#  Electrical Magnetochiral current in Tellurium

L. E. Golub  Terahertz Center, University of Regensburg, 93040 Regensburg,
Germany  E. L. Ivchenko  Ioffe Institute, 194021 St. Petersburg, Russia  B.
Spivak  University of Washington, Seattle, WA 98195, USA

###### Abstract

We have studied theoretically the effect of Electrical Magneto-Chiral
Anisotropy (eMChA) in $p$-type tellurium crystals. It is shown that the terms
$k_{i}B_{j}$ in the hole Hamiltonian, linear both in the wave vector
${\bm{k}}$ and the magnetic field ${\bm{B}}$, do not lead to the eMChA and one
needs to include the higher-order terms like $k_{i}^{3}B_{j}$. Two microscopic
mechanisms of the effect are considered. In the first one only elastic
scattering of holes by impurities or imperfections are taken into
consideration only. In the second mechanism, besides the elastic scattering
processes the hole gas heating and its energy relaxation are taken into
account. It is demonstrated that he both contributions to the magneto-induced
rectification are comparable in magnitude. The calculation is performed by
using two independent approaches, namely, in the time relaxation approximation
and in the limit of of small chiral band parameter $\beta$. A bridge is thrown
between the eMChA and magneto-induced photogalvanic effects.

##  I Introduction

Tellurium is an elemental chiral crystal with a D3 point symmetry. It has a
natural optical activity [1, 2], and it is tellurium where the Circular
Photogalvanic effect [3, 4], electric-current induced optical activity [5, 6]
and bulk Circular Photon Drag effect [7] were discovered; Sakano et al. has
for the first time verified experimentally the spin texture of the right- and
left-handed tellurium by the ARPES and SARPES measurements [8]. Recently,
Rikken and Avarvari observed the effect of Electrical Magneto-Chiral
Anisotropy (eMChA) in Te crystals [9]. This effect manifests itself as an
additional contribution to the sample resistance $R=R_{0}(1+\gamma BI)$, where
$R_{0}$ is a constant, $B$ is the magnetic field strength, $I$ is the electric
current, and the coefficient of bilinear magneto-electric resistance $\gamma$
describes a rectification by the sample, see Refs. [10, 11] for reviews.
Earlier, the effect of chirality (or non-reciprocity) in magnetotransport has
been observed in a number of other gyrotropic materials: distorted bismuth
wires [12], carbon nanotubes [13, 14], crystals of chiral salt (DM-EDT-
TTF)2ClO4 [15], polar semiconductor crystal BiTeBr [16], topological
insulators [17, 18, 19, 20, 21], semimetals ZrTe5 [22], WTe2 [23] and
$\alpha$-Sn [24], and on the surface of SrTiO3(111) [25].

Theoretically, the eMChA effect has been considered for carbon nanotubes [26,
27], Weyl semimetals of TaAs type [28], semimetal ZrTe5 [22] (Supplemental
Material), surface states in topological insulators [29] and molecular
conductors [30]. In the works [29, 22], a calculation of the correction to the
electric current $\delta j\propto E^{2}B$, proportional to the squared
electric field strength $E$ and linear in the magnetic field $B$, has been
performed in the simplest approximation of a general relaxation time
($\tau$-approximation). This approach does not take into account a difference
between quasimomentum and energy relaxations, or between elastic and inelastic
relaxation processes of free charge carriers. In this paper we show that, with
account for this difference, there are two independent microscopic mechanisms
of eMChA. In a simplified form, the presence of two mechanisms can be
explained as follows: Let us divide a correction to the charge carrier
distribution function $\delta f_{\bm{k}}\propto E^{2}$ in two terms, $\delta
f(\varepsilon_{\bm{k}})$ and $\delta f^{as}_{\bm{k}}$, where the first
function depends on the carrier energy $\varepsilon_{\bm{k}}$ (${\bm{k}}$ is a
wavevector), and the second function, $\delta f^{as}_{\bm{k}}$, is an
asymmetric correction with zero average over the directions of the wavevector
${\bm{k}}$ at constant energy. The correction $\delta f^{as}_{\bm{k}}$ is
controlled by the momentum relaxation time $\tau_{p}$, while in order to
calculate $\delta f(\varepsilon_{\bm{k}})$ one must account for inelastic
processes of carrier-phonon interaction and, hence, introduce the energy
relaxation time $\tau_{\varepsilon}$ which can be much longer than $\tau_{p}$.
As noticed in Ref. [26], although the correction $\delta
f(\varepsilon_{\bm{k}})\propto\tau_{\varepsilon}$ by itself does not result in
the electric current, its relaxation through interaction with phonons produces
an asymmetric distribution of carriers in the $\bm{k}$ space with an extra
multiplier $\tau_{p}/\tau_{\varepsilon}$. As a result, the mechanisms related
to $\delta f^{as}_{\bm{k}}$ and $\delta f(\varepsilon_{\bm{k}})$ lead to
comparable contributions to the electrical magneto-chiral current $\delta
j\propto\tau_{p}^{2}E^{2}B$.

Here we consider both mechanisms resulting in eMChA of holes in the Te valence
band. The paper is organized as follows. In Sec. 2, macroscopic equations are
presented. General consideration of eMChA effect in Te is given in Sec. III.
In Sec. IV, the eMChA current is estimated in the relaxation-time
approximation. Sections V and VII are devoted to rigorous calculations of the
contributions caused by the elastic and inelastic relaxation processes,
respectively. The perturbative results in the lowest order in the chirality
parameter are presented in Sec. VI. In Sec. VIII, discussion of results is
given, and Sec. IX summarizes the paper.

##  II Macroscopic equations

The phenomenon under study is described by a fourth-rank tensor in the
expansion of the electric current density in powers of the electric field
strength ${\bm{E}}$ and magnetic field ${\bm{B}}$

$j_{i}=\sigma_{ij}E_{j}+\sigma^{(H)}_{ijk}E_{j}B_{k}+G_{ijkl}E_{j}E_{k}B_{l}\>.$
(1)

The first two terms are allowed by any point symmetry, ${\bm{\sigma}}$ is the
tensor of linear conductivity and $\sigma^{(H)}_{ijk}$ is the Hall
conductivity tensor. The eMChA effect is represented by the magnetochiral
tensor ${\bm{G}}$ symmetrical in indices $j$ and $k$. It is related by

$G_{ijkl}\propto\gamma_{ij^{\prime}k^{\prime}l}\sigma_{j^{\prime}j}\sigma_{k^{\prime}k}$

with the tensor ${\bm{\gamma}}$ which is introduced in Eq. (1) in Ref. [9] and
describes the second-harmonics generation

$E^{2\omega}_{i}=\gamma_{ijkl}j^{\omega}_{j}j^{\omega}_{k}B_{l}\>,$

under conditions where the modulation period $T=2\pi/\omega$ exceeds by far
all the microscopic times of the system.

In crystals of D3 symmetry there are ten linearly-independent components of
the $G_{ijkl}$ tensor with indices $zzzz$, $xxxx$, $zzxx$, $xxzz$, $zxxz$,
$xzzx$, $xyyx$, $zxxy$, $xxyz$ and $xzxy$ [31]. Note that, in this point
group, the component $G_{xxyy}$ equals to $(G_{xxxx}-G_{xyyx})/2$. In Ref.
[9], the following estimates are given: $12G_{zzzx}\approx G_{xxxy}\approx
3G_{xxxx}$, and the inequality $G_{zzzz}\ll G_{zzzx}$ is presented. This
contradicts the point symmetry D3 where the nonzero components $G_{zzzx}$ and
$G_{xxxy}$ are forbidden. In our work the attention is focused on the
components $G_{zzzz}$ and $G_{xxxx}$ allowed by the symmetry, i.e., on the
geometries ${\bm{j}}\parallel{\bm{E}}\parallel{\bm{B}}\parallel z$ (shortly
$z$-eMChA geometry) and ${\bm{j}}\parallel{\bm{E}}\parallel{\bm{B}}\parallel
x$ ($x$-eMChA). In these cases the Hall effect does not appear and hence is
not discussed here. We use the notation $\delta{\bm{j}}$ for the electric
magnetochiral (eMCh) current, or the third term in Eq. (1). The paper is
devoted to consideration of this particular current.

With acoount for the symmetry, the macroscopic relation between the correction
to the current $\delta{\bm{j}}$ and the electric and magnetic vectors can be
written in the following convenient form

$\displaystyle\delta j_{z}$ $\displaystyle=$ $\displaystyle
G^{(1)}E_{z}^{2}B_{z}+G^{(2)}(E_{x}^{2}+E_{y}^{2})B_{z}+G^{(3)}[(E_{x}^{2}-E_{y}^{2})B_{y}+2E_{x}E_{y}B_{x}]+G^{(4)}E_{z}(E_{x}B_{x}+E_{y}B_{y})\>,$
(2) $\displaystyle\delta j_{x}$ $\displaystyle=$ $\displaystyle
G^{(5)}(E_{x}^{2}+E_{y}^{2})B_{x}+G^{(6)}E_{z}^{2}B_{x}+G^{(7)}[(E_{x}^{2}-E_{y}^{2})B_{x}+2E_{x}E_{y}B_{y}]+G^{(8)}2E_{x}E_{y}B_{z}$
$\displaystyle+\
G^{(9)}E_{x}E_{z}B_{z}+G^{(10)}E_{z}(E_{x}B_{y}+E_{y}B_{x})\>,$
$\displaystyle\delta j_{y}$ $\displaystyle=$ $\displaystyle
G^{(5)}(E_{x}^{2}+E_{y}^{2})B_{y}+G^{(6)}E_{z}^{2}B_{y}+G^{(7)}[-(E_{x}^{2}-E_{y}^{2})B_{y}+2E_{x}E_{y}B_{x}]+G^{(8)}(E_{x}^{2}-E_{y}^{2})B_{z}$
$\displaystyle+\
G^{(9)}E_{y}E_{z}B_{z}+G^{(10)}E_{z}(E_{x}B_{x}-E_{y}B_{y})\>,$

where $G^{(n)}~{}(n=1\ldots 10)$ are macroscopic parameters. The material
relation between $\delta j_{x}$, $\delta j_{y}$ and the transverse components
of vectors ${\bm{E}}$ and ${\bm{B}}$ has an axial symmetry and preserves its
form at any orientation of the $x,y$ axes relative to the second-order
symmetry axes C2.

