# Intrinsic anomalous spin Hall effect

Ping Li  State Key Laboratory for Mechanical Behavior of Materials, Center for
Spintronics and Quantum System, School of Materials Science and Engineering,
Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, China  Key Laboratory for
Computational Physical Sciences (Ministry of Education), Fudan University,
Shanghai, 200433, China  Jing-Zhao Zhang  State Key Laboratory for Mechanical
Behavior of Materials, Center for Spintronics and Quantum System, School of
Materials Science and Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi,
710049, China  Zhi-Xin Guo<EMAIL_ADDRESS>State Key Laboratory for
Mechanical Behavior of Materials, Center for Spintronics and Quantum System,
School of Materials Science and Engineering, Xi’an Jiaotong University, Xi’an,
Shaanxi, 710049, China  Key Laboratory for Computational Physical Sciences
(Ministry of Education), Fudan University, Shanghai, 200433, China  Tai Min
State Key Laboratory for Mechanical Behavior of Materials, Center for
Spintronics and Quantum System, School of Materials Science and Engineering,
Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, China  X. R. Wang
<EMAIL_ADDRESS>Physics Department, The Hong Kong University of Science and
Technology, Clear Water Bay, Kowloon, Hong Kong.  HKUST Shenzhen Research
Institute, Shenzhen 518057, China.

###### Abstract

Charge-spin interconversion in magnetic materials is investigated by using
first-principles calculations. In addition to the conventional spin Hall
effect (SHE) that requires mutual orthogonality of the charge current, spin-
flow direction, and spin polarization, the recently proposed anomalous SHE
(ASHE) is confirmed in Mn2Au and WTe2. The interaction of the order parameter
with conduction electrons leads to sizeable nonzero spin Berry curvatures that
give rise to nonzero anomalous spin Hall conductivity (ASHC). Our calculations
show that the ASHE is intrinsic and originates from the order-parameter-
controlled spin-orbit interaction, which generates an extra anomalous
effective field. A useful relationship among the order parameter, spin Berry
curvature and ASHC is revealed. Our findings open a new avenue for arbitrary-
type spin current generation and detection.

Keywords: anomalous spin Hall effect, spin Berry curvature, order parameter,
spin-orbital coupling, spin-dependent electric field

PACS: 71.10.-d, 74.25.Ha, 75.30-m, 75.40.Cx

## INTRODUCTION

The spin Hall effect (SHE) is related to spin current generation or spin
accumulation from an electric currentZhang ; MacDonald ; Jungwirth ; Hirsch .
Different from an electric current, which is a flow of charges and a vector, a
spin current, or an angular momentum flow, is a tensor of rank two with nine
components in Cartesian coordinates. The SHE has attracted great attention and
is widely used to generate a spin current, which is an indispensable component
of spintronics Zhang ; MacDonald ; Jungwirth ; Hirsch . The inverse SHE is a
powerful method for detecting spin currents Chien ; Deranlot .

In the SHE, a charge current along the $\hat{\beta}$ direction in a material
with a strong spin-orbit interaction (SOI) can generate a spin current
$j^{\gamma}_{\alpha}$ whose propagation and spin polarization are along the
$\hat{\alpha}$ and $\hat{\gamma}$ directions, respectively. The conventional
SHE requires $\hat{\alpha}$, $\hat{\beta}$, and $\hat{\gamma}$ to be mutually
orthogonal to each other. Recently, Wang proposed that interactions of
conduction electrons with order parameters, such as magnetization for
ferromagnets and the N$\acute{e}$el order for antiferromagnets, give rise to
the anomalous SHE (ASHE) Wang . In linear response theory, spin current
$j^{j}_{i}$ due to charge current $\bf J$ is given by

$j^{j}_{i}=\frac{\hbar}{2e}\theta^{SH}_{ijk}J_{k},$ (1)

where $\theta^{SH}_{ijk}$ is the spin Hall angle tensor of rank 3.
$i,j,k=1,2,3$ denote the x, y, and z directions, and the Einstein summation
notation is used. i, j, and k denote the spin current direction, spin
polarization direction, and charge current direction. Based on the general
tensor requirement of a physical quantity, one has Wang

