# Revealing magnetoelectric coupling effect in polar antiferromagnet Fe2Mo3O8
by static and time resolved second harmonic generation

L. Y. Shi  International Center for Quantum Materials, School of Physics,
Peking University, Beijing 100871, China  D. Wu  Songshan Lake Materials
Laboratory, Dongguan, Guangdong 523808, China  T. Lin  International Center
for Quantum Materials, School of Physics, Peking University, Beijing 100871,
China  S. J. Zhang  International Center for Quantum Materials, School of
Physics, Peking University, Beijing 100871, China  Q. M. Liu  International
Center for Quantum Materials, School of Physics, Peking University, Beijing
100871, China  Z. X. Wang  International Center for Quantum Materials, School
of Physics, Peking University, Beijing 100871, China  T. C. Hu  International
Center for Quantum Materials, School of Physics, Peking University, Beijing
100871, China  T. Dong  International Center for Quantum Materials, School of
Physics, Peking University, Beijing 100871, China  N. L. Wang
<EMAIL_ADDRESS>International Center for Quantum Materials, School of
Physics, Peking University, Beijing 100871, China  Beijing Academy of Quantum
Information Sciences, Beijing 100913, China

###### Abstract

We present both static and time-resolved second harmonic generation (SHG)
measurements on polar antiferromagnet Fe2Mo3O8 to monitor the evolution of the
electric polarization change and its coupling to magnetic order. We find that
only one of the second order tensor elements, $\chi_{ccc}^{(2)}$ ,shows a
prominent change below the Neel temperature $T_{N}=60$ K, indicating a
magnetic order induced electric polarization change along the c-axis. Time-
resolved SHG measurement reveals an ultrafast recovery of the second order
tensor element upon the ultrashort laser excitation with fluence above 0.3
$mJ/cm^{2}$, yielding evidence for a photoinduced ultrafast phase transition
from the AFM ordered state to the paramagnetic state. Our work will help
understand the spin induced polarization and the ultrafast optical tuning
effect in Fe2Mo3O8.

##  I Introduction

Multiferroic materials that contain coupled ferroelectricity and magnetism
have attracted much attention in condensed matter physics and materials
science due to the rich and fascinating physical phenomena and potential
application in novel electronic devices. The strong magnetoelectric coupling
effect enables the mutual control of the macroscopic magnetization
(polarization) with external electric (magnetic) fields Kimura  _et al._
(2003); Tokura (2007); Fiebig  _et al._ (2016). It also provides a platform
for ultrafast switching and manipulation of different orders by ultrashort
laser pulses. In optical approach, time resolved Kerr rotation and time
resolved second harmonic generation (SHG) are two important techniques. Kerr
rotation is often used to study the magnetic interaction of ferromagnetic
ordered materials Beaurepaire  _et al._ (1996); Kise  _et al._ (2000);
Ogasawara  _et al._ (2005a, b). However, directly probing physical effect in
antiferromagnetism (AFM) via table-top optical technique is challenging
because of the lack of net magnetization. On the other hand, the SHG
spectroscopy is very sensitive to the lattice inversion symmetry. It is a
powerful tool to study the electric polarization in the crystallographic
structure. Besides the strong signal of SHG arising from broken inversion
symmetry, the breaking of time-inversion symmetry in magnetic ordered
structure may also contribute to SHG. Therefore SHG is often used to
investigate noncentrosymmetric AFM and multiferroics Fiebig  _et al._ (2000);
Nordlander  _et al._ (2018); Matsubara  _et al._ (2009); Sheu  _et al._
(2018); Cheong (2019).