##  III General consideration

The current $\delta j_{z}\propto G_{zzzz}$ is induced in the magnetic field
${\bm{B}}\parallel z$. In presence of this field, the effective 2$\times$2
valence-band Hamiltonian in Te has the following form [4, 32, 33]

$\mathcal{H}=\mathcal{A}_{1}k_{z}^{2}+\mathcal{A}_{2}k_{\perp}^{2}+(\beta
k_{z}+gB_{z})\sigma_{z}+\Delta_{2}\sigma_{x}\>.$ (3)

Here $\bm{k}$ is a wavevector, $k_{\perp}^{2}=k_{x}^{2}+k_{y}^{2}$,
$\sigma_{x}$ and $\sigma_{z}$ are the pseudospin Pauli matrices in the basis
$\pm 3/2$ (the reducible representation ${\cal D}=H_{4}+H_{5}$), $\Delta_{2}$
is the spin-orbit half-splitting of the valence-band states

$(|3/2\rangle\pm|-3/2\rangle)/\sqrt{2}$

at the $H$ point of the Brillouin zone, the parameter $g$ describes the Zeeman
effect, the parameters $\mathcal{A}_{1},\mathcal{A}_{2}$ are responsible for
parabolic scalar terms, and the coefficient $\beta$ determines strength of
$k_{z}$-linear term, it has opposite signs in the two Te enantiomorphs
D${}_{3}^{4}$ and D${}_{3}^{6}$ (or P3121 and P3221). Hereafter we use the
hole representation and take ${\mathcal{A}_{1,2}>0}$.

We study magnetoelectric transport of holes occupying the lowest valence band
of Te (uppermost in the electron representation). According to Eq. (3) its
energy dispersion relation is given by

$\varepsilon_{\bm{k}}=\mathcal{A}_{1}k_{z}^{2}+\mathcal{A}_{2}k_{\perp}^{2}-\sqrt{\Delta_{2}^{2}+(\beta
k_{z}+gB_{z})^{2}}+\Delta_{2}.$ (4)

Since we are interested in linear-${\bm{B}}$ effects, we make an expansion
$\varepsilon_{\bm{k}}\approx\varepsilon_{\bm{k}}^{0}+\delta\varepsilon_{\bm{k}}$,
where the zero-field energy is

$\varepsilon_{\bm{k}}^{0}=\mathcal{A}_{1}k_{z}^{2}+\mathcal{A}_{2}k_{\perp}^{2}-\sqrt{\Delta_{2}^{2}+\beta^{2}k_{z}^{2}}+\Delta_{2}\>,$
(5)

and the correction

$\delta\varepsilon_{\bm{k}}=-gB_{z}\eta(k_{z}),\qquad\eta={\beta
k_{z}\over\sqrt{\Delta_{2}^{2}+\beta^{2}k_{z}^{2}}}\>.$ (6)

The hole energy dispersion at zero magnetic field and at $B_{z}\neq 0$ is
illustrated in Fig. 1.

Figure 1:  The lowest valence subband of tellurium in the hole representation
in the vicinity of the $H$ point. The curves show the hole energy dispersion
$\varepsilon_{\bm{k}}$ vs. $k_{z}$ at $k_{\perp}=0$ in the absence (dashed)
and presence (solid) of the magnetic field $\bm{B}\parallel z$.

At $B_{z}=0$ the eigenvectors of the Hamiltonian (3) are two-component columns

$u_{k_{z}}^{0}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{c}\sqrt{1+\eta(k_{z})}\\\
\sqrt{1-\eta(k_{z})}\end{array}\right]\>.$ (7)

The dispersion $\varepsilon_{\bm{k}}^{0}$ has the camel’s back shape with the
energy minimum $\varepsilon_{m}=-\Delta^{2}{\cal A}_{1}/\beta^{2}$. At fixed
hole energy $\varepsilon_{\bm{k}}^{0}=\varepsilon\geq\varepsilon_{m}$ the
values of $k_{\perp}^{2}$ lie in the range between 0 and
$\sqrt{(\varepsilon-\varepsilon_{m})/{\cal A}_{2}}$ while the values of
$k_{z}$ fill the the range $K_{z}(\varepsilon)$ containing two intervals
$[-\kappa(\varepsilon),-\kappa^{\prime}(\varepsilon)]$ and
$[\kappa^{\prime}(\varepsilon),\kappa(\varepsilon]$, where

$\displaystyle\kappa(\varepsilon)=\sqrt{\varepsilon-\Delta+\sqrt{\Delta^{2}+\beta^{2}\varepsilon/\mathcal{A}_{1}}\over\mathcal{A}_{1}}\>,$
(8)
$\displaystyle\kappa^{\prime}(\varepsilon)=\sqrt{\varepsilon-\Delta-\sqrt{\Delta^{2}+\beta^{2}\varepsilon/\mathcal{A}_{1}}\over\mathcal{A}_{1}}\>,$

and $\Delta=\Delta_{2}-\beta^{2}/(2\mathcal{A}_{1})$. For $\varepsilon>0$, the
value of $\kappa^{\prime}$ should be set to 0 and the range
$K_{z}(\varepsilon)=[-\kappa(\varepsilon),\kappa(\varepsilon)]$.

For calculation of the $G_{xxxx}$ component one should add to the Hamiltonian
(3) scalar terms linear in $B_{x}$ and odd in $k_{x}$, see the next section.

The eMCh current density is calculated in the standard way with the help of
the hole distribution function $f_{\bm{k}}$ as follows

$\delta\bm{j}=2e\sum_{\bm{k}}\bm{v}(\bm{k})f_{\bm{k}}\>,$ (9)

where $e>0$ is the elementary charge, the factor 2 accounts for the two
valleys $H$ and $H^{\prime}$, and $\bm{v}(\bm{k})$ is the hole velocity
$\hbar^{-1}\partial\varepsilon_{\bm{k}}/\partial\bm{k}$.

In the $z$-eMChA geometry, the distribution function $f_{\bm{k}}$ is dependent
on $k_{z}$ and $k^{2}_{\perp}$ and independent of the azimuth angle between
${\bm{k}}_{\perp}$ and the $x$ axis. It is helpful to change variables from
$(k_{z},k^{2}_{\perp})$ to $(k_{z},\varepsilon^{0}_{\bm{k}})$ bearing in mind
that

$k^{2}_{\perp}=\frac{1}{{\cal
A}_{2}}\quantity(\varepsilon^{0}_{\bm{k}}+\sqrt{\Delta_{2}^{2}+\beta^{2}k^{2}_{z}}-\Delta_{2})\>.$

Thus, all functions $k_{z}$ and $k^{2}_{\perp}$ are treated as dependent on
$k_{z}$ and energy $\varepsilon^{0}_{\bm{k}}$,
$\mathcal{F}(k_{z},\varepsilon^{0}_{\bm{k}})$. A sum of any function
$\mathcal{F}(k_{z},\varepsilon^{0}_{\bm{k}})$ over ${\bm{k}}$ is calculated as
follows

$\sum_{\bm{k}}\mathcal{F}(k_{z},\varepsilon^{0}_{\bm{k}})=g_{2D}\int\limits_{\varepsilon_{m}}^{\infty}\differential\varepsilon^{0}_{\bm{k}}\int\limits_{K_{z}(\varepsilon^{0}_{\bm{k}})}\differential
k_{z}\mathcal{F}(k_{z},\varepsilon^{0}_{\bm{k}})\>,$ (10)

where $g_{2D}={1/(8\pi^{2}\mathcal{A}_{2})}$ is the density of states for two-
dimensional motion in the $(xy)$ plane. In the following we consider the case
where the energy minimum $\varepsilon_{m}$ is small compared with the average
hole energy and use the limits
$[-\kappa(\varepsilon_{\bm{k}}),\kappa(\varepsilon_{\bm{k}})]$ and
$(0,\infty)$ of integration over $k_{z}$ and $\varepsilon^{0}_{\bm{k}}$ in
equations like Eq. (10).

The distribution function obeys the Boltzmann kinetic equation

${e\over\hbar}\bm{E}\cdot\partialderivative{f_{\bm{k}}}{\bm{k}}+\hat{\mathcal{I}}^{(\rm
el)}_{\bm{k}}\quantity[f]+\hat{\mathcal{I}}^{(\rm
inel)}_{\bm{k}}\quantity[f]=0\>,$ (11)

where the left-hand side contains the force and collision terms respectively.
The collision integral consists of two contributions describing elastic and
inelastic hole scattering. We will solve Eq. (11) by iterations up to the
second order in $\bm{E}$ and, therefore, present $f_{\bm{k}}$ as a sum
$f_{0}(\varepsilon_{\bm{k}})+f_{1}(\bm{k})+f_{2}(\bm{k})$ with
$f_{0}(\varepsilon_{\bm{k}})$ being the Fermi-Dirac distribution and
$f_{n}\propto E^{n}$.

##  IV Relaxation time approximation

We use the relaxation-time approximation taking the collision integral in the
form

$\hat{\mathcal{I}}_{\bm{k}}\quantity[f]={f_{\bm{k}}-f_{0}(\varepsilon_{\bm{k}})\over\tau}$
(12)

with $\tau$ being a constant. This corresponds to fast energy relaxation with
a rate equal to the elastic scattering rate.