$\begin{split}\theta^{SH}_{ijk}=\theta_{0}\epsilon_{ijk}+[(\theta_{1}+\theta_{2})\delta_{ij}\delta_{kl\neq
i}+\theta_{1}\delta_{ik}\delta_{jl\neq i}+\theta_{2}\delta_{il}\delta_{jk\neq
i}]M_{l},\end{split}$ (2)

where _M l_ and $\epsilon_{ijk}$ are the lth component of M and the Levi-
Civita symbols, respectively. $\theta_{0}$ is the usual spin Hall angle that
does not depend on M, while $\theta_{1}$ and $\theta_{2}$ are two ASHE
coefficients. Thus, the conventional SHE is nonzero only when $i,j,$ and $k$
take different values. However, the terms involving _M l_ can be nonzero,
which leads to nonzero ASHE spin currents, as indicated in Figs. 1f-1h. This
newly proposed ASHE was not only directly verified by experiments of the
anomalous spin torques generated in Mn2Au Pan  but also agreed with the
details of the observed magnetization-dependent anomalous inverse SHE (AISHE)
in a Pt/Co/Pt heterostructure Chuang , TbCo alloys Yagmur , and Co/Pd
multilayers Dinghai . The ASHE overcomes the limitation of the SHE, and the
generated spin current can be controlled by order parameters. In particular,
the ASHE allows the spin polarization and spin propagation directions to be
collinear when the applied electric current is along the order parameter
direction. Such a spin current can generate an out-of-plane anti-damping spin-
orbit torque (SOT) that is crucial to realizing field-free switching of
perpendicular magnetization in high-density SOT devices such as SOT-MRAM
Gaudin ; Buhrman ; Yang ; Ralph ; yin ; Chen .

Although the ASHE concept has been widely recognized, its microscopic origin
remains to be revealed. Recently, first-principles calculations Haney ; Felser
were used to study the anomalous spin torques observed in experiments
involving ferromagnets and antiferromagnets. However, the discussions are
limited to qualitative symmetry analysis Ebert , and the intrinsic mechanism
of the ASHE is still elusive Park . One cannot quantitatively compute the
anomalous spin torques of specific materials without knowing the intrinsic
mechanism. Such an investigation is particularly desirable in the field of
high-performance SOT devices Ralph ; Chen ; Park .

Using the local spin density approximation theory with the SOI, we investigate
the intrinsic mechanism of the ASHE, with the following findings. 1) The ASHE
originates from the order-parameter-induced extra spin-dependent electric
field, which generates anomalous spin currents via the SOI. 2) The anomalous
spin Hall conductivity (ASHC), which quantitatively represents the strength of
the ASHE, can be well described by the spin Berry curvature. 3) There is an
intrinsic connection among the order parameter, spin Berry curvature, and
ASHC.

## METHODS

Our spin-polarized density functional theory (DFT) calculations are performed
using the Vienna ab initio Simulation Package (VASP) 26 ; 27  and QUANTUM
ESPRESSO (QE) QE1 ; QE2 , in which the projector augmented wave method and a
plane-wave basis set are applied. We use the VASP for Mn2Au and WTe2 and QE
for Fe. All calculations are performed using spin polarization within the
framework of the local spin density approximation and generalized gradient
approximation for the exchange-correlation energy. The electron exchange-
correlation functional is described by the Perdew-Burke-Ernzerhof functional
of the generalized gradient approximation 28 . The plane-wave cutoff energy is
chosen to be 500 eV for Mn2Au and WTe2 and 120 Ry for Fe. Moreover,
$\Gamma$-centered $k$ meshes of $20\times 20\times 20$ (Pt), $16\times
16\times 16$ (Fe), $18\times 9\times 3$ (WTe2), and $24\times 24\times 8$
(Mn2Au) are adopted in the self-consistent calculations. To better describe
the van der Waals (vdW) interaction, the optB86b-vdW functional is adopted for
WTe2 29 ; 30 . Then, DFT wave functions are projected onto maximally localized
Wannier functions using the WANNIER90 package 31 , and the Kubo formula is
applied to calculate the spin Hall conductivity (SHC) Guo ; Zhao . Dense
k-point meshes of $150\times 150\times 150$, $600\times 600\times 600$,
$200\times 200\times 200$ and $150\times 150\times 150$ are employed for Pt,
Fe, WTe2 and Mn2Au, respectively, to perform the Brillouin zone (BZ)
integration for the SHC calculations with a careful convergence test.

## RESULTS

According to local spin density approximation theory LSDA1 ; LSDA2 , a
position-dependent spin-dependent electric field $\vec{E}^{p}(\vec{r})$, which
is proportional to the difference in the charge density gradients of spin-up
and spin-down electrons, [$\nabla n^{\uparrow}(\vec{r})$] and [$\nabla
n^{\downarrow}(\vec{r})$], can be induced by the magnetic order parameter $\bf
M$ (see Supporting Information). Note that the magnetization-induced spin-
dependent electric field differs from the conventional electric field. The
conventional electric field is defined as $\vec{E}(\vec{r})=-[\nabla
H^{\uparrow}(\vec{r})+\nabla H^{\downarrow}(\vec{r})]$, while the spin-
dependent electric field is defined as $\vec{E}^{p}(\vec{r})=-[\nabla
H^{\uparrow}(\vec{r})-\nabla H^{\downarrow}(\vec{r})]$, where $H^{\uparrow}$
and $H^{\downarrow}$ are the Hamiltonians for spin-up and spin-down electrons,
respectively. Hence, the spin-dependent electric field represents the
difference in the electric fields between spin-up and spin-down electrons.

