# Spin pumping by a moving domain wall at the interface of an
antiferromagnetic insulator and a two-dimensional metal

A. G. Mal’shukov  Institute of Spectroscopy, Russian Academy of Sciences,
Troitsk, Moscow, 108840, Russia

###### Abstract

A domain wall (DW) which moves parallel to a magnetically compensated
interface between an antiferromagnetic insulator (AFMI) and a two-dimensional
(2D) metal can pump spin polarization into the metal. It is assumed that
localized spins of a collinear AFMI interact with itinerant electrons through
their exchange interaction on the interface. We employed the formalism of
Keldysh Green’s functions for electrons which experience potential and spin-
orbit scattering on random impurities. This formalism allows a unified
analysis of spin pumping, spin diffusion and spin relaxation effects on a 2D
electron gas. It is shown that the pumping of a nonstaggered magnetization
into the metal film takes place in the second order with respect to the
interface exchange interaction. At sufficiently weak spin relaxation this
pumping effect can be much stronger than the first-order effect of the Pauli
magnetism which is produced by the small nonstaggered exchange field of the
DW. It is shown that the pumped polarization is sensitive to the geometry of
the electron’s Fermi surface and increases when the wave vector of the
staggered magnetization approaches the nesting vector of the Fermi surface. In
a disordered diffusive electron gas the induced spin polarization follows the
motion of the domain wall. It is distributed asymmetrically around the DW over
a distance which can be much larger than the DW width.

##  I Introduction

Antiferromagnets (AFM) have drawn growing interest recently due to their
potential use for various spintronic applications. One of the most important
characteristics of spintronic devices is their ability to transmit and control
spin polarization. From this point of view AFM materials demonstrate numerous
interesting features. The progress made in this field was presented in several
reviews (see, for instance Baltz ; Gomonay ; Yan ; Wadley ). Considerable
progress has been achieved in understanding of mechanisms for angular moment
transfer between spins of localized and itinerant electrons in metallic AFM,
as well as interface spin transfer between a normal metal and a metallic or
insulating AFM Zelezny ; Cheng ; Saidaoui ; Swaving ; Takei ; Nunez ; Ohnuma .
These mechanisms allow to control localized spins of AFM, as well as spins of
itinerant electrons. For instance, the spin current of electrons produces the
torque effect on the staggered AFM magnetization. Recent experimental studies
have demonstrated that this torque results in rotation of the magnetization
and its switching Zhang ; Cogulu ; Omari ; Wadley . Alternatively, when the
Néel order varies in time the magnetization can be pumped through the
interface into the electron gas of a paramagnetic metal which makes a contact
with an AFM Cheng ; Frangou . In this case the spin polarization may be
delivered to the interface by spin waves Cheng ; Vaidya ; Li ; Wang , or by
moving topological defects, like DWs and skyrmions. Compared with
ferromagnets, spin waves and topological spin textures exhibit much faster
dynamics in AFMs. For instance, in antiferromagnetic insulators spin waves can
propagate over relatively large (submicron) distances Lebrun  and DWs can move
much faster than in ferromagnets Gomonay2 ; Kim . These outstanding features
of conducting and insulating antiferromagnets form the basis for their future
applications in spintronic devices.

So far, the activity in studying the spin pumping from an AFMI into a normal
metal was focused on three dimensional (3D) metals. On the other hand, there
is a great interest in heterostructures which are combined of magnetic systems
and 2D metals. In particular, this interest is caused by recent success in
creating of various 2D van der Waals metallic and insulating systems. However,
the problem of spin pumping from an AFMI’s space-time dependent spin texture
into a 2D metal film was not addressed in literature. At the same time, there
are some significant distinctions between 3D and 2D cases. First of all, 2D
electrons undergo scattering from an interfacial spin texture which has the
same 2D dimensionality. Therefore, constructive interference of spin dependent
scattering amplitudes from two AFM sublattices can result in a strong
enhancement of the scattering probability. This effect becomes important when
the Fermi surface reveals nesting parts with roughly the same wave vector as
that of the staggered magnetization. In contrast, in 3D systems such an
interference effect is smeared out due to integration over $k_{z}$, where
$k_{z}$ is the component of the electron’s wave-vector which is perpendicular
to the interface. One more specific feature of 2D systems is that electronic
transport takes place along the interface, so that electrons are always in
contact with localized spins of the AFM, while in 3D systems the angular
moment, which electrons obtain from a dynamic AFMI texture, is carried away
from the interface. Therefore, with a good accuracy a 3D metal can be
considered as a spin sink for electrons, that is not true for 2D systems. In
the former case, the angular moment transfer is controlled by the so called
interface spin mixing conductance Tserkovnyak ; Cheng  which is simply a local
characteristic of a given interface. Such an approach can not be applied for
the analysis of the spin transfer across the AFMI/2D metal interface, because
it is closely related to the lateral transport of 2D electrons. Therefore, one
needs a unified theory which combines quantum dynamics of 2D electrons with
their scattering from a space-time dependent spin texture of AFMI.

In this context the problem of spin pumping by a domain wall, which moves
along the magnetically compensated interface, has not been addressed so far.
To solve this problem we employ the Keldysh Keldysh  formalism of
nonequilibrium Green’s functions for a disordered 2D electron gas which
interacts with localized spins of an adjacent AFMI by means of the exchange
interaction $J$. Besides the potential scattering from random impurities, the
spin-orbit scattering of electrons will also be taken into account. The latter
gives rise to relaxation of the spin polarization of itinerant electrons. As a
result, in the diffusive regime the spin density distribution will diffusively
evolve in space and decrease in time with some spin relaxation rate.

This problem will be considered within a simple tight binding model where 2D
and AFMI lattices form a commensurate contact. The exchange interaction is
treated within the perturbation theory which is valid when $J\ll E_{F}$, where
$E_{F}$ is the Fermi energy of conduction electrons. The Fermi level is placed
not too close to the van Hove singularity, where a gap in the electron band
energy is formed due to the interface exchange interaction with AFMI. Baltz
Within the perturbation theory the pumping of (nonstaggered) spin polarization
into a normal metal by the staggered Néel magnetization takes place only in
even orders of the perturbational expansion with respect to $J$. On the other
hand, besides the staggered magnetization, a time dependent spin texture of a
moving DW carries a small nonstaggered component Baryakhtar  which is
localized near the DW. In turn, due to the interface exchange interaction such
a ”ferromagnetic” magnetization polarizes spins of itinerant electrons in
adjacent normal metal already in the first order by $J$. However, as it will
be shown, in a reasonable range of parameters the effect of second order
perturbational terms may exceed considerably that of the first order ones,
because these competing effects involve very different physical mechanisms.
Indeed, in the former case the angular moment, which is carried by a DW, is
transferred through the interface to electrons. This process leads to
accumulation of the electron’s spin polarization near DW and, as it will be
shown, the latter increases with the spin relaxation time. In contrast, the
first-order effect is simply the Pauli magnetization. In the absence of spin
relaxation it does not increase in time. Instead, it simply redistributes in
space, by following the motion of the DW.

The article is organized in the following way. In Sec.II a general formalism
of the spin density response in a 2D disordered electron gas to a moving DW is
expressed in terms of Green’s functions, up to the second-order with respect
to the exchange interaction. Sec. III is devoted to calculations of the spin
polarization. The results are discussed in Sec. IV. Two section are added in
the Appendix in order to clarify some details of calculations.

