#  Electric field effect on electron gas spins in two-dimensional magnets with
strong spin-orbit coupling

K.S. Denisov<EMAIL_ADDRESS>Ioffe Institute, 194021
St.Petersburg, Russia

###### Abstract

The recent rise of material platforms combining magnetism and two-
dimensionality of mobile carriers reveals a diverse spectrum of spin-orbit
phenomena and stimulates its ongoing theoretical discussions. In this work we
use the density matrix approach to provide a unified description of subtle
microscopic effects governing the electron gas spin behavior in the clean
limit upon electric perturbations in two-dimensional magnets with strong spin-
orbit coupling. We discuss that an inhomogeneity of electrostatic potential
generally leads to the electron gas spin tilting with the subsequent formation
of equilibrium skyrmion-like spin textures and demonstrate that several
microscopic mechanisms of 2DEG spin response are equally important for this
effect. We analyze the dynamics of 2DEG spin upon an oscillating electric
field with a specific focus on the emergent electric dipole spin resonance. We
address the resonant enhancement of magneto-optical phenomena from the spin
precession equation perspective and discuss it in terms of the resonant spin
generation. We also clarify the connection of both static and dynamic spin
phenomena arising in response to a scalar perturbation with the electronic
band Berry curvature.

##  I Introduction

The recent advances in the development of spintronics devices extensively use
relativistic spin-orbit properties of free carriers interacting with magnetic
layers. The spin-orbit coupling (SOC) of charge carriers generally opens up
the possibility to deal with the magnetization purely by electrical means; the
magnetization orientation can be detected electrically by virtue of the
anisotropic magnetoresistance effect [1, 2, 3], while electric current-induced
spin-orbit torque occurs to be a highly effective tool for switching its
direction [4, 5, 6, 7]. Nonstationary dynamics of carriers in presence of SOC
can result in stimulated photon emission, as in case of terahertz spintronic
light emitter [8, 9, 10, 11] and spin Hall nano-oscillators [12, 13]. Apart
from kinetic phenomena spin-orbit effects can modify equilibrium spin
configurations via indirect RKKY exchange interaction [14, 15, 16] and lead to
the formation of magnetic skyrmions [17, 18] due to Dzyaloshinskii–Moriya
terms [19, 20]. An efficient charge-to-spin conversion wanted for modern
spintronics needs is often realized when turning to a two-dimensional electron
gas, as the reduction of the dimensionality tends to be accompanied by the
lowering of symmetry and by the subsequent increase in SOC [19, 21]. There are
an increasing number of different material platforms that allow one to combine
systematically stronger SOC magnitudes of 2D electrons directly with a
magnetic component, the examples include van der Waals heterostructures [22]
either proximitized by magnetic layer [23, 24, 25, 26, 27] or being intrinsic
ferromagnets [28, 29, 30, 31], semiconductor nanostructures doped by magnetic
dopants [32, 33], surface states of magnetic topological insulators [34, 35],
or layered magnetic heterostructures [36, 19]. Moreover, combining magnetism
with 2D conductive channels additionally offers new functionalities, such as
spin tunnel field-effect transistors [37], spin inversion effect [24] or novel
class of spinterfaces [38].

In order to fully benefit from two-dimensional magnetic systems it is of key
importance to have a comprehensive understanding of how the spin density of
electron gas in a 2D channel responds to an applied electric field, that is
the understanding of free electron gas magnetoelectric properties. However, a
complete microscopic treatment of the related phenomena appears to be
extremely challenging, even despite there is a few theoretical approaches
effectively dealing with multiband systems (e.g. wave-packet dynamics theory
[39, 40, 41, 42, 43], diagrammatic and ab-initio calculations [44, 45, 46,
47]). The difficulty lies in the fact that in spin-orbital systems multiple
microscopic mechanisms of quite a subtle character often contribute on the
equal footing, which hinders a simplified consideration. In particular, an
exchange interaction induced spin splitting in combination with strong spin-
orbit coupling generally lead to a geometrical structure of electronic band
states featured by nonzero Berry curvature in k-space. Treating different
spin-related phenomena with account for the electronic band geometry remains
an ongoing discussion. It covers, for instance, the issues of the Liouville’s
theorem with account for the Berry phase [48, 49, 50], the Hall conductivity
modifications in presence of real-space magnetic textures [51], or, concerning
the anomalous and spin Hall effects, the interplay between Karplus-Luttinger
anomalous velocity and disorder-induced mechanisms [52, 53, 54, 55]; the
latters have recently been enriched by the electron scattering on a pair of
impurities [55, 56]. Moreover, when calculating spin-related quantities a
specific class of coarse graining effects should be taken into account, as is
clearly demonstrated in [40, 42].

In this paper we respond to an ever-growing role that two-dimensional magnetic
systems plays for spintronics and consider in detail a complex pattern of
microscopic effects relevant for the magnetoelectric behavior of 2DEG in the
clean limit. Based on the density matrix approach we describe the most
significant spin-response mechanisms of two-dimensional spin-orbital systems
within the unified framework, reveal the interconnection between different
microscopic effects and clarify its relation to an electronic band geometry.

The theoretical model and the density matrix description are formulated in
Sec. II. In Sec. III we analyze a magnetoelectric effect in thermal
equilibrium, namely we consider the formation of equilibrium spin textures and
local persistent electric currents arising due to an inhomogeneous
electrostatic potential. We discuss in detail semiclassical electron dynamics
with account for a spin-to-momentum locking and identify microscopic
mechanisms responsible for the magnetoelectric response. Namely, we attribute
the generation of an extra-spin density directed within 2DEG plane both to the
non-adiabatic correction to the electron spin precession and to the correlated
change of charge and spin electron densities, the latter scenario is sometimes
referred as spin-dipole effect [40]. We provide a unified treatment of these
mechanisms using the density matrix, derive general equations governing the
contribution due to each mechanism independently and reveal the role that the
Berry curvature plays for the emergent phenomena.

In Sec. IV we turn to the dynamical regime and investigate the 2DEG spin
dynamics upon an oscillating electric field. We focus specifically on spin
resonance phenomena due to electric dipole transitions, also referred as the
electric dipole spin resonance (EDSR). We derive the precession equation for
2DEG spin density capturing the spin resonance scenario, and clarify the
relation of the band states Berry curvature with the spin response
susceptibility. We also discuss the spin resonance in terms of optical
conductivity and describe the associated magneto-optical properties of 2DEG.
In particular, we describe how the EDSR induced generation of the in-plane
spin density is accompanied by the resonant enhancement of the Hall
conductivity, the latter is responsible for magneto-optical Kerr and Faraday
effects. We classify different spin polarizations emerging in the dynamical
regime and present analytic expressions for the spin resonance related optical
conductivity.

##  II Theoretical framework

###  II.1 Model band structure

We consider a two-dimensional electron gas with parabolic bands affected both
by the Rashba effect and by an exchange interaction with a magnetic host. We
assume that the magnetization responsible for the spin splitting is directed
along $z$-axis perpendicular to the electron motion plane. The so-called
Rashba ferromagnet model covers all the physics relevant for our consideration
and allows one address the related spin phenomena in the most transparent way.
The effective Hamiltonian describing this model is given by

$\mathcal{H}=\frac{\bm{k}^{2}}{2m}+\bm{\Omega}_{k}\cdot\hat{\bm{S}},$ (1)

here the first term describes the parabolic dispersion with an effective mass
$m$, and $\bm{\Omega}_{k}$ is an effective $k$-space magnetic field acting on
the electron spin $\hat{\bm{S}}=\hat{\bm{\sigma}}/2$; $\hat{\bm{\sigma}}$ is
the vector of Pauli matrices. The field $\bm{\Omega}_{k}$ leads to a spin
splitting of the electronic subbands, in our model $\bm{\Omega}_{k}$ consists
of two parts

$\displaystyle\bm{\Omega}_{k}=\bm{\Omega}^{so}(\bm{k})-\Omega_{0}\bm{e}_{z},\qquad\bm{\Omega}^{so}(\bm{k})=2\lambda_{so}\left[\bm{e}_{z}\times\bm{k}\right],$
(2)

where $\bm{\Omega}^{so}(\bm{k})$ describes the spin-orbit Rashba interaction
with the coupling constant $\lambda_{so}$, and the second term is due to an
exchange interaction with a magnetic background, the parameter $\Omega_{0}$
describes the corresponding splitting of spin subbands at zero momentum. The
eigenstates of Eq. 1 Hamiltonian can be written in the following form

$\psi_{k}^{\pm}=e^{i\bm{kr}}|u_{k}^{\pm}\rangle,\qquad|u_{k}^{+}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}b_{k}\\\
-ie^{i\varphi}a_{k}\end{pmatrix},\quad|u_{k}^{-}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}-ie^{-i\varphi}a_{k}\\\
b_{k}\end{pmatrix},$ (3)

where $(a_{k},b_{k})=(1\pm\Omega_{0}/\Omega_{k})^{1/2}$. We use the notation
$\eta=(\pm)$ for two electron spin subbands. The states $\psi_{k}^{\eta}$ are
characterized by the electron spin $\bm{s}_{k}^{\eta}=\langle
u_{k}^{\eta}|\hat{\bm{S}}|u_{k}^{\eta}\rangle$ directed either parallel or
antiparallel to $\bm{\Omega}_{k}$

$\displaystyle\bm{s}_{k}^{\pm}=\pm\frac{1}{2}\bm{n}_{k},\qquad\bm{n}_{k}=\frac{\bm{\Omega}_{k}}{{\Omega}_{k}},\qquad\Omega_{k}=\sqrt{\Omega_{0}^{2}+(2\lambda_{so}k)^{2}},$
(4)

where the unit vector $\bm{n}_{k}$ points along the direction of
$\bm{\Omega}_{k}$.

The energy dispersion corresponding to $\eta$-subband is
$\varepsilon_{k}^{\eta}=k^{2}/2m+\eta\Omega_{k}/2$. The presence of
$k$-dependent spin splitting leads to the renormalization of effective masses
nearby $k\approx 0$, namely $m^{\pm}=m/(1\pm\xi)$, where the parameter
$\xi\equiv 2m\lambda_{so}^{2}/\Omega_{0}$. We focus on systems with
sufficiently strong exchange interaction, when $\Omega_{0}$ greatly exceeds
the spin-orbital coupling. We thus take the parameter $\xi<1$, at that the
effective mass $m^{-}>0$ is positive and the lower energy branch is a
monotonic function of the momentum, see Fig. 4b.

Let us discuss the role of the spin splitting terms. The presence of the
Rashba effect induced spin-momentum locking directly manifests itself in the
velocity operator

$\hat{\bm{v}}=\frac{i}{\hbar}\left[\mathcal{H},{\bm{r}}\right]=\frac{\hat{\bm{k}}}{m}+2\lambda_{so}[\bm{e}_{z}\times\hat{\bm{S}}],$
(5)

where the second term is sensitive to the instantaneous direction of the
electron spin. While the average velocity for the eigen spin states is
determined by the unperturbed spin vector $\bm{s}_{k}^{\eta}$

$\bm{v}_{k}^{\eta}\equiv\langle\psi_{k}^{\eta}|\hat{\bm{v}}|\psi_{k}^{\eta}\rangle=\frac{\bm{k}}{m}+2\lambda_{so}[\bm{e}_{z}\times\bm{s}_{k}^{\eta}],$
(6)

the changes in the direction of an electron spin caused by external fields can
directly affect the average of the velocity operator and, correspondingly,
influence the orbital motion.