In a collinear magnetic structure with inversion symmetry, the effective spin-
dependent electric field (averaged over the unit cell) defined as
$\vec{E}_{eff}^{p}=\int\vec{E}^{p}(\vec{r})d\vec{r}=0$ is zero. However,
$\vec{E}_{eff}^{p}$ becomes nonzero if the inversion symmetry is broken. Such
an $\vec{E}_{eff}^{p}$ can induce an effective Hamiltonian $H^{eff}_{SOC}$ via
the SOI (see Supporting Information),

$H^{eff}_{SOC}=\frac{\hbar
e}{2mc^{2}}\left(\vec{\sigma}\times\vec{E}_{eff}^{p}\right)\cdot\vec{v},$ (3)

where $\vec{\sigma}$ and $\vec{v}$ are the Pauli matrices and electron
velocity, respectively. The nonzero $H^{eff}_{SOC}$ means that the spin-up and
spin-down electrons experience different forces and thus have different
velocities. Hence, Eq. (3) leads directly to the ASHE as long as
$\vec{\sigma}$ is perpendicular to $\vec{E}_{eff}^{p}$. For example, in a
magnetic material without inversion symmetry in the $\vec{z}$ direction, i.e.,
$E_{eff}^{p_{z}}\neq 0$, one has nonzero $H^{eff}_{SOC}=\frac{\hbar
e}{2mc^{2}}E_{eff}^{p_{z}}\left({\sigma_{y}}v_{x}-\sigma_{x}v_{y}\right)$.
Thus, energy splitting between spin-up and spin-down electrons is generated by
the SOI, leading to different velocities. As a result, anomalous spin currents
with spin polarization perpendicular to $\vec{z}$, as predicted in Wang’s
theory Wang , will be generated (see Supporting Information). Note that since
the spin-dependent electric field directly correlates to the magnetic order
parameter $\bf M$, nonzero $E_{eff}^{p}$ is naturally expected in the
direction of broken inversion symmetry induced by $\bf{M}$, rather than in the
direction of the magnetic moment.

We find that the strength of the ASHE can be quantitatively described by the
spin Berry curvature. According to the Kubo formula, the SHC
$\sigma^{\gamma}_{\alpha\beta}$ of spin polarization direction $\hat{\gamma}$
and spin flow direction $\hat{\alpha}$ under an external electric field
(charge current) along $\hat{\beta}$ can be expressed as Guo ; Zhao ; Luqi ;
Leandro

$\sigma^{\gamma}_{\alpha\beta}=-\frac{e^{2}}{\hbar}\frac{1}{VN^{3}_{k}}\sum_{k}\sum_{n}f_{nk}\Omega^{\gamma}_{n,\alpha\beta}(\bf{k}),$
(4)

where V is the cell volume and $N^{3}_{k}$ is the number of $k$ points in the
BZ. $f_{nk}$ represents the Fermi-Dirac distribution function, and
$\Omega^{\gamma}_{n,\alpha\beta}(\emph{{k}})$ is the band-projected spin Berry
curvature, defined as

$\Omega^{\gamma}_{n,\alpha\beta}({\bf k})=\hbar^{2}\sum_{m\neq
n}\frac{-2\mathrm{Im}[\left\langle n{\bf
k}\mid\frac{1}{2}\\{\hat{\sigma}_{\gamma},\hat{\upsilon}_{\alpha}\\}\mid m{\bf
k}\right\rangle\left\langle m{\bf k}\mid\hat{\upsilon}_{\beta}\mid n{\bf
k}\right\rangle]}{(\epsilon_{n\bf k}-\epsilon_{m\bf k})^{2}}.$ (5)

From Eqs. (4) and (5), one can see that the Hall conductivity
$\sigma^{\gamma}_{\alpha\beta}$ is intrinsically determined by the band
structure through the spin Berry curvature
$\Omega^{\gamma}_{n,\alpha\beta}(\bf{k})$ and density of states. In the
conventional SHE, nonzero $\Omega^{\gamma}_{n,\alpha\beta}(\emph{{k}})$ is
expected only when $\alpha$, $\beta$ and $\gamma$ are mutually orthogonal to
each other. However, as discussed below for the case involving the order
parameter, nonzero $\Omega^{\gamma}_{n,\alpha\beta}(\emph{{k}})$ and thus a
sizeable ASHE can also be obtained when only two of them are mutually
perpendicular. In addition, the general Berry curvature formula, which
requires $\alpha\neq\beta$, was derived by simply considering the spin
direction perpendicular to the velocities of the spin current and charge
current ($\alpha\neq\beta\neq\gamma$) Niu . In this case, the spin degree of
freedom is not effectively considered, and the Berry curvature is in fact a
tensor of rank 2. However, in the definition of spin Berry curvature shown in
Eq. (5), the interaction between the spin degree of freedom and spin current
is effectively considered via the anticommutator operator. As a result, the
spin Berry curvature becomes a tensor of rank 3, which is nonzero even for
$\alpha=\beta$.

