# Interface disorder as a cause for kinetic Rashba-Edelstein effect and
interface Spin-Hall effect at the metal - insulator boundary.

A. V. Shumilin, V. V. Kabanov  Jozef Stefan Institute, 1000 Ljubljana,
Slovenia

###### Abstract

The spin phenomena observed at a clean metall-insulator interface are
typically reduced to Rashba-Edelstein effect, that leads to spin accumulation
over a few monolayers. We demonstrate that the presence of interface disorder
significantly expands the range of potential phenomena. Specifically, the skew
scattering at the metal - insulator boundary gives rise to the "kinetic
Rashba-Edelstein effect", where spin accumulation occurs on a much larger
length scale comparable to mean free path. Moreover, at higher orders of spin-
orbit interaction, skew scattering is accompanied with spin relaxation
resulting in the interface spin-Hall effect - a conversion of electrical
current to spin current at the metal surface. Unlike the conventional spin-
Hall effect, this phenomenon persists even within the Born approximation.
These two predicted phenomena can dominate the spin density and spin current
in devices of intermediate thickness.

##  I Introduction

Spin-orbit interaction is a fundamental phenomenon that enables the exchange
of angular momentum between orbital and spin degrees of freedom. One of the
promising applications of this interaction is in the development of spin-orbit
torque devices, which offer a scalable and field-free solution for magnetic
memory that can be controlled by electrical currents [1, 2, 3]. Experimental
studies have already demonstrated the magnetization switching [4, 5, 6] and
domain wall motion [7] in such devices, with other potential applications
including the control of magnetic skyrmions [8] and spin-waves [9].

The basic setup for an spin-orbit torque device involves a magnetic bilayer
composed of a heavy metal layer with strong spin-orbit coupling and a
ferromagnetic layer that acts as the detector for spin polarization and spin
current from the heavy metal. Although this simple bilayer composition is
already functional [10], additional layers are often added to enhance or
modify the device properties. These additional layers can be added from the
side of the ferromagnet [11], from the side of heavy metal [12], or between
them [13]. They are usually composed from materials possessing specific spin
properties, such as antiferromagnets [14] and topological insulators [15, 16].

Surprisingly, recent research has shown that even insulating non-magnetic
molecules can significantly enhance the spin torque when added from the side
of a heavy metal, for the heavy metal thicknesses up to 5nm [17]. These
results have drawn our attention to the spin phenomena occurring at the
interface between heavy metals and insulators.

Figure 1: The three phenomenological effects leading to the spin polarization
in the heavy metal layer. (A) Rashba-Edelstein effect, (B) Spin-Hall effect,
(C) Interface spin-Hall effect. The blue arrow stands for the direction of
electrical current. Grey arrows show the direction of spin flow. The red
arrows correspond to the direction spin polarization. Green spheres depict the
impurities.

Spin generation in magnetic bilayers is typically attributed to either the
Rashba-Edelstein surface effect or the Spin-Hall effect (see Fig. (1)).
Although it can be difficult to distinguish between these effects in a given
experiment [4], they differ significantly in terms of the device engineering.
The Rashba-Edelstein effect results in spin polarization at the surface of
heavy metal where inversion symmetry is broken. However, this polarization is
confined to only a few monolayers near the interface [18, 19, 20, 21], meaning
that the spin generated at the heavy metal interface with a third material can
affect the ferromagnetic layer only in very thin devices. When the heavy metal
layer is thick, the spin-torque in the ferromagnetic layer is usually
attributed to Spin-Hall effect, which is the conversion of electrical to spin
current in the heavy metal layer [22, 23]. However, the Spin-Hall effect is
related to the bulk properties of the heavy metal [24, 25, 26] and is not
expected to be significantly influenced by the heavy metal - insulator
interface. In relatively clean samples Spin-Hall effect is dominated by skew
scattering at impurities in the bulk, that is absent in the Born
approximation, resulting in a suppression factor of $V_{0}/\varepsilon_{F}$
where $V_{0}$ is the potential of single impurity and $\varepsilon_{F}$ is
Fermi energy.

The recently predicted interface Spin-Hall effect combines the properties of
Rashba-Edelstein and conventional Spin-Hall effects. It refers to the
electrical to spin current conversion at the interface. Its phenomenological
possibility is demonstrated in [27, 28, 29]. However, it has only been studied
for the interface of two metals and was attributed to spin filtering, which
involves different probabilities for spin-up and spin-down electrons to
traverse the interface between metals [30, 31, 2]. Similar spin-filtering was
also predicted for tunneling through semiconductor barriers [32, 33, 34].

Here we investigate the spin kinetics near a disordered heavy metal-insulator
interface. We demonstrate that the disorder significantly increases the
variety of interface spin phenomena. Skew scattering at the interface
impurities causes spin accumulation over a distance comparable to the mean
free path from the interface. This phenomenologically corresponds to the
Rashba-Edelstein effect; however, the thickness of the spin accumulation layer
is significantly larger than that predicted in [19, 20]. Combined with spin
relaxation, this leads to the interface Spin-Hall effect, which is absent at a
clean metal-insulator interface. Both of these phenomena are sensitive to the
materials that make up the interface and their properties, as well as their
disorder.