The presence of a magnetic gap due to the magnetization directed perpendicular
to 2DEG plane leads additionally to the fact that electron band states acquire
a geometric structure. Indeed, the electron spin direction in $\bm{k}$-space
forms a hedgehog pattern which underlies the appearance of the Berry curvature
$\mathcal{F}_{k}^{\eta}=i\langle\nabla_{\bm{k}}u_{k}^{\eta}|\times|\nabla_{\bm{k}}u_{k}^{\eta}\rangle$.
For a spin-$1/2$ Hamiltonian this Berry curvature can be expressed as follows

$\mathcal{F}_{k}^{\eta}=\eta\frac{1}{4\pi}\bm{n}_{k}\cdot\left[\frac{\partial\bm{n}_{k}}{\partial{k_{x}}}\times\frac{\partial\bm{n}_{k}}{\partial{k_{y}}}\right]=\eta~{}2\lambda_{so}^{2}\frac{\Omega_{0}}{\Omega_{k}^{3}},$
(7)

and we keep the notation $\mathcal{F}_{k}=|\mathcal{F}_{k}^{\eta}|$ for its
absolute value. The total Berry flux $Q_{F}^{\eta}$ accumulated by electrons
from $\eta$ subband up to the Fermi energy $\mu$ is given by

$Q_{F}^{\eta}=\sum_{k<k_{F}^{\eta}}\mathcal{F}_{k}^{\eta}=\eta~{}\frac{1}{4\pi}\left(1-\frac{\Omega_{0}}{\Omega_{F}^{\eta}}\right),\qquad\Omega_{F}^{\eta}=\sqrt{\Omega_{0}^{2}+(2\lambda_{so}k_{F}^{\eta})^{2}},$
(8)

where $\Omega_{F}^{\pm}$ is the spin splitting energy for $\eta=(\pm)$
subbands at the Fermi energy, see Fig. 3b. The strong spin-orbit coupling
considered in our work means that we do not account for the disorder-induced
smearing of SOC features of electronic bands.

###  II.2 Density matrix approach

Let us firstly discuss the structure of the density matrix ${f}^{0}$ for 2DEG
in thermal equilibrium without external perturbations. The general form is
${f}^{0}=(e^{\beta(\hat{\mathcal{H}}-\mu)}+1)^{-1}$, where $\hat{\mathcal{H}}$
is given by Eq. 1, $\beta$ is the inverse temperature and $\mu$ is the Fermi
energy. In this work we focus on zero temparature limit $\beta\to\infty$. The
density matrix $\hat{f}_{k}^{0}$ in the momentum representation is a $2\times
2$ matrix which can be presented as follows (we keep hats for spin indices
only)

$\hat{f}_{k}^{0}=\frac{1}{2}n_{k}^{0}+\bm{S}_{k}^{0}\cdot\hat{\bm{\sigma}}.$
(9)

We note that $\hat{f}_{k}^{0}$ is diagonal in the basis of eigen states
$\psi_{k}^{\pm}$, so we can present it as a sum of $\eta=(\pm)$ spin subband
contributions $\hat{f}_{k}^{\eta}$

$\displaystyle\hat{f}_{k}^{0}=\hat{f}_{k}^{+}+\hat{f}_{k}^{-},\qquad\hat{f}_{k}^{\eta}=n_{k}^{\eta}\left(\frac{1}{2}+\bm{{s}}_{k}^{\eta}\cdot\hat{\bm{\sigma}}\right),$
(10)

where $n_{k}^{\eta}=(e^{\beta(\varepsilon_{k}^{\eta}-\mu)}+1)^{-1}$ is the
Fermi-Dirac distribution function of electrons in the spin subband with energy
$\varepsilon_{k}^{\eta}$. The terms in Eq. 9 are given
$n_{k}^{0}=n_{k}^{+}+n_{k}^{-}$, and
$\bm{S}_{k}^{0}=n_{k}^{+}\bm{s}_{k}^{+}+n_{k}^{-}\bm{s}_{k}^{-}$, here
$\bm{s}_{k}^{\eta}$ corresponds to the eigen spin states from Eq. 4. The
equilibrium spin density $\bm{S}_{0}$ is directed perpendicular to the 2DEG
plane

$\bm{S}_{0}=\frac{1}{2}\sum_{k}{\rm
Sp}\left(\hat{f}_{k}^{0}\cdot\hat{\bm{\sigma}}\right)=\sum_{k}\left(n_{k}^{+}\bm{s}_{k}^{+}+n_{k}^{-}\bm{s}_{k}^{-}\right)=\bm{e}_{z}~{}\frac{\Omega_{F}^{-}-\Omega_{F}^{+}}{16\pi\lambda_{so}^{2}}.$
(11)

We note that when both spin subbands are populated ($\mu>\Omega_{0}/2$) the
equilibrium spin density takes value ${S}_{0}=m\Omega_{0}/4\pi$ independent of
the Fermi energy, this is specific for Hamiltonian from Eq. 1.

The application of a scalar potential $U(\bm{r},t)$ deviates the electron
distribution from Eq.10. In this paper we focus on spatially smooth
perturbations ($k_{F}\cdot\nabla_{k}\ll 1$ and $\lambda_{F}\cdot\nabla_{r}\ll
1$) and study the electron gas response in the classical limit. For this
purpose we introduce the Wigner density matrix $\hat{f}_{k}(\bm{r},t)$ in the
following form

$\displaystyle\hat{f}_{k}(\bm{r},t)=\frac{1}{2}n_{k}(\bm{r},t)+\bm{S}_{k}(\bm{r},t)\cdot\hat{\bm{\sigma}},$
(12)

where $n_{k}(\bm{r},t),\bm{S}_{k}(\bm{r},t)$ can be treated as particle and
spin distribution functions locally in real space. In particular, the 2DEG
spin density perturbation emerging in the real space at point $\bm{r}$ can be
found from

$\delta\bm{S}(\bm{r},t)=\frac{1}{2}\sum_{k}{\rm
Sp}\left(\hat{f}_{k}(\bm{r},t)\cdot\hat{\bm{\sigma}}\right)-\bm{S}_{0}=\sum_{k}\bm{S}_{k}(\bm{r},t)-\bm{S}_{0}.$
(13)

In the clean limit $\hat{f}_{k}(\bm{r},t)$ satisfies the kinetic equation [57]

$\displaystyle\frac{\partial\hat{f}_{k}}{\partial
t}+\frac{1}{2}\Bigl{\\{}\left(\hat{\bm{v}}\cdot\nabla_{\bm{r}}\right);\hat{f}_{k}\Bigr{\\}}-\left[\bm{\Omega}_{k}\times\bm{S}_{k}\right]\cdot\hat{\bm{\sigma}}+\left(\bm{F}\cdot\nabla_{\bm{k}}\right)\hat{f}_{k}=0,$
(14)

where $\\{;\\}$ stands for the anticommutator, $\nabla_{\bm{r},\bm{k}}$ are
the nabla operators, $\bm{F}(\bm{r},t)=-\nabla_{\bm{r}}U(\bm{r},t)$ describes
the dynamical force acting on electrons, and the third term takes into account
the precession of the electron spin in the effective magnetic field
$\bm{\Omega}_{k}$. Let us draw the attention to the anticommutator type of
ordering between $\hat{\bm{v}}$ and $\hat{f}_{k}$ that appears in the second
term. This ordering directly stems from the Wigner transformation procedure
[58] and it is especially important to describe accurately the response in the
inhomogeneous regime.

##  III Static spin textures

We start our analysis by inspecting the redistribution of the 2DEG charge and
spin densities nearby smooth electrostatic defects, such as Coulomb centres or
gating potential perturbations. The geometric character of electronic band
states and the associated nonzero Berry curvature underline the appearance of
chiral spin textures and adjoint persistent electric currents that surround
electrostatic potential inhomogeneity, see Fig. 1. In [59] we used the Kubo
formalism to address the nonlocal regime of the spin density response due to
short-range impurities. In this section, instead, we provide a detailed
semiclassical description of this phenomenon and accompany it by the
comprehensive physical analysis.

Figure 1: Formation of skyrmion-like spin textures and the distribution of the
persistent electric currents nearby electrostatic defects.

###  III.1 General mechanisms of the intrinsic spin generation

Let us qualitatively discuss the effect of the electron spin non-adiabatic
rotation upon the precession in a slowly varying magnetic field [60, 61, 62].
We start by considering the precession equation for an electron spin $\bm{s}$
rotating upon a time-dependent frequency $\bm{\Omega}(t)$

$\frac{d\bm{s}}{dt}=\left[\bm{\Omega}(t)\times\bm{s}\right].$ (15)

Assuming the adiabatically slow rotation of $\bm{\Omega}(t)$, i.e. that the
characteristic time $\tau$ of its variation satisfies $\Omega\tau\gg 1$, the
zero-order solution of the precession equation simply describes the electron
spin $\bm{s}^{0}(t)=\bm{\Omega}(t)/2|\bm{\Omega}(t)|$ remaining co-aligned
with the instant direction of $\bm{\Omega}(t)$. However, the adiabatic
rotation of $\bm{s}^{0}(t)$ can be maintained only due to the appearance of
the non-adiabatic correction $\delta\bm{s}(t)$ directed perpendicular to the
instant vector $\bm{\Omega}(t)$. Naturally, this correction exists in the
first order in $(\Omega\tau)^{-1}$ and it can be found from the precession
equation keeping only the leading term due to $\bm{s}^{0}(t)$ in the time
derivative

$\frac{d\bm{s}^{0}}{dt}=\left[\bm{\Omega}(t)\times\delta\bm{s}(t)\right]\quad\rightarrow\quad\delta{\bm{s}}(t)=\frac{1}{2\Omega^{3}}\left[\bm{\Omega}\times\frac{d\bm{\Omega}}{dt}\right].$
(16)

The appearance of $\delta{s}\propto(\Omega\tau)^{-1}{s}^{0}$ is a general
property of the precession equation. Naturally, this is also valid when a
Larmor frequency stems from an effective magnetic field in k-space due to a
spin-orbit coupling. In this case, however, the vector $\bm{\Omega}_{k}$ that
governs the spin dynamics of an electron with momentum $\bm{k}$ varies in time
only provided that the electron momentum does not remain constant along its
trajectory $\dot{\bm{k}}\neq 0$, which is the case if $\bm{F}\neq 0$. The non-
adiabatic spin component acquired by an electron can be estimated from Eq. 16
by replacing the time derivative by $d/dt\to\dot{\bm{k}}\cdot\nabla_{\bm{k}}$

$(\dot{\bm{k}}\cdot\nabla_{\bm{k}})\bm{s}_{k}^{0}=\left[\bm{\Omega}_{k}\times\delta\bm{s}_{k}\right]\quad\rightarrow\quad\delta\bm{s}_{k}=\frac{1}{2\Omega_{k}^{3}}\left[\bm{\Omega}_{k}\times(\dot{\bm{k}}\cdot\nabla_{\bm{k}})\bm{\Omega}_{k}\right].$
(17)

We conclude that an electron moving along its classical trajectory with finite
acceleration has its spin always slightly tilted compared to the instantaneous
direction of $\bm{\Omega}_{k}$. Moreover, in view of the spin-momentum locking
such an intrinsically generated extra-spin leads to the change in the electron
velocity $\delta\bm{v}_{k}=2\lambda_{so}(\bm{e}_{z}\times\delta\bm{s}_{k})$.

