# Strongly nonlinear antiferromagnetic dynamics in high magnetic fields

Pavel Stremoukhov  FELIX Laboratory, Radboud University, Toernooiveld 7, 6525
ED Nijmegen, The Netherlands  Ansar Safin  Kotel’nikov Institute of
Radioengineering and Electronics, Russian Academy of Sciences, 125009, Moscow,
Russia  National Research University “Moscow Power Engineering Institute”,
111250, Moscow, Russia  Casper F. Schippers  Reinoud Lavrijsen  Department of
Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB
Eindhoven, Netherlands  Maurice Bal  Uli Zeitler  High Field Magnet Laboratory
(HFML – EMFL), Radboud University, Toernooiveld 7, 6525 ED Nijmegen, The
Netherlands  Alexandr Sadovnikov  Kotel’nikov Institute of Radioengineering
and Electronics, Russian Academy of Sciences, 125009, Moscow, Russia  Saratov
State University, 410071, Saratov, Russia  Kamyar Saeedi Ilkhchy  FELIX
Laboratory, Radboud University, Toernooiveld 7, 6525 ED Nijmegen, The
Netherlands  Sergey Nikitov  Kotel’nikov Institute of Radioengineering and
Electronics, Russian Academy of Sciences, 125009, Moscow, Russia  Saratov
State University, 410071, Saratov, Russia  Moscow Institute of Physics and
Technology, Dolgoprudny, 141701, Moscow Region, Russia  Andrei Kirilyuk  FELIX
Laboratory, Radboud University, Toernooiveld 7, 6525 ED Nijmegen, The
Netherlands

###### Abstract

Antiferromagnetic (AFM) materials possess a well-recognized potential for
ultrafast data processing thanks to their intrinsic ultrafast spin dynamics,
absence of stray fields, and large spin transport effects. The very same
properties, however, make their manipulation difficult, requiring frequencies
in THz range and magnetic fields of tens of Teslas. Switching of AFM order
implies going into the nonlinear regime, a largely unexplored territory. Here
we use THz light from a free electron laser to drive antiferromagnetic NiO
into a highly nonlinear regime and steer it out of nonlinearity with magnetic
field from a 33-Tesla Bitter magnet. This demonstration of large-amplitude
dynamics represents a crucial step towards ultrafast resonant switching of AFM
order.

The dynamics of nonlinear systems has become a very active field of research
in various branches of science [1]. In condensed matter physics, the nonlinear
behavior reveals itself in multiple aspects, from phase transitions and
shifting resonance frequencies, to solitons and chaos. In particular, in the
case of magnetic materials, nonlinear dynamics is essential for the operation
of spintronic devices whenever magnetic state switching via large amplitude
precession is involved. It is well-known that, in ferromagnets, spin-wave
instabilities destroy the homogeneous precession at high amplitudes [2] and
lead to chaotic magnetization dynamics [3]. Striking example of this is the
irreproducibility of ultrafast precessional magnetization reversal at
picosecond time scales [4, 5].

While the fundamental understanding of magnetization dynamics resulted in vast
applications of ferromagnetic materials [6], antiferromagnetic (AFM) materials
for a long time were of academic interest only. Only recently the
fundamentally different and appealing features of AFMs started to attract wide
attention [7, 8, 9]. These are the absence of net magnetization and stray
fields eliminating cross-talk in dense arrays and making AFM materials
extraordinary stable. The fact that the strong exchange interaction is
involved in even homogeneous precession of an AFM pattern, also unlike in
ferromagnets, results in orders of magnitude faster spin dynamics in AFMs [10,
11, 12]. However, it is the robustness of the AFM order that makes it far more
difficult to control and to read out [13]. In particular, control of large-
amplitude AFM dynamics remains a major challenge. In addition to the
fundamental understanding of the enabling interaction mechanisms, strong THz
fields are required to excite the AFM order into the nonlinear regime [14,
15], while high magnetic fields must be applied to affect the equilibrium
energy landscape. The combination of these two stimuli in a single setup could
only be classified as an unmatched experimental tool.

Here we report the results of strong excitation of AFM resonance in nickel
oxide NiO as a function of applied magnetic fields. For the experiments, a
combination of 33 Tesla Bitter magnet with high-intensity THz radiation from a
free-electron laser was used. Interestingly, we observe a highly nonlinear
dynamics of AFM spins, when the observed amplitude of precession does not
depend on the excitation intensity. In addition, non-monotonous behavior of
the spin-pumping into the adjacent Pt layer is observed, explained by the
counter-acting frequency shifts resulting from the large amplitude and from
the applied magnetic field. We believe that our results will be useful for the
development of theoretical description of nonlinear AFM behavior, which may
result in tunable and highly sensitive THz-frequency AFM devices such as
detectors[16], emitters [17] and spectrum analyzers [18].

