# Supplementary Information: Ultra-high Spin Emission from Antiferromagnetic
Metal FeRh

Dominik Hamara  Microelectronics Group, Cavendish Laboratory, University of
Cambridge, J.J.Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Mara
Strungaru  0000-0003-4606-7131 School of Physics, Engineering and Technology,
University of York, York, YO10 5DD, United Kingdom  Jamie R. Massey
0000-0002-7793-7008 School of Physics and Astronomy, University of Leeds,
Leeds LS2 9JT, United Kingdom  Laboratory for Mesoscopic Systems, Department
of Materials, ETH Zurich, 8093 Zurich, Switzerland.  Paul Scherrer Institute,
5232 Villigen PSI, Switzerland.  Quentin Remy  0000-0002-4801-3199 Department
of Physics, Freie Universität Berlin, 14195 Berlin, Germany  Guillermo Nava
Antonio  0000-0003-2813-2841 Microelectronics Group, Cavendish Laboratory,
University of Cambridge, J.J.Thomson Avenue, Cambridge CB3 0HE, United Kingdom
Obed Alves Santos  0000-0002-0192-8236 Microelectronics Group, Cavendish
Laboratory, University of Cambridge, J.J.Thomson Avenue, Cambridge CB3 0HE,
United Kingdom  Michel Hehn  0000-0002-4240-5925 Université de Lorraine, CNRS,
IJL, F-54000 Nancy, France  Richard F.L. Evans  0000-0002-2378-8203 School of
Physics, Engineering and Technology, University of York, York, YO10 5DD,
United Kingdom  Roy W. Chantrell  0000-0001-5410-5615 School of Physics,
Engineering and Technology, University of York, York, YO10 5DD, United Kingdom
Stéphane Mangin  0000-0001-6046-0437 Institut Jean Lamour (UMR 7198),
Université de Lorraine, Vandoeuvre-lès-Nancy, France  Christopher H. Marrows
0000-0003-4812-6393 School of Physics and Astronomy, University of Leeds,
Leeds LS2 9JT, United Kingdom  Joseph Barker  0000-0003-4843-5516
<EMAIL_ADDRESS>School of Physics and Astronomy, University of Leeds,
Leeds LS2 9JT, United Kingdom  Chiara Ciccarelli  0000-0003-2299-3704
<EMAIL_ADDRESS>Microelectronics Group, Cavendish Laboratory, University of
Cambridge, J.J.Thomson Avenue, Cambridge CB3 0HE, United Kingdom

## S1. Temperature dependence of the correction function

The temperature dependence of the THz emission in FeRh-Pt is the central part
of our study. It is therefore very important to understand how the optical and
electrical properties of our sample depend on temperature to correctly account
for changes in the absorbed pump fluence, $A(T)$, and the efficiency of THz
field outcoupling $C(T)$. To filter out the contribution of these two
variables from our temperature analysis, we introduce a correction function,
$G(T)=1/(A(T)C(T))$, and define a renormalised THz emission amplitude
$S^{*}_{y}(T)=G(T)S_{y}(T)$, where $S_{y}(T)$ is the measured amplitude.

Fig. S 1:  Temperature dependence of the correction function parameters in the
top pumping geometry: optical absorption fraction $A(T)$ (a) in
FeRh(30)/Pt(3.5) and (b) uncapped FeRh (30); normalised THz outcoupling,
$C(T)$ in (c) FeRh(30)/Pt(3.5) and (d) uncapped FeRh (30).

To obtain $A(T)$ we calculate the total absorption of an optical laser light
at 800 nm using the standard Transfer Matrix Method Byrnes2016. To do so, we
require the refractive indices of each of the constituents of the sample stack
for the corresponding optical wavelength. We take the refractive indices of
MgO and Pt to have negligible dependence on temperature as
$n_{\mathrm{MgO}}($800 nm$)=1.7$ Stephens_JResNatlBureauStand_49_249_1952 and
$n_{\mathrm{Pt}}($800 nm$)=0.95+4.71i$ Werner_JPhysChemRefData_38_1013_2009.
The temperature dependence of the refractive index of FeRh at the fixed
angular frequency $\omega=2\pi c/\lambda$ with $\lambda=800$ nm, is given by
Saidl_NewJPhys_18_083017_2016:

$n_{FeRh}^{2}(T)=1-\frac{\omega_{p}(T)^{2}}{\omega^{2}+1/\tau(T)^{2}}+\frac{i\omega_{p}(T)^{2}/\omega\tau(T)}{\omega^{2}+1/\tau(T)^{2}}+\frac{i\sigma_{\mathrm{inter}}(T)}{\varepsilon_{0}\omega}.$
(1)