##  II General formalism

###  II.1 Basic equations

In this section we express the spin polarization in 2D electron gas as an
expansion over the exchange interaction between itinerant electrons and
localized spins of an adjacent AFMI. By assuming that the 2D lattice of the
normal metal is commensurate with the lattice of localized spins on the AFMI
interface and that metal atoms make an on-top contact with atoms of the AFMI,
the exchange interaction can be written in the form

$M=J\sum_{i}c^{+}_{i}\mathbf{S}_{i}(t)\cdot\bm{\sigma}c_{i}\,.$ (1)

where $c^{+}_{i}=(c^{+}_{i\uparrow},c^{+}_{i\downarrow})$ is the two-component
creation operator of an electron whose spin projections are $\uparrow$, or
$\downarrow$ and $c_{i}$ is the conjugate to $c^{+}_{i}$ destruction operator.
The vector $\mathbf{S}_{i}(t)$ represents a spin which is localized on the
lattice site $\mathbf{r}_{i}$ and
$\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the vector of Pauli
matrices. Since there are two sublattices, the lattice sites will be denoted
as $i1$ and $i2$. Correspondingly, spins localized on these sublattices will
be denoted as $\mathbf{S}_{i1}$ and $\mathbf{S}_{i2}$. These spins are treated
as classical variables satisfying the constraint $|\mathbf{S}_{i}(t)|=S$. In
many practical situations $\mathbf{S}_{i}$ varies slowly within each of two
AFM sublattices. Therefore, one may introduce two vector fields
$\mathbf{m}_{1}(\mathbf{r},t)$ and $\mathbf{m}_{2}(\mathbf{r},t)$, which are
defined on sublattices 1 and 2, respectively, where
$\mathbf{m}_{1(2)}(\mathbf{r}_{i},t)=\mathbf{S}_{i1(2)}/S$. The Néel order is
given by the unit vector field
$\mathbf{n}(\mathbf{r},t)=(\mathbf{m}_{1}(\mathbf{r},t)-\mathbf{m}_{2}(\mathbf{r},t))/|\mathbf{m}_{1}(\mathbf{r},t)-\mathbf{m}_{2}(\mathbf{r},t)|$.
Due to the strong exchange coupling of spins in different sublattices we have
$\mathbf{m}_{1}(\mathbf{r},t)\simeq-\mathbf{m}_{2}(\mathbf{r},t)$. Therefore,
the nonstaggered field
$\mathbf{m}(\mathbf{r},t)=(\mathbf{m}_{1}(\mathbf{r},t)+\mathbf{m}_{2}(\mathbf{r},t))/2\ll
1$. In this case by using the Landau-Lifshitz-Gilbert equation $\mathbf{m}$
can be expressed Baryakhtar  in terms of $\mathbf{n}$, as

$\mathbf{m}(\mathbf{r},t))=\frac{1}{E_{\mathrm{ex}}}(\partial_{t}\mathbf{n}(\mathbf{r},t))\times\mathbf{n}(\mathbf{r},t)))\,,$
(2)

where $E_{\mathrm{ex}}$ is the exchange energy of near-neighbor spins in AFM.
Further, by expressing the operators $c_{i}$ as
$c_{i}=\sum_{\mathbf{k}}c_{\mathbf{k}}\exp i\mathbf{k}\mathbf{r}_{i}$, matrix
elements of the exchange interaction in Eq.(1) can be written in terms of
$\mathbf{n}_{\mathbf{f}}(t)$ and $\mathbf{m}_{\mathbf{f}}(t)$, which are
spatial Fourier transforms of these fields with the wave vector $\mathbf{f}$.
These matrix elements are given by

$M_{\mathbf{k}+\mathbf{f},\mathbf{k}}(t)=JS[c^{+}_{\mathbf{k}+\mathbf{f}+\mathbf{G}}\mathbf{n}_{\mathbf{f}}(t)\bm{\sigma}c_{\mathbf{k}}+c^{+}_{\mathbf{k}+\mathbf{f}}\mathbf{m}_{\mathbf{f}}(t)\bm{\sigma}c_{\mathbf{k}}]\,,$
(3)

where $\mathbf{G}=(G_{x},G_{y})$ is the umklapp vector which is associated
with the AFM’s staggered magnetization, so that for the square lattice
$G_{x}=\pm\pi/a$ and $G_{y}=\pm\pi/a$. In contrast, the vector $\mathbf{f}$ is
small, namely, $f\ll 1/a$, where $a$ is the lattice constant. It is because
relatively slow spatial variations of $\mathbf{n}(\mathbf{r},t)$ and
$\mathbf{m}(\mathbf{r},t)$ are determined by the spin texture of DW whose
width is assumed to be much larger than $a$.

Figure 1: (Color online) A bilayer system is composed of an antiferromagnetic
insulator and a 2D paramagnetic metal film. A domain wall which moves from
left to right with the velocity $v$ leads to spin accumulation in the metal
film. The spin is oriented parallel to the $y$-axis. The corresponding spin
density (shown by color) is asymmetrically distributed around DW over
distances which can be much larger than the width of the DW.

The induced spin density of electrons can be expressed in terms of the Keldysh
Keldysh ; Rammer  function
$G^{K}(\mathbf{r}.t;\mathbf{r}^{\prime},t^{\prime})$, which is a 2$\times$2
matrix in the spin space. This expression has the form

$\mathbf{S}(\mathbf{r},t)=-\frac{i}{4}\mathrm{Tr}[\bm{\sigma}\langle
G^{K}(\mathbf{r},t;\mathbf{r},t)\rangle_{\text{imp}}]\,,$ (4)

where $\langle...\rangle_{\text{imp}}$ denotes averaging over impurity
positions. The Keldysh function, in turn, is given by the perturbational
expansion over the exchange interaction $M$. The corresponding correction
$\delta G^{K}(\mathbf{q},\omega)$ to the space-time Fourier transform of
$G^{K}(\mathbf{r},t;\mathbf{r},t)$ can be written in terms of the unperturbed
Green’s functions $G^{K}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega)$,
$G^{r}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega)$ and
$G^{a}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega)$, which are, respectively, the
Keldysh, retarded and advanced ones. These functions are not averaged over
impurity positions. Therefore, they depend on the two wave vectors
$\mathbf{k}$ and $\mathbf{k}^{\prime}$. They may be represented by the
2$\times$2 matrix $\hat{G}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega)$ which is
given by

$\hat{G}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega)=\left[\begin{matrix}G^{r}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega)&G^{K}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega)\\\
0&G^{a}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega)\end{matrix}\right]\,,$ (5)

where in thermal equilibrium $G^{K}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega)$
has the form

$G^{K}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega)=\left(G^{r}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega)-G^{a}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega)\right)\tanh(\omega/2k_{B}T)$
(6)

Within the Keldysh formalism Keldysh ; Rammer  the correction $\delta
G^{K}(\mathbf{q},\omega)$ is given by

$\displaystyle\delta
G^{K}(\mathbf{q},\omega)=\int\frac{d\omega^{\prime}}{2\pi}\sum_{\mathbf{k,p,p}^{\prime}}\left[\hat{G}_{\mathbf{k}^{+},\mathbf{p}^{+}}\left(\omega^{\prime+}\right)\times\right.$
$\displaystyle\left.\hat{\Sigma}_{\mathbf{p},\mathbf{p}^{\prime}}(\omega^{\prime},\omega,\mathbf{q})\hat{G}_{\mathbf{p}^{\prime-},\mathbf{k}^{-}}\left(\omega^{\prime-}\right)\right]^{K}\,,$
(7)

where $\omega^{\prime\pm}=\omega^{\prime}\pm\omega/2$,
$\mathbf{k}^{\pm}=\mathbf{k}\pm\mathbf{q}/2$ and the superscript ”K” denotes
the Keldysh component of the matrix product in Eq.(II.1). By performing the
time Fourier transform of $\mathbf{n}_{\mathbf{f}}(t)$ and
$\mathbf{m}_{\mathbf{f}}(t)$ in Eq.(3) one can express $\hat{\Sigma}$ in the
form

$\displaystyle\hat{\Sigma}_{\mathbf{p},\mathbf{p}^{\prime}}(\omega^{\prime},\omega,\mathbf{q})$
$\displaystyle=$
$\displaystyle\hat{\Sigma}^{(1)}(\omega,\mathbf{q})\delta_{\mathbf{p},\mathbf{p}^{\prime}}+$
(8)
$\displaystyle\int\frac{d\nu}{2\pi}\sum_{\mathbf{f}}\hat{\Sigma}^{(2)}_{\mathbf{p},\mathbf{p}^{\prime}}(\omega^{\prime},\omega,\mathbf{q};\nu,\mathbf{f})\,,$

where the functions $\hat{\Sigma}^{(1)}$ and $\hat{\Sigma}^{(2)}$ correspond
to the first-order and second-order corrections, respectively. They are given
by

$\hat{\Sigma}^{(1)}(\omega,\mathbf{q})=\hat{1}JS\bm{\sigma}\mathbf{m}_{\mathbf{q},\omega}$
(9)

and

$\displaystyle\hat{\Sigma}^{(2)}_{\mathbf{p},\mathbf{p}^{\prime}}(\omega^{\prime},\omega,\mathbf{q};\nu,\mathbf{f})=2J^{2}S^{2}\left(\bm{\sigma}\mathbf{n}_{\mathbf{f}^{+},\nu^{+}}\right)\times$
$\displaystyle\hat{G}_{\mathbf{p}-\mathbf{f}+\mathbf{G},\mathbf{p}^{\prime}-\mathbf{f}+\mathbf{G}}(\omega^{\prime}-\nu)(\bm{\sigma}\mathbf{n}^{*}_{\mathbf{f}^{-},\nu^{-}})\,,$
(10)

where $\nu^{\pm}=\nu\pm\omega/2$, $\mathbf{f}^{\pm}=\mathbf{f}\pm\mathbf{q}/2$
and $\hat{1}$ is the unit matrix in the Keldysh space.