The second spin-related phenomenon being important for the collective response
of 2DEG concerns the spin-dipole effect [40]. This mechanism is relevant when
the single electron density $|\psi(\bm{r})|^{2}$ deviates from the homogeneous
distribution and acquires some finite $\bm{r}$-dependence nearby an
inhomogeneity. Let us consider an electron at the unperturbed plane-wave state
$\psi_{k}^{\pm}$ from Eq. 3 with the momentum $\bm{k}$, its spin
$\bm{s}_{k}^{\pm}$ is determined by $\bm{\Omega}_{k}$. The corresponding
density $|\psi_{k}^{\pm}|^{2}$ is spatially homogeneous. In fact, the smooth
spatial variation of the density for such electron is possible only provided
that its wave-function gets an admixture of other plane-wave band states
$\psi_{k^{\prime}}^{\pm}$ with momenta $\bm{k}^{\prime}$ slighty differing
from $\bm{k}$. Essentially, the added states have different spin orientation
$\bm{s}_{k^{\prime}}^{\pm}\neq\bm{s}_{k}^{\pm}$, so the resulting average spin
density appears to be slightly tilted. In terms of the wave-packet dynamics
[63, 40] the mixing of spin-orbital states leads to the fact that the charge
and spin centers of the electron wave-packet do not coincide, which creates an
additional spin polarization. This scenario is specifically important for
localized electron states [64, 65]. We emphasize that the spin-dipole effect
is essentially connected with the spatial variation of the electron density.
In particular, if a given external field keeps an electron gas in the
homogeneous state, the spin-dipole contribution will be absent. The appearance
of the non-adiabatic correction from Eq. 17, on the contrary, is not connected
with the change of an electron density, it simply tracks the exact electron
spin dynamics along quasiclassical trajectories.

###  III.2 Density matrix in a static inhomogeneous setting

We proceed with giving a rigorous description of the outlined phenomena based
on the kinetic equation for the density matrix. Let us consider an electron
gas subjected to an electrostatic potential $U(\bm{r})$ smoothly varying in
space. Since the unperturbed density matrix
$\hat{f}_{k}^{0}=\hat{f}_{k}^{+}+\hat{f}_{k}^{-}$ given by Eq. 10 has two
parts corresponding to $\eta=(\pm)$ subband states, the linear response
correction $\delta\hat{f}_{k}(\bm{r})=\hat{f}_{k}(\bm{r})-\hat{f}_{k}^{0}$
will be determined independently by two subband terms
$\delta\hat{f}_{k}(\bm{r})=\delta\hat{f}_{k}^{+}(\bm{r})+\delta\hat{f}_{k}^{-}(\bm{r})$.
We present the corresponding correction $\delta\hat{f}_{k}^{\eta}$ as follows

$\displaystyle\delta\hat{f}_{k}^{\eta}(\bm{r})=\frac{1}{2}\delta
n_{k}^{\eta}(\bm{r})+\delta\bm{S}_{k}^{\eta}(\bm{r})\cdot\hat{\bm{\sigma}},$
(18)

where $\delta n_{k}^{\eta}(\bm{r}),\delta\bm{S}_{k}^{\eta}(\bm{r})$ are the
perturbations of the electron density and spin distribution functions,
respectively.

The key suggestion implemented in this paper is to use the following ansats
for the linear response spin density

$\delta\bm{S}_{k}^{\eta}(\bm{r})=\delta
n_{k}^{\eta}(\bm{r})\bm{s}_{k}^{\eta}+n_{k}^{\eta}\delta\bm{s}_{k}^{\eta}(\bm{r})+\delta\bm{\mathcal{S}}_{k}^{\eta}(\bm{r}),$
(19)

where we took into account all possible types of
$\delta\bm{S}_{k}^{\eta}(\bm{r})$ variation. Indeed, the first term describes
the change of the electron spin distribution due to the change in the density
$\delta n_{k}^{\eta}$. The second term corresponds to the change of the spin
vector $\delta\bm{s}_{k}^{\eta}$ for each individual electron independently of
the electron number distribution. The third term is the remaining linear-order
variation, which is essentially neither due to $\delta n_{k}^{\eta}(\bm{r})$
or $\delta\bm{s}_{k}^{\eta}(\bm{r})$ separately; thus
$\delta\bm{\mathcal{S}}_{k}^{\eta}$ describes the correlated change of both
the electron spin and charge densities. Naturally, the second and the third
terms in this expansion turn out to describe the non-adiabatic spin tilting
and the spin-dipole effects, respectively.

We proceed with calculating $\delta\bm{S}_{k}^{\eta}(\bm{r})$ from the kinetic
equation 14. In what follows we keep in Eq. 14 only the terms linear in $U$
and $\bm{F}=-\nabla_{\bm{r}}U$. In this limit the change of the electron
density $\delta n_{k}^{\eta}$ can be determined independently from the scalar
part of Eq. 14. Taking the trace over Eq. 14 we get

$\left(\bm{v}_{k}^{\eta}\cdot\nabla_{r}+\bm{F}(\bm{r})\cdot\nabla_{k}\right)n_{k}^{\eta}(\bm{r})=0.$
(20)

Here $\bm{v}_{k}^{\eta}$ is the electron group velocity given by Eq. 6. In the
linear response regime the correction $\delta n_{k}^{\eta}$ is given by:
$\delta n_{k}^{\eta}(\bm{r})=U(\bm{r})(\partial
n_{k}^{\eta}/\partial\varepsilon)$, where $\varepsilon$ is the electron
energy. The change in the overall 2DEG density is $\delta n(\bm{r})=\delta
n^{+}(\bm{r})+\delta n^{-}(\bm{r})$, where $\delta
n^{\eta}(\bm{r})=-\nu_{F}^{\eta}~{}U(\bm{r})$ and $\nu_{F}^{\eta}$ is the
density of states in $\eta$ subbands taken at the Fermi energy.
Correspondingly, the perturbation of the spin density Eq. 13 due to the first
term in Eq. 19 is given by

$\delta\bm{S}^{(1)}(\bm{r})=\sum_{k,\eta}\bm{s}_{k}^{\eta}\cdot\delta
n_{k}^{\eta}(\bm{r})=\bm{e}_{z}\Omega_{0}\left(\frac{\nu_{F}^{+}}{\Omega_{F}^{+}}-\frac{\nu_{F}^{-}}{\Omega_{F}^{-}}\right)U(\bm{r}).$
(21)

The term $\delta\bm{S}^{(1)}(\bm{r})$ is responsible for the change in the
out-of-plane spin density component and it appears even if there is no spin-
orbit interaction. A complex spin-orbital electron dynamics is responsible for
an extra spin response described by $\delta{\bm{s}}_{k}^{\eta}$ and
$\delta\bm{\mathcal{S}}_{k}^{\eta}$. We notice that
$\delta\bm{s}_{k}^{\eta},\delta\bm{\mathcal{S}}_{k}^{\eta}$ are absent in a
homogeneous setting, thus the expansion of
$\delta\bm{s}_{k}^{\eta},\delta\bm{\mathcal{S}}_{k}^{\eta}$ starts with the
linear term $\nabla_{r}U$. Taking the trace over Eq. 14 multiplied by
$\hat{\bm{\sigma}}$ and keeping only the terms linear in $\nabla_{\bm{r}}$
gradient we get

$\displaystyle\left[\bm{\Omega}_{k}\times\delta\bm{s}_{k}^{\eta}(\bm{r})\right]-\left(\bm{F}(\bm{r})\cdot\nabla_{k}\right)\bm{s}_{k}^{\eta}=0,$
(22)
$\displaystyle\left[\bm{\Omega}_{k}\times\delta\bm{\mathcal{S}}_{k}^{\eta}(\bm{r})\right]+\left[\bm{s}_{k}^{\eta}\times\left(\bm{s}_{k}^{\eta}\times{\bm{\Omega}^{so}({\nabla}_{\bm{r}}n_{k}^{\eta})}\right)\right]=0,$
(23)

where $\bm{\Omega}^{so}(\nabla_{r}n_{k}^{\eta})$ is obtained from Eq. 2 by
replacing $\bm{k}\to\nabla_{\bm{r}}n_{k}^{\eta}(\bm{r})$.

Let us comment on the relation between
$\delta\bm{s}_{k}^{\eta},\delta\bm{\mathcal{S}}_{k}^{\eta}$ and the previously
described kinematic effects. The first equation Eq. 22 can be satisfied by
changing the electron spin vector $\delta\bm{s}_{k}^{\eta}$ independently of a
particular density distribution $n_{k}^{\eta}$, it thus indeed describes the
spin rotation of individual electrons due to the precession in the effective
magnetic field $\bm{\Omega}_{k}$. Naturally, the nonzero term
$\delta\bm{s}_{k}^{\eta}$ is exactly the non-adiabatic correction to the
instant spin vector $\bm{s}_{k}^{\eta}$ which follows adiabatically the local
direction of $\bm{\Omega}_{k}$. The solution of the equation 22 replicates the
result from Eq.17

$\delta\bm{s}_{k}^{\eta}(\bm{r})=\eta\frac{1}{2\Omega_{k}^{3}}\Bigl{[}\bm{\Omega}_{k}\times(\bm{F}(\bm{r})\cdot\nabla_{\bm{k}})\bm{\Omega}_{k}\Bigr{]}.$
(24)

It is worth noting that $\delta\bm{s}_{k}^{\eta}$ is nonlinear with respect to
$\bm{\Omega}_{k}$. The second equation Eq. 23 describes the appearance of
$\delta\bm{\mathcal{S}}_{k}^{\eta}$, the general form of the solution is given
by

$\delta\bm{\mathcal{S}}_{k}^{\eta}(\bm{r})=-\frac{1}{4\Omega_{k}^{2}}\left[\bm{\Omega}_{k}\times\bm{\Omega}^{so}({\nabla}_{r}n_{k}^{\eta})\right].$
(25)

Importantly, the additional spin density $\delta\bm{\mathcal{S}}_{k}^{\eta}$
responds directly to the spatial gradient of the electron density
$\nabla_{r}n_{k}^{\eta}(\bm{r})$ entering in $\bm{\Omega}^{so}$. In fact, this
allows us to refer $\delta\bm{\mathcal{S}}_{k}^{\eta}$ as the correlational
term: it is neither due to the independent change in the number of electrons
or due to the individual electron spin rotation. Instead,
$\delta\bm{\mathcal{S}}_{k}^{\eta}$ describes the simultaneous change in the
electron spin due to the variation in its spatial density, it is indeed
relevant to the spin-dipole effect.

