# Helicity-dependent Ultrafast Photocurrents in Weyl Magnet Mn3Sn

Dominik Hamara§ Cavendish Laboratory, University of Cambridge, J.J. Thomson
Avenue, Cambridge CB3 0HE, United Kingdom  Gunnar F. Lange§ Cavendish
Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE,
United Kingdom  Farhan Nur Kholid  Cavendish Laboratory, University of
Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Anastasios
Markou  Max Planck Institute for Chemical Physics of Solids, 01187 Dresden,
Germany  Physics Department, University of Ioannina, 45110 Ioannina, Greece
Claudia Felser  Max Planck Institute for Chemical Physics of Solids, 01187
Dresden, Germany  Robert-Jan Slager<EMAIL_ADDRESS>Cavendish Laboratory,
University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, United
Kingdom  Chiara Ciccarelli<EMAIL_ADDRESS>Cavendish Laboratory, University
of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom

###### Abstract

§ These authors contributed equally.

We present an optical pump-THz emission study on non-collinear antiferromagnet
Mn3Sn. We show that Mn3Sn acts as a source of THz radiation when irradiated by
femtosecond laser pulses. The polarity and amplitude of the emitted THz fields
can be fully controlled by the polarisation of optical excitation. We explain
the THz emission with the photocurrents generated via the photon drag effect
by combining various experimental measurements as a function of pump
polarisation, magnetic field, and sample orientation with thorough symmetry
analysis of response tensors.

##  I Introduction

Mn3Sn is a noncollinear antiferromagnet (AF) and a Weyl semimetal (WSM). It
crystallises in a hexagonal P63/mmc structure. Below the Néel temperature
($T_{N}$ $\approx$ 420 K for bulk Mn3Sn Sung  _et al._ (2018)) the geometrical
frustration of the Mn atoms in the $a$-$b$ plane of the Kagome lattice leads
to an inverse triangular spin structure, with 120° ordering Brown  _et al._
(1990); Sung  _et al._ (2018); Yang  _et al._ (2017); Li  _et al._ (2018);
Cheng  _et al._ (2019). Despite a vanishingly small net magnetisation, Mn3Sn
displays phenomena that conventionally occur in ferromagnets, such as a large
anomalous Hall effect Kübler and Felser (2014); Nakatsuji  _et al._ (2015);
Nayak  _et al._ (2016), anomalous Nernst effect Ikhlas  _et al._ (2017), and
magneto-optical Kerr effect Higo  _et al._ (2018). This is possible due to the
unique material topology and a nonzero Berry curvature resulting from the
inverse triangular spin structure Sung  _et al._ (2018); Nayak  _et al._
(2016). Ab-initio band structure calculations have reported the existence of
multiple Weyl points in the bulk and corresponding Fermi arcs on the surface
of Mn3Sn Yang  _et al._ (2017). An effect associated with WSMs is the presence
of helicity-dependent photocurrents arising from non-linear optical effects.
These have been observed using both electrical and THz techniques Chan  _et
al._ (2017), and have been linked to the topological charge of the Weyl nodes
de Juan  _et al._ (2017), via the circular photogalvanic effect.

In this work, we present the experimental observation of helicity-dependent
ultrafast photocurrents in an 80 nm Mn3Sn film at room temperature (RT) using
optical pump-THz emission spectroscopy. The magnitude and direction of the
photocurrents depend on the polarisation of the pump pulse and the direction
of its wavevector relative to the surface of the film, but have no dependence
on magnetic field. These currents cannot be attributed to a bulk photogalvanic
effect as this requires the breaking of inversion symmetry Le and Sun (2021).
Mn3Sn, however, respects inversion symmetry even when accounting for the
magnetic ordering. This suggests that our signal originates either from a
different bulk mechanism such as the inverse spin Hall effect Hirai  _et al._
(2020) or photon-drag effect Ribakovs and Gundjian (1977); Maysonnave  _et
al._ (2014), or from a surface photogalvanic effect Steiner  _et al._ (2022).
Our symmetry analysis of response tensors suggests that the helicity-dependent
photocurrents arise predominantly due to the circular photon drag effect.

##  II Experimental Results

###  II.1 The sample

The subject of the study is an MgO(111)(0.5 mm)/Ru(5 nm)/Mn3Sn(80 nm)/Si(3 nm)
sample. Epitaxial Mn3Sn films were grown using magnetron sputtering in a
BESTEC ultra-high vacuum (UHV) system with a base pressure less than $2\times
10^{-9}$ mbar and a process gas (Ar 5 N) pressure of $3\times 10^{-3}$ mbar.
The target to substrate distance was fixed at $20$ cm and the substrates were
rotated during deposition to ensure homogeneous growth. The underlayer was
deposited using a Ru (5.08 cm) target by applying 40 W DC power with the
substrate held at 400∘ C. Following cooling back to room temperature, Mn3Sn
was grown from Mn (7.62 cm) and Sn (5.08 cm) sources in confocal geometry,
using 47 W and 11 W DC power respectively. The stack was then annealed in-situ
under UHV at 350∘ C for 10 minutes. The stoichiometry is Mn75Sn25 (± 2 at. %),
estimated by using energy dispersive x-ray spectroscopy (see SI). Finally, a
Si capping layer was deposited at room temperature using an Si (5.08 cm)
target at 60 W RF power to protect the film from oxidation. Magnetotransport
studies on films grown under the same conditions and with comparable crystal
quality are presented in Taylor  _et al._ (2020) and show a large anomalous
Hall effect at room temperature and a transition to topological Hall effect
below 50 K.

###  II.2 Experimental layout

In Fig. 1(a) we show the optical pump-THz emission geometries used in our
experiments. When presenting a data set we will refer to the experiment
geometry in which this data was collected as configuration 1, 2 or 3. To
indicate different directions we introduce two Cartesian coordinate systems,
also indicated in Fig. 1(a): ($x$, $y$, $z$) - fixed with respect to the
experimental setup; ($a$, $b$, $c$) - fixed with respect to the sample. Laser
pulses of 50 fs duration with a central wavelength of 800 nm propagate along
the $z$ axis. The optical fluence is fixed at 2.9 mJ/cm2. A quarter-wave plate
(QWP) placed in the pump path is used to control the polarisation and helicity
of the pulses. In-plane rotations of the QWP by an angle $\varphi$ allow
changing between linear ($\varphi=0\degree\pm n\pi/2$), left-handed circular
(LHCP) ($\varphi=45\degree\pm n\pi$), right-handed circular (RHCP)
($\varphi=-45\degree\pm n\pi$), and intermediate elliptical polarisations. The
angle between the laser beam and the sample surface can be varied by rotating
the sample away from normal incidence about the $x$ axis (configuration 2) or
$y$ axis (configuration 3). We define the tilting angles as $\theta_{x}$ and
$\theta_{y}$ respectively. Additionally, the sample can be rotated in-plane,
about the $c$ axis by an angle defined as $\theta_{c}$. An external magnetic
field up to $\mu_{0}H_{x}\approx$ 860 mT can be applied along the $x$
direction. Unless stated otherwise, the experiments presented in the main body
of the paper were performed at room temperature (RT).

