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# Spin anomalous-Hall unidirectional magnetoresistance

M. Mehraeen  Department of Physics, Case Western Reserve University,
Cleveland, OH 44106, USA  Steven S.-L. Zhang  Department of Physics, Case
Western Reserve University, Cleveland, OH 44106, USA

###### Abstract

We predict a spin anomalous-Hall unidirectional magnetoresistance (AH-UMR) in
conducting bilayers composed of a ferromagnetic layer and a nonmagnetic layer,
which does not rely on the spin Hall effect in the normal metal layer—in stark
contrast to the well-studied unidirectional spin-Hall magnetoresistance—but,
instead, arises from the spin anomalous Hall effect in the ferromagnetic
layer. Physically, it is the charge-spin conversion induced by the spin
anomalous Hall effect that conspires with the structural inversion asymmetry
to generate a net nonequilibrium spin density in the ferromagnetic layer,
which, in turn, modulates the resistance of the bilayer when the direction of
the applied current or the magnetization is reversed. The dependences of the
spin AH-UMR effect on materials and geometric parameters are analyzed and
compared with other nonlinear magnetoresistances. In particular, we show that,
in magnetic bilayers where spin anomalous Hall and spin Hall effects are
comparable, the overall UMR may undergo a sign change when the thickness of
either layer is varied, suggesting a scheme to quantify the spin Hall or spin
anomalous Hall angle via a nonlinear transport measurement.

Suggested keywords

##  I Introduction

Originating from the interplay between magnetism and relativistic spin-orbit
interaction, the anomalous Hall (AH) effect in solid-state systems with broken
time-reversal symmetry has been of enduring interest for more than a century
[1, 2]. One class of materials that has received particular attention in
studies of this effect are conducting ferromagnets [2], such as ferromagnetic
metals.

The AH effect in ferromagnetic metals has several salient properties. Due to
the coupling of spin and orbital degrees of freedom, the effect not only
generates a transverse charge current—which is perpendicular to both the
magnetization and the applied electric field, but also gives rise to a
transverse spin current. And, in ferromagnetic metals with strong exchange
interaction, conduction-electron spins are well aligned with the local
magnetization, making the coupled spin and charge currents controllable by
varying the direction of the magnetization. Furthermore, the mobility of
conduction electrons in a ferromagnetic metal is, in general, spin-dependent,
enabling mutual conversion between spin and charge currents mediated by the AH
effect.

These properties have been shown to spawn unconventional magnetoresistances in
the linear response regime. For instance, both a bulk anisotropic
magnetoresistance and planar Hall resistance may result from two consecutive
transverse scatterings of spin-polarized conduction electrons, due to the AH
effect [3]. In geometrically confined systems—such as ferromagnetic-metal thin
films or layered structures, the anomalous-Hall induced anisotropic
magnetoresistances may acquire distinctive angular dependences, owing to the
modulation of the bulk spin and charge currents caused by interfacial spin
accumulation and the resulting diffusive spin current [4, 5, 6, 7, 8, 9].

In the nonlinear response regime—where the Onsager reciprocal relations no
longer hold, the role of the AH effect is yet to be explored. In this work, we
unveil a unidirectional magnetoresistance (UMR) driven by the spin AH effect
in conducting ferromagnet$\mid$nonmagnet bilayer systems, whereby the
resistance can be altered by reversing the direction of either the
magnetization or the applied electric field. Hereafter, we shall refer to this
nonlinear magnetoresistance as spin anomalous-Hall unidirectional
magnetoresistance (AH-UMR).

The underlying physics of the spin AH-UMR can be understood as follows. In a
single ferromagnetic-metal layer, the spin current induced by the spin AH
effect creates spin accumulations of opposite orientations at the top and
bottom surfaces, but there is no net nonequilibrium spin density due to
inversion symmetry, as shown schematically in Fig. 1a. This, however, is no
longer the case when a nonmagnetic-metal layer is attached to the
ferromagnetic-metal layer, as the spin accumulation at the interface would
“leak” into the nonmagnetic layer, resulting in a net nonequilibrium spin
density in the ferromagnetic layer, which conspires with the spin asymmetry in
electron mobility to produce the spin AH-UMR effect, as depicted in Figs. 1b
and 1c.

There is a key difference between the spin AH-UMR and other types of UMR
effects previously studied in various magnetic systems [10, 11, 12, 13, 14,
15, 16, 17, 18, 19]: for the spin AH-UMR, the nonequilibrium spin density in
the presence of an external electric field emanates from the spin AH effect in
the ferromagnetic layer itself, whereas for other UMRs—such as the
unidirectional spin Hall magnetoresistance (USMR) [10] and the unidirectional
Rashba magnetoresistance [18]—the nonequilibrium spin density is engendered by
the spin Hall (SH) effect in the nonmagnetic layer [10, 11, 12, 13, 15, 19] or
by the Rashba-Edelstein effect due to the spin-momentum locked surface states
[14, 16, 17, 18, 20].