Then the corrections of the first and second orders in $E_{z}$ are given by

$\displaystyle f_{1}(\bm{k})$ $\displaystyle=$ $\displaystyle-e\tau
E_{z}f_{0}^{\prime}(\varepsilon_{\bm{k}})v_{z}\>,$ (13) $\displaystyle
f_{2}(\bm{k})$ $\displaystyle=$ $\displaystyle-{e\tau
E_{z}\over\hbar}\partialderivative{f_{1}(\bm{k})}{k_{z}}\>,$

where $f_{0}^{\prime}(\varepsilon)=\partial
f_{0}(\varepsilon)/\partial\varepsilon$. Substitution of
$f_{\bm{k}}=f_{2}({\bm{k}})$ into Eq. (9) and integration by parts yields for
the eMCh current density

$\delta j_{z}=2{e^{3}\quantity(\tau
E_{z})^{2}\over\hbar^{3}}\sum_{\bm{k}}f_{0}(\varepsilon_{\bm{k}})\partialderivative[3]{\varepsilon_{\bm{k}}}{k_{z}}\>.$
(14)

An analogous result was obtained previously for 1D transport in quantum wires
[20]. Expanding

$f_{0}(\varepsilon_{\bm{k}}^{0}+\delta\varepsilon_{\bm{k}})\approx
f_{0}(\varepsilon_{\bm{k}}^{0})+f_{0}^{\prime}(\varepsilon_{\bm{k}}^{0})\delta\varepsilon_{\bm{k}}\>,$
(15)

calculating the third derivatives of $\varepsilon_{\bm{k}}^{0}$ and
$\delta\varepsilon_{\bm{k}}$ and integrating by parts we obtain

$G_{zzzz}=-12{e^{3}\tau^{2}\mathcal{A}_{1}g\beta\over\hbar^{3}\Delta_{2}}\sum_{\bm{k}}\eta^{2}(k_{z})\zeta^{3}(k_{z})f_{0}^{\prime}(\varepsilon_{\bm{k}}^{0})\>,$
(16)

where

$\zeta(k_{z})={\Delta_{2}\over\sqrt{\Delta_{2}^{2}+\beta^{2}k_{z}^{2}}}=\sqrt{1-\eta^{2}(k_{z})}.$
(17)

For degenerate hole gas with the Fermi energy $\varepsilon_{F}>0$ we come to
the final equation

$G_{zzzz}=g{\mathcal{A}_{1}\over\mathcal{A}_{2}}{e^{3}\tau^{2}\over\pi^{2}\hbar^{3}}\eta^{3}(\kappa_{\text{F}})\>,$
(18)

where $\kappa_{\text{F}}=\kappa(\varepsilon_{\text{F}})$ and
$\kappa(\varepsilon)$ is defined by Eq. (8).

An important result to stress is that, at small $\beta$, the eMCh current is
not linear but cubic function of $\beta$. This can be understood taking
$\varepsilon^{0}_{\bm{k}}$ as $\mathcal{A}_{1}k_{z}^{2}+A_{2}k_{\perp}^{2}$
and the magnetic-field induced correction to the energy as
$\delta\varepsilon=Pk_{z}$, where $P=-gB_{z}\beta/\Delta_{2}$, and shifting
the origin of the ${\bm{k}}$-space by $k^{0}_{z}=P/(2{\cal A}_{1})$. In the
new frame $k^{\prime}_{z}=k_{z}+k^{0}_{z}$ we obtain a fully parabolic
dispersion $\varepsilon_{{\bm{k}}^{\prime}}=\mathcal{A}_{1}k_{z}^{\prime
2}+\mathcal{A}_{2}k_{\perp}^{2}$ as in a centrosymmetric crystal where an eMCh
current is forbidden.

A similar calculation of the $G_{xxxx}$ component with the following energy
dispersion in the magnetic field $\bm{B}\parallel x$

$\varepsilon_{\bm{k}}=\varepsilon_{\bm{k}}^{0}+B_{x}k_{x}(\Xi_{\perp}k_{\perp}^{2}+\Xi_{z}k_{z}^{2})\>,$
(19)

yields

$G_{xxxx}=2{\mathcal{A}_{1}\over\mathcal{A}_{2}}{e^{3}\tau^{2}\over\pi^{2}\hbar^{3}}\Xi_{\perp}\kappa^{3}_{\text{F}}.$
(20)

Since the presence of a linear-$k_{x}$ term is not enough to get the
magnetochiral current, we took into account cubic-$\bm{k}$ terms in Eq. (19).
Note, however, that $\Xi_{z}$ makes no contribution to the eMCh current
because the third derivative $\partial^{3}(k_{x}k_{z}^{2})/\partial
k_{x}^{3}=0$, see Eq. (14).

###  IV.1 Microscopic interpretation of eMChA

We give here the simplest interpretation of the eMChA current (14). In the
external electric field $E_{z}$ the equilibrium hole distribution is shifted
in the ${\bm{k}}$-space by $\delta k_{z}=eE_{z}\tau/\hbar$. Then in the
simplest description one can present the nonequilibrium distribution function
as $f_{0}({\bm{k}}_{\perp},k_{z}-\delta k_{z})$. Let us expand this function
in powers of $\delta k_{z}$ as follows

$f_{0}({\bm{k}}_{\perp},k_{z}-\delta k_{z})=f_{0}({\bm{k}})-\frac{\partial
f_{0}}{\partial k_{z}}\delta
k_{z}+\frac{1}{2}\frac{\partial^{2}f_{0}}{\partial k_{z}^{2}}(\delta
k_{z})^{2}\>.$

The linear term contributes to the Ohmic current while the nonlinear
contribution is

$\delta j_{z}=e\sum\limits_{\bm{k}}v_{z}\frac{\partial^{2}f_{0}}{\partial
k_{z}^{2}}(\delta
k_{z})^{2}=\frac{e^{3}\tau^{2}E_{z}^{2}}{\hbar^{3}}\sum\limits_{\bm{k}}\frac{\partial^{3}\varepsilon_{\bm{k}}}{\partial
k_{z}^{3}}f_{0}(\varepsilon_{\bm{k}})\>.$ (21)

This equation differs from Eq. (14) only by a factor of 2 which reflects the
simplified character of the latter consideration.

##  V Mechanism due to elastic scattering

Now we consider the eMCh current formed in the process of elastic scattering
by short-range impurities. In this case the collision integral reads

$\hat{\mathcal{I}}^{(\rm
el)}_{\bm{k}}[f]={2\pi\over\hbar}\mathcal{N}_{i}\sum_{\bm{k}^{\prime}}\absolutevalue{V_{\bm{k}^{\prime}\bm{k}}}^{2}\delta\quantity(\varepsilon_{\bm{k}}-\varepsilon_{\bm{k}^{\prime}})(f_{\bm{k}}-f_{\bm{k}^{\prime}}),$
(22)

where $\mathcal{N}_{i}$ is the impurity concentration, and
$V_{\bm{k}^{\prime}\bm{k}}$ is the matrix element of scattering by an
individual impurity potential $V(\bm{r})=V_{0}\delta(\bm{r})$ given by
$V_{\bm{k}^{\prime}\bm{k}}=V_{0}\innerproduct{u_{k_{z}^{\prime}}}{u_{k_{z}}}$,
with $u_{k_{z}}$ being the eigenvectors of the Hamiltonian (3). For the
mechanism under consideration all the noneqilibrium corrections to the
distribution function $f_{\bm{k}}$ vanish after averaging over ${\bm{k}}$ at
the fixed energy. The role of corrections $\delta f$ dependent on the energy
$\varepsilon_{\bm{k}}$ is analyzed in Sect. VII.

###  V.1 Inversion of the collision integral at $\bf{B=0}$

At $\bm{B}=0$ we obtain from Eq. (7):

$\absolutevalue{V_{\bm{k}^{\prime}\bm{k}}}^{2}={V_{0}^{2}\over
2}\quantity[1+\eta(k_{z})\eta(k_{z}^{\prime})+\zeta(k_{z})\zeta(k_{z}^{\prime})].$
(23)

Below we use for brevity the notation $\mathcal{I}_{\bm{k}}[f]$ instead of
$\mathcal{I}^{(\rm el)}_{\bm{k}}[f]$ for the elastic collision integral at
${\bm{B}}=0$.

It follows from Eq. (23) that, for the short-range scattering potential, the
kernel of the elastic collision integral (22) is degenerate: It is a sum of
products of functions depending solely on $k_{z}$ or $k_{z}^{\prime}$. This
allows us to invert the operator $\hat{\mathcal{I}}_{\bm{k}}[f]$ by reducing
the following integral equation to the algebraic one

$G(k_{z},\varepsilon^{0}_{\bm{k}})+\hat{\mathcal{I}}_{\bm{k}}[f]=0\>,$ (24)

where the source function $G(k_{z},\varepsilon^{0}_{\bm{k}})$ satisfies the
integral condition

$\sum\limits_{\bm{k}}G(k_{z},\varepsilon^{0}_{\bm{k}})\delta(\varepsilon^{0}_{\bm{k}}-\varepsilon)\propto\int\limits_{-\kappa(\varepsilon)}^{\kappa(\varepsilon)}\differential
k_{z}G(k_{z},\varepsilon)=0\>,$ (25)

which means that the number of particles of a given energy are conserved under
elastic scattering. If the source function $G(k_{z},\varepsilon^{0}_{\bm{k}})$
in the kinetic equation (24) does not satisfy the condition (25) it should be
presented as a sum of the function satisfying this condition and the function
$G(\varepsilon^{0}_{\bm{k}})$ dependent purely on $\varepsilon^{0}_{\bm{k}}$.
In order to find the solution of the kinetic equation with the source
$G(\varepsilon^{0}_{\bm{k}})$ one must replace $\hat{\mathcal{I}}_{\bm{k}}[f]$
in Eq. (24) by the inelastic collision integral $\hat{\mathcal{I}}^{(\rm
inel)}_{\bm{k}}[f]$, see Section VII. For an odd source term,
$G(k_{z},\varepsilon^{0}_{\bm{k}})=-G(-k_{z},\varepsilon^{0}_{\bm{k}})$, we
obtain for the inverse operator

$\hat{\mathcal{I}}_{\bm{k}}^{-1}\quantity[G]={\hbar[G(k_{z},\varepsilon^{0}_{\bm{k}})+\eta(k_{z})L_{G\eta}/\quantity(1-L_{\eta^{2}})]\over
2\pi g_{2D}\mathcal{N}_{i}V_{0}^{2}C(k_{z},\varepsilon^{0}_{\bm{k}})}\>,$ (26)

and for an even function
$G(k_{z},\varepsilon^{0}_{\bm{k}})=G(-k_{z},\varepsilon^{0}_{\bm{k}})$ we have