$\tau(T)=\sigma_{\mathrm{DC}}(T)/(\varepsilon_{0}\omega_{p}^{2}(T))$ is the
scattering time Saidl_NewJPhys_18_083017_2016. The temperature dependence of
the DC conductivity of FeRh, $\sigma_{\mathrm{DC}}(T)$, is obtained from van
der Pauw measurements. $\omega_{p}(T)$ is the plasma frequency, which we take
as $\hbar\omega_{p}^{\mathrm{AFM}}=1.8$ eV and
$\hbar\omega_{p}^{\mathrm{FM}}=5.5$ eV Saidl_NewJPhys_18_083017_2016 in the AF
and FM phases of FeRh, respectively. $\sigma_{\mathrm{inter}}(T)$ is the
interband conductivity, which we take as
$\sigma_{\mathrm{inter}}^{\mathrm{AFM}}=-i\omega\varepsilon_{0}\times(0.74+0.58i)$
and
$\sigma_{\mathrm{inter}}^{\mathrm{FM}}=-i\omega\varepsilon_{0}\times(18.26+32.77i)$
Vyborny.

To describe $\omega_{p}(T)$ and $\sigma_{\mathrm{inter}}(T)$ in the phase-
transition region, we introduce a temperature dependent scaling parameter
$\alpha(T)$, with $\alpha(T\ll T_{\mathrm{(AF-FM)}})=1$ in the AFM phase and
$\alpha(T\gg T_{\mathrm{(AF-FM)}})=0$ in the FM phase. The temperature
dependence of $\alpha$ is obtained from the renormalisation of the THz
transmission data around the AFM/FM transition. The full temperature
dependence of the plasma frequency and interband conductivity is then:

$\omega_{p}(T)=\alpha(T)\omega_{p}^{\mathrm{AFM}}+(1-\alpha(T))\omega_{p}^{\mathrm{FM}},$
(2)

and

$\sigma_{\mathrm{inter}}(T)=\alpha(T)\sigma_{\mathrm{inter}}^{\mathrm{AFM}}+(1-\alpha(T))\sigma_{\mathrm{inter}}^{\mathrm{FM}}.$
(3)

The optical absorption temperature dependencies for the FeRh(30)/Pt(3.5) and
uncapped FeRh(30) samples in the top-pumping geometry are plotted in Fig.
S1(a) and (b) respectively. These $A(T)$ find that for the maximum pump
incident fluence of 9.52 mJ/cm2, the absorbed fluences are in the ranges of
2.38-2.86 mJ/cm2 and 2.67-3.28 mJ/cm2 for FeRh(30)/Pt(3.5) and FeRh(30)
respectively.

To find the temperature dependence of the THz field outcoupling at 1 THz, we
use the expression given in Ref. Seifert_NatPhotonics_10_483_2016:

$C(T)\propto\frac{1}{1+n_{\mathrm{MgO}}+Z_{0}\int_{0}^{d}{dz\ \sigma(T,z)}}.$
(4)

Here, $n_{\mathrm{MgO}}=3.1$ Nenno_SciRep_9_13348_2019 is the refractive index
of the MgO substrate at 1 THz, considered to be temperature-independent.
$Z_{0}=377$ $\Omega$ is the vacuum impedance, $d$ is the metal stack
thickness, and $\sigma(T,z)$ is the conductivity distribution of the metal
stack. At THz frequencies $\int_{0}^{d}{dz\
\sigma(T,z)}=d_{\mathrm{FeRh}}\sigma_{\mathrm{FeRh}}(T)+d_{\mathrm{Pt}}\sigma_{\mathrm{Pt}}(T)$,
with $d_{\mathrm{FeRh}}=30$ nm, and $d_{\mathrm{Pt}}=3.5$ nm. The temperature
dependence of the DC conductivities in FeRh Vries and Pt
Wang_ApplPhysLett_105_152412_2014 are obtained from the literature. The
normalised $C(T)$ for the FeRh(30)/Pt(3.5) and FeRh(30) samples are plotted in
Fig. S1(c) and (d) respectively.

## S2. Temperature dependence of the magnetisation

###  .1 Magnetic hysteresis loops of FeRh(30)/Pt(3.5)

Fig. S2 shows hysteresis loops of FeRh(30)/Pt(3.5) measured using a SQUID
magnetometer at temperatures between 400 K and 20 K. The MgO(001) background
was removed using data sets obtained for a bare substrate. The paramagnetic
contributions were subtracted. From these measurements we extract the
temperature dependence of the saturation magnetisation $M_{\mathrm{sat}}$,
shown in Fig. 1(b) and 2(d) of the main text and of the coercive $B_{c}$ and
exchange bias $B_{\mathrm{ex}}$ fields in Fig. 2(d) of the main text.

Fig. S 2:  Magnetic hysteresis loops of FeRh(30)/Pt(3.5) measured by SQUID at
400 K (a), 300 K (b), 200 K (c), 100 K (d), 50 K (e) and 20 K (f), after
subtracting the paramagnetic contribution and the contribution of the MgO(001)
substrate. The saturation magnetisation and coercive field values in the main
article were acquired from this data set.