###  II.3 THz emission from Mn3Sn

The optically induced charge currents $J_{i}(t)$ result in broadband THz
electro-dipole emission $s_{i}(t)$. The subscript $i$ indicates the
polarisation components $i=x$ or $i=y$, along the two axes of the experimental
setup $x$ and $y$. $S_{i}$ is defined as the integrated peak amplitude of the
emitted THz pulse.

Fig. 1(b) shows the $\varphi$-dependence of $S_{y}$, measured at zero magnetic
field ($\mu_{0}H_{x}=0$) and at normal pump incidence (configuration 1). The
different polarisations of the pump pulse that correspond to the different
values of $\varphi$ are also indicated to facilitate the reading. The data is
decomposed into different harmonics by fitting with the equation Ji  _et al._
(2019):

$S_{i}(\varphi)=H_{i}(\varphi)+L_{i}(\varphi)+B_{i}$ (1)

Here, $H_{i}(\varphi)=h_{i}\sin(2\varphi+\varphi_{1})$ is the magnitude of the
circular polarisation helicity-dependent component with a phase shift
$\varphi_{1}$, $L_{i}(\varphi)=l_{i}\sin(4\varphi+\varphi_{2})$ is the linear
polarisation-dependent component with a phase shift $\varphi_{2}$, and $B_{i}$
is the polarisation-independent background component. As displayed in Fig.
1(c), the decomposition of $S_{y}$ shows that $H_{y}(\varphi)$,
$L_{y}(\varphi)$, and $B_{y}$ all contribute to the measured signal. For
comparison, $S_{x}$ measured in the same experimental geometry (configuration
1) displays a relatively smaller contribution from the helicity-dependent
$H_{x}(\varphi)$ component, and is dominated by $L_{x}(\varphi)$ (Fig. 6 in
Supplementary Information). We attribute this difference to a small
unintentional rotation around the $y$ axis, as it will be further justified in
the analysis that follows. In our set-up, the sample’s mount orientation can
be freely adjusted around the $y$ axis, but not around the $x$ axis, so an
unintentional tilting around the $y$ axis is more likely.

###  II.4 The effect of experiment geometry

Here we study how both polarisation and amplitude of the photocurrent-
generated THz emission depend on the direction of the optical wavevector
$q_{z}$ relative to the sample surface. For this purpose we tilt the sample
whilst leaving the direction of the pump wavevector $q_{z}$ unchanged with
respect to the laboratory frame of reference, along the $z$-direction. For
$\theta_{x}$ (configuration 2) or $\theta_{y}$ (configuration 3) different
from zero, $q_{z}$ has a non-zero projection along the $b$ and $a$ directions
on the plane of the Mn3Sn film, which we label $q_{b}$ and $q_{a}$
respectively. Consequently, the components of the wavevector relative to the
sample frame are $[0,q_{b},q_{c}]$ in configuration 2 and and
$[q_{a},0,q_{c}]$ in configuration 3.

Fig. 2(a) and (b) show the components of the THz emission polarised along the
$x$ and the $y$ directions as a function of $\varphi$, measured in
configuration 2 (a) and configuration 3 (b). $S_{x}(\varphi)$ and
$S_{y}(\varphi)$ are fitted with Eq. (1) to extract the coefficients $h_{x}$
and $h_{y}$. While in Fig. 2(a) $h_{x}\gg h_{y}$, in Fig. 2(b) the trend is
inverted and $h_{x}\ll h_{y}$. This suggests that the direction of the
helicity-dependent photocurrent, and therefore of the THz polarisation,
depends on the projection of the pump wavevector on the sample plane and is
perpendicular to it. Due to finer control of $\theta_{y}$ in comparison to
$\theta_{x}$ in our setup, we restrict the following analysis to configuration
3 only. In Fig. 2(c) we show that the THz emission amplitude increases with
tilting angle $\theta_{y}$. We now study the symmetry of the THz emission for
opposite tilting angles by decomposing $S_{i}(\varphi)$ into even and odd
contributions as:

$\begin{split}S_{i}(\varphi)_{even}(\theta_{y}=\pm
15\degree)=S_{i}(\varphi)(\theta_{y}=+15\degree)+S_{i}(\varphi)(\theta_{y}=-15\degree)\end{split}$
(2)

$\begin{split}S_{i}(\varphi)_{\text{odd}}(\theta_{y}=\pm
15\degree)=S_{i}(\varphi)(\theta_{y}=+15\degree)-S_{i}(\varphi)(\theta_{y}=-15\degree)\end{split}$
(3)

In Fig. 2(d) We observe that the odd component of $S_{y}$ is dominant.
Analogous behaviour is presented in the Supplementary Information (SI) for
$S_{x}$. Our observations suggest that the helicity-dependent photocurrents
are generated in the direction perpendicular to the in-plane projection of the
pump wavevector and are proportional to it.

###  II.5 Magnetic field dependence

In this section we want to understand whether the photocurrents have a
magnetic origin and depend on the magnetic phase of Mn3Sn. Fig. 3(a) shows
$S_{y}$ measured for two opposite directions of the magnetic field $\pm 0.860$
mT. According to Reichlova et al. Reichlova  _et al._ (2019), $0.860$ mT may
be too low to switch the magnetic ordering in Mn3Sn thin films at RT, while it
is sufficient to reverse the spins at temperatures close to $T_{N}$. Hence,
for these measurements we followed a cool down procedure with the magnetic
field continuously applied from 420 K to 380 K, at which the experiment was
performed. No significant dependence on magnetic field is measured. We also do
not observe a qualitative difference in the time-domain THz transients
measured at different fields as shown in the SI.

We further confirm that the polarisation and amplitude of the emitted THz
pulse is not correlated with the magnetic phase of Mn3Sn by repeating the
measurement after rotating the sample by $90\degree$ around the $c$ axis. Fig.
3(b) shows $S_{x}(\varphi)$ and $S_{y}(\varphi)$ prior and after the rotation
by $\theta_{c}=+90\degree$. If the direction of the photoinduced currents,
hence the polarisation of the THz emission, were correlated with the
orientation of the spins we would have expected a rotation of the THz
polarisation plane by $90\degree$, which we do not observe. Instead, the two
graphs of $S_{x}(\varphi)$ and $S_{y}(\varphi)$ overlap, as is discussed
further in the SI.

##  III Theoretical analysis

###  III.1 Nonlinear optical effects

In this section, we investigate nonlinear optical effects in Mn3Sn as an
explanation of the observed signal. Other possible sources of helicity-
dependent photocurrents are discussed in the subsequent section.