##  II Model of the heavy metal surface

We consider a clean heavy metal interface with insulator described by the
model Hamiltonian

$\widehat{H}=\frac{\widehat{\bf p}^{2}}{2m}+U(z)-\gamma\frac{\partial
U}{\partial
z}\left(\sigma_{x}\widehat{p}_{y}-\sigma_{y}\widehat{p}_{x}\right)$ (1)

Here $\widehat{\bf p}=-i\hbar\nabla$ is the momentum operator, $m$ is the
effective mass, $U(z)=U_{0}\theta(z)$ is the potential energy describing the
abrupt barrier at $z=0$ with the height $U_{0}$. $\gamma$ is effective spin-
orbit interaction inside heavy metal. It’s solution in Appendix A leads to the
following electron wavefunction on the metal side of the interface

$\Psi_{\alpha}({\bf k})=\frac{e^{i{\bf k_{\perp}}{\bf
r}_{\perp}}}{\sqrt{2V}}\left(e^{i|k_{z}|z}+\hat{r}({\bf
k})e^{-i|k|_{z}z}\right)u_{\alpha}.$ (2)

Here ${\bf k}$ is electron wavevector, $k_{z}$ is its component along $z$,
${\bf k_{\perp}}$ is its $xy$-component and $u_{\alpha}$ is arbitrary spinor.
Eq. (2) includes the spin-dependent reflection amplitude $\hat{r}({\bf k})$

$\hat{r}({\bf
k})=-e^{2i\phi_{0}}\left[\cos\left(\Delta\phi\right)\hat{1}+i\sin\left(\Delta\phi\right)\hat{\sigma}_{\bf
k}\right],$ (3)

where $\phi_{0}=(\phi_{+}+\phi_{-})/2$ is the average phase change during the
reflection and $\Delta\phi=\phi_{+}-\phi_{-}$ shows its spin dependence.
$\phi_{\pm}=\arctan[k_{z}/(\kappa\mp 2m\gamma U_{0}k_{\perp}/\hbar)]$,
$\kappa=\sqrt{2mU_{0}/\hbar^{2}-k_{z}^{2}}$. $\hat{\sigma}_{\bf k}$ is the
combination of Pauli matrices.

$\hat{\sigma}_{\bf
k}=\frac{k_{y}}{|k_{\perp}|}\hat{\sigma}_{x}-\frac{k_{x}}{|k_{\perp}|}\hat{\sigma}_{y}$
(4)

It is demonstrated in Appendix A that in agreement with previous studies [19,
20] the clean interface does not exhibit an interface spin Hall effect and the
spin polarization is confined to several monolayers.

The primary objective of this study is to incorporate interface disorder into
the theory. The most common approach to the electron kinetics near disordered
interface involves finite probabilities of specular and non-specular
reflection [35]. However, to address the new spin phenomena it is crucial to
consider a microscopic mechanism responsible for non-specular reflection. Two
conventional approaches exist for modeling reflection from disordered
interfaces: roughness of the interface [36, 37] and interface impurities [38,
39, 40]. Both approaches allow for the possibility of non-specular reflection,
and we consider them to be interchangeable. In this work we adopt the latter
one.

A single impurity can be described with potential energy $V_{0}({\bf r})$.
With respect to the spin-orbit interaction it leads to the additional term in
the electron Hamiltonian

$V({\bf r})=V_{0}({\bf r})+\gamma\bm{\sigma}\left[\frac{\partial
V_{0}}{\partial{\bf r}}\times{\bf p}\right]$ (5)

In this work we consider small impurities with $V_{0}({\bf
r})=V_{I}\delta({\bf r})$ where $V_{I}$ stands for the magnitude of impurity
potential.

Figure 2: (a) The four possibilities for the impurity scattering from state
${\bf k}$ to the state ${\bf k}^{\prime}$ in the presence of the interface.
(b) the spin dependence of scattering amplitude. (c) asymmetric spin rotation
angle. The results presented in panels (b) and (c) correspond to
$U_{0}/\varepsilon_{F}=4$, $\gamma p_{F}^{2}/\hbar=0.2$ and to the geometry
shown in panel (d).

The scattering from the impurities is characterized by the matrix elements
$\hat{V}({\bf k}_{1},{\bf k}_{2})=V_{\alpha\beta}({\bf k}_{1},{\bf
k}_{2})=\langle\Psi_{\alpha}({\bf k}_{1})|V({\bf r})|\Psi_{\beta}({\bf
k}_{2})\rangle$, where $\Psi_{\alpha}({\bf k}_{1})$ and $\Psi_{\beta}({\bf
k}_{2})$ correspond to the electron states at the clean interface, as
described by Eq. (2). In order to analyze the scattering process, it is useful
to decompose the scattering elements into the two terms.

$\hat{V}({\bf k}_{1},{\bf k}_{2})=\hat{V}^{(N)}({\bf k}_{1},{\bf
k}_{2})+\hat{V}^{(SO)}({\bf k}_{1},{\bf k}_{2})$ (6a)
$V_{\alpha\beta}^{(N)}({\bf k}_{1},{\bf k}_{2})=\\\
\frac{1}{2}\frac{V_{I}}{V}(1+\hat{r}^{+}({\bf k}_{1}))(1+\hat{r}({\bf
k}_{2}))$ (6b) $\hat{V}^{(SO)}({\bf k}_{1},{\bf
k}_{2})=\frac{iV_{I}\gamma}{2\hbar V}\left({\bm{\sigma}}[{\bf
p}_{1}^{(inc)}\times{\bf p}_{2}^{(inc)}]\right.\\\ +\widehat{r}^{+}{({\bf
k}_{1})}{\bm{\sigma}}[{\bf p}_{1}^{(ref)}\times{\bf
p}_{2}^{(inc)}]+{\bm{\sigma}}[{\bf p}_{1}^{(inc)}\times{\bf
p}_{2}^{(ref)}]\widehat{r}({\bf k}_{2})\\\ \left.+\widehat{r}^{+}{({\bf
k}_{1})}{\bm{\sigma}}[{\bf p}_{1}^{(ref)}\times{\bf
p}_{2}^{(ref)}]\widehat{r}({\bf k}_{2})\right)$ (6c)

Here $V_{\alpha\beta}^{(N)}({\bf k}_{1},{\bf k}_{2})$ corresponds to the first
term in r.h.s. of Eq. (5) and $V_{\alpha\beta}^{(SO)}({\bf k}_{1},{\bf
k}_{2})$ is related to the spin-orbit correction and to the second term in
r.h.s. of Eq. (5). ${\bf p}^{(inc)}=\hbar(k_{x},k_{y},|k_{z}|)$ is the
momentum of the incident electron and ${\bf
p}^{(ref)}=\hbar(k_{x},k_{y},-|k_{z}|)$ is the momentum of reflected electron.
$\widehat{r}^{+}{({\bf k})}$ is the Hermitian conjugate of the reflection
amplitude described with Eq. (3).