Fe2Mo3O8 is a polar magnet recently under intensive investigations Le Page and
Strobel (1982); Wang  _et al._ (2015); Kurumaji  _et al._ (2015); Li  _et al._
(2017); Kurumaji  _et al._ (2017a, b); Li  _et al._ (2017); Yu  _et al._
(2018); Sheu  _et al._ (2019); Wang  _et al._ (2021); Csizi  _et al._ (2020);
Reschke  _et al._ (2020); Stanislavchuk  _et al._ (2020). The structure can be
viewed as the stacking of honeycomb Fe layers and Kagome Mo layers along the
c-axis. The crystal structure of Fe2Mo3O8 is illustrated in Fig. 1 (a). The
magnetism is dominated by the partially occupied 3d electrons in Fe ions at
two sites, Fe(I) and Fe(II), which are surrounded by tetrahedral and
octahedral coordination of oxygen, respectively. The lattice structure is
characterized by the space group P63mc with a crystalline polarization
pointing along the c-axis. Below $T_{N}$ = 60 K, the spin moments are ordered
along the c-axis. The moments of Fe at two different sites have different
magnitudes and opposite directions, which makes the layer ferrimagnetic.
Adjacent layers have opponent magnetic moment arrangement, which makes the
material antiferromagnetic. Accompanied by the magnetic ordering below
$T_{N}$, a lattice structure transition occurs. The oxygen atoms shifts along
the c-axis, inducing additional electric polarization change $\Delta P$ along
the c-axis. The shift also causes an increase of the Fe-O-Fe angle and the
associated in-plane AFM coupling J Wang  _et al._ (2015). Therefore, a strong
magnetic-electric coupling effect exists in the material. The ionic shifts as
well as the accompanied polarization change is attributed to the exchange
striction in the AFM state Wang  _et al._ (2015); Kurumaji  _et al._ (2015).
Recently, it was found that, when the compound is pumped by circularly
polarized laser pulse, a huge magneto-optical Kerr rotation can be observed
Sheu  _et al._ (2019).

In this work, we present both static and time-resolved SHG probe on Fe2Mo3O8.
The temperature dependent static SHG measurement reveals a prominent reduction
in one of the second order tensor elements, $\chi_{ccc}^{(2)}$, below the Neel
tempreature $T_{N}=60$ K, suggesting a magnetic order induced change of
electric polarization along the c-axis. Time-resolved SHG measurement
indicates an ultrafast recovery of the second order tensor element when the
pump fluence exceeds a threshold value in the low temperature AFM state,
yielding evidence for a photoinduced ultrafast phase transition from the AFM
ordered state to the paramagnetic state. The results provide new information
about the electric polarization and its coupling to magnetization in Fe2Mo3O8.

##  II Experiments

The single crystal of Fe2Mo3O8 was grown by chemical vapor transport method,
similar to previous reports Strobel and Page (1983); Strobel  _et al._ (1982);
Satoh  _et al._ (2007). The crystal sample is characterised by magnetic
susceptibility measurement via physical property measurement system (PPMS,
Quantum Design) and time-domain terahertz measurement. A sharp
antiferromagnetic phase transition is observed at 60 K, and an electric-active
magnon is observed at frequency of 1.25 THz Shi  _et al._ (2020), being the
same as the reported measurements Wang  _et al._ (2015); Kurumaji  _et al._
(2017a). Static and time resolved SHG measurement is preformed on the ac-
surface of the sample using a standard optical pump-probe system. Pulses from
a Ti:sapphire amplifier with 800 nm wavelength, 35 fs duration and 1 kHz
repetition are separated into pump and probe beams. The pump light is normally
incident to the sample and the probe light is reflected from the sample with a
small incident angle, as shown in Fig. 1 (b). The polarization angle $\theta$
of the incident probe laser deviated from the c-axis is controlled by half-
wave plate. The SHG signal for either p or s polarization is detected by a
photomultiplier tube after filtering out the fundamental photon. In the time
resolved SHG measurement, the probe light is kept much weaker than the pump
light.

##  III Static SHG

In magnetic materials, the second order nonlinear polarization
$P_{i}(2\omega)$ can be expressed as
$P_{i}(2\omega)=\epsilon_{0}\underset{j,k}{\sum}(\chi_{ijk}^{i(2)}+\chi_{ijk}^{c(2)})E_{j}(\omega)E_{k}(\omega)$,
where $\chi_{ijk}^{i(2)}$ is the time invariant nonlinear tensor corresponding
to the lattice symmetry and $\chi_{ijk}^{c(2)}$ is the time noninvariant
nonlinear tensor corresponding to magnetic symmetry group, respectivelyFiebig
_et al._ (2005); Matsubara  _et al._ (2009). It should be noticed that while
the electric SHG is usually dominant, the magnetic dipole may also contribute
SHG signal. However, the magnetic dipole effect is negligible due to the
symmetry of Fe2Mo3O8 as we shall explain below.