Figure 1:  Atomic structures, schematic diagrams of the BZs, and schematic
diagrams of the ASHE. (a), (c) and (d) show the atomic structures of TET
Mn2Au, BCC Fe and FCC Pt, respectively. (b) and (e) show the corresponding
BZs. Note that BCC Fe and FCC Pt have the same BZ structure, shown in (e). (f)
illustrates the ASHE with charge current $\vec{J}$ parallel to order parameter
$\bf M$, and (g, h) present the ASHE with charge current $\vec{J}$
perpendicular to $\bf M$.

We first consider tetragonal (TET) Mn2Au as an illustration, whose nonzero
ASHE has been experimentally observed Pan . We systematically show below an
intrinsic connection among the order parameter, spin Berry curvature and ASHE.
DFT calculations were performed to obtain the spin Berry curvature and SHC
(see Supporting Information). The crystal structure and schematic BZ diagram
of Mn2Au are illustrated in Figs. 1a and 1b, which verify that Mn2Au is an
antiferromagnetic material with a N$\acute{e}$el order parameter $\bf M$. In
this study, $\bf M$ is fixed in the $\vec{x}$ direction. Hence, as indicated
in Figs. 1e-1g, six nonzero ASHCs are expected, that is, $\sigma^{y}_{yx}$ and
$\sigma^{z}_{zx}$ for charge current $\bf J$ parallel to $\bf M$ and
$\sigma^{x}_{zz}$, $\sigma^{z}_{xz}$, $\sigma^{x}_{yy}$, and $\sigma^{y}_{xy}$
for $\bf J$ perpendicular to $\bf M$ Wang .

The calculated components of both the conventional SHC (CSHC) and ASHC are
shown in Table I. In addition to the large CSHCs, there are also sizeable
ASHCs in Mn2Au. Moreover, both CSHCs and ASHCs exhibit significant anisotropy,
depending on the combination of the spin current, spin polarization, and
charge current directions. This characteristic is quite different from that of
the conventional SHE, whose anisotropy arises solely from the crystalline
structure Ebert . For instance, the symmetry of Mn2Au is expected to give rise
to the same CSHCs for $\sigma^{z}_{xy}$ and $\sigma^{z}_{yx}$. However, the
former [270.58 $(\hbar/e)S/cm$] is more than 3 times larger than the latter
[-81.11 $(\hbar/e)S/cm$] because of the broken fourfold rotational symmetry
due to $\bf M$ along the $\vec{x}$ direction.

Moreover, $\bf M$ along the $\vec{x}$ direction in Mn2Au can lead to an
additional nonzero spin-dependent electric field in the $\vec{z}$ direction
($E^{p}_{z}$) due to its broken inversion symmetry. According to Eq. (3), the
nonzero $E^{p}_{z}$ would induce a spin current with spin polarization in both
the $\vec{x}$ and $\vec{y}$ directions via the SOI and thus leads to sizeable
ASHCs. The details are given in the Supporting Information. Our DFT
calculations indeed show that the absolute values of ASHCs with spin
polarization in the $\vec{x}$ direction ($\sigma^{x}_{zz}$, $\sigma^{x}_{yy}$)
and $\vec{y}$ direction ($\sigma^{y}_{yx}$, $\sigma^{y}_{xy}$) are
significantly larger than those with spin polarization in the $\vec{z}$
direction ($\sigma^{z}_{zx}$, $\sigma^{z}_{xz}$) (Table I). The nonzero values
of $\sigma^{z}_{zx}$ and $\sigma^{z}_{xz}$ may be from other ASHE mechanisms
that are beyond the scope of effective field theory discussed above. Among the
six ASHC components, the largest component is $\sigma^{y}_{yx}$ [15.57
$(\hbar/e)S/cm$], the magnitude of which reaches 19% of that of its
conventional counterpart $\sigma^{z}_{yx}$ [-81.11 $(\hbar/e)S/cm$]. Thus, a
sizeable spin current, whose propagation direction is collinear to the spin
polarization direction, can be effectively generated in Mn2Au. This result is
consistent with the experiments Pan . The good agreement between our
theoretical derivations and DFT calculations verifies the intrinsic connection
among the order parameter, spin Berry curvature and ASHE.

