# Anisotropic magnetoresistance and memory effect
in bulk systems with extended defects

K. S. Denisov, K. A. Baryshnikov, P. S. Alekseev, N. S. Averkiev  Ioffe
Institute, Politekhnicheskaya 26, St. Petersburg 194021, Russia
<EMAIL_ADDRESS>
###### Abstract

Memory effects can have a profound impact on the resistivity of semiconductor
systems, resulting in giant negative magnetoresistance and MIRO phenomena.
This work opens the discussion of the memory effects in 3D conducting systems
featured by the presence of the extended one-dimensional defects, such as
screw dislocations or static charge stripes. We demonstrate that accounting
for the memory effect, that is the capture of electrons on collisionless
spiral trajectories winding around extended defects, leads to the strong
negative magnetoresistance in case when the external magnetic field direction
becomes parallel to the defects axis. This effect gives rise to a significant
magnetoresistance anisotropy already for an isotropic Fermi surface and no
spin-orbit effects. The proposed resistivity feature can be used to detect
one-dimensional scattering defects in these systems.

††: J. Phys.: Condens. Matter

Keywords: memory effects, AMR, magnetoresistance anisotropy, one-dimensional
defects, screw dislocations, charged stripes

##  1 Introduction

Among a broad family of the magnetotransport phenomena, a particularly
important one concerns the anisotropic magnetoresistance (AMR), that is the
dependence of the longitudinal resistivity on the direction of the applied
magnetic field. AMR can have different microscopic origins and it commonly
allows one to analyze rather subtle features of a conductive system. For
instance, the AMR of ferromagnets is generally induced by spin-orbit effects
[1-4], and it can serve to determine the magnetization easy axis and the
exchange anisotropy of magnetic films [5-7]. Alternatively, the AMR in
metallic systems points out at the opened character of the underlying Fermi
surface [8-11], or even at the topological structure of band states, as in the
case of the chiral anomaly in topological semimetals [12-14].

In this paper, we theoretically predict strong anisotropic magnetoresistance
of a three-dimensional nonmagnetic metal with an isotropic Fermi surface in
case when the electron transport is influenced by the scattering on extended
(one-dimensional) defects. The role of this type of defects can be played by
screw dislocations [15], charged stripes in some dodecaboride crystals [10] or
even artificially made tunnels [16]. We demonstrate that accounting for the
Baskin-Magarill-Entin (BME) mechanism of the memory effect [17] leads to the
significant enhancement of the AMR in the classically strong range of magnetic
fields even despite that the underlying electron spectrum had isotropic
character.

Essentially, the BME-mechanism-based memory effect takes into account the
appearance of collisionless electron cyclotron trajectories in a random static
pattern of short-range impurities. In particular, this phenomenon is
responsible for the strong negative magnetoresistance in 2D conducting films
[18] as well as for the renormalization of the electron scattering time at
ultra-high magnetic fields [19]. Starting from the pioneering work [17] and
continuing through Bobylev with coauthors and other works [18,20,21] (see full
review of the works in the field in [19]) the mechanisms of reduction of
static electrical resistance in strong magnetic fields in 2D materials was
theoretically discussed. Moreover, many other mechanisms for the memory
effects have been recently proposed to clarify the understanding of MIRO
effects [22-24].

Importantly, the BME-mechanism-based memory effect is restricted to a two-
dimensional electron gas motion, as the cyclotron collisionless electrons
appear only in 2D geometry. The memory effects in 3D samples generally have
other microscopic origins, as firstly considered in [25,26] they are mostly
connected with multiple returns between scattering events either due to long-
ranged and short-ranged scattering potentials. In this manuscript we extend
the topic of BME-based memory effect in bulk metals with residential extended
defects (EOD). The key finding of our work is that in case when the external
magnetic field is directed along EOD axis the electron motion in the plane
perpendicular to EOD mimics the two-dimensional geometry, thus allowing the
electrons to be captured on the spiral trajectories which winds around EOD
without experiencing any collisions. Similarly to the two-dimensional systems
[18], this effect leads to the significant negative magnetoresistance at
classically strong magnetic fields. Naturally, the enhanced magnetoresistance
exists only in one particular geometry and it remains absent for other
directions of magnetic field, thus giving rise to the strong AMR. Since this
effect is the direct consequence of scattering processes on EOD, the AMR
constitutes a purely electrical tool to evidence the presence of EOD in a bulk
system.

The paper is organized as follows. In Section 2 we provide a general
discussion of the magnetoresistance anisotropy in EOD-based systems with
isotropic Fermi surface and no spin-orbit effects. In Section 3 we describe
the electron kinetics in materials with EOD-scatterers. In this section we
focus on the Boltzmann equation approximation and neglect any memory effects.
In Section 4 we develop our original approach to the memory effect description
in 3D materials with EODs, which is based on the direct calculation of the
electron velocity correlation functions with account for the classical
electron trajectories. In Section 5 we discuss the obtained results, and we
also study the effect from a deviation of the magnetic field direction from
the direction of EODs on the memory effect and discuss other possible memory
effects in the considered system.