$\hat{\mathcal{I}}_{\bm{k}}^{-1}\quantity[G]={\hbar[G(k_{z},\varepsilon^{0}_{\bm{k}})-\zeta(k_{z})L_{G}/L_{\zeta}]\over
2\pi g_{2D}\mathcal{N}_{i}V_{0}^{2}C(k_{z},\varepsilon^{0}_{\bm{k}})}\>,$ (27)
$C(k_{z},\varepsilon^{0}_{\bm{k}})=\kappa(\varepsilon^{0}_{\bm{k}})+{\Delta_{2}\over\beta}\zeta(k_{z})\operatorname{Arctanh}\quantity{\eta[\kappa(\varepsilon^{0}_{\bm{k}})]}\>,$
(28)

where $\operatorname{Arctanh}(z)=[\ln(1+z)-\ln(1-z)]/2$, and the function
$L_{F}(\varepsilon^{0}_{\bm{k}})$ is defined for any even-$k_{z}$ function
$F(k_{z},\varepsilon^{0}_{\bm{k}})$ as follows

$L_{F}(\varepsilon^{0}_{\bm{k}})=\int\limits_{0}^{\kappa(\varepsilon^{0}_{\bm{k}})}\differential
k_{z}{F(k_{z},\varepsilon^{0}_{\bm{k}})\over
C(k_{z},\varepsilon^{0}_{\bm{k}})}\>.$ (29)

By using the inverse collision integral we can calculate the conductivity

$\sigma_{zz}=2e\sum_{\bm{k}}v^{0}_{z}(\bm{k})\hat{\mathcal{I}}_{\bm{k}}^{-1}\quantity[-ev^{0}_{z}f_{0}^{\prime}]\>,$
(30)

where $v_{z}^{0}=\hbar^{-1}\partial\varepsilon_{\bm{k}}^{0}/\partial k_{z}$ is
the hole velocity in the absence of magnetic field. For degenerate hole
statistics we have

$\sigma_{zz}={2e^{2}\hbar\over\pi\mathcal{N}_{i}V_{0}^{2}}\quantity(L_{v^{2}}+{L_{v\eta}^{2}\over
1-L_{\eta^{2}}})\>,$ (31)

where the functions $L_{F}$ are taken at
$\varepsilon^{0}_{\bm{k}}=\varepsilon_{\rm F}$.

###  V.2 Allowance for linear-$\bf B$ term in collision integral

Scattering by impurities is affected by the magnetic field. In the
linear-$\bm{B}$ approximation we obtain

$\displaystyle\delta\hat{\mathcal{I}}_{\bm{k}}\quantity[f]=$
$\displaystyle{2\pi\over\hbar}\mathcal{N}_{i}\sum_{\bm{k}^{\prime}}\biggl{[}\delta\absolutevalue{V_{\bm{k}^{\prime}\bm{k}}}^{2}\delta\quantity(\varepsilon_{\bm{k}}^{0}-\varepsilon_{\bm{k}^{\prime}}^{0})$
(32)
$\displaystyle+\absolutevalue{V_{\bm{k}^{\prime}\bm{k}}}^{2}\delta^{\prime}\quantity(\varepsilon_{\bm{k}}^{0}-\varepsilon_{\bm{k}^{\prime}}^{0})\quantity(\delta\varepsilon_{\bm{k}}-\delta\varepsilon_{\bm{k}^{\prime}})\biggr{]}\quantity(f_{\bm{k}}-f_{\bm{k}^{\prime}}).$

Here $\delta\varepsilon_{\bm{k}}$ is given by Eq. (6), and, since the
Hamiltonian (3) in the presence of $B_{z}$ is obtained from its zero-field
value by the sustitution $k_{z}\to k_{z}+gB_{z}/\beta$, we have

$\delta\absolutevalue{V_{\bm{k}^{\prime}\bm{k}}}^{2}={gB_{z}\over\beta}\quantity(\partialderivative{k_{z}}+\partialderivative{k_{z}^{\prime}})\absolutevalue{V_{\bm{k}^{\prime}\bm{k}}}^{2},$
(33)

which yields

$\displaystyle\hskip
28.45274pt\delta\absolutevalue{V_{\bm{k}^{\prime}\bm{k}}}^{2}=V_{0}^{2}{gB_{z}\over
2\Delta_{2}}$ (34)
$\displaystyle\times\quantity[\eta^{2}(k_{z}^{\prime})-\eta^{2}(k_{z})]\quantity[\eta(k_{z}^{\prime})\zeta(k_{z})-\eta(k_{z})\zeta(k_{z}^{\prime})]\>.$

Passing from summation to integration over the variables
$(k_{z}^{\prime},\varepsilon_{\bm{k}^{\prime}}^{0})$ and integrating the term
with

$\delta^{\prime}(\varepsilon_{\bm{k}}^{0}-\varepsilon_{{\bm{k}}^{\prime}}^{0})=-\frac{\partial\delta(\varepsilon_{\bm{k}}^{0}-\varepsilon_{{\bm{k}}^{\prime}}^{0})}{\partial\varepsilon_{{\bm{k}}^{\prime}}^{0}}$

by parts, we get

$\delta\hat{\mathcal{I}}_{\bm{k}}\quantity[f]={2\pi\over\hbar}\mathcal{N}_{i}g_{2D}\\\
\times\Biggl{\\{}\int\limits_{-\kappa(\varepsilon_{\bm{k}}^{0})}^{\kappa(\varepsilon_{\bm{k}}^{0})}\differential
k_{z}^{\prime}\delta\absolutevalue{V_{\bm{k}^{\prime}\bm{k}}}^{2}\quantity[f(\varepsilon_{\bm{k}}^{0},k_{z})-f({\varepsilon_{\bm{k}}^{0},k_{z}^{\prime}})]\\\
+f_{\bm{k}}\derivative{\varepsilon_{\bm{k}}^{0}}\int\limits_{-\kappa(\varepsilon_{\bm{k}}^{0})}^{\kappa(\varepsilon_{\bm{k}}^{0})}\differential
k_{z}^{\prime}\absolutevalue{V_{\bm{k}^{\prime}\bm{k}}}^{2}\quantity(\delta\varepsilon_{\bm{k}}-\delta\varepsilon_{\bm{k}^{\prime}})\\\
-\derivative{\varepsilon_{\bm{k}}^{0}}\int\limits_{-\kappa(\varepsilon_{\bm{k}}^{0})}^{\kappa(\varepsilon_{\bm{k}}^{0})}\differential
k_{z}^{\prime}\absolutevalue{V_{\bm{k}^{\prime}\bm{k}}}^{2}\quantity(\delta\varepsilon_{\bm{k}}-\delta\varepsilon_{\bm{k}^{\prime}})f({\varepsilon_{\bm{k}}^{0},k_{z}^{\prime}})\Biggr{\\}}.$
(35)

Here we took into account that both
$\absolutevalue{V_{\bm{k}^{\prime}\bm{k}}}^{2}$ and
$\delta\varepsilon_{\bm{k}}$ are independent of $\varepsilon_{\bm{k}}^{0}$ and
dependent on $k_{z},k_{z}^{\prime}$ only.

###  V.3 Procedure to calculate the eMCh current

According to Eq. (32) or (35), at nonzero magnetic field the kinetic equation
takes the form

${e\over\hbar}E_{z}\partialderivative{f_{\bm{k}}}{k_{z}}+\hat{\mathcal{I}}_{\bm{k}}\quantity[f]+\delta\hat{\mathcal{I}}_{\bm{k}}\quantity[f]=0\>.$
(36)

The equilibrium hole gas is described by the Fermi–Dirac distribution function
(15) satisfying the identity

$\delta\hat{\mathcal{I}}_{\bm{k}}\quantity[f_{0}(\varepsilon_{\bm{k}}^{0})]+\hat{\mathcal{I}}_{\bm{k}}\quantity[f_{0}^{\prime}(\varepsilon_{\bm{k}}^{0})\delta\varepsilon_{\bm{k}}]=0\>,$
(37)

which is Eq. (36) at $E_{z}=0$.

The correction to the distribution function proportional to $E_{z}^{2}B_{z}$
can be found by iterations of the kinetic equation (36). First of all, we find
a linear-$E_{z}$ correction $f_{\bm{k}}^{(E)}$ at $B_{z}=0$. It is given by
$f_{\bm{k}}^{(E)}=-eE_{z}\hat{\mathcal{I}}_{\bm{k}}^{-1}\quantity[v_{z}f_{0}^{\prime}]$,
see Eq. (30). The required solution $\delta f_{\bm{k}}\propto E_{z}^{2}B_{z}$
is sought as a sum of two corrections labeled $f_{\bm{k}}^{(E^{2}B)}$ and
$f^{(EBE)}_{\bm{k}}$.