Figure 1: Schematics of the spin AH-UMR effect. (a) In a single ferromagnetic-
metal (FM) layer, the spin accumulation $\mu_{s,F}$ (indicated by the small
yellow arrows) has an antisymmetric distribution about the center line of the
layer, with no net nonequilibrium spin density (spatially-averaged
$\mu_{s,F}$) induced. So the overall spin AH-UMR is also zero. (b) The
presence of a neighboring nonmagnetic-metal (NM) layer induces a net
nonequilibrium spin density $\delta\mu_{s,F}$ (indicated by the large yellow
arrow) in the FM layer, giving rise to a finite spin AH-UMR. (c) Reversing the
electric field direction flips the direction of the AH current
$\mathbf{j}_{\text{AH}}$ and thus the sign of $\delta\mu_{s,F}$, thereby
changing the sign of the spin AH-UMR.

In what follows, we will first examine the coupled nonlinear transport of spin
and charge induced by the spin AH effect in a ferromagnetic metal that is
adjacent to a nonmagnetic-metal layer with negligible SH effect. Based on a
generalized drift-diffusion model, an analytical expression of the UMR
coefficient—a proper characterization of the nonlinear transport
phenomenon—will be derived, which reveals the dependences of the spin AH-UMR
effect on specific materials and geometric parameters of the bilayer. We will
then generalize our results to bilayer structures comprised of a ferromagnetic
metal and a heavy metal, wherein both the spin AH-UMR and USMR are present.
And we predict, for particular choices of materials combinations, that the
total UMR would exhibit a sign reversal when the thickness of either the
ferromagnetic-metal or the heavy-metal layer is varied, which suggests a
scheme to quantify the SH and spin AH angles experimentally through a UMR
measurement. We will conclude with some materials considerations on both
direct and indirect detections of the spin AH-UMR as well as the enhancement
of the effect.

##  II Spin-dependent drift-diffusion model

Consider a thin-film ferromagnetic layer located at $0<z<d_{F}$ placed on top
of a nonmagnetic layer at $-d_{N}<z<0$, as shown in Figs. 1b,c. Applying an
in-plane electric field $\mathbf{E}$, the coupled drift-diffusion equations
for charge and spin densities—taking into account the AH and the SH effects
[21] in the ferromagnetic layer—can be written as [22, 3, 23]

$\displaystyle j_{i}$
$\displaystyle=\sigma\left(\mathcal{E}_{i}+p_{\sigma}\mathcal{E}^{s}_{ij}m_{j}\right)-\theta_{\text{SH}}^{I}\epsilon_{ijk}\mathcal{J}_{jk}+\theta_{\text{SH}}^{A}\epsilon_{ijk}m_{j}m_{l}\mathcal{J}_{kl}\,,$
(1a) $\displaystyle\mathcal{J}_{ij}$
$\displaystyle=\sigma\left(\mathcal{E}^{s}_{ij}+p_{\sigma}\mathcal{E}_{i}m_{j}\right)+\theta_{\text{SH}}^{I}\epsilon_{ijk}j_{k}+\theta_{\text{SH}}^{A}\epsilon_{ilk}m_{j}m_{l}j_{k}\,,$
(1b)

where $j_{i}(z)$ and $\mathcal{J}_{ij}(z)$ are the local charge and spin
current densities 111By assuming spatial homogeneity in the $xy$ plane, all
local quantities will have only a $z$ dependence., respectively, with the
index $i$ indicating the direction of momentum flow and $j$ the spin
polarization direction. $\theta_{\text{SH}}^{I}$ ($\theta_{\text{SH}}^{A}$)
characterizes the strength of the isotropic (anisotropic) SH effect in the
ferromagnet and $\mathbf{m}$ is a unit vector denoting the direction of the
magnetization. In Eq. (1), we have defined an effective local electric field
felt by the electrons (in units with $e=1$) as
$\boldsymbol{\mathcal{E}}(z)\equiv\mathbf{E}+\boldsymbol{\nabla}\mu_{c}(z)$,
as well as an effective local ‘spin electric field’,
$\mathcal{E}^{s}_{ij}(z)\equiv\partial_{i}\mu_{s,j}(z)$. Here, $\mu_{c}$ and
$\boldsymbol{\mu}_{s}$ are the charge and spin chemical potentials,
respectively, whose gradients determine effective fields generated by local
variations in the carrier densities 222Note that, owing to the SH effect in
the ferromagnet, the spin accumulation therein is, in general, in an arbitrary
direction. This constitutes a generalization of the formalism presented in
Ref. [12]. Furthermore, $\sigma(z)$ is the local conductivity and $p_{\sigma}$
the spin asymmetry in the linear conductivity in the ferromagnet.

To leading order in the Hall angles, the components of the spin current
density polarized transverse to the magnetization—as well as the relevant
boundary conditions—are decoupled from the charge current density [23], the
details of which are presented in Appendix A. Thus, Eq. (1) reduces to

$\displaystyle\mathbf{j}$
$\displaystyle=\sigma\mathbf{E}+\sigma_{0}\left(\boldsymbol{\nabla}\mu_{c}+p_{\sigma}\boldsymbol{\nabla}\mu_{s}+p_{\sigma}\theta_{\text{SAH}}\mathbf{m}\times\mathbf{E}\right),$
(2a) $\displaystyle\boldsymbol{\mathcal{J}}$
$\displaystyle=p_{\sigma}\sigma\mathbf{E}+\sigma_{0}\left(p_{\sigma}\boldsymbol{\nabla}\mu_{c}+\boldsymbol{\nabla}\mu_{s}+\theta_{\text{SAH}}\mathbf{m}\times\mathbf{E}\right),$
(2b)

where $\sigma_{0}$ is the equilibrium conductivity,
$\theta_{\text{SAH}}(\equiv\theta_{\text{SH}}^{I}+\theta_{\text{SH}}^{A}$) is
the spin AH angle with $p_{\sigma}\theta_{\text{SAH}}$ being its charge
counterpart, $\mathcal{J}_{i}\equiv\mathcal{J}_{ij}m_{j}$ and
$\mu_{s}\equiv\boldsymbol{\mu}_{s}\cdot\mathbf{m}$ are the longitudinal
components of the spin current density and spin chemical potential,
respectively.