At high temperature, the space group of Fe2Mo3O8 is P63mc (point group 6mm).
The polar symmetry provides nonzero components of the i-type tensor
$\chi_{ccc}^{(2)}$, $\chi_{caa}^{(2)}$ = $\chi_{aca}^{(2)}$, while the c-type
tensor is zero and the magnetic dipole effect is forbidden. Below $T_{N}$ = 60
K, the time inversion symmetry is broken, the space group of Fe2Mo3O8 changes
from 6mm to $\underline{6}m\underline{m}$. The c-type tensor of magnetic group
$\underline{6}m\underline{m}$ has nonzero components $\chi_{bbb}^{c(2)}$ =
-$\chi_{baa}^{c(2)}$ = -$\chi_{aba}^{c(2)}$ = -$\chi_{aab}^{c(2)}$.
Considering the light is nearly normal incident to the ac plane of the sample,
all these components are undetectable. So all the SHG signals we measured in
the ac-plane are contributed from the time invariant electric dipole. We also
measured the SHG signal in an ab-cut sample. Within our measurement
resolution, no observable signal emerges at the magnetic phase transition. The
magnetic SHG is negligibly small compared to the electric SHG.

Figure 2 (a) (b) shows the polar angle $\theta$ (the angle of the light
polarization rotated from the c-axis) dependent patterns of s-out and p-out
SHG signals, measured at 20 K (below $T_{N}$) and 70 K (above $T_{N}$),
respectively. The SHG signal is proportional to the nonlinear tensor. The
generated SHG intensity can be expressed as
$I_{p}\propto\left|2\chi_{aca}^{(2)}\cos\theta\sin\theta\right|^{2}$ and
$I_{s}\propto\left|\chi_{ccc}^{(2)}\cos^{2}\theta+\chi_{caa}^{(2)}\sin^{2}\theta\right|^{2}$
(see the appendix). Indeed, the $\theta$-dependent polar patterns for both
s-out and p-out SHG signals can be well reproduced by the expressions, shown
as the lines in Fig.2 (a) and (b). From the fitting to s-out SHG signal
$I_{s}$ as displayed in Fig.2 (a), we find that $\chi_{ccc}^{(2)}$ must have
the opposite sign with $\chi_{caa}^{(2)}$. The values at $\theta$=0 and
$\theta$=90∘ are related only to $\chi_{ccc}^{(2)}$ and $\chi_{caa}^{(2)}$
respectively. Obviously, the absolute value of $\chi_{caa}^{(2)}$ is larger
than $\chi_{ccc}^{(2)}$. With the temperature drops down to 20 K, a prominent
decrease of the SHG is observed in $I_{s}$, at the petal around $\theta=0$.
The other petal of $I_{s}$ and all the petals of $I_{p}$ remain unchanged.
According to the equation of $I_{s}$, the change of the SHG signal is
originated from $\chi_{ccc}^{(2)}$. These results reveal the $\chi_{ccc}$ is
sensitive to the phase transition while $\chi_{caa}^{(2)}$ (or
$\chi_{aca}^{(2)}$) is not sensitive.

Figure 1: (a) Crystal and magnetic structure of Fe2Mo3O8. (b) Schematic
illustration of measurement configuration of the TR-SHG.