Table 1:  Spin Hall Conductivity. Calculated CSHCs and ASHCs for Fe, Mn2Au and Pt. Each SHC is denoted as $\sigma^{\gamma}_{\alpha\beta}$, where $\alpha$ is the direction of the spin current, $\beta$ is that of the applied electrical field and $\gamma$ is that of the spin polarization of the spin current (unit: $(\hbar/e)S/cm$). The magnetic order parameter $\bf M$ is fixed to be along the $\vec{x}$ direction for Fe and Mn2Au. | SHC | Mn2Au | Fe | Pt
---|---|---|---|---
| $\sigma^{z}_{xy}$ | 270.58 | 120.76 | 2239.42
CSHC | $\sigma^{z}_{yx}$ | -81.11 | -683.71 | -2239.58
| $\sigma^{y}_{yx}$ | 15.57 | 2.87 | 0.00
| $\sigma^{z}_{zx}$ | 0.02 | -2.89 | 0.00
| $\sigma^{x}_{zz}$ | -10.46 | -0.66 | 0.00
ASHC | $\sigma^{z}_{xz}$ | -2.95 | -3.14 | 0.00
| $\sigma^{x}_{yy}$ | -6.72 | 0.49 | 0.00
| $\sigma^{y}_{xy}$ | -9.09 | 1.37 | 0.00

As a comparison, we also calculated the SHCs of bulk Fe in the body-centered
cubic (BCC) structure. As shown in Fig. 1(c), although a large $\bf M$ exists
along the $\vec{x}$ direction, $\vec{E}^{p}$ in this system is zero due to the
inversion symmetry (see Supporting Information). This means that the order
parameter does not directly induce the ASHE in BCC Fe. Our DFT calculations
show that despite its considerably large CSHCs, all the ASHCs in Fe are very
small, with an ASHC/CSHC ratio of less than 0.5% (Table I). We also explored
the SHCs of nonmagnetic Pt in the face-centered cubic (FCC) structure. As
shown in Table I, although the CSHCs of Pt are larger than those of Mn2Au by
an order of magnitude, all its ASHCs are negligible [$\textless 10^{-3}$
$(\hbar/e)S/cm$]. This result confirms the intrinsic connection between the
order parameter and ASHE.

Figure 2: Spin Berry curvature in BZ slices. Density plots of the spin Berry
curvature in the BZ slice of kz = 0 for bulk Mn2Au (a, b) and Fe (c, d) and
the 2D BZ of kx = 0 for Pt (e, f). (a), (c), and (e) are the spin Berry
curvature for CSHCs: $\sigma^{z}_{yx}$ of Mn2Au, $\sigma^{z}_{xy}$ of Fe, and
$\sigma^{z}_{xy}$ of Pt, respectively. (b), (d), and (f) are the spin Berry
curvature for ASHCs: $\sigma^{y}_{yx}$ of Mn2Au (b), $\sigma^{y}_{yx}$ of Fe
(d), and $\sigma^{y}_{yx}$ of Pt (f). The black lines denote the intersections
of the Fermi surface with the slices. The color code is on the log scale.

As discussed in the Supporting Information, the exchange-correlation potential
between spin-up and spin-down electrons in the Kohn-Sham equations depends on
the magnetic order parameter (see Supporting Information). Thus, the spin-
polarized DFT calculations under the local spin density approximation can be
used to study order parameter effects on exchange interactions and SOIs LSDA1
; LSDA2 . Hence, the calculated electronic structures, such as band structures
and wave functions, contain the effects of the order parameter. Since the spin
Berry curvature is directly calculated on the basis of electronic structures
[Eq. (5)], this shows an intrinsic connection of the ASHE to the spin Berry
curvature. In fact, the intrinsic connection between the spin Berry curvature
and conventional SHE was also well justified in previous studies Yao ; Nagaosa
; Niu .

We further computed the $k$-resolved spin Berry curvatures for both CSHCs (a,
c, e in Fig. 2) and ASHCs (b, d, f in Fig. 2) of bulk Mn2Au, Fe, and Pt. A
common feature is that the spin Berry curvatures for ASHCs are much smaller
than those for CSHCs, which agrees with the experimental fact that the ASHCs
in these materials are smaller than the CSHCs. In addition, positive and
negative spin Berry curvatures coexist in the BZ, and both CSHC and ASHC
values are determined by the sum of all $k$ points. Thus, although the spin
Berry curvature value of Fe is larger than that of Mn2Au throughout the BZ,
its ASHC is much smaller than that of Mn2Au due to the cancellation of
positive and negative spin Berry curvatures. This result also agrees well with
the symmetry analysis above. For Pt, the spin Berry curvatures for the CSHC
are dominated by regions with large positive values, while those for ASHC are
tiny with equal weights on the positive and negative values, leading to the
largest CSHC and a negligible ASHC in comparison with those of Mn2Au and Fe.
The calculated Hall conductivities shown in Table I are in good agreement with
these spin Berry curvatures profiles. The above results clearly show that the
spin Berry curvature is intrinsically connected not only to the conventional
SHE but also to the ASHE.

