# Light-induced Nonlinear Spin Hall Current in Single-layer WTe2

Pankaj Bhalla  Department of Physics, School of Engineering and Sciences, SRM
University AP, Amaravati, 522240, India  Habib Rostami  Department of Physics,
University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom  Nordita, KTH
Royal Institute of Technology and Stockholm University, Hannes Alfvéns väg 12,
10691 Stockholm, Sweden

###### Abstract

In this theoretical investigation, we analyze light-induced nonlinear spin
Hall currents in a gated single-layer 1T′-WTe2, flowing transversely to the
incident laser polarization direction. Our study encompasses the exploration
of the second and third-order rectified spin Hall currents using an effective
low-energy Hamiltonian and employing the Kubo’s formalism. We extend our
analysis to a wide frequency range spanning both transparent and absorbing
regimes, investigating the influence of light frequency below and above the
optical band gap. Additionally, we investigate the influence of an out-of-
plane gate potential on the system, disrupting inversion symmetry and
effectively manipulating both the strength and sign of nonlinear spin Hall
responses. We predict a pronounced third-order spin Hall current relative to
its second-order counterpart. The predicted nonlinear spin currents show
strong anisotropic dependence on the laser polarization angle. The outcomes of
our study contribute to a generalized framework for nonlinear response theory
within the spin channel will impact the development of emerging field of opto-
spintronic.

##  I Introduction

Electronic transport in two-dimensional (2D) quantum materials has emerged as
a central topic for new physics and novel technologies in the spintronic
industry and future energy-saving devices Keimer and Moore (2017); Giustino
_et al._ (2020). The second-order nonlinear photocurrent in novel quantum
materials is attracting interest from both applied and fundamental point of
view Sodemann and Fu (2015a); Morimoto and Nagaosa (2016); de Juan  _et al._
(2017); Rostami and Polini (2018); Wang and Qian (2019a); Mueller and Malic
(2018); Soavi  _et al._ (2018); Principi  _et al._ (2019); Bhalla  _et al._
(2020); Wang  _et al._ (2021); Klimmer  _et al._ (2021); Bhalla (2021); Sinha
_et al._ (2022); Bhalla and Rostami (2022); Chakraborty  _et al._ (2022);
Bhalla  _et al._ (2022a); Lahiri  _et al._ (2022); Bhalla  _et al._ (2021);
Zeng  _et al._ (2022); Bhalla  _et al._ (2023, 2022b); Ikeda  _et al._ (2023).
In particular, one of the main aims is to realize topological protection of
nonlinear current in topological quantum materials such as Dirac and Weyl
systems de Juan  _et al._ (2017); Rostami and Polini (2018); Shvetsov  _et
al._ (2019); Tiwari  _et al._ (2021). Experimental measurements in 2D WTe2
show that the nonlinear Hall effect arises due to the momentum derivative of
the Berry curvature, so-called the Berry curvature dipole Sodemann and Fu
(2015a) that can be tuned with the out-of-plane potential bias via the top and
bottom gate voltages Ma  _et al._ (2019); Kang  _et al._ (2019). The counter
propagation of electrons with opposite spins Sun and Xie (2005) conceives the
celebrated spin Hall effect on applying a bias voltage between two contacts
Kane and Mele (2005); Bernevig  _et al._ (2006); Bernevig and Zhang (2006); Qi
_et al._ (2006). The phenomenon arises due to the strong spin-orbit coupling,
a key element in their electronic band structure, and generating a spin
current Bernevig  _et al._ (2006); Xu and Moore (2006). A generalization to
the nonlinear regime is a rapidly growing field of research. For instance, we
recall theoretical and experimental studies on photoinduced second-order spin
current in quantum materials with spin-orbit coupling Ishizuka and Sato
(2022); Sakimura  _et al._ (2014); Nair  _et al._ (2020); Hamamoto  _et al._
(2017a); Bhat  _et al._ (2005); Lihm and Park (2022); Wang and Qian (2019a).

Figure 1: Schematic setups for the light-induced nonlinear rectified spin
current in a 2D WTe2 device. The left panel refers to the second-order spin
current that arises in response to the single-color light beam. The right
panel stems from the interference of two-color light beams and represents the
third-order rectified spin current. Here ${\bm{E}}_{1}(t)$ and
${\bm{E}}_{2}(t)$ refer to the electric fields, and $U$ corresponds to the
out-of-plane gate potential.

Measurement of nonlinear Hall effect Du  _et al._ (2021a) using infrared pulse
excitation, and high-temperature intrinsic spin-Hall effect Garcia  _et al._
(2020a) in two-dimensional WTe2 motivates to look for the nonlinear
generalization of the spin Hall current in this exotic quantum material with a
rich ground state phase diagram. The generation of the rectified spin current
has been widely studied in traditional semiconductors and quantum well
structures, where a linear wave vector term in the Bloch Hamiltonian stemming
from the spin-orbit interaction leads to the spin splitting in the band
structure and plays the significant role Zhou and Shen (2007); Ivchenko and
Tarasenko (2008); Tarasenko and Ivchenko (2005); Zhao  _et al._ (2005); Wang
_et al._ (2010); Hamamoto  _et al._ (2017b). The nonlinear spin current has
yet to be discussed or explored for monolayer WTe2 having different
crystalline symmetry, which is the main aim of the present work. The time-
reversal 1T′-WTe2 structure yields second-order current when the inversion
symmetry is broken Bhalla and Rostami (2022); Du  _et al._ (2018) by an out-
of-plane potential bias that allows tuning the gap associated with each valley
and spin Xu  _et al._ (2018). This results in a phase transition driven by the
interplay of the spin-orbit coupling and the out-of-plane bias Qian  _et al._
(2014).

The second-order rectified current $j^{(2)}_{dc}\sim
E(\omega)E^{\ast}(\omega)$ can be generated in non-centrosymmetric materials.
There is, however, a growing interest in the third-order rectified current
that can be achieved in the presence of an extra direct electric field
$j^{(3)}_{dc}\sim E_{dc}E(\omega)E^{\ast}(\omega)$ which can induce rectified
current in centrosymmetric systems Fregoso (2019); Fregoso  _et al._ (2018);
Ahn  _et al._ (2022). Another third-order rectification process is obtained in
response to a two-color driving field. The two-color field is formally written
as

$\displaystyle{\bm{E}}(t)$
$\displaystyle={\bm{E}}_{1}e^{i(\omega_{1}t+\phi_{1})}+{\bm{E}}_{2}e^{i(\omega_{2}t+\phi_{2})}+c.c.~{},$
(1)

where ${\bm{E}}_{1}$, and ${\bm{E}}_{2}$ are the electric fields associated
with frequency $\omega_{1}$ and $\omega_{2}$, and $\phi_{1}$ and $\phi_{2}$
phases respectively. Two-color optical rectification is a process that
involves the interference of single frequency beam with the second-harmonic
laser beam $j^{(3)}_{dc}\sim
E_{1}(\omega_{1})E_{1}(\omega_{1})E^{\ast}_{2}(\omega_{2})$ with
$\omega_{2}=2\omega_{1}$ which was first introduced by Manykin and Alfanasev
Manykin and Afanasev (1967). The process has been extensively exploited in
semiconductors and two-dimensional materials by illuminating a combination of
monochromatic beams of frequency $\omega$ and $2\omega$, leads to one-photon
and two-photon absorption transitions Bhat and Sipe (2000); Rioux  _et al._
(2011); Khurgin (1995); Totero Gongora  _et al._ (2020); Cheng  _et al._
(2014, 2015); Atanasov  _et al._ (1996); Haché  _et al._ (1997).

In addition to the interest in the Fermi-surface contribution to the low-
frequency (e.g. THz range) photocurrent generation Principi  _et al._ (2019);
Sodemann and Fu (2015b); Du  _et al._ (2021b); Gao  _et al._ (2021); Zhang and
Fu (2021), recent studies have also predicted the presence of photocurrent in
the transparent frequency range based on various mechanisms Kaplan  _et al._
(2020); Onishi  _et al._ (2022); Golub and Glazov (2022); Shi  _et al._
(2023); Matsyshyn  _et al._ (2023). This process is highly counter-intuitive
since the typical photocurrent generation mechanisms in semiconductors, namely
second-order shift and injection photocurrents, require a real optical
transition of electrons from the valence to the conduction band. Therefore,
the linear optical absorption is accompanied with the conventional
photocurrent generation in semiconductors. The rationale behind the generation
of photocurrent relies on various mechanisms, including time-reversal symmetry
breaking Kaplan  _et al._ (2020), Raman scattering of light Golub and Glazov
(2022), residual intraband absorption Shi  _et al._ (2023) with a small but
finite carrier relaxation rate Onishi  _et al._ (2022). Using the Floquet
model, it is shown that the occupation number becomes a “staircase” form,
allowing for a finite rectified electric current within the optical gap
Matsyshyn  _et al._ (2023). It is worth highlighting that the second-order
current cannot generate Joule heating, as the time-averaged dissipated power,
represented as
$W^{(2)}=-\langle{\bm{j}}^{(2)}(t)\cdot{\bm{E}}(t)\rangle_{\text{time-
average}},$ is always zero for a periodic electric field Bassani and Scandolo
(1991). This is because, for both the second-harmonic and rectification
currents, the transient power density will oscillate in time leading to a
vanishing time-averaged value.