With the experimental configration setup, we can separately pick up the
signals of $\chi_{ccc}^{(2)}$ and $\chi_{caa}^{(2)}$. Monitoring the
temperature evolution of different tensor elements can help identify their
relation to the order parameters. To measure the temperature evolution of
$\chi_{ccc}^{(2)}$, the polarization of the probe light is set to s-in
($E^{\omega}\parallel c$) and s-out ($E^{2\omega}\parallel c$). For
comparison, the signal related to $\chi_{caa}^{(2)}$ is measured by setting of
p-in and s-out, and signal related to $\chi_{aca}^{(2)}$ can be measured by
p-out with $\theta=45^{\circ}$. The temperature dependence of the
$\chi_{ccc}^{(2)}$ is shown in Fig.2 (c). In the paramagnetic state, SHG
intensity decreases slightly as temperature is reduced from 140 K due to
anharmonic effects of the lattice. The prominent change happened around TN.
The intensity from $\chi_{ccc}^{(2)}$ drops sharply by 20 percent, while the
SHG signal from $\chi_{caa}^{(2)}$ shows no significant change. The electric
dipole SHG signal is usually considered to be proportional to the electric
polarization. Fe2Mo3O8 is supposed to contain a magnetic induced polarization
modulation at $T_{N}$. The reduction of the SHG signal indicates that the
magnetic order induced polarization is negative with the primal lattice
polarization. It means that the magnetic ordered state has smaller inversion
symmetry broken than the paramagnetic state. Usually, with the decrease of the
electric polarization, all the components of the nonlinear tensor are supposed
to decrease. However, the experimental results show that only the
$\chi_{ccc}^{(2)}$ changes with the polarization. This results may be related
to the fact that magnetic order induced polarization change $\Delta P$ arising
from the spin exchange striction effect is along the positive direction of
c-axis Wang  _et al._ (2015).

We notice that previous report revealed presence of antiphase magnetic domains
of the antiferromagnetic order Kim  _et al._ (2018). Magnetic SHG signal is
sensitive to magnetic domain Cheong  _et al._ (2020). We performed measurement
on different points of the sample, as well as different samples, and obtained
the same result. The intensity of $\chi_{ccc}^{(2)}$ always drops at $T_{N}$,
independence of magnetic domain. This is consistent with the fact that the
signals are all generated from electric dipole effect. The results indicate
the magnetism induced polarization is independent of the magnetic antiphase
domain.

Figure 2: (a)(b)Polar plot of the SHG signal from Fe2Mo3O8, detected for both
s-out and p-out polarizations. The lines are fits to the data using the
formula descried in text. (c) Temperature dependence of the SHG signal related
to $\chi_{ccc}^{(2)}$ and $\chi_{caa}^{(2)}$, normalized at 140 K. Clear
anomaly near the magnetic phase transition temperature ($T_{N}$ = 60 K) is
observed for $\chi_{ccc}^{(2)}$.
Figure 3: (a)Transient dynamics of SHG change(normalized to $I_{65K}-I_{40K}$)
at different fluence. The solid lines show the fitting from equation (1). (b)
Polarization dependence of SHG from Fe2Mo3O8 at -5 ps (before pump), 0 ps (at
the dip in (a)) and 30 ps with s-out configuration.
Figure 4: (a) Fluence dependence of the time scale of the slow rise component
$\tau_{m}$, indicating the AFM order melting. (b) Fluence dependence of the
magnitude of the slow rise component for 10 K and 40 K, respectively. The
dashed line presents the saturation model fit.

##  IV Time - resolved SHG

Since only $\chi_{ccc}^{(2)}$ has relation to the magnetic order induced
polarization, the time resolved SHG measurement is mainly performed on this
tensor element. With excitation by 800 nm femtosecond laser pulse, we observed
a significant change of the SHG signal in picosecond time scale. Figure 3 (a)
shows the time evolution of the pump induced transient SHG signal of
$\chi_{ccc}^{(2)}$, measured at 40 K with different pump fluences. The time
trace of SHG contains two main features, a sharp decrease followed by a fast
relaxation, and a slower rise to a plateau. The first feature is observed for
all the nonlinear tensors, while the second feature is only observed for
$\chi_{ccc}^{(2)}$. We will discuss later that these two features are related
to the excitation of electrons and suppression of spin order, respectively. To
quantitatively extract the time scales of different processes, we fit the
transient SHG by following expression,

$\Delta
SHG/SHG=\Theta(t)[a_{1}\exp(-t/\tau_{1})+a_{m}(1-\exp(-t/\tau_{m}))+c]$ (1)

where $\Theta(t)$ is the Heaviside step with 35 fs rise time, $a_{1}$ and
$a_{m}$ are the magnitude of changes of spike and plateau, respectively, and
the $\tau_{m}$ is the rise time associated to the suppression of the magnetic
order, which was found to vary from 8 ps at low fluence to 2 ps at saturated
fluence. The solid lines in Fig.3 (a) show the fitting curves. The fitting
parameters are shown in Fig. 4.