We consider a phenomenological expression for the induced photocurrent $J_{i}$
that in turn generates THz emission via electro-dipole interaction. Using the
notation from Ref. Hamh  _et al._ (2016) and expanding to second order in the
light wave amplitude $E(\omega)$ for a frequency $\omega$ we write:

$J_{i}=\chi_{ijk}^{(2)}E_{j}E_{k}^{*}+\chi_{ijkl}^{(3)}q_{j}E_{k}E_{l}^{*}$
(4)

where all indices run over the Cartesian coordinates of the sample
$i,j,k,l\in(a,b,c)$ and $\boldsymbol{q}$ is the momentum of the incoming
light. At normal incidence (configuration 1), only $q_{c},E_{a}$ and $E_{b}$
are non-zero. The first term describes the photogalvanic effect (PGE), whereas
the second term describes the photon-drag effect (PDE). We further decompose
each of these tensors into symmetric and antisymmetric components with respect
to the light wave amplitude which respectively give rise to the linear
photogalvanic/photon drag effect (LPGE/LPDE) and the circular
photogalvanic/photon drag effect (CPGE/CPDE) Karch  _et al._ (2010). Note that
all quantitative features of these effects depend crucially on the details of
the band structure. We show the ab-initio bulk and surface band structures in
Fig. 4, for an energy-window of $\hbar\omega\approx 1.55$ eV, corresponding to
the central wavelength of the laser pulses. Due to the large number of bands
involved, we focus on a qualitative phenomenological symmetry analysis of the
tensors in Eq. (4).

####  III.1.1 Symmetry-constrained model

The spatial symmetries of the material constrain the tensors in Eq. (4) and
topology Kruthoff  _et al._ (2017); Schrunk  _et al._ (2022). Mn3Sn has space-
group symmetry P$6_{3}/$mmc when ignoring magnetism and magnetic space-group
symmetry C$m^{\prime}$c$m^{\prime}$ when including the AFM ordering Brown  _et
al._ (1990). For the tensor symmetry analysis, we focus on the unitary point-
group symmetries as detailed in the SI, where also complete expressions for
the symmetry-allowed forms of the PGE/PDE tensors in Eq. (4) are given. Here,
we only summarise the number of independent coefficients for the various
symmetry settings as shown in Tab. 1. In general, there are too many possible
terms, making a quantitative model infeasible. However, by carefully comparing
with our experimental results, it is possible to identify effects are the most
relevant as detailed in the next section and in the SI.

PG | Relevance | LPGE | CPGE | LPDE | CPDE
---|---|---|---|---|---
6/mmm | NM bulk | 0 | 0 | 7 | 3
3m | NM surface | 4 | 1 | 10 | 4
2/m | M bulk | 0 | 0 | 28 | 13
1 | M surface | 18 | 9 | 54 | 27
Table 1: Number of independent elements for the linear/circular photogalvanic
effect (LPGE/CPGE) and the linear/circular photon drag effect (LPDE/CPDE), for
various unitary point-group (PG) symmetries relevant to the non-
magnetic/magnetic (NM/M) bulk/surface of the material.

####  III.1.2 Interpretation of results

We begin by considering the effect of magnetism. As shown in Fig. 3(a) our
results are insensitive to the direction and magnitude of the external
magnetic field, as well as to the intrinsic spin ordering of the material [see
Fig. 3(b)]. This suggests that the generated photocurrents do not arise as a
result of the magnetic ordering in the material. We therefore focus on the
non-magnetic symmetry analysis in what follows. Because the bulk of the sample
respects inversion symmetry we should not expect any bulk contribution from
the PGE in this case.

We address here the result of tilting the sample away from the normal pump
incidence, as shown in Fig. 2. As discussed in detail in the SI, the tilting
changes both what currents are generated in the material, and which part of
the resultant THz radiation is measured at the detector. Tilting in
$\theta_{x}$ (configuration 2) or $\theta_{y}$ (configuration 3) changes the
geometry of the sample relative to the detector. In particular, we are able to
resolve currents generated in the $c$-direction Ni  _et al._ (2021). Thus, the
detected integrated amplitude of the THz transient pulse in configuration 2 is
given by:

$\begin{split}\begin{gathered}S_{x}\propto J_{a}(\theta_{x})\\\ S_{y}\propto
J_{b}(\theta_{x})\cos\theta_{x}+J_{c}(\theta_{x})\sin\theta_{x}\end{gathered}\end{split}$
(5)

And in configuration 3:

$\begin{split}\begin{gathered}S_{x}\propto
J_{a}(\theta_{y})\cos\theta_{y}+J_{c}(\theta_{y})\sin\theta_{y}\\\
S_{y}\propto J_{b}(\theta_{y})\end{gathered}\end{split}$ (6)

Where the expressions for $J_{i}(\theta_{k})$ arising from PGE/PDE are give in
the SI. When considering the contribution from the bulk photon drag effect and
the surface photogalvanic effect to $\boldsymbol{J}$, we find that all terms
in the detected amplitude $S_{i}$ arising from a bulk photon-drag effect are
odd under tilting in both $\theta_{x}$ and $\theta_{y}$, whereas the linear
surface photogalvanic effect contains lowest-order terms that are even under
tilting. As the odd components make a significant contribution to our results
[see e.g. Fig. 1(c)], we interpret our signal to arise predominantly from the
bulk photon drag effect. As discussed in the SI, we further find that the
circular photon drag contribution to $S_{i}$ normal to the rotation axis is
suppressed, in agreement with the experimental results, and that only the bulk
non-magnetic photon drag contribution is invariant under in-plane rotation
[see Fig. 3(b)]. We therefore interpret our signal to predominantly arise from
a bulk photon drag effect.

We note finally that the photon drag effect is also appealing from a bulk
band-structure perspective. As shown in Fig. 4(b), Mn3Sn has a flat bulk band
below the Fermi energy around the K-point, with a corresponding band at a
distance of $\hbar\omega$. This may lead to a large joint density of states,
and allow for non-vertical transitions with finite $\boldsymbol{q}$.

###  III.2 Other possible sources of photocurrent

Another way in which helicity-dependent photocurrents can be generated is
through a combination of the inverse Faraday effect (IFE) and inverse spin
Hall effect (ISHE), as described recently for Bismuth thin-films in Ref. Hirai
_et al._ (2020). This could be an important mechanism in Mn3Sn, where effects
related to the nonzero Berry curvature are significant Nayak  _et al._ (2016)
and may result in strong responses of the Faraday effect Yang  _et al._
(2014). Furthermore, Mn3Sn has been shown to have a large spin Hall angle
Matsuda  _et al._ (2020). However, we believe this mechanism does not play a
significant role in our experimental results. Firstly, for an efficient
conversion of IFE-induced spin currents into charge currents and THz electric
fields the spins must travel relatively long distances Hirai  _et al._ (2020).
This is not the case in Mn3Sn, where the reported spin propagation length is
below 1 nm Muduli  _et al._ (2019). Secondly, a Berry curvature related
effect, such as the IFE, would show a strong dependence on the magnetic phase
of Mn3Sn. We do not observe this behaviour in our temperature-resolved
measurements (see Fig. 9 in the SI). Previous studies have reported that at
temperatures above 420 K Mn3Sn becomes paramagnetic and that upon cooling
below RT the material can undergo transitions into the spiral and spin glass
phases Sung  _et al._ (2018). Our investigation in the temperature range of
$50-400$ K does not reveal any abrupt changes in the magnitude of THz signals
that could indicate the role of magnetic phase-dependent mechanisms. We do
not, however, rule out that our sample remains in the same magnetic state over
the entire investigated temperature range. Finally, the mechanism relying on
the IFE and ISHE could only explain the helicity-dependent photocurrents and
cannot account for the generation of photocurrents that show linear dependence
on the pump polarisation. As shown in Fig. 1(c), the magnitudes of $H_{y}$ and
$L_{y}$ are comparable, and therefore we suggest their main contributions
originate from related mechanisms, namely the CPDE and LPDE.