In this article, we theoretically investigate the nonlinear rectified spin
current in monolayer WTe2–see Fig. 1, based on second-order and third-order
response theories. We utilize a many-body diagrammatic perturbation technique.
We discuss that the transverse nonlinear charge current vanishes due to the
symmetry conditions, but a nonlinear spin current remains finite in both
second and third-order rectification mechanisms. The current injection
mechanism dominantly drives the second-order spin current, which is finite in
the interband frequency regime $\hbar\omega>\Delta_{op}$ where $\Delta_{op}$
is the optical gap based on the Pauli exclusion principle at zero electronic
temperature. However, the two-color spin current mechanism is more complex
with non-trivial features. Intriguingly, we obtain two-color spin response
that can be finite in both $\hbar\omega<\Delta_{op}$ and
$\hbar\omega>\Delta_{op}$ regimes. In simpler terms, the effect below the gap
does not stem from the transport of the photoexcited carrier population;
instead, it originates from virtual excitations in transparent regime, akin to
the references mentioned Kaplan  _et al._ (2020); Onishi  _et al._ (2022);
Golub and Glazov (2022); Shi  _et al._ (2023); Matsyshyn  _et al._ (2023). In
the remainder of this paper, we present a quantitative analysis of second and
third-order spin susceptibilities and discuss their dependence on laser
frequency, external displacement field (gate potential), chemical potential,
and laser polarization angle.

##  II Theory and Method

Figure 2: Panel (a) and (b) refers to the spin-up energy dispersion of
1T′-WTe2 with the back gate potential $U=0$ and $U=2\delta_{{\rm SOC}}$. Here,
$\Delta_{\pm}=2|U\pm\delta_{\rm SOC}|$ is the energy gap around two valley
points $Q$ and $Q^{\prime}$ with $2\delta_{\rm SOC}$ being the spin-orbit gap.
The horizontal dotted black line refers to the chemical potential $\mu=0$.
Panel (c) and (d) illustrate the spin-up Berry curvature variation for $U=0$
and $U=\delta_{{\rm SOC}}$, respectively.

We utilize an effective continuum ${\bm{k}}\cdot{\bm{p}}$ model Hamiltonian
for the 1T′ phase of the monolayer WTe2 which incorporates p-orbital of Te and
d-orbital of W. This Hamiltonian can nicely describe the low-energy bands,
which has the following representation Xu  _et al._ (2018); Garcia  _et al._
(2020b):

$\displaystyle\mathcal{\hat{H}}=$
$\displaystyle\\{Ak^{2}{\hat{\sigma}}_{0}+(\delta+Bk^{2})\hat{\sigma}_{z}+v_{y}k_{y}\hat{\sigma}_{y}+U\hat{\sigma}_{x}\\}\otimes\hat{s}_{0}+v_{x}k_{x}\hat{\sigma}_{x}\otimes\hat{s}_{y},$
(2)

where $\hat{\sigma}_{i=x,y,z}$ and $\hat{s}_{i=x,y,z}$ are Pauli matrices in
the orbital and spin basis, respectively. The $\hat{\sigma}_{0}$
($\hat{s}_{0}$) is the identity matrix in the orbital (spin) basis. The
parameter $2\delta$ stands for the gap at $\Gamma$-point, and $v_{y}$
characterizes the anisotropy in the momentum space. The parameters
$2A=1/m_{p}-1/m_{d}$ and $2B=1/m_{p}+1/m_{d}$, where $m_{p}$ and $m_{d}$
define the effective masses of p-orbital of Te and d-orbital of W
respectively. The parameter $U$ represents the displacement field, the
coupling between the out-of-plane electric field and the orbitals which breaks
the inversion symmetry of the system, $v_{x}$ is the spin-orbit coupling
strength, and the wave vector ${\bm{k}}=(k_{x},k_{y})$ having $k=|{\bm{k}}|$.
The parameter values are considered from the ab initio band structure
calculations Qian  _et al._ (2014) and fitted with the experimental
predictions to obtain the spin-orbit coupling gap
$\delta_{{\rm{SOC}}}=v_{x}Q=45$ meV at $Q=\sqrt{|\delta|/2B}$ Dirac points in
the momentum space Tang  _et al._ (2017).

For the above model Hamiltonian considered, the Berry curvature ($U=0$ case)
reads

$\displaystyle\Omega^{c}_{{\bm{k}},s}=-\Omega^{v}_{{\bm{k}},s}$
$\displaystyle=-\frac{sv_{x}v_{y}}{N}\bigg{(}h_{z}(v_{x}^{2}k_{x}^{2}-v_{y}^{2}k_{y}^{2})-|h_{12}|^{2}2B(k_{x}^{2}-k_{y}^{2})\bigg{)},$
(3)

where $N=|h_{12}|\sqrt{[h_{z}]^{2}+[h_{12}]^{2}}$,
$h_{12}=sv_{x}k_{x}-iv_{y}k_{y}$ and $h_{z}=\delta+Bk^{2}$. As seen the Berry
curvature is proportional to the spin index, thus flips sign for up and down
spin indices. The energy dispersion and the Berry curvature corresponding to
the spin-up component of the Hamiltonian given in Eq. (2) are depicted in Fig.
2.

Figure 3: Feynman diagrams for the second-order response. (a) correspond to
the paramagnetic contributions, (b)-(d) to the diamagnetic contributions. Here
solid lines indicate the fermionic propagators, dashed lines refer to external
photons, and solid circles denote current vertices.

To compute the nonlinear rectified spin Hall response of 1T′-WTe2 to an
external homogeneous vector potential ${\bm{A}}(t)$, we consider the
corresponding driving electric field ${\bm{E}}(t)=-\partial_{t}{\bm{A}}(t)$.
In a spin-diagonal basis, the nonlinear spin response is given as the
difference of susceptibility for two spin polarisations:

$\displaystyle\chi^{(n),{\rm
spin}}=\chi^{(n)}(\uparrow)-\chi^{(n)}(\downarrow).$ (4)

The second-order rectified spin current thus follows

$\displaystyle j_{a}^{\rm PG}$
$\displaystyle=\sum_{bc}\frac{\chi_{abc}^{(2),\rm{spin}}(\omega,-\omega)}{(i\omega)(-i\omega)}E_{b}(\omega)E_{c}^{*}(\omega).$
(5)

Similarly, the two-color third-order rectified spin current reads

$\displaystyle j_{a}^{\rm 2c-PG}$
$\displaystyle=\sum_{bcd}\frac{\chi_{abcd}^{(3),\rm{spin}}(\omega,\omega,-2\omega)}{(i\omega)(i\omega)(-i2\omega)}E_{b}(\omega)E_{c}(\omega)E_{d}^{*}(2\omega),$
(6)

where $\chi_{abc}^{(2),{\rm spin}}(\omega_{1},\omega_{2})$ and
$\chi_{abcd}^{(3),{\rm spin}}(\omega_{1},\omega_{2},\omega_{3})$ respectively
stand for the second- and third-order spin-resolved susceptibility tensor
elements in the two-dimensional $x-y$ coordinate. In addition,
${\bm{E}}(\omega)$ and ${\bm{E}}(2\omega)$ stand for the field elements at two
colors with frequency $\omega$ and $2\omega$. We employ the diagrammatic
perturbative approach, and the response functions are represented by the
Feynman diagrams in Fig. 3 and 4.

The second-order spin-resolved current given by the correlation of three
paramagnetic current operators (or one-photon current coupling)
$\hat{j}_{a}=(-e/\hbar)\partial_{k_{a}}\hat{\cal H}$ as
$\chi^{(2),P}_{abc}\sim\langle\hat{j}_{a}\hat{j}_{b}\hat{j}_{c}\rangle$ and
can be diagrammatically depicted as in Fig. 3(a) which formally reads

$\displaystyle\chi_{abc}^{(2),P}(\omega_{1},\omega_{2},s)$
$\displaystyle=\sum_{\mathcal{P}}\sum_{\\{\lambda_{i}\\},\bm{k}}\frac{{j}_{a,{\bm{k}},s}^{\lambda_{1}\lambda_{2}}~{}{j}_{b,{\bm{k}},s}^{\lambda_{2}\lambda_{3}}~{}{j}_{c,{\bm{k}},s}^{\lambda_{3}\lambda_{1}}}{\hbar\omega_{\Sigma}+\varepsilon_{\bm{k},s}^{\lambda_{2}\lambda_{1}}}\left\\{\frac{f^{\lambda_{2}\lambda_{3}}_{{\bm{k}},s}}{\hbar\omega_{1}+\varepsilon_{{\bm{k}},s}^{\lambda_{2}\lambda_{3}}}-\frac{f^{\lambda_{3}\lambda_{1}}_{{\bm{k}},s}}{\hbar\omega_{2}+\varepsilon_{{\bm{k}},s}^{\lambda_{3}\lambda_{1}}}\right\\},$
(7)

where $\omega_{\Sigma}=\omega_{1}+\omega_{2}$,
$\varepsilon_{\bm{k},s}^{\lambda_{i}\lambda_{j}}=\varepsilon_{\bm{k},s}^{\lambda_{i}}-\varepsilon_{\bm{k},s}^{\lambda_{j}}$
is the energy difference between different bands with given spin index $s=\pm
1$ ($+1$ for up-spin and $-1$ for down-spin) and
$f^{\lambda_{i}\lambda_{j}}_{{\bm{k}},s}=f(\varepsilon_{\bm{k},s}^{\lambda_{i}})-f(\varepsilon_{\bm{k},s}^{\lambda_{j}})$
refers to the difference between the Fermi function corresponding to distinct
bands. Here $f(\varepsilon)=[1+e^{\beta(\varepsilon-\mu)}]^{-1}$ is the Fermi-
Dirac distribution function, $\mu$ is the chemical potential, and
$\beta=1/k_{B}T$ having $k_{B}$ as the Boltzmann constant, $T$ as an electron
temperature. The one-photon coupling vertex is
${j}_{a,{\bm{k}},s}^{\lambda_{i}\lambda_{j}}=\langle
u_{{\bm{k}},s}^{\lambda_{i}}|\hat{j}_{a}|u_{{\bm{k}},s}^{\lambda_{j}}\rangle$
with $|u_{{\bm{k}},s}^{\lambda_{i}}\rangle$ being an eigen vector of the
Hamiltonian. Note that $\sum_{\mathcal{P}}$ stands for the intrinsic
permutation symmetry $(b,\omega_{1})\Longleftrightarrow(c,\omega_{2})$ and
$\hbar\omega_{i}\rightarrow\hbar\omega_{i}+i\eta$ with the parameter
$\eta\rightarrow 0^{+}$. The diamagnetic contribution to the second-order
response (Fig. 3(b)-(d)) is given by