##  2 Magnetoresistance

###  2.1 General consideration

We consider a bulk metallic system with isotropic electron Fermi surface. The
system is randomly filled by extended 1D defects directed along $z$ axis, see
Fig 1. The presence of co-aligned EOD lowers the system symmetry by selecting
$z$ axis and, thus, makes the media effectively anisotropic. The
magnetoresistance, however, does not appear simply due to the anisotropy of an
effective media. Indeed, the equation of motion for an average drift carrier
velocity ${\bf v}$ upon the external electric ${\bf E}$ and magnetic ${\bf B}$
fields with the anisotropic friction term is given by

$\frac{d{\bf v}}{dt}=\frac{e}{m}{\bf E}+\frac{e}{mc}\left[{\bf v}\times{\bf
B}\right]-\hat{\Gamma}\cdot{\bf v},$ (1)

here $\hat{\Gamma}={\rm diag}(\tau_{0},\tau_{0},\tau_{z})$ is the anisotropic
tensor of the scattering times, $\tau_{z}\neq\tau_{0}$. The tensor of
resistivity is determined by the emergent electric field ${\bf
E}=\hat{\rho}\cdot{\bf j}$, which appears due to the applied electric current
density ${\bf j}=ne{\bf v}$. In the steady state

${\bf E}=\frac{m}{ne^{2}}\hat{\Gamma}\cdot{\bf j}-R_{H}\left[{\bf j}\times{\bf
B}\right]\equiv\hat{\rho}\cdot{\bf j},$ (2)

where $R_{H}=(nec)^{-1}$ is the Hall coefficient. It follows from (2) that the
${\bf B}$-dependent term contributes to the $\alpha$ component of ${\bf E}$
only via the perpendicular components of the electrical current. Thus, in such
a simplified model the magnetic field affects only the Hall resistivity and
has no effect on the longitudinal elements of $\hat{\rho}$, hence no
magnetoresistance.

A general mechanism giving rise to the magnetoresistance concerns the
spreading of the electron drift velocity. Generally speaking, instead of using
the basic relation ${\bf j}=ne{\bf v}$ we should firstly calculate the total
electric conductivity by summing over different electrons, and only after that
should we inverse the conductivity tensor. For instance, the energy dependence
of the electron scattering rates $\Gamma(\varepsilon)$ leads to the non-
uniform velocity distribution which directly results in the magnetoresistance
[27]. This mechanisms leads to the positive quadratic magnetoresistance at
weak magnetic field and it becomes suppressed in the strong limit.

In our paper we focus on several microscopic mechanisms of the
magnetoresistance, which are specifically relevant for the considered setting
and which stem from the spreading of the electron drift velocity due to the
electron scattering on EOD. Firstly, we are going to consider the electron
kinetics with account for the angular dependence of the electron scattering
rates $\Gamma({\bf v})$ on EOD, this mechanism is mostly relevant at weak
magnetic field. Secondly, we investigate the Baskin-Magarill-Entin mechanism
of the memory effect due to the formation of collision-free spiral electron
trajectories winding around EOD in case when the magnetic field is directed
along their direction. This mechanism is highly effective at strong magnetic
fields and it leads to a significant increase of AMR.

###  2.2 Model

Let us describe the considered model in more detail (see Fig. 1). We assume
that the bulk system contains two types of elastic scatterers, namely the EOD
directed along $z$-axis and additional short-range impurities. The electron
scattering on short-range impurities is isotropic (the electron velocity
directions before and after the scattering event are totally uncorrelated), it
is described by a single parameter, which is the scattering time $\tau_{i}$.
The electron scattering on EOD is more complex, see Fig. 1. In view of the
transnational invariance along z axis the electron momentum
$p_{z}=p_{F}\cos{\theta}$ is conserved upon the scattering on EOD, here
$p_{F}$ is the Fermi momentum. Also, the scattering on EOD with respect to the
electron momentum projection ${\bf p}_{\perp}=p_{x}{\bf e}_{x}+p_{y}{\bf
e}_{y}$ onto ($xy$) plane, is assumed to be isotropic.

Figure 1: Scattering on short-range impurities and extended 1D defects.