###  .2 Four-spin atomistic spin model

To model FeRh we used the atomistic spin dynamics software package vampire
Evans_JPhysCondensMatter_26_103202_2014; *vampire that describes the dynamics
of each spin, $\mathbf{S}_{i}$. The metamagnetic transition of FeRh is
modelled in terms of the competition between the higher-order exchange
interaction (four-spin) responsible for the antiferromagnetic ordering (AFM)
at low temperatures and the bilinear exchange interaction, responsible for
ferromagnetic ordering (FM) at elevated temperatures. The model has been
developed by Barker and Chantrell barker and further used in the literature
ostler2017modeling; strungaru_physrevmat.

The spin Hamiltonian used in the atomistic simulations includes contributions
from the bilinear exchange in form of nearest and next-nearest neighbors
exchange and four-spin interaction and has the following form:

$\begin{split}{\mathcal{H}}=&-\frac{1}{2}\,\sum_{i,j}J_{ij}\,(\mathbf{S}_{i}\,\cdot\mathbf{S}_{j}\>)-k_{u}\sum_{i}(\mathbf{S}_{i}\,\cdot\mathbf{e})^{2}\\\
&-\frac{1}{4}\,\sum_{i,j,k,l}\frac{1}{3}D_{ijkl}\,\left((\mathbf{S}_{i}\,\cdot\mathbf{S}_{j}\>)(\mathbf{S}_{k}\,\cdot\mathbf{S}_{l}\>)+(\mathbf{S}_{i}\,\cdot\mathbf{S}_{k}\>)(\mathbf{S}_{j}\,\cdot\mathbf{S}_{l}\>)+(\mathbf{S}_{i}\,\cdot\mathbf{S}_{l}\>)(\mathbf{S}_{k}\,\cdot\mathbf{S}_{j}\>)\right),\end{split}$
(5)

where $J_{ij}$ and $D_{ijkl}$ represent the bilinear and four-spin exchange
interaction between Fe atomic sites, respectively, $k_{\mathrm{u}}$ represents
the uniaxial anisotropy constant, with $\mathbf{e}$ representing the easy axis
direction. The factor $\frac{1}{2}$ accounts for the double summation
$i\rightarrow j$ and $j\rightarrow i$ since numerically both interactions are
included explicitly. Similarly, for the four-spin exchange term, the factor
$\frac{1}{4}$ appears due to the explicit inclusion of all interactions for
atoms $i$, $j$, $k$, $l$.

The parameters used in the simulations are presented in Table 1, with the
exchange parameters being summarised in Table 2. The value of the uniaxial
anisotropy was extracted from Ref. ostler2017modeling. We calculate the
equilibrium magnetisation of the system by employing Monte-Carlo simulations,
where we equilibrate the system at each temperature for 50,000 Monte-Carlo
steps, and then average the properties for 50,000 more steps.

We modelled a system of 65 $\times$ 32 $\times$ 32 atomic sites, periodic in
the $yz$ directions, where at the FeRh/Pt interface we assume a FM monolayer
of FePd. A reduced exchange ($J_{\mathrm{FePd}}$) compared to the bulk FePd is
used for the monolayer, an effect occuring due to the disordered state of the
interface. The magnetic moment at the interface is also reduced to
$0.45~{}\mu_{B}$, a reduction in the moment being also observed in literature
fan_ferh. From the SQUID data $M_{\mathrm{sat}}$ (20 K) is about 20 times
smaller than $M_{\mathrm{sat}}$ (400 K), renormalising by the total volume of
FeRh. If we consider that the FM interfaces extend over a combined thickness
of 10 nm, we obtain that the magnetisation of the interfaces at low
temperature is about 7 times smaller than in the ferromagnetic phase, hence
the value of $0.45~{}\mu_{B}$. This is just a rough approximation because the
ferromagnetic interface with the MgO substrate is of a different nature and
its magnetic moment might be different. A coupling of $10\%$ of the exchange
value of FePd is used across the interface. The inclusion of the FM monolayer
leads to a similar temperature dependence of magnetisation as observed
experimentally, with a low Curie temperature of the monolayer (of 220K) -
green curve in Fig. S3.