The local conductivity in Eq. (2) may be expressed as the sum of the
conductivities of each spin channel, given by

$\sigma^{\alpha}(z)=\nu^{\alpha}\left[n_{0}^{\alpha}+n^{\alpha}(z)\right]\,,$
(3)

where $\alpha=+(-)$ denotes the spin moment parallel (antiparallel) to the
local magnetization, $n_{0}^{\alpha}$ and $\nu^{\alpha}$ are the equilibrium
electron density and mobility of spin-$\alpha$ electrons, respectively, which
in combination give rise to the longitudinal conductivity in the linear
response regime ( i.e.,
$\sigma_{0}=\sum_{\alpha}\sigma_{0}^{\alpha}~{}\text{with}~{}\sigma_{0}^{\alpha}=\nu^{\alpha}n_{0}^{\alpha}$)
as well as the spin asymmetry in the conductivity by
$p_{\sigma}\equiv(\sigma_{0}^{+}-\sigma_{0}^{-})/(\sigma_{0}^{+}+\sigma_{0}^{-})$.
And $n^{\alpha}(z)$ are the current-induced nonequilibrium carrier densities
that are responsible for the UMR. Local charge neutrality is assumed, i.e.,
$\sum_{\alpha}n^{\alpha}(z)=0$, for the metallic system.

The charge and spin chemical potentials may also be defined as
$\mu_{c}\equiv(\mu^{+}+\mu^{-})/2$ and $\mu_{s}\equiv(\mu^{+}-\mu^{-})/2$,
where $\mu^{\alpha}(z)$ is the spin-dependent chemical potential parallel to
the magnetization, which is related to the nonequilibrium electron density
through

$\mu^{\alpha}(z)=\left[N^{\alpha}(\epsilon_{F})\right]^{-1}n^{\alpha}(z)-\phi(z)\,,$
(4)

with $N^{\alpha}(\epsilon_{F})$ the density of states of spin-$\alpha$
electrons at the Fermi level and $\phi(z)$ the spin-independent part of the
chemical potential.

In the presence of spin-flip scattering, the charge- and spin-current
densities satisfy, respectively, the following continuity equations [26]

$\boldsymbol{\nabla}\cdot\mathbf{j}=0~{}~{}\text{and}~{}~{}\boldsymbol{\nabla}\cdot\boldsymbol{\mathcal{J}}=\frac{2}{\tau_{sf}}n_{s}\,,$
(5)

where $\tau_{sf}$ is the spin-flip relaxation time and
$n_{s}(z)=\left(1-p_{N}^{2}\right)N\left(\epsilon_{F}\right)\mu_{s}(z)$ is the
local spin density at the Fermi level, with
$N(\epsilon_{F})=\sum_{\alpha}N^{\alpha}(\epsilon_{F})$ the total density of
states and $p_{N}\equiv(N^{+}-N^{-})/(N^{+}+N^{-})$.

Inserting Eq. (2) into Eq. (5), we obtain a set of differential equations for
the charge and spin chemical potentials

$\displaystyle\frac{d^{2}}{dz^{2}}\mu_{c,F}(z)+p_{\sigma}\frac{d^{2}}{dz^{2}}\mu_{s,F}(z)=0\,,$
(6a)
$\displaystyle\frac{d^{2}}{dz^{2}}\mu_{s,F}(z)-\frac{\mu_{s,F}(z)}{\lambda_{F}^{2}}=0\,,$
(6b)

where
$\lambda_{F}=\sqrt{\sigma_{0,F}(1-p_{\sigma}^{2})\tau_{sf}/2N_{F}(\epsilon_{F})(1-p_{N}^{2})}$
is the spin diffusion length of the ferromagnetic metal.

In a nonmagnetic metal, where $p_{\sigma},p_{N}=0$, the charge and spin
chemical potentials satisfy

$\displaystyle\frac{d^{2}}{dz^{2}}\mu_{c,N}(z)=0\,,$ (7a)
$\displaystyle\frac{d^{2}}{dz^{2}}\mu_{s,N}(z)-\frac{\mu_{s,N}(z)}{\lambda_{N}^{2}}=0\,,$
(7b)

where $\lambda_{N}=\sqrt{\tau_{sf}\sigma_{0,N}/2N_{N}(\epsilon_{F})}$ is the
spin diffusion length of the normal metal with $N_{N}(\epsilon_{F})$ the
density of states of electrons at the Fermi level.

At the interface of the bilayer ($z=0$), we assume both the current density
and chemical potential for each conduction channel are continuous (see
Appendix B for a more detailed discussion on the boundary conditions), i.e.,
$\mu^{\alpha}_{N}(0^{-})=\mu^{\alpha}_{F}(0^{+})$ and
$j_{z}^{\alpha}(0^{-})=j_{z}^{\alpha}(0^{+})$. And open boundary conditions
are imposed at the two outer surfaces, i.e.,
$j_{z}^{\alpha}(-d_{N})=j_{z}^{\alpha}(d_{F})=0$.