We focus on 3D degenerate electron gas and present the conductivity tensor in
the following way

$\hat{\sigma}({\bf B})=e^{2}\nu_{F}\hat{\mathcal{D}}({\bf B}),$ (3)

where $\nu_{F}$ and $\mathcal{D}_{\alpha\beta}$ are the electron density of
states and the diffusion tensor respectively, both are taken at the Fermi
energy. The resistivity tensor is obtained by inverting
$\hat{\rho}=\hat{\sigma}^{-1}$

$\hat{\rho}({\bf B})=\frac{1}{e^{2}\nu_{F}}\hat{\mathcal{D}}^{-1}({\bf B}).$
(4)

In what follows we are going to consider the anisotropy of $\hat{\rho}$ in the
static limit ($\omega\to 0$) for two geometries with respect to the mutual
orientation of the applied magnetic field and the EOD axis: ${\bf
B}\parallel{\bf e}_{z}$ and ${\bf B}\parallel{\bf e}_{y}$. The anisotropy of
magnetoresistance is described by a difference of resistivities in these two
types of the considered geometries

$\Delta_{AM}(B)\equiv\frac{\rho_{xx}({\bf B}\parallel{\bf
e}_{y})-\rho_{xx}({\bf B}\parallel{\bf e}_{z})}{\rho_{xx}(B=0)}.$ (5)

The AMR within the kinetic equation approximation due to the
$\theta$-dependence of EOD scattering rates is described in Sec. 3. The strong
enhancement of $\Delta_{AM}$ due to the memory effect is further considered in
Sec. 4.

##  3 Electron kinetics

Let us consider the electron transport on the basis of the Boltzmann kinetic
equation. The electron distribution function $f_{\bf p}=f_{\bf p}^{0}+\delta
f_{\bf p}$, which contains the equilibrium $f_{\bf p}^{0}$ and nonequilibirum
$\delta f_{\bf p}$ parts, satisfies

$e({\bf E}\cdot{\bf v})\frac{\partial f_{\bf
p}^{0}}{\partial\varepsilon}+\frac{e}{c}\left[{\bf v}\times{\bf
B}\right]\cdot\frac{\partial f_{\bf p}}{\partial{\bf p}}=-\frac{\delta f_{\bf
p}}{\tau_{i}}-\frac{\delta f_{\bf p}-\langle\delta f_{\bf
p}\rangle_{\varphi}}{\tau_{s}},$ (6)

where ${\bf E},{\bf B}$ are the external electric and magnetic fields, ${\bf
v}={\bf p}/m$ is the electron velocity, and we presented the collision
integral as two terms. The first one corresponds to the isotropic electron
scattering on short-range impurities described by the scattering time
$\tau_{i}$. The second term describes the electron scattering on EOD with the
magnitude of rates controlled by the parameter $\tau_{s}$, here
$\langle\dots\rangle_{\varphi}$ stands for the average over the polar angle
$\varphi$ in $(xy)$ plane perpendicular to EOD axis. This form of the
collision integral is the direct consequence of the implied in-plane isotropic
model for the electron scattering on EOD and it does not lead to the
relaxation of $\varphi$-independent parts of the distribution function. In
particular, it preserves $v_{z}=v_{F}\cos{\theta}$ component of the electron
drift velocity. We justify this expression and connect the scattering time
$\tau_{s}$ with microscopic parameters of EOD in the A.

It is important to note that depending on the microscopic model the time
$\tau_{s}$ can become dependent on the azimuthal angle $\theta$ of the
electron velocity. For instance, one can estimate the scattering time on
extended defects as follows: $\tau_{s}^{-1}=n_{s}v_{\perp}\sigma_{s}$, where
$n_{s},\sigma_{s}$ are the sheet density and the scattering cross-section of
EOD, and $v_{\perp}=v_{F}\sin{\theta}$ is the electron velocity component in
$(xy)$ plane. Assuming the $\theta$-independent value of $\sigma_{s}$ we get
the $\theta$-dependent scattering time
$\tau_{s}^{-1}(\theta)\propto\sin{\theta}$, which is due to the dependence of
the electron incident flux intensity on its velocity in ($xy$) plane. In
particular, this model takes into account that the electrons moving strictly
parallel to EOD and having $v_{z}=v_{F}$ are not scattered by the EOD.
Naturally, the microscopic structure of the electron scattering on EOD is
important for the appearance of the magnetoresistance. We also note that an
angular dependence of the scattering rates was previously implemented for the
description of the paramagnetic gas kinetics [28].

The general form of $\delta f_{\bf p}$ solving the kinetic equation can be
written in the following way

$\delta f_{\bf p}=e\left(-\frac{\partial f_{\bf
p}^{0}}{\partial\varepsilon}\right){\bf E}\cdot\left(\hat{\mathcal{T}}_{{\bf
B}}{\bf
v}\right),\qquad\mathcal{D}_{\alpha\beta}=\Bigl{\langle}v_{\alpha}(\hat{\mathcal{T}}_{{\bf
B}}{\bf v})_{\beta}\Bigr{\rangle},$ (7)

where $\mathcal{D}_{\alpha\beta}$ are the corresponding diffusion
coefficients, the average goes over the Fermi momentum direction ${\bf n}$,
the notation $\hat{\mathcal{T}}_{\bf B}$ stands for the operator acting on the
electron velocity ${\bf v}$, it depends both on the scattering times and on
the direction of the applied magnetic field. The explicit form of
$\hat{\mathcal{T}}_{\bf B}$ and the corresponding expressions for the
diffusion tensor depend crucially on the considered geometry.