Fig. S 3:  Temperature dependence of magnetisation as extracted from atomistic spin dynamics simulations. The green curve shows the magnetisation of only the FePd monolayer. The blue, orange and yellow curves show the temperature dependence of the total magnetisation of 2, 5 and 10 monolayes of FeRh respectively plus a monolayer of FePd.  Table 1: Parameters used for the simulation of FeRh/Pt. Quantity | Symbol | Value | Units
---|---|---|---
FeRh anisotropy energy | $K_{\mathrm{FeRh}}$ | $1.404\times 10^{-23}$ | J
FePd anisotropy energy | $K_{\mathrm{FePd}}$ | $2.63\times 10^{-22}$ | J
FeRh magnetic moment | $\mu_{\mathrm{FeRh}}$ | $3.15$ | $\mu_{\mathrm{B}}$
FePd magnetic moment | $\mu_{\mathrm{FePd}}$ | $0.45$ | $\mu_{\mathrm{B}}$
Table 2:  Exchange constants for the four-spin exchange Hamiltonian (5). symbol | types | number | distance (lattice constants) | value (meV)
---|---|---|---|---
$J_{2}^{\mathrm{FeRh}}$ | Fe-Fe | $6$ | $1$ |  $2.49$111Ref. barker
$J_{3}^{\mathrm{FeRh}}$ | Fe-Fe | $12$ | $\sqrt{2}$ |  $17.16$111Ref. barker
$J_{2}^{\mathrm{FePd}}$ | Fe-Fe | $6$ | $1$ | $2.49$
$J_{2}^{\mathrm{FeRh-FePd}}$ | Fe-Fe | $6$ | $1$ | $0.249$
$D_{ijkl}^{\mathrm{FeRh}}$ | Fe-Fe-Fe-Fe | 32 | 1 |  $1.43$111Ref. barker

We next investigate the magnetisation behaviour arising from the various
thickness of the considered interface. For an interface consisting of 2 layers
(monolayer of FePd and monolayer of FeRh) blue curve in Fig. 3) we observe
that the magnetisation in the low temperature regime has a similar amplitude
that the one arising after the first order phase transition in FeRh. By
distancing from the FePd interface, a decrease in the low temperature
magnetisation is observed (as shown for 5 and 10 layers, orange and yellow
curve).

The field dependence observed experimentally can appear as an effect of
magnetic domains. For the simulation results presented above, the system size
considered allows only for monodomain configurations at low temperature.
However, for larger system sizes (which go beyond our computational
capabilities), magnetic domains can be present at low temperature. Applying a
magnetic field will saturate these domains in the direction of the field,
hence the increased in the overall magnetisation along field direction can
lead to an increse in the spin current with enhanced field as shown
experimentally.

## S3. Additional experimental data

###  .3 Raw THz emission data of FeRh(30)/Pt(3.5)

Fig. S4 presents the temperature and magnetic field dependence of the raw THz
emission data, before renormalising by the temperature dependent optical
absorption and temperature dependent THz outcoupling. The THz emission
amplitude undergoes an abrupt decrease across the phase transition from
ferromagnetic to antiferromagnetic. As temperature is further decreased, the
THz emission amplitude starts increasing again up to 60% of the value in the
ferromagnetic phase.

Fig. S 4: (a)-(c) THz transient time-traces measured at different temperatures
in the range 420 K-360 K (a), 290 K-180 K (b) and 180 K - 20 K (c). (d)
Temperature dependence of the THz emission amplitude measured at 860 mT for
two pumping geometries, pumping the sample from the Pt side and measuring the
THz emission behind the substrate (red), or pumping the sample from the
substrate side and measuring the THz emission behind the Pt layer (blue). (e)
Temperature dependence of the THz emission amplitude for different values of
the in-plane magnetic field. (f) Dependence of the THz emission amplitude on
the in-plane magnetic field at different temperatures. For these measurements
a top pumping geometry was adopted, with the pump hitting the sample from the
Pt side and the THz emission being measured behind the substrate.

###  .4 Magnetic properties and raw THz signal of uncapped FeRh(30 nm)

The SQUID data in Fig. S5 (a) shows that in the uncapped FeRh(35) sample,
residual ferromagnetism exists also in the antiferromagnetic phase and its
saturation magnetisation reaches values comparable to the FeRh(30 nm)/Pt(3.5
nm) sample at low temperature. Similarly to the FeRh/Pt sample, we see that
the residual magnetisation is never zero but starts increasing slightly below
200 K. Polarized neutron reflectivity measurements have shown that this
residual magnetism mainly resides at the interface with the MgO substrate and
the top interface. Residual magnetism at the bottom interface does not have a
strong dependence on temperature Almeida_PhysRevMater_4_034410_2020. Residual
magnetism at the top surface, which is of interest for the interpretation of
our THz emission measurements, can be caused by alloying and therefore depends
on the capping material and unsually has a stronger dependence on temperature
fan_ferh; Baldasseroni_ApplPhysLett_100_262401_2012;
Ding_JApplPhys_103_07B515_2008. In our case, however, we don not observe a
significant difference in the SQUID data of capped and uncapped FeRh and we
therefore tend to associate the ferromagnetic top-layer with ferromagnetic
FePd that forms independently on capping. Previous works have shown that for
uncapped FeRh and FeRh capped with Pt, ferromagnetism at the top-surface is
negligible Baldasseroni2014; Baldasseroni_ApplPhysLett_100_262401_2012.
Differently from these works, our FeRh is doped with 2.8% Pd at the top
interface.

Fig. S5 (b) shows the THz emission amplitude as a function of temperature. As
discussed in the main text, in the FM phase the THz emission has a magneto-
dipole nature due to the reduced contribution from the spin-Hall effect. In
the AF phase we still observe a non-zero THz emission, odd with respect to
sample flipping. We explain this with the non-negligible spin-Hall angle of
AF-FeRh.