Figure 2: Spatial dependences of the spin accumulation and spin AH-UMR
coefficient in the bilayer system. (a) z-dependences of the spin accumulation
$\mu_{s}$ and spin AH-UMR coefficient $\zeta_{\text{SAH}}$ in the FM and NM
layers. Note the absence of the spin AH-UMR in the NM. (b) Plots of the spin
accumulation and spin AH-UMR coefficient in the FM for different thicknesses
of the NM layer. Parameters used: $\lambda_{F}=10$ nm, $\lambda_{N}=20$ nm,
$\theta_{\text{SAH}}=0.05$, $\epsilon_{F}=5$ eV,
$\sigma_{0,F}=\sigma_{0,N}=0.033$ (µ cm)-1, $p_{\sigma}=0.7$, and $p_{N}=0.2$.

##  III Spin accumulation and nonlinear transport

Without loss of generality, let us fix the electric field $\mathbf{E}$ in the
$x$ direction, $\mathbf{E}=E_{x}\mathbf{x}$, and set the magnetization to
point in the $y$ direction—in which case the magnitude of the longitudinal UMR
would reach the maximum. The total charge current density, given by
$\sum_{\alpha}j^{\alpha}(z)$, can be divided into two parts:
$\mathbf{j}=\mathbf{j}^{(1)}+\mathbf{j}^{(2)}$, with a linear component
$\mathbf{j}^{(1)}$ that is proportional to $E_{x}$ and a nonlinear one that is
quadratic in $E_{x}$. The latter can be expressed as

$j_{x}^{(2)}(z)=(\nu^{+}-\nu^{-})\mu_{s}(z)\bar{\mathcal{H}}(N^{+},N^{-})E_{x}$
(8)

where $\bar{\mathcal{H}}(N^{+},N^{-})$ is the harmonic mean of the density of
states. And the spin accumulation $\mu_{s}$ is linear in
$\theta_{\text{SAH}}E_{x}$, resulting in $j_{x}^{(2)}\propto E_{x}^{2}$. Note
that $j_{x}^{(2)}$ only emerges in the ferromagnetic layer wherein
$\nu^{+}\neq\nu^{-}$.

In order to properly quantify the nonlinear charge current, we introduce a UMR
coefficient $\zeta(z)$ as

$\zeta(z)\equiv\frac{\sigma_{xx}(z,E_{x})-\sigma_{xx}(z,-E_{x})}{\sigma_{0}E_{x}}\;,$
(9)

where $\sigma_{ij}=j_{i}/E_{j}$ is the conductivity tensor, and $\sigma_{0}$
is the linear longitudinal conductivity. The UMR coefficient $\zeta$ is so
defined that its magnitude is independent of the electric field. The dimension
of $\zeta$ is length per Volt, the inverse of which sets the scale of the
electric field for which the nonlinear longitudinal conductivity—given by
$j_{x}^{(2)}/E_{x}$—becomes comparable to its linear counterpart.

The spatial distribution of the spin AH-UMR coefficient $\zeta_{\text{SAH}}$
is displayed in Fig. 2a, along with that of the spin chemical potential
$\mu_{s}$ (or the spin accumulation) in the bilayer structure. There is a
clear correlation between the two quantities: $\zeta_{\text{SAH}}$ goes to
zero wherever $\mu_{s}$ vanishes. Furthermore, the spin AH-UMR completely
comes from the ferromagnetic layer, in which the electron mobility is spin-
dependent. Within the nonmagnetic layer, for which $\nu^{+}=\nu^{-}$,
$\zeta_{\text{SAH}}$ vanishes everywhere despite the remnant $\mu_{s}$ near
the interface. These observations are in full agreement with Eq. (8).

Although the nonmagnetic layer in question neither plays an active role as a
spin polarizer nor accommodates any nonlinear charge transport, it is still
indispensable to the generation of a net spin AH-UMR in its neighboring
ferromagnetic layer. In the absence of the nonmagnetic layer, the spin
accumulation $\mu_{s}$ and thus the local spin AH-UMR coefficient
$\zeta_{\text{SAH}}$ have antisymmetric distributions about the center line of
the ferromagnetic layer, as shown by the dashed lines in Fig. 2b. In this
case, the total (spatially-averaged) spin AH-UMR is zero, as a result of the
lack of a net nonequilibrium spin density.

From a symmetry perspective, a net current-induced spin density is allowed,
when and only when a system lacks inversion symmetry. For the present case,
the nonmagnetic layer introduces structural inversion asymmetry, and makes a
net nonequilibrium spin density achievable in the ferromagnetic layer next to
it. Physically, it “absorbs” spin accumulation at the interface from the
ferromagnetic layer, leaving a net nonequilibrium spin density in the latter,
as illustrated by the dash-dotted and solid lines in Fig. 2b.