To calculate $f_{\bm{k}}^{(E^{2}B)}$ we perform the next iteration and find
the correction $f_{\bm{k}}^{(E^{2})}\propto E_{z}^{2}$ at $B_{z}=0$ from the
equation

${eE_{z}\over\hbar}\quantity(\partialderivative{f_{\bm{k}}^{(E)}}{k_{z}}-\overline{\partialderivative{f_{\bm{k}}^{(E)}}{k_{z}}})+\hat{\mathcal{I}}_{\bm{k}}\quantity[f^{(E^{2})}]=0\>.$
(38)

Here the bar denotes averaging over $k_{z}$ at a fixed energy
$\varepsilon^{0}_{\bm{k}}$, namely,

$\overline{F}={1\over
2\kappa(\varepsilon^{0}_{\bm{k}})}\int\limits_{-\kappa(\varepsilon^{0}_{\bm{k}})}^{\kappa(\varepsilon^{0}_{\bm{k}})}\differential
k_{z}F(k_{z},\varepsilon^{0}_{\bm{k}})\>.$ (39)

Then we include into consideration the magnetic field $B_{z}$ and find
$f_{\bm{k}}^{(E^{2}B)}$ as a solution of the linear equation

$\delta\hat{\mathcal{I}}_{\bm{k}}\quantity[f_{\bm{k}}^{(E^{2})}]+\hat{\mathcal{I}}_{\bm{k}}\quantity[f^{(E^{2}B)}]=0\>.$
(40)

In order to determine the second contribution, $f_{\bm{k}}^{(EBE)}$, we first
find the correction $f_{\bm{k}}^{(EB)}\propto E_{z}B_{z}$. It satisfies the
equation

$\displaystyle\hskip
19.91684pt{e\over\hbar}E_{z}\quantity[\partialderivative{\quantity(f_{0}^{\prime}\delta\varepsilon_{\bm{k}})}{k_{z}}-\overline{\partialderivative{\quantity(f_{0}^{\prime}\delta\varepsilon_{\bm{k}})}{k_{z}}}]$
(41) $\displaystyle+\
\delta\hat{\mathcal{I}}_{\bm{k}}\quantity[f^{(E)}]-\overline{\delta\hat{\mathcal{I}}_{\bm{k}}\quantity[f^{(E)}]}+\hat{\mathcal{I}}_{\bm{k}}\quantity[f^{(EB)}]=0\>.$

Finally we substitute the correction $f_{\bm{k}}^{(EB)}$ to

${eE_{z}\over\hbar}\partialderivative{f_{\bm{k}}^{(EB)}}{k_{z}}+\hat{\mathcal{I}}_{\bm{k}}\quantity[f^{(EBE)}]=0$
(42)

and find $f_{\bm{k}}^{(EBE)}$.

It should be noted that both $f_{\bm{k}}^{(E)}$ and the resulting functions
$f_{\bm{k}}^{(E^{2}B)},f_{\bm{k}}^{(EBE)}$ are odd in $k_{z}$, whereas those
obtained at intermediate iteration steps, $f_{\bm{k}}^{(E^{2})}$ and
$f^{(EB)}$, are even functions of $k_{z}$. For the mechanism due to elastic
scattering all these functions satisfy the integral condition (25).

The eMCh current is calculated according to Eq. (9) as follows

$\delta
j_{z}=2e\sum_{\bm{k}}\quantity[v_{z}^{0}\quantity(f_{\bm{k}}^{(EBE)}+f_{\bm{k}}^{(E^{2}B)})+\delta
v_{z}f_{\bm{k}}^{(E^{2})}].$ (43)

Here, following Eq. (4) we present the hole velocity as
$v_{z}=v^{0}_{z}+\delta v_{z}$ with

$\displaystyle
v^{0}_{z}(k_{z})=\frac{1}{\hbar}\left[2\mathcal{A}_{1}k_{z}+\beta\eta(k_{z})\right]\>,$
(44) $\displaystyle\delta
v_{z}(k_{z})=gB_{z}{\beta\over\hbar\Delta_{2}}\zeta^{3}(k_{z})\>.$ (45)

##  VI The magnetochiral current in the small $\beta$ limit

At small $\beta$, the equation (18) derived in the relaxation-time
approximation reduces to

$G_{zzzz}\approx
g{\mathcal{A}_{1}\over\mathcal{A}_{2}}{e^{3}\tau^{2}\over\pi^{2}\hbar^{3}}z^{3}\>,$
(46)

where $z=\beta\kappa_{F}/\Delta_{2}$.

Here we go beyond the relaxation-time approximation, apply the scheme
developed in the previous Section and calculate each of three contributions to
the eMCh current (43) assuming the constant $\beta$ to be small.

In the limit $\beta\to 0$, we have approximately

$\displaystyle C(k_{z},\varepsilon^{0}_{\bm{k}})\approx
2\kappa(\varepsilon_{\bm{k}}^{0})\>,\>\kappa(\varepsilon_{\bm{k}}^{0})\approx\sqrt{\varepsilon_{\bm{k}}^{0}\over\mathcal{A}_{1}}\>,\>\absolutevalue{V_{\bm{k}^{\prime}\bm{k}}}^{2}\approx
V_{0}^{2}\>,$ $\displaystyle\hskip
31.2982pt\varepsilon_{\bm{k}}^{0}\approx\mathcal{A}_{1}k_{z}^{2}+\mathcal{A}_{2}k_{\perp}^{2}\>,\>\quad
v_{z}^{0}\approx{2\mathcal{A}_{1}k_{z}\over\hbar}\>,$ (47)

and the inverted collision integral is given by

$\hat{\mathcal{I}}_{\bm{k}}^{-1}[G]\approx-\tau(\varepsilon_{\bm{k}}^{0})G(k_{z},\varepsilon^{0}_{\bm{k}})\>,\>\tau(\varepsilon_{\bm{k}}^{0})={2\pi\mathcal{A}_{2}\hbar\over\mathcal{N}_{i}V_{0}^{2}\kappa(\varepsilon_{\bm{k}}^{0})}\>.$
(48)

In the magnetic-field induced correction to the energy spectrum we take into
account the cubic-$\beta$ term because, as discussed above, the linear-$\beta$
correction does not result in eMChA. Therefore we take

$\delta\varepsilon_{\bm{k}}\approx-{gB_{z}\over 2}\quantity({\beta
k_{z}\over\Delta_{2}})^{3},\quad\delta v_{z}\approx-{3gB_{z}\over
2\hbar}\quantity({\beta\over\Delta_{2}})^{3}k_{z}^{2}.$ (49)

The $B_{z}$-linear correction to the scattering matrix element squared reads

$\delta\absolutevalue{V_{\bm{k}^{\prime}\bm{k}}}^{2}\approx{V_{0}^{2}\over
2}{gB_{z}\over\Delta_{2}}\quantity({\beta
k_{z}\over\Delta_{2}})^{3}\quantity(k_{z}^{\prime
2}-k_{z}^{2})\quantity(k_{z}^{\prime}-k_{z}).$ (50)

It can be neglected in the following because its contribution to the current
is parametrically smaller by a factor of $\varepsilon_{\rm F}/\Delta_{2}\ll 1$
compared with other contributions coming from the $B_{z}$-linear correction
(49). As a result, only two last lines of Eq. (35) contibute to
$\delta\hat{\mathcal{I}}_{\bm{k}}\quantity[f]$:

$\delta\hat{\mathcal{I}}_{\bm{k}}\quantity[f]={1\over\kappa\tau}\quantity[f_{\bm{k}}\delta\varepsilon_{\bm{k}}\derivative{\kappa}{\varepsilon_{\bm{k}}^{0}}+{1\over
2}\int\limits_{-\kappa(\varepsilon_{\bm{k}}^{0})}^{\kappa(\varepsilon_{\bm{k}}^{0})}\differential
k_{z}^{\prime}\delta\varepsilon_{\bm{k}^{\prime}}\partialderivative{f({\varepsilon_{\bm{k}}^{0},k_{z}^{\prime}})}{\varepsilon_{\bm{k}}^{0}}].$
(51)

We start from calculation of the third term in the right-hand side of Eq.
(43). The correction $f_{\bm{k}}^{(E^{2})}$ found from Eq. (38) with
$f_{\bm{k}}^{(E)}=-eE_{z}\tau v_{z}f_{0}^{\prime}$ is given by

$f_{\bm{k}}^{(E^{2})}=\quantity(2\mathcal{A}_{1}{eE_{z}\over\hbar})^{2}\tau(\tau
f_{0}^{\prime})^{\prime}\quantity(k_{z}^{2}-{\kappa^{2}\over 3}).$ (52)

Substituting this function into the last term in Eq. (43) we find its
contribution to the eMChA effect

$G^{(v)}_{zzzz}=-{32\mathcal{A}_{1}^{2}ge^{3}\tau\over
15\hbar^{3}}\quantity({\beta\over\Delta_{2}})^{3}g_{2D}\partialderivative{[\kappa_{\rm
F}^{5}\tau(\varepsilon_{\rm F})]}{\varepsilon_{\rm F}}.$ (53)

Using the relations $\kappa_{\rm F}\propto\varepsilon_{\rm F}^{1/2}$,
$\tau(\varepsilon_{\rm F})\propto\varepsilon_{\rm F}^{-1/2}$,
$\mathcal{A}_{1}\kappa_{\rm F}^{2}=\varepsilon_{\rm F}$, we arive at

$G^{(v)}_{zzzz}=-{8\over
15}g{\mathcal{A}_{1}\over\mathcal{A}_{2}}{e^{3}\tau^{2}\over\pi^{2}\hbar^{3}}z^{3}\>.$
(54)

Next, we search for the correction $f_{\bm{k}}^{(E^{2}B)}$. It is found from
Eq. (40) to be as follows, see Eq. (51)),

$f_{\bm{k}}^{(E^{2}B)}=gB_{z}\quantity({\beta
k_{z}\over\Delta_{2}})^{3}{1\over
2\kappa}\derivative{\kappa}{\varepsilon_{\bm{k}}^{0}}f_{\bm{k}}^{(E^{2})}\>,$
(55)

where $f_{\bm{k}}^{(E^{2})}$ is given by Eq. (52). This allows us to calculate
the second contribution in Eq. (43):

$G^{(E^{2}B)}_{zzzz}={16\over
105}g{\mathcal{A}_{1}\over\mathcal{A}_{2}}{e^{3}\tau^{2}\over\pi^{2}\hbar^{3}}z^{3}\>.$
(56)

Finally we calculate the contribution related to the function
$f_{\bm{k}}^{(EBE)}$. According to Eq. (42) this function has the form

$f_{\bm{k}}^{(EBE)}=-\tau{eE_{z}\over\hbar}\partialderivative{f_{\bm{k}}^{(EB)}}{k_{z}}\>.$
(57)

It allows us to rewrite the first contribution in Eq. (43) as

$j_{z}^{(EBE)}=2e^{2}E_{z}\quantity({2\mathcal{A}_{1}\over\hbar})^{2}\sum_{\bm{k}}f_{\bm{k}}^{(EB)}\tau^{\prime}k_{z}^{2}\>.$
(58)

While deriving this equation we took into account that the function
$f_{\bm{k}}^{(EB)}$ satisfies Eq. (25).