Eqs. (6b,6c) can be represented as the four different reflection possibilities
shown in Fig. 2(a). The electron can undergo scattering from the impurity
without interaction with the surface, or can be specularly reflected before or
after the scattering or both.

The probability of scattering depends on the quantum interference between the
different reflection possibilities, which can be constructive or destructive
depending on the phase $\varphi_{0}\pm\Delta\varphi$ and the relation between
$\hat{V}^{(N)}$ and $\hat{V}^{(SO)}$. This probability varies for different
spin projections, leading to skew scattering and spin separation at the
surface. Fig.2(b) shows the probability for electrons with the incident angle
of $\pi/4$ in the xz-plane to scatter to the yz-plane. The scattering rate
depends on the spin projection to the y-axis. The calculation details are
presented in the Appendix B. When spin-orbit interaction is present, electron
scattering also results in spin rotation, leading to spin relaxation after
multiple random scattering events [37]. Fig.2(c) demonstrates that in our
case, this rotation becomes asymmetric, causing the spin-up to rotate
differently than the spin-down. This asymmetric spin rotation produces spin
polarization of reflected electrons, even when the incident electrons are not
spin-polarized, resulting in the interface spin-Hall effect. The geometry
corresponding to Figs. 2(b,c) is shown in Fig.2(d).

##  III Spin current and spin polarization

To understand impact of the skew scattering and asymmetric spin rotation on
the electrons in the bulk of heavy metal we introduce Boltzmann equation

$\frac{\partial\hat{f}({\bf r},{\bf p})}{\partial t}+{\bf
v}\frac{\partial\hat{f}({\bf r},{\bf p})}{\partial{\bf r}}+{\bf
F}\frac{\partial\hat{f}({\bf r},{\bf p})}{\partial{\bf p}}=I(\hat{f}({\bf
r},{\bf p}))$ (7)

Here the distribution function $\hat{f}({\bf r},{\bf p})$ is a $2\times 2$
matrix in spin space that depends on the coordinate and momentum as usual.
$I(\hat{f}({\bf r},{\bf p}))$ is the scattering operator.

We consider the following ansatz for the distribution function:
$\hat{f}=\hat{f}_{0}+\hat{f}_{1}+\hat{f}_{2}$. Here $\hat{f}_{0}$ is the
equilibrium electron distribution. It is proportional to the unit matrix
$\hat{1}$ in the spin space.

$\hat{f}_{1}=-\frac{j}{e}\frac{p_{x}}{n}\frac{\partial\hat{f}_{0}}{\partial\varepsilon}$
(8)

represents the electric current density $j$ which is assumed to flow along the
$x$-axis. $\hat{f}_{2}$ describes the spin polarization. We assume
$\hat{f}_{2}\ll\hat{f}_{1}\ll\hat{f}_{0}$, which corresponds to a relatively
small spin polarization due to the skew scattering. In this case it is
possible to neglect $f_{2}$ for the incident electrons, because it would lead
only to a small correction for $f_{2}$ of scattered electrons (that is
responsible for kinetic Rashba-Edelstein and interface spin-Hall effects).

The skew scattering should be introduced into Boltzmann equation as a boundary
condition. It is derived in Appendix C and reads

$\hat{f}_{2}({\bf p},z=0)=\frac{mV}{S|p_{z}|}\hat{r}({\bf
p})\left(\int\frac{Vd{\bf p}^{\prime}}{(2\pi\hbar)^{3}}\times\right.\\\
\left.\widehat{\cal W}\left(\frac{{\bf p}}{\hbar},\frac{{\bf
p}^{\prime}}{\hbar}\right)\bigl{(}f_{1}({\bf p}^{\prime})-f_{1}({\bf
p})\bigr{)}\right)\hat{r}({\bf p})^{+}$ (9)

Here

$\widehat{\cal W}({\bf k},{\bf
k}^{\prime})=\frac{2\pi}{\hbar}N_{I}\widehat{V}({\bf k},{\bf
k}^{\prime})\widehat{V}({\bf k}^{\prime},{\bf
k})\delta(\varepsilon_{k}-\varepsilon_{k^{\prime}})$ (10)

$N_{I}$ is the total number of impurities.

Eqs. (38,39) show that $f_{2}$ is proportional to the two-dimensional impurity
concentration $N_{I}/S$ which controls the probability $P_{nsp}$ that an
incident electron with Fermi energy is reflected non-specularly. This
probability, averaged over incident electron momenta, can be expressed as
follows:

$P_{nsp}=\frac{4\pi^{2}\hbar^{3}}{Sp_{F}^{2}}{\cal I}_{N}$ (11a) ${\cal
I}_{N}=\int\int\frac{V^{2}d{\bf p}d{\bf p}^{\prime}}{(2\pi\hbar)^{6}}{\rm
Tr}\widehat{\cal W}\left(\frac{\bf p}{\hbar},\frac{{\bf
p}^{\prime}}{\hbar}\right)\delta(\varepsilon-\varepsilon_{F})$ (11b)

In our model the macroscopic symmetry in $xy$-plane is not broken, and the
only possible spin current density $j_{s}$ in $z$ direction describes the flow
of y-polarized spins. It is conventionally expressed with the interface spin-
Hall angle