The rapid decrease of the SH intensity immediately after the photo-excitation
recovered in less than 1 ps, labeled as the fast component $\tau_{1}$. We
attribute the fast process to the ultrafast modulation of the spontaneous
polarization. The ultrafast laser pulse excites the electrons from the valence
band to conduction bands. These delocalized electrons can not contribute to
dipole moments due to the screening effect. The polarization recover with the
electron decay in $\tau_{1}$ by electron phonon scattering. Thus the laser
pulse transiently modulates the polarization, and directly affect the
intensity of all the nonlinear tensors as shown below. This ultrafast
polarization modulation process is an effective way to generate terahertz
emission.

The subsequent slowly rising signal reaches the maximum value in several
picoseconds, and then stays at a plateau that last for more than 100 ps. The
maximum of the pump induced SH intensity grows with increasing fluence and
saturates ($\Delta I/(I_{65K}-I_{40K})=100\%$) at 0.4 $\textrm{mJ/cm}^{2}$. At
saturated value, this component compensates the nagetive SHG signal induced by
magnetic order. In order to clarify the observed dynamics, we did the polar
angle scan of the s-out SHG signal at -5 ps, 0 ps and 30 ps, as shown in Fig.
3 (b). The signal of -5 ps is before the pump pulse, consistent with the
static SHG. At 0 ps, different tensors are simultaneously reduced with the
electron excitations. This is consistent with the expectation that all the
second order tensors are dependent on the spontaneous polarization. At 30 ps,
the photoexcitation only dramatically changes the s-in s-out signal from
$\chi_{ccc}^{(2)}$, which is sensitive to magnetic order. The results show
that at saturation condition, the pump induced increase of the SH intensity
exactly offsets the negative signal below TN. Obviously, the light excitations
melt the magnetic order and eliminate the exchange striction effect. To
determine the critical fluence required to suppress magnetic order, a simple
saturated model is used to fit the fluence dependent increase of the SHG
intensity at long live time delays (30 ps) for base temperature of 10 K and 40
K, respectively, as shown in Fig.4 (b). The critical fluence($F_{c}$) for
melting magnetic order at 10 K is only 0.3 $mJ/cm^{2}$, which is about an
order of magnitude lower than the pump fluence in previously reported light
induced demagnetization in isolating multiferroics Johnson  _et al._ (2015);
Bowlan  _et al._ (2016); Baldini  _et al._ (2018).

In previous report about the pump induced Kerr rotation in Fe2Mo3O8, the
circularly polarized light can induce prominent Kerr rotation, while the
linear polarized light can not induce Kerr rotation. Kerr rotation is a
signature of magnetization. In antiferromagnetic systems, the magnetization
from different magnetic sublattices counteracts each other. The light induced
Kerr signal indicates the circularly polarized light can selectivity flip the
spins in the antiferromagntic sublattice and induce net magnetic momentSheu
_et al._ (2019). Our SHG measurement of Fe2Mo3O8 is sensitive to the lattice
polarization rather than the magnetic order. However, the signal from the
exchange striction induced polarization can be taken as a sign of the
antiferromagntic spin order. In our experiments, the SHG signal is independent
on the polarization of the pump light. We have used circularly polarized light
and linear polarized light with different polarizations to excite the sample,
and obtained the same result. We also changed the frequency of pump light to 2
$\mu m$ by using an optical parametric amplifier, the pump induced phase
transition is essentially unchanged (see appendix). The results indicate that
the light excitation can easily melt the primary antiferromagntic order, but
we can not identify the magnetization of the light induced state. Considering
the fact that the linear polarized light may drive the system to a disorder
state while the circularly polarized light induce net magnetization Sheu  _et
al._ (2019), our measurements imply that the exchange striction induced
polarization can be eliminated in both cases.

##  V Summary

To summarize, we performed static and time-resolved SHG on Fe2Mo3O8 to study
the evolution of the lattice polarization. The equilibrium SHG results
confirmed the lattice symmetry of Fe2Mo3O8. At $T_{N}$, we observed prominent
change of $\chi_{ccc}^{(2)}$, which should be directly related to the magnetic
induced polarization along the c-axis. On the other hand, the other second
order tensor elements are not sensitive to the magnetic order. The time-
resolved SHG reveals the ultrafast dynamics of the polarization. The magnetic
order induced electric polarization is eliminated in several picoseconds at
relatively high fluence, indicating an ultrafast melting of the
antiferromagnetic order.

