin\omega_{\mathrm{HF}}t$

We conclude that the IFE in GdYb-BIG operates in opposite directions for each
sublattice (Case 1). In special cases, $\mathbf{H}_{\mathrm{IFE}}$ is
generated only in $\mathbf{M}^{\text{Fe}}$ or $\mathbf{M}^{\text{RE}}$. This
is consistent with the sublattice selectivity of the FE, in which the signs of
the contributions of Fe and RE ions to the FE are opposite and the Fe ion’s
contribution is dominant in the near-infrared range.[22] This is in contrast
to antiferromagnets, in which the magnitude and direction of
$\mathbf{H}_{\mathrm{IFE}}$ acting on each sublattice are identical.[7] The
initial phase analysis can be applied to other ferrimagnets by measuring below
and above the compensation temperatures. A clarification of the sublattice
selectivity of the IFE is expected to promote the development of devices that
utilize ferrimagnetic materials and magnetooptical effects, such as the study
of magnetization reversal using the IFE.[25]

###### Acknowledgements.

We are grateful to Kouki Mikuni for his technical assistance. This study was
financially supported by the Japan Society for the Promotion of Science
KAKENHI (grant Nos. JP19H01828, JP19H05618, JP21H01032, JP22H01154, and
JP22K14588), the Frontier Photonic Sciences Project (NINS grant Nos. 01212002
and 01213004), and OML Project (NINS grant No. OML012301) of the National
Institutes of Natural Sciences (NINS), and MEXT Initiative to Establish NeXt-
generation Novel Integrated Circuits CenterS (X-NICS) (grant No. JPJ011438).

## References

  * Kimel  _et al._ [2005] A. V. Kimel, A. Kirilyuk, P. A. Usachev, R. V. Pisarev, A. M. Balbashov, and Th. Rasing, Nature 435, 655 (2005).
  * Pitaevskii [1961] L. P. Pitaevskii, Sov. Phys. JETP 12, 1008 (1961).
  * Pershan [1963] P. S. Pershan, Phys. Rev. 130, 919 (1963).
  * van der Ziel  _et al._ [1965] J. P. van der Ziel, P. S. Pershan, and L. D. Malmstrom, Phys. Rev. Lett. 15, 190 (1965).
  * Kalashnikova  _et al._ [2007] A. M. Kalashnikova, A. V. Kimel, R. V. Pisarev, V. N. Gridnev, A. Kirilyuk, and Th. Rasing, Phys. Rev. Lett. 99, 167205 (2007).
  * Satoh  _et al._ [2010] T. Satoh, S.-J. Cho, R. Iida, T. Shimura, K. Kuroda, H. Ueda, Y. Ueda, B. A. Ivanov, F. Nori, and M. Fiebig, Phys. Rev. Lett. 105, 077402 (2010).
  * Tzschaschel  _et al._ [2017] C. Tzschaschel, K. Otani, R. Iida, T. Shimura, H. Ueda, S. Günther, M. Fiebig, and T. Satoh, Phys. Rev. B 95, 174407 (2017).
  * Kaplan and Kittel [1953] J. Kaplan and C. Kittel, J. Chem. Phys. 21, 760 (1953).
  * Wangsness [1953] R. K. Wangsness, Phys. Rev. 91, 1085 (1953).
  * Lax and Button [1962] B. Lax and K. J. Button,  _Microwave Ferrites and Ferrimagnetics_ (McGraw-Hill, 1962).
  * Ivanov [2019] B. A. Ivanov, Low Temp. Phys. 45, 935 (2019).
  * Finley and Liu [2020] J. Finley and L. Liu, Appl. Phys. Lett. 116, 110501 (2020).
  * Kim  _et al._ [2022] S. K. Kim, G. S. Beach, K.-J. Lee, T. Ono, Th. Rasing, and H. Yang, Nat. Mater. 21, 24 (2022).
  * Stanciu  _et al._ [2007] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Tsukamoto, A. Itoh, A. Kirilyuk, and Th. Rasing, Phys. Rev. Lett. 98, 207401 (2007).
  * Reid  _et al._ [2010] A. H. M. Reid, A. V. Kimel, A. Kirilyuk, J. F. Gregg, and Th. Rasing, Phys. Rev. Lett. 105, 107402 (2010).
  * Satoh  _et al._ [2012] T. Satoh, Y. Terui, R. Moriya, B. A. Ivanov, K. Ando, E. Saitoh, T. Shimura, and K. Kuroda, Nat. Photon. 6, 662 (2012).
  * Parchenko  _et al._ [2013] S. Parchenko, A. Stupakiewicz, I. Yoshimine, T. Satoh, and A. Maziewski, Appl. Phys. Lett. 103, 172402 (2013).
  * Parchenko  _et al._ [2016] S. Parchenko, T. Satoh, I. Yoshimine, F. Stobiecki, A. Maziewski, and A. Stupakiewicz, Appl. Phys. Lett. 108, 032404 (2016).
  * Deb  _et al._ [2016] M. Deb, P. Molho, B. Barbara, and J.-Y. Bigot, Phys. Rev. B 94, 054422 (2016).
  * Stupakiewicz and Satoh [2021] A. Stupakiewicz and T. Satoh, J. Phys. Soc. Jpn 90, 081008 (2021).
  * S. Parchenko  _et al._ [2014] S. Parchenko, M. Tekielak, I. Yoshimine, T. Satoh, A. Maziewski, and A. Stupakiewicz, IEEE Trans. Magn. 50, 6000904 (2014).
  * Khorsand  _et al._ [2013] A. R. Khorsand, M. Savoini, A. Kirilyuk, A. V. Kimel, A. Tsukamoto, A. Itoh, and Th. Rasing, Phys. Rev. Lett. 110, 107205 (2013).
  * Khan  _et al._ [2019] P. Khan, M. Kanamaru, W.-H. Hsu, M. Kichise, Y. Fujii, A. Koreeda, and T. Satoh, J. Phys.: Condens. Matter 31, 275402 (2019).
  * Yoshimine  _et al._ [2014] I. Yoshimine, T. Satoh, R. Iida, A. Stupakiewicz, A. Maziewski, and T. Shimura, J. Appl. Phys. 116, 043907 (2014).
  * Dannegger  _et al._ [2021] T. Dannegger, M. Berritta, K. Carva, S. Selzer, U. Ritzmann, P. M. Oppeneer, and U. Nowak, Phys. Rev. B 104, L060413 (2021).