$\tan\theta_{sh}=\frac{j_{s}}{j/e}=\frac{3}{4}P_{nsc}\frac{{\cal
I}_{sh}}{{\cal I}_{N}}$ (12a) ${\cal I}_{sh}=\int\frac{V^{2}d{\bf p}d{\bf
p}^{\prime}}{(2\pi\hbar)^{6}}\left(\frac{p_{x}^{\prime}}{p_{F}}-\frac{p_{x}}{p_{F}}\right)\delta(\varepsilon-\varepsilon_{F})\times\\\
{\rm Tr}\hat{\sigma}_{y}\left[\hat{r}({\bf p})\widehat{\cal W}\left(\frac{\bf
p}{\hbar},\frac{{\bf p}^{\prime}}{\hbar}\right)\hat{r}^{+}({\bf p})\right]$
(12b)

The interface spin-Hall angle calculated with Eq. (12) is shown in Fig. 3 (a)
as a function of spin-orbit interaction for different barrier heights $U_{0}$.
Interestingly, $\theta_{sh}$ is not an odd function of $\gamma$. It is related
to the electron momentum ${\bf p}$ being an operator that does not commute
with impurity potential $V_{0}({\bf r})$. Fig. 3 (b) shows
$\theta_{sh}(\gamma)$ dependence in the double logarithm scale for
$U_{0}=10\varepsilon_{F}$ and small positive $\gamma$. The spin-Hall angle is
a high-order function of $\gamma$, $\theta_{SH}\propto\gamma^{3}$ at very
small $\gamma$ and becomes steeper with the increase of $\gamma$, up to
$\theta_{SH}\propto\gamma^{6}$. As shown in Fig. 3 (a) there is a maximum in
this dependence corresponding to $\gamma
p_{F}^{2}/\hbar\sim\sqrt{\varepsilon_{F}/U_{0}}$.

The physical mechanism of interface spin Hall effect consists of the two
parts. The first one is the skew scattering shown in Fig. 2(b). However, the
skew scattering alone is insufficient to generate the spin current if spin is
conserved during the scattering. Therefore, the interface spin Hall requires
also the asymmetric spin rotation shown in Fig. 2(c) that leads to the "spin
relaxation" which depends on spin projection. The spin relaxation is the
second order in $\gamma$ phenomenon [37], meaning that the interface spin-Hall
effect is absent in the first order approximation over $\gamma$.

Figure 3: (a) Interface Spin-Hall angle as a function of spin-orbit
interaction for various relationships between interface potential and Fermi
energy $U_{0}/\varepsilon_{F}$ as indicted in the legend. (b) The
$\theta_{sh}(\gamma)$ dependence for $U_{0}/\varepsilon_{F}=10$ in the double
logarithm scale. The blue dashed line and the black dashed-dotted line
correspond to $\theta_{sh}\propto\gamma^{6}$ and
$\theta_{sh}\propto\gamma^{3}$ respectively.

To show that the spin separation itself is a first order effect, we examine
the spin accumulation near the interface. We solve the Boltzmann equation
using the minimal model, which assumes that the scattering operator in the
bulk can be described by a single relaxation time $\tau$:
$I(\hat{f})=(\hat{f}_{0}-\hat{f})/\tau$. By applying this assumption to Eq.
(7), we obtain the solution

$\hat{f}_{2}({\bf p},z)=\hat{f}_{2}({\bf
p},0)\exp\left(-\frac{zp_{f}}{p_{z}l_{free}}\right)$ (13)

Here $l_{free}=v_{F}\tau$ is the mean free path.

The spin polarization calculated from Eq. (13) reads

$m_{y}(z)=\frac{3}{4}\frac{j}{ev_{F}}P_{nsp}\frac{{\cal I}_{s}(z)}{{\cal
I}_{N}}$ (14a) ${\cal I}_{s}(z)=\int\frac{V^{2}d{\bf p}d{\bf
p}^{\prime}}{(2\pi\hbar)^{6}}\times\\\ {\rm Tr}\sigma_{y}\hat{r}({\bf
p})\widehat{\cal W}\left(\frac{\bf p}{\hbar},\frac{{\bf
p}^{\prime}}{\hbar}\right)\hat{r}^{+}({\bf p})\times\\\
\left(\frac{p_{x}^{\prime}}{|p_{z}|}-\frac{p_{x}}{|p_{z}|}\right)\exp\left(-\frac{zp_{f}}{l_{free}|p_{z}|}\right)\delta(\varepsilon-\varepsilon_{F})$
(14b)

Figure 4: Spin accumulation $m_{y}$ due to kinetic Rashba-Edelstein effect
calculated for $U_{0}=10\varepsilon_{F}$. Panels (a) and (b) show the
dependence $m_{y}(z)$ for different values of $\gamma$. Panel (c) shows the
dependence of accumulated spin on the spin orbit interaction parameter
$\gamma$ for three different values $z/l_{free}$.

Fig. 4 shows the spin polarization $m_{y}$ calculated for
$U_{0}=10\varepsilon_{F}$. Similarly to the interface spin-Hall effect, the
polarization obtained from the Boltzmann equation appears only due to the
possibility of non-specular reflection and is normalized by $P_{nsc}$. Fig.
4(a) and (b) show the $m_{y}$ distribution over $z$ for $\gamma
p_{F}^{2}/\hbar=\pm 0.05$ and $\gamma p_{F}^{2}/\hbar=\pm 0.2$
correspondingly. For small values of $\gamma$, $m_{y}$ changes its sign as a
function of $z$ indicating the difference in the average reflection angle for
different spin projections. This change of the sign disappears at larger
values of $\gamma$ when interface spin-Hall effect becomes sufficiently strong
to dominate the spin accumulation.