###  III.3 Interplay between microscopic mechanisms and the role of Berry
curvature

The explicit evaluation of extra-spin density terms from Eq. 24,25 for the
Rashba ferromagnet model gives the following expressions

$\displaystyle\delta\bm{s}_{k}^{\eta}=\eta\frac{e\mathcal{F}_{k}}{2\lambda_{so}}\cdot\bm{E}(\bm{r})-\eta\frac{2e\lambda_{so}^{2}}{\Omega_{k}^{3}}\left[\bm{k}\times\bm{E}(\bm{r})\right],$
(26)
$\displaystyle\delta\bm{\mathcal{S}}_{k}^{\eta}=-\mathcal{F}_{k}\cdot\frac{\Omega_{k}}{4\lambda_{so}}\nabla_{\bm{r}}n_{k}^{\eta}(\bm{r})+\eta\frac{\lambda_{so}}{2\Omega_{k}^{2}}\bm{e}_{z}\left(\bm{\Omega}_{k}\cdot\nabla_{\bm{r}}\right)n_{k}^{\eta}(\bm{r}),$
(27)

where $\mathcal{F}_{k}$ is the magnitude of the Berry curvature from Eq. 7,
and the density gradient
$\nabla_{r}n_{k}^{\eta}(\bm{r})=-e\bm{E}(\bm{r})(\partial
n_{k}^{\eta}/\partial\varepsilon)$ is due to the redistribution of electrons
in the vicinity of an electrostatic potential inhomogeneity.

We note that various terms from Eqs. 26, 27 give rise to quite different spin
phenomena. For instance, the second terms in
$\delta\bm{s}_{k}^{\eta},\delta\bm{\mathcal{S}}_{k}^{\eta}$ depend on the
electron momentum direction and they are particularly important for the
generation of spin currents in nonmagnetic systems (they survive at
$\Omega_{0}\to 0$); the second term in $\delta\bm{s}_{k}^{\eta}$ is
responsible for the universal spin Hall conductivity [62]. Alternatively, it
keeps significance for spin dynamics, see the details in Sec. IV. Below we
focus on the local magnetoelectric effect, that is the appearance of an
equilibrium spin density in response to the local electric field. This
phenomenon stems from the first terms in
$\delta\bm{s}_{k}^{\eta},\delta\bm{\mathcal{S}}_{k}^{\eta}$; they can directly
generate an additional spin density at a given point in a space as they
survive averaging over the electron momentum direction. Moreover, these terms
can be explicitly expressed in terms of the Berry curvature, thus they are
specific for topological systems.

The equilibrium spin density perturbations coupled with the Berry curvature of
electronic states have only in-plane components; substituting Eqs. 26, 27 to
the spin density perturbation from Eq. 13 we get

$\displaystyle\delta\bm{S}_{\parallel}(\bm{r})=\sum_{k,\eta}n_{k}^{\eta}\delta\bm{s}_{k}^{\eta}(\bm{r})+\delta\bm{\mathcal{S}}_{k}^{\eta}(\bm{r})\equiv\left(\chi_{t}+\chi_{d}\right)\cdot\bm{E}(\bm{r}),$
(28)

where the magnetoelectric susceptibilities $\chi_{t,d}$ correspond to the non-
adiabatic spin tilting and spin-dipole effects, respectively. The evaluated
expressions for $\chi_{t},\chi_{d}$ are given by

$\chi_{t}=\frac{e}{2\lambda_{so}}\left(Q_{F}^{+}+Q_{F}^{-}\right),\qquad\chi_{d}=-e\frac{\lambda_{so}\Omega_{0}}{2}\left(\frac{\nu_{F}^{+}}{\Omega_{F+}^{2}}+\frac{\nu_{F}^{-}}{\Omega_{F-}^{2}}\right),$
(29)

where $Q_{F}^{\pm}$ is the total Berry flux from Eq. 8. It is important to
emphasize that both the non-adiabatic spin tilting and the spin-dipole effects
are equally important to describe correctly the emergent spin patterns in
2DEG. In Fig. 2 we plot the dependence of the overall spin-response
coefficient $\chi\equiv\chi_{t}+\chi_{d}$ (solid lines) along with the partial
contributions from $\chi_{t}$ and $\chi_{d}$ (dotted lines) on the electron
gas Fermi energy $\mu$. We note that the terms $\chi_{t}$ and $\chi_{d}$ are
generally of the same order of magnitude. Moreover, in case when the electron
gas populates both spin subbands $\mu>\Omega_{0}/2$ the overall response
entirely disappears $\chi_{t}+\chi_{d}=0$ (this feature was previously noted
by [59, 66]). In the opposite case when electrons fill only the lowest spin-
subband $\mu<\Omega_{0}/2$ the terms $\chi_{t},\chi_{d}$ have opposite signs,
which results in the sign-altering dependence of $\chi$ on the Fermi energy.
We finally note that when either the spin-orbit coupling or the exchange
interaction is absent, the coefficients $\chi_{t}=\chi_{d}=0$ turn to zero and
the corresponding equilibrium spin patterns disappear.

Figure 2:  The dependence of the susceptibility $\chi=\chi_{t}+\chi_{d}$ on
the Fermi energy for two values of $\xi$ parameter: (a) $\xi=0.6$ and (b)
$\xi=0.2$.

###  III.4 Discussion

Let us discuss the physical significance of the described phenomena. We
firstly comment on the role that intrinsic mechanisms described by Eqs. 26, 27
play for the charge and spin transport on distances that greatly exceed the
mean free path. The non-adiabatic spin precession lies in the basis of the
Karplus-Luttinger mechanism of the anomalous Hall effect (AHE) [67, 68, 69],
of the so-called intrinsic mechanisms of the spin Hall (SHE) [62] and spin-
galvanic effects [70]. However, in order to estimate correctly the overall
electron gas response one has to additionally examine the disorder effects. In
particular, the intrinsic contribution to AHE, which is due to the anomalous
velocity term $\delta\bm{v}_{k}^{\eta}\propto
e\mathcal{F}_{k}^{\eta}\cdot\left[\bm{e}_{z}\times\bm{E}\right]$, is generally
cancelled out by the contributions due to side-jump scattering processes [57,
53, 56]. Alternatively, considering the generation of spin currents upon the
applied homogeneous electric field one has to carefully account for the
emergent nonequilibrium phenomena [71, 70, 72, 73]; e.g. the spin Hall current
due to the intrinsic mechanism is often compensated by the nonequilibrium spin
current arising nearby the sample boundaries [58, 74, 75].

However, the contributions
$\delta\bm{s}_{k}^{\eta},\delta\bm{\mathcal{S}}_{k}^{\eta}$ preserve the
importance in the nondissipative regime, when the underlying electrostatic
perturbation varies at the distances much smaller than the mean free path. In
particular, this matters for 2DEG charge and spin distribution around an
ionized impurity, at that the typical spatial scale under consideration is the
Thomas-Fermi screening length. The distribution of an excessive 2DEG spin
density emerging around an axially symmetric perturbation forms a skyrmion-
like vortex pattern which is schematically shown in Fig. 1. One concludes that
a smooth electrostatic potential disorder in topological spin polarized 2DEG
inevitably generates chiral spin textures, which can be particularly important
for the transport properties of the corresponding system; the formation of
non-collinear spin order generally leads to the topological Hall effect [76,
77, 78]. Moreover, in view of the spin-velocity coupling the formation of a
mesoscopic in-plane spin density is accompanied by the generation of the
persistent electrical current density
$\bm{j}(\bm{r})=e~{}2\lambda_{so}\left[\bm{e}_{z}\times\delta{\bm{S}(\bm{r})}\right]$.
In this regard an axially symmetric perturbation from Fig. 1 is additionally
featured by radially propagating electric currents. The presence of local
equilibrium currents also maintains the orbital magnetization, this effect has
been considered in [66].

It is worth mentioning that the considered magnetoelectric susceptibility of
free electrons generally opens up a possibility to directly affect the host
magnetization by a mesoscopic electric perturbation. The electric field-
induced 2DEG spin density lies in 2D channel plane and it is perpendicular to
the orientation of host magnetization, thus it is able to produce torque-like
effects. However, these issues remain poorly investigated, even despite its
importance for the magnetization control at nanoscales.

The microscopic mechanisms under consideration are general for multiband
systems. In the appendix A we present the connection of our method with the
wave-packet quasiclassical technique used in [39, 40, 41]. In the appendix B
we relate $\delta\bm{s}_{k}^{\eta},\delta\bm{\mathcal{S}}_{k}^{\eta}$ to the
Kubo formula method for the charge-spin correlation functions used in [59]. In
particular, we show that the non-adiabatic spin precession is described by the
interband correlation functions, while the spin-dipole effect stems from the
intraband ones.

##  IV Spin dynamics and magneto-optical effects

###  IV.1 Electric dipole spin resonance

In this section we focus on the electron gas spin dynamics in presence of an
oscillating electric field and describe the corresponding optical properties
of a magnetic two-dimensional system. The optical response of a 2D conductive
channel is generally encoded in the optical conductivity $\sigma(\omega)$. In
particular, the absorption coefficient $\alpha(\omega)=(4\pi/c){\rm
Re}[\sigma_{xx}(\omega)]$ is connected with the longitudinal part of
conductivity $\sigma_{xx}$. Also, since the time-reversal symmetry is broken
in presence of magnetism, different magneto-optical effects are possible, e.g.
the magneto-optical Kerr effect (MOKE), that is the rotation of the reflected
light polarization by the complex Kerr angle $\phi_{K}$. MOKE generally
appears in a conductive media due to nonzero optical Hall conductivity
$\sigma_{H}(\omega)$, for a 2D layer and normal incidence [79] one can expess
$\phi_{K}=\sigma_{H}/\sigma_{xx}\sqrt{1+(4\pi i/\omega)\sigma_{xx}}$.
Importantly, the considered geometry opens up the possibility to realize the
resonant enhancement of the Hall conductivity and, thus, of the related
magneto-optical effects.

Commonly, MOKE is seen to acquire a resonance structure due to interband
transitions affected by the combined effect of the spin-orbit coupling and the
electron spin polarization; the corresponding intrinsic contributions to the
Hall conductivity at finite frequencies have been investigated in a number of
papers [80, 81, 44, 82]. The general idea that we are going to explore in this
paper and which stands in the basis for the enhancement of magneto-optical
phenomena is that the optical properties of magnetic 2D systems can be
understood in terms of the electric dipole spin resonance (EDSR).
Correspondingly, the part of the optical conductivity responsible for the
resonant features can be directly related to the resonantly generated spin
density of 2DEG.