##  IV Summary

In conclusion, using optical pump-THz emission spectroscopy we demonstrate the
generation of helicity-dependent ultrafast photocurrents in a Mn3Sn thin film.
The magnitude and direction of these can be fully controlled by the
polarisation and incidence angle of the optical pump and are not affected by
external magnetic fields. We combine the experimental results with theoretical
analysis to suggest that the bulk photon drag effect is the main mechanism
responsible for the generation of the helicity-dependent photocurrents.

## Methods

The electronic band structure was calculated using density-functional theory
(DFT) as implemented in Quantum Espresso Giannozzi  _et al._ (2009, 2017) with
a fully-relativistic norm-conserving pseudopotential, generated using the
ONCVPSP package Hamann (2013). We used the experimental crystal parameters
$a=b=5.67$ Å and $c = 4.53$ Å, with an $8\times 8\times 8$ k-grid and a
kinetic-energy cutoff of 870 eV. The magnetic structure was relaxed by
constraining the total direction of the magnetization. The bands were then
Wannierised using Wannier90 Mostofi  _et al._ (2014), with all d-orbitals of
Mn considered in the projector. Finally, the slab band structure was computed
using WannierTools Wu  _et al._ (2018).

###### Acknowledgements.

C. C. and D. H. acknowledge support from the Royal Society. G. F. L is funded
by the Aker Scholarship. R. J. S acknowledges funding from a New Investigator
Award, EPSRC grant EP/W00187X/1, as well as Trinity college, Cambridge. G. F.
L thanks B. Peng, S. Chen, U. Haeusler and M.A.S. Martínez for useful
discussions.

##  V figures

Fig. 1: Optical pump-THz emission from Mn3Sn. (a) Experiment schemes in
different geometries. Ultrafast laser pulses travelling along $z$ axis are
incident on the sample, leading to the emission of $E_{i}(t)$ fields. In
experiments, we resolve their two orthogonal components of transient THz
signals: $s_{x}(t)\propto E_{x}(t)$ and $s_{y}$(t)$\propto E_{y}(t)$.
Configuration 1 corresponds to a measurement in the standard geometry with
normal laser incidence, in which $\theta_{x}=0$ and $\theta_{y}=0$. In
configuration 2 and configuration 3 the sample is rotated about the $x$
($\theta_{x}\neq 0$ and $\theta_{y}=0$) or $y$ ($\theta_{x}=0$ and
$\theta_{y}\neq 0$) axes respectively. Please note that the scheme of
configuration 2 is presented in a different perspective than configuration 1
and 3. (b) shows THz signals detected along $y$, emitted from Mn3Sn optically
pumped with laser pulses of different polarisations. The measurements were
performed in configuration 1 without an external magnetic field. The
experimental data set is fitted with function introduced in Eq. 1 to find
$H_{y}$, $L_{y}$, and $B_{y}$. These are presented in (c) with the extracted
values of $|h_{y}|$, $|l_{y}|$, and $B_{y}$. Fig. 2: Study of various
experiment geometries in optical pump-THz emission from Mn3Sn. Normalised
$S_{x}(\varphi)$ and $S_{y}(\varphi)$ data sets measured in configuration 2
and configuration 3 are plotted in (a) and (b) respectively. The sample was
rotated by $\theta_{x}=+25\degree\pm 5\degree$ or $\theta_{y}=+15\degree\pm
2\degree$ respectively. (c) shows THz signals measured along $y$ in
configuration 3 for tilting angles $\theta_{y}$ between 0 and 30°. The data
sets were fitted with Eq. 1 to extract the values of $|h_{y}|$ and $|l_{y}|$
parameters. Extracted $|h_{x}|$ and $|h_{y}|$ are displayed in the graphs. (d)
shows even (see Eq. 2) and odd (see Eq. 3) responses in respect to the
direction of the rotation around the $\theta_{y}$ axis. The figure displays
normalised THz signals measured along $y$ in configuration 3. All measurements
in (a-d) were performed at RT with no magnetic field. Data sets shown in (a),
(b) and (d) were fitted with Eq. 1 prior to normalisation to subtract the
polarisation-independent backgrounds, $B_{i}$. Fig. 3: Magnetic field
dependence of THz emission from Mn3Sn. (a) THz signals measured at 380 K in
configuration 3 after field-cooling from 420 K with opposite directions of
external magnetic field continuously applied along $x$. (b) Normalised
$S_{x}(\varphi)$ and $S_{y}(\varphi)$ data sets obtained prior, and after an
in-plane rotation of the film by $90\degree$. Prior to the measurements the
sample was cooled down from 420 K to RT with $\mu_{0}H_{x}=+860\text{mT}$
applied. The field was switched off at RT, and $S_{i}(\varphi)(\theta_{c}=0)$
were measured. Without repeating the field-cooling procedure and at
$\mu_{0}H_{x}=0$, the film was rotated i-p by $+90\degree$ to obtain
$S_{i}(\varphi)(\theta_{c}=+90\degree)$. The experiment was performed in
configuration 3. The plots show normalised signal values. Prior to
normalisation the data sets were fitted with Eq. 1 to subtract the
polarisation-independent backgrounds, $B_{i}$. Fig. 4: Bulk ((a) & (b)) and
surface ((c) & (d)) band structure for Mn3Sn. In (a) we show the bulk band
structure, for an energy window corresponding to the frequency of the laser.
In (b) we show the same plot, but with the color and opacity of the bands
indicating the value of
$\min_{m}\\{E_{n}(\boldsymbol{k})-(E_{m}(\boldsymbol{k})\pm\hbar\omega)\\}$
for band $n$, with $\pm$ indicating whether the band is above/below the Fermi
surface. This is a rough indicator of the possibility of a vertical transition
from band $n$ occurring. The flat bands around K may also allow non-vertical
transitions. In (c) and (d) we show the surface band structure for the
$c$-directed top and bottom surface respectively, with the color indicating
how well-localised the states are on the surface.

##  Appendix A Sample characterisation by X-ray diffraction analysis

Fig. 5:  X-ray diffraction of the MgO(111)(0.5 mm)/Ru(5 nm)/Mn3Sn(50 nm)/Si(3
nm) sample.

##  Appendix B Components of THz signal at normal incidence

Fig. 6:  THz components emitted at normal laser incidence. THz signals
measured for two orthogonal THz polarisations (along $x$ and along $y$) are
plotted as a function of $\varphi$. The data sets were fitted with Eq. (1) in
the main text to extract the $h_{i}$ and $l_{i}$ parameters.