$\displaystyle\chi_{abc}^{(2),D}(\omega_{1},\omega_{2},s)$
$\displaystyle=-\sum_{\mathcal{P}}\sum_{\\{\lambda_{i}\\},\bm{k}}\bigg{\\{}{\xi}_{abc,{\bm{k}},s}^{\lambda_{1}\lambda_{1}}f(\varepsilon_{\bm{k},s}^{\lambda_{1}})$
$\displaystyle+{j}_{a,{\bm{k}},s}^{\lambda_{1}\lambda_{2}}~{}{\kappa}_{bc,{\bm{k}},s}^{\lambda_{2}\lambda_{1}}\frac{f^{\lambda_{1}\lambda_{2}}_{{\bm{k}},s}}{\hbar\omega_{1}+\varepsilon_{{\bm{k}},s}^{\lambda_{1}\lambda_{2}}}+\frac{f^{\lambda_{1}\lambda_{2}}_{{\bm{k}},s}}{2}\frac{{j}_{b,{\bm{k}},s}^{\lambda_{2}\lambda_{1}}~{}{\kappa}_{ac,{\bm{k}},s}^{\lambda_{1}\lambda_{2}}+{j}_{c,{\bm{k}},s}^{\lambda_{2}\lambda_{1}}~{}{\kappa}_{ba,{\bm{k}},s}^{\lambda_{1}\lambda_{2}}}{\hbar\omega_{\Sigma}+\varepsilon_{{\bm{k}},s}^{\lambda_{1}\lambda_{2}}}\bigg{\\}}.$
(8)

Figure 4: Feynman diagrams for the third-order response. (a) refer to the
paramagnetic contributions, (b)-(h) to the diamagnetic contributions.

Notice that the two-photon and three-photon current couplings are
$\hat{\kappa}_{ab}=(-e/\hbar)^{2}\partial_{k_{a}}\partial_{k_{b}}\hat{\mathcal{H}}$
and
$\hat{\xi}_{abc}=-(1/2)(e/\hbar)^{3}\partial_{k_{a}}\partial_{k_{b}}\partial_{k_{c}}\hat{\cal
H}$, respectively. It is important to emphasize that our diagrammatic method
is highly systematic, allowing us to include all contributions, including both
paramagnetic and diamagnetic terms, to second and third-order currents.
Consequently, the final result adheres to gauge-invariant sum-rules Rostami
_et al._ (2021). Furthermore, our low-energy model Hamiltonian approach
produces physically meaningful results. This is due to the rapid convergence
of wave-vector integration for large ${\bm{k}}$ values, making our final
result independent of the choice of the ultraviolet cutoff.

Similarly, following the diagrammatic scheme in Fig. 4 (a), the third-order
spin response originating from the paramagnetic current operator is formally
given by
$\chi^{(3),P}_{abcd}\sim\langle\hat{j}_{a}\hat{j}_{b}\hat{j}_{c}\hat{j}_{d}\rangle$.
In principle, there are seven distinct Feynman diagrams involving multi-photon
current couplings Rostami  _et al._ (2021). However, for our Hamiltonian
model, only four additional diagrams can contribute, as depicted in Fig.
4(b,c,d,f), which can be written in terms of the following correlation
functions $\langle\hat{\kappa}_{ab}\hat{\kappa}_{cd}\rangle$,
$\langle\hat{j}_{a}\hat{j}_{b}\hat{\kappa}_{cd}\rangle$,
$\langle\hat{\kappa}_{ab}\hat{j}_{c}\hat{j}_{d}\rangle$ and
$\langle\hat{j}_{a}\hat{\kappa}_{bc}\hat{j}_{d}\rangle$. We provide the
explicit form of the third-order response in A for brevity.

###  II.1 Nonlinear spin Hall current

The $n^{\rm th}$-order nonlinear response functions to external fields are
dictated by the susceptibility tensor components $\chi_{abc\cdots}^{(n)}$. The
tensor components are restricted by space inversion, time-reversal, and
rotational and mirror symmetries of the crystalline quantum material.
Specifically, for a system having mirror symmetry $\mathcal{M}_{a}$ along the
plane perpendicular to an arbitrary spatial axis $a$ and the spin polarization
direction, the susceptibility tensor components with an odd number of ‘$a$’
spatial indices do not contribute to the charge current. This arises due to
the cancellation of the up and down spin components as
$\chi^{(n)}=\chi^{(n)}_{\uparrow}+\chi^{(n)}_{\downarrow}=0$. But, these
tensor components contribute to the spin response function $\chi^{(n),{\rm
spin}}=\chi^{(n)}_{\uparrow}-\chi^{(n)}_{\downarrow}=2\chi^{(n)}_{\uparrow}$.

In the time-reversal symmetric WTe2 with broken inversion symmetry due to
vertical gate potential, the mirror symmetry with respect to $x$-axis, i.e.,
$\mathcal{M}_{x}$ reduces non-vanishing tensor elements to four and eight for
the second-order and third-order responses respectively. These second-order
components are
$\chi_{xyy}^{(2)},\chi_{xxx}^{(2)},\chi_{yyx}^{(2)},\chi_{yxy}^{(2)}$. For the
third-order response, the non-vanishing tensor elements are the following

$\displaystyle\chi_{yxxx}^{(3)},\chi_{xyyy}^{(3)},\chi_{xxxy}^{(3)},\chi_{xyxx}^{(3)},$
$\displaystyle\chi_{xxyx}^{(3)},\chi_{yxyy}^{(3)},\chi_{yyxy}^{(3)},\chi_{yyyx}^{(3)}.$
(9)

To demonstrate the polarization dependence, we decompose the nonlinear spin
Hall current in the longitudinal and transverse basis as ${\bm{j}}^{\rm
spin}=j^{\rm spin}_{||}\hat{\bm{\epsilon}}_{||}+j^{\rm
spin}_{\perp}\hat{\bm{\epsilon}}_{\perp}$. Here, the unit vectors are
orthogonal $\hat{\bm{\epsilon}}_{||}\cdot\hat{\bm{\epsilon}}_{\perp}=0$.

Accordingly, the longitudinal (parallel) and transverse (perpendicular)
components of the spin current can be formulated using the relation $j^{\rm
spin}_{||}={\bm{j}}^{\rm spin}\cdot\hat{\bm{\epsilon}}_{||}$ and $j^{\rm
spin}_{\perp}={\bm{j}}^{\rm spin}\cdot\hat{\bm{\epsilon}}_{\perp}$
respectively. In this study, we focus on the transverse (Hall) components of
the nonlinear rectified spin current due to single-color driving field
${\bm{E}}(t)=|E_{1}|e^{i\phi_{1}}\hat{\bm{\epsilon}}(\theta)e^{i\omega
t}+c.c.$ and two-color light field
${\bm{E}}(t)=\hat{\bm{\epsilon}}(\theta)\left[|E_{1}|e^{i(\omega
t+\phi_{1})}+|E_{2}|e^{i(2\omega t+\phi_{2})}+c.c.\right]$ where
$\hat{\bm{\epsilon}}_{||}=\hat{\bm{\epsilon}}(\theta)=\cos\theta\hat{\bm{x}}+\sin\theta\hat{\bm{y}}$,
with $\theta$ being the polarization angle, is the linear polarization unit
vector. The corresponding light-induced single-color and two-color nonlinear
spin-current are given by $j_{\perp}^{\rm PG}$ and $j_{\perp}^{\rm 2c-PG}$
respectively. Symmetry-based argument (as discussed earlier) leads to the
following polarisation dependence of the transverse component of the nonlinear
spin currents:

$\displaystyle j_{\perp}^{\rm PG}=\frac{|E_{1}|^{2}}{\omega^{2}}{\rm
Re}\left[\sin^{2}\theta\chi_{xyy}^{(2)}+\cos^{2}\theta\chi_{xxx}^{(2)}\right]\sin\theta,$
(10)

and the third-order spin Hall current is given by

$\displaystyle j_{\perp}^{\rm 2c-PG}=\frac{|E_{1}|^{2}|E_{2}|}{\omega^{3}}{\rm
Im}\Big{[}e^{i\Delta\phi}\Big{(}\chi^{(3)}_{1}\sin^{4}\theta-\chi^{(3)}_{2}\cos^{4}\theta+\chi^{(3)}_{3}\sin^{2}2\theta\Big{)}\Big{]},$
(11)

where $\chi^{(3)}_{1}=\chi_{xyyy}^{(3)}$, $\chi^{(3)}_{2}=\chi_{yxxx}^{(3)}$,
and

$\displaystyle\chi^{(3)}_{3}=\frac{\chi_{xxxy}^{(3)}+\chi_{xxyx}^{(3)}+\chi_{xyxx}^{(3)}-\chi_{yxyy}^{(3)}-\chi_{yyxy}^{(3)}-\chi_{yyyx}^{(3)}}{4}.$
(12)

The derivation of the nonlinear spin currents is provided in B.