At zero external magnetic field $B=0$ the action of $\hat{\mathcal{T}}_{\bf
B}$ operator is reduced to the multiplication of the velocity by the
corresponding scattering time $\hat{\mathcal{T}}_{\bf B}{\bf
v}=(\tau_{0}v_{x},\tau_{0}v_{y},\tau_{i}v_{z})$, where we introduced the
overall scattering time

$\tau_{0}^{-1}=\tau_{i}^{-1}+\tau_{s}^{-1},$ (8)

which is responsible for the relaxation of ($xy$) velocity components. The
diffusion coefficients and the components of the resistivity tensor at $B=0$
read as

$\displaystyle\mathcal{D}_{xx}=\mathcal{D}_{yy}=\langle
v_{x}^{2}\tau_{0}\rangle,\quad\mathcal{D}_{zz}=\langle
v_{z}^{2}\tau_{i}\rangle,\quad\mathcal{D}_{xy}=\mathcal{D}_{xz}=0,$ (9)
$\displaystyle\rho_{xx}=\rho_{yy}=\frac{1}{e^{2}\nu_{F}\mathcal{D}_{xx}},\quad\rho_{zz}=\frac{1}{e^{2}\nu_{F}\mathcal{D}_{zz}},\quad\rho_{xy}=\rho_{xz}=0.$
(10)

Naturally, the resistivity is anisotropic $\rho_{xx}\neq\rho_{zz}$, which is
due to the different relaxation times $\tau_{i}\neq\tau_{0}$. We note that for
the $\theta$-dependent model the time $\tau_{0}(\theta)$ also depends on the
velocity projection onto EOD axis, so the average over the velocity directions
${\bf n}$ in formulas for $\mathcal{D}_{\alpha\beta}$ is sensitive to its
particular form.

The solution of the kinetic equation at finite magnetic field directed
parallel to EOD (${\bf B}\parallel{\bf e}_{z}$) can be obtained analytically
for an arbitrary $\tau_{s}(\theta)$ model. In this case the distribution
function mimics the two-dimensional geometry and it can be presented in the
following form $\delta f_{\bf
p}=A(\theta)\cos{\varphi}+C(\theta)\sin{\varphi}$, where $A,C$ are some
functions of $\theta$-angle. The corresponding diffusion coefficients are
given by

$\mathcal{D}\equiv\mathcal{D}_{xx}+i\mathcal{D}_{yx}=\Bigl{\langle}\frac{v_{x}^{2}\tau_{0}}{1-i\omega_{c}\tau_{0}}\Bigr{\rangle},\quad\mathcal{D}_{zz}=\langle
v_{z}^{2}\tau_{i}\rangle.$ (11)

We note that $\mathcal{D}_{zz}$ coincides with its value at zero magnetic
field; in this geometry ${\bf B}$ does not affect the electron motion along
EOD. The in-plane coefficients, on the contrary, are modified accounting for
the cyclotron motion in $(xy)$ plane. For the $\theta$-dependent model
$\tau_{s}(\theta)$ it is the account for the angular averaging in
$\mathcal{D}$ that leads to the magnetoresistance.

The solution of the kinetic equation for the geometry with ${\bf
B}\parallel{\bf e}_{y}$ is much more complicated and it essentially depends on
the particular $\tau_{s}(\theta)$ function. In fact, the analytical solution
can be obtained only for the simplest $\theta$-independent model. In this case
the diffusion coefficients have the following form

$\displaystyle\mathcal{D}_{zz}=\frac{1}{3}\frac{v_{F}^{2}\tau_{i}}{1+\omega_{c}^{2}\tau_{0}\tau_{i}},\quad\mathcal{D}_{xx}=\frac{1}{3}\frac{v_{F}^{2}\tau_{0}}{1+\omega_{c}^{2}\tau_{0}\tau_{i}},$
(12)
$\displaystyle\mathcal{D}_{zx}=-\mathcal{D}_{xz}=\frac{1}{3}\frac{v_{F}^{2}\omega_{c}\tau_{0}\tau_{i}}{1+\omega_{c}^{2}\tau_{0}\tau_{i}}.$
(13)

In order to account for some $\theta$-dependence of $\tau_{s}(\theta)$ and to
describe the associated magnetoresistance we made a numerical calculations of
the Boltzmann equation; this case is described in details in the B.

Figure 2: The dependence $\rho_{xx}(B)$ for $B||z$ (red solid line) and $B||y$
(blue dash-dotted line) geometries in $\theta$-dependent model of $\tau_{s}$,
the
ratio $\tau_{i}/\tau_{s}^{0}=1$.