##  Appendix C Even and odd THz responses measured along $x$

Fig. 7:  Even and odd THz responses measured along $x$ as a function of
$\varphi$. The responses were obtained using Eq. (2) and Eq. (3) in the main
text respectively. The experiment was performed in configuration 3.

##  Appendix D THz transients measured at different magnetic fields

Fig. 8:  Transients THz fields detected for different directions of magnetic
field and with no field applied. The measurement was performed at RT at a
fixed position of the quarter wave plate ($\varphi=45\degree$), corresponding
to RHCP.

##  Appendix E Temperature dependence of THz signal components

Fig. 9:  Temperature dependence of THz signal components. (a) shows peak THz
signals measured along $y$ as a function of $\varphi$ at different
temperatures between 50 K and 400 K. The measurement was performed in
configuration 1 and with no magnetic field applied. The values of $|l_{y}|$
and $|h_{y}|$ were extracted from these data sets and are displayed in (b).

##  Appendix F Further theory details

###  F.1 Relevant space-groups

Mn3Sn crystallizes in a layered hexagonal lattice. When ignoring the magnetic
moments, the bulk space group is the non-magnetic (gray) group
P$6_{3}/$mmc$1^{\prime}$ (No.194). Accounting for magnetism, the triangular
antiferromagnetic phase stabilized between approximately 250 K and 420 K, is
described by the magnetic space-group C$m^{\prime}$c$m^{\prime}$ (No. 63.464
in the BNS convention) albeit in a non-standard setting as shown in Ref. Brown
_et al._ (1990).

For the tensor symmetry analysis, only the point-groups (PG) matter. The non-
magnetic bulk PG is 6/mmm$1^{\prime}$, whereas the bulk magnetic PG is
$m^{\prime}m^{\prime}m$. Because the photogalvanic effect (PGE) vanishes in
the bulk, we also analyze the symmetry properties of the surface of the
material. For all configurations, the light impinges on the surface
perpendicular to the $c$-axis of the material. The non-magnetic PG associated
with this surface is 3m$1^{\prime}$, whereas the magnetic PG is simply
$m^{\prime}$. Note that this assumes that there is no reordering of the
surface. We will argue that the observed signal arises predominantly from a
bulk effect and as such this is not a major limitation.

###  F.2 Symmetry analysis of non-linear optical tensors

This section discusses the full symmetry analysis for the optical response
tensors considered in the main text. We begin by discussing the role of time-
reversal symmetry, and then go on to discussing spatial symmetries.

####  F.2.1 Role of time-reversal symmetry

The role of time-reversal in tensor symmetry analysis requires special care as
described in Refs. El-Batanouny and Wooten (2008); Eremenko  _et al._ (1992);
Gallego  _et al._ (2019). There are two distinct microscopic mechanisms
(usually called ”shift” and ”injection” currents) Sipe and Shkrebtii (2000)
leading to induced currents in the PGE. These behave differently under time-
reversal. As we are only interested in phenomenological expressions for the
induced current, we do not consider the microscopic mechanisms. We therefore
neglect antiunitary symmetries entirely when constraining our tensors, so that
the analysis is agnostic to the underlying microscopic mechanism. The unitary
PG in the non-magnetic case are 6/mmm in the bulk and 3m on the surface. In
the magnetic case, the unitary part of the bulk PG is 2/m, whereas the unitary
part of the surface PG is 1.

####  F.2.2 Role of spatial symmetries

To understand how the remaining unitary point-group symmetries constrain the
tensors $\chi_{ijk}^{(2)}$ and $\chi_{ijk}^{(3)}$ in Eq. (4) of the main text,
we first decompose the tensors describing the induced currents into symmetric
and antisymmetric components with respect to the light field:

$J_{i}=\chi^{(2),s}_{ijk}[E_{j},E_{k}^{*}]+\chi^{(2),a}_{ik}P_{\mathrm{circ}}\hat{e}_{k}+\chi_{ijkl}^{(3),s}q_{j}[E_{k},E_{l}^{*}]+\chi^{(3),a}_{ijk}q_{j}P_{\mathrm{circ}}\hat{e}_{k}$

Where the symmetric (s) and antisymmetric (a) terms corresponds to the
linear/circular PGE/PDE respectively. Following notation from Ref. Karch  _et
al._ (2010) we have defined:

$[E_{j},E_{k}^{*}]=\frac{E_{j}E_{k}^{*}+E_{k}E_{j}^{*}}{2}$ (7)

$P_{\mathrm{circ}}\hat{e}_{k}=i(E\times E^{*})_{k}$ (8)

The permutation symmetry of the tensors can be accounted for by using Jahn’s
symbols as discussed in Gallego  _et al._ (2019). The Jahn’s symbol for each
tensor is respectively $\chi_{i[jk]}^{(2),s}$: V[V2], $\chi_{ik}^{(2),a}$:
V{V2}, $\chi_{ij[kl]}^{(3),s}$: V2[V2] and $\chi_{ijk}^{(3),a}$: V2{V2}. To
find all symmetry-allowed terms, we use the MTENSOR tool Gallego  _et al._
(2019) hosted on the Bilbao Crystallographic Server (BCS) Aroyo  _et al._
(2006). We write the resultant currents in the $(a,b,c)$ Cartesian coordinate
system of the material. The full expressions for the induced current for all
combinations of magnetic/non-magnetic, PDE/PGE and surface/bulk are in Sec. G.

###  F.3 Dependence of signal on polarization and sample geometry

####  F.3.1 Dependence of incoming fields on angle and sample geometry

We assume the laser is linearly polarized before passing through the quarter-
wave plate, corresponding to a Jones vector Yariv and Yeh (2003) of $\hat{x}$.
After passing through a quarter-wave plate at angle $\varphi$, we find in the
laboratory frame:

$|E_{x}|^{2}\propto\cos 4\varphi$ (9) $|E_{y}|^{2}\propto\cos 4\varphi$ (10)
$[E_{x},E_{y}]\propto\sin 4\varphi$ (11) $P_{circ}\hat{e}_{k}\propto[\sin
2\varphi]\hat{e}_{k}$ (12)

We assume that the THz emission follows the same polarization dependence, and
therefore associate components in the measured signal with frequency
$4\varphi$ with the LPGE/LPDE and components with frequency $2\varphi$ with
the CPGE/CPDE.