Phenomenologically, the calculated spin accumulation is the Rashba-Edelstein
effect because it is confined near the interface. However, unlike the
conventional Rashba-Edelstein effect at clean interfaces studied in [19, 20],
the polarization spans up to the mean free path $l_{free}$. In the case where
the sample only contains impurities at the surface and the bulk is clean,
$l_{free}$ can be arbitrary large. Thus, the resulting spin polarization can
be referred as the kinetic Rashba-Edelstein effect.

##  IV discussion

The interface spin Hall and kinetic Rashba-Edelstein effects are closely
related to the interference of different reflected waves shown in Fig. 2 (a).
Both the effects should disappear if this interference is somehow suppressed.
For example one can consider bulk impurities near the interface as a possible
reason for interface spin Hall. However, in this case the phase of the
reflection amplitude $\hat{r}$ would be modified by the random distance
between impurity and the surface. If this distance exceeds $\hbar/p_{f}$, the
interference and the spin generation would be suppressed. Nevertheless, if the
impurities are located precisely at the interface and are small compared to
$p_{F}$, their potential $V_{I}$ is not significant: it is absorbed into non-
specular reflection probability $P_{nsp}$ in our final expressions. This is a
consequence of the calculations performed within Born approximation. We
presume that introduction of complex impurities with large size and high
potential energy can significantly modify the spin accumulation and spin
current.

The existence of interface spin-Hall effect in Born approximation makes it
fundamentally different from the ordinary spin-Hall effect that appears only
as a higher-order corrections over impurity potential $V_{I}$ and is
suppressed if the disorder is Gaussian. Although there are some predicted
mechanisms for its existence in Gaussian disorder, namely the combined
scattering from impurities and phonons [41] and the scattering at close
impurity complexes [41, 42], all these mechanisms require high orders of
perturbation theory. Interface spin Hall is possible in the Born approximation
because of the large interface potential energy that exists already in zeros
order.

The unusual properties of the kinetic Rashba-Edelstein effect and interface
spin-Hall effect suggest that in some cases the predicted interface phenomena
should dominate the spin accumulation and spin current. It happens when the
thickness $d$ of the sample is intermediate $\hbar/p_{F}\ll d\lesssim
l_{free}$ and bulk impurities that control conductivity have small potential
energy $V_{0}\ll\varepsilon_{F}$. If these conditions are met, the control of
interface properties is important to optimize the spin-torque. The effect of
interface roughness is sometimes reported in experiments [17, 43], however, to
the best of our knowledge, there is no its systematic study. We predict that
the perfect interface is not necessarily the clean one. The controlled
disorder can serve as a source of both spin polarization and spin current. It
allows to control the functionality of spin-torque devices by surface
engineering, i.e. by manipulating surface impurity concentration and their
type. Furthermore, the sensitivity of interface spin-Hall to the interface
potential $U_{0}$ provides a means of controlling spin-torque with gating.

In conclusion, our study demonstrates that the impurities at the metal-
insulator interface significantly increase the variety of surface spin
effects. The skew scattering from these impurities induces spin accumulation
that extends up to the mean free path. When combined with spin relaxation, it
gives rise to the interface spin Hall effect, which converts charge currents
to spin currents at the metal-insulator boundary.

##  V Acknowledgements

We have received funding from the European Community’s H2020 Programme under
Grant Agreement INTERFAST (H2020-FET-OPEN-965046), and from the Slovenian
Research Agency Program No.P1-0040.

##  Appendix A spin polarization near the clean interface

Our starting point is the Hamiltonian described by Eq. (1). We are interested
in its solutions when electron energy $\varepsilon$ is below the barrier
$U_{0}$. We start with rotation in the spin space with the matrix:

$U_{\sigma}=\begin{pmatrix}\tilde{k}_{\perp}&-\tilde{k}_{\perp}\\\
1&1\end{pmatrix}$ (15)

here $\tilde{k}_{\perp}=(ik_{x}+k_{y})/k_{\perp}$,
$k_{\perp}=\sqrt{k_{x}^{2}+k_{y}^{2}}$. After canonical transformation
$U^{-1}_{\sigma}HU_{\sigma}$ the Hamiltonian becomes diagonal:

$H=-{\hbar^{2}\over{2m}}{d^{2}\over{dz^{2}}}+{\hbar^{2}k_{\perp}^{2}\over{2m}}+U_{0}\theta(z)\mp\gamma
U_{0}\delta(z)k_{\perp}$ (16)

Using Eq.(16) we find the eigenfunctions:

$\psi_{\pm}(\mathbf{r}_{\perp},z)=\begin{cases}A_{\pm}\begin{pmatrix}\pm\tilde{k}_{\perp}\\\
1\end{pmatrix}\sin{(k_{\pm}z-\phi_{\pm})}\exp{(i\mathbf{k}_{\perp}\mathbf{r}_{\perp})},z<0\\\
-A_{\pm}\begin{pmatrix}\pm\tilde{k}_{\perp}\\\
1\end{pmatrix}\sin{(\phi_{\pm})}\exp{(i\mathbf{k}_{\perp}\mathbf{r}_{\perp})}\exp{(-\kappa
z)},z>0\end{cases}$ (17)

Here $A_{\pm}=\sqrt{{\kappa_{\pm}\over{\kappa_{\pm}L+1}}}$,
$\kappa=\sqrt{{2m(U_{0}-\epsilon_{\pm})\over{\hbar^{2}}}}$,
$\kappa_{\pm}=\kappa\mp{2m\gamma{U_{0}}k_{\perp}\over{\hbar^{2}}}$,
$\epsilon_{\pm}={\hbar^{2}k_{\pm}^{2}\over{2m}}$, $L$ is the size of the
sample in z direction that corresponds to the thickness of heavy metal layer.
$k_{\pm}$ is electron wavevector in $z$-direction that will become spin-
dependent after effects of finite $L$ will be taken into account. The phase
shift $\phi_{\pm}$ is determined by the equation
$\tan{\phi_{\pm}}=k_{\pm}/\kappa_{\pm}$. The energy is