Figure 3: The electric dipole spin resonance scheme and the appearance of MOKE
due to the resonant Hall current generation
$\bm{j}_{\omega}\propto\left[\bm{e}_{z}\times\delta\bm{S}_{\omega}\right]$ .

Let us illustrate this process in more detail, see Fig. 3. The exchange
interaction field gives rise to a momentum-independent Zeeman splitting of the
electron spin subbands, for the considered geometry it is directed
perpendicular to the 2DEG plane. In fact, the spin-orbit interaction can be
viewed as $\bm{k}$-dependent effective magnetic field
$\bm{\Omega}^{so}(\bm{k})$ acting on electron spins. The applied in-plane ac-
electric field $\bm{E}_{\omega}e^{-i\omega t}$ causes the electron’s momentum
oscillations $\delta\bm{k}\propto\bm{E}_{\omega}e^{-i\omega t}$, so the
associated spin-orbital field also oscillates with frequency $\omega$. We note
that $\bm{\Omega}^{so}(\bm{k})$ is perpendicular to the out-of-plane exchange
interaction component $\bm{\Omega}_{0}$. Naturally, this makes it possible to
induce spin transitions when the electric field frequency coincides with the
magnitude of the Zeeman spin splitting $\hbar\omega=\Omega_{0}$, which is
exactly the EDSR scheme [83]. This spin resonance causes the equilibrium
electron spin density $\bm{S}_{0}\parallel\bm{e}_{z}$ from Eq. 11 to rotate
onto 2DEG plane, thus resonantly generating an excessive in-plane spin density
$\delta\bm{S}_{\omega}$. In view of the spin-orbit coupling Eq. 5 between the
velocity and spin operators, the accumulation of $\delta\bm{S}_{\omega}$
immediately leads to a resonant enhancement of the associated electric current
density
$\delta\bm{j}_{\omega}=2e~{}\lambda_{so}\left[\bm{e}_{z}\times\delta\bm{S}_{\omega}\right]$
and of the corresponding contribution to the optical conductivity.
Importantly, the in-plane spin density appears in tilted polarization with
respect to the vector of the electric field, see Fig 3. In particular, the
manifestation of the nonzero Berry curvature lies in the fact, that there
exists the ”perpendicular” polarization of the spin density, which gives rise
to the anomalous velocity $\delta\bm{v}_{k}^{\eta}\propto
e\mathcal{F}_{k}^{\eta}\cdot\left[\bm{e}_{z}\times\bm{E}\right]$ directed
perpendicular to $\bm{E}_{\omega}$ and responsible for the the magneto-optical
effects. The resonant generation of the spin density in this polarization
leads to the enhancement of $\sigma_{H}(\omega)$.

###  IV.2 Density matrix in the dynamical regime

Let us consider an oscillating electric field $\bm{E}_{\omega}e^{-i\omega t}$
applied in plane of the electron gas. We assume that the system remains
homogeneous and present $\hat{f}_{k}$ in the following form

$\hat{f}_{k}(t)=\frac{1}{2}n_{k}(t)+\bm{S}_{k}(t)\cdot\hat{\bm{\sigma}}.$ (30)

We keep to the high-frequency regime when $\omega$ greatly exceeds the typical
inverse relaxation time $\tau_{sc}^{-1}$ due to the scattering processes. The
distribution function $n_{k}(t)=n_{k}+\delta n_{k}(\omega)e^{-i\omega t}$
satisfies the scalar part of the kinetic equation Eq. 14

$\frac{\partial n_{k}(t)}{\partial
t}-e\left(\bm{E}(t)\cdot\nabla_{k}\right)n_{k}(t)=0,$ (31)

Since the equilibrium part contains terms from both spin subbands
$n_{k}=n_{k}^{+}+n_{k}^{-}$, the linear response perturbation $\delta
n_{k}(\omega)=\delta n_{k}^{+}(\omega)+\delta n_{k}^{-}(\omega)$ generally
contains two contributions

$\delta
n_{k}^{\eta}(\omega)=-\frac{e\bm{E}\cdot\bm{v}_{k}^{\eta}}{i\omega}\left(-\frac{\partial
n_{k}^{\eta}}{\partial\varepsilon}\right).$ (32)

The equation governing 2DEG spin dynamics is obtained similarly to Eq. 14 and
reads as

$\displaystyle\frac{\partial\bm{S}_{k}(t)}{\partial
t}-\left[\bm{\Omega}_{k}\times\bm{S}_{k}(t)\right]+e\left(\bm{E}(t)\cdot\nabla_{k}\right)\bm{S}_{k}(t)=0.$
(33)

At zero electric field this equation describes the electron spin precession
around $\bm{\Omega}_{k}$. The static regime solution in this case corresponds
to the equilibrium spin distribution
$\bm{S}_{k}^{\pm}\parallel\bm{\Omega}_{k}$ directed parallel or antiparallel
to the spin splitting field, while the non-stationary solution describes the
electron spin precession around $\bm{\Omega}_{k}$ with an eigenfrequency
$\Omega_{k}$. The nonzero $\bm{E}$, in its turn, drives the spin dynamics due
to the spin transfer in the momentum space. Naturally, when the frequency of
an external field $\omega$ coincides with the precession frequency of the
$\bm{k}$-electrons, the EDSR conditions are fulfilled leading to the resonant
rotation. This rotation occurs with the Rabi frequency
$\omega_{R}\propto\lambda_{so}E$, which goes to zero at small electric fields.
Naturally, in case of vanishing $\omega_{R}$ we can consider the linear
response regime with
$\bm{S}_{k}(t)=\bm{S}_{k}^{0}+\delta\bm{S}_{k}(\omega)e^{-i\omega t}$
differing from the equilibrium value
$\bm{S}_{k}^{0}=n_{k}^{+}\bm{s}_{k}^{+}+n_{k}^{-}\bm{s}_{k}^{-}$ by the
linear-order correction $\delta\bm{S}_{k}(\omega)$. This is justified when the
ongoing evolution of $\bm{S}_{k}(t)$ due to the Rabi oscillations is
interrupted by the spin relaxation processes. We thus introduce the
phenomenological spin relaxation rate $\Gamma$ and assume
$\omega_{R}\ll\Gamma\ll\Omega_{0}$.

In the linear response regime we can consider the spin response
$\delta\bm{S}_{k}(\omega)=\delta\bm{S}_{k}^{+}(\omega)+\delta\bm{S}_{k}^{-}(\omega)$
independently for each spin subband (recall that
$\bm{S}_{k}^{0}=n_{k}^{+}\bm{s}_{k}^{+}+n_{k}^{-}\bm{s}_{k}^{-}$). It is
convenient to present the linearized part in the following way
$\delta\bm{S}_{k}^{\eta}(\omega)=\delta
n_{k}^{\eta}(\omega)\bm{s}_{k}^{\eta}+n_{k}^{\eta}\delta\bm{s}_{k}^{\eta}(\omega)$,
where $\delta n_{k}^{\eta}(\omega)$ is determined by Eq. 32 and the equation
for $\delta\bm{s}_{k}^{\eta}(\omega)$ is given by

$\left(-i\omega+\Gamma\right)\delta\bm{s}_{k}^{\eta}(\omega)-\left[\bm{\Omega}_{k}\times\delta\bm{s}_{k}^{\eta}(\omega)\right]+e\left(\bm{E}_{\omega}\cdot\nabla_{k}\right)\bm{s}_{k}^{\eta}=0.$
(34)

Let us introduce the notation
$\delta\bm{s}_{k0}^{\eta}\equiv\delta\bm{s}_{k}^{\eta}(\omega\to 0)$ for the
additional electron spin density from Eq. 24 emerging in the static limit, we
note that $(\delta\bm{s}_{k0}^{\eta}\cdot\bm{\Omega}_{k})=0$. The third term
in this equation can be presented as follows
$e\left(\bm{E}_{\omega}\cdot\nabla_{k}\right)\bm{s}_{k}^{\eta}=\left[\bm{\Omega}_{k}\times\delta\bm{s}_{k0}^{\eta}\right]$.
The spin density perturbation $\delta\bm{s}_{k}^{\eta}(\omega)$ lies in the
plane perpendicular to $\bm{\Omega}_{k}$, the two independent polarizations
for $\delta\bm{s}_{k}^{\eta}(\omega)$ are given by $\delta\bm{s}_{k0}^{\eta}$
and $\left[\bm{n}_{k}\times\delta\bm{s}_{k0}^{\eta}\right]$, where
$\bm{n}_{k}=\bm{\Omega}_{k}/\Omega_{k}$ from Eq. 4. The solution of the
precession equation can be written in terms of these two vectors as follows

$\delta\bm{s}_{k}^{\eta}(\omega)=-\frac{\Omega_{k}^{2}}{\left(\omega-\Omega_{k}+i\Gamma\right)\left(\omega+\Omega_{k}+i\Gamma\right)}\left(\delta\bm{s}_{k0}^{\eta}+\frac{-i\omega+\Gamma}{\Omega_{k}}\left[\bm{n}_{k}\times\delta\bm{s}_{k0}^{\eta}\right]\right).$
(35)

The first term is directly due to the finite-frequency evolution of the non-
adiabatic spin tilt mechanism. The second term exists only at finite
frequencies and it arises from the electron spin retardation in the momentum
space. The denominator has a pole structure which reflects the EDSR with the
multiple resonances determined by $\omega=\Omega_{k}$.

The resulting correction to the density matrix can be presented as a sum of
two terms $\delta\hat{f}_{k}=e^{-i\omega t}(\delta\hat{f}_{k}^{\rm
den}+\delta\hat{f}_{k}^{\rm spin})$, where $\delta\hat{f}_{k}^{\rm den,spin}$
take the following form

$\displaystyle\delta\hat{f}_{k}^{\rm den}=\frac{1}{2}\left(\delta
n_{k}^{+}(\omega)+\delta n_{k}^{-}(\omega)\right)+\left(\delta
n_{k}^{+}(\omega)\bm{s}_{k}^{+}+\delta
n_{k}^{-}(\omega)\bm{s}_{k}^{-}\right)\cdot\hat{\bm{\sigma}},$ (36)
$\displaystyle\delta\hat{f}_{k}^{\rm
spin}=\left(n_{k}^{+}\delta\bm{s}_{k}^{+}(\omega)+n_{k}^{-}\delta\bm{s}_{k}^{-}(\omega)\right)\cdot\hat{\bm{\sigma}}.$
(37)

###  IV.3 Resonant spin response and optical conductivity

We start the discussion of the optical conductivity. The contribution
$\delta\hat{f}_{k}^{\rm den}$ is related specifically to the perturbation of
the electron density and it gives rise to the dominant part of the
longitudinal conductivity

$\bm{j}_{\omega}=e\sum_{k,\eta}\delta
n_{k}^{\eta}(\omega)\cdot\bm{v}_{k}^{\eta}=\sigma_{xx}^{0}(\omega)\bm{E}_{\omega},\qquad\sigma_{xx}^{0}(\omega)=\frac{ie^{2}}{\omega}~{}\frac{{v_{F+}^{2}\nu_{F}^{+}}+{v_{F-}^{2}\nu_{F}^{-}}}{2}.$
(38)

This is simply the Drude conductivity at finite frequency and it describes
nondissipative retardation of the 2DEG density in ac-electric field. On the
contrary, the term $\delta\hat{f}_{k}^{\rm spin}$ is due to the spin rotation
only. This contribution is responsible for the spin resonance related
phenomena and below we consider its role in more detail.