Figure 2: (Color online) a) A second-order Feynman diagram for the spin
density $\mathbf{S}(\mathbf{q},\omega)$, which is induced by a moving domain
wall. The DW perturbation Eq.(3) is shown by red arrows. b) The Bethe-Salpeter
equation for the spin diffusion vertex $\mathbf{V}$, where $\bm{\sigma}$ is
the vector of Pauli matrices. Elastic scattering from random impurirties is
shown by the dashed line.

The second order contribution to $\mathbf{S}(\mathbf{r},t)$ is shown as a
Feynman diagram in Fig.2, while the first-order term is given by a usual
fermion loop. The averaging in Eq.(4) over random positions of impurities
results Altshuler  in the occurrence of average Green functions and the vertex
$\mathbf{V}$ in Fig.2. This vertex describes multiple scattering processes
which result in diffusion of particles and their spin relaxation. Within the
Born approximation the calculation of $\mathbf{V}$ is reduced to a solving of
the Bethe-Salpeter equation which is graphically shown in Fig.2. The multiple
scattering processes are important when the frequency and momentum transfer in
the vertex are much smaller than $\Gamma=1/2\tau$ and $1/l$, where $\tau$ and
$l$ are the elastic scattering time and electron mean free path, respectively.
Therefore, from Fig.2 it is seen that such a diffusion regime takes place when
$\omega\ll\Gamma$ and $q\ll 1/l$. At the same time, the vertices which are
associated with the exchange interaction of electrons with the staggered
magnetization retain unrenormalized, because of the large momentum transfer
$\sim\mathbf{G}$ caused by such a magnetization. By using Eqs.(5,6) and
Eqs.(9,II.1) one may express $\langle\delta G^{K}\rangle_{\mathrm{imp}}$ in
Eq.(II.1), as well as the space-time Fourier transform
$\mathbf{S}(\mathbf{q},\omega)$ of Eq.(4), in terms of averaged retarded and
advanced Green’s functions. The total spin polarization may be represented by
a sum of terms which are renormalized by $\mathbf{V}$ and those which are not.
The former will be denoted as $\mathbf{S}_{V}(\mathbf{q},\omega)$. It includes
only vertices which involve the product of retarded and advanced functions in
the ladder series. Otherwise, the renormalization is not important Altshuler .
At the same time the unrenormalized term is given by the bare vertex
$\bm{\sigma}$ instead of $\mathbf{V}$. The corresponding bare contribution to
$\mathbf{S}(\mathbf{q},\omega)$ will be denoted as
$\mathbf{S}_{0}(\mathbf{q},\omega)$.

It is convenient to express the products of Green’s functions in Eq.(II.1) in
terms of retarded and advanced functions by using Eqs.(5) and (6). For
example, the ”Keldysh” component (12 matrix component in Eq.(5)) of a product
of three functions can be written as

$\displaystyle(G_{1}G_{2}G_{3})^{K}=(G^{r}_{1}-G^{a}_{1})G^{a}_{2}G^{a}_{3}\tanh\frac{\omega_{1}}{2k_{B}T}+$
$\displaystyle
G^{r}_{1}(G^{r}_{2}-G^{a}_{2})G^{a}_{3}\tanh\frac{\omega_{2}}{2k_{B}T}+$
$\displaystyle
G^{r}_{1}G^{r}_{2}(G^{r}_{3}-G^{a}_{3})\tanh\frac{\omega_{3}}{2k_{B}T}\,,$
(11)

where the subscripts 1,2,3 denote variables of the Green functions. By
combining all terms, which are generated by the product of Keldysh matrices in
Eq.(II.1), we arrive at
$\mathbf{S}=\mathbf{S}^{(1)}_{V}+\mathbf{S}^{(2)}_{V}+\mathbf{S}^{(1)}_{0}+\mathbf{S}^{(2)}_{0}$,
where

$\displaystyle\mathbf{S}^{(1)}_{V}(\mathbf{q},\mathbf{\omega})=-\frac{i}{4}\sum_{\mathbf{k}}\int\frac{d\omega^{\prime}}{2\pi}\mathrm{Tr}[\mathbf{V}(\omega,\mathbf{q})G^{r}_{\mathbf{k}^{+}}\left(\omega^{\prime+}\right)\times$
$\displaystyle\Sigma^{(1)}(\omega,\mathbf{q})G^{a}_{\mathbf{k}^{-}}\left(\omega^{\prime-}\right)]\left(\tanh\frac{\omega^{\prime+}}{2k_{B}T}-\tanh\frac{\omega^{\prime-}}{2k_{B}T}\right)\,,$
(12)
$\displaystyle\mathbf{S}^{(1)}_{0}(\mathbf{q},\mathbf{\omega})=-\frac{i}{4}\sum_{\mathbf{k}}\int\frac{d\omega^{\prime}}{2\pi}\mathrm{Tr}\left[\bm{\sigma}G^{r}_{\mathbf{k}^{+}}\left(\omega^{\prime+}\right)\Sigma^{(1)}(\omega,\mathbf{q})\times\right.$
$\displaystyle\left.G^{r}_{\mathbf{k}^{-}}\left(\omega^{\prime-}\right)\tanh\frac{\omega^{\prime-}}{2k_{B}T}-\bm{\sigma}G^{a}_{\mathbf{k}^{+}}\left(\omega^{\prime+}\right)\Sigma^{(1)}(\omega,\mathbf{q})\times\right.$
$\displaystyle\left.G^{a}_{\mathbf{k}^{-}}\left(\omega^{\prime-}\right)\tanh\frac{\omega^{\prime+}}{2k_{B}T}\right]\,,$
(13)

$\displaystyle\mathbf{S}_{V}^{(2)}(\mathbf{q},\mathbf{\omega})=-\frac{i}{4}\sum_{\mathbf{k},\mathbf{f}}\int\frac{d\omega^{\prime}}{2\pi}\frac{d\nu}{2\pi}\mathrm{Tr}\left[\mathbf{V}(\omega,\mathbf{q})(P^{raa}(\omega^{\prime+})\right.$
$\displaystyle\left.-P^{rra}(\omega^{\prime-})+P^{rra}(\omega^{\prime}-\nu)-P^{raa}(\omega^{\prime}-\nu))\right]$
(14)

and

$\displaystyle\mathbf{S}_{0}^{(2)}(\mathbf{q},\mathbf{\omega})=-\frac{i}{4}\sum_{\mathbf{k},\mathbf{f}}\int\frac{d\omega^{\prime}}{2\pi}\frac{d\nu}{2\pi}\mathrm{Tr}\left[\bm{\sigma}(P^{rrr}(\omega^{\prime-})-\right.$
$\displaystyle\left.P^{aaa}(\omega^{\prime+}))\right]\,,$ (15)

where the functions $P^{ijk}(\Omega)$, with $\Omega=\omega^{\prime\pm}$ and
$\Omega=\omega^{\prime}-\nu$, are given by

$\displaystyle
P^{ijk}(\Omega)=G^{i}_{\mathbf{k}^{+}}\left(\omega^{\prime+}\right)\Sigma^{(2)j}_{\mathbf{k}}(\omega^{\prime},\omega,\mathbf{q};\nu,\mathbf{f})\times$
$\displaystyle
G^{k}_{\mathbf{k}^{-}}\left(\omega^{\prime-}\right)\tanh\left(\frac{\Omega}{2k_{B}T}\right)\,,$
(16)

where $G^{i}_{\mathbf{k}}(\omega)=\langle
G^{i}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega)\rangle_{\mathrm{imp}}\delta_{\mathbf{k},\mathbf{k}^{\prime}}$,
$\Sigma^{(2)i}_{\mathbf{k}}(\omega^{\prime},\omega,\mathbf{q};\nu,f)=\langle\Sigma^{(2)i}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega^{\prime},\omega,\mathbf{q};\nu,f)\rangle_{\mathrm{imp}}\delta_{\mathbf{k},\mathbf{k}^{\prime}}$
and $i=r,a$. It follows from Eq.(II.1) that after the averaging over impurity
positions the function
$\Sigma^{(2)i}_{\mathbf{k},\mathbf{k}^{\prime}}(\omega^{\prime},\omega,\mathbf{q};\nu,f)$
becomes