In the laboratory $(x,y,z)$ Cartesian frame, $\boldsymbol{E}=(E_{x}.E_{y},0)$
and $\boldsymbol{q}=(0,0,1)$. Writing
$\boldsymbol{\tilde{E}},\boldsymbol{\tilde{q}}$ for the quantities measured in
the material $(a,b,c)$ Cartesian frame then gives for a rotation along the
$x$-axis (configuration 2):

$\boldsymbol{\tilde{E}}=(E_{x},E_{y}\cos\theta_{x}E_{y},\sin\theta_{x})$ (13)
$\boldsymbol{\tilde{q}}=(0,q_{z}\sin\theta_{x},q_{z}\cos\theta_{x})$ (14)
$P_{\mathrm{circ}}\hat{e}_{a}=0$ (15)

Whereas rotating along the $y$-axis (configuration 3) gives:

$\boldsymbol{\tilde{E}}=(E_{x}\cos\theta_{y},E_{y},E_{x}\sin\theta_{y})$ (16)
$\boldsymbol{\tilde{q}}=(q_{z}\sin\theta_{y},0,q_{z}\cos\theta_{y})$ (17)
$P_{\mathrm{circ}}\hat{e}_{b}=0$ (18)

####  F.3.2 Expressions at normal incidence, $\theta_{x}=0,\theta_{y}=0$
(configuration 1)

At normal incidence (configuration 1), we can ignore optical corrections from
refraction. From the full expression in Sec. G, we find for the induced bulk
current when ignoring the magnetic ordering:

$\begin{split}J_{a}=0\\\ J_{b}=0\\\ J_{c}\propto q_{z}\cos
4\varphi\end{split}$ (19)

Whereas when we include the magnetic ordering, we find the bulk current:

$J_{a}\propto q_{z}\cos 4\varphi$ (20) $J_{b}\propto q_{z}(\sin
4\varphi+C_{1}\sin 2\varphi)$ (21) $J_{c}\propto q_{z}\cos 4\varphi$ (22)

The allowed surface current at normal incidence when ignoring magnetic
ordering are given by:

$J_{a}\propto\cos 4\varphi+C_{2}q_{z}\cos 4\varphi$ (23) $J_{b}\propto\sin
4\varphi+C_{3}q_{z}\sin 4\varphi$ (24) $J_{c}\propto\cos
4\varphi+C_{4}q_{z}\cos 4\varphi$ (25)

When taking into account the magnetic ordering of the surface, all terms are
symmetry-allowed. Here $C_{i}$ are constants that are independent of $\varphi$
and $\boldsymbol{q}$. We note that the non-magnetic analysis cannot explain
the appearance of a circular effect at normal incidence, even when taking the
surface into account. This is discussed further in the next section.

####  F.3.3 Expressions away from normal incidence (configuration 2 & 3)

As can be seen from the full expressions in Sec. G, away from normal
incidence, all cases (magnetic/non-magnetic and surface/bulk) allow for a
linear and a circular photocurrent, though all bulk contributions still arise
exclusively from the PDE due to bulk inversion symmetry. Because the measured
signal does not depend on the magnetic field (see Fig. 3 and discussion in the
main text as well as Fig. 8), we discuss only the non-magnetic symmetry
settings in what follows.

As the detector is in a fixed position, the detected signal will display a
purely geometric variation under rotation away from normal incidence, as
various faces of the crystal are exposed. The generated photocurrents will
induce a dipole of strength $\boldsymbol{d}$, and the associated emitted THz
field can be written in terms of unit vectors as:

$\boldsymbol{S}(\boldsymbol{r})\propto\boldsymbol{E}^{\mathrm{THz}}(\boldsymbol{r)}\propto\hat{\boldsymbol{r}}\times\hat{\boldsymbol{d}}\times\hat{\boldsymbol{r}}$
(26)

Note that this ignores reflection and refraction effects, which will play a
role away from normal incidence. As the sample is rotated by the same angles
in both $x$ and $y$, however, these effects should be irrelevant when
comparing configuration 2 and configuration 3. The THz field is measured along
the $z$-axis. The measured signal in configuration 1 is then:

$S_{x}\propto J_{a}$ (27) $S_{y}\propto J_{b}$ (28)

Whereas in configuration 2 it is :

$S_{x}\propto J_{a}$ (29) $S_{y}\propto
J_{b}\cos\theta_{x}+J_{c}\sin\theta_{x}$ (30)

An in configuration 3:

$S_{x}\propto J_{a}\cos\theta_{y}+J_{c}\sin\theta_{y}$ (31) $S_{y}\propto
J_{b}$ (32)

Where the induced currents $\boldsymbol{J}$ also depend on the angles
$\theta_{x,y}$. This dependence can be written out explicitly, using the
expressions in Sec. F.3.1 and Sec. G. This was shown for configuration 1 in
all symmetry settings in the previous section. For the non-magnetic bulk
signal in configuration 2, we find:

$S_{x}\propto\sin\theta_{x}\cos\theta_{x}(C_{1}\sin 4\varphi+C_{2}\sin
2\varphi)$ (33)
$S_{y}\propto(C_{3}\cos\theta_{x}\sin\theta_{x}+C_{4}\cos^{3}\theta_{x}\sin\theta_{x}+C_{5}\cos\theta_{x}\sin^{3}\theta_{x})\cos
4\varphi+(C_{6}\cos^{3}\theta_{x}\sin\theta_{x}+C_{7}\cos\theta_{x}\sin^{3}\theta_{x})\sin
4\varphi$ (34)

And in configuration 3:

$S_{x}\propto(D_{1}\cos\theta_{y}\sin\theta_{y}+D_{2}\cos^{3}\theta_{y}\sin\theta_{y}+D_{3}\cos\theta_{y}\sin^{3}\theta_{y})\cos
4\varphi+(D_{4}\cos^{3}\theta_{y}\sin\theta_{y}+D_{5}\cos\theta_{y}\sin^{3}\theta_{y})\sin
4\varphi$ (35) $S_{y}\propto\sin\theta_{y}\cos\theta_{y}(D_{6}\sin
4\varphi+D_{7}\sin 2\varphi)$ (36)

Turning to the non-magnetic surface effects, the lowest order surface effects
arises from the photogalvanic effect. For the surface photogalvanic effect,
the angular dependence in configuration 2 is given by:

$S_{x}\propto(C_{1}^{s}+C_{2}^{s}\cos^{2}\theta_{x})\cos
4\varphi+C_{3}^{s}\sin\theta_{x}\sin 4\varphi+C_{4}^{s}\sin\theta_{x}\sin
2\varphi$ (37)
$S_{y}\propto(C_{5}^{s}\sin\theta_{x}+C_{6}^{s}\cos^{2}\theta_{x}\sin\theta_{x}+C_{7}^{s}\sin^{3}\theta_{x})\cos
4\varphi+(C_{8}^{s}\cos^{2}\theta_{x}+C_{9}^{s}\cos^{2}\theta_{x}\sin\theta_{x})\sin
4\varphi$ (38)

And in configuration 3:

$S_{x}\propto(D_{1}^{s}\cos\theta_{y}+D_{2}^{s}\sin\theta_{y}+D_{3}^{s}\cos^{3}\theta_{y}+D_{4}^{s}\sin^{3}\theta_{y}+D_{5}^{s}\cos^{2}\theta_{y}\sin\theta_{y})\cos
4\varphi+D_{6}^{s}\cos^{2}\theta_{y}\sin\theta_{y}\sin 4\varphi$ (39)
$S_{y}\propto(D_{7}^{s}\sin\theta_{y}+D_{8}^{s}\cos\theta_{y})\cos
4\varphi+D_{9}^{s}\sin\theta_{y}\sin 2\varphi$ (40)

In the above expressions, $C_{i}$ and $D_{i}$ ($C_{i}^{s}$ and $D_{i}^{s}$)
are constant independent of $\theta_{x,y}$ and $\varphi$ associated with the
generated bulk (surface) currents.