Nickel oxide is the classical collinear type-II AFM, with approximately easy-
plane magnetic anisotropy and Néel temperature of $T_{N}$=523 K [19]. In the
paramagnetic phase above $T_{N}$, NiO has the NaCl-type structure (point group
m$\overline{3}$m). At temperatures below $T_{N}$, in contrast, spins are
coupled ferromagnetically within the {111} planes with neighboring planes
being coupled antiferromagnetically by the exchange interaction mediated by
$O^{2-}$ ions [20]. In our experiments, a bi-layer structure consisting of an
antiferromagnetic NiO layer and a heavy metal layer (Pt) is used, as sketched
in Fig. 1a. Magnetic field is applied in the easy plane of NiO and thus also
parallel to the NiO/Pt interface.

Figure 1: Excitation of antiferromagnetic resonance in NiO. (a) The
experimental geometry, here $J_{s}$ \- spin current, $J_{c}$ \- charge
current, $H_{0}$ \- the direction of externally applied magnetic field,
$\theta_{0}$ \- an angle between the easy plane of the sample and $H_{0}$. (b)
Evolution of the AFMR frequency as a function of amplitude of the steady state
precession of the Néel vector and the applied magnetic field. The pink line is
a contour line where AFMR frequency is 1.1 THz, as in small-amplitude
approximation and in zero applied field.

The magnetic structure of NiO is described by two antiferromagnetic
sublattices $\mathbf{M_{1}}$ and $\mathbf{M_{2}}$ shown in Fig. 1(a) as red
and blue arrows correspondingly. The magnetization dynamics of an
antiferromagnet under action of both DC $\mathbf{H}_{0}$ and AC
$\mathbf{h}_{AC}(t)$ magnetic fields is usually described in terms of Néel
vector
$\mathbf{l}=(\mathbf{M_{1}}-\mathbf{M_{2}})/|\mathbf{M}_{1}+\mathbf{M}_{2}|$,
by the so-called “sigma-model” [21, 22, 15]:

$\displaystyle\mathbf{l}\times\left(\frac{d^{2}\mathbf{l}}{dt^{2}}+\Gamma_{\mathrm{eff}}\frac{d\mathbf{l}}{dt}-2\gamma\left[\frac{d\mathbf{l}}{dt}\times\mathbf{H}_{0}\right]+\frac{\partial
W_{AFM}}{\partial\mathbf{l}}\right)$
$\displaystyle=\left[\mathbf{l}\times\gamma\frac{d\mathbf{h}_{AC}}{dt}\right]\times\mathbf{l}.$
(1)

Here $\Gamma_{\mathrm{eff}}$ is the spectral linewidth of the AFM resonance at
the zero DC magnetic field [23], $\gamma$ is the gyromagnetic ratio. Here the
vector product $d\mathbf{l}/dt\times\mathbf{H}_{0}$ is the gyroscopic torque
[15] and $W_{AFM}(\mathbf{l},\mathbf{H}_{0})$ is the magnetic energy density
in the presence of the DC magnetic field, which can be expressed in the form
(for more details see [15, 21]):

$W_{AFM}(\mathbf{l},\mathbf{H}_{0})=W_{a}(\mathbf{l})+\frac{\gamma^{2}}{2}\left(\mathbf{H}_{0}\cdot\mathbf{l}\right)^{2},$
(2)

where $W_{a}(\mathbf{l})$ describes the effective anisotropy energy and
determines the nonlinearity of the system. The resonance frequencies of the
NiO crystal in the absence of magnetic field are determined by the exchange
constant as well as the anisotropy constants and are equal to
$\omega_{1}~{}=~{}2\pi\cdot~{}0.2$ THz and $\omega_{2}~{}=~{}2\pi\cdot~{}1.1$
THz [24].

Figure 2: Nonlinear behavior of high-amplitude antiferromagnetic resonance.
(a) The antisymmetric shape of the iSHE voltage as a function of bias magnetic
field at the excitation frequency of 1.1 THz. (b) Evolution of the detected
voltage in magnetic field for various input light intensities and for the 1.1
THz excitation frequency. (c) Resonance behavior of the observed voltages:
blue symbols – derivative near 0 T field, red – results of theoretical
modeling, green – results of Brillouin light scattering experiment. (d)
Evolution of the detected voltage for the 0.72 THz excitation and in low
magnetic fields.