We proceed with analyzing the anisotropic magnetoresistance due to the
$\theta$-dependent electron scattering on the extended defects. In Fig. 2 we
present the dependence of $\rho_{xx}$ on the magnetic field magnitude for two
geometries (${\bf B}\parallel{\bf e}_{z}$ and ${\bf B}\parallel{\bf e}_{y}$);
the EOD scattering time was modeled $1/\tau_{s}=\sin{\theta}/\tau_{s}^{0}$ as
due to the velocity independent scattering cross-section. The obtained
magnetoresistance is positive for all directions of the magnetic field. It
demonstrates the initial increase for $\omega_{c}\tau_{s}\lesssim 2$ with the
subsequent saturation in the strong limit $\omega_{c}\tau_{s}\gg 1$. The
overall effect remains small, the magnetoresistance in the saturation regime
barely approaches $1\%$. Importantly, we draw the attention to the appearance
of the magnetoresistance anisotropy: as follows from Fig. 2 the resistivity in
two geometries indeed differs by several percents.

##  4 Memory effect

###  4.1 General consideration

In this section we investigate the memory effect based mechanism for the
anisotropic magnetoresistance of 3D system with extended defects. We
demonstrate that accounting for the classical collisionless trajectories,
which naturally appear when the magnetic field is directed along EOD, leads to
the significant enhancement of AMR at strong magnetic fields. This effect is
closely related to the classical mechanism of the negative magnetoresistance
in two-dimensional systems with static disorder due to the Baskin-Magarill-
Entin memory effect [18]. Let us illustrate the latter in more detail. The
close inspection of possible electron 2D trajectories in a given static
disorder upon the magnetic field perpendicular to the motion plane reveals
that there exist two groups of electrons labeled as the ”wandering” and
”circling”. The ”wandering” electrons experience occasional scattering and
thus behave similarly to the Drude-like electron gas (at least while
$\omega_{c}$ remains not large enough neither to induce the localization [18]
the Shubnikov De Haas oscillations), they contribute both to the longitudinal
and transverse electric conductivities. The ”circling” electrons, on the
contrary, move along the collision-free circles, which are inevitably
presented in case of static distribution of scalar impurities. Importantly,
the ”circling” electrons are deflected by the crossed electric and magnetic
fields only in the perpendicular to ${\bf E}$ direction, thus contributing
entirely to the Hall conductivity. These two electron groups thus are
effectively described by different drift velocities, which leads to the
magnetoresistivity that increases at strong magnetic field ($\omega_{c}\tau\gg
1$) and has an essentially negative value, hence the negative
magnetoresistance.

We draw the attention that a bulk systems with 3D electrons moving in a static
pattern of the extended defects which are co-aligned with the external
magnetic field naturally imitates the physics of the memory effect in two-
dimensions. Indeed, the scattering of electrons by EOD does not lead to the
change of $v_{z}=v_{F}\cos{\theta}$ velocity component. Thus the electrons
with a fixed projection $v_{z}(\theta)$ can be viewed effectively as a two-
dimensional electron gas having the Fermi velocity
$v_{\perp}=v_{F}\sin{\theta}$ and experiencing the two-dimensional diffusion
in ($xy$) plane at $B=0$. The nonzero magnetic field applied along EOD does
not change the magnitude of an effective Fermi velocity $v_{\perp}$, thus each
two-dimensional slice of $\theta$-electrons becomes inevitably separated onto
the ”wandering” electrons, which continue to experience the 2D diffusion in
($xy$) plane, and ”circling” electrons, which become captured on EOD
collisions-free circles. The actual trajectory of these ”circling” electrons
is, however, the 3D spiral line winding around EOD along its axis. We conclude
that for each group of electrons with different $v_{\perp}$ the memory effect
due to classical collisionless trajectories gives rise to the negative
magnetoresistance. As a result, the overall resistivity in $(xy)$ plane
becomes $B$-dependent and exhibits the strong decrease at large magnetic
fields. Importantly, this effect can exist only in a particular geometry of
${\bf B}\parallel{\bf e}_{z}$; for other orientations of ${\bf B}$ the
electron velocity component $v_{z}$ varies in time, thus suppressing the role
of the static correlations between EOD positions. Thus the BME-mechanism of
the memory effect in bulk systems with static EOD pattern manifests itself as
an enhanced anisotropic magnetoresistance.

###  4.2 Correlation function technique

Let us proceed with describing the memory effect in more detail. The memory
effects arise from the electron motion along some classical trajectories whose
evolution is fully determined by the initial conditions and, thus, appears to
be free of stochastic changes. Such a deterministic behavior of an electron
motion naturally violates the treating of the electron kinetics purely in
terms of the kinetic equation (6). One common approach to account for the
deterministic trajectories is to artificially modify the collision integral in
the kinetic equation to include possible returns to the same scatterers or
non-scattering trajectories in consideration [20-22].