$\displaystyle\Sigma^{(2)i}_{\mathbf{k}}(\omega^{\prime},\omega,\mathbf{q};\nu,f)=2J^{2}S^{2}\left(\bm{\sigma}\mathbf{n}_{\mathbf{f}^{+},\nu^{+}}\right)\times$
$\displaystyle
G^{i}_{\mathbf{k}-\mathbf{f}+\mathbf{G}}(\omega^{\prime}-\nu)(\bm{\sigma}\mathbf{n}^{*}_{\mathbf{f}^{-},\nu^{-}})\,.$
(17)

Eqs.(II.1)-(II.1) form a basis for calculation of the spin density created by
DW in 2D gas. The sum of the terms, that are given by Eqs.(II.1) and (II.1),
represents the spin density induced by the interaction
$JS\bm{\sigma}\mathbf{m}_{\mathbf{q},\omega}$ of electron spins with the
nonstaggered Zeeman field $JS\mathbf{m}_{\mathbf{q},\omega}$. It is expressed
in terms of the space-time dependent Pauli susceptibility, which is given by a
single fermion loop, where the multiple scattering from impurities is taken
into account in a standard way through the vertex function $\mathbf{V}$.
Rammer ; Altshuler  At the same time, Eqs.(II.1) and (II.1) represent the
effect whose nature is quite different from the Pauli magnetism. They describe
second-order processes where the staggered magnetization, whose wave vector is
$\mathbf{G}$, gives rise to quantum transitions of electrons between states
with the wave vectors $\mathbf{k}$ and $\mathbf{k}+\mathbf{G}$. The DW, whose
relatively smooth profile is characterized by the wave vector $\mathbf{f}$,
such that $f\ll G$, adds $\mathbf{f}$ to $\mathbf{G}$. Therefore, the second-
order scattering amplitude of electrons from a DW in an antiferromagnet
carries terms of the form
$(E_{\mathbf{k}+\mathbf{G}-\mathbf{f}}-E_{\mathbf{k}})^{-1}$, which are
represented by $G^{i}_{\mathbf{k}-\mathbf{f}+\mathbf{G}}$ in Eq.(II.1). In
this expression $\mathbf{k}$ is close to the Fermi surface. Therefore,
$(E_{\mathbf{k}+\mathbf{G}-\mathbf{f}}-E_{\mathbf{k}})^{-1}$ becomes large if
this surface is close to the nesting condition and $\mathbf{G}$ coincides with
the nesting vector. This leads to the enhancement of the effect of second-
order terms Eqs.(II.1) and (II.1). Moreover, this expression becomes strongly
dependent on $\mathbf{f}$, although the latter is much smaller than
$\mathbf{G}$. On the other hand, when the Fermi surface is far from the
nesting conditions this dependence is weak and the scattering amplitude may be
expanded in powers of $\mathbf{v}_{F}\mathbf{f}$, where $\mathbf{v}_{F}$ is
the Fermi velocity. Since $\mathbf{f}$ is associated with the coordinate
dependence of the Néel order these terms generate spatial gradients of the
form $\mathbf{n}(\mathbf{r},t)\times\nabla^{i}\mathbf{n}(\mathbf{r},t)$ in the
induced spin density. Such sort of terms, with $\mathbf{n}(\mathbf{r},t)$
substituted for the ferromagnetic order parameter $\mathbf{m}(\mathbf{r},t)$,
were discussed in connection with the spin pumping from a ferromagnetic DW
into a 3D metal film. Duine

###  II.2 Disorder effects

In this subsection we shall consider effects of disorder on 2D electrons.
Besides the usual potential scattering from impurities the spin-dependent
scattering will also be taken into account. The latter leads to spin
relaxation of electrons. A short-range impurity scattering potential will be
assumed. In this case, the scattering amplitude from a single impurity has the
form Abrikosov

$f(\mathbf{k},\mathbf{k}^{\prime})=a+ib\bm{\sigma}\cdot(\mathbf{k}\times\mathbf{k}^{\prime})$
(18)

The first term in this expression is the isotropic spin-independent amplitude,
while the second one represents the spin-orbit scattering. In a 2D system both
incident and scattered wave vectors, $\mathbf{k}$ and $\mathbf{k}^{\prime}$,
respectively, lie in the same $xy$ plane. Therefore, only the $\sigma_{z}$
Pauli matrix enters in the amplitude of the spin-orbit scattering. Therefore,
the scattering probability, which is presumably given by the second-order Born
approximation over the scattering amplitude, is spin-independent. As follows
from Ref.[Abrikosov, ], the total elastic scattering rate of electrons can be
expressed as a sum of spin dependent and spin independent scattering channels.
Accordingly, it can be written as Abrikosov

$\Gamma=1/2\tau=\pi N_{F}(a^{2}+\frac{k^{4}_{F}}{2}b^{2}),$ (19)

where $\tau$ is the elastic scattering time, while $N_{F}$ and $k_{F}$ are the
state density and the Fermi wave-vector, respectively. At the same time, the
retarded and advanced unperturbed Green’s functions take the form Abrikosov

$G^{r}_{\mathbf{k}}(\omega)=G^{a}_{\mathbf{k}}(\omega)^{*}=\frac{1}{\omega-
E_{\mathbf{k}}-\mu+i\Gamma}\,,$ (20)

where $\mu$ is the chemical potential.

Multiple scattering events should be taken into account in order to study the
particle diffusion and spin relaxation effects. These effects are determined
by the vertex function $\mathbf{V}(\omega,\mathbf{q})$ in Eqs.(II.1) and
(II.1). With spin-independent Green functions Eq.(20) in hand the vertex
$\mathbf{V}$ can be easy calculated by summation of ladder diagrams. It is
convenient to use its vector components $V^{l}$ ($l=x,y,z$) which are given by
$V^{l}\mathbf{e}^{l}=(1/2)\mathrm{Tr}[\mathbf{V}\sigma^{l}]$ ($l=x,y,z$),
where $\mathbf{e}^{l}$ is the unit vector in the $l$-direction. In more detail
the calculation of $V^{l}$ is presented in Appendix A. Within the diffusion
approximation, which is valid at $\omega\ll\Gamma$ and $qv_{F}\ll\Gamma$, the
sum of the ladder diagrams is given by the diffusion propagator

$V^{l}(\omega,\mathbf{q})=\frac{2\Gamma}{i\omega-Dq^{2}-\Gamma_{s}^{l}}\,,$
(21)

where the spin relaxation rates $\Gamma_{s}^{l}$ are
$\Gamma_{s}^{x}=\Gamma_{s}^{y}=\pi N_{F}k^{4}_{F}b^{2}$, $\Gamma_{s}^{z}=0$
and the diffusion constant $D=v_{F}^{2}\tau/2$. It should be noted that the
spin relaxation turns to zero when $l=z$ in Eq.(21). It occurs because the
spin projection on the $z$-axis is conserved due to a specific form of the
spin-orbit scattering amplitude, which is proportional to $\sigma_{z}$ in a 2D
gas. In this situation other mechanisms of the spin relaxation should be taken
into account. However, such a strong spin relaxation channel as the scattering
on AFM magnons can be efficient only at high enough temperatures. At low
temperatures, due to the Fermi-liquid character of the electron gas, such an
inelastic mechanism is weak, even in the absence of a gap in the excitation
spectrum of magnons. The same can be said about the spin-lattice relaxation.
The spin-orbit splitting of the conduction band might result in the spin
relaxation through the D’yakonov-Perel mechanism DP . In the considered here
simple model, however, such sort of the spin-orbit coupling does not take
place. Nevertheless, a weak $\Gamma_{s}^{z}$ can be taken into account as a
phenomenological parameter.