If one of the antiferromagnetic frequency modes of NiO [25] is excited, a non-
zero spin current can flow into the adjacent Pt layer. Cheng et al. developed
a theoretical framework to understand dynamical spin injection from an AFM
material undergoing coherent precession into an adjacent nonmagnetic material
[23]. Contrary to the initial idea that spin pumping from antiparallel
sublattice spins would cancel out, it was shown that coherent resonant
rotations of different sublattice spins contribute constructively to the
pumped spin current [23].

Typically, in a uniaxial antiferromagnet and in zero applied magnetic field,
the antiferromagnetic modes are degenerate $\omega_{1}=\omega_{2}$. As the
result of this degeneracy, spin pumping resulting from these two modes cancel.
That makes spin current transport possible in uniaxial AFMs such as MnF2 and
Cr2O3 only in applied magnetic field for circular polarization of the incident
electromagnetic wave [26]. However, NiO represents the class of easy-plane
antiferromagnets. The presence of the additional, though weak, in-plane
anisotropy in this material lifts the degeneracy $\omega_{1}\neq\omega_{2}$
and allows for magnonic spin current without external magnetic field [27, 28].
We observe however that this contribution is small compared to the field-
induced signals, as can be seen in Fig. 2a where zero-field contribution is
practically invisible as compared with the field-induced part.

The DC component of the spin current density $\mathbf{j}_{s}$ is described as
[29]

$\mathbf{j}_{s}=\frac{\omega}{2\pi}\int_{0}^{2\pi/\omega}\frac{\hbar}{4\pi}g_{r}^{\uparrow\downarrow}\frac{1}{M_{s}^{2}}\Big{[}\mathbf{l}(t)\times\frac{d\mathbf{l}(t)}{dt}\Big{]}\,dt\,$
(3)

where $\omega$ and $g_{\uparrow\downarrow}$ are the angular frequency of
magnetization precession and the real part of the mixing conductance,
respectively [30]. This shows that the excitation of dynamics of the Néel
vector will result in a DC spin current $\mathbf{j}_{s}$. The inverse spin-
Hall effect in Pt layer converts this spin current into a DC voltage [31, 32]
due to the spin-dependent scattering [33, 34], which is thus used in our
experiments as the direct fingerprint of AFM dynamics.

The AFMR frequency can be found by solving numerically the nonlinear equation
(Strongly nonlinear antiferromagnetic dynamics in high magnetic fields) taking
into account Eq. (2) and magnetic field component of the AC electromagnetic
field created by the THz source. The corresponding dependencies are shown in
Fig. 1b. The pink line here serves as a guide to the eye that indicates the
combination of magnetic field and precession amplitude that results in the
same value of the AFMR frequency. The corresponding dependence of the
frequency $\omega_{res}$ of the resonant mode of NiO on the amplitude of
stationary oscillations $A_{0}$ can be approximated by the equation:

$\omega_{res}=\sqrt{\omega_{2}^{2}+\kappa\omega_{\text{H}}^{2}-N\cdot
A_{0}^{2}}.$ (4)

Here $\kappa$ and $N$ are phenomenological fit parameters that have been
obtained by comparing experimental data with numerical simulations of model
described by Eq. (Strongly nonlinear antiferromagnetic dynamics in high
magnetic fields) and $\omega_{\text{H}}=\gamma\cdot H_{0}$. As can be seen
from Eq. (4), while the application of magnetic field increases the resonance
frequency, the latter decreases with the amplitude of precession $A_{0}$. The
two effects can thus balance each other.

Detuning the AFMR frequency away from its resonant value (for NiO it is thus
1.1 THz at room temperature), still results in AFM precession of large
amplitude, however smaller than in resonance. In particular, for experiments,
this means that an AFMR excited in NiO further away from 1.1 THz will show a
lower signal than an AFMR excited closer to 1.1 THz.

Fig. 2a shows the dependence of the measured ISHE voltage as a function of the
externally applied magnetic field, in the presence of radiation with central
frequency 1.1 THz incident on the sample. First of all, the measured curve is
antisymmetric with respect to the magnetic field, which happens because of the
sign reversal of the spin current flowing into the Pt layer. This is in
agreement with the behavior of the AFMR mode as a function of field [26] and
time-reversal symmetry. However, strongly non-monotonic behavior of ISHE
voltage as a function of magnetic field and excitation intensity, is much less
obvious.