##  VI Appendix

###  VI.1 SHG pattern fitting

In this Appendix, we present details of fitting the polar pattern. We consider
the second order polarization $P_{s}$ and the electric field $E$ are vectors
with directions. The SHG process can be expressed as

$P_{s}=\left(\begin{array}[]{c}P_{1}\\\ P_{2}\\\ P_{3}\\\
\end{array}\right)=d_{ijk}\left(\begin{array}[]{c}E_{1}^{2}\\\ E_{2}^{2}\\\
E_{3}^{2}\\\ 2E_{2}E_{3}\\\ 2E_{1}E_{3}\\\ 2E_{1}E_{2}\\\ \end{array}\right)$
(2)

The nonlinear coefficient $d_{ijk}$ is a third rank tensor. For second
harmonic generation, $E_{1}E_{2}=E_{2}E_{1}$, $d_{ijk}=d_{ikj}$. So in general
the subscripts 23 or 32 can be substituted by subscript 4, 31 or 13 by 5, and
12 or 21 by 6. The equation 1 can be write as

$\left(\begin{array}[]{c}P_{1}\\\ P_{2}\\\ P_{3}\\\
\end{array}\right)=\left(\begin{array}[]{cccccc}d_{11}&d_{12}&d_{13}&d_{14}&d_{15}&d_{16}\\\
d_{21}&d_{22}&d_{23}&d_{24}&d_{25}&d_{26}\\\
d_{31}&d_{32}&d_{33}&d_{34}&d_{35}&d_{36}\\\
\end{array}\right)\left(\begin{array}[]{c}E_{1}^{2}\\\ E_{2}^{2}\\\
E_{3}^{2}\\\ 2E_{2}E_{3}\\\ 2E_{1}E_{3}\\\ 2E_{1}E_{2}\\\ \end{array}\right)$
(3)

In the experiment set up as shown in figure 1, the lab axis 1, 2, 3 are
consistent to the crystal axis a, b, c. The light normally incident to the ac
plane of the sample, so $E_{1}=E\cos\theta,E_{2}=0,E_{3}=E\sin\theta$.
According to the symmetry of the sample P63mc, the nonzero components of the
tensor are $d_{33}$ and $d_{15}=d_{31}=d_{32}=d_{26}$.

$\left(\begin{array}[]{c}P_{1}\\\ P_{2}\\\ P_{3}\\\
\end{array}\right)=\left(\begin{array}[]{cccccc}0&0&0&0&d_{15}&0\\\
0&0&0&0&0&d_{26}\\\ d_{31}&d_{32}&d_{33}&0&0&0\\\
\end{array}\right)\left(\begin{array}[]{c}E\cos\theta\\\ 0\\\ E\sin\theta\\\
0\\\ 2E\cos\theta E\sin\theta\\\ 0\\\ \end{array}\right)\\\ $ (4)

$=\left(\begin{array}[]{c}d_{15}2E\cos\theta E\sin\theta\\\ 0\\\
d_{31}E\cos\theta+d_{33}E\sin\theta\\\ \end{array}\right)$ (5)

Therefore the generated SHG intensity can be expressed as
$I_{p}\propto\left|2\chi_{aca}^{(2)}\cos\theta\sin\theta\right|^{2}$ and
$I_{s}\propto\left|\chi_{ccc}^{(2)}\cos^{2}\theta+\chi_{caa}^{(2)}\sin^{2}\theta\right|^{2}$.

In the optical pump SHG probe measurement, the probe light reflects from the
sample with a very small angle, rather than normal incident. The incident
angle dose not affect the i-type SHG measurement. The incident angle is in the
horizontal plane, which is the ab-plane. The angle would project a small part
of $E_{a}$ to $E_{b}$, since a-axis is equal to b-axis for i-type tensor. The
result with a small incident angle is the same as the result with normal
incident.

###  VI.2 Pumped at 2 $\mu m$

The light with different wavelength may excite different electron transitions.
We tune the pump pulse to 2 $\mu m$ by using an optical parametric amplifier.
The time traces of the SHG signal related to $\chi_{ccc}^{(2)}$ are shown in
Fig.5. The spike like fast process disappears while the slow process is
qualitatively unchanged. The results are not dependent on the polarization of
the pump pulse.