Fig. S 5:  (a) Temperature dependence of the saturation magnetisation
$M_{sat}$ extracted from SQUID magnetometry measurements for an uncapped
FeRh(30 nm) film. The inset represents a zoom-in of the low temperature region
below 300 K. (b) Temperature dependence of the THz emission amplitude in
uncapped FeRh(30 nm). An optical fluence of 1.2 mJ/cm2 was used and an in-
plane field of 860 mT was applied during the measurement. The two curves
represent measurements performed for the sample pumped from the substrate side
(blue curve) and top surface (red curve).

###  .5 Optical fluence dependence of THz emission from FeRh(30)/Pt(3.5)

Fig. S6 shows how the THz emission from FeRh(30)/Pt(3.5) depends on the
optical pump fluence. Normalised temperature dependence data presented in (c)
demonstrates that the low temperature THz emission enhancement is
qualitatively identical for all studied fluences as all curves overlap in the
20-300 K range. In (b), we show that the magnitude of $S_{y}^{*}$ is linear
with the fluence at temperatures in the 20-300 K range, and that no threshold
behaviour is observed. These results exclude that the THz enhancement at low
temperatures is due to a phase transition induced by transient pump heating.
In Fig. S6(d) we focus instead on the FM-AF transition region and show that as
the pump fluenece is increased the hysteresis closes, in agreement with
previous works Awari_ApplPhysLett_117_122407_2020.

Fig. S 6:  Optical fluence dependence of THz emission from FeRh(30)/Pt(3.5).
(a) Temperature dependence of $S_{y}^{*}$ for different pump fluences between
$0.28$ and $2.38$ mJ/cm2. (b) $S_{y}^{*}$ plotted as a function of pump
fluence at selected temperatures between 20 and 300 K. (c) Data sets from (a)
normalised with respect to the values of $S_{y}^{*}$ at 20 K. (d) Temperature
dependence of normalised $S_{y}^{*}$ for different pump fluences. Both cooling
and heating branches are shown.

###  .6 Dependence of THz emission on FeRh doping in FeRh(30)/Pt(3.5)
structures

According to our microscopic model, residual ferromagnetism at the interface
with Pt plays a key role in triggering spin pumping from antiferromagnetic
FeRh. This ferromagnetism will depend on the chemical composition and doping
level of the interface PdDoping. We have therefore compared spin pumping in
two different samples with the following composition:

FeRh35-Pt:
MgO/Fe_50Rh_46.8Pd_1.7Ir_1.5(5)/Fe_50Rh_47.1Pd_2.2Ir_0.7(10)/Fe_50Rh_47.2Pd_2.8(15)/Pt(3.5)
FeRh36-Pt: MgO/Fe_50Rh_47.2Pd_2.8(15)/ Fe_50Rh_47.1Pd_2.2Ir_0.7(10)/
Fe_50Rh_46.8Pd_1.7Ir_1.5(5)/Pt(3.5)
The two samples have identical doping levels but the doping gradient is
inverted. In this way we guarantee very similar bulk properties (crystal
quality and conductivity, which determines the THz outcoupling), but different
interface properties. In Fig. S7 we compare the temperature dependence of the
THz emission amplitude $S_{y}$ and spin pumping $S^{*}_{y}$ in the two
samples, normalised to the values in the ferromagnetic phase. First, we notice
that in sample FeRh35-Pt, the phase transition is shifted to higher
temperatures. This is in agreement with the fact that Ir-rich FeRh has a
higher transition temperature than Pd-rich FeRh 10.1063/1.4907282 and confirms
that only the volume closer to the Pt interface is involved in the spin-
pumping. Also, we see that as temperature is decreased below the transition
temperature, we measure a lower value of the spin pumping in FeRh35-Pt. This
shows that the chemical composition of the interface with Pt is important. The
fact that spin pumping decreases with Pd concentration provides further
evidence that residual magnetism resides in FePd alloying.

Fig. S 7: Temperature dependence of $S_{y}$ (a) and $S_{y}^{*}$ (b) when
pumping from the Pt side in sample FeRh35-Pt (black) and FeRh36-Pt (red). The
pump fluence is 2.38 mJ/cm2 and a magnetic field of 860 mT is applied.

###  .7 THz transmission through FeRh(30)/Pt(3.5

Fig. S8 shows the THz transmissivity of the FeRh-Pt sample as a function of
temperature. $\theta$ is defined as the ratio between the transmitted THz
radiation and the reference signal measured with no sample in the propagation
path at 1 THz. Changes in the structural and electronic properties of FeRh
across the AF-FM transition result in a steep variation of $\theta$ around
$T_{\mathrm{(AF-FM)}}$. The phase transition region displays a clear
temperature hysteresis between the cooling and heating branches, as expected
for a first-order phase transition. Below $T_{\mathrm{(AF-FM)}}$, $\theta$
continuously decreases, which is attributed due to the increasing
conductivities of FeRh and Pt Glover_PhysRev_108_243_1957;
Huisman_PhysRevB_92_104419_2015. However, if we compare the value of
$T_{\mathrm{(AF-FM)}}$ measured via THz transmission and optical pump-THz
emission (red dotted line) we observe that this is about 10 K lower in the
second case, which could be explained by a small amount of accumulated heat
when the sample is pumped optically.