Figure 3: Atomic structure, BZ, and spin Berry curvature of WTe2. Atomic
structure (a), schematic diagram of the BZ (b), and $k$-resolved spin Berry
curvature on a log scale in a slice of the 2D BZ at ky = 0 (C) for bulk WTe2.
The left and right panels in (c) show the spin Berry curvature for
$\sigma^{z}_{zx}$ and $\sigma^{z}_{zy}$, respectively. The black lines show
the intersections of the Fermi surface with a slice of the BZ.

## DISCUSSION AND CONCLUSION

The ASHE is not limited to the magnetic order parameter; any order parameter
breaking the inversion symmetry can lead to nonzero ASHC via the SOI (such
effect in nonmagnetic materials is denoted as the unconventional SHE). For
example, the crystal structure of nonmagnetic material Td-WTe2 [Fig. 3(a)]
breaks the mirror symmetry with respect to the xz plane and thus leads to
broken inversion symmetry. The broken symmetry in this case induces a nonzero
conventional electric field in the $\vec{y}$ direction ($E_{eff}^{y}$).
According to the SOI Hamiltonian presented in Eq. (3), a sizable
unconventional SHE spin current can still be generated when the spin
polarization direction is perpendicular to $\vec{y}$ (see Supporting
Information). To verify this, we performed DFT calculations on the spin Berry
curvature and ASHC of WTe2 and found $\sigma^{z}_{zx}$=18.99 $(\hbar/e)S/cm$,
reaching 18% of its conventional SHC counterpart $\sigma^{y}_{zx}$ [103
($\hbar/e$)S/cm] Zhao11 . However, we found that $\sigma^{z}_{zy}$ has a much
smaller value [2.08 $(\hbar/e)S/cm$], which may be attributed to the indirect
interaction between $E_{eff}^{y}$ and $v_{y}$. This result is consistent with
experimental observations, where field-free switching of perpendicular
magnetization was observed only with charge current in the $\vec{x}$ direction
and not in the $\vec{y}$ direction Ralph ; Dash ; Singh . In addition, as
shown in Fig. 3, the positive spin Berry curvature for $\sigma^{z}_{zx}$ is
much larger than that for $\sigma^{z}_{zy}$. This result confirms the
intrinsic connection between the spin Berry curvature and unconventional SHE
in nonmagnetic materials.

In addition, we would like to discuss the novelty of this work in comparison
with previous studies Ebert ; Tomohiro ; Das ; Haney ; Qu . The notion of the
ASHE was first proposed for spin current generation in a ferromagnet from an
AHE-induced charge current through the conventional SHE (a phenomenon of
SHE+AHE) Tomohiro ; Das ; Haney . However, the ASHE in our study resulted from
the combined effect of the order parameter and SOI. The physical origins of
the two theories are fundamentally different. Moreover, the two theories make
distinct predictions: The ASHE in Refs. Tomohiro, ; Das, ; Haney,  predicts a
nonzero spin current only for the noncollinear magnetic order parameter and
external electric field. In contrast, the ASHE in this work additionally
predicts that the collinear order parameter and charge current generate a spin
current with collinear propagation direction and polarization. Unconventional
components of the SHC tensor were also discussed in Ref. Ebert,  from the
symmetry restriction point of view. Their analysis is valid only when the
order parameter is not involved in spin-charge conversion, in contradiction to
the ASHE in this work. In fact, the analysis of Ref. Ebert,  fails to give
rise to the ASHC component in Mn2Au observed in experiments Pan . Note also
that order parameter dependence of the conventional SHE and AHE in
ferromagnets was recently observed in ferromagnets Qu .

Before closing, we would like to emphasize that a dense k-point mesh setting
is necessary to guarantee the reliability of the SHC and ASHC calculations.
For example, in Fig. S1 of the Supporting Information, we show the ASHC
$\sigma^{y}_{xy}$ of Fe calculated with various k-point mesh settings.
Clearly, $\sigma^{y}_{xy}$ is as large as 183.48 $(\hbar/e)S/cm$ for a small
k-point mesh of $50\times 50\times 50$ while its converged value is
approximately 1.0 $(\hbar/e)S/cm$, which can be obtained when the k-point mesh
is larger than $400\times 400\times 400$. Therefore, the k-point mesh
convergence test is crucial to the Hall conductivity calculations.