##  III Spin polarization of electrons

###  III.1 Spin pumping by the staggered magnetization

Based on the general formalism presented in the previous section, let us
consider the spin polarization which is produced in the normal metal by the
exchange field of AFMI in the presence of a moving DW. As it was discussed in
Sec.II there are two contributions to the spin polarization. Namely, the
first-order effect due to the nonstaggered ”ferromagnetic” magnetization and
the second-order one produced by the staggered exchange field. The latter
effect, which is given by Eqs.(II.1) and (II.1), will be considered in this
subsection. Let us assume that within the classical theory the corresponding
Néel vector $\mathbf{n}(\mathbf{r},t)$ is given by the well known solution of
the equation of motion for a one-dimensional DW in an uniaxial AFM. Walker
The precession of $\mathbf{n}(\mathbf{r},t)$ around the easy axis is assumed
to be absent. It depends, however, on the method which is employed for the
excitation of DW motion. For instance, the precession may be produced by the
magnon’s impact on the DW Tveten ; Kim2 . Otherwise, the azimuthal angle of
the Néel vector retains fixed during DW motion. The spin polarization of
electrons, which can be induced by such DW, depends strongly on this angle. It
becomes evident from the following consideration. Since $\hat{G}$ in Eq.(II.1)
is given by the unit matrix in the spin space (retarded and advanced functions
are given by the spin-independent $G^{r}$ and $G^{a}$ in Eq.(20)), the spin
structure of the function $\hat{\Sigma}$ in Eq.(II.1) is determined by the
product
$(\bm{\sigma}\mathbf{n}_{\mathbf{f}^{+},\nu^{+}})(\bm{\sigma}\mathbf{n}^{*}_{\mathbf{f}^{-},\nu^{-}})$.
Its spin-dependent part is given by
$i\bm{\sigma}(\mathbf{n}_{\mathbf{f}^{+},\nu^{+}}\times\mathbf{n}^{*}_{\mathbf{f}^{-},\nu^{-}})$.
Therefore, $\bm{\sigma}$ in this equation is perpendicular to the plane where
the Néel vector of the DW resides. At the same time, this plane is fixed by
the azimuthal angle of $\mathbf{n}$. In turn, as it follows from
Eqs.(II.1),(II.1) and (21) the trace in Eq.(II.1) dictates that this spin
direction must coincide with that of the vertex $\mathbf{V}$, whose vector
components strongly depend on respective spin relaxation times. For example,
if $\mathbf{n}(\mathbf{r},t)$ belongs to the $xy$ plane, the spin relaxation
is given by $\Gamma^{z}_{s}$, which can be much smaller than $\Gamma^{x}_{s}$
and $\Gamma^{y}_{s}$ when the spin-orbit impurity scattering is a dominating
mechanism of the spin relaxation. Below, let us assume that the easy axis of
AFM is oriented parallel to the $z$-direction, while the DW moves in the
$x$-direction with the velocity $v$. In this case the vector
$(\mathbf{n}_{\mathbf{f}^{+},\nu^{+}}\times\mathbf{n}^{*}_{\mathbf{f}^{-},\nu^{-}})$
lies in the $xy$-plane. Therefore, $\Gamma_{s}^{x}$ and $\Gamma_{s}^{y}$ enter
in diffusion propagator Eq.(21) . In the absence of precession around the easy
axis the Néel vector is given in spherical coordinates by

$\mathbf{n}(\mathbf{r},t)=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\,,$
(22)

where $\cos\theta=\tanh[(x-vt)/\lambda]$ and $\lambda$ is the width of the DW,
while $\phi$ is fixed. Below, for simplicity we choose $\phi=0$. In this case
the functions $P^{ijk}$ in Eq.(II.1) are proportional to the Pauli matrix
$\sigma^{y}$. As a result, by taking the trace in Eqs.(II.1) and (II.1) one
obtains $\mathbf{Tr}[\mathbf{V}\sigma^{y}]=2V^{y}\mathbf{e}^{y}$. Therefore,
spins which are pumped into the 2D metal are oriented in the $y$-direction.

Parameters of the considered system are chosen in such a way that the wave
vectors $q$ and $f$ in Eqs.(II.1)-(II.1) are much smaller than $k_{F}$.
Similarly, the frequencies $\omega$ and $\nu$ are much smaller than the Fermi
energy. For a steady moving DW the variables $\omega$ and $q$, as well as
$\nu$ and $f$, are related to each other. Indeed, the Néel vectors
$\mathbf{n}_{\mathbf{f}^{\pm},\nu^{\pm}}$ in Eq.(II.1) can be written as

$\mathbf{n}_{\mathbf{f}^{\pm},\nu^{\pm}}=2\pi\delta(\nu^{\pm}-vf^{\pm}_{x})\int
d\xi e^{-if^{\pm}_{x}\xi}\mathbf{n}(\xi)\,,$ (23)

where $\xi=x-vt$. The delta-function in this relation fixes the frequencies
$\nu^{\pm}=vf^{\pm}_{x}$. By combining $\nu^{\pm}=\nu\pm\omega/2$ and
$f_{x}^{\pm}=f_{x}\pm q_{x}/2$ we find that $\nu=vf_{x}$ and $\omega=vq_{x}$.
Note, that $\omega$ and $q$ characterize the diffusion of electrons, because
the diffusion propagator Eq.(21) depends on these variables. At the same time,
$\nu$ and $f$ are associated with variations of $\mathbf{n}(\mathbf{r},t)$
within a DW. The integration over wavevectors and frequencies in Eq.(II.1) may
be simplified at small $\omega$ and $\nu$ by replacing $\tanh(\Omega/k_{B}T)$
in Eq.(II.1) with $\tanh(\Omega/k_{B}T)-\tanh(\omega^{\prime}/k_{B}T)$. Such a
replacement does not change the result, because the terms in Eq.(II.1) that
are proportional to $\tanh(\omega^{\prime}/k_{B}T)$ cancel each other.
Therefore, since $\Omega=\omega^{\prime}\pm\omega/2$, or
$\Omega=\omega^{\prime}+\nu$, at the low temperature such a replacement
restricts the integration over $\omega^{\prime}$ to the range of small
frequencies. As a result, the main contribution to the sum in Eq.(II.1) is
given by vectors $k$ which are close to the Fermi surface. Let us consider a
simple tight binding model with the electronic band energy
$E_{\mathbf{k}}=-\alpha(\cos k_{x}a+\cos k_{y}a)$. Note, that in this case
$E_{\mathbf{k}}=-E_{\mathbf{k}+\mathbf{G}}$. Consequently, by integrating
Eq.(II.1) over $\mathbf{k}$, in the leading approximation with respect to the
small parameters $\omega/\Gamma,\nu/\Gamma,v_{F}q/\Gamma,v_{F}f/\Gamma$ and
$\Gamma/\mu$, we obtain the $y$-component $S^{y(2)}_{V}(\omega,\mathbf{q})$ of
the vector $\mathbf{S}^{(2)}_{V}(\omega,\mathbf{q})$ in the form

$\displaystyle S^{y(2)}_{V}(\omega,\mathbf{q})$ $\displaystyle=$
$\displaystyle-i\frac{J^{2}S^{2}}{\mu^{2}}V^{y}(\omega,\mathbf{q})N_{F}(\mu)\times$
(24)
$\displaystyle\int\frac{d\nu}{2\pi}\sum_{\mathbf{f}}\nu(\mathbf{n}_{\mathbf{f}^{+},\nu^{+}}\times\mathbf{n}^{*}_{\mathbf{f}^{-},\nu^{-}})\,,$

where $N_{F}(\mu)$ is electronic state density at the chemical potential.
Details of this calculation can be found in Appendix B. It is seen that
$S^{y(2)}_{V}$ gives the main contribution in the spin density
$\mathbf{S}^{(2)}$ which is a sum of $S^{y(2)}_{0}(\omega,\mathbf{q})$ and
$S^{y(2)}_{V}(\omega,\mathbf{q})$. Indeed, it follows from a comparison of
Eqs.(II.1) and (II.1) that these functions differ from each other by the
absence in $S^{y(2)}_{0}(\omega,\mathbf{q})$ of the diffusion propagator
$V^{y}(\omega,\mathbf{q})$. The latter, however, is much larger than one,
because in the diffusion regime $\Gamma$ is large in comparison with the
denominator of Eq.(21). Therefore, $S^{y(2)}_{0}$ is small compared with
$S^{y(2)}_{V}$, so that the total spin density
$\mathbf{S}^{(2)}=\mathbf{S}^{(2)}_{0}+\mathbf{S}^{(2)}_{V}\simeq\mathbf{S}^{(2)}_{V}$.
It is important to note that $\mathbf{S}^{(2)}$ is proportional to $\mu^{-2}$.
Since the chemical potential is measured from the middle of the band, this
dependence means that the pumping effect increases when $\mu$ approaches to
the van Hove singularity. Such a dependence agrees with the discussed above
role of the Fermi surface nesting in the spin pumping.