To investigate this further, Figs. 2b,d show positive-field parts of this
dependence measured for various intensities of THz radiation as well as for
excitation frequencies on- and off resonance. Note that qualitatively the
shape of signals is very similar in the case of resonant excitation (1.1 THz,
Fig. 2b) as well as in the case of the off-resonant one (0.72 THz, Fig. 2d).
Several features should be pointed out. First of all, note that in moderate
applied magnetic fields and in high excitation intensity (see for example Fig.
2d, field values $B\leq 1.5$ T), the observed ISHE signal does not actually
depend on the intensity. Thus, curves for 23.5 and 47 mJ pulse energy follow
exactly the same trajectory up to above 2 T field. Also the dependence at
lower pulse energy, 14.9 mJ, follows the same path until almost 1T field. This
is a clear indication, that the observed excitation process is strongly
nonlinear. On a side note, such behavior also directly rules out thermal
effects such as thermally-induced spin-Seebeck effect [35, 36], as the total
thermal load on the sample is still low enough as to warrant linear heating
regime.

Second, after the initial linear increase of the ISHE voltage, signal reaches
a maximum in the field range of 1–6 T, depending on the excitation intensity,
followed by a slower decrease. Both the value of the signal at maximum as well
as the magnetic field in which the maximum is achieved, depend on the THz
radiation intensity.

To prove that the observed ISHE signals are actually related to the expected
AFMR, Fig. 2c shows the derivative of the signals in low fields $dI/dH$ as a
function of the excitation frequency (blue dots). The data clearly show a peak
that is very similar to the one calculated theoretically. In addition, we show
here the data of Brillouin light scattering experiment on the very same NiO
film showing thermal magnons with $k\approx 0$ (for details, see Supplementary
Information). The overlap is obvious, although the resonance measured with
these high excitation levels is clearly broadened as well as shifted to low
frequencies, as actually expected for large-amplitude behavior. This indicates
that similar to Ref. [26], we detect spin currents induced by the excitation
of AFMR in the antiferromagnetic layer.

Figure 3: Features of non-monotonic ISHE behavior reflecting non-linear AFMR.
(a) AFMR peak position in bias magnetic field as a function of incident
radiation power. Pink line is the same line that is shown in Fig. 1b. (b) Peak
of ISHE signal normalized to the input pulse power in AFMR regime, where dark
yellow line represents result of mathematical modeling using Eq. (1)

The observed non-trivial behavior of the ISHE signal as a function of the
applied magnetic field can be understood as a direct consequence of strongly
nonlinear excitation of AFMR. It derives from the combination of two factors
that drive the system. First, the behavior of large-amplitude
antiferromagnetic precession is essentially nonlinear in the sense that its
frequency depend on the amplitude. As the amplitude grows, the frequency
reduces first slowly, then faster, as illustrated in Fig. 1b. Therefore at
some amplitude the frequency will be shifted out of the excitation bandwidth,
and the precession saturates. Experimentally, we observe this saturation as an
independence of the signal on the incident THz power in weak magnetic fields,
see Fig. 2d.

Applying magnetic field to this system has double effect. On the one hand, it
creates a non-zero spin current into the adjacent Pt layer, allowing the
detection of the antiferromagnetic precession via ISHE [37, 26]. Because of
this, in low magnetic fields, we see an approximate proportionality of the
observed ISHE signals to the applied field. On the other hand, the applied
field also changes the AFMR frequency [25], see Fig. 1b. In strong fields,
this change becomes sufficient to bring the system out of resonance. When this
happens, the excitation becomes less efficient and results in a decrease of
AFMR amplitude and also of the ISHE voltages. The field value at which this
happens actually depends on the intensity of the excitation: the higher the
intensity, the larger is the power broadening of the resonance line, and thus
stronger field is required to move past the resonance condition. In
experiment, we thus observe a shift of the maximum of the ISHE signal to
higher fields (Figs. 2b,d), exactly in line with this expectation. The shape
of the curves and the shift of the maximum are very well described by theory
of nonlinear AFMR, which is proven by fitting using our model Eq. (Strongly
nonlinear antiferromagnetic dynamics in high magnetic fields).

The measurement were thus performed for a number of frequencies in the
vicinity of NiO AFMR. Qualitatively, the same trend is observed for different
values of the central frequency of incident THz light. However, the
significant shift of local maximum in biasing magnetic field was registered.