$\varepsilon_{\pm}={\hbar^{2}k_{\pm}^{2}\over{2m}}+{\hbar^{2}k_{\perp}^{2}\over{2m}}$
(18)

To describe the boundary conditions for Boltzmann equation it is enough to
consider the infinite sample $L\rightarrow\infty$. In this case
$k_{+}=k_{-}=k_{z}$, $\varepsilon_{+}=\varepsilon_{-}=\varepsilon_{k}$ and the
solution of Schrodinger equation exists for an arbitrary incident wave.

$\begin{pmatrix}a\\\
b\end{pmatrix}\exp{(ik_{z}z)}\exp{(i\mathbf{k}_{\perp}\mathbf{r}_{\perp})}$
(19)

Here $(a,b)$ is an arbitrary spinor. According to Eq. (17) its reflected wave
is described as follows

$\hat{r}(k)\begin{pmatrix}a\\\
b\end{pmatrix}\exp{(-ik_{z}z)}\exp{(i\mathbf{k}_{\perp}\mathbf{r}_{\perp})}$
(20)

Here

$\hat{r}(k)=\exp{(i\pi+2i\phi_{0})}\times\\\
(\cos{(\Delta\phi)}\hat{\sigma}_{0}+i\sin{(\Delta\phi)}\hat{\sigma}_{k})$ (21)
$\hat{\sigma}_{k}={k_{y}\over{|k_{\perp}|}}\hat{\sigma}_{x}-{k_{x}\over{|k_{\perp}|}}\hat{\sigma}_{y}$
(22)

Interestingly, to account for Rashba-Edelstein effect at the clean interface,
it is important to consider $L$ to be finite. Here we assume the boundary
conditions at $z=-L$: $d\psi(\mathbf{r}_{\perp},z)/dz|_{z=-L}=0$. It leads the
following equations for $k_{\pm}$ and $\phi_{\pm}$:

$k_{\pm}={\pi\over{2L}}(2n+1)-{\phi_{\pm}\over{L}},\\\
\phi_{\pm}=\arctan{\left[{(n+1/2)\pi-\phi_{\pm}\over{L\kappa_{\pm}}}\right]},$
(23)

where $n$ is arbitrary integer. Eqs. (18,23) show that when $L$ is finite the
discrete levels for spin up and down are different, that will later lead to
the current-induced spin polarization.

To account for in-plane electric current we introduce the distribution
function of the electrons $f({\bf k})=f_{0}(\varepsilon-v\hbar k_{x})$. Here
$f_{0}$ is the Fermi function, $v$ is the drift velocity that is assumed to be
along $x$-direction. The current density $j_{x}(z)$ is defined as follows:

$j_{x}(z)=\frac{e}{L}\sum_{n}\int{d\mathbf{k}_{\perp}\over{(2\pi)^{2}}}{2\hbar
k_{x}\over{m}}\times\\\
\bigl{(}\sin{(k_{+}z-\phi_{+})}^{2}f_{0}(\varepsilon_{+}-\hbar vk_{x})\\\
+\sin{(k_{-}z-\phi_{-})}^{2}f_{0}(\varepsilon_{-}-\hbar vk_{x})\bigr{)}$ (24)

Note, that although the distribution $f({\bf k})$ does not depend on spin, the
energies $\varepsilon_{\pm}$ do.

Eq. (24) can be simplified by expanding Fermi functions over small $vk_{x}$
and integrating over the angle of ${\bf k}_{\perp}$.

$j_{x}(z)={ev{\hbar^{2}}\over{8\pi mTL}}\sum_{n}\int
dk_{\perp}k_{\perp}^{3}\times\\\
\left[\frac{\sin^{2}(k_{+}z-\phi_{+})}{{\cosh^{2}{({\varepsilon_{+}-\mu\over{2T}})}}}+\frac{\sin^{2}(k_{-}z-\phi_{-})}{{\cosh^{2}{({\varepsilon_{-}-\mu\over{2T}})}}}\right]$
(25)

Here $T$ is temperature and $\mu$ is the chemical potential. Eq. (25)
describes the distribution of current density near the interface.

To describe the spin polarization we take into account that

$(\pm\tilde{k}^{*}_{\perp},1)\sigma_{y}\begin{pmatrix}\pm\tilde{k}_{\perp}\\\
1\end{pmatrix}=\mp 2k_{x}/k_{\perp}$ (26)

It allows us to derive the expression for the distribution of spin
polarization $m_{y}(z)$

$m_{y}(z)={\frac{1}{L}}\sum_{n}\int{d\mathbf{k}_{\perp}\over{(2\pi)^{2}}}{2k_{x}\over{k_{\perp}}}\times\\\
\Bigl{[}\sin{(k_{+}z-\phi_{+})}^{2}f_{0}(\varepsilon_{+}-\hbar vk_{x})\\\
-\sin{(k_{-}z-\phi_{-})}^{2}f_{0}(\varepsilon_{-}-\hbar vk_{x})\Bigr{]}$ (27)

After expanding it over small $vk_{x}$ and performing the angle integration we
obtain:

$m_{y}(z)={v\hbar\over{8\pi TL}}\sum_{n}\int dk_{\perp}k_{\perp}^{2}\times\\\
\left[\frac{\sin^{2}{(k_{+}z-\phi_{+})}}{{\cosh^{2}{({\varepsilon_{+}-\mu\over{2T}})}}}-\frac{\sin^{2}{(k_{-}z-\phi_{-})}}{{\cosh^{2}{({\varepsilon_{-}-\mu\over{2T}})}}}\right]$
(28)

This formula describes the spin accumulation near the interface when current
is flowing parallel to the surface. Fig. 5 represents calculated current
density and the accumulated spin density as a function of $z$ (Rashba-
Edelshtein effect). The current density is normalized to
$j_{x,0}=evk_{F}^{3}/3\pi^{2}$ which is the current density far from
interface. Fig. 5 shows that in a clean sample the spin polarization exists
only at the distances $\sim\hbar/p_{F}$ from the surface where the current is
modified by quantum effects.