The density of an electric current $\delta\bm{j}_{\omega}$ emerging due to the
spin part of the density matrix $\delta\hat{f}_{k}^{\rm spin}$ is coupled with
an induced in-plane spin density $\delta\bm{S}_{\omega}$ of 2DEG

$\displaystyle\delta\bm{j}_{\omega}=2e\cdot\lambda_{so}\left[\bm{e}_{z}\times\delta\bm{S}_{\omega}\right],$
(39) $\displaystyle\delta\bm{S}_{\omega}=\frac{1}{2}\sum_{k}{\rm
Sp}\left(\delta\hat{f}_{k}^{\rm
spin}\cdot\hat{\bm{\sigma}}\right)=\sum_{k,\eta}n_{k}^{\eta}~{}\delta\bm{s}_{k}^{\eta}(\omega).$
(40)

Since $\delta\bm{s}_{k}^{\eta}(\omega)$ generally has two polarizations, see
Eq. 35, the overall spin $\delta\bm{S}_{\omega}$ and correspondingly the
associated current $\delta\bm{j}_{\omega}$ are also featured by two
independent polarizations

$\displaystyle\delta\bm{S}_{\omega}=\chi_{l}(\omega)\left[\bm{e}_{z}\times\bm{E}_{\omega}\right]+\chi_{H}(\omega)\bm{E}_{\omega},$
(41)
$\displaystyle\delta\bm{j}_{\omega}=\sigma_{l}(\omega)\bm{E}_{\omega}+\sigma_{H}(\omega)\left[\bm{e}_{z}\times\bm{E}_{\omega}\right],$
(42)

where $\sigma_{l,H}(\omega)=2e\lambda_{so}\chi_{l,H}(\omega)$. By this we
identified the contributions to the optical conductivity related to the
magnetoelectric spin susceptibility.

The correction to the longitudinal conductivity $\sigma_{l}(\omega)$ is
related to the retardation term
$\left[\bm{n}_{k}\times\delta\bm{s}_{k0}^{\eta}\right]$ in Eq. 35. Using the
formula Eq. 26 for $\delta\bm{s}_{k0}^{\eta}$ and averaging over momentum
directions we get (below we restore the Planck constant $\hbar$)

$\sigma_{l}(\omega)=-ie^{2}\cdot\sum_{k}\frac{\left(n_{k}^{-}-n_{k}^{+}\right)\hbar\omega}{\left(\hbar\omega-\Omega_{k}+i\Gamma\right)\left(\hbar\omega+\Omega_{k}+i\Gamma\right)}\frac{\lambda_{so}^{2}}{\Omega_{k}}\left(1+\frac{\Omega_{0}^{2}}{\Omega_{k}^{2}}\right).$
(43)

The straightforward calculation of this integral gives

$\sigma_{l}(\omega)=-\frac{ie^{2}}{16\pi\hbar}\left[\frac{2\Omega_{0}^{2}}{\hbar\omega}\left(\frac{1}{\Omega_{\rm
min}}-\frac{1}{\Omega_{F}^{-}}\right)+\left(1+\frac{\Omega_{0}^{2}}{(\hbar\omega)^{2}}\right)\ln{\left(\frac{\hbar\omega+\Omega_{F}^{-}}{\hbar\omega-\Omega_{F}^{-}}\cdot\frac{\hbar\omega-\Omega_{\rm
min}}{\hbar\omega+\Omega_{\rm min}}\right)}\right],$ (44)

where $\Omega_{\rm min}=\Omega_{0}$ for $\mu<\Omega_{0}/2$ and $\Omega_{\rm
min}=\Omega_{F}^{+}$ for $\mu>\Omega_{0}/2$. The expression from above remains
well-defined at $\Gamma\to 0$. In fact, the poles $\hbar\omega=\Omega_{k}$ in
the denominator of $\delta\bm{s}_{k}^{\eta}(\omega)$ lie in the continuum
spectrum, so the overall response of closely lying resonances merges onto the
$\omega$-regular curve featured by the Van Hove singularities at the edges of
the spin splittins $\hbar\omega=(\Omega_{0},\Omega_{F}^{\pm})$.

The real part of the longitudinal conductivity describes the energy
dissipation. The presence of the resonant poles in Eq. 35 reflects the
appearance of a finite absorption. Indeed, the absorption coefficient is
nonzero in the frequency range $\Omega_{\rm min}<\hbar\omega<\Omega_{F}^{-}$
(see Fig. 4b) corresponding to EDSR, the expression is given by

$\alpha(\omega)=\frac{4\pi}{c}{\rm
Re}\left[\sigma_{l}(\omega)\right]=\frac{\pi e^{2}}{4\hbar
c}\left[1+\left(\frac{\Omega_{0}}{\hbar\omega}\right)^{2}\right],\qquad\Omega_{\rm
min}<\hbar\omega<\Omega_{F}^{-}.$ (45)

Figure 4:  (a) The dependence of optical conductivities $\sigma_{l,H}$ on
frequency exhibits a resonant structure due to EDSR. (b) Electron band
structure and the transition energies $\Omega_{F}^{\pm}$ at the Fermi level.

The Hall conductivity $\sigma_{H}(\omega)$ stems from the Berry curvature
related term in $\delta\bm{s}_{k0}^{\eta}$. Taking into account Eqs. 26, 35
and averaging over the momentum direction we express $\sigma_{H}(\omega)$

$\sigma_{H}(\omega)=-\frac{e^{2}}{\hbar}\sum_{k}\frac{(n_{k}^{-}-n_{k}^{+})~{}\Omega_{k}^{2}}{\left(\hbar\omega-\Omega_{k}+i\Gamma\right)\left(\hbar\omega+\Omega_{k}+i\Gamma\right)}\cdot\mathcal{F}_{k}.$
(46)

The evaluation of this expression gives the following result

$\displaystyle\sigma_{H}(\omega)=-\frac{e^{2}}{4\pi\hbar}\frac{\Omega_{0}}{\hbar\omega}\ln{\left(\frac{\hbar\omega+\Omega_{F}^{-}}{\hbar\omega-\Omega_{F}^{-}}\cdot\frac{\hbar\omega-\Omega_{\rm
min}}{\hbar\omega+\Omega_{\rm min}}\right)}.$ (47)

Importantly, the Hall conductivity has the same resonance-aware logarithmic
term as $\sigma_{l}(\omega)$. Fig. 4 demonstrates the resonant enhancement of
the Hall conductivity in the EDSR absorption frequency range. Namely, we plot
the dependence of ${\rm Re}[\sigma_{l}(\omega)]$ and the absolute value
$|\sigma_{H}(\omega)|$ on the electric field frequency. It is clearly seen
from Fig. 4 that the increase in $|\sigma_{H}(\omega)|$ magnitude occurs
exactly in the same frequency range where ${\rm Re}[\sigma_{l}(\omega)]\neq 0$
is nonzero. In Fig. 5 we plot the dependences of real and imaginary parts of
the spin-resonance related optical conductivities $\sigma_{l,H}(\omega)$ on
frequency. The parameters are the same as in Fig. 4. The Van Hove
singularities give rise to the pronounced peaks in $|\sigma_{l,H}(\omega)|$ at
the boundary of the absorption band
$\hbar\omega=\Omega_{F}^{+},\Omega_{F}^{-}$. For the parameters taken in this
plot ($\mu=1.3\Omega_{0}$) the lower boundary is determined by
$\Omega_{F}^{+}$, see Fig. 4, as the electrons populate both spin subbands. We
also note that the behavior of $\sigma_{l,H}(\omega)$ when approaching the
static limit $\omega\to 0$ is different, see Fig. 5. While the longitudinal
part goes to zero $\sigma_{l}\to 0$, the Hall conductivity has a finite
nonzero limit $\sigma_{H}\to(e^{2}/\hbar)(Q_{F}^{+}+Q_{F}^{-})$ determined by
the total Berry flux $Q_{F}^{\pm}$ from Eq. 8 and reflecting the appearance of
persistent electric currents associated with the magnetoelectric
susceptibility. In the static limit, however, the accurate calculation of
$\sigma_{H}$ for a macroscopic sample requires one to take into account the
disorder effect [84].

Figure 5:  The dependence of the optical conductivities $\sigma_{l}(\omega)$
(frame a) and $\sigma_{H}(\omega)$ (frame b) on the frequency $\omega$, the
parameters $\xi=0.5,\mu=1.3~{}\Omega_{0}$.

###  IV.4 Discussion

The calculations of the optical conductivity of multiband systems is typically
performed using the Kubo formula [80, 81, 44, 82]. In the Appendix C we relate
the spin polarization and the density contributions from the density matrix
approach with different terms from the Kubo formalism. In Table 1 we summarize
the correspondence between these approaches; naturally the spin resonance
related terms are connected with the interband contributions $\sigma^{\rm
inter}$ to the conductivity.

Let us comment on the role of spin relaxation and electron scattering. The
multiple-peak structure of $\sigma_{l,H}(\omega)$ visible in Fig. 4 can be
well resolved only provided that the spin-orbit interaction splitting
($|\Omega_{F}^{\pm}-\Omega_{0}|\gg\tau_{sc}^{-1}$) exceeds the energy
broadening due to scattering processes. This requires rather strong spin-orbit
coupling. In the opposite case, the resonance profile of
$\sigma_{l,H}(\omega)$ will merge onto the single resonant-peak structure
centered at $\Omega_{0}$ with the line-shape sensitive to particular
scattering and spin relaxation processes, in analogy with EDSR due to an
electron gas in nonmagnetic semiconductors [85]. Interestingly, the Hall
conductivity can possess an additional information on spin relaxation times.

Kubo formula |  $\sigma_{xx}^{\rm intra}$ |  $\sigma_{xx}^{\rm inter}$ |  $\sigma_{H}^{\rm inter}$
---|---|---|---
Density matrix |  $\displaystyle\delta\hat{f}_{k}^{\rm den}$ |  $\displaystyle n_{k}^{\eta}\left[\bm{n}_{p}\times\delta\bm{s}_{k0}^{\eta}\right]\cdot\hat{\bm{\sigma}}$ |  $\displaystyle n_{k}^{\eta}\delta\bm{s}_{k0}^{\eta}\cdot\hat{\bm{\sigma}}$
Table 1: Density matrix and Kubo formula correspondence

We note that the finite absorption due to the electric dipole spin resonance
in 2DEG is not strictly limited to the case when the Zeeman field has an out-
of-plane component. In fact, most of the EDSR experiments with 2DEG in
nonmagnetic semiconductors [86, 87, 88] were carried out for the in-plane
magnetic field geometry. This is particularly useful when one aims to suppress
the orbital quantization effects and to focus on the spin-related response
only. On the contrary, combining spin-orbital electronic channels with
magnetism allows one to orient the Zeeman field perpendicular to the 2DEG
plane without breaking the spectrum onto Landau levels. Moreover, in this
setting the electron band states are featured by the appearance of a
topological structure. Studying experimentally the electronic spin resonance
phenomena in these systems seems of high interest as EDSR has an extra degree
of freedom that is the strong enhancement of the adjoint magneto-optical
effects.