For further discussion of the nonlinear behavior of AFMR we show in Fig.3 the
normalized height of the signal maximum as well as the field value where the
maximum is achieved, as a function of macropulse energy. The data are compiled
from the data sets obtained at different excitation frequencies. Surprisingly,
on- and off-resonance data roughly follow a single dependence, for both
normalized maximum and the field where it is achieved. This can only be
explained assuming that because of the very strong excitation, the resonance
is pumped to the saturation from the very beginning. With that, the value of
the AFMR frequency shifts according the Fig. 1b. The summary of the
experimental data in Fig. 3a is reasonably fitted by the theoretical
prediction of Eq. (4). Note that the fitting line here is taken directly from
Fig. 1b, where it shows the constant-frequency cross-section on the field-
amplitude diagram. Thus interplay between the counteracting trends on the
frequency introduced by the field and precession amplitude (see Fig. 1b) can
only be weakly affected by the excitation power as the amplitude of the Néel
vector precession nonlineary depends on the excitation amplitude. This is yet
another convincing proof of the drastically nonlinear behavior of AFM spin
dynamics in our experiments.

To conclude, we here demonstrate the realization of the first steps directed
on the understanding of ultrafast large-amplitude dynamics of
antiferromagnets. The next crucial step would be a demonstration of a complete
reversal of the AFM order driven by its resonance excitation.

###### Acknowledgements.

We gratefully acknowledge the Nederlandse Organisatie voor Wetenschappelijk
Onderzoek (NWO-I) for their financial contribution, including the support of
the FELIX Laboratory. This study was supported in part by the Russian
Foundation for Basic Research (projects numbers 18-29-27020, 19-29-03015,
18-52-16006) and by the grant from the Government of the Russian Federation
for state support of scientific research conducted under the guidance of
leading scientists (project number 075-15-2019-1874). A part of the study was
carried out within a state assignment for the Kotel’nikov Institute of Radio
Engineering and Electronics of RAS. We also acknowledge support from the COST
Action CA17123 “Ultrafast opto-magneto-electronics for non-dissipative
information technology” (MAGNETOFON).

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## Supplementary Information

###  .1 Samples

The NiO/Pt bi-layered structure was deposited on MgO(111) substrates with
dimensions $\approx$4$\times$4 mm2. First, the substrates were cleaned by ion-
beam milling to ensure high quality, crystalline interface. Without breaking
the vacuum, layers of NiO and Pt were deposited. NiO was deposited by reactive
magnetron sputtering from a pure Ni target in a 10:1 argon:oxygen mixture
$(1\times 10^{-3}$ mbar) at a temperature of 430∘C. Pt was grown by standard
magnetron sputtering at room temperature in a pure argon environment $(1\times
10^{-2}$ mbar). Afterwards, the samples were diced and wire-bonded to a chip-
carrier. Over the course of the measurements, it became apparent that some
considerations about the sample’s geometry and area are essential for the
proper investigation of spin-pumping from NiO to Pt. For these purposes, the
samples were patterned in a Hall bar geometry [38, 39].

MgO was chosen as the substrate to grow the sample on due to reasonable
transparency of the material for THz radiation in order to efficiently
illuminate the NiO layer. Before doing so we performed TDS (time-domain
spectroscopy) of the MgO (500 $\mu m$) sample described above. For this
purpose TERA-K15-NL - Terahertz Time-Domain Spectrometer was employed. The
transmissivity of the substrate is shown in Fig. 5 and is in a good agreement
with the previous studies[40, 41], see below. From the plot, it follows that
MgO substrate is reasonably transparent $(\approx 55\%-60\%)$ near the area of
the interest where AFM of NiO is expected to be observed [25].

###  .2 High magnetic field measurements

The high magnetic field setup is shown in Fig. 4. The sample itself sits
inside the insert, that is a cylindrical tube that works as an extension of
the beamline inside the magnet and holds the sample. The insert is situated
inside the continuous-flow cryostat that consist of a liquid helium bath (blue
color) and a liquid nitrogen bath (green color).

Figure 4: A schematic view of the experimental setup. The setup includes both
the magnet at HFML and the FLARE free-electron laser. 1 - electron gun, 2 -
accelerator, 3 - resonantor mirror, 4 - outcoupling resonator mirror, 5 -
undulator magnets, 6 - electron bunches, 7 - focusing parabolic mirror, 8 -
beamline, 9 - laser beam, 10 - sample holder stick/waveguide, 11 - sample
area, 12 - liquid helium bath, 13 - liquid nitrogen bath, 14 - Bitter magnet
coils, 15 - sample.