Note, that in the case of a 1D domain wall, which moves in the $x$-direction,
the second line in Eq.(24) may be expressed as

$\displaystyle\int\frac{d\nu}{2\pi}\sum_{\mathbf{f}}\nu(\mathbf{n}_{\mathbf{f}^{+},\nu^{+}}\times\mathbf{n}^{*}_{\mathbf{f}^{-},\nu^{-}})=2\pi
iv\times$ $\displaystyle\delta(\omega-vq_{x})\delta_{q_{y}}\int d\xi
e^{-iq_{x}\xi}\left(\mathbf{n}(\xi)\times\nabla_{\xi}\mathbf{n}(\xi)\right)\,.$
(25)

Therefore, by setting $\cos\theta=\tanh(\xi/\lambda)$ in Eq.(22) we obtain
$(\mathbf{n}(\xi)\times\nabla_{\xi}\mathbf{n}(\xi))=(1/\lambda)\cosh^{-1}(\xi/\lambda)$.
Further, from Eqs.(21), (24) and (III.1) the spatial dependence of the induced
spin density can be written as
$S^{y(2)}(\mathbf{r},t)=S^{y(2)}(\xi)=S_{V}^{y(2)}(\xi)$, where

$S^{y(2)}(\xi)=\frac{J^{2}S^{2}}{\mu^{2}}\int\frac{dqd\xi^{\prime}}{2\pi\lambda}\frac{N_{F}v\Gamma}{\cosh\frac{\xi^{\prime}}{\lambda}}\frac{\exp
iq(\xi-\xi^{\prime})}{Dq^{2}-ivq+\Gamma_{s}^{y}}\,.$ (26)

By calculating the integral over $q$ we arrive at

$\displaystyle S^{y(2)}(\xi)=\frac{J^{2}S^{2}}{2\mu^{2}}\frac{\Gamma
N_{F}v}{\sqrt{v^{2}+4\Gamma_{s}^{y}D}}\int\frac{d\xi^{\prime}}{\lambda}\frac{1}{\cosh\frac{\xi^{\prime}}{\lambda}}\times$
$\displaystyle\left(e^{-p_{1}|\xi-\xi^{\prime}|}\theta(\xi-\xi^{\prime})+e^{p_{2}|\xi-\xi^{\prime}|}\theta(\xi^{\prime}-\xi)\right)\,,$
(27)

where $p_{1}=(1/2D)(v+\sqrt{v^{2}+4\Gamma_{s}^{y}D})$ and
$p_{2}=(1/2D)(v-\sqrt{v^{2}+4\Gamma_{s}^{y}D})$. These parameters determine
widths of forward ($1/p_{1}$) and backward ($1/p_{2}$) diffuse propagations of
the spin density with respect to the DW center. In most realistic cases these
widths are much larger than the DW width $\lambda$. By taking into account
that $D\sim v_{F}^{2}\tau$ it follows from above expressions for $p_{1}$ and
$p_{2}$ that $p_{1}\lambda\ll 1$ and $p_{2}\lambda\ll 1$, if
max$[v_{F}/v,1/\sqrt{\Gamma_{s}\tau}]\gg\lambda/l$, where $l$ is the
electron’s elastic mean free path. Typically $v_{F}/v\gg 1$ and
$\Gamma_{s}\tau\ll 1$. Therefore, $p_{1}\lambda$ and $p_{2}\lambda$ are small,
if the ratio $\lambda/l$ is not too large. In this case the integration over
$\xi^{\prime}$ in Eq.(III.1) is restricted to a small interval around
$\xi^{\prime}=0$. Hence, by setting $\xi^{\prime}=0$ in Eq.(III.1) it may be
simplified to

$S^{y(2)}(\xi)=\frac{J^{2}S^{2}}{2\mu^{2}}\frac{\pi\Gamma
N_{F}v}{\sqrt{v^{2}+4\Gamma_{s}^{y}D}}\left(e^{-p_{1}\xi}\theta(\xi)+e^{p_{2}|\xi|}\theta(-\xi)\right)\,.$
(28)

Note, that the singularity of the derivative of $S^{y}(\xi)$ at $\xi=0$
vanishes, when the actual profile of the DW in the integral over
$\xi^{\prime}$ is taken into account. Such fine details, however, are not
important, because the above expression is valid only in the range of
distances from DW which are much larger than its width.

###  III.2 Spin polarization due to the nonstaggered magnetization of the
domain wall

The polarization which is induced in the normal metal by the ”ferromagnetic”
part of the exchange field is given by Eqs.(II.1) and (II.1). For a one-
dimensional DW moving with the velocity $v$ in the $x$-direction
$\mathbf{m}(\mathbf{r},t)=\mathbf{m}(x-vt)\equiv\mathbf{m}(\xi)$. By
performing Fourier transformation of $\mathbf{m}(\mathbf{r},t)$ in Eq.(2) one
can see that its Fourier transform $\mathbf{m}_{\mathbf{q},\omega}$ is given
by the right-hand side of Eq.(III.1) multiplied by the factor
$-i/E_{\mathrm{ex}}$. This expression should be substituted in Eq.(9) which,
in turn, enters in Eqs.(II.1) and (II.1) for the spin density. By taking into
account Eq.(21), in the leading approximation with respect to the small
parameters $\omega/\Gamma$ and $v_{F}q_{x}/\Gamma$ the spin density
$S^{(1)}=S_{V}^{(1)}+S_{0}^{(1)}$ can be written in the form

$\displaystyle
S^{y(1)}(\xi)=-i\frac{JS}{E_{\mathrm{ex}}}N_{F}v\int\frac{dqd\xi^{\prime}}{2\pi}\frac{\exp
iq(\xi-\xi^{\prime})}{\lambda\cosh\frac{\xi^{\prime}}{\lambda}}\times$
$\displaystyle\left[\frac{vq}{Dq^{2}-ivq+\Gamma_{s}^{y}}-i\right]\,.$ (29)

Similar to the previous subsection, at distances from DW which are larger than
$\lambda$ the integration over $\xi^{\prime}$ may be simplified, that results
in

$\displaystyle S^{y(1)}(\xi)=\frac{\pi JSN_{F}v}{E_{\mathrm{ex}}}\times$
$\displaystyle\left(\delta(\xi)+\frac{vp_{1}e^{-p_{1}\xi}\theta(\xi)+vp_{2}e^{p_{2}|\xi|}\theta(-\xi)}{\sqrt{v^{2}+4\Gamma_{s}^{y}D}}\right)\,.$
(30)

The first term in this expression is represented by the delta-function, as
long as large distances are of interest, while in the range of DW it is given
by the function $\pi^{-1}\lambda^{-1}\cosh^{-1}(\xi/\lambda)$, as it follows
from the second term in the square brackets of Eq.(III.2).

##  IV Discussion

It was shown above that a moving DW in AFMI induces a macroscopic spin density
in a 2D electron gas, which is in the epitaxial contact with the compensated
surface of the AFMI. This spin density is a sum of two parts
$\mathbf{S}^{(1)}$ and $\mathbf{S}^{(2)}$ whose $y$-components are given by
Eqs.(III.1-28) and Eqs.(III.2-III.2). The spin polarization is perpendicular
to the plane where the Néel vector of DW evolves. In the considered case it is
the zx-plane, so that the polarization is parallel to the $y$-axis. The spin
densities $\mathbf{S}^{(1)}$ and $\mathbf{S}^{(2)}$ have very different
physical origins. Thus, $\mathbf{S}^{(1)}$ is induced by the nonstaggered part
of the exchange field of a moving DW. It is determined by the Pauli magnetism
in the first order with respect to the exchange interaction. At the same time,
$\mathbf{S}^{(2)}$ is produced by the staggered Néel order in the second order
with respect to $J$. It is instructive to compare these competing
contributions to the total spin polarization $\mathbf{S}$. Let us consider the
total polarization which is accumulated in the metal (per unit length of DW in
the y-direction). By integrating spin densities Eq.(28) and Eq.(III.2) over
$x$ we obtain $S_{\mathrm{tot}}^{y(1)}=\pi(JS/E_{\mathrm{ex}})N_{F}v$ and
$S_{\mathrm{tot}}^{y(2)}=(J^{2}S^{2}/\mu^{2})(\Gamma/\Gamma_{s}^{y})N_{F}v$.
First, it should be noted a fundamental difference between these two spin
densities. The effect of the staggered magnetization $S_{\mathrm{tot}}^{(2)}$
diverges when the spin relaxation rate $\Gamma_{s}\rightarrow 0$, feedback
while $S_{\mathrm{tot}}^{(1)}$ does not depend on the spin relaxation. The
reason is that the former effect is based on spin pumping from AFM into the
metal. In the absence of spin relaxation such a pumping leads to steady
increase of the spin polarization in the metal, until it saturates due to the
spin relaxation. At the same time, the nonstaggered exchange field gives rise
to the spin polarization through the Pauli mechanism. The motion of DW results
only in redistribution of this polarization over the 2D metal. From a
comparison of $S_{\mathrm{tot}}^{y(1)}$ and $S_{\mathrm{tot}}^{y(2)}$ it is
seen that the pumping mechanism dominates at
$JS>(\Gamma_{s}^{y}/\Gamma)(\mu^{2}/E_{\mathrm{ex}})$. For instance, at
$JS/\mu=0.1$ and $\mu\sim E_{\mathrm{ex}}$ the above inequality is fulfilled
at $\Gamma_{s}^{y}/\Gamma\lesssim 0.1$, which always takes place if the spin
relaxation is determined by the spin-orbit impurity scattering. This is
because $\Gamma_{s}$ is a relativistic correction to $\Gamma$, which is given
by the nonrelativistic potential scattering. Also, $\mu$ must be close enough
to zero, where the van Hove singularity just in the middle of the band is
placed. In the above evaluation $\mu$ was assumed to be comparable with the
exchange energy of spins in AFM, but much larger than the exchange interaction
of itinerant and AFM interface spins. In turn, $\mu$ is much less than the
Fermi energy that is counted from the bottom of the electronic energy band. As
long as the effect of the staggered DW’s magnetization dominates, let us
further focus on a discussion of this effect.