##  IV Spatially-averaged spin AH-UMR

By taking the spatial average of the overall UMR coefficient over the
thickness of the bilayer,
$\bar{\zeta}\equiv\int_{-d_{N}}^{d_{F}}dz\;\zeta(z)/(d_{N}+d_{F})$, we find
that, up to $\mathcal{O}(\theta_{\text{SAH}})$, the spatially-averaged spin
AH-UMR coefficient reads

$\bar{\zeta}_{\text{SAH}}=p_{F}\theta_{\text{SAH}}\left(\frac{\lambda_{F}}{\epsilon_{F}}\right)\mathcal{G}\left(\frac{d_{F}}{\lambda_{F}},\frac{d_{N}}{\lambda_{N}};\frac{\sigma_{0,F}}{\sigma_{0,N}},\frac{\lambda_{F}}{\lambda_{N}}\right)\,,$
(10)

where $p_{F}(=p_{\sigma}-p_{N})$ characterizes the overall spin asymmetry of
electron mobility for the ferromagnetic layer, and the thickness dependence of
the spin AH-UMR is encapsulated in the dimensionless $\mathcal{G}$ function as

$\mathcal{G}\left(s,t;u,v\right)=\frac{3\left(\frac{uv}{(uv)\cdot
s+t}\right)\tanh(s)\tanh(\frac{s}{2})}{1+\left(1-p_{\sigma}^{2}\right)\left(\frac{u}{v}\right)\tanh(s)\coth(t)}.$
(11)

For simplicity, we have adopted the free-electron model whereby
$N_{F}^{\alpha}=3n_{0,F}^{\alpha}/2\epsilon_{F}$ with $\epsilon_{F}$ the Fermi
energy of conduction electrons in the ferromagnet. Equations (10) and (11) are
the main results of the paper.

Several remarks regarding the spin AH-UMR are in order.

1) The spin AH-UMR coefficient, to leading order, is proportional to the spin
AH angle $\theta_{\text{SAH}}$, in contrast to the USMR effect, which is
proportional to the SH angle of the heavy-metal layer [12].

2) The spin AH-UMR coefficient is also linear in $p_{F}$, as is the USMR [12].
This is not surprising, as the conversion of a net nonequilibrium spin density
to a (nonlinear) charge current relies entirely on the spin asymmetry in
electron scatterings.

3) The ratio $\frac{\lambda_{F}}{\epsilon_{F}}$ has the same dimension as the
spin AH-UMR coefficient (with $e=1$). In fact, the prefactor of the averaged
UMR coefficient—the thickness independent part in Eq. (10)—sets the scale of
the maximum spin AH-UMR that one can obtain for a given ferromagnet. For a
typical transition metal with $p_{F}=0.3$, $\theta_{\text{SAH}}=0.02$,
$\lambda_{F}=100$ nm, and $\epsilon_{F}=5$ eV, the upper bound of the spin AH-
UMR coefficient, $\bar{\zeta}_{\text{SAH}}$, is of the order of 1 Å/V.

4) Information about how other geometric and materials parameters of a
magnetic bilayer would shape the spin AH-UMR is all encoded in the
dimensionless $\mathcal{G}$ function given by Eq. (11). The first two
variables, $d_{F}/\lambda_{F}$ and $d_{N}/\lambda_{N}$, indicate that the
dependences of the spin AH-UMR on the thicknesses of the ferromagnetic and
nonmagnetic layers must scale with their respective spin diffusion lengths, as
plotted in Fig. 3.

5) Another remarkable property of the spin AH-UMR—revealed by the
$\mathcal{G}$ function—is that it increases monotonically with the ratio
$\frac{\lambda_{F}}{\lambda_{N}}$, which would be useful for guiding the
search for magnetic bilayers with a sizable spin AH-UMR effect.

Figure 3: Thickness dependence of the spin AH-UMR coefficient. Dependence of
$\bar{\zeta}_{\text{SAH}}$ on the thickness ($i=F,N$) of the FM layer for
$d_{N}=15$ nm (solid) and on the thickness of the NM layer for $d_{F}=15$ nm
(dashed). See the caption of Fig. 2 for the list of materials parameters used.

##  V UMR sign reversal and SH/spin-AH angle quantification

In a magnetic bilayer consisting of a ferromagnetic metal and a heavy metal,
both spin AH and SH effects, in principle, may contribute to the total UMR
measured in the bilayer. And their contributions turn out to be additive,
i.e., $\bar{\zeta}=\bar{\zeta}_{\text{SAH}}+\bar{\zeta}_{\text{SH}}$ (see
Appendix B for the full expression of $\bar{\zeta}$). The ratio of the two UMR
contributions due to spin-dependent scattering—provided electron-magnon
scattering is suppressed by applying a magnetic field or lowering the
temperature [15]—takes a rather neat form

$\frac{\bar{\zeta}_{\text{SAH}}}{\bar{\zeta}_{\text{SH}}}=\frac{\theta_{\text{SAH}}\lambda_{F}\tanh\left(\frac{d_{F}}{2\lambda_{F}}\right)}{\theta_{\text{SH}}\lambda_{N}\tanh\left(\frac{d_{N}}{2\lambda_{N}}\right)}\,.$
(12)

It is worthy to note that this ratio depends on only a few parameters, namely
the SH/spin-AH angle, the spin diffusion length, and the thickness of each
layer.