The cryostat with the insert is placed inside a 33 Tesla Bitter magnet. The
top of the insert is connected to a waveguide of the FLARE free-electron
laser. The sample inside the insert is placed in the Voight geometry see Fig.
4. This is achieved by placing the chip holder with the sample inside a custom
made brass insert with two 45∘ oriented polished brass surfaces. The surfaces
serve as flat mirrors reflecting the light 90∘ and directing it through the
sample. The external magnetic field is oriented in the sample plane. To the
bottom of the insert, an InSb terahertz light detector (QMC Instruments) is
connected in order to have a constant track of the light power that is
transmitted through the sample. The whole setup is required to be in a vacuum
because the region of interest [25] includes a number of water vapor
absorption lines for THz radiation. In the case of the particular experiment
the sample, the insert, and the waveguide were evacuated using a number of
pumps, the pressure inside the system was $10^{-6}$ mBar. The signal path
fully reproduces the one in the table-top stage of experiment.

Here we address some peculiarities of the setup that are important for
experimental data presentation. Due to certain technological restrictions,
direct measurement of the effective light power illuminating the sample
becomes challenging. Another feature of the setup is that the free-electron-
laser source for the THz region (FLARE) has nonconstant power distribution for
a given frequency range (see attachments). However, there is still a
possibility to measure stick input power in a place of beamline/cryostat
coupling, see Fig. 4. The setup includes a set of attenuators in the beamline
to control FLARE output macropulse energy.

###  .3 THz pulse time structure

The time evolution of the THz radiation provided by a free-electron laser
defines the detection scheme and the setup layout. In this case, the FLARE
light source of the FELIX laboratory [42] was used. The source covers the
range from 0.24 to 3 THz (8 - 100 cm-1). The source has the following pulse
structure: macropulses with a repetition rate of 5 Hz and duration 10 $\mu$s,
each macropulse is filled with micropulses with a repetition rate of 3 GHz.
The duration of each micropulse is defined by cavity detuning of the free-
electron laser as well as . Further in this work, we treat the macropulse as a
continuous pulse of THz light with a duration of 10 $\mu$s with an average
macropulse energy of 100 mJ ignoring the micropulse structure.

###  .4 Model

In Eq. (2) of the main text, the anisotropy energy of the ”easy plane” biaxial
NiO is modeled by the function:

$W_{a}(\mathbf{l})=-\frac{\omega_{\text{ex}}\omega_{\text{EA}}}{2}\left(\mathbf{l}\cdot\mathbf{e}_{\text{EA}}\right)^{2}+\frac{\omega_{\text{ex}}\omega_{\text{HA}}}{2}\left(\mathbf{l}\cdot\mathbf{e}_{\text{HA}}\right)^{2},$
(5)

where characteristic frequencies are defined as follows:
$\omega_{ex}~{}=~{}\gamma H_{ex}$, $\omega_{EA}~{}=~{}\gamma H_{EA}$,
$\omega_{HA}=\gamma H_{HA}$, $H_{ex}$ is the AFM internal exchange magnetic
field, $H_{e},H_{h}$ and $\mathbf{e}_{EA}=\mathbf{x}$,
$\mathbf{e}_{HA}=\mathbf{z}$ are the AFM anisotropy easy and hard fields and
unit vectors, respectively [24, 43]. The corresponding values of all the
constants listed in the paper are given in the table below.

The spectral linewidth of the AFM resonance
$\gamma_{eff}~{}=~{}\alpha_{eff}\omega_{ex}$ is determined by the effective
damping $\alpha_{eff}$ including Gilbert constant and spin-pumping terms [23].
Thus, the quality factor of the AFMR for the high frequency mode 1.1 THz and
effective Gilbert damping $\alpha_{eff}=3.5\cdot 10^{-3}$ is equal to
$Q=11.4$, which is typical for such AFM crystals [11].

The magnetic field component of the AC electromagnetic field
$\mathbf{h}_{AC}=h_{AC}\mathbf{e_{\text{AC}}}\cdot\sin(\omega t+\phi_{0})$
created by a laser source, where $h_{AC}$, $\omega$ and $\phi_{0}$ are the
amplitude, frequency and initial phase of the input signal, respectively. The
$\mathbf{e}_{AC}$ is the polarization vector of the electromagnetic wave,
which in our case is oriented in the easy plane $\mathbf{xy}$.