Note that there is no circular current being generated perpendicular to the
rotation axis, independent of where the current is generated, which explains
the large suppression in the circular current seen in Fig. 2 of the main text.
We attribute the appearance of a very small circular effect perpendicular to
the rotation axis to an imperfect sample alignment. All terms generated in the
bulk are odd in the rotation angle, whereas some of the linear terms arising
from the surface photogalvanic effect are even in rotation angle. By contrast,
all circular terms are odd in rotation angle for both the surface and bulk
effect. As shown in Fig. 2 in the main text and Fig. 7, we see that the odd
response dominates, even though the linear and circular components have
comparable magnitudes. This suggests that the signal originates predominantly
from the bulk photon drag effect.

At normal incidence, we would expect to not detect any signal from the bulk
photon drag effect. As shown in Fig. 2(c) of the main text, the detected
signal at normal incidence is very weak compared to the signal at larger
rotation angles. This suggests that the measured signal arises from an
imperfect sample alignment, resulting in a small non-zero rotation angle. This
also explains the observed signal in Fig. 6.

####  F.3.4 Effect of in-plane rotation, $\theta_{c}\neq 0$

Rotating by $90\degree$ in-plane (along the $c$-axis) keeps all optical
parameters the same, so that all changes in signal arise as a result of the
photocurrent generation mechanisms in the material. As shown in Fig. 3(b) in
the main text, an in-plane rotation by $90\degree$ has negligible impact on
the measured signal. Writing $R_{xa}$ for the rotation matrix relating the lab
$(x,y,z)$ coordinates to the material $(a,b,c)$ coordinates, the measured
signals are given by:

$\begin{split}S_{x}\propto\sum_{i\in(a,b,c)}R^{-1}_{xi}J_{i}(R\mathbb{E},R\boldsymbol{q})\\\
S_{y}\propto\sum_{i\in(a,b,c)}R^{-1}_{yi}J_{i}(R\mathbb{E},R\boldsymbol{q})\end{split}$
(41)

Inserting this into the equations found in Sec. G, we find that the non-
magnetic bulk PDE is invariant under this transformation (in both the linear
and circular component), whereas none of the other contributions are invariant
under this transformation. This adds further credence to the result that our
signal arises predominantly from a non-magnetic bulk PDE.

##  Appendix G Full expression for non-linear optical tensors

Here we provide full expressions for the symmetry-allowed currents, written in
the material $(a,b,c)$ Cartesian coordinate system. These are found using the
MTENSOR functionality Gallego  _et al._ (2019) on the BCS Aroyo  _et al._
(2006).

###  G.1 Non-magnetic bulk

####  G.1.1 Linear photon drag effect

$\begin{split}J_{a}=q_{a}(\chi^{(3),s}_{aaaa}|E_{a}|^{2}+\chi^{(3),s}_{aabb}|E_{b}|^{2}+\chi^{(3),s}_{aacc}|E_{c}|^{2})+\\\
\frac{q_{b}}{2}\big{(}\chi^{(3),s}_{aaaa}-\chi^{(3),s}_{aabb}\big{)}[E_{a},E_{b}]+q_{c}\chi^{(3),s}_{acac}[E_{a},E_{c}]\end{split}$
(42)
$\begin{split}J_{b}=q_{b}(\chi^{(3),s}_{aabb}|E_{a}|^{2}+\chi^{(3),s}_{aaaa}|E_{b}|^{2}+\chi^{(3),s}_{aacc}|E_{c}|^{2})+\\\
\frac{q_{a}}{2}\big{(}\chi^{(3),s}_{aaaa}-\chi^{(3),s}_{aabb}\big{)}[E_{a},E_{b}]+q_{c}\chi^{(3),s}_{acac}[E_{b},E_{c}]\end{split}$
(43)
$\begin{split}J_{c}=q_{c}(\chi^{(3),s}_{ccaa}|E_{a}|^{2}+\chi^{(3),s}_{ccaa}|E_{b}|^{2}+\chi^{(3),s}_{cccc}|E_{c}|^{2})+\\\
q_{a}\chi^{(3),s}_{caac}[E_{a},E_{c}]+q_{b}\chi^{(3),s}_{caac}[E_{b},E_{c}]\end{split}$
(44)

For a total of $7$ independent components.

####  G.1.2 Circular photon drag effect

$J_{a}=q_{b}\chi^{(3),a}_{abc}P_{\mathrm{circ}}\hat{e}_{c}-q_{c}\chi^{(3),a}_{acb}P_{\mathrm{circ}}\hat{e}_{b}$
(45)
$J_{b}=q_{c}\chi^{(3),a}_{acb}P_{\mathrm{circ}}\hat{e}_{a}-q_{a}\chi^{(3),a}_{abc}P_{\mathrm{circ}}\hat{e}_{c}$
(46)
$J_{c}=q_{b}\chi^{(3),a}_{cab}P_{\mathrm{circ}}\hat{e}_{a}-q_{a}\chi^{(3),a}_{cab}P_{\mathrm{circ}}\hat{e}_{b}$
(47)

For a total of 3 independent components

###  G.2 Magnetic bulk

####  G.2.1 Linear photon drag effect

$\begin{split}J_{a}=q_{a}(\chi^{(3),s}_{aaaa}|E_{a}|^{2}+\chi^{(3),s}_{aabb}|E_{b}|^{2}+\chi^{(3),s}_{aacc}|E_{c}|^{2})+\\\
q_{a}\chi^{(3),s}_{aaac}[E_{a},E_{c}]+q_{b}\big{(}\chi^{(3),s}_{abbc}[E_{b},E_{c}]+\chi^{(3),s}_{abab}[E_{a},E_{b}]\big{)}+\\\
q_{c}(\chi^{(3),s}_{acaa}|E_{a}|^{2}+\chi^{(3),s}_{acbb}|E_{b}|^{2}+\chi^{(3),s}_{accc}|E_{c}|^{2}+\chi^{(3),s}_{acac}[E_{a},E_{c}])\end{split}$
(48)

$\begin{split}J_{b}=q_{b}(\chi^{(3),s}_{bbaa}|E_{a}|^{2}+\chi^{(3),s}_{bbbb}|E_{b}|^{2}+\chi^{(3),s}_{bbcc}|E_{c}|^{2})+\\\
q_{b}\chi^{(3),s}_{bbac}[E_{a},E_{c}]+q_{a}\big{(}\chi^{(3),s}_{babc}[E_{b},E_{c}]+\chi^{(3),s}_{baab}[E_{a},E_{b}]\big{)}+\\\
q_{c}(\chi^{(3),s}_{bcbc}[E_{b},E_{c}]+\chi^{(3),s}_{bcab}[E_{a},E_{b}])\end{split}$
(49)

$\begin{split}J_{c}=q_{a}(\chi^{(3),s}_{caaa}|E_{a}|^{2}+\chi^{(3),s}_{cabb}|E_{b}|^{2}+\chi^{(3),s}_{cacc}|E_{c}|^{2})+\\\
q_{a}\chi^{(3),s}_{caac}[E_{a},E_{c}]+q_{b}\big{(}\chi^{(3),s}_{cbbc}[E_{b},E_{c}]+\chi^{(3),s}_{cbab}[E_{a},E_{b}]\big{)}+\\\
q_{c}(\chi^{(3),s}_{ccaa}|E_{a}|^{2}+\chi^{(3),s}_{ccbb}|E_{b}|^{2}+\chi^{(3),s}_{cccc}|E_{c}|^{2}+\chi^{(3),s}_{ccac}[E_{a},E_{c}])\end{split}$
(50)

For a total of 28 independent components.