Finally, the presented interpretation of the magneto-optical effects
enhancement in terms of spin resonance is equally relevant for other two-
dimensional models beyond Rashba ferromagnets. For instance, e.g. massive
Dirac metals [89], honeycomb lattices [90] or Haldane model [91] demonstrate
similar resonant features of the Hall conductivity.

## Summary

In summary, we have considered various spin-orbital phenomena leading to a
nontrivial behavior of an electron gas spin density upon application of the
electric field in two-dimensional magnets. Based on the density matrix
formalism we identified different microscopic mechanisms responsible for the
2DEG spin tilting in presence of an inhomogeneous electrostatic potential, and
described microscopic features of spin resonance upon oscillating electric
field with specific focus on optical conductivity and magneto-optical
phenomena. We traced the connection of the considered spin phenomena with the
Berry curvature of electronic band states thereby specifying the role of
electrons band topology. The presented analysis clarifies the basics of the
electron gas magnetoelectric response in two-dimensional magnets and
contributes to the ongoing discussion of its spintronics applications.

## Acknoledgments

The Author thanks I.V. Rozhansky, M.M. Glazov, P.S. Alekseev and N.S. Averkiev
from the Ioffe Institute for a very fruitful discussion of the results and for
giving useful advices. The work has been carried out with the financial
support of the Russian Science Foundation (project 18-72-10111). K.S.D. also
thanks the Foundation for the Advancement of Theoretical Physics and
Mathematics “BASIS”.

##  Appendix A Wave-packet dynamics semiclassical approach

The semiclassical theory of band electrons moving in a spatially varying
adiabatic perturbation $U(\bm{r})$ can be built by considering the wave-packet
dynamics [42]. Let us introduce the wave packet $|W_{k}^{n}\rangle$ consisting
of the $n$-th band Bloch states $|u_{k}^{n}\rangle$, its centre of mass
coordinates in real and momentum spaces are located at $(\bm{r}_{c},\bm{k})$.
The average of the physical quantity ${Q}$ described by the operator $\hat{Q}$
can be expressed in the following way [40]

${Q}=\sum_{k,n}f_{n}(\bm{k},\bm{r})\cdot\langle
W_{k}^{n}|\hat{Q}|W_{k}^{n}\rangle|_{\bm{r}=\bm{r}_{c}}-\nabla_{\bm{r}}\cdot\sum_{k,n}f_{n}(\bm{k},\bm{r})\cdot\langle
W_{k}^{n}|\hat{Q}\cdot\left(\hat{\bm{r}}-\bm{r}\right)|W_{k}^{n}\rangle|_{\bm{r}=\bm{r}_{c}},$
(48)

where the first term treats the wave packet as a point particle with the
distribution function $f_{n}(\bm{k},\bm{r})$, and the second term is the
first-order correction due to the wave-packet finite size effects. The great
advantage of this consideration is that it allows one to describe the electron
dynamics in terms of semiclassical equations. For instance, in the
nondissipative regime $f_{n}(\bm{k},\bm{r})$ satisfies the Liouville’s
equation

$\frac{df_{n}}{dt}=\frac{\partial f_{n}}{\partial
t}+\left\\{f_{n};\mathcal{H}\right\\}=0,$ (49)

where $\mathcal{H}=\varepsilon_{k}^{n}+U(\bm{r})$ is the classical Hamiltonian
function in $n$-th electron band with energy $\varepsilon_{k}^{n}$. The
Poisson bracket $\\{A;B\\}$ for $A,B$ physical quantities depending on
($\bm{r},\bm{k}$) takes into account the kinematic Berry phase [43, 49, 50]

$\displaystyle\left\\{A;B\right\\}=\omega_{\alpha\beta}\cdot(\partial_{\alpha}A)(\partial_{\beta}B),\qquad\omega_{\alpha\beta}=\begin{pmatrix}\varepsilon_{\alpha\beta\gamma}\Omega_{\gamma}^{n}&\delta_{\alpha\beta}\\\
-\delta_{\alpha\beta}&0\end{pmatrix},\qquad(\alpha,\beta)=({\bm{r},\bm{k}}),$
(50)

where $\omega_{\alpha\beta}$ is the antisymmetric Poisson matrix,
$\varepsilon_{\alpha\beta\gamma}$ is the Levi-Civita tensor, and
$\bm{\Omega}^{n}$ is the Berry curvature in $n$-th Bloch band defined as
follows
$\bm{\Omega}^{n}=\nabla_{\bm{k}}\times\mathcal{A}_{k}^{n}=i\langle\nabla_{\bm{k}}u_{k}^{n}|\times|\nabla_{\bm{k}}u_{k}^{n}\rangle,$
where $\mathcal{A}_{k}^{n}$ is the Berry connection. The expression for the
Liouville’s equation with account for the explicit form of
$\omega_{\alpha\beta}$ is given by:

$\displaystyle\frac{\partial f_{n}}{\partial
t}+\left(\frac{\partial\varepsilon_{k}^{n}}{\partial\bm{k}}+\left[\dot{\bm{k}}\times\bm{\Omega}_{n}\right]\right)\cdot\frac{\partial
f_{n}}{\partial\bm{r}}+\dot{\bm{k}}\cdot\frac{\partial
f_{n}}{\partial\bm{k}}=0$ (51)

where $\dot{\bm{k}}=-\nabla_{\bm{r}}\mathcal{H}=-\nabla_{\bm{r}}U(\bm{r})$.
The second term in brackets describes a full electron velocity
${\bm{v}}=\\{\mathcal{H};\bm{r}\\}=\bm{v}_{k}^{n}-[\nabla_{\bm{r}}U,\bm{\Omega}^{n}]$,
here $\bm{v}_{k}^{n}=\nabla_{k}\varepsilon_{k}^{n}$.

Let us apply this technique to calculate the emerging spin density nearby the
electrostatic inhomogeneity. We focus on the linear response regime. Following
Eq. 48 we present the spin density $\bm{S}(\bm{r})$ as follows

$\displaystyle\bm{S}(\bm{r})=\sum_{k,n}f_{n}(\bm{k},\bm{r})\cdot\langle
W_{k}^{n}|\hat{\bm{S}}|W_{k}^{n}\rangle-\nabla_{\bm{r}}\cdot\sum_{k,n}f_{n}(\bm{k},\bm{r})\langle
u_{k}^{n}|\hat{\bm{S}}\left(i\nabla_{\bm{k}}-\mathcal{A}_{k}^{n}\right)|u_{k}^{n}\rangle.$
(52)

In the second term we took into account that the wave packet
$|W_{k}^{n}\rangle$ is strongly localized nearby $\bm{k}$ in the momentum
space and we can approximate it as follows $|W_{k}^{n}\rangle\approx
e^{i\bm{kr}}|u_{k}^{n}\rangle$, which leads us directly to the expression in
Eq. 52. The unperturbed spin density $\bm{S}_{0}$ corresponds to
$U(\bm{r})=0$, at that $f_{n}(\bm{k},\bm{r})=f_{n}^{0}(\bm{k})$ and
$\bm{S}_{0}$ is given by

$\bm{S}_{0}=\sum_{k,n}f_{n}^{0}(\bm{k})\langle
u_{k}^{n}|\hat{\bm{S}}|u_{k}^{n}\rangle.$ (53)

The linear order deviations from $\bm{S}_{0}$ arise from three different
origins. Firstly, the distribution function
$f_{n}(\bm{k},\bm{r})=f_{n}^{0}(\bm{k})+\delta f_{n}(\bm{k},\bm{r})$ in
presence of $U$ is modified according to Eq. 51

$\left(\bm{v}_{k}^{n}\cdot\nabla_{\bm{r}}\right)\delta
f_{n}(\bm{k},\bm{r})+\bm{F}(\bm{r})\cdot\frac{\partial
f_{n}^{0}}{\partial\bm{k}}=0,\qquad\delta
f_{n}(\bm{k},\bm{r})=-U(\bm{r})\left(-\frac{\partial
f_{n}^{0}}{\partial\varepsilon}\right).$ (54)

Taking into account the redistribution of the electron density in the first
term in Eq. 52 and approximating $\langle
W_{k}^{n}|\hat{\bm{S}}|W_{k}^{n}\rangle\approx\langle
u_{k}^{n}|\hat{\bm{S}}|u_{k}^{n}\rangle$ we obtain the contribution identical
with Eq. 21 in the density matrix approach

$\displaystyle\delta\bm{S}^{(1)}(\bm{r})=\sum_{k,n}\delta
f_{n}(\bm{k},\bm{r})\langle u_{k}^{n}|\hat{\bm{S}}|u_{k}^{n}\rangle.$ (55)

Also, the inhomogeneous structure of $f_{n}$ gives rise to the spin-dipole
contribution, that is the second term in Eq. 52

$\delta\bm{\mathcal{S}}(\bm{r})=-\bm{F}(\bm{r})\cdot\sum_{k,n}\left(-\frac{\partial
f_{n}^{0}}{\partial\varepsilon}\right)\langle
u_{k}^{n}|\hat{\bm{S}}\left(i\nabla_{\bm{k}}-\mathcal{A}_{k}^{n}\right)|u_{k}^{n}\rangle.$
(56)

The straightforward evaluation of this expression for the Rashba ferromagnet
model leads to the susceptibility $\chi_{d}$ given by Eq. 29. Finally, there
is also the linear order perturbation which is not associated with the change
in the electron distribution. In fact, the first term in Eq. 52 is determined
by the average spin of an electron wave packet $\bm{s}_{k}^{n}(t)=\langle
W_{k}^{n}|\hat{\bm{S}}|W_{k}^{n}\rangle$, which satisfies the precession
equation

$\frac{d\bm{s}_{k}^{n}}{dt}=\left[\bm{\Omega}_{k}\times\bm{s}_{k}^{n}\right].$
(57)

According to our discussion from III.1, the wave-packet spin acquires a non-
adiabatic correction $\delta\bm{s}_{k}^{n}$ linear in $\bm{F}$ and given by
Eq. 17. This term gives rise to the spin perturbation
$\delta\bm{S}=\sum_{(k,n)}f_{n}^{0}\delta\bm{s}_{k}^{n}$ identical to
$\chi_{t}$ contribution to the spin susceptibility from Eq. 29.