We rewrite the nonlinear Eq. (1) in a spherical coordinate system with in-
plane polar angle $\varphi(t)$ and azimuthal out-of-plane angle $\theta(t)$ in
a form:

$\left\\{\begin{array}[]{c}\begin{array}[]{c}\sin^{2}\theta\cdot\left(\begin{array}[]{c}\frac{d^{2}\varphi}{dt^{2}}+\gamma_{eff}\frac{d\varphi}{dt}+\\\
\frac{\sin
2\varphi}{2}\bigl{(}\omega_{EA}\omega_{ex}+\omega_{y}^{2}-\omega_{x}^{2}\bigr{)}\end{array}\right)+\\\\[4.0pt]
2\gamma(\mathbf{l}\cdot\mathbf{H}_{0})\sin\theta\frac{d\theta}{dt}+\omega_{x}\omega_{y}\sin^{2}\theta\cos
2\varphi+\\\\[4.0pt] \sin
2\theta\frac{d\varphi}{dt}\frac{d\theta}{dt}=\gamma(\mathbf{l}\cdot\frac{d\mathbf{h}_{AC}}{dt})\cos\theta,\\\\[4.0pt]
\end{array}\\\\[20.0pt]
\begin{array}[]{c}\frac{d^{2}\theta}{dt^{2}}+\gamma_{eff}\frac{d\theta}{dt}-\\\\[4.0pt]
\frac{\sin
2\theta}{2}\left(\begin{array}[]{c}\omega_{HA}\omega_{ex}+\omega_{EA}\omega_{ex}\cos^{2}\varphi-\\\\[4.0pt]
-(\omega_{x}^{2}\cos^{2}\varphi+\omega_{y}^{2}\sin^{2}\varphi)\end{array}\right)-\\\\[4.0pt]
2\gamma\left(\mathbf{l}\cdot\mathbf{H}_{0}\right)\sin\theta\frac{d\varphi}{dt}+\omega_{x}\omega_{y}\frac{\sin
2\theta}{2}\sin 2\varphi=\\\\[4.0pt]
\gamma\left(\frac{dh_{AC,x}}{dt}\sin\varphi-\frac{dh_{AC,y}}{dt}\cos\varphi\right),\end{array}\end{array}\right.$
(6)

where
$\omega_{x,y}=\gamma(\left[\mathbf{x},\mathbf{y}\right]\cdot\mathbf{H}_{0})$
and $h_{AC,\left[x,y\right]}$ are the projections of a vector
$\mathbf{h}_{AC}$ into $\mathbf{x,y}$, respectively. We numerically solved the
system of non-linear equations (6) for different values of amplitude $h_{AC}$
and frequency $\omega$ of the EM wave. It was also solved for a number of
different parameters using Powell’s dog leg method, in particular its Python
implementation.

The output ISHE voltage can be calculated as a result of finding the
projection of the Eq. (2) on the $x$ axis and multiplying it by the
phenomenological constant, which was selected as a result of fitting the
experimental data.

###  .5 THz-range transmission from TDS

The data from time-domain spectroscopy (TDS) are shown in Fig. 5 and confirm
the transparency of the MgO substrate in the required frequency range.

Figure 5: TDS characterization results for MgO(111) substrate crystal

###  .6 Signals in the absence of magnetic field

This experiment was carried out in two stages. The main experimental stage was
carried out in the presence of a high magnetic field for the observation of
non-linearly excited AFMR. There was however the preparation stage that was
carried out in absence of a magnetic field. The goal of these measurements was
to find, optimize and study the origin of the ISHE signal in the Pt layer.

The employed setup is depicted on Fig. 6. The main goal of the setup is to
expose the NiO layer of the sample to THz excitation and detect the resulting
electric signals in Pt.

The THz radiation propagates through the optical attenuator consisting of a
polarizer and motorized analyzer. This allows to smoothly control optical
power according to Malus’s law[44]. The signal path is the following: the Pt
layer is electrically wired to the chip holder, the chip holder is connected
to SR560 Stanford Research Preamplifier with coaxial cable, amplified signal
from the preamplifier goes to National Instruments PXIe-5162 digital
oscilloscope. No external magnetic field was being applied at this part of the
experiment.

Figure 6: Table-top setup Figure 7: Voltages measured from different sides of
the sample.

One of the most important tasks of the experiment was to show the possibility
of electrical detection of a spin-current by means of ISHE. However, it was
important to make sure that the signal we have observed is not just a pure
thermoelectric effect (Joule heating) as the sample is being exposed to a
significant amount of optical power. To do so a series of tests were
performed. As a reference, the following sample was used
MgO(500$\mu$m)/Pt(10nm). The results were compared with the result obtained
from the following sample MgO(500$\mu$m)/NiO(25nm)/Pt(10nm). Each sample was
exposed to the light from the MgO side and the Pt side. The electrical signal
was measured from the Pt side. The results can be seen in Fig. 7.