Further, $\Delta\phi=(2\phi_{1}-\phi_{2})$ is the phase difference between the
single- and second-harmonic frequency beams. Here, we get two sets of terms
that are proportional to ${\rm Re}[\chi^{(3)}]\sin\Delta\phi$ and ${\rm
Im}[\chi^{(3)}]\cos\Delta\phi$ contribute to the third-order rectified spin
Hall current depending on the choice of phase difference. It has been found
that for the linearly polarized beams with $\Delta\phi=\pm\pi/2$, the
nonlinear Hall current Eq. (11) yields maximum value as discussed in Ref.
Rioux  _et al._ (2011) for the case of graphene. In this study, we consider
zero phase difference $\Delta\phi=0$.

The following section presents numerical results in gated single-layer
1T′-WTe2 for the second and third-order spin susceptibilities versus laser
frequency and the gate potential $U$. Afterward, we analyze the anisotropic
polarisation dependence of the nonlinear spin Hall susceptibilities.

##  III Numerical Results and Discussion

Figure 5: Top: (a)-(c) Density plots for the second-order and third-order spin
Hall response as a function of the normalized incident energy and normalized
out-of-plane gate potential with respect to the spin-orbit coupling gap. The
dotted lines in plot (c) refer to the values for the cross-section plots in
the second and third-row panels, and the color bar refers to the magnitude of
the spin response. Second row: (d)-(f) represent the rectified spin Hall
response $\chi_{xyy}^{(2)}$, $\chi_{yxxx}^{(3)}$ and $\chi_{xyyy}^{(3)}$ at
out-of-plane gate potential $U=\delta_{\rm{SOC}}$ and $U=2\delta_{\rm{SOC}}$.
Third row: (g)-(i) represent the response at an incident energy
$\hbar\omega=\delta_{\rm{SOC}}$ and $\hbar\omega=2\delta_{\rm{SOC}}$. Here, we
set the chemical potential $\mu=0$ eV and set other parameters based on the
DFT calculations Qian  _et al._ (2014) as $v_{x}=0.09$ eVÅ, $v_{y}=2.6$ eVÅ,
$\delta=-0.33$ eV, $B=5.28$ eVÅ2, $A=-2.64$ eVÅ2, and $T=1$ K. Further, we
normalize the response by setting $\chi_{0}^{(2)}=e^{3}/\hbar$ and
$\chi_{0}^{(3)}=e^{4}/\hbar$.

We find that there is no second-order spin current ($\propto\chi^{(2)}_{yxx}$)
in the Hall measurement geometry along $\hat{y}$-direction with the field
along $\hat{x}$-direction due to the cancellation of the current induced by up
and down spins, but the second-order Hall charge current will remain finite.
Unlike the second-order current, the third-order spin Hall current
$j_{a}^{(3),\text{spin}}$ follows different symmetry properties, thus can be
obtained in both $\hat{x}$ and $\hat{y}$ Hall measurement geometries.
Specifically, the third-order spin Hall effect can be observed in both
inversion symmetric and time-reversal symmetric materials. Here, we discuss
the two Hall components of susceptibility $\chi_{yxxx}^{(3),\text{spin}}$, and
$\chi_{xyyy}^{(3),\text{spin}}$, subjected to the mirror symmetry possessed by
1T′-WTe2, to the linearly polarized two-color optical beam of light.

In Fig. 5 (a)-(c), we demonstrate the density plot for the second- and third-
order spin Hall susceptibility tensor components
$\chi_{xyy}^{(2),{\rm{spin}}}$, $\chi_{yxxx}^{(3),{\rm{spin}}}$ and
$\chi_{xyyy}^{(3),{\rm{spin}}}$ for 1T′-WTe2 having spin texture in the
momentum space, as a function of $U/\delta_{\rm{SOC}}$ and
$\hbar\omega/\delta_{\rm{SOC}}$, where $\delta_{\rm SOC}$ is the spin-orbit
coupling gap. Here, we mainly focus on the rectified (dc) spin current with
single-color and two-color laser beams, contributed by the real part of the
second-order and imaginary part of the third-order response functions,
respectively. We observe that the second-order spin Hall susceptibility
component $\chi_{xyy}^{(2),{\rm spin}}$ in response to the electric field
along $\hat{y}$ direction gives interband resonances at an incident energy
$\hbar\omega\geq 2\Delta_{\pm}$, with $\Delta_{\pm}=2|U\pm\delta_{\rm{SOC}}|$
being the band gap at $Q$ and $Q^{\prime}$, as indicated by the dark red and
dark blue colors in Fig. 5(a). However, resonances’ strength and position
change with the out-of-plane gate potential, thus shifting in different
frequency regimes.

The resulting feature is intriguing because of the band closing and reopening
around the Dirac points at valleys $Q$ and $Q^{\prime}=-Q$ due to the gate
potential $U$. Furthermore, the spin current generation occurs due to the
imbalance of carrier excitation at different valleys. Since spin-up and spin-
down currents are equal in magnitude and flow in opposite directions, the
charge Hall current vanishes and we obtain a pure second-order spin Hall
current by irradiating the linearly polarized single-color beam of light. For
the case of zero phase difference, the two-color rectified spin current is
given by the real part of the third-order conductivity (see B)

$\displaystyle j_{\perp}^{\rm 2c-PG}$ $\displaystyle={\rm
Re}\left[\hat{\bm{\epsilon}}_{\perp}\cdot{\bm{\sigma}}^{(3)}(\omega,\omega,-2\omega)\vdots\hat{\bm{\epsilon}}_{||}\hat{\bm{\epsilon}}_{||}\hat{\bm{\epsilon}}_{||}\right]|E(\omega)|^{2}|E(2\omega)|.$
(13)

One of the most striking results of this study is the presence of light-
induced nonlinear spin current when $\hbar\omega<\Delta_{\pm}$. Unlike usual
shift and injection current mechanisms which are related to the imaginary part
${\rm Im}[1/(\Delta_{\pm}-\hbar\omega-i0^{+})]$, the current below the optical
gap is given by the real part ${\rm Re}[1/(\Delta_{\pm}-\hbar\omega-i0^{+})]$.
In the clean limit, the imaginary part leads to a delta function
$\delta(\Delta_{\pm}-\hbar\omega)$ implying that we can induce a current after
absorbing the photon and generating a photo-excited carrier. However, the
current below the optical gap is related to the principal value of
$1/(\Delta_{\pm}-\hbar\omega-i0^{+})$ and thus can be finite even without
generating a photo-excited carrier density. This mechanism has been discussed
for the second-order charge current in the time-reversal symmetry breaking
systems Kaplan  _et al._ (2020); Gao  _et al._ (2021), and its thermodynamics
ground is discussed in detail Shi  _et al._ (2023).

In Ref. Kaplan  _et al._ (2020), the second-order charge conductivity below
the optical gap regime has been predicted for the time-reversal symmetry
broken system. The conductivity is given by the expression
$\sigma^{(2)}_{aaa}\sim\sum_{m\neq
n}\sum_{{\bm{k}},s}f^{mn}_{{\bm{k}},s}|r^{mn}_{a,{\bm{k}},s}|^{2}v^{m}_{a,{\bm{k}},s}$,
where
$f^{mn}_{{\bm{k}},s}=f(\varepsilon^{m}_{{\bm{k}},s})-f(\varepsilon^{n}_{{\bm{k}},s})$.
In this expression, $r^{mn}_{a,{\bm{k}},s}$ represents the inter-band matrix
element of the position vector Cartesian component labeled by the subscript
$a$, and
$v^{m}_{a,{\bm{k}},s}=\hbar^{-1}\partial_{k_{a}}\varepsilon^{m}_{{\bm{k}},s}$
denotes the band velocity component. It is worth noting that the current
vanishes in the presence of time-reversal symmetry, as the velocity is odd
under time-reversal.

Our numerical results in Fig. 5 show that we can extend the above result of
Ref. Kaplan  _et al._ (2020) to the third-order regime in the spin channel
without breaking the time-reversal symmetry, offering an intriguing extension
to this exotic behavior. To shed further light, we also provide the explicit
analysis in C. Here, we present an intuitive sketch of the analytical
expression for the third-order spin current within the optical gap regime. The
concept is based on replacing the band velocity with the anomalous velocity
$(-e/\hbar){\bm{E}}\times{\bm{\Omega}}^{m}_{{\bm{k}}}$, which allows us to
obtain the third-order charge current. In this case,
${\bm{\Omega}}^{m}_{\bm{k}}={\bm{\Omega}}^{m}_{{\bm{k}}\uparrow}+{\bm{\Omega}}^{m}_{{\bm{k}}\downarrow}$
represents the Berry curvature vector. To obtain the spin current, we can
roughly replace the Berry curvature with the spin-Berry curvature
${\bm{\Omega}}_{{\bm{k}}}^{m,{\rm
spin}}={\bm{\Omega}}^{m}_{{\bm{k}},\uparrow}-{\bm{\Omega}}^{m}_{{\bm{k}},\downarrow}$.
Consequently, the third-order spin current can be finite in the presence of
time-reversal symmetry. The explicit calculation given in C is slightly
different but qualitatively consistent with the above intuitive picture. Below
the gap, the third-order effect is given by the spin Hall current $j^{\rm
2c-PG}_{y}=\sigma^{\rm spin}_{yxxx}E^{2}_{x}(\omega)E^{\ast}_{x}(2\omega)$
where the nonlinear spin-Hall conductivity for inversion symmetric case
($U=0$) in our two-band model is given in terms of conduction band spin-Berry
curvature:

$\displaystyle\sigma^{\rm spin}_{yxxx}\approx
C(\omega)\sum_{\bm{k}}\Omega^{c,\rm spin}_{\bm{k}}\alpha^{cv}_{\bm{k}}$ (14)

where $C(\omega)=e^{3}/(2\hbar^{4}\omega^{3})$ and the real-valued parameter
$\alpha_{\bm{k}}^{cv}$ is given by

$\displaystyle\alpha^{cv}_{\bm{k}}=\frac{1}{2}\sum_{s=\pm}\bigg{\\{}|r^{cv}_{x,{{\bm{k}},s}}|^{2}\varepsilon_{{\bm{k}},s}^{cv}+\tilde{\kappa}^{cv}_{xx,{\bm{k}},s}\bigg{\\}}f^{cv}_{{\bm{k}},s},$
(15)

in which
$\tilde{\kappa}^{cv}_{xx,{\bm{k}},s}=(\hbar^{2}/e^{2})\kappa^{cv}_{xx,{\bm{k}},s}$,
and the position vector matrix element is given in terms of the paramagnetic
current matrix element:
$r^{cv}_{a,{\bm{k}},s}=({i\hbar}/{e}){{j}^{cv}_{a,{\bm{k}},s}}/{\varepsilon^{cv}_{{\bm{k}},s}}$
Considering time-reversal and inversion symmetries, we have
$\Omega^{c}_{\bm{k},\uparrow}=-\Omega^{c}_{\bm{k},\downarrow}=-\Omega^{c}_{-\bm{k},\downarrow}$,
therefore $\Omega^{c}_{\bm{k}}=0$, and
$\Omega^{c,\mathrm{spin}}_{\bm{k}}=\Omega^{c,\mathrm{spin}}_{-\bm{k}}$.
Similarly, we have
$\varepsilon^{cv}_{\bm{k},\uparrow}=\varepsilon^{cv}_{-\bm{k},\downarrow}$,
which leads to $\alpha^{cv}_{\bm{k}}=\alpha^{cv}_{-\bm{k}}$. Accordingly, both
$\alpha^{cv}_{\bm{k}}$ and spin-Berry curvature
$\Omega^{c,\mathrm{spin}}_{\bm{k}}$ are even function under
$\bm{k}\to-\bm{k}$. Consequently, the third-order spin Hall current below the
optical gap is expected to be finite, which is consistent with our numerical
results shown in Fig. 5.

In the following, we show that one can generate third-order spin current below
the gap in a time-reversal symmetric system. However, there is no such effect
in the second-order spin response due to the cancellation of the effect for
spin-up and spin-down components.

To illustrate features with more detail, we plot the cross-section curves at
different values of the scaled back-gate potential $U/\delta_{\rm{SOC}}$ and
the incident photon energy $\hbar\omega/\delta_{\rm{SOC}}$. For $U=0$, the
inversion symmetry of the system remains intact, which results in the
vanishing second-order response $\chi_{xyy}^{(2)}$ consistent with the
symmetry argument. For finite but small $U\ll\delta_{\rm SOC}$, the inversion
symmetry breaks down that leads to finite $\chi_{xyy}^{(2)}$.

Figure 6: Polarization dependence of the second-order and third-order tangent
component of the spin response at fixed normalized incident energy and
different normalized out-of-plane gate potential with respect to the spin-
orbit coupling gap. The red color curve correspond to $j_{\perp}^{{\rm PG,{\rm
spin}}}/j_{0}^{(2)}$ and $0.1j_{\perp}^{{\rm 2c-PG,{\rm spin}}}/j_{0}^{(3)}$
at $U=2\delta_{\rm SOC}$ and the blue refers to $0.1j_{\perp}^{{\rm PG,{\rm
spin}}}/j_{0}^{(2)}$ and $j_{\perp}^{{\rm 2c-PG,{\rm spin}}}/j_{0}^{(3)}$ at
$U=\delta_{\rm SOC}$.

As seen in Fig. 5(d), the effective band gap $\Delta_{-}$ vanishes, and the
bottom of the conduction and top of the valence bands touch each other at $Q$
point when $U=\delta_{\rm SOC}$ and therefore optical response reveals the
gapless nature of the system that is finite even at the tiny value of the
light frequency. By further increasing the frequency, we see a step-like
feature in the second-order response at $\hbar\omega=2\Delta_{+}=4U$. We have
generated the same curve for the case of $U=2\delta_{\rm SOC}$ where the
effective gaps at different valleys are $\Delta_{+}/\delta_{\rm{SOC}}=3$ and
$\Delta_{-}/\delta_{\rm{SOC}}=1$. This leads to the resonant jumps at
$\hbar\omega/\delta_{\rm{SOC}}=6$ and $\hbar\omega/\delta_{\rm{SOC}}=2$.

In Fig. 5(e)-(f), we find the emergence of resonant peaks in the third-order
response due to the two-photon and one-photon absorption processes which
correspond to $\hbar\omega=\Delta_{\pm}$ and $\hbar\omega=2\Delta_{\pm}$,
respectively. Among these, the two peaks associated with a larger effective
gap $\Delta_{+}$ show similar behavior, such as resonances for different back-
gate potentials. However, the other two associated with $\Delta_{-}$ reveal
the different feature that is the presence and absence of the resonance peak
at $U=\delta_{\rm{SOC}}$ and $U=2\delta_{\rm{SOC}}$.

We have made vertical cross-section plots to investigate the behavior of
nonlinear spin susceptibilities concerning fixed values of photon energy
$\hbar\omega$ and varying values of $U$. The second-order and third-order
effects are depicted in Fig. 5(g) and Fig. 5(h)-(i), respectively. Contrary to
the second-order case, the third-order response remains finite at $U=0$.
Furthermore, we observe a strong and non-monotonic dependence of the nonlinear
spin current on the back-gate potential $U$. This behavior implies that small
changes in the back-gate potential can induce significant changes in the
resulting nonlinear spin response, which could have important implications for
the design and optimization of spin-based devices.

Further, we present the numerical results for the polarisation dependence of
the spin transverse currents for a fixed value of $\hbar\omega=\delta_{\rm
SOC}$ in polar plots in Fig. 6. The polarization dependence of nonlinear spin
current highly depends on the crystalline symmetry and orientation of the
monolayer WTe2. The anisotropic behavior of the second-order spin Hall current
that follows $j^{{\rm PG}}_{\perp}\propto{\rm
Re}[\chi^{(2)}_{xyy}]\sin^{3}\theta$. The third-order spin current is
proportional to the imaginary part of $\chi^{(3)}$ components for zero phase
difference between two color laser pulses. The anisotropic polarization
dependence of third-order spin current is more complex due to the competing
behavior of different tensor components of the susceptibility multiplied by
$\sin^{4}\theta$, $\cos^{4}\theta$, and $\sin^{2}2\theta$.

Finally, we provide the numerical estimation of the spin Hall photocurrents at
a particular incident energy and using the microscopic parameter obtained from
DFT calculations and experiments Xu  _et al._ (2018); Garcia  _et al._
(2020b); Tang  _et al._ (2017). At $\hbar\omega=90$ meV, which is twice the
spin-orbit coupling gap $\delta_{\rm{SOC}}=45$ meV, the out-of-plane gate
potential $U=90$ meV, and the electric field $E$ of the order of
$(10^{6}-10^{9})$ V/m, the magnitude of the second and third-order rectified
spin Hall conductivity comes out to be of the order of
$(10^{-1}-10^{2})\sigma_{0}$ and $(10^{-4}-10^{2})\sigma_{0}$ respectively,
where $\sigma_{0}=2e/h$ is the unit of spin conductivity. Here, the back-gate
potential $U$ is obtained using the relation $U\approx e\epsilon E_{z}d$ and
setting vertical field $E_{z}\approx 0.1$V/nm, dielectric constant
$\epsilon\approx 3$, and the separation between the top and bottom contacts
$d\approx 0.31$ nm aturia  _et al._ (2018); Shahnazaryan  _et al._ (2019).
Compared to the linear dc spin conductivity, the presented rectified nonlinear
spin Hall conductivity estimation is feasible to measure experimentally.

Experimentally, the observation of nonlinear spin currents can be achieved
through two distinct techniques: the inverse-spin Hall setup and magneto-
optical Kerr rotation measurements. In the inverse-spin Hall setup, the flow
of the spin current can be observed by measuring the transverse voltage drop
(charge current flow) due to the inverse spin Hall effect Saitoh  _et al._
(2006); Ando  _et al._ (2011); Bottegoni  _et al._ (2013). The alternative
approach involves magneto-optical spectroscopy of spin current, employing
spatially-resolved Kerr rotation measurements– as utilized to observe the
spin-Hall effect in a 2D electron liquid Kato  _et al._ (2004). Light-induced
nonlinear spin current induces an electron spin accumulation at the edges of
the sample. The spatial distribution of this spin accumulation (local
magnetization) can be experimentally probed via the Kerr rotation
spectroscopy.