####  G.2.2 Circular photon drag effect

$\begin{split}J_{a}=-q_{a}\chi^{(3),a}_{aab}P_{\mathrm{circ}}\hat{e}_{b}-q_{c}\chi^{(3),a}_{acb}P_{\mathrm{circ}}\hat{e}_{b}\\\
+q_{b}(\chi^{(3),a}_{abc}P_{\mathrm{circ}}\hat{e}_{c}+\chi^{(3),a}_{aba}P_{\mathrm{circ}}\hat{e}_{a})\end{split}$
(51)

$\begin{split}J_{b}=q_{a}(\chi^{(3),a}_{bac}P_{\mathrm{circ}}\hat{e}_{c}+\chi^{(3),a}_{baa}P_{\mathrm{circ}}\hat{e}_{a})\\\
+q_{c}(\chi^{(3),a}_{bcc}P_{\mathrm{circ}}\hat{e}_{c}+\chi^{(3),a}_{bca}P_{\mathrm{circ}}\hat{e}_{a})\\\
-q_{b}\chi^{(3),a}_{bbb}P_{\mathrm{circ}}\hat{e}_{b}\end{split}$ (52)

$\begin{split}J_{c}=-q_{a}\chi^{(3),a}_{cab}P_{\mathrm{circ}}\hat{e}_{b}-q_{c}\chi^{(3),a}_{ccb}P_{\mathrm{circ}}\hat{e}_{b}\\\
+q_{b}(\chi^{(3),a}_{cbc}P_{\mathrm{circ}}\hat{e}_{c}+\chi^{(3),a}_{cba}P_{\mathrm{circ}}\hat{e}_{a})\end{split}$
(53)

For a total of 13 independent coefficients.

###  G.3 Non-magnetic surface

####  G.3.1 Linear photogalvanic effect

$J_{a}=\chi^{(2),s}_{aaa}(|E_{a}|^{2}-|E_{b}|^{2})+\chi^{(2),s}_{aac}[E_{a},E_{c}]$
(54) $J_{b}=\chi^{(2),s}_{aac}[E_{b},E_{c}]-\chi^{(2),s}_{aaa}[E_{a},E_{b}]$
(55)
$J_{c}=\chi^{(2),s}_{caa}(|E_{a}|^{2}+|E_{b}|^{2})+\chi^{(2),s}_{ccc}|E_{c}|^{2}$
(56)

For a total of 4 independent coefficients.

####  G.3.2 Circular photogalvanic effect

$J_{a}=-\chi^{(2),a}_{ab}P_{\mathrm{circ}}\hat{e}_{b}$ (57)
$J_{b}=\chi^{(2),a}_{ab}P_{\mathrm{circ}}\hat{e}_{a}$ (58) $J_{c}=0$ (59)

With a single independent parameter.

####  G.3.3 Linear photon drag effect

$\begin{split}J_{a}=q_{a}(\chi^{(3),s}_{aaaa}|E_{a}|^{2}+\chi^{(3),s}_{aabb}|E_{b}|^{2}+\chi^{(3),s}_{aacc}|E_{c}|^{2})\\\
+q_{a}\chi^{(3),s}_{aaac}[E_{a},E_{c}]+\frac{q_{b}}{2}\big{(}\chi^{(3),s}_{aaaa}-\chi^{(3),s}_{aabb}\big{)}[E_{a},E_{b}]\\\
-q_{b}\chi^{(3),s}_{aaac}[E_{b},E_{c}]+q_{c}\chi^{(3),s}_{acaa}\big{(}|E_{a}|^{2}-|E_{b}|^{2}\big{)}+q_{c}\chi^{(3),s}_{acac}[E_{a},E_{c}]\end{split}$
(60)

$\begin{split}J_{b}=q_{b}(\chi^{(3),s}_{aabb}|E_{a}|^{2}+\chi^{(3),s}_{aaaa}|E_{b}|^{2}+\chi^{(3),s}_{aacc}|E_{c}|^{2})\\\
-q_{b}\chi^{(3),s}_{aaac}[E_{a},E_{c}]-q_{a}\chi^{(3),s}_{aaac}[E_{b},E_{c}]\\\
+\frac{q_{a}}{2}\big{(}\chi^{(3),s}_{aaaa}-\chi^{(3),s}_{aabb})[E_{a},E_{b}]+q_{c}(\chi^{(3),s}_{acac}[E_{b},E_{c}]-\chi^{(3),s}_{acaa}[E_{a},E_{b}])\end{split}$
(61)

$\begin{split}J_{c}=q_{c}(\chi^{(3),s}_{ccaa}(|E_{a}|^{2}+|E_{b}|^{2})+\chi^{(3),s}_{cccc}|E_{c}|^{2})\\\
+q_{a}\big{(}\chi^{(3),s}_{caaa}(|E_{a}|^{2}-|E_{b}|^{2})+\chi^{(3),s}_{caac}[E_{a},E_{c}]\big{)}\\\
+q_{b}(\chi^{(3),s}_{caac}[E_{b},E_{c}]-\chi^{(3),s}_{caaa}[E_{a},E_{b}])\end{split}$
(62)

For a total of 10 independent components.

####  G.3.4 Circular photon drag effect

$\begin{split}J_{a}=q_{b}P_{\mathrm{circ}}(\chi^{(3),a}_{abc}\hat{e}_{c}-\chi^{(3),a}_{aab}\hat{e}_{a})\\\
-\chi^{(3),a}_{aab}q_{a}P_{\mathrm{circ}}\hat{e}_{b}-\chi^{(3),a}_{acb}q_{c}P_{\mathrm{circ}}\hat{e}_{b}\end{split}$
(63)

$\begin{split}J_{b}=-q_{a}P_{\mathrm{circ}}(\chi^{(3),a}_{abc}\hat{e}_{c}+\chi^{(3),a}_{aab}\hat{e}_{a})\\\
+q_{b}P_{\mathrm{circ}}\chi^{(3),a}_{aab}\hat{e}_{b}+q_{c}P_{\mathrm{circ}}\chi^{(3),a}_{acb}\hat{e}_{a}\end{split}$
(64)

$J_{c}=\chi^{(3),a}_{cab}P_{\mathrm{circ}}(q_{b}\hat{e}_{a}-q_{a}\hat{e}_{b})$
(65)

For a total of 4 independent components.

###  G.4 Magnetic surface

As the unitary part of the symmetry group is $1$ on the surface when
considering magnetic symmetries, all terms are allowed.

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