As it follows from Eqs.(III.1) and (28), the pumped spin density is
distributed asymmetrically with respect to DW. A tail of spin polarized
electrons extends behind the moving DW over the distance $\sim p_{2}^{-1}$,
which increases up to $\infty$ when $v\gg v_{F}\sqrt{\Gamma_{s}\tau}$. At the
same time, ahead of DW the spin density extends up to $\sim p_{1}^{-1}$ which
is always smaller than $l(\sqrt{(v^{2}/v_{F}^{2})+\Gamma_{s}\tau})^{-1/2}$.
The magnitude of the induced magnetization is largest at the center of the DW.
From Eq.(28) one can evaluate it as

$S^{y}(0)=N_{F}\Gamma\frac{J^{2}S^{2}}{\mu^{2}}\frac{v}{\sqrt{v^{2}+4\Gamma_{s}^{y}D}}\,.$
(31)

This expression turns to 0 at $v=0$ and reaches its maximum when the DW
velocity $v\gg\sqrt{\Gamma_{s}^{y}D}\sim v_{F}\sqrt{\Gamma_{s}^{y}\tau}$, by
taking into account that $D=v_{F}^{2}\tau/2$. Further, we notice that $S^{y}$
increases as $\mu^{-2}$ at $\mu\rightarrow 0$ where the Fermi level approaches
the middle of the band. This position of $\mu$ corresponds to the nesting
condition for the Fermi surface, when
$E_{\mathbf{k}_{F}}=E_{\mathbf{k}_{F}+\mathbf{G}}$. It agrees with the
qualitative behavior of the second-order scattering of 2D electrons from an
AFMI spin texture, which was discussed in SecIIA. On the other hand, one
should take into account that in a 2D electronic system, which makes a contact
with AFM, the exchange interaction with localized spins results in the energy
gap $2JS$ at $\mu=0$. Baltz  Therefore, within the chosen tight binding model
the chemical potential can not be placed too close to the middle of the band.
Hence, it will be assumed that $|\mu|\gg JS$. In order to evaluate
quantitatively the effect of spin pumping by a DW, it is convenient to compare
it with the spin polarization which could be produced in the electron gas by
an external static magnetic field $H_{\mathrm{eff}}$. Since this spin
polarization is $N_{F}\mu_{B}H_{\mathrm{eff}}$, we get
$\mu_{B}H_{\mathrm{eff}}=S^{y}(0)/N_{F}$, where $S^{y}(0)$ is given by
Eq.(31). From this equation one can see that in the case when
$\Gamma_{s}^{y}D/v^{2}\gg 1$ the effective magnetic field
$\mu_{B}H_{\mathrm{eff}}\sim\Gamma(J^{2}S^{2}/\mu^{2})(v/v_{F})(\Gamma_{s}^{y}\tau)^{-1/2}$
(it was taken into account that $D=v^{2}_{F}\tau/2$). By assuming
$\tau=10^{-13}$s, $\Gamma_{s}^{y}=10^{9}$s-1, $v_{F}=10^{5}$m/s, $v=100$m/s,
$(J^{2}S^{2}/\mu^{2})$=0.01, and $\Gamma=(\hbar/2\tau)\sim 3$meV, we obtain
$\mu_{B}H_{\mathrm{eff}}=0.003$meV, or $H_{\mathrm{eff}}\sim 0.05$T. In this
parameter range the induced spin polarization linearly increases with the
velocity of DW. As shown in Refs.[Tveten, ; Kim2, ], in AFM insulators spin
waves provide an efficient mechanism for DW propulsion. A more efficient
effect may be realized due to the so called staggered torque effect, which is
produced by electric current in a metallic AFM. Gomonay2  Such a mechanism,
however, cannot be implemented in AFM insulators. On the other hand, it might
be efficient in bilayers of 2D AFM insulators and metals. Regarding the
magnon’s effect, the relatively high value $v\sim 100$m/s was calculated
Tveten  for circularly polarized magnons, while linearly polarized magnons
produce much weaker effect. It should be noted that in the former case magnons
cause precession of DW, so that the axial angle in Eq.(22) varies as
$\phi=\Omega t$. As a result, the induced spin density will oscillate, in
contrast to the stationary spin density soliton given by Eq.(28). Moreover,
the spin polarization vector will have not only $S^{y}$ components. Such a
situation was not analyzed in this work. Also, a further analysis is necessary
of the physics close to the nesting point, as it was discussed in the end of
Sec.IIA.

The spin polarization, which is pumped by DW, can be detected by measuring the
electric current in a heavy metal contact. The contact may be placed at the
right edge of the junction which is shown in Fig.1. The current can be
produced by the inverse spin Hall effect, or spin galvanic effect due to the
strong spin-orbit coupling of electrons in the heavy metal. This method is
usually applied for detection of the pumped spin polarization. Baltz  In the
considered set up a DW arriving at the edge of AFM will inject the spin
polarization into the contact and produce a pulse of the electric current
there.

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##  Appendix A Calculation of the vertex function
$\mathbf{V}(\mathbf{q},\omega)$

A single element of the ladder array, which corresponds to the Bethe-Salpeter
equation in Fig. 2, is given by

$\displaystyle\psi_{0}^{ij}=\frac{1}{2}\sum_{\mathbf{k}}\mathrm{Tr}[\sigma^{i}G^{r}(\omega^{\prime+},\mathbf{k}^{+})f(\mathbf{k}^{+},\mathbf{k}^{+\prime})\sigma^{j}\times$
$\displaystyle
f(\mathbf{k}^{-\prime},\mathbf{k}^{-})G^{a}(\omega^{\prime-},\mathbf{k}^{-})]\,,$
(32)

where $G^{r}$ and $G^{a}$ are given by Eq.(20), Since $\omega^{\prime}\ll\mu$
the integration in Eq.(A) is restricted to $k\simeq k_{F}$. Therefore, one may
set $|\mathbf{k}^{\pm}|=k_{F}$ and $|\mathbf{k}^{\pm\prime}|=k_{F}^{\prime}$
in the scattering amplitude $f$, which is given by Eq.(18). Note, that in
general, for a nonspherical Fermi surface $k_{F}\neq k_{F}^{\prime}$. Further,
as was noted in the main text, the spin-dependent part of $f$ is proportional
to $\sigma_{z}$. Hence, by calculating the trace in Eq.(A) one may express it
in the form

$\displaystyle\frac{1}{2}\mathrm{Tr}[\sigma^{i}f(\mathbf{k}^{+},\mathbf{k}^{+\prime})\sigma^{j}f(\mathbf{k}^{-\prime},\mathbf{k}^{-})]=a^{2}\delta^{ij}+$
$\displaystyle b^{2}k_{F}^{2}k_{F}^{\prime
2}(1-(\mathbf{\hat{k}}\mathbf{\hat{k}}^{\prime})^{2})(\delta^{iz}\delta^{jz}-\delta^{ix}\delta^{jx}-\delta^{iy}\delta^{jy})\,,$
(33)

where $\mathbf{\hat{k}}$ and $\mathbf{\hat{k}}^{\prime}$ are unit vectors
which are parallel to $\mathbf{k}$ and $\mathbf{k}^{\prime}$, respectively.
Let us assume, for simplicity, that the Fermi line has an approximately
circular form, which takes place if the chemical potential is sufficiently far
from the middle of the considered tight binding band. In this case after
integration in Eq.(A) over $|\mathbf{k}|$ and by taking into account Eq.(A) we
obtain