The solution of Eq. (41) for $f_{\bm{k}}^{(EB)}$ reads

$\displaystyle f^{(EB)}_{\bm{k}}=$
$\displaystyle\frac{gB_{z}eE_{z}\tau}{2\hbar}\quantity({\beta\over\Delta_{2}})^{3}\left[3f_{0}^{\prime}\quantity(k_{z}^{2}-{\kappa^{2}\over
3})\right.$ (59) $\displaystyle\left.\hskip 28.45274pt+\
\mathcal{A}_{1}\quantity(2f_{0}^{\prime\prime}-{f_{0}^{\prime}\over\varepsilon_{\bm{k}}^{0}})\quantity(k_{z}^{4}-{\kappa^{4}\over
5})\right]\>.$

Substitution of this expression to Eq. (58) leads to

$G_{zzzz}^{(EBE)}=-{2\over
105}g{\mathcal{A}_{1}\over\mathcal{A}_{2}}{e^{3}\tau^{2}\over\pi^{2}\hbar^{3}}z^{3}\>.$
(60)

The sum of three contributions (53), (56) and (60) yields

$G_{zzzz}=-{2\over
5}g{\mathcal{A}_{1}\over\mathcal{A}_{2}}{e^{3}\tau^{2}\over\pi^{2}\hbar^{3}}z^{3}\>,$
(61)

where $\tau$ is defined by Eq. (48). Comparing with the relaxation-time
approximation result (46) we see a difference both in the sign and a factor of
2/5.

##  VII Mechanism involving inelastic scattering

Now we turn to the mechanism of magnetochiral current involving the inelastic
scattering. Compared to the previous section we change the attention from the
asymmetric part of $\delta f_{\bm{k}}$ satisfying condition (25) to the
energy-dependent part $\delta f(\varepsilon_{\bm{k}})$ of the correction to
the hole distribution function. Assuming the hole-hole collisions to be more
effective than the hole energy relaxation on acoustic phonons we can describe
the energy-dependent sum $f_{0}(\varepsilon_{\bm{k}})+\delta
f(\varepsilon^{0}_{\bm{k}})$ as the Fermi-Dirac distribution function
$f_{0}(\varepsilon_{\bm{k}},T_{h})$ characterized by the hole temperature
$T_{h}$ different from the bath temperature $T$. Here we first briefly
describe the procedure to calculate $T_{h}$ and then show how the inelastic
relaxation of hole nonequilibrium distribution
$f_{0}(\varepsilon_{\bm{k}},T_{h})$ gives rise to an electric current
proportional to $(T_{h}-T)B_{z}$.

###  VII.1 Estimation of the hole effective
temperature $\propto E_{z}^{2}$

The effective temperature $T_{h}$ can be found from the heat balance equation

$\sigma_{zz}E_{z}^{2}={\cal J}\>.$ (62)

The left-hand side represents Joule heating produced by the passage of an
electric current with $\sigma_{zz}$ being the conductivity. The right-hand
side describes the energy relaxation of the holes following acoustic-phonon
scattering and has the form

${\cal
J}=\sum_{{\bm{k}}^{\prime}{\bm{k}}}\left(\varepsilon_{\bm{k}}-\varepsilon_{{\bm{k}}^{\prime}}\right)\left(W^{({\rm
ab})}_{{\bm{k}}^{\prime},{\bm{k}}}-W^{({\rm
em})}_{{\bm{k}},{\bm{k}}^{\prime}}\right)\>,$ (63)

where $W^{({\rm ab})}_{{\bm{k}}^{\prime},{\bm{k}}}$, $W^{({\rm
em})}_{{\bm{k}},{\bm{k}}^{\prime}}$ are the hole scattering rates for phonon
absorption and emission processes. Their difference is given by

$\displaystyle W^{({\rm ab})}_{{\bm{k}}^{\prime},{\bm{k}}}-W^{{\rm
em}}_{{\bm{k}},{\bm{k}}^{\prime}}=$
$\displaystyle\frac{2\pi}{\hbar}|M_{{\bm{k}}^{\prime}{\bm{k}}}|^{2}\delta(\varepsilon_{{\bm{k}}^{\prime}}-\varepsilon_{\bm{k}}-\hbar\Omega_{\bm{q}})$
(64)
$\displaystyle\times\left[(f_{\bm{k}}-f_{{\bm{k}}^{\prime}})N_{\bm{q}}-f_{{\bm{k}}^{\prime}}(1-f_{\bm{k}})\right]\>.$

Here ${\bm{q}}={\bm{k}}^{\prime}-{\bm{k}}$, $\Omega_{\bm{q}}$ and $N_{\bm{q}}$
are the phonon wave vector, frequency and occupation number

$N_{\bm{q}}=\frac{1}{\exp(\hbar\Omega_{\bm{q}}/k_{B}T)-1}\>,$ (65)

$M_{{\bm{k}}^{\prime}{\bm{k}}}$ is the scattering matrix element. For the
energy-dependent distribution function $f_{0}(\varepsilon_{\bm{k}},T_{h})$ the
term in the brackets in Eq. (64) reduces to

$\displaystyle\frac{{\rm
e}^{(\varepsilon-\varepsilon_{F})/k_{B}T_{h}}\left[{\rm
e}^{(\varepsilon^{\prime}-\varepsilon)/k_{B}T_{h}}-{\rm
e}^{(\varepsilon^{\prime}-\varepsilon)/k_{B}T}\right]}{({\rm
e}^{(\varepsilon-\varepsilon_{F})/k_{B}T_{h}}+1)({\rm
e}^{(\varepsilon-\varepsilon^{\prime})/k_{B}T_{h}}+1)({\rm
e}^{(\varepsilon^{\prime}-\varepsilon)/k_{B}T}-1)}$
$\displaystyle\approx-\frac{T_{h}-T}{T}\frac{\hbar\Omega_{\bm{q}}/k_{B}T}{{\rm
e}^{\hbar\Omega_{\bm{q}}/k_{B}T}-1}f_{0}(\varepsilon)\left[1-f_{0}(\varepsilon^{\prime})\right]\>,$

where $\varepsilon=\varepsilon^{0}_{\bm{k}}$,
$\varepsilon^{\prime}=\varepsilon^{0}_{{\bm{k}}^{\prime}}$.

For the degenerate statistics, $\varepsilon_{F}\gg k_{B}T$, a reasonable
estimation of ${\cal J}$ in Eqs. (62), (63) is

${\cal
J}\sim\Delta\varepsilon\frac{T_{h}-T}{T}\frac{\rho(\varepsilon_{F})k_{B}T}{\tau_{in}}=k_{B}(T_{h}-T)\frac{\rho(\varepsilon_{F})\Delta\varepsilon}{\tau_{in}}\>,$
(66)

where $\Delta\varepsilon=\min{(\hbar\Omega_{k_{F}},k_{B}T)}$, and
$\rho(\varepsilon)$ is the 3D density of states. The characteristic inelastic-
scattering time $\tau_{in}$ is defined by

$\frac{1}{\tau_{in}}=\frac{2\pi}{\hbar}\sum_{{\bm{k}^{\prime}}}|M_{{\bm{k}}^{\prime}{\bm{k}}}|^{2}\delta(\varepsilon^{0}_{{\bm{k}}^{\prime}}-\varepsilon^{0}_{\bm{k}})$
(67)

for $\varepsilon^{0}_{\bm{k}}=\varepsilon_{F}$. Equations (62) and (66) allow
one to estimate the heating of the hole gas.

###  VII.2 Current driven by energy relaxation

The energy-dependent nonequilibrium function $f(\varepsilon_{\bm{k}},T_{h})$
makes no contribution to the current (9). However, an electric current appears
due to the inelastic relaxation of this distribution to
$f(\varepsilon_{\bm{k}},T)\equiv f_{0}(\varepsilon_{\bm{k}})$. The current is
given by

$\delta j_{z}=-e\sum_{\bm{k}}\tau v^{(0)}_{z}(k_{z}){\cal
I}^{(ne)}_{\bm{k}}\\{f\\}\>,$ (68)

where the inelastic collision integral has the form

$\displaystyle{\cal
I}^{(ne)}_{\bm{k}}\\{f\\}=\frac{2\pi}{\hbar}\sum_{{\bm{k}}^{\prime}}|M_{{\bm{k}}^{\prime}{\bm{k}}}|^{2}$
$\displaystyle\left\\{\left[(f_{\bm{k}}-f_{{\bm{k}}^{\prime}})N_{\bm{q}}+f_{\bm{k}}(1-f_{{\bm{k}}^{\prime}})\right]\delta(\varepsilon_{{\bm{k}}^{\prime}}-\varepsilon_{\bm{k}}+\hbar\Omega_{\bm{q}})\right.$
$\displaystyle\left.+\left[(f_{\bm{k}}-f_{{\bm{k}}^{\prime}})N_{\bm{q}}-f_{{\bm{k}}^{\prime}}(1-f_{\bm{k}})\right]\delta(\varepsilon_{{\bm{k}}^{\prime}}-\varepsilon_{\bm{k}}-\hbar\Omega_{\bm{q}})\right\\}\>,$

with $f_{\bm{k}}=f_{0}(\varepsilon_{\bm{k}},T_{h})$ and
$\varepsilon_{\bm{k}}=\varepsilon^{0}_{\bm{k}}+\delta\varepsilon_{\bm{k}}$,
see Eqs. (5) and (6). It is clear that the current is contributed by the odd-
in-$k_{z}$ part of ${\cal I}^{(ne)}_{\bm{k}}\\{f\\}$. For simplicity we used
the relaxation time approximation for deriving the antisymmetric component of
the hole distribution function $f^{(2)}_{\bm{k}}$ and get
$f^{(2)}_{\bm{k}}=-\tau\mathcal{I}^{(ne)}_{\bm{k}}\\{f\\}$.