Fig. S 8: Transmission of 1 THz field through FeRh(30)/Pt(3.5) for Pt
incidence. Cooling (grey) and heating (black) branches are indicated. The
experiment was performed with an external magnetic field of 860 mT
continuously applied in-plane. The red dotted line indicates the temperature
dependence of $S^{*}_{y}$ when cooling from high to low temperature. The pump
fluence is 2.38 mJ/cm2 and a magnetic field of 860 mT is applied.

###  .8 Comparison with MgO/CoFeB(30)/Pt(5) emitter

Here we compare the temperature dependence of the THz emission in
FeRh(30)/Pt(3.5) and CoFeB(30)/Pt(5). The magnetisation of CoFeB varies only
slightly with temperature in the explored temperature range (Fig. S9 (a)),
while the decrease in THz transmissivity as temperature is lowered reflects
the increase in conductivity, similarly to the FeRh sample (Fig. S8). The THz
emission (Fig. S 9 (c)-(d)), although comparable in magnitude, follows a very
different temperature dependence in the two samples, decreasing with
decreasing temperature in the CoFeB sample.

Fig. S 9: (a) Magnetic hysteresis loops of CoFeB(30)/Pt(5) measured by SQUID
at 300 K (black) and 20 K (red). (b) Transmission of 1 THz field through
CoFeB(30)/Pt(5) for Pt incidence. The experiment was performed with an
external magnetic field of 860 mT continuously applied in-plane. (c)
Temperature dependence of the THz emission amplitude $S^{_{y}}$ in
CoFeB(30)/Pt(5) (red) and FeRh(30/Pt(3.5) (black). The measurement was done
applying a 860 mT magnetic field and pumping the sample from the Pt side and
measuring the THz emission behind the substrate. (d) Temperature dependence of
the THz emission amplitude rescaled by the optical absorption and THz
outcoupling, $S^{*}_{y}$, in top-pumping geometry with an applied magnetic
field of 860 mT for CoFeB(30)/Pt(5) (red) and FeRh(30)/Pt(3.5) (black).

## S4. Generalised Atomistic Spin Dynamics Model

In the four-spin FeRh model in S2 .2 the Rh moments are not explicitly
represented, only the Fe moments are simulated. This means we cannot simulate
the effect of Rh moments being induced by the laser pulse. We therefore
develop a second spin model within the JAMS software package, where the
dynamics of both the Fe and Rh moments are solved. The FeRh crystal has the
space group Pm$\overline{3}$m (221). We perform calculations using the cubic
cell with the basis vectors

$\displaystyle\begin{split}\mathbf{a}&=a_{0}(1,0,0),\\\
\mathbf{b}&=a_{0}(0,1,0),\\\ \mathbf{c}&=a_{0}(0,0,1),\end{split}$ (6)

with the lattice constant $a_{0}=3.0$ Å. In the cubic cell, there is one Fe
atom and one Rh atom. These are located at (in fractional coordinates)

$\displaystyle\begin{split}\text{Fe}&=(0.0,0.0,0.0),\\\
\text{Rh}&=(0.5,0.5,0.5).\end{split}$ (7)

The Hamiltonian is

$\mathcal{H}=-\frac{1}{2}\sum_{ij}J_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\frac{1}{2}\sum_{ij\in
Fe}B_{ij}(\mathbf{S}_{i}\cdot\mathbf{S}_{j})^{2}-\sum_{i\in\mathrm{Fe}}k(\hat{\mathbf{e}}\cdot\mathbf{S}_{i})^{2}+\sum_{i}L_{2,i}|S_{i}|^{2}+L_{4,i}|S_{i}|^{4}+L_{6,i}|S_{i}|^{6},$
(8)

where $\vec{S}_{i}$ are dimensionless classical vectors which can change in
length. The first term is the Heisenberg exchange, where $J_{ij}$ are the
interaction energies. We used values from Polesya et al.
Polesya_PhysRevB_93_024423_2016 for Fe-Fe interactions and Fe-Rh interactions.
Rh-Rh interactions are considered to be negligible. The second term in the
Hamiltonian is a biquadratic exchange between Fe sites, which is necessary for
the metamagnetic phase transition Polesya_PhysRevB_93_024423_2016;
Derlet_PhysRevB_85_174431_2012. The factors of $1/2$ account for the double
counting of each bond. The third term is the uniaxial anisotropy energy of the
Fe sites with an axis $\hat{\mathbf{e}}$. The final term is a Landau
Hamiltonian that describes the energetics of changes in the length of spin
moments, where $L_{2},L_{4},L_{6}$ are parameters with units of energy. In our
model of FeRh the Fe moments have a stable minimum at unit length. The Rh
moments are unstable and contain only an $L_{2}$ term. This means that the
minimum energy for the Rh Landau term is for zero spin length. However, the
ferromagnetic Fe-Rh interactions compete with this and cause a Rh moment to
form if there is a net Fe magnetisation. The values we used in our model
Hamiltonian are given in Table 4. We use the generalised Langevin spin
equation Ma_PhysRevB_86_054416_2012