In this paper we are going to consider this problem based on the correlation
function technique. According to the fluctuation-dissipation relation, the
diffusion coefficients in the static limit ($\omega\to 0$) can be expressed as
the time integral

$\mathcal{D}_{\alpha\beta}=\int\limits_{0}^{\infty}dt\langle
v_{\alpha}(t)v_{\beta}(0)\rangle_{eq}$ (14)

over the electron velocity correlation function $\langle
v_{\alpha}(t)v_{\beta}(0)\rangle_{eq}$ in the thermodynamic equilibrium. In
order to prepare the ground for the description of the memory effect, we
firstly re-consider the computation of the diffusion tensor in the kinetic
equation approximation. For ${\bf B}\parallel{\bf e}_{z}$ the electron
trajectory consists of the spiral lines segments featured by the deterministic
cyclotron rotation in ($xy$) plane (and the uniform motion along $z$-axis),
which are interrupted by the occasional scattering. Importantly, the kinetic
equation approximation signifies that the probability density regulating the
location of kinks due to scattering events on a trajectory is determined by
the Poisson stochastic process. The latter suggests that the probability of an
electron to be scattered remains constant in time. Naturally, assuming the
independent Poisson processes both for the scattering on short-range
impurities and EOD, we present the probability density for being scattered for
the first time at the interval $(t,t+dt)$ either by short-range impurities
$P_{i}(t)$ or by EOD $P_{s}(t)$ in the following way:

$P_{i}(t)=\frac{1}{\tau_{i}}e^{-t/\tau_{0}},\qquad
P_{s}(t)=\frac{1}{\tau_{s}}e^{-t/\tau_{0}},$ (15)

here $\tau_{i},\tau_{s}$ are the corresponding scattering times, and
$\tau_{0}^{-1}=\tau_{i}^{-1}+\tau_{s}^{-1}$ is an overall scattering time,
which accounts for both scattering events. Naturally, the parameters
($\tau_{i},\tau_{s},\tau_{0}$) are essentially the same that we used in the
collision integral in (6). Moreover, since the electron $v_{z}$ component
remains constant between collisions, the formula for $P_{s}$ can be also
applied for the $\theta$-dependent models. The overall density
$P_{0}=P_{i}+P_{s}$ describes the probability of being scattered for the first
time disregard to the particular type of the scattering event.

We calculate the diffusion coefficients by summing separately the
contributions to the time integral (14) from different segments of the
electron trajectories. We also take into account the specifics of our model,
which is that the scattering on short-range impurities is isotropic and the
scattering on EOD is such that it preserves the correlation of only $v_{z}$
velocity component. We present here the consideration of the ($xy$) plane
diffusion coefficients, the calculation of $\mathcal{D}_{zz}$ is given in the
C. The correlations of ($xy$) velocity components are destroyed already after
the very first collision disregard to the type of scatterer. Thus the only
nonzero contribution to $\mathcal{D}_{xx},\mathcal{D}_{xy}$ stems from the
very first segment of all trajectories that collects the correlations before
the first scattering event

$\mathcal{D}\equiv\mathcal{D}_{xx}+i\mathcal{D}_{yx}=\Bigl{\langle}\int\limits_{0}^{\infty}P_{0}(\tau_{1})d\tau_{1}\int\limits_{0}^{\tau_{1}}dt\cdot
e^{i\omega_{c}t}v_{x}^{2}(0)\Bigr{\rangle}=\Bigl{\langle}\frac{v_{x}^{2}\tau_{0}}{1-i\omega_{c}\tau_{0}}\Bigr{\rangle},$
(16)

where we explicitly took into account the time evolution of an electron
velocity due to the cyclotron motion, that means
$v_{x}(t)=v_{F}\sin{\theta}\cos{(\omega_{c}t+\varphi_{0})}$,
$v_{y}(t)=v_{F}\sin{\theta}\sin{(\omega_{c}t+\varphi_{0})}$, and the average
goes over the initial velocity direction $(\theta,\varphi_{0})$; this also
implies $\langle v_{y}(0)v_{x}(0)\rangle=0$. Naturally, we obtained
essentially the same expressions for the diffusion coefficient ${\mathcal{D}}$
as we did using the kinetic equation approach, see (11).

Let us discuss the modifications we should make in this calculation in order
to take into account the existence of the ”circling” electrons. The
integration in (16) can be presented in the following way

$\mathcal{D}=\Bigl{\langle}\int\limits_{0}^{T_{c}}P_{0}(\tau_{1})d\tau_{1}\int\limits_{0}^{\tau_{1}}dt\cdot
e^{i\omega_{c}t}v_{x}^{2}\Bigr{\rangle}+\Bigl{\langle}\int\limits_{T_{c}}^{\infty}P_{0}(\tau_{1})d\tau_{1}\int\limits_{T_{c}}^{\tau_{1}}dt\cdot
e^{i\omega_{c}t}v_{x}^{2}\Bigr{\rangle},$ (17)

where we divided the integral over the time of the first scattering event
$\tau_{1}$ over two parts, before and after the first cyclotron period
$T_{c}=2\pi/\omega_{c}$ is finished. In fact, one should identify the
”circling” trajectory as such, for which the electron has not yet experienced
a scattering after one full cyclotron period $T_{c}$ is completed. Indeed, if
no short-range impurities were additionally presented, the fact that an
electron have succeeded to accomplish its cyclotron circle without
experiencing the collisions with any EOD would suggest that its cyclic motion
is guaranteed to linger forever. The relative number of so-defined ”circling”
trajectories $\mathcal{P}_{\theta}$ for electrons with fixed $\theta$ is thus
simply given by