##  IV Conclusion

Our study focuses on the investigation of the rectified nonlinear spin Hall
current in the 1T′-WTe2 material, utilizing both single-color and two-color
laser beams. We have conducted an extensive analysis on the effects of the
displacement field, light intensity, and polarization direction on the
nonlinear spin Hall current to gain a comprehensive understanding of this
phenomenon. Our findings indicate that the nonlinear response exhibits
interband resonances arising from one-photon and two-photon absorption
processes. It is worth noting that the two-color rectified response was found
to be significantly stronger in magnitude than the single-color response when
the displacement field was twice the spin-orbit coupling gap. Our study
provides valuable insights into the intrinsic nonlinear spin Hall effect in
the 1T′-WTe2 material and offers a path toward designing advanced
optoelectronic devices capable of operating in nonlinear regimes for
spintronics applications. Moreover, the present study can be extended to
bilayer, trilayer and in general to few-layered WTe2 systems. In particular,
the intriguing ferroelectric transition observed in few-layered WTe2 Fei  _et
al._ (2018); Wang and Qian (2019b); Xiao  _et al._ (2020); Sharma  _et al._
(2019) can impact the nonlinear spin Hall current. In such systems, the
ferroelectric polarization will possibility enhance the effect by
spontaneously breaking the inversion symmetry. The quantitative analysis of
the nonlinear spin Hall effect in bilayer WTe2 is left as the subject of a
future study.

## Acknowledgment

This work was supported by Nordita and the Swedish Research Council (VR
Starting Grant No. 2018-04252). Nordita is partially supported by Nordforsk.
PB acknowledges the computational facilities provided by SRM-AP and is
financially supported by Science and Engineering Research Board-State
University Research Excellence under project number SUR/2022/000289.

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##  Appendix A Third-order susceptibility expression

In a similar spirit of the second-order response, the paramagnetic
contribution to the third-order response corresponding to the Feynman diagram
Fig. 4(a) is given by Ahn  _et al._ (2022); Säynätjoki  _et al._ (2017)

$\displaystyle\chi_{abcd}^{P,(3)}(\omega_{1},\omega_{2},\omega_{3},s)$
$\displaystyle=\sum_{\mathcal{P}}\sum_{{\bm{k}}}\sum_{\lambda_{i}}\frac{{j}_{a,{\bm{k}},s}^{\lambda_{4}\lambda_{3}}~{}{j}_{b,{\bm{k}},s}^{\lambda_{3}\lambda_{2}}~{}{j}_{c,{\bm{k}},s}^{\lambda_{2}\lambda_{1}}~{}{j}_{d,{\bm{k}},s}^{\lambda_{1}\lambda_{4}}}{\hbar\omega_{\Sigma}+\varepsilon_{{\bm{k}},s}^{\lambda_{1}\lambda_{4}}}$
$\displaystyle\times\bigg{\\{}\frac{1}{\hbar\omega_{1}+\hbar\omega_{2}+\varepsilon_{{\bm{k}},s}^{\lambda_{1}\lambda_{3}}}\bigg{(}\frac{f^{\lambda_{1}\lambda_{2}}_{{\bm{k}},s}}{\hbar\omega_{1}+\varepsilon_{{\bm{k}},s}^{\lambda_{1}\lambda_{2}}}-\frac{f^{\lambda_{2}\lambda_{3}}_{{\bm{k}},s}}{\hbar\omega_{2}+\varepsilon_{{\bm{k}},s}^{\lambda_{2}\lambda_{3}}}\bigg{)}$
$\displaystyle-\frac{1}{\hbar\omega_{2}+\hbar\omega_{3}+\varepsilon_{{\bm{k}},s}^{\lambda_{2}\lambda_{4}}}\bigg{(}\frac{f^{\lambda_{2}\lambda_{3}}_{{\bm{k}},s}}{\hbar\omega_{2}+\varepsilon_{{\bm{k}},s}^{\lambda_{2}\lambda_{3}}}-\frac{f^{\lambda_{3}\lambda_{4}}_{{\bm{k}},s}}{\hbar\omega_{3}+\varepsilon_{{\bm{k}},s}^{\lambda_{3}\lambda_{4}}}\bigg{)}\bigg{\\}},$
(16)

where $\omega_{\Sigma}=\omega_{1}+\omega_{2}+\omega_{3}$ and the diamagnetic
part of the third-order susceptibility associated with Figs. 4(b)-(h) is given
by Rostami  _et al._ (2021); Säynätjoki  _et al._ (2017)

$\displaystyle\chi_{abcd}^{D,(3)}(\omega_{1},\omega_{2},\omega_{3},s)$
$\displaystyle=\sum_{\mathcal{P}}\sum_{{\bm{k}}}\sum_{\lambda_{i}}\bigg{\\{}\frac{{\kappa}_{ab,{\bm{k}},s}^{\lambda_{1}\lambda_{2}}~{}{\kappa}_{cd,{\bm{k}},s}^{\lambda_{2}\lambda_{1}}~{}f^{\lambda_{1}\lambda_{2}}_{{\bm{k}},s}}{\hbar\omega_{1}+\hbar\omega_{2}+\varepsilon_{{\bm{k}},s}^{\lambda_{1}\lambda_{2}}}$
$\displaystyle-\frac{{j}_{a,{\bm{k}},s}^{\lambda_{3}\lambda_{2}}~{}{j}_{b,{\bm{k}},s}^{\lambda_{2}\lambda_{1}}~{}{\kappa}_{cd,{\bm{k}},s}^{\lambda_{1}\lambda_{3}}}{\hbar\omega_{1}+\hbar\omega_{3}+\varepsilon_{{\bm{k}},s}^{\lambda_{1}\lambda_{3}}}\bigg{(}\frac{f^{\lambda_{1}\lambda_{2}}_{{\bm{k}},s}}{\hbar\omega_{1}+\varepsilon_{{\bm{k}},s}^{\lambda_{1}\lambda_{2}}}-\frac{f^{\lambda_{2}\lambda_{3}}_{{\bm{k}},s}}{\hbar\omega_{2}+\varepsilon_{{\bm{k}},s}^{\lambda_{2}\lambda_{3}}}\bigg{)}$
$\displaystyle-\frac{{\kappa}_{ab,{\bm{k}},s}^{\lambda_{3}\lambda_{2}}~{}{j}_{c,{\bm{k}},s}^{\lambda_{2}\lambda_{1}}~{}{j}_{d,{\bm{k}},s}^{\lambda_{1}\lambda_{3}}}{\hbar\omega_{\Sigma}+\varepsilon_{{\bm{k}},s}^{\lambda_{1}\lambda_{3}}}\bigg{(}\frac{f^{\lambda_{1}\lambda_{2}}_{{\bm{k}},s}}{\hbar\omega_{1}+\hbar\omega_{2}+\varepsilon_{{\bm{k}},s}^{\lambda_{1}\lambda_{2}}}-\frac{f^{\lambda_{2}\lambda_{4}}_{{\bm{k}},s}}{\hbar\omega_{2}+\varepsilon_{{\bm{k}},s}^{\lambda_{2}\lambda_{4}}}\bigg{)}\bigg{\\}}.$
(17)

##  Appendix B Nonlinear rectified currents

###  B.1 Second-order rectified current

The second-order rectified current is obtained by setting
$\omega_{1}+\omega_{2}=0$. In response to a time-dependent external electric
field ${\bm{E}}(t)=\sum_{\omega}{\bm{E}}_{1}(\omega)e^{i(\omega
t+\phi_{1})}+{\rm c.c.}$, the current can be expressed as follows

$\displaystyle j_{a}^{\rm PG}$
$\displaystyle=\sum_{b,c}\sigma_{abc}^{(2)}(\omega,-\omega)E_{1,b}(\omega)E_{1,c}^{*}(\omega)+{\rm
c.c.}$ (18)

Here, the conductivity is $\sigma_{abc}^{(2)}=-\chi_{abc}^{(2)}/\omega^{2}$.
Further, on considering the spatial indices $b=c$ we get

$\displaystyle j_{a}^{\rm PG}$
$\displaystyle=\sum_{b}\left(\sigma_{abb}^{(2)}(\omega,-\omega)E_{1,b}(\omega)E_{1,b}^{*}(\omega)+\sigma_{abb}^{(2)}(-\omega,\omega)E_{1,b}^{*}(\omega)E_{1,b}(\omega)\right)$
$\displaystyle j_{a}^{\rm PG}$
$\displaystyle=\sum_{b}\left(\sigma_{abb}^{(2)}(\omega,-\omega)+\sigma_{abb}^{(2)}(-\omega,\omega)\right)E_{1,b}(\omega)E_{1,b}^{*}(\omega).$
(19)

Note that $E_{b}^{*}(\omega)=E_{b}(-\omega)$. Expressing the conductivity in
real and imaginary parts as $\sigma_{abb}^{(2)}={\rm
Re}[\sigma_{abb}^{(2)}]+i{\rm Im}[\sigma_{abb}^{(2)}]$ and using the relation
$\sigma_{abb}^{(2)}(\omega,-\omega)=[\sigma_{abb}^{(2)}(-\omega,\omega)]^{\ast}$,
we obtain

$\displaystyle j_{a}^{(2),{\rm PG}}$ $\displaystyle=\sum_{b}{\rm
Re}[\sigma_{abb}^{(2)}(\omega,-\omega)]E_{1,b}(\omega)E_{1,b}^{*}(\omega).$
(20)

###  B.2 Third-order rectified current

The third-order rectified current is obtained by setting
$\omega_{1}+\omega_{2}+\omega_{3}=0$. In response to a time-dependent external
electric field ${\bm{E}}(t)=\sum_{\omega}[{\bm{E}}_{1}(\omega)e^{i(\omega
t+\phi_{1})}+{\bm{E}}_{2}(\omega)e^{i(2\omega t+\phi_{2})}]+{\rm c.c.}$, the
current can be expressed as follows

$\displaystyle j_{a}^{\rm 2c-PG}$
$\displaystyle=\sum_{b,c,d}\sigma_{abcd}^{(3)}(\omega,\omega,-2\omega)E_{1,b}(\omega)E_{1,c}(\omega)E_{2,d}^{*}(2\omega)+{\rm
c.c.}$ (21)