Note also, that the wavefunctions (17) do not include density flux ${\rm
Im}\psi_{\pm}^{*}\nabla\psi_{\pm}=0$ and the spin density (28) is not
accompanied by spin current.

##  Appendix B spin-dependent scattering at the interface impurities

In the general case, impurity scattering requires density matrix formalism for
its description. The usual procedure for such a description starts with the
density matrix $\hat{\rho}({\bf k})$ diagonal in the space of wavefunctions
$\Psi_{\bf k}$ and considers non-diagonal terms $\hat{\rho}({\bf k},{\bf
k}^{\prime})$ as a small perturbation. However, the terms $\hat{\rho}({\bf
k})$ diagonal in the space of $\Psi_{\bf k}$ are still $2\times 2$ matrices in
spin space.

The master equation for the density matrix reads

$\frac{\partial\hat{\rho}}{\partial
t}=\frac{i}{\hbar}(\hat{\rho}\hat{V}-\hat{V}\hat{\rho}).$ (29)

Here $V$ is the impurity energy that includes both potential energy and spin-
orbital correction (6).

Figure 5: The distribution of the current density near the surface (a) and
spin density as a function of z (b).

We assume the slow variance of diagonal matrix elements compared to the
oscillation frequencies $(\varepsilon_{\bf k}-\varepsilon_{{\bf
k}^{\prime}})/\hbar$. It allows us to derive expression for the perturbation
$\hat{\rho}({\bf k},{\bf k}^{\prime})$ from Eq. (29)

$\hat{\rho}({\bf k},{\bf k}^{\prime})=\frac{\exp\left(i\frac{\varepsilon_{\bf
k}-\varepsilon_{\bf k^{\prime}}}{\hbar}t\right)}{\varepsilon_{\bf
k}-\varepsilon_{\bf k^{\prime}}-i\delta}\times\left(\hat{\rho}({\bf
k})\hat{V}({\bf k}{\bf k}^{\prime})-\hat{V}({\bf k}{\bf
k}^{\prime})\hat{\rho}({\bf k}^{\prime})\right)$ (30)

This equation should be substituted back to Eq. (29) where we now keep only
the non-oscillating terms

$\frac{\partial\hat{\rho}({\bf k})}{\partial t}=\sum_{\bf
k^{\prime}}\frac{i/\hbar}{\varepsilon_{{\bf k}^{\prime}}-\varepsilon_{\bf
k}-i\delta}\times\left(\hat{V}({\bf k}{\bf k}^{\prime})\hat{V}({\bf
k}^{\prime}{\bf k})\hat{\rho}({\bf k})-\hat{V}({\bf k}{\bf
k}^{\prime})\hat{\rho}({\bf k}^{\prime})\hat{V}({\bf k}^{\prime}{\bf
k})\right)-\\\ \frac{i/\hbar}{\varepsilon_{{\bf k}}-\varepsilon_{{\bf
k}^{\prime}}-i\delta}\times\left(\hat{V}({\bf k}{\bf
k}^{\prime})\hat{\rho}({\bf k}^{\prime})\hat{V}({\bf k}^{\prime}{\bf
k})-\hat{\rho}({\bf k})\hat{V}({\bf k}{\bf k}^{\prime})\hat{V}({\bf
k}^{\prime}{\bf k})\right)$ (31)

Here the real part of $1/(\varepsilon_{{\bf k}^{\prime}}-\varepsilon_{\bf
k}-i\delta)$ corresponds to the small modification of electron states due to
impurity potential that can be neglected. Imaginary part leads to actual
transitions between states. They can be described with the equations

$\frac{\partial\rho_{ij}({\bf k})}{\partial t}=\int\frac{Vd{\bf
k^{\prime}}}{(2\pi)^{3}}\left(W_{ij,lm}^{(in)}({\bf k},{\bf
k^{\prime}})\rho_{lm}({\bf k}^{\prime})\right.\left.-W_{ij,lm}^{(out)}({\bf
k},{\bf k^{\prime}})\rho_{lm}({\bf k})\right)$ (32) ${W}_{ij,lm}^{(in)}({\bf
k},{\bf k^{\prime}})\rho_{lm}({\bf k}^{\prime})=\frac{2\pi}{\hbar}N_{I}\times
V_{il}({\bf k},{\bf k}^{\prime})\rho_{lm}({\bf k}^{\prime})V_{mj}({\bf
k}^{\prime},{\bf k})\delta(\varepsilon_{k}-\varepsilon_{k^{\prime}})$ (33)
${W}_{ij,lm}^{(out)}({\bf k},{\bf k^{\prime}})\rho_{lm}({\bf
k})=\frac{\pi}{\hbar}N_{I}\times\Bigl{[}V_{in}({\bf k},{\bf
k}^{\prime})V_{nl}({\bf k}^{\prime},{\bf k})\rho_{lm}({\bf
k})\delta_{jm}+\rho_{lm}({\bf k})V_{mn}({\bf k},{\bf k}^{\prime})V_{nj}({\bf
k}^{\prime},{\bf
k})\delta_{il}\Bigr{]}\delta(\varepsilon_{k}-\varepsilon_{k^{\prime}})$ (34)