$\mathcal{P}_{\theta}(T_{c})=\int\limits_{T_{c}}^{\infty}P_{0}(\tau)d\tau=e^{-2\pi/\omega_{c}\tau_{0}}.$
(18)

The first term in (17) collects the contributions from all electrons which did
experience the collision at $t\leq T_{c}$, those are actually the ”wandering”
electrons and we should keep this term as it is. The second term in (17) in
its current form describes the stochastic process of an electron scattering
for times greater than $t>T_{c}$. Naturally, the inclusion of the memory
effect suggests to modify the second term in an appropriate way so that it
excludes the scattering events associated with EOD. To do this we note that
for times satisfying $\tau_{1}>T_{c}$ the stochastic probability density
$P_{0}(\tau_{1})$ entering the second integral can be interpreted as
$P_{0}(\tau_{1}>T_{c})=\tilde{P}(\tau_{1}|T_{c})\cdot\mathcal{P}_{\theta}(T_{c})$,
where $\tilde{P}(\tau_{1}|T_{c})$ is the conditional probability that an
electron will experience some scattering event at time $\tau_{1}$ provided
that it was not scattered for as long as $T_{c}$ starting from $t=0$; the
quantity $\mathcal{P}_{\theta}(T_{c})$ here counts the overall probability
that $\theta$-electrons were not scattered by the time $T_{c}$. We further
express the diffusion coefficient in the following way

$\mathcal{D}=\Bigl{\langle}v_{x}^{2}\tau_{0}\cdot\frac{1-\mathcal{P}_{\theta}}{1-i\omega_{c}\tau_{0}}\Bigr{\rangle}+\Bigl{\langle}v_{x}^{2}\mathcal{P}_{\theta}\int\limits_{T_{c}}^{\infty}\tilde{P}(\tau_{1}|T_{c})d\tau_{1}\int\limits_{T_{c}}^{\tau_{1}}dt\cdot
e^{i\omega_{c}t}\Bigr{\rangle}.$ (19)

For the kinetic equation approximation the conditional probability entering in
the second term is given by
$\tilde{P}(\tau_{1}|T_{c})=\tau_{0}^{-1}e^{-(\tau_{1}-T_{c})/\tau_{0}}$, thus
it takes into account both the short-range impurities and EOD. In order to
exclude the subsequent scattering on EOD for $\tau_{1}>T_{c}$, it is natural
to replace the Poisson conditional probability density by
$\tilde{P}^{\prime}(\tau_{1}|T_{c})=\tau_{i}^{-1}e^{-(\tau_{1}-T_{c})/\tau_{i}}$,
which now takes into account that at times $\tau_{1}>T_{c}$ an electron can be
scattered only by short-range impurities, hence the time $\tau_{i}$ instead of
$\tau_{0}$. Making this assumption we get the following expression for the
diffusion coefficients

$\mathcal{D}=\Bigl{\langle}v_{x}^{2}\tau_{0}\frac{1-\mathcal{P}_{\theta}}{1-i\omega_{c}\tau_{0}}\Bigr{\rangle}+\langle
v_{x}^{2}\mathcal{P}_{\theta}\rangle\frac{\tau_{i}}{1-i\omega_{c}\tau_{i}}.$
(20)

The first and second terms here can be approximately attributed to the
”wandering” and ”circling” electrons, respectively. It is now quite clear that
these two groups of electrons cannot be described by the same drift velocity
as their contributions to the diffusion coefficients are completely different;
the parameter discriminating between them is $\mathcal{P}_{\theta}$. This
leads to the magnetoresistance featured by the $B$-dependence similar to the
classical negative magnetoresistance of two-dimensional electron gas; its
behavior is considered in detail in the next section.

##  5 Results and Discussion

###  5.1 Anisotropic magnetoresistance: calculation and analysis

We proceed with discussing the anisotropic magnetoresistance emerging due to
the memory effect. We note that in order to describe $(xy)$ components of the
resistivity tensor and, thus, to analyze the emerging magnetoresistance
$\rho_{xx}(B)$ for ${\bf B}\parallel{\bf e}_{z}$ it is sufficient to evaluate
a single parameter $\mathcal{D}$. Indeed, as soon as
$\mathcal{D}_{zx}=\mathcal{D}_{zy}=0$, the ($xy$) components of the tensor
$\hat{\rho}$ obtained from (4) by inverting $\hat{\mathcal{D}}$-matrix are
decoupled from $\mathcal{D}_{zz}$. The calculation of $\rho_{xx}$ for ${\bf
B}\parallel{\bf e}_{y}$, on the contrary, generally requires the knowledge of
other components $\mathcal{D}_{zx},\mathcal{D}_{zy}$.