Here, the conductivity is
$\sigma_{abcd}^{(3)}=-i\chi_{abcd}^{(3)}/2\omega^{3}$. Further, on considering
the spatial indices $b=c=d$ we get

$\displaystyle j_{a}^{\rm 2c-PG}$
$\displaystyle=\sum_{b}\left(\sigma_{abbb}^{(2)}(\omega,\omega,-2\omega)E_{1,b}(\omega)E_{1,b}(\omega)E_{2,b}^{*}(2\omega)+\sigma_{abbb}^{(3)}(-\omega,-\omega,2\omega)E_{1,b}^{*}(\omega)E_{1,b}^{*}(\omega)E_{2,b}(2\omega)\right).$
(22)

Considering real-valued field amplitudes, we obtain

$\displaystyle j_{a}^{\rm 2c-PG}$
$\displaystyle=\sum_{b}\left(\sigma_{abbb}^{(2)}(\omega,\omega,-2\omega)e^{i(2\phi_{1}-\phi_{2})}+\sigma_{abbb}^{(3)}(-\omega,-\omega,2\omega)e^{-i(2\phi_{1}-\phi_{2})}\right)|E_{1,b}|^{2}|E_{2,b}|.$
(23)

Writing the conductivity in real and imaginary parts as
$\sigma_{abbb}^{(3)}={\rm Re}[\sigma_{abbb}^{(3)}]+i{\rm
Im}[\sigma_{abbb}^{(3)}]$ and using the relation
$\sigma_{abbb}^{(3)}(\omega,\omega,-2\omega)=[\sigma_{abbb}^{(3)}(-\omega,-\omega,2\omega)]^{\ast}$,
we obtain

$\displaystyle j_{a}^{\rm 2c-PG}$ $\displaystyle=\sum_{b}{\rm
Re}[\sigma_{abbb}^{(3)}(\omega,\omega,-2\omega)e^{i(2\phi_{0}-\phi_{1})}]|E_{1,b}|^{2}|E_{2,b}|.$
(24)

###  B.3 Transverse component of the nonlinear spin currents

For the single-color driving field
${\bm{E}}(t)=|E_{1}|e^{i\phi_{1}}\hat{\bm{\epsilon}}(\theta)e^{i\omega
t}+c.c.$, the transverse nonlinear rectified current is given by

$j^{\rm PG}_{\perp}={\bm{j}}^{\rm PG}\cdot\hat{\bm{\epsilon}}_{\perp},$ (25)

where the unit vector
$\hat{\bm{\epsilon}}_{\perp}=-\sin\theta\hat{\bm{x}}+\cos\theta\hat{\bm{y}}$
and $\hat{\bm{\epsilon}}_{||}=\cos\theta\hat{\bm{x}}+\sin\theta\hat{\bm{y}}$
is the linear polarization unit vector having $\theta$ an angle of
polarization. Using this property, the transverse current becomes

$j^{\rm PG}_{\perp}=-j_{x}^{\rm PG}\sin\theta+j_{y}^{\rm PG}\cos\theta.$ (26)

Using Eq. (20) and expressing
$\sigma_{abc}(\omega_{1},\omega_{2})=\chi_{abc}(\omega_{1},\omega_{2})/(i\omega_{1}i\omega_{2})$
and considering the symmetry-permitted components of the second-order response
in gated single-layer WTe2, we obtain

$j^{\rm PG}_{\perp}=\frac{1}{\omega^{2}}{\rm
Re}\left[\chi_{xyy}^{(2)}E_{1,y}E_{1,y}^{*}+\chi_{xxx}^{(2)}E_{1,x}E_{1,x}^{*}\right]\sin\theta.$
(27)

On further simplifications, the rectified second-order current yields

$j^{\rm PG}_{\perp}=\frac{|E_{1}|^{2}}{\omega^{2}}{\rm
Re}\left[\sin^{2}\theta\chi_{xyy}^{(2)}+\cos^{2}\theta\chi_{xxx}^{(2)}\right]\sin\theta.$
(28)

This derivation leads to Eq. (10) in the main text. Similarly, for the two-
color light field
${\bm{E}}(t)=\hat{\bm{\epsilon}}(\theta)\left[|E_{1}|e^{i(\omega
t+\phi_{1})}+|E_{2}|e^{i(2\omega t+\phi_{2})}+c.c.\right]$ the third-order
rectified transverse current following Eq. (24) gives

$\displaystyle j^{\rm 2c-PG}_{\perp}=\frac{1}{\omega^{3}}{\rm
Im}\bigg{[}\big{\\{}\chi_{xyyy}^{(3)}e^{i\Delta\phi}E_{1,y}^{2}E_{2,y}^{*}+(\chi_{xyxx}^{(3)}+\chi_{xxyx}^{(3)}+\chi_{xxxy}^{(3)})e^{i\Delta\phi}E_{1,x}^{2}E_{2,y}^{*}\big{\\}}\sin\theta$
$\displaystyle-\big{\\{}\chi_{yxxx}^{(3)}e^{i\Delta\phi}E_{1,x}^{2}E_{2,x}^{*}+(\chi_{yxyy}^{(3)}+\chi_{yyxy}^{(3)}+\chi_{yyyx}^{(3)})e^{i\Delta\phi}E_{1,y}^{2}E_{2,x}^{*}\big{\\}}\cos\theta\bigg{]}.$
(29)

On substituting the field, the above equation reduces to

$\displaystyle j^{\rm 2c-PG}_{\perp}$ $\displaystyle=\frac{1}{\omega^{3}}{\rm
Im}\bigg{[}\big{\\{}\chi_{xyyy}^{(3)}\sin^{4}\theta+(\chi_{xyxx}^{(3)}+\chi_{xxyx}^{(3)}+\chi_{xxxy}^{(3)})\sin^{2}\theta\cos^{2}\theta$
$\displaystyle-\chi_{yxxx}^{(3)}\cos^{4}\theta+(\chi_{yxyy}^{(3)}+\chi_{yyxy}^{(3)}+\chi_{yyyx}^{(3)})\sin^{2}\theta\cos^{2}\theta\big{\\}}e^{i\Delta\phi}|E_{1}|^{2}|E_{2}|\bigg{]},$
(30) $j^{\rm 2c-PG}_{\perp}=\frac{|E_{1}|^{2}|E_{2}|}{\omega^{3}}{\rm
Im}\bigg{[}\big{\\{}\chi_{1}^{(3)}\sin^{4}\theta-\chi_{3}^{(2)}\cos^{4}\theta+\chi_{3}^{(3)}\sin^{2}2\theta\big{\\}}e^{i\Delta\phi}\bigg{]}.$
(31)

where the quantities $\Delta\phi=2\phi_{1}-\phi_{2}$,
$\chi^{(3)}_{1}=\chi_{xyyy}^{(3)}$, $\chi^{(3)}_{2}=\chi_{yxxx}^{(3)}$, and
$\chi^{(3)}_{3}=\\{\chi_{xxxy}^{(3)}+\chi_{xxyx}^{(3)}+\chi_{xyxx}^{(3)}-\chi_{yxyy}^{(3)}-\chi_{yyxy}^{(3)}-\chi_{yyyx}^{(3)}\\}/{4}$.
This derives the Eq. (11) of the main text.

##  Appendix C Proof of Eq. (13)

Here, we provide a detailed derivation for the third-order spin Hall current
in the transparent (low-frequency) regime. The third-order conductivity due to
the two-color excitation field is related to the following third-order
susceptibility

$\sigma^{(3)}(\omega,\omega,-2\omega)=-i\frac{\chi^{(3)}(\omega,\omega,-2\omega)}{2\omega^{3}}.$
(32)

Using the many-body perturbation theory, the susceptibility $\chi^{(3)}$ is
expressed as the sum of paramagnetic and diamagnetic contributions to the
susceptibilities. The paramagnetic part of the third-order susceptibility due
to two-color field is

$\displaystyle\chi_{abcd}^{P,(3)}(\omega,\omega,-2\omega)$
$\displaystyle=\sum_{\mathcal{P}}\sum_{{\bm{k}},s}\sum_{\lambda_{i}}\frac{{j}_{a,{\bm{k}},s}^{\lambda_{4}\lambda_{3}}~{}{j}_{b,{\bm{k}},s}^{\lambda_{3}\lambda_{2}}~{}{j}_{c,{\bm{k}},s}^{\lambda_{2}\lambda_{1}}~{}{j}_{d,{\bm{k}},s}^{\lambda_{1}\lambda_{4}}}{\varepsilon_{{\bm{k}},s}^{\lambda_{1}\lambda_{4}}}$
$\displaystyle\bigg{\\{}\frac{1}{2(\hbar\omega+i\eta)+\varepsilon_{{\bm{k}},s}^{\lambda_{1}\lambda_{3}}}\bigg{(}\frac{f^{\lambda_{1}\lambda_{2}}_{{\bm{k}},s}}{(\hbar\omega+i\eta)+\varepsilon_{{\bm{k}},s}^{\lambda_{1}\lambda_{2}}}-\frac{f^{\lambda_{2}\lambda_{3}}_{{\bm{k}},s}}{(\hbar\omega+i\eta)+\varepsilon_{{\bm{k}},s}^{\lambda_{2}\lambda_{3}}}\bigg{)}$
$\displaystyle-\frac{1}{-(\hbar\omega+i\eta)+\varepsilon_{{\bm{k}},s}^{\lambda_{2}\lambda_{4}}}\bigg{(}\frac{f^{\lambda_{2}\lambda_{3}}_{{\bm{k}},s}}{(\hbar\omega+i\eta)+\varepsilon_{{\bm{k}},s}^{\lambda_{2}\lambda_{3}}}-\frac{f^{\lambda_{3}\lambda_{4}}_{{\bm{k}},s}}{-2(\hbar\omega+i\eta)+\varepsilon_{{\bm{k}},s}^{\lambda_{3}\lambda_{4}}}\bigg{)}\bigg{\\}},$
(33)

where $\eta\rightarrow 0^{+}$. We use the following identity to separate real
and imaginary parts