The simple relation between $\bar{\zeta}_{\text{SAH}}$ and
$\bar{\zeta}_{\text{SH}}$ nonetheless has a remarkable physical consequence:
the total UMR coefficient of such a magnetic bilayer may exhibit qualitatively
different thickness dependences, depending on the relative signs of the spin
AH and SH angles. When $\theta_{\text{SAH}}$ and $\theta_{\text{SH}}$ have the
same sign, the associated contributions simply add up (see the blue curves in
Fig. 4). It becomes more intriguing when $\theta_{\text{SAH}}$ and
$\theta_{\text{SH}}$ are of opposite signs. In this case, the total UMR
inevitably undergoes a sign reversal as the thickness of either layer is
varied (see the red curves in Fig. 4).

Figure 4: Thickness dependence of the total UMR coefficient. Dependence of
$\bar{\zeta}$ on the thickness ($i=F,N$) of the FM layer for $d_{N}=5$ nm
(solid) and on the thickness of the NM layer for $d_{F}=10$ nm (dashed). Other
parameters used: $\lambda_{F}=20$ nm, $\lambda_{N}=5$ nm, $\epsilon_{F}=5$ eV,
$\sigma_{0,F}=\sigma_{0,N}=0.033$ (µ cm)-1, $p_{\sigma}=0.7$, and $p_{N}=0.2$.

At the sign reversal point, the ratio of the Hall angles fulfills the
following condition:

$\frac{\theta_{\text{SH}}}{\theta_{\text{SAH}}}=-\frac{\lambda_{F}\tanh\left(\frac{d_{F}}{2\lambda_{F}}\right)}{\lambda_{N}\tanh\left(\frac{d_{N}}{2\lambda_{N}}\right)}\,,$
(13)

which can be used to experimentally quantify the SH (spin AH) angle of the
heavy-metal (ferromagnetic-metal) layer, provided the Hall angle and spin
diffusion length of the other layer—which serves as a reference layer—are
known.

##  VI Proposal for detecting spin AH-UMR

As compared to the USMR, there are more options of materials systems for
probing the spin AH-UMR effect. For the USMR, the nonmagnetic heavy-metal
layer plays a central role in creating nonequilibrium spin density—via the SH
effect—in the adjacent ferromagnetic layer. But this is not the case for the
spin AH-UMR. For the spin AH-UMR, it is the ferromagnetic layer that serves as
the spin polarizer and, hence, the choice of its neighboring layer is not
necessarily limited to nonmagnetic materials with a strong SH effect.

For instance, the spin AH-UMR effect, in principle, can also be hosted in
bilayers comprised of a ferromagnetic metal and a normal metal, such as Cu,
Al, or Ag. Given the weak SH effect in the normal-metal layer, the spin AH-UMR
is expected to dominate over the USMR in these systems, making the detection
of the former more straightforward. There is, perhaps, also a downside to
these metallic bilayers—the spin AH-UMR therein is likely to be much smaller
than that in magnetic bilayers with heavy-metals, as normal metals with a weak
SH effect are oftentimes also poor “spin sinks” with long spin diffusion
lengths [27], which would diminish the spin AH-UMR effect (especially when the
ratio $\frac{\lambda_{F}}{\lambda_{N}}$ is small 333For instance, a UMR was
not detected in Cu$|$Co bilayers [34] (or, at least, was demonstrated to be
much smaller than that in W$|$Co bilayers), which is not surprising, as the
$\frac{\lambda_{F}}{\lambda_{N}}$ ratio for the former is about two orders of
magnitude smaller than that for the latter., as was discussed in a previous
section).

The shortcoming of normal metals with long spin diffusion lengths may be
compensated for by choosing a ferromagnetic layer with low carrier density. To
see this, let us insert Eq. (8) together with
$\sigma_{0}=\sum_{\alpha}n_{0}^{\alpha}\nu^{\alpha}$ into Eq. (9), which
yields

$\zeta_{\text{SAH}}\propto\left(\frac{\nu^{+}-\nu^{-}}{\nu^{+}+\nu^{-}}\right)\frac{\theta_{\text{SAH}}}{n_{0}}\,.$
(14)

The above relation conveys a valuable piece of information: the lower the
equilibrium carrier density of the ferromagnet, the larger the UMR
coefficient. Thus, a more sizable spin AH-UMR is expected to arise in bilayers
consisting of a normal-metal and a ferromagnetic semiconductor [e.g.,
(Ga,Mn)As] [11] whose carrier density is usually two to three orders of
magnitude smaller than that of a ferromagnetic metal.

In magnetic bilayers comprised of nonmagnetic materials with strong spin-orbit
coupling, the coexistence of the spin AH-UMR and USMR poses a challenge to
differentiate the two. But, interestingly, they may also conspire to bring
about a sign reversal of the overall UMR when the thickness of either layer is
varied, a distinct transport signature that would not appear when either
effect stands alone. This can be experimentally verified by contrasting the
thickness dependences of the total UMR in two ferromagnetic-metal$|$heavy-
metal bilayers, either with different heavy-metal layers whose SH angles are
of opposite signs (such Pt and $\beta$-Ta [29]) or with different
ferromagnetic-metal layers whose spin AH angles are of opposite signs (such as
Fe and Gd [30]). We anticipate that the results of such comparative
measurements will resemble what are shown in Fig. 4, with one bearing a sign
change and the other not. On a related note, a sign change of the UMR was
recently observed in single-crystalline Fe$|$Pt bilayers as the thickness of
the Fe layer was increased [31], implying possible competition between the
spin AH-UMR and the USMR.