It would be reasonable to compare the green trace (Fig. 7) with the blue trace
to estimate the input of the Joule heating. From these plots, it is clear that
the Joule heating contribution is about $\approx 8\%$, points out the that
$\approx 92\%$ of the green trace are of the non-heating origin. That can be
explained with two following scenarios. The first one is that the signal is a
result of a non-resonant spin dynamics being excited in the magnetic lattice
of NiO (due to the spin-Seebeck effect) that is being converted into
electrical current in Pt via ISHE on the interface NiO/Pt. The second scenario
is connected with a resonant dynamics being excited by THz pulse (due to
antiferromagnetic resonance in NiO) that is being converted into electrical
current in Pt via ISHE on the interface NiO/Pt.

The pink trace (Fig. 7), in this case, is explained with pure thermoelectrical
effect (the Joule heating) in Pt. The effect is significantly more pronounced
as compared to the blue trace that is explained by energy losses by absorption
in the MgO together with back-reflection on the MgO/Pt interface in the case
of the blue trace.

The orange trace (Fig. 7) is logically followed from the pink trace with
additional contribution from a magnetic effect that originates in NiO.
However, this contribution is smaller than in the case of the green term due
to the energy losses on heating the Pt layer together with back reflection
from Pt/air interface and back reflection from Pt/NiO interface.

###  .7 The fitting of experimental data

For the experimental data fitting on Figs. 2(b,d) of the main text, the
following parameters of the model (Eq. (6)) were used:

Model parameters list
---
Parameter |  values for Fig. 2(b) |  values for Fig. 2(d)
$\theta$ |  10° |  10°
$\omega_{in}$ |  1.1 THz |  0.72 THz
$\gamma_{eff}$ |  28.024 $\frac{MHz}{T}$ |  28.024 $\frac{MHz}{T}$
$\omega_{HA}$ |  43.9 GHz |  43.9 GHz
$\omega_{ex}$ |  27.5 THz |  27.5 THz
$\alpha$ |  $3.4\cdot 10^{-3}$ |  $1.1\cdot 10^{-2}$
$h_{AC}$ |  |
green |  0.2067 rad |  0.4921 rad
orange |  0.1934 rad |  0.3107 rad
red |  0.1741 rad |  0.2237 rad
blue |  0.168 rad |  0.1777 rad
black |  0.162 rad |
weighting |  |
coefficients |  |
green |  0.6 |  0.46
orange |  0.45 |  0.565
red |  0.28 |  0.63
blue |  0.17 |  0.6
black |  0.135 |

The main fitting parameter for the data is $h_{AC}$ which is a direct
representation of effective laser pulse energy. The $h_{AC}$ is used to adjust
the peak position of the dependencies on Figs. 2(b.d). The effective damping
$\alpha$ parameter was adjusted for the case of the 0.72 THz excitation.
Possibly due to the high power of the laser light, additional non-linear
mechanisms of effective damping were introduced. In case of the 1.1 THz
excitation we kept $\alpha$ close to theoretically predicted value [24].
Additionally to these parameters weighting coefficients were used to match
theoretical curves with the experimental data in amplitude.

For Fig. 3(b) all the same parameter values as for Fig. 2(b) were used. The
$h_{AC}$ parameter is spanned in interval [0:1000].

For the Fig. 1(b) Eq. (4) was solved keeping all the parameter values the
same, $N\approx 1.28\cdot 10^{25}rad/s$ and $\kappa=4.5$.

###  .8 Brillouin light scattering data

Brillouin light scattering (BLS) experiments were performed in the
backscattering geometry using the Sandercock tandem interferometer in “Two-
free-spectral-ranges” (2FSR) operation regime. This corresponds to the
selected frequency range of inelastically scattered light from -1.5 THz to 1.5
THz. Fig. 8 demonstrates the BLS spectra in this regime and three well
pronounced modes are indicated with AFM1, AFM2 and AFM3. In the resulting
spectrum, three peaks in the Stokes and three peaks in the anti-Stokes parts
can be distinguished, having frequencies of 110 GHz, 366 GHz and 1140 GHz,
respectively. Peak AFM2 has the center frequency 366 GHz and FWHM of 27 GHz.
Also in the literature there are peak frequencies around 390 GHz [45] and
frequencies of the order of 356 GHz [46]. Similar to the studies using the
Raman spectroscopy method, BLS makes it possible to determine a high-frequency
peak, which is the result of light scattering by antiferromagnetic magnons,
and has a frequency of about 1100 GHz, as shown in Ref. 47. Thus, in addition
to the determination of the magnon frequencies, we also prove the structural
and magnetic quality of our samples.

Figure 8: Backscattering Brillouin spectra of NiO taken in 2FSR operational
mode. Identified AFM modes are indicated with AFM1, AFM2 and AFM3.