$\frac{\partial\mathbf{S}_{i}}{\partial
t}=-\gamma\mathbf{S}_{i}\times\mathbf{B}_{i}+\gamma\lambda\mathbf{B}_{i}+\bm{\xi}_{i}$
(9)

where $\mathbf{B}_{i}=-(1/\mu_{i})\nabla_{\mathbf{S}_{i}}\mathcal{H}$ is the
effective field in teslas with $\mu_{i}$ the magnetic moment on site $i$ in
Bohr magnetons, $\mu_{\mathrm{B}}$. $\mathbf{\xi}_{i}$ describes stochastic
processes of the Langevin thermostat. $\lambda$ is a dimensionless damping
parameter which describes the coupling to the bath. Here we use a quantum
thermostat, the correlations of $\xi_{i}$ obey

$\langle\xi_{i,a}(t)\rangle=0;\quad\langle\xi_{i,a}(t)\xi_{j,b}(t^{\prime})\rangle_{\omega}=\delta_{ij}\delta_{ab}\frac{2\lambda
k_{\mathrm{B}}T}{\gamma\mu_{i}}\frac{\hbar\omega}{\exp{(\hbar\omega/k_{\mathrm{B}}T)-1}},$
(10)

where $a,b$ are cartesian components ($x,y,z$), $k_{\mathrm{B}}$ is
Boltzmann’s constant, $T$ is the temperature in kelvins.
$\langle\cdots\rangle_{\omega}$ indicates the correlation function is defined
in frequency ($\omega$) space. Further details of the quantum thermostat
implementation can be found in Ref. barker. We integrate (9) numerically using
the Runge-Kutta fourth order method with a timestep of $\Delta t=0.1$ fs. The
values of the material parameters used for FeRh are given in Table 3. To
simulate the effect of a transient spin current inducing a Rh moment, we
increase the size of the Rh moment at the beginning of each time step using a
temporal Gaussian pulse

$\mathbf{S}_{i,\mathrm{Rh}}^{\prime}=\mathbf{S}_{i,\mathrm{Rh}}+\frac{\Delta}{2\pi\sigma}\exp{\left(-\frac{(t-t_{0})^{2}}{2\sigma^{2}}\right)\hat{\mathbf{J}}},$
(11)

which has a width of $\sigma$ in dimensions of time, a temporal center of
$t_{0}$, a dimensionless amplitude of $\Delta$ and $\hat{\mathbf{J}}$ is a
unit vector in the direction of the polarisation of the incoming spin current.
Within the model we assume that the FeRh Néel vector is at 45 degrees to the
[100] (along $x$ in Fig. 1(a) of the main text) direction
Xie_AIPAdv_7_056314_2017; Xie_NPGAsiaMater_12_67_2020 along which the field is
applied and the ferromagnetic regions will align. Therefore,
$\hat{\mathbf{e}}=(\sqrt{2}/2,\sqrt{2}/2,0)$ and the transient spin current is
polarised along $\hat{\mathbf{J}}=(1,0,0)$.