Other nonlinear effects that may intertwine with the spin AH-UMR are the
anomalous Nernst [32] and spin Seebeck [33] effects: the Joule heating may
induce a vertical temperature gradient across the ferromagnetic layer, which
would in turn give rise to a nonlinear current in the direction of
$\mathbf{m}\times\boldsymbol{\nabla}T$. However, one can separate the UMR
contribution from the thermal contribution based on their different
dependences on the direction of the applied electric field: the former is
proportional to $\hat{\mathbf{z}}\cdot(\mathbf{m}\times\mathbf{E})$, whereas
the latter is independent of the relative orientation between $\mathbf{E}$ and
$\mathbf{m}$, as the temperature gradient only relies on the magnitude of
electric field (i.e.,
$\boldsymbol{\nabla}T\propto\lvert\mathbf{E}\rvert^{2}$).

## Acknowledgments

We thank Chong Bi, Kirill Belashchenko, and Xin Fan for helpful discussions.
This work was supported by the College of Arts and Sciences, Case Western
Reserve University.

##  Appendix A Decoupling of Transverse Spin Chemical Potentials

Recall Eq. (1) in the main text

$\displaystyle j_{i}$
$\displaystyle=\sigma\left(\mathcal{E}_{i}+p_{\sigma}\mathcal{E}^{s}_{ij}m_{j}\right)-\theta_{\text{SH}}^{I}\epsilon_{ijk}\mathcal{J}_{jk}+\theta_{\text{SH}}^{A}\epsilon_{ijk}m_{j}m_{l}\mathcal{J}_{kl}\,,$
(15a) $\displaystyle\mathcal{J}_{ij}$
$\displaystyle=\sigma\left(\mathcal{E}^{s}_{ij}+p_{\sigma}\mathcal{E}_{i}m_{j}\right)+\theta_{\text{SH}}^{I}\epsilon_{ijk}j_{k}+\theta_{\text{SH}}^{A}\epsilon_{ilk}m_{j}m_{l}j_{k}\,.$
(15b)

Here, $\mathcal{J}_{ij}$ is the spin current density tensor, in which the
index $i$ indicates the direction of electron momentum flow and $j$ the spin
polarization direction. Let us introduce the following decomposition of the
spin current density

$\mathcal{J}_{ij}=\mathcal{J}_{i}m_{j}+\mathcal{J}_{i}^{\perp_{1}}p_{j}+\mathcal{J}_{i}^{\perp_{2}}q_{j}\,,$
(16)

where $\mathbf{m}$, $\mathbf{p}$ and $\mathbf{q}$ are three mutually
orthogonal unit vectors. $\mathcal{J}_{i}\equiv\mathcal{J}_{ij}m_{j}$ is the
spin current density vector with the spins polarized along the magnetization,
while $\mathcal{J}_{i}^{\perp_{1}}\equiv\mathcal{J}_{ij}p_{j}$ and
$\mathcal{J}_{i}^{\perp_{2}}\equiv\mathcal{J}_{ij}q_{j}$ are two spin current
density vectors with spins polarized perpendicular to the magnetization and to
each other. Using the definitions of the effective local electric and spin
electric fields,
$\boldsymbol{\mathcal{E}}\equiv\mathbf{E}+\boldsymbol{\nabla}\mu_{c}$ and
$\mathcal{E}^{s}_{ij}\equiv\partial_{i}\mu_{s,j}$, to leading order in the
Hall angles, the charge and spin current density vectors read

$\displaystyle\mathbf{j}=\sigma\mathbf{E}+\sigma_{0}\left(\boldsymbol{\nabla}\mu_{c}+p_{\sigma}\boldsymbol{\nabla}\mu_{s}+p_{\sigma}\theta_{\text{SAH}}\mathbf{m}\times\mathbf{E}\right),$
(17a)
$\displaystyle\boldsymbol{\mathcal{J}}=p_{\sigma}\sigma\mathbf{E}+\sigma_{0}\left(p_{\sigma}\boldsymbol{\nabla}\mu_{c}+\boldsymbol{\nabla}\mu_{s}+\theta_{\text{SAH}}\mathbf{m}\times\mathbf{E}\right),$
(17b)
$\displaystyle\boldsymbol{\mathcal{J}}^{\perp_{1}}=\sigma_{0}\left(\boldsymbol{\nabla}\mu_{s}^{\perp_{1}}+\theta_{\text{SH}}^{I}\mathbf{p}\times\mathbf{E}\right),$
(17c)
$\displaystyle\boldsymbol{\mathcal{J}}^{\perp_{2}}=\sigma_{0}\left(\boldsymbol{\nabla}\mu_{s}^{\perp_{2}}+\theta_{\text{SH}}^{I}\mathbf{q}\times\mathbf{E}\right),$
(17d)

where $\theta_{\text{SAH}}=\theta_{\text{SH}}^{I}+\theta_{\text{SH}}^{A}$ is
the spin AH angle,
$\mu_{s}^{\perp_{1}}\equiv\boldsymbol{\mu}_{s}\cdot\mathbf{p}$ and
$\mu_{s}^{\perp_{2}}\equiv\boldsymbol{\mu}_{s}\cdot\mathbf{q}$.