$\lim_{~{}~{}~{}\eta\to 0^{+}}\frac{1}{\hbar\omega+i\eta}={\rm
P}\bigg{(}\frac{1}{\hbar\omega}\bigg{)}-i\frac{\pi}{\hbar}\delta(\omega),$
(34)

where ${\rm P}(\dots)$ stands for the principal value part. The imaginary part
of the paramagnetic contribution of the third-order susceptibility below the
gap for the two-band case gives

$\displaystyle{\rm Im}[\chi_{yxxx}^{P,(3)}(\omega,\omega,-2\omega)]$
$\displaystyle=\sum_{\mathcal{P}}\sum_{{\bm{k}},s}\bigg{\\{}\frac{{\rm
Im}[{j}_{y,{\bm{k}},s}^{cv}~{}{j}_{x,{\bm{k}},s}^{vc}]|{j}_{x,{\bm{k}},s}^{cv}|^{2}}{\varepsilon_{{\bm{k}},s}^{cv}}\frac{f^{cv}_{{\bm{k}},s}}{\varepsilon_{{\bm{k}},s}^{cv}}\bigg{[}{\rm
P}\bigg{(}\frac{1}{\hbar\omega+\varepsilon_{{\bm{k}},s}^{cv}}\bigg{)}+{\rm
P}\bigg{(}\frac{1}{2\hbar\omega+\varepsilon_{{\bm{k}},s}^{cv}}\bigg{)}\bigg{]}$
$\displaystyle-\frac{{\rm
Im}[{j}_{y,{\bm{k}},s}^{vc}~{}{j}_{x,{\bm{k}},s}^{cv}]|{j}_{x,{\bm{k}},s}^{vc}|^{2}}{\varepsilon_{{\bm{k}},s}^{vc}}\frac{f^{vc}_{{\bm{k}},s}}{\varepsilon_{{\bm{k}},s}^{vc}}\bigg{[}{\rm
P}\bigg{(}\frac{1}{\hbar\omega-\varepsilon_{{\bm{k}},s}^{cv}}\bigg{)}+{\rm
P}\bigg{(}\frac{1}{2\hbar\omega-\varepsilon_{{\bm{k}},s}^{cv}}\bigg{)}\bigg{]}\bigg{\\}}.$
(35)

On further simplifications and using the relations ${\rm
Im}[{j}_{y,{\bm{k}},s}^{cv}{j}_{x,{\bm{k}},s}^{vc}]=-{\rm
Im}[{j}_{y,{\bm{k}},s}^{vc}{j}_{x,{\bm{k}},s}^{cv}]$,
$f^{cv}_{{\bm{k}},s}=-f^{vc}_{{\bm{k}},s}=f^{c}_{{\bm{k}},s}-f^{v}_{{\bm{k}},s}$
and
$\varepsilon_{{\bm{k}},s}^{cv}=-\varepsilon_{{\bm{k}},s}^{vc}=\varepsilon_{{\bm{k}},s}^{c}-\varepsilon_{{\bm{k}},s}^{v}$,
Eq. (C) reduces to

$\displaystyle{\rm Im}[\chi_{yxxx}^{P,(3)}(\omega,\omega,-2\omega)]$
$\displaystyle=\sum_{{\bm{k}},s}\frac{{\rm
Im}[{j}_{y,{\bm{k}},s}^{cv}~{}{j}_{x,{\bm{k}},s}^{vc}]|{j}_{x,{\bm{k}},s}^{cv}|^{2}}{\varepsilon_{{\bm{k}},s}^{cv}}\frac{f^{cv}_{{\bm{k}},s}}{\varepsilon_{{\bm{k}},s}^{cv}}\bigg{\\{}\bigg{[}{\rm
P}\bigg{(}\frac{1}{\hbar\omega+\varepsilon_{{\bm{k}},s}^{cv}}\bigg{)}+{\rm
P}\bigg{(}\frac{1}{2\hbar\omega+\varepsilon_{{\bm{k}},s}^{cv}}\bigg{)}\bigg{]}$
$\displaystyle-\bigg{[}{\rm
P}\bigg{(}\frac{1}{\hbar\omega-\varepsilon_{{\bm{k}},s}^{cv}}\bigg{)}+{\rm
P}\bigg{(}\frac{1}{2\hbar\omega-\varepsilon_{{\bm{k}},s}^{cv}}\bigg{)}\bigg{]}\bigg{\\}}.$
(36)

In the regime $\hbar\omega\ll\Delta$, the third-order susceptibility yields

$\displaystyle{\rm
Im}[\chi_{yxxx}^{P,(3)}(\omega,\omega,-2\omega)]\approx\sum_{{\bm{k}},s}\frac{{\rm
Im}[{j}_{y,{\bm{k}},s}^{cv}~{}{j}_{x,{\bm{k}},s}^{vc}]|{j}_{x,{\bm{k}},s}^{cv}|^{2}}{(\varepsilon_{{\bm{k}},s}^{cv})^{2}}\frac{f^{cv}_{{\bm{k}},s}}{\varepsilon_{{\bm{k}},s}^{cv}}.$
(37)

Utilizing the definitions of the Berry curvature in the conduction band Xiao
_et al._ (2010), denoted as ${\Omega}^{c}_{{\bm{k}},s}$, and the valence band,
denoted as ${\Omega}^{v}_{{\bm{k}},s}$, we can write

$\displaystyle{\Omega}^{c}_{{\bm{k}},s}=-{\Omega}^{v}_{{\bm{k}},s}=2\left(\frac{\hbar}{e}\right)^{2}{\rm
Im}\frac{[{j}_{y,{\bm{k}},s}^{cv}~{}{j}_{x,{\bm{k}},s}^{vc}]}{(\varepsilon_{{\bm{k}},s}^{cv})^{2}},$
(38)

and the current matrix element in terms of the position vector matrix element
reads

$\displaystyle
r^{cv}_{a,{\bm{k}},s}=\frac{i\hbar}{e}\frac{j^{cv}_{a,{\bm{k}},s}}{\varepsilon^{cv}_{{\bm{k}},s}}.$
(39)

Therefore, the third-order spin Hall susceptibility becomes

$\displaystyle{\rm
Im}[\chi_{yxxx}^{P,(3)}(\omega,\omega,-2\omega)]\approx\frac{e^{4}}{2\hbar^{4}}\sum_{{\bm{k}}}{\Omega}^{c,{\rm
spin}}_{{\bm{k}}}\sum_{s=\pm}|r^{cv}_{x,{\bm{k}},s}|^{2}\varepsilon_{{\bm{k}},s}^{cv}f^{cv}_{{\bm{k}},s},$

where ${\Omega}^{c,{\rm
spin}}_{{\bm{k}}}={\Omega}^{c}_{{\bm{k}},\uparrow}-{\Omega}^{c}_{{\bm{k}},\downarrow}$.
Therefore, the third-order Hall conductivity yields

$\displaystyle\sigma_{yxxx}^{(3),P}(\omega,\omega,-2\omega)\approx
C(\omega)\sum_{{\bm{k}}}{\Omega}^{c,{\rm
spin}}_{{\bm{k}}}\sum_{s=\pm}|r^{cv}_{x,{{\bm{k}},s}}|^{2}\varepsilon_{{\bm{k}},s}^{cv}f^{cv}_{{\bm{k}},s},$
(40)

where $C(\omega)=e^{4}/2\hbar^{4}\omega^{3}$ for charge conductivity and
$C(\omega)=e^{3}/2\hbar^{4}\omega^{3}$ for spin conductivity. This is the
paramagnetic part of the third-order conductivity expression for the
contribution at below gap. On the other hand, the diamagnetic part of the spin
conductivity in the regime below the band gap yields

$\displaystyle\sigma_{yxxx}^{(3),D}(\omega,\omega,-2\omega)\approx\frac{C(\omega)\hbar^{2}}{e^{2}}\sum_{{\bm{k}}}{\Omega}^{c,{\rm
spin}}_{{\bm{k}}}\sum_{s=\pm}\kappa^{cv}_{xx,{\bm{k}},s}f^{cv}_{{\bm{k}},s}.$
(41)

The total third-order spin conductivity thus follows

$\displaystyle\sigma_{yxxx}^{(3)}(\omega,\omega,-2\omega)\approx
C(\omega)\sum_{{\bm{k}}}{\Omega}^{c,{\rm
spin}}_{{\bm{k}}}\sum_{s=\pm}\bigg{\\{}|r^{cv}_{x,{{\bm{k}},s}}|^{2}\varepsilon_{{\bm{k}},s}^{cv}+\tilde{\kappa}^{cv}_{xx,{\bm{k}},s}\bigg{\\}}f^{cv}_{{\bm{k}},s},$
(42)

where
$\tilde{\kappa}^{cv}_{xx,{\bm{k}},s}=(\hbar/e)^{2}\kappa^{cv}_{xx,{\bm{k}},s}$.
This derives the equation (15). Therefore, as expected, the nonlinear spin
Hall conductivity depends on the Berry curvature.