In summary, we have revealed the nature of the intrinsic ASHE: The ASHE
originates from the order-parameter-induced spin-dependent electric field,
which generates a spin current via the SOI. The intrinsic relationship among
the order parameter, spin Berry curvature and ASHE is also revealed. The
effects of the order parameter on the electronic structure lead to a nonzero
spin Berry curvature and thus a sizeable ASHE. The order parameter that gives
rise to the ASHE is not limited to the magnetic order parameter. Any order
parameter that can lead to broken inversion symmetry should contribute to the
ASHE. This study is expected to provide an efficient way to search for and
optimize materials with large ASHE, which is particularly important in the
design of high-performance SOT spintronic devices.

We thank Prof. Jian Zhou and Dr. Yongliang Shi for valuable discussions. This
work is supported by the National Natural Science Foundation of China (No.
12074301, 12004295, and 11974296) and Hong Kong RGC Grants (No. 16300522,
16301619, and 16302321). P. Li thanks China’s Postdoctoral Science Foundation
funded project (No. 2020M673364) and the Open Project of the Key Laboratory of
Computational Physical Sciences (Ministry of Education). This research used
the resources of the HPCC platform in Xi’an Jiaotong University.

## Supporting Information

The supporting information is available online. The supporting materials are
published as submitted, without typesetting or editing. The responsibility for
scientific accuracy and content remains entirely with the authors.