Substituting the collision integral into Eq. (68) we can reduce this equation
to

$\displaystyle\delta j_{z}=-\frac{2\pi
e\tau}{\hbar}\sum_{{\bm{k}},{\bm{k}}^{\prime}}|M_{{\bm{k}}^{\prime}{\bm{k}}}|^{2}\left[v^{(0)}_{z}(k_{z})-v^{(0)}_{z}(k^{\prime}_{z})\right]$
(70)
$\displaystyle\times\left[(f_{\bm{k}}-f_{{\bm{k}}^{\prime}})N_{\bm{q}}-f_{{\bm{k}}^{\prime}}(1-f_{\bm{k}})\right]\delta(\varepsilon_{{\bm{k}}^{\prime}}-\varepsilon_{\bm{k}}-\hbar\Omega_{\bm{q}})\>.$

For an estimation of the current magnitude we simplify in the collision
integral the dispersion (5) to $\varepsilon^{0}_{\bm{k}}={\cal A}k^{2}$ and
take into account only the cubic term in the expansion of
$\delta\varepsilon(k_{z})$, see Eq. (49). Then the expressions in the sums
(63) and (70) differ by the multipliers
$(\varepsilon_{\bm{k}}-\varepsilon_{{\bm{k}}^{\prime}})$ and

$\displaystyle{gB_{z}\over\varepsilon_{F}}\left[v^{(0)}_{z}(k_{z})\left(\frac{\beta
k_{z}}{\Delta_{2}}\right)^{3}-v^{(0)}_{z}(k^{\prime}_{z})\left(\frac{\beta
k^{\prime}_{z}}{\Delta_{2}}\right)^{3}\right]$ $\displaystyle\hskip
42.67912pt\sim{gB_{z}\over\varepsilon_{F}}\left(\frac{\beta}{\Delta_{2}}\right)^{3}\frac{k^{2}}{\hbar}\left(\varepsilon_{\bm{k}}-\varepsilon_{{\bm{k}}^{\prime}}\right)\>.$

It follows then that the current (70) can be estimated as

$\delta j_{z}\sim
e\tau{gB_{z}\over\varepsilon_{F}}\left(\frac{\beta}{\Delta_{2}}\right)^{3}\frac{\kappa_{F}^{2}}{\hbar}{\cal
J}\>.$

For the simplified energy dispersion the conductivity reads

$\sigma_{zz}\sim\frac{e^{2}\tau\varepsilon_{F}\kappa_{F}}{\hbar^{2}}$

and we finally obtain

$\delta j_{z}\sim
g\frac{e^{3}\tau^{2}}{\hbar^{3}}\left(\frac{\beta\kappa_{F}}{\Delta_{2}}\right)^{3}E_{z}^{2}B_{z}\>.$
(71)

One can see that the obtained estimation of the magnetochiral current for the
second mechanism has the same order as the contribution (61).

##  VIII Discussion

We begin the discussion with a general symmetry analysis of the eMChA effect
studied in this paper. Tellurium is a crystal with chiral (or enantiomorphic)
structure. By definition, a chiral periodic solid (or molecule) is non-
superimposable with its mirror image and has a ‘‘handedness’’. Two
modifications of a chiral structure that are mirror-like to each other are
called enantiomorphic. In tellurium crystals, the two mirror modifications are
characterized by the space groups $D_{3}^{4}~{}(P3_{1}21)$ and
$D_{3}^{6}~{}(P3_{2}21)$. Among 32 crystallographic point groups, 11 are
enantiomorphic, namely, ${\cal F}=C_{1},C_{2},D_{2},C_{4},D_{4},C_{3},D_{3}$
(quartz, tellurium), $C_{6},D_{6},T$ and $O$.

In this regard, the question arises which of the coefficients $G^{(n)}$ in
Eqs. (2) coincide and which differ in sign for the two enantiomorphs. To
answer this question, consider the achiral point group $D_{3h}$, which differs
from the $D_{3}$ group by the presence of a symmetry plane $\sigma_{h}$ and
includes 12 operations $g\in D_{3h}$. The $D_{3h}$ symmetry allows nonzero
terms in (2) with coefficients $G^{(3)},G^{(8)}$ and $G^{(10)}$. Consequently,
these three coefficients describe an electric current nonlinear in ${\bm{E}}$
and linear in ${\bm{B}}$ with its sign independent of the enantiomorphic
modification. The remaining seven coefficients $G^{(n)}$ describe
magnetochiral currents with opposite directions for the $D_{3}^{4}$ and
$D_{3}^{6}$ phases. This way of separating the chiral and achiral
contributions to the electric current is applicable for ten enantiomorphic
crystal classes ${\cal F}$, except for the $O$ class. Each of them can be
associated with an achiral point group ${\cal F}_{a}\ni{\cal F}$, which has no
spatial inversion center and which admits nonzero coefficients $G_{ijkl}$ in
Eq. (1). These coefficients describe achiral transport, whereas the additional
coefficients arising in the ${\cal F}$ group are chiral. Chiral and achiral
nature of the coefficients can be readily determined from the behavior of
physical quantities in the left and right parts of Eqs. (2) under reflection
in the $\sigma_{h}$ plane. Indeed, under this operation the component $\delta
j_{z}$ changes sign, but the product $E_{z}^{2}B_{z}$ is invariant which means
that the coefficient $G^{(1)}$ is chiral. At the same time, the product
$E_{x}^{2}B_{y}$ changes sign upon reflection $\sigma_{h}$, as does the
component $\delta j_{x}$. Therefore, the coefficient $G^{(3)}$ describes
achiral transport.

Let us list the achiral groups corresponding to the above ten chiral groups:
${\cal F}_{a}=C_{s},C_{2v},D_{2d},C_{4v},D_{4d},C_{3v},D_{3h},C_{6v},D_{6d}$
and $T_{d}$. As another example, consider the chiral group $T$ (sillenite
Bi12SiO20, bismuth germanate Bi12GeO20) and the corresponding achiral group
$T_{d}$. The $T_{d}$ symmetry allows for the current $\delta j_{x}$ terms
proportional to $(E_{y}^{2}-E_{z}^{2})B_{x}$ and
$(E_{y}B_{y}-E_{z}B_{z})E_{x}$. In addition, the $T$ group has chiral
contributions proportional to
$|{\bm{E}}|^{2}B_{x},({\bm{E}}\cdot{\bm{B}})E_{x}$ and $E_{x}^{2}B_{x}$. The
symmetry point transformation which can be used to divide between chiral and
achiral coefficients is the reflection in the plane
$\sigma_{v}\parallel(110)$.

As for the enantiomorphic group $O$, it has no partner ${\cal F}_{a}\ni O$
without an inversion center. Adding a reflection plane to the group $O$ leads
to the $O_{h}$ group in which all the coefficients $G_{ijkl}$ are equal to
zero. Hence, for the $O$ group, all the coefficients $G_{ijkl}$ in the
expansion (1) are chiral.

Note that the BiTeI crystal has the achiral trigonal symmetry $C_{3v}$ and
allows a nonreciprocal rectification effect $\delta j_{x}\propto
E_{x}^{2}B_{y}$ [16], which however is not a magnetochiral effect.

In Sections IV–VII, we have considered successively various models and
mechanisms of the eMChA effect: the approximation of a constant relaxation
time, the general procedure for calculating the magnetochiral current for
different elastic and inelastic relaxation times, and the approximation of a
small chiral parameter $\beta$. A derivation of the exact expression for the
current beyond the fixed relaxation time approximation cannot be obtained
analytically and is outside the scope of this work. However, the carried-out
study shows that the magneto-chiral current $\delta j_{z}$ in tellurium for a
degenerate hole gas can be described by $\delta j_{z}=G_{zzzz}E_{z}^{2}B_{z}$
with

$G_{zzzz}=cg\frac{{\cal A}_{1}}{{\cal
A}_{2}}\frac{e^{3}\tau^{2}}{\pi^{2}\hbar^{3}}\left(\frac{\beta\kappa_{F}}{\Delta_{2}}\right)^{3}\>,$
(72)

where $c$ is a factor of the order of unity. This means that the resistance of
tellurium $R$ has the nonreciprocal chiral contribution

$R=R_{0}(1+\gamma j_{z}B_{z}),\qquad\gamma={G_{zzzz}\over\sigma^{2}},$ (73)

where $R_{0}$ is the resistance in the absence of magnetic field, and $\sigma$
is the conductivity. For an estimation we ignore the difference between
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$. Taking the conductivity as
$\sigma=pe^{2}\tau/m^{*}$ where $m^{*}=2\mathcal{A}_{1}/\hbar^{2}$, the
coefficient $g$ in Eq. (3) as $g=g^{*}\mu_{\rm B}$, where $\mu_{\rm B}$ is the
Bohr magneton and $g^{*}$ is the effective $g$-factor, and noting that the
Fermi wavevector is related to the hole concentration as $\kappa_{\rm
F}^{3}=3\pi^{2}p$, we get

$\gamma\approx{3\mu_{\rm B}g^{*}{m^{*}}^{2}\over
ep}\quantity({\beta\over\hbar\Delta_{2}})^{3}.$ (74)

We use the parameters suitable for Te: $\beta=2.5\times 10^{-8}$ eV cm,
$\Delta_{2}=63$ meV, $m^{*}=0.2m_{0}$, $g^{*}=1$, and the hole concentration
$p=10^{16}$ cm-3. Then we obtain that the ratio $\beta\kappa_{\rm
F}/\Delta_{2}\approx 0.27\ll 1$, and one can apply the approximate equation
(72) for the estimation. The result yields 3$\times 10^{-7}$ cm2 T-1A-1 for
the magneto-induced rectification coefficient $\gamma$.