Figure 5:  Time traces of the SHG signal related to $\chi_{ccc}^{(2)}$ ,
pumped by 2$\mu m$ light. The linear polarized light and circular polarized
light induce similar effect.

##  VII Acknowledgments

We would like to thank Profs. S. Dong, S. W. Cheong, J. Demsar for helpful
discussions. This work was supported by National Natural Science Foundation of
China (No. 11888101), the National Key Research and Development Program of
China (No. 2017YFA0302904).

## References

  * Kimura  _et al._ (2003) T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y. Tokura, Nature 426, 55 (2003).
  * Tokura (2007) Y. Tokura, Journal of Magnetism and Magnetic Materials 310, 1145 (2007), proceedings of the 17th International Conference on Magnetism.
  * Fiebig  _et al._ (2016) M. Fiebig, T. Lottermoser, D. Meier, and M. Trassin, Nature Reviews Materials 1, 16046 (2016).
  * Beaurepaire  _et al._ (1996) E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Phys. Rev. Lett. 76, 4250 (1996).
  * Kise  _et al._ (2000) T. Kise, T. Ogasawara, M. Ashida, Y. Tomioka, Y. Tokura, and M. Kuwata-Gonokami, Phys. Rev. Lett. 85, 1986 (2000).
  * Ogasawara  _et al._ (2005a) T. Ogasawara, K. Ohgushi, Y. Tomioka, K. S. Takahashi, H. Okamoto, M. Kawasaki, and Y. Tokura, Phys. Rev. Lett. 94, 087202 (2005a).
  * Ogasawara  _et al._ (2005b) T. Ogasawara, K. Ohgushi, Y. Tomioka, K. S. Takahashi, H. Okamoto, M. Kawasaki, and Y. Tokura, Phys. Rev. Lett. 94, 087202 (2005b).
  * Fiebig  _et al._ (2000) M. Fiebig, D. Fröhlich, K. Kohn, S. Leute, T. Lottermoser, V. V. Pavlov, and R. V. Pisarev, Phys. Rev. Lett. 84, 5620 (2000).
  * Nordlander  _et al._ (2018) J. Nordlander, G. De Luca, N. Strkalj, M. Fiebig, and M. Trassin, Applied Sciences 8 (2018), 10.3390/app8040570.
  * Matsubara  _et al._ (2009) M. Matsubara, Y. Kaneko, J.-P. He, H. Okamoto, and Y. Tokura, Phys. Rev. B 79, 140411 (2009).
  * Sheu  _et al._ (2018) Y. M. Sheu, N. Ogawa, Y. Tokunaga, H. C. Chan, and Y. Tokura, Phys. Rev. B 98, 100301 (2018).
  * Cheong (2019) S.-W. Cheong, npj Quantum Materials 4, 53 (2019).
  * Le Page and Strobel (1982) Y. Le Page and P. Strobel, Acta Crystallographica Section B 38, 1265 (1982).
  * Wang  _et al._ (2015) Y. Wang, G. L. Pascut, B. Gao, T. A. Tyson, K. Haule, V. Kiryukhin, and S.-W. Cheong, Scientific Reports 5, 12268 (2015).
  * Kurumaji  _et al._ (2015) T. Kurumaji, S. Ishiwata, and Y. Tokura, Phys. Rev. X 5, 031034 (2015).
  * Li  _et al._ (2017) Y. Li, G. Gao, and K. Yao, EPL (Europhysics Letters) 118, 37001 (2017).
  * Kurumaji  _et al._ (2017a) T. Kurumaji, Y. Takahashi, J. Fujioka, R. Masuda, H. Shishikura, S. Ishiwata, and Y. Tokura, Phys. Rev. B 95, 020405 (2017a).
  * Kurumaji  _et al._ (2017b) T. Kurumaji, Y. Takahashi, J. Fujioka, R. Masuda, H. Shishikura, S. Ishiwata, and Y. Tokura, Phys. Rev. Lett. 119, 077206 (2017b).