Here $N_{I}$ is the total number of impurities. $W_{ij,lm}^{(in)}({\bf k},{\bf
k^{\prime}})$ describes the electrons scattering from the state ${\bf
k}^{\prime}$ to the state ${\bf k}$. Because both the initial and the final
states of electrons are described with $2\times 2$ density matrix,
$W_{ij,lm}^{(in)}({\bf k},{\bf k^{\prime}})$ is, therefore, a four-dimensional
$2\times 2\times 2\times 2$ matrix. ${W}_{ij,lm}^{(out)}({\bf k},{\bf
k^{\prime}})$ is the out-scattering term that stands for the backward
transition from ${\bf k}$ to ${\bf k}^{\prime}$. It is different from
${W}_{ij,lm}^{(in)}({\bf k},{\bf k^{\prime}})$ because the spin is not
conserved during the scattering.

For a given density matrix $\hat{\rho}({\bf k}^{\prime})$ of incident
electrons,

${W}_{ij,lm}^{(in)}({\bf k},{\bf k^{\prime}})\rho_{lm}({\bf k}^{\prime})$ (35)

represents the properties of the scattered ones. In particular the results
shown in Fig. 2(b,c) are calculated with Eq. (35) where $\hat{\rho}({\bf
k}^{\prime})=(\hat{1}\pm\sigma_{y})/2$ and ${\bf
k}^{\prime}=(p_{F}/\sqrt{2}\hbar)(1,0,1)$. The scattering probabilities
correspond to ${\rm Tr}\widehat{W}^{(in)}({\bf k},{\bf
k^{\prime}})\hat{\rho}({\bf k}^{\prime})={W}_{ii,lm}^{(in)}({\bf k},{\bf
k^{\prime}})\rho_{lm}({\bf k}^{\prime})$ and the spin polarization vector
${\bf s}$ to ${\rm Tr}\bm{\sigma}r({\bf k})\widehat{W}^{(in)}({\bf k},{\bf
k^{\prime}})\hat{\rho}({\bf k}^{\prime})r^{+}({\bf k})=r^{+}_{ii_{1}}({\bf
k})\bm{\sigma}_{i_{1}i_{2}}r_{i_{2}j}({\bf k}){W}_{ji,lm}^{(in)}({\bf k},{\bf
k^{\prime}})\rho_{lm}({\bf k}^{\prime})$.

##  Appendix C Boundary conditions for the Boltzmann equation

The approach (32) is based on the wavefunctions $\Psi_{\alpha}({\bf k})$
defined in the main text. They correspond to the coherence of incident and
specularly reflected electrons. However, in the bulk of the film this
coherence is lost due to the scattering at the bulk impurities and The
Boltzmann equation approach is based on the distribution function
$\hat{f}({\bf p})$ that neglects such a coherence.

To relate the two approaches we consider the spin polarization generated per
unit time. Because the kinetic equation is linear we can decouple it into
$G_{\alpha}({\bf p})d{\bf p}$ \- the spin polarization in the
$\alpha$-direction related to the reflected electrons with the momentum ${\bf
p}$. In terms of Boltzmann equation it is expressed as follows

$G_{\alpha}({\bf p})d{\bf p}=\frac{d{\bf p}}{(2\pi\hbar)^{3}}{\rm
Tr}\hat{\sigma}_{\alpha}\hat{f}_{2}({\bf p})\frac{|p_{z}|}{m}S$ (36)

Here $\hat{f}_{2}({\bf p})$ is taken at the interface and is assumed not to
depend on the exact point of the interface. $S$ is the interface area. Note
that $p_{z}$ is negative for the reflected electrons.

In terms of Eq. (32) $G_{\alpha}({\bf p})d{\bf p}$ is equal to

$G_{\alpha}({\bf k})d{\bf k}=\frac{Vd{\bf k}}{(2\pi)^{3}}{\rm Tr}\hat{r}_{\bf
k}^{+}\hat{\sigma}_{\alpha}\hat{r}_{\bf k}\times\int\frac{Vd{\bf
k}^{\prime}}{(2\pi)^{3}}\left(W_{ij,lm}^{(in)}({\bf k},{\bf
k}^{\prime})\rho_{lm}({\bf k}^{\prime})-\right.\left.W_{ij,lm}^{(out)}({\bf
k},{\bf k}^{\prime})\rho_{lm}({\bf k})\right)$ (37)

When $\hat{f}_{2}\ll\hat{f}_{1}\ll\hat{f}_{0}$ one can substitute
$\hat{\rho}({\bf k})$ with $\hat{f}_{0}(\hbar k_{x},\hbar
k_{y},\hbar|k_{z}|)+\hat{f}_{1}(\hbar k_{x},\hbar k_{y},\hbar|k_{z}|)$ in the
r.h.s. of Eq. (37). However, $\hat{f}_{0}$ does not lead to spin polarization
and can be dropped.

It allows us to derive the following boundary condition for $\hat{f}_{2}$ at
$z=0$.

$\hat{f}_{2}({\bf p})=\frac{mV}{S|p_{z}|}\hat{r}({\bf
p})\left(\int\frac{Vd{\bf
p}^{\prime}}{(2\pi\hbar)^{3}}\times\right.\left.\widehat{\cal
W}\left(\frac{{\bf p}}{\hbar},\frac{{\bf
p}^{\prime}}{\hbar}\right)\bigl{(}f_{1}({\bf p}^{\prime})-f_{1}({\bf
p})\bigr{)}\right)\hat{r}({\bf p})^{+}$ (38)

Here

${\cal W}_{ij}=W_{ij,ll}^{(in)}=W_{ij,ll}^{(out)},$ (39)

and we took into account that $\hat{f}_{1}=f_{1}\hat{1}$.

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