## References

  * (1) S. Murakami, N. Nagaosa, and S. C. Zhang, _Science_ 301, 1348 (2003).
  * (2) J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, _Phys. Rev. Lett._ 92, 126603 (2004).
  * (3) J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, _Rev. Mod. Phys._ 87, 1213 (2015).
  * (4) J. E. Hirsch, _Phys. Rev. Lett._ 83, 1834 (1999).
  * (5) B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, _Phys. Rev. Lett._ 111, 066602 (2013).
  * (6) J. C. Rojas-Sanchez, N. Reyren, P. Laczkowski, W. Savero, J. P. Attane, C. Deranlot, M. Jamet, J. M. George, L. Vila, and H. Jaffres, _Phys. Rev. Lett._ 112, 106602 (2014).
  * (7) X. R. Wang, _Commun. Phys._ 4, 55 (2021).
  * (8) X. Chen, S. Shi, G. Shi, X. Fan, C. Song, X. Zhou, H. Bai, L. Liao, Y. Zhou, H. Zhang, A. Li, Y. Chen, X. Han, S. Jiang, Z. Zhu, H. Wu, X. Wang, D. Xue, H. Yang, and F. Pan, _Nat. Mater._ 20, 800 (2021).
  * (9) T. C. Chuang, D. Qu, S. Y. Huang, and S. F. Lee, _Phys. Rev. Research_ 2, 032053(R) (2020).
  * (10) A. Yagmur, S. Sumi, H. Awano, and K. Tanabe, _Phys. Rev. B_ 103, 214408 (2021).
  * (11) M. Yang, B. Miao, J. Cheng, K. He, X. Yang, Y. Zeng, Z. Wang, L. Sun, X. Wang, A. Azevedo, S. Bedanta, and H. Ding, _Phys. Rev. B_ 105, 224426 (2022).
  * (12) M. Cubukcu, O. Boulle, M. Drouard, K. Garello, C. A. Onur, I. M. Miron, J. Langer, B. Ocker, P. Gambardella, and G. Gaudin, _Appl. Phys. Lett._ 104, 042406 (2014).
  * (13) L. Liu, C. F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, _Science_ 336, 555 (2016).
  * (14) D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, R. A. Buhrman, J. Park, and D. C. Ralph, _Nat. Phys._ 13, 330 (2017).
  * (15) R. Ramaswamy, J. M. Lee, K. Cai, and H. Yang, _Appl. Phys. Rev._ 5, 031107 (2018).
  * (16) L. Liu, C. Zhou, X. Shu, C. Li, T. Zhao, W. Lin, J. Deng, Q. Xie, S. Chen, J. Zhou, R. Guo, H. Wang, J. Yu, S. Shi, P. Yang, S. Pennycook, A. Manchon, and J. Chen, _Nat. Nanotechnol._ 16, 277 (2021).
  * (17) Y. Zhang, H. Y. Yuan, X. S. Wang, and X. R. Wang, _Phys. Rev. B_ 97, 144416 (2018).
  * (18) Y. Zhang, Y. Sun, H. Yang, J. Železnỳ, S. P. P. Parkin, C. Felser, and B. Yan, _Phys. Rev. B_ 95, 075128 (2017).
  * (19) V. P. Amin, J. Li, M. Stiles, and P. Haney, _Phys. Rev. B_ 99, 220405(R) (2019).
  * (20) M. Seemann, D. Ködderitzsch, S. Wimmer, and H. Ebert, _Phys. Rev. B_ 92, 155138 (2015).
  * (21) S. Lee, J. Ryu, and B.-G. Park, _Nat. Nanotechnol._ 16, 226 (2021).
  * (22) G. Kresse, and J. Hafner, _Phys. Rev. B_ 47, 558 (1993).
  * (23) G. Kresse, and D. Joubert, _Phys. Rev. B_ 59, 1758 (1999).
  * (24) P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso, S. Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Koalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. Seitsonen, A. Smogunov, P. Umari, and R. Wentzcovitch, _J. Phys.: Condens. Matter_ 21, 395502 (2009).
  * (25) P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau. M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Cococcioni, N. Colonna, I. Carnimeo, A. Corso, S. Gironcoli, P. Delugas, R. DiStasio Jr, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H-Y Ko, A. Kokalj, E. Kucukbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. Nguyen, H-V Nguyen, A. Otero-de-la-Roza, L. Paulatto, S. Ponce, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. Seitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu, and S. Baroni, _J. Phys.: Condens. Matter_ 29, 465901 (2017).
  * (26) J. P. Perdew, K. Burke, and M. Ernzerhof, _Phys. Rev. Lett._ 77, 3865 (1996).
  * (27) M. Dion, H. Rydberg, E. Schroder; D. C. Langreth, and B. I. Lundqvist, _Phys. Rev. Lett._ 92, 246401 (2014).
  * (28) J. Klimes, D. R. Bowler, and A. Michaelides, _Phys. Rev. B_ 83, 195131 (2011).
  * (29) G. Pizzi, V. Vitale, R. Arita, S. Blugel, F. Freimuth, G. Geranton, M. Gibertini, D. Gresch, C. Jognson, T. Koretsune, J. I. Azpiroz, H. Lee, J. M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, S. Ponce, T. Ponweiser, J. Qiao, F. Thole, S. S. Tsirkin, M. Wierzbowska, N. Marzari, D. Vanderbilt, I. Souza, A. A. Mostofi, and J. R. Yates, _J. Phys.: Condens. Matter_ 32, 165902 (2020).
  * (30) G. Y. Guo, Y. Yao, and Q. Niu, _Phys. Rev. Lett._ 94, 226601 (2005).
  * (31) J. Qiao, J. Zhou, Z. Yuan, and W. Zhao, _Phys. Rev. B_ 98, 214402 (2018).
  * (32) R. Zeller, _Comput. Nanosci._ 31, 419 (2006).
  * (33) H.-A. Engle, E. I. Rashba, B. I. Halperin, in _Handbook of Magnetism and Advanced Magnetic Materials_ edited by H. Kronm$\ddot{u}$ller and S. Parkin (Wiley, Chichester, UK, 2007), Vol. V, pp. 3-4.
  * (34) Q. Lu, P. Li, Z. Guo, G. Dong, B. Peng, X. Zha, T. Min, Z. Zhou, and M. Liu, _Nat. Commun._ 13, 1650 (2022).
  * (35) L. Salemi, and P. M. Oppeneer, _Phys. Rev. B_ 106, 024410 (2022).
  * (36) D. Xiao, M. C. Chang, and Q. Niu, _Rev. Mod. Phys._ 82, 1959 (2010).
  * (37) Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova, T. Jungwirth, D. S. Wang, E. Wang, and Q. Niu, _Phys. Rev. Lett._ 92, 037204 (2004).
  * (38) N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, _Rev. Mod. Phys._ 82, 1539 (2010).
  * (39) J. Zhou, J. Qiao, A. Bournel, and W. Zhao, _Phys. Rev. B_ 99, 060408(R) (2019).
  * (40) B. Zhao, B. Karpiak, D. Khokhriakov, A. Johansson, A. M. Hoque, X. Xu, Y. Jiang, I. Mertig, and S. P. Dash, _Adv. Mater._ 32, 2000818 (2020).
  * (41) I. Kao, R. Muzzio, H. Zhang, M. Zhu, J. Gobbo, D. Weber, R. Rao, J. Li, J. H. Edgar, J. E. Goldberger, J. Yan, D. G. Mandrus, J. Hwang, R. Cheng, J. Katoch, and S. Singh, _arXiv:_ 2021.12388 (2021).
  * (42) T. Taniguchi, J. Grollier, and M. D. Stiles, _Phys. Rev. Appl._ 3, 044001 (2015).
  * (43) K. S. Das, W. Y. Schoemaker, B. J. van Wees, and I. J. Vera-Marun, _Phys. Rev. B_ 96, 220408(R) (2017).
  * (44) G. Qu, K. Nakamura, and M. Hayashi, _Phys. Rev. B_ 102, 144440 (2020).