  * Yu  _et al._ (2018) S. Yu, B. Gao, J. W. Kim, S.-W. Cheong, M. K. L. Man, J. Madéo, K. M. Dani, and D. Talbayev, Phys. Rev. Lett. 120, 037601 (2018).
  * Sheu  _et al._ (2019) Y. M. Sheu, Y. M. Chang, C. P. Chang, Y. H. Li, K. R. Babu, G. Y. Guo, T. Kurumaji, and Y. Tokura, Phys. Rev. X 9, 031038 (2019).
  * Wang  _et al._ (2021) W. Wang, P. Z. Li, Y. T. Chang, M. F. Liu, C. L. Lu, X. B. Lu, M. Zeng, and J.-M. Liu, Applied Physics Letters 118, 112901 (2021), https://doi.org/10.1063/5.0044565 .
  * Csizi  _et al._ (2020) B. Csizi, S. Reschke, A. Strinić, L. Prodan, V. Tsurkan, I. Kézsmárki, and J. Deisenhofer, Phys. Rev. B 102, 174407 (2020).
  * Reschke  _et al._ (2020) S. Reschke, A. A. Tsirlin, N. Khan, L. Prodan, V. Tsurkan, I. Kézsmárki, and J. Deisenhofer, Phys. Rev. B 102, 094307 (2020).
  * Stanislavchuk  _et al._ (2020) T. N. Stanislavchuk, G. L. Pascut, A. P. Litvinchuk, Z. Liu, S. Choi, M. J. Gutmann, B. Gao, K. Haule, V. Kiryukhin, S.-W. Cheong, and A. A. Sirenko, Phys. Rev. B 102, 115139 (2020).
  * Strobel and Page (1983) P. Strobel and Y. L. Page, Journal of Crystal Growth 61, 329 (1983).
  * Strobel  _et al._ (1982) P. Strobel, Y. L. Page, and S. McAlister, Journal of Solid State Chemistry 42, 242 (1982).
  * Satoh  _et al._ (2007) T. Satoh, B. B. Van Aken, N. P. Duong, T. Lottermoser, and M. Fiebig, Phys. Rev. B 75, 155406 (2007).
  * Shi  _et al._ (2020) L. Y. Shi, D. Wu, Z. X. Wang, T. Lin, C. M. Hu, and N. L. Wang,  1, 1 (2020), arXiv:2004.05823 .
  * Fiebig  _et al._ (2005) M. Fiebig, V. V. Pavlov, and R. V. Pisarev, J. Opt. Soc. Am. B 22, 96 (2005).
  * Kim  _et al._ (2018) M. G. Kim, H. Miao, B. Gao, S.-W. Cheong, C. Mazzoli, A. Barbour, W. Hu, S. B. Wilkins, I. K. Robinson, M. P. M. Dean, and V. Kiryukhin, Nature Communications 9, 5013 (2018).
  * Cheong  _et al._ (2020) S.-W. Cheong, M. Fiebig, W. Wu, L. Chapon, and V. Kiryukhin, npj Quantum Materials 5, 3 (2020).
  * Johnson  _et al._ (2015) J. A. Johnson, T. Kubacka, M. C. Hoffmann, C. Vicario, S. de Jong, P. Beaud, S. Grübel, S.-W. Huang, L. Huber, Y. W. Windsor, E. M. Bothschafter, L. Rettig, M. Ramakrishnan, A. Alberca, L. Patthey, Y.-D. Chuang, J. J. Turner, G. L. Dakovski, W.-S. Lee, M. P. Minitti, W. Schlotter, R. G. Moore, C. P. Hauri, S. M. Koohpayeh, V. Scagnoli, G. Ingold, S. L. Johnson, and U. Staub, Phys. Rev. B 92, 184429 (2015).
  * Bowlan  _et al._ (2016) P. Bowlan, S. A. Trugman, X. Wang, Y. M. Dai, S.-W. Cheong, E. D. Bauer, A. J. Taylor, D. A. Yarotski, and R. P. Prasankumar, Phys. Rev. B 94, 184429 (2016).
  * Baldini  _et al._ (2018) E. Baldini, T. Kubacka, B. P. P. Mallett, C. Ma, S. M. Koohpayeh, Y. Zhu, C. Bernhard, S. L. Johnson, and F. Carbone, Phys. Rev. B 97, 125149 (2018).