$\psi_{0}^{zz}=2\pi
iN_{F}\int\frac{d\phi}{2\pi}\frac{a^{2}+b^{2}k_{F}^{4}(1-(\mathbf{\hat{k}}\mathbf{\hat{k}}^{\prime})^{2})}{\omega-\mathbf{v}_{F}\mathbf{q}+2i\Gamma}$
(34)

and

$\psi_{0}^{xx}=\psi_{0}^{yy}=2\pi
iN_{F}\int\frac{d\phi}{2\pi}\frac{a^{2}-b^{2}k_{F}^{4}(1-(\mathbf{\hat{k}}\mathbf{\hat{k}}^{\prime})^{2})}{\omega-\mathbf{v}_{F}\mathbf{q}+2i\Gamma}\,,$
(35)

where $\phi$ is the polar angle of the vector $|\mathbf{\hat{k}}|$. These
functions depend on the angle between $\mathbf{k}^{\prime}$ and $\mathbf{q}$.
This dependence originates from the second term in integrands of Eqs.(34) and
(35), which, in turn, is proportional to the spin-orbit scattering amplitude
$b$. The latter is much weaker than the usual potential scattering $a$.
Therefore, in the leading approximation the vertex function is angular
independent. Hence, within this approximation Eqs.(34) and (35) can be
averaged over directions of $k^{\prime}$. By expanding them over $\omega$, $q$
and $b$ up to their respective leading orders, after averaging over
$\mathbf{\hat{k}}^{\prime}$ and by substituting $\Gamma$ from Eq.(19), we
arrive at

$\displaystyle\psi_{0}^{xx}=\psi_{0}^{yy}=1+\frac{1}{2\Gamma}(i\omega-
Dq^{2}-2\pi N_{F}b^{2}k_{F}^{4})\,;$
$\displaystyle\psi_{0}^{zz}=1+\frac{1}{2\Gamma}(i\omega-Dq^{2})\,.$ (36)

By summing the ladder diagrams the vertex function $V^{l}$ can be expressed in
terms of $\psi_{0}$ as

$V^{l}=\frac{1}{1-\psi_{0}^{ll}}\,.$ (37)

Finally, by substituting $\psi_{0}^{ll}$ from Eq.(A) we obtain Eq.(21).

##  Appendix B Calculation of the spin density

In this section the calculation is presented of the spin density, starting
from Eq.(II.1). There are four terms, namely
$P^{raa}(\omega^{\prime+}),P^{rra}(\omega^{\prime-}),P^{rra}(\omega^{\prime}+\nu)$,
and $P^{raa}(\omega^{\prime}+\nu)$ in the right hand side of this equation.
Each of these terms is proportional to the Fermi statistical factors in the
form of
$\tanh[(\omega^{\prime}+\omega_{i})/2k_{B}T]-\tanh[\omega^{\prime}/2k_{B}T]$,
where $\omega_{i}=\pm\omega$, or $\omega_{i}=\nu$. Since these frequencies are
small, all these terms give small contribution to the spin density. However,
the first two of them are much smaller than the third one. The reason is that
the former are proportional to $\omega$, while the latter $\sim\nu$. However,
the function $\Sigma^{i}_{\mathbf{k}}(\omega^{\prime},\omega;\nu,\mathbf{f})$
in Eq.(II.1) is an odd function with respect to change of signs of $\nu$ and
$\mathbf{f}$, because two Néel vectors in Eq.(II.1) form a cross product, as
it was explained in subsection III A. Indeed,
$(\mathbf{n}_{\mathbf{f}^{+},\nu^{+}}\times\mathbf{n}^{*}_{\mathbf{f}^{-},\nu^{-}})$
changes sign when $\nu\rightarrow-\nu,\mathbf{f}\rightarrow-\mathbf{f}$,
because at such sign reversal $\mathbf{f}^{\pm}\rightarrow-\mathbf{f}^{\mp}$
and $\nu^{\pm}\rightarrow-\nu^{\mp}$. Hence,
$\mathbf{n}_{\mathbf{f}^{+},\nu^{+}}\leftrightarrows\mathbf{n}^{*}_{\mathbf{f}^{-},\nu^{-}}$.
Therefore, those terms which are proportional to $\omega$ have an additional
small factor, because they must turn to zero with $\nu$ and $f$. In contrast,
the term $\sim\nu$ is initially an odd function of $\nu$. On this reason, it
dominates in Eq.(II.1), as long as
$\nu\Gamma\gg\omega\cdot\mathrm{max}(\nu,v_{F}f)$. By using Eq.(II.1) the
contribution of this term in Eq.(II.1) can be written in the form

$\displaystyle\mathbf{S}^{(2)}_{V}(\mathbf{q},\mathbf{\omega})=J^{2}S^{2}\frac{1}{2}\sum_{\mathbf{f}}\int\frac{d\nu}{2\pi}I(\omega,\mathbf{q};\nu,\mathbf{f})\times$
$\displaystyle\mathrm{Tr}\left[\mathbf{V}(\omega,\mathbf{q})\left(\bm{\sigma}\cdot(\mathbf{n}_{\mathbf{f}^{+},\nu^{+}}\times\mathbf{n}^{*}_{\mathbf{f}^{-},\nu^{-}})\right)\right]\,,$
(38)

where

$\displaystyle
I(\omega,\mathbf{q};\nu,\mathbf{f})=\sum_{\mathbf{k}}\int\frac{d\omega^{\prime}}{2\pi}G^{r}_{\mathbf{k}^{+}}(\omega^{\prime+})G^{a}_{\mathbf{k}^{-}}(\omega^{\prime-})\left(G^{r}_{\mathbf{k}+\mathbf{f}+\mathbf{G}}(\omega^{\prime}+\nu)-G^{a}_{\mathbf{k}+\mathbf{f}+\mathbf{G}}(\omega^{\prime}+\nu)\right)\left(\tanh\frac{\omega^{\prime}+\nu}{k_{B}T}-\tanh\frac{\omega^{\prime}}{k_{B}T}\right)\,.$
(39)

By substituting in this equation the expressions for Green functions from
Eq.(5) and by expanding there the electron energy as
$E_{\mathbf{k}+\mathbf{Q}}=E_{\mathbf{k}}+\mathbf{v}_{k}\cdot\mathbf{Q}$,
where $\mathbf{Q}=\mathbf{q}$, or $\mathbf{f}$ and $\mathbf{v}_{\mathbf{Q}}$
is the velocity, the integral $I$ can be written as

$\displaystyle
I(\omega,\mathbf{q};\nu,\mathbf{f})=\int\frac{d^{2}k}{4\pi^{2}}\int\frac{d\omega^{\prime}}{2\pi}\frac{1}{(\omega^{\prime}+\frac{\omega}{2}-E_{\mathbf{k}}-\frac{\mathbf{v}_{\mathbf{k}}\mathbf{q}}{2}+\mu+i\Gamma)}\frac{1}{(\omega^{\prime}-\frac{\omega}{2}-E_{\mathbf{k}}+\frac{\mathbf{v}_{\mathbf{k}}\mathbf{q}}{2}+\mu-i\Gamma)}\times$
$\displaystyle\left(\frac{1}{(\omega^{\prime}+\frac{\nu}{2}-E_{\mathbf{k+G}}-\frac{\mathbf{v}_{\mathbf{k+G}}\mathbf{f}}{2}+\mu+i\Gamma)}-\frac{1}{(\omega^{\prime}+\frac{\nu}{2}-E_{\mathbf{k+G}}-\frac{\mathbf{v}_{\mathbf{k+G}}\mathbf{f}}{2}+\mu-i\Gamma)}\right)\left(\tanh\frac{\omega^{\prime}+\nu}{k_{B}T}-\tanh\frac{\omega^{\prime}}{k_{B}T}\right)\,.$
(40)

One should take into account that within the tight binding model
$E_{\mathbf{k+G}}=-E_{\mathbf{k}}$ and $v_{\mathbf{k+G}}=-v_{\mathbf{k}}$.
Then, it is seen that since $|\omega^{\prime}|\sim|\nu|\ll\mu$, as well as
$\omega$, $\Gamma$, $v_{\mathbf{k}}q$ and $v_{\mathbf{k+G}}f\ll\mu$ the major
contribution to the integral at small $T$ is given by two poles, whose wave
vectors are close to the Fermi surface. Therefore, one can set
$v_{\mathbf{k}}\simeq v_{\mathbf{k}_{F}}\equiv v_{F}$, where
$E_{\mathbf{k}_{F}}=\mu$. In the leading approximation
$\mu\gg\Gamma\gg[v_{F}q,v_{F}f,\nu,\omega]$ this integration gives

$I(\omega,\mathbf{q};\nu,\mathbf{f})=-i\frac{\nu}{\mu^{2}}N_{F}\,.$ (41)

By substituting this expression in Eq.(B) we arrive at Eq.(24).