In the presence of spin-flip scattering, and assuming local charge neutrality,
the charge and spin current densities obey the following continuity equations

$\displaystyle\boldsymbol{\nabla}\cdot\mathbf{j}$ $\displaystyle=0\,,$ (18a)
$\displaystyle\partial_{i}\mathcal{J}_{ij}$
$\displaystyle=\frac{2}{\tau_{sf}}n_{s,j}\,,$ (18b)

where $\tau_{sf}$ is the spin-flip relaxation time and
$\mathbf{n}_{s}(z)=\left(1-p_{N}^{2}\right)N\left(\epsilon_{F}\right)\boldsymbol{\mu}_{s}(z)$
is the local spin density vector. Inserting Eq. (17) into Eq. (18), we obtain
the following set of differential equations for the charge and spin chemical
potentials

$\displaystyle\frac{d^{2}}{dz^{2}}\mu_{c}(z)+p_{\sigma}\frac{d^{2}}{dz^{2}}\mu_{s}(z)=0\,,$
(19a)
$\displaystyle\frac{d^{2}}{dz^{2}}\mu_{s}(z)-\frac{\mu_{s}(z)}{\lambda^{2}}=0\,,$
(19b)
$\displaystyle\frac{d^{2}}{dz^{2}}\mu_{s}^{\perp_{1}}(z)-\frac{\mu_{s}^{\perp_{1}}(z)}{\lambda_{\perp}^{2}}=0\,,$
(19c)
$\displaystyle\frac{d^{2}}{dz^{2}}\mu_{s}^{\perp_{2}}(z)-\frac{\mu_{s}^{\perp_{2}}(z)}{\lambda_{\perp}^{2}}=0\,,$
(19d)

where
$\lambda=\sqrt{\sigma_{0}(1-p_{\sigma}^{2})\tau_{sf}/2N(\epsilon_{F})(1-p_{N}^{2})}$
and $\lambda_{\perp}=\sqrt{\sigma_{0}\tau_{sf}/2N(\epsilon_{F})(1-p_{N}^{2})}$
are the spin diffusion lengths parallel and perpendicular to the
magnetization, respectively. We thus find that the transverse spin chemical
potentials are decoupled from $\mu_{c}$ and $\mu_{s}$ to leading order in the
Hall angles, as are their boundary conditions [23]. Therefore, for the purpose
of UMR analysis, we may disregard transverse spin chemical potentials.

##  Appendix B Contribution of Boundary Resistance

The presence of a boundary resistance at the bilayer interface modifies the
boundary conditions on the charge and spin chemical potentials as [26]

$\displaystyle\mu_{c,F}\left(0^{+}\right)-\mu_{c,N}\left(0^{-}\right)$
$\displaystyle=-\gamma r_{b}\mathcal{J}_{z}\left(0\right),$ (20a)
$\displaystyle\mu_{s,F}\left(0^{+}\right)-\mu_{s,N}\left(0^{-}\right)$
$\displaystyle=r_{b}\mathcal{J}_{z}\left(0\right),$ (20b)

where $\gamma$ is the interfacial spin asymmetry coefficient and $r_{b}$ is
the boundary resistance for a unit surface of the interface.

Resolving the continuity equations presented in the main text with the
modified boundary conditions, the total UMR coefficient $\zeta$ of the system
may be calculated. This is comprised of the SH- and spin AH-UMR coefficients,
$\zeta=\zeta_{\text{SAH}}+\zeta_{\text{SH}}$. Taking the spatial average
defined as $\bar{\zeta}\equiv\int_{-d_{N}}^{d_{F}}dz\;\zeta(z)/(d_{N}+d_{F})$,
to first order in the SH and spin AH angles, we obtain

$\bar{\zeta}=\frac{3\left(p_{\sigma}-p_{N}\right)}{\epsilon_{F}}\left(\frac{\sigma_{0,F}\lambda_{F}}{\sigma_{0,F}d_{F}+\sigma_{0,N}d_{N}}\right)\frac{\tanh\left(\frac{d_{F}}{\lambda_{F}}\right)\left[\theta_{\text{SAH}}\lambda_{F}\tanh\left(\frac{d_{F}}{2\lambda_{F}}\right)+\theta_{\text{SH}}\lambda_{N}\tanh\left(\frac{d_{N}}{2\lambda_{N}}\right)\right]}{1+\left(1-p_{\sigma}^{2}\right)\tanh\left(\frac{d_{F}}{\lambda_{F}}\right)\left[\left(\frac{\sigma_{0,F}\lambda_{N}}{\sigma_{0,N}\lambda_{F}}\right)\coth\left(\frac{d_{N}}{\lambda_{N}}\right)+\frac{r_{b}\sigma_{0,F}}{\lambda_{F}}\right]}\,,$
(21)

where, as in the main text, we note the relation between the spin AH- and SH-
UMR coefficients

$\frac{\bar{\zeta}_{\text{SAH}}}{\bar{\zeta}_{\text{SH}}}=\frac{\theta_{\text{SAH}}\lambda_{F}\tanh\left(\frac{d_{F}}{2\lambda_{F}}\right)}{\theta_{\text{SH}}\lambda_{N}\tanh\left(\frac{d_{N}}{2\lambda_{N}}\right)}\,.$
(22)

Taking into account the interfacial resistance, we have verified that, for a
typical boundary resistance of $r_{b}\sim 1\text{f}\Omega\;\text{m}$ [26, 27],
the effect on the transport coefficients is negligible. Thus, we may safely
neglect the interfacial resistance in the present study.

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