Table 3:  Model parameters for the generalised Langevin spin equation (9). symbol | value | units
---|---|---
$\gamma$ | $0.17608596$ | $\mathrm{rad}\cdot\mathrm{ps}^{-1}\cdot\mathrm{T}^{-1}$
$\lambda$ | $0.1$ | (dimensionless)
$\mu_{\mathrm{Fe}}$ |  $3.2$444Ref. Polesya_PhysRevB_93_024423_2016 | $\mu_{\mathrm{B}}$
$\mu_{\mathrm{Rh}}$ |  $1.0$444Ref. Polesya_PhysRevB_93_024423_2016 | $\mu_{\mathrm{B}}$
Table 4:  Exchange constants for the Hamiltonian (8). A single interaction vector is given in fractional coordinates; equivalent vectors can be generated from the m$\overline{3}$m point group symmetry operations. The interaction distances are given in units of the lattice constant ($a$). symbol | types | vector (fractional coordinates) | number | distance (lattice constants $a_{0}$) | value (meV)
---|---|---|---|---|---
$J_{1}$ | Fe-Rh | $(1/2,1/2,1/2)$ | $8$ | $0.866025$ |  $27.16$111Ref. Polesya_PhysRevB_93_024423_2016, Fig. 4b
$J_{2}$ | Fe-Fe | $(\quad 0,\quad 0,\quad 1)$ | $6$ | $1.000000$ |  $-3.90$222Ref. Polesya_PhysRevB_93_024423_2016, Fig. 4a, ‘DLM’ dataset
$J_{3}$ | Fe-Fe | $(\quad 0,\quad 1,\quad 1)$ | $12$ | $1.414214$ |  $5.84$222Ref. Polesya_PhysRevB_93_024423_2016, Fig. 4a, ‘DLM’ dataset
$J_{4}$ | Fe-Rh | $(1/2,1/2,3/2)$ | $24$ | $1.658312$ |  $1.54$111Ref. Polesya_PhysRevB_93_024423_2016, Fig. 4b
$J_{5}$ | Fe-Fe | $(\quad 1,\quad 1,\quad 1)$ | $8$ | $1.732051$ |  $-16.34$222Ref. Polesya_PhysRevB_93_024423_2016, Fig. 4a, ‘DLM’ dataset
$J_{6}$ | Fe-Fe | $(\quad 0,\quad 0,\quad 2)$ | $6$ | $2.000000$ |  $2.76$222Ref. Polesya_PhysRevB_93_024423_2016, Fig. 4a, ‘DLM’ dataset
$J_{7}$ | Fe-Fe | $(\quad 0,\quad 1,\quad 2)$ | $24$ | $2.236068$ |  $0.80$222Ref. Polesya_PhysRevB_93_024423_2016, Fig. 4a, ‘DLM’ dataset
$J_{8}$ | Fe-Fe | $(\quad 1,\quad 1,\quad 2)$ | $24$ | $2.449490$ |  $-1.32$222Ref. Polesya_PhysRevB_93_024423_2016, Fig. 4a, ‘DLM’ dataset
$B_{1}$ | Fe-Fe | $(\quad 0,\quad 0,\quad 1)$ | $6$ | $1.000000$ | $5.00$
$k$ | Fe | - | - | - | $0.06242$
$L_{2}$ | Fe | - | - | - |  $-440.987$333Ref. Ellis_PhysRevB_100_214434_2019
$L_{4}$ | Fe | - | - | - |  $150.546$333Ref. Ellis_PhysRevB_100_214434_2019
$L_{6}$ | Fe | - | - | - |  $50.6794$333Ref. Ellis_PhysRevB_100_214434_2019
$L_{2}$ | Rh | - | - | - | $100.64$
$L_{4}$ | Rh | - | - | - | $0.0$
$L_{6}$ | Rh | - | - | - | $0.0$

## S5. Estimate of Rh exchange field

The sudden appearance of the Rh moment couples to the Fe through a strong
exchange interaction. For the Fe in the AF state to generate a net moment the
system must overcome the critical field for a spin-flop transition. The
exchange field exists for such a short time that the spin-flop state is only
transient, but a net magnetisation is produced (Fig.4b main text – there is a
net Fe magnetisation) in the AF and a corresponding spin pumping. We can put
this on a more quantitative footing. The FeRh AFM state is dominated by the
antiferromagnetic $J_{5}$ interaction, and the effective exchange field felt
by the Fe sublattices has a field strength,

$B_{E}=\frac{8|J_{5}|}{\mu_{\mathrm{Fe}}}.$ (12)

The anisotropy field strength is,

$B_{A}=\frac{2k}{\mu_{\mathrm{Fe}}}.$ (13)

The lower bound for the critical field of a two sublattice antiferromagnet
with easy axis (at $T=0$) is gurevich_book,

$B_{0}>\sqrt{B_{A}(2B_{E}-B_{A})}$ (14)

where $B_{0}$ is a field strength applied along the easy axis. For our model
of FeRh $B_{0}\approx 30$ Tesla. Our applied field is much lower than this,
meaning that the two AFM modes can be split only a small amount, leading to
only a small spin current, proportional to the applied field as seen in normal
AFs. However, the spontaneously generated Rh moment causes the sudden
appearance of an additional exchange field inside of the AFM. This field is
not staggered (alternating between sublattices) but directional, due to the
spin polarised current. Hence, it acts on the antiferromagnetic Fe sublattices
in a similar way to an applied field, but is much stronger. The field strength
is approximately,

$B_{\mathrm{Rh}}=\frac{8|J_{1}|}{\mu_{\mathrm{Fe}}}S_{\mathrm{Rh}}.$ (15)

Our results show that we only generate
$S_{\mathrm{Rh}}=0.2\mu_{\mathrm{B}}/1.0\mu_{\mathrm{B}}$ (Fig. 4b main text),
but nevertheless the peak Fe-Rh exchange field is therefore
$B_{\mathrm{Rh}}=235$ Tesla, much greater than the spin-flop field. This
explains why the Fe develops a net magnetisation, it is trying to spin-flop,
although does not have enough time to fully transition in the 1 ps the field
appears for. This also explains why the spin pumping is far larger than can be
achieved from an applied field below the spin-flop field of an
antiferromagnet, and more similar to the large change seen in spin pumping
once the field is above the spin flop transition
Wu_PhysRevLett_116_097204_2016.