The eMCh current measurements presented in Figs. 3 and 4 in Ref. [9] were
performed in the following two geometries: (i) the electric current measured
in the $x$ direction at ${\bm{E}}\parallel x$ and the magnetic field vector
lying in the $(xy)$ plane, (ii) ${\bm{j}},{\bm{E}}\parallel z$, and the
magnetic field in the $(xz)$ plane. It follows from the general equations (2)
that, in these two setups, one has

$\displaystyle\delta
j_{x}=G_{xxxx}E_{x}^{2}B\sin{\theta_{y}}=(G^{(5)}+G^{(7)})E_{x}^{2}B\sin{\theta_{y}}\>,$
(75) $\displaystyle\delta
j_{z}=G_{zzzz}E_{z}^{2}B\sin{\theta_{x}}=G^{(1)}E_{z}^{2}B\sin{\theta_{x}}\>,$
(76)

where $B=\absolutevalue{\bm{B}}$, $\theta_{y}$ is the angle between the vector
${\bm{B}}$ lying in the $(xy)$ plane and the $y$ axis, $\theta_{x}$ is the
angle between the vector ${\bm{B}}$ lying in the $(xz)$ plane and the $x$
axis.

According to Rikken and Avarvari [9] their measurements on tellurium show that
$3\gamma_{xxxx}\approx\gamma_{xxxy}$ and
$12\gamma_{zzzx}\approx\gamma_{xxxy}$, and $\gamma_{zzzz}\ll\gamma_{zzzx}$.
These results are in complete contradiction to the phenomenological equations
(2), (75) and (76) derived for D3 symmetry crystals. Indeed, the symmetry
predicts that $\gamma_{xxxy}=\gamma_{zzzx}=0$ while the component
$\gamma_{zzzz}$ is allowed. This is a key difficulty in comparing the derived
theory with the experiment [9] and an additional experimental work is needed
on the study of the chiral transport in tellurium crystals.

In Ref. [9], a theoretical estimate of the $\gamma$ value is also given. The
equation for $\gamma$ is derived in the framework of a model where the linear
in $k_{z}B_{z}$ term in the hole energy dispersion is taken into account only.
As stressed in Section IV this term does not lead to the eMChA and one needs
to include the higher-order term $k_{z}^{3}B_{z}$ in the hole Hamiltonian, as
unambiguously follows from Eq. (14) for $\delta j_{z}$. Moreover, the
$k_{z}B_{z}$-linear term $\delta\varepsilon_{\bm{k}}=\chi k_{z}B_{z}$ in the
hole dispersion is given by $\chi=-g\beta/\Delta_{2}$ with an estimate for
tellurium $\absolutevalue{\chi}=3.7\times 10^{-32}$ J m/T. The value of
$\absolutevalue{\chi}$ assumed in Ref. [9] is $\sim 40$ times larger.

So far, we have examined the effect of a static electric field ${\bm{E}}$. It
is easiest to generalize the theory to the case of a time-dependent field
$E_{z}(t)=E^{(0)}_{z}\cos{\omega t}$ in the constant-time approximation for
frequencies satisfying the condition $\omega\ll\varepsilon_{F}/\hbar$, while
the product $\omega\tau$ may be arbitrary. To find the distribution function
$f_{\bm{k}}(t)$, the derivative $\partial f_{\bm{k}}/\partial t$ must be added
to the left side of the kinetic equation (11). Omitting calculations, we
present the result. The formula (14) for $\omega\neq 0$ becomes

$\delta j_{z}(\omega)=\frac{\delta j_{z}(0)}{1+\omega^{2}\tau^{2}}\>,$ (77)

where $\delta j_{z}(0)$ is the magnetochiral current in a static electric
field. The alternating electric field induces not only a dc current (77), but
also a current at double frequency $2\omega$. For the second harmonic
generation we have

$j_{z}^{2\omega}(t)=j_{z;2\omega}{\rm e}^{-2{\rm i}\omega
t}+j_{z;-2\omega}{\rm e}^{2{\rm i}\omega t}\>,$ (78)

where the complex amplitude is given by

$j_{z;2\omega}=j^{*}_{z;-2\omega}=\frac{1}{2}\frac{\delta j_{z}(0)}{(1-{\rm
i}\omega\tau)(1-2{\rm i}\omega\tau)}\>.$

It should be mentioned that experimentally it is convenient to detect the
magneto-chiral current by measuring the amplitude of the second harmonic at
$\omega\tau\ll 1$ [9].

In fact, Eq. (77) describes the phenomenon is called magneto-photogalvanic
effect (MPGE). In general, it is described by the following phenomenological
equation [34, 35, 36, 37, 38]

$j_{i}=G_{ijkl}(\omega)\\{E_{j}E^{*}_{k}\\}B_{l}+G^{(\rm
circ)}_{klm}R_{l}B_{m}\>,$ (79)

where ${\bm{E}}$ is the complex amplitude of the radiation electric field and

$\\{E_{j}E^{*}_{k}\\}=\frac{1}{2}(E_{j}E_{k}^{*}+E^{*}_{j}E_{k})\>,\quad{\bm{R}}={\rm
i}({\bm{E}}\times{\bm{E}}^{*})\>.$

The first and second contributions in the right-hand side of Eq. (79)
represent the so-called linear and circular MPGE. At zero frequency (static
electric field) the coefficients $G_{ijkl}(\omega)$ coincide with the
coefficients $G_{ijkl}$ in Eq. (1). The electric field of the electromagnetic
wave is complex and the magneto-photogalvanic current contains an additional
contribution described by the pseudotensor ${\bm{G}}^{(\rm circ)}$ if the
circular polarization of the exciting light is nonzero. The circular MPGE has
first been observed in the achiral GaAs crystal [39]. Similarly to the dc
effect (2) the coefficients $G_{ijkl}(\omega)$ and ${\bm{G}}^{(\rm circ)}$ can
be divided into chiral and achiral ones. Let us consider the products
$R_{i}B_{j}$ which transform in $D_{3h}$ according to
$(A^{\prime}_{2}+E^{\prime\prime})\times(A^{\prime}_{2}+E^{\prime\prime})=A^{\prime}_{1}+2E^{\prime\prime}+(A^{\prime}_{1}+A^{\prime}_{2}+E^{\prime})$:

$\displaystyle R_{z}B_{z}~{}(A^{\prime}_{1});\quad
R_{x}B_{x}+R_{y}B_{y}~{}(A^{\prime}_{1});$ (80) $\displaystyle
R_{x}B_{y}-R_{y}B_{x}~{}(A^{\prime}_{2});$ $\displaystyle
R_{x}B_{x}-R_{y}B_{y},-R_{x}B_{y}-R_{y}B_{x}~{}(E^{\prime});$ $\displaystyle
R_{z}B_{y},-R_{z}B_{x}~{}(E^{\prime\prime});\quad
R_{y}B_{z},-R_{x}B_{z}~{}(E^{\prime\prime})\>.$

Thus, an achiral contribution to the current is given by

$\displaystyle\delta j_{x}=G_{1}^{({\rm circ})}(R_{x}B_{x}-R_{y}B_{y})\>,$
$\displaystyle\delta j_{y}=-G_{1}^{({\rm circ})}(R_{x}B_{y}+R_{y}B_{x})\>.$

In the $D_{3}$ symmetry, additional chiral terms appear

$\displaystyle\delta j_{x}=G_{2}^{({\rm circ})}R_{z}B_{y}+G_{3}^{({\rm
circ})}R_{y}B_{z}\>,$ (81) $\displaystyle\delta j_{y}=-G_{2}^{({\rm
circ})}R_{z}B_{x}-G_{3}^{({\rm circ})}R_{x}B_{z}\>,$ $\displaystyle\delta
j_{z}=G_{4}^{({\rm circ})}(R_{x}B_{y}-R_{y}B_{x})\>.$

It is instructive to describe the hierarchical sequence of point-group
categories: among 21 crystal classes lacking inversion symmetry, 18 are
gyrotropic and, as mentioned above, 11 are enantiomorphic. All
noncentrosymmetric crystals allow nonzero coefficients $G_{ijkl}$ in Eq. (1)
and $G_{ijkl}(\omega),G^{(\rm circ)}_{klm}$ in Eq. (79). We remind that the
gyrotropic classes allow nonzero components of the rank 3 tensors
$\gamma_{ijk}$ antisymmetric under exchange of one pair of its indices or,
equivalently, the rank 2 pseudotensors. In the gyrotropic crystals, there
exist coefficients in Eq. (79) that relate the current vector components with
pseudovector combinations of the products of $E_{j}E^{*}_{k}B_{l}$ and
describe the magneto-gyrotropic photogalvanic effects [35, 37, 40]. And
finally, in the chiral crystals there are coefficients which have different
signs for the different enantiomorphic modifications. Recently the magneto-
chiral photogalvanic current $\bm{j}\propto\bm{B}\times\bm{R}$ has been
studied in bulk tellurium [41] in both terahertz and infrared ranges at
indirect intraband and direct intersubband optical transitions in the valence
band, respectively.

##  IX Summary

We have derived the theory of eMChA effect in tellurium which shows an
intricate combination of chirality and magnetism. Macroscopic phenomenological
relationship is established between the electric current density and products
of the magnetic field and bilinear combinations of the electric field
strength. Two microscopic mechanisms of the effect are considered, one with
allowance for elastic scattering processes only and the other where the eMChA
current is formed in the course of hole gas heating and its energy relaxation.
In the purely elastic mechanism, the general formalism is developed to
calculate the eMChA current at arbitrary ratio between the camel-back
dispersion parameter $\beta$, Fermi energy and valence-band splitting
$2\Delta_{2}$.

The exact result is obtained in the limit of small $\beta$. It shows the same
order of magnitude of the magneto-induced rectification coefficient $\gamma$
as that obtained in the simple relaxation-time approximation; however, the
value and even the sign of $\gamma$ are different.

An attention is attracted to the difference between the achiral and chiral
contributions to the magneto-induced rectification which, respectively,
coincide and are opposite in sign in the two enantiomorphic modifications of
chiral crystals.

Relationship between the eMChA and magneto-induced photogalvanic effects is
discussed and the chiral and achiral coefficients describing these effects in
tellurium are identified. The developed theory of eMChA is compared with the
available experimental data.

###### Acknowledgements.

L. E. G. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) Project No. Ga501/19-1. E. L. I. acknowledges
support from the Russian Science Foundation (Project No. 22-12-00211).

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