##  Appendix B Kubo formula in the static limit

In this appendix we relate the semiclassical description of magnetoelectric
susceptibility in terms of the density matrix with the Kubo formula for the
charge-spin correlation functions, considered in detail in [59]. The spin
density induced in 2DEG by the change in the potential energy $U(\bm{r})$ is
given in linear response by

$\delta\bm{S}(\bm{r})=\int\frac{d\bm{q}}{(2\pi)^{2}}e^{i\bm{qr}}\bm{\mathcal{Q}}(\bm{q})U(\bm{q}),$
(58)

where $U(\bm{q})$ is the Fourier component of $U(\bm{r})$ and the static
charge-spin correlation function $\bm{\mathcal{Q}}(\bm{q})$ can be computed
from the Kubo formula

$\displaystyle\bm{\mathcal{Q}}(\bm{q})=\sum_{m,n}\bm{\mathcal{Q}}^{mn}(\bm{q}),$
(59)
$\displaystyle\bm{\mathcal{Q}}^{mn}(\bm{q})=\sum_{k}f_{k}^{m}\frac{\langle
u_{k}^{m}|\bm{\hat{S}}|u_{k+q}^{n}\rangle\langle
u_{k+q}^{n}|u_{k}^{m}\rangle}{\varepsilon_{k}^{m}-\varepsilon_{k+q}^{n}+i0}-f_{k+q}^{m}\frac{\langle
u_{k}^{n}|\bm{\hat{S}}|u_{k+q}^{m}\rangle\langle
u_{k+q}^{m}|u_{k}^{n}\rangle}{\varepsilon_{k}^{n}-\varepsilon_{k+q}^{m}+i0}.$

The terms $\bm{\mathcal{Q}}^{nn}$ with $m=n$ describe the intraband
contributions, while $\bm{\mathcal{Q}}^{mn}$ with $m\neq n$ correspond to the
interband ones.

The Kubo formula 59 has been explicitly evaluated for an arbitrary wavevector
$\bm{q}$ in [59] for Rashba ferromagnet and Dirac models. Here we focus on the
semiclassical regime when the potential $U$ changes smoothly on the Fermi
wavelength $\lambda_{F}$ scale, so the following relation is fulfilled
$\lambda_{F}\cdot\nabla_{r}U\ll U$. In this case the spin response becomes
local and the correlation function for the Rashba ferromagnet model takes the
following form $\bm{\mathcal{Q}}=i\bm{q}\cdot\chi$, where $\chi$ is the
$q$-independent coefficient describing the susceptibility
$\delta\bm{S}(\bm{r})=\chi\cdot\bm{E}(\bm{r})$.

We now proceed with considering the role of intra- and interband terms. In the
intraband contribution $\bm{\mathcal{Q}}^{nn}$ we replace
$(f_{k}^{n}-f_{k+q}^{n})/(\varepsilon_{k}^{n}-\varepsilon_{k+q}^{n}+i0)\approx\partial
f_{k}^{n}/\partial\varepsilon$ and keep only the ${q}$-linear terms in the
matrix elements. At that the expression takes the following form

$\bm{\mathcal{Q}}^{nn}=-i\bm{q}\cdot\sum_{k}\left(-\frac{\partial
f_{n}^{0}}{\partial\varepsilon}\right)\langle
u_{k}^{n}|\hat{\bm{S}}\left(i\nabla_{\bm{k}}-\mathcal{A}_{k}^{n}\right)|u_{k}^{n}\rangle,$
(60)

where $\mathcal{A}_{k}^{n}=i\langle u_{k}^{n}|\nabla_{{k}}u_{k}^{n}\rangle$ is
the Berry connection. When taking the Fourier transform Eq. 58
$\bm{\mathcal{Q}}^{nn}$ gives exactly the spin perturbation
$\delta\bm{\mathcal{S}}$ in form of Eq. 56 corresponding to the spin-dipole
term within the semiclassical wave-packet approach. We thus conclude that the
spin-dipole effect from Eq. 29 is related to the intraband terms in the Kubo
formula.

In the interband contributions $\bm{\mathcal{Q}}^{mn}$ we also keep only the
linear terms with respect to $\bm{q}$, which brings us to the following
expression

$\displaystyle\bm{\mathcal{Q}}^{mn}(\bm{q})=i\bm{q}\cdot\sum_{k}f_{k}^{m}~{}{\rm
Re}\left(\frac{\langle
u_{k}^{n}|\hat{\bm{\sigma}}|u_{k}^{m}\rangle\cdot{\mathcal{A}}_{k}^{mn}}{\varepsilon_{k}^{m}-\varepsilon_{k}^{n}}\right),$
(61)

where $\mathcal{A}_{k}^{mn}=i\langle u_{k}^{m}|\nabla_{k}u_{k}^{n}\rangle$.
The straightforward calculations for the Rashba ferromagnet model gives

$\bm{\mathcal{Q}}^{mn}(\bm{q})=i\bm{q}\sum_{k}f_{k}^{m}\cdot\frac{\mathcal{F}_{k}^{mn}}{2\lambda_{so}},$
(62)

where $\mathcal{F}_{k}^{mn}=\nabla_{{k}}\times\mathcal{A}_{k}^{mn}$ is the
Berry curvature. The interband terms are related exactly to the non-adiabatic
spin tilt effect described by $\delta\bm{s}_{k}^{\eta}$ in the density matrix
formalism and given by $\chi_{t}$ susceptibility from Eq. 29.

##  Appendix C Kubo formula in the dynamical regime

In this appendix we relate the Kubo formula calculations of the optical
conductivity with the spin resonance related terms emerging in the density
matrix approach. Kubo formula for the conductivity is given by

$\sigma_{\alpha\beta}(\omega)=\frac{ie^{2}}{S}\sum_{k,m,n}\frac{f_{k}^{m}-f_{k+q}^{n}}{\varepsilon_{k}^{m}-\varepsilon_{k+q}^{n}}\cdot\frac{v_{(k,m),(k+q,n)}^{\alpha}v_{(k+q,n),(k,m)}^{\beta}}{\varepsilon_{k}^{m}-\varepsilon_{k+q}^{n}+\hbar\omega+i0},$
(63)

where $\bm{q}\to 0$ and ${\bm{v}}_{ij}$ is the proper matrix element of the
velocity operator between $i,j$ states. We consider firstly the longitudinal
conductivity $\sigma_{xx}(\omega)$. The contribution to $\sigma_{xx}(\omega)$
due to intraband terms has the form

$\sigma_{xx}^{\rm intra}(\omega)=\frac{ie^{2}}{\hbar\omega}\sum_{m}\int
d\varepsilon~{}\nu_{m}(\varepsilon)\left(-\frac{\partial
f_{k}^{m}}{\partial\varepsilon}\right)\langle|v_{{k},m}^{x}|^{2}\rangle,$ (64)

where $\nu_{m}$ is the density of states in the corresponding band $m$ and
$\langle|v_{{k},m}^{x}|^{2}\rangle$ is the angular averaged square of the
matrix element modulus. This part describes the Drude conductivity at
$\omega\tau_{sc}\gg 1$ due to the perturbation of the electron density and it
corresponds to Eq. 38 from the main text. For the Rashba ferromagnet model the
evaluation of the integral gives

$\sigma_{xx}^{\rm
intra}(\omega)=i\frac{e^{2}}{\omega}\cdot\frac{{v_{F+}^{2}\nu_{F}^{+}}+{v_{F-}^{2}\nu_{F}^{-}}}{2}.$
(65)

The contribution to $\sigma_{xx}(\omega)$ due to interband terms in case of
the Rashba ferromagnet model has the following form

$\sigma_{xx}^{\rm
inter}(\omega)=\frac{ie^{2}}{S}\sum_{k}\frac{f_{k}^{-}-f_{k}^{+}}{-\Omega_{k}}\cdot\frac{\langle|v_{(k,-),(k,+)}^{x}|^{2}\rangle}{\hbar\omega-\Omega_{k}+i0}+\frac{f_{k}^{+}-f_{k}^{-}}{\Omega_{k}}\cdot\frac{\langle|v_{(k,-),(k,+)}^{x}|^{2}\rangle}{\hbar\omega+\Omega_{k}+i0}.$
(66)

The angular averaged term is
$\langle|v_{(k,-),(k,+)}^{x}|^{2}\rangle=(\lambda_{so}^{2}/2)(1+\Omega_{0}^{2}/\Omega_{k}^{2})$.
Using this formula and combining the denominators in $\sigma_{xx}^{\rm inter}$
we get the following expression

$\sigma_{xx}^{\rm
inter}(\omega)=-ie^{2}\cdot\sum_{k}\frac{\left(f_{k}^{-}-f_{k}^{+}\right)\hbar\omega}{\left(\hbar\omega-\Omega_{k}+i\Gamma\right)\left(\hbar\omega+\Omega_{k}+i\Gamma\right)}\frac{\lambda_{so}^{2}}{\Omega_{k}}\left(1+\frac{\Omega_{0}^{2}}{\Omega_{k}^{2}}\right),$
(67)

which repeats Eq. 43 for $\sigma_{l}(\omega)$. We thus conclude that
$\sigma_{xx}^{\rm inter}(\omega)$ is related to
$\left[\bm{n}_{k}\times\delta\bm{s}_{k0}^{\eta}\right]$ polarization in terms
of the in-plane spin density (see Eqs. 35, 26 from the main text). It is
instructive to analyze the energy absorption due to the spin resonance. For
this purpose we write down explicitly the expression for the real part of the
longitudinal conductivity due to the interband terms

${\rm Re}[\sigma_{xx}^{\rm inter}(\omega)]=\frac{\pi e^{2}}{\hbar\omega
S}\sum_{k}\left(f_{k}^{-}-f_{k}^{+}\right)\left|v_{(k,-),(k,+)}^{x}\right|^{2}\cdot\delta\left(\varepsilon_{k}^{-}-\varepsilon_{k}^{+}+\hbar\omega\right).$
(68)

The expression has the form of the Fermi golden rule, its straigthforward
calculation leads to the Eq. 45.

We now turn to the transversal component of the conductivity. The interband
contribution can be expressed as:

$\sigma_{yx}^{\rm
inter}(\omega)=\frac{ie^{2}}{S}\sum_{k}\frac{f_{k}^{-}-f_{k}^{+}}{-\Omega_{k}}\frac{v_{-+}^{y}v_{+-}^{x}}{\hbar\omega-\Omega_{k}+i0}+\frac{f_{k}^{+}-f_{k}^{-}}{\Omega_{k}}\frac{\left(v_{-+}^{y}v_{+-}^{x}\right)^{\ast}}{\hbar\omega+\Omega_{k}+i0}$
(69)

The angular averaged combination of matrix elements $\langle
v_{-+}^{y}v_{+-}^{x}\rangle=-i\lambda_{so}\Omega_{0}/\Omega_{k}$ is purely
imaginary. Combining both terms we obtain

$\sigma_{yx}^{\rm
inter}(\omega)=-\frac{e^{2}}{S}\sum_{k<k_{F}^{-}}\frac{\Omega_{k}^{2}}{(\hbar\omega-\Omega_{k}+i0)(\hbar\omega+\Omega_{k}+i0)}\cdot\mathcal{F}_{k},$
(70)

which is same expression Eq. 46 that we get via the density matrix formalism
considering $\bm{s}_{\bm{k}0}$ contribution to the spin density (see Eqs. 35,
26 from the main text).

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