Let us consider the $\theta$-independent EOD scattering model $\tau_{s}={\rm
const}(\theta)$ and assume that the scattering time on short-range impurities
is long compared with all other times $\tau_{i}\gg\tau_{s},\omega_{c}^{-1}$.
In this case the diffusion coefficient in ${\bf B}\parallel{\bf e}_{z}$
geometry takes the form

$\mathcal{D}=(1-\mathcal{P})\mathcal{D}_{0}(\omega_{c})+\mathcal{P}\mathcal{D}_{c},$

where $\mathcal{P}=\exp{(-2\pi/\omega_{c}\tau_{s})}$. The first term is due to
the contribution of $(1-\mathcal{P})$ ”wandering” electrons described by the
ordinary diffusion coefficient $\mathcal{D}_{0}(\omega_{c})=\langle
v_{x}^{2}\tau_{s}/(1-i\omega_{c}\tau_{s})\rangle$ at finite magnetic field.
The second term is due to the pure Hall current of the remaining $\mathcal{P}$
”circling” electrons described by $\mathcal{D}_{c}=\langle
v_{x}^{2}\rangle/(-i\omega_{c})$. Inverting $\hat{\mathcal{D}}$-matrix gives
the following expression for the resisitivity

$\rho_{xx}(B)\approx\rho_{0}\left(1-\mathcal{P}\right),\qquad\rho_{0}=\frac{1}{e^{2}\nu_{F}\mathcal{D}_{0}(\omega_{c}=0)}.$
(21)

This result naturally coincides with the resistivity expression obtained in
[18] for 2D systems.

Let us draw the attention to the most prominent feature of the considered
memory effect mechanism. We note that the magnetoresistance in ${\bf
B}\parallel{\bf e}_{y}$ geometry is very weak (see Fig. 2), implying
$\rho_{xx}\approx\rho_{0}$. Therefore, the anisotropy of magnetoresistance
acquires an exponential dependence on the magnetic field

$\Delta_{AM}\approx\mathcal{P}=\exp{(-2\pi/\omega_{c}\tau_{s})}.$ (22)

We conclude that the magnitude of the effect increases with the increase of
the magnetic field and becomes significantly important at
$\omega_{c}\tau_{s}\geq 1$. For instance, the magnetoresistane achieves $25\%$
at $\omega_{c}\tau_{s}\approx 3$, see Fig. 3.

Figure 3: Comparison of isotropic (red dash-dotted line) and
$\theta$-dependent (blue solid line) models at $\tau_{i}/\tau_{s}=10^{20}$.
The black
dashed line is $\theta$-dependent model implementation without me-
mory effects (${\cal P}$ is set to be equal zero).

Considering $\theta$-dependent models of the electron scattering on EOD does
not noticeably change this situation. Indeed, the expression for the diffusion
coefficient in the limit $\tau_{i}/\tau_{s}\to\infty$ can be presented as

$\mathcal{D}=\Bigl{\langle}v_{x}^{2}\tau_{s}\frac{1-\mathcal{P}_{\theta}}{1-i\omega_{c}\tau_{s}}\Bigr{\rangle}+\langle
v_{x}^{2}\mathcal{P}_{\theta}\rangle\frac{1}{-i\omega_{c}}.$ (23)

The difference with the two-dimensional case and $\theta$-independent model
lies in the fact, that the relative number of the ”circling” electrons
$\mathcal{P}_{\theta}=\exp{(-2\pi/\omega_{c}\tau_{s}(\theta))}$ now depends on
the electron velocity projection $v_{z}=v_{F}\cos{{\theta}}$, hence the
averaging over different $\theta$-electrons. The calculated resisitivity
$\rho_{xx}(B)$ is shown in Fig. 3. The red and blue curves in Fig. 3
correspond to the isotropic $\tau_{s}={\rm const}(\theta)$ and
$\theta$-dependent $1/\tau_{s}=\sin{\theta}/\tau_{s}^{0}$ models,
respectively, the green curve demonstrates the resistivity of the
$\theta$-dependent model without accounting for the memory effect. The AMR-
parameter for the $\theta$-dependent model can be estimated as the difference
between the curves with and without accounting for the memory effect.

Figure 4: The memory effect at finite $\tau_{i}$ in nonisotropic
($\theta$-depen-
dent) model of scattering on EOD
($\tau_{s}^{-1}=(\tau_{s}^{0})^{-1}\cdot\sin{{\theta}}$).

It is worth emphasizing that when both mechanisms of the magnetoresistance are
presented, the magnetoresistance in ${\bf B}\parallel{\bf e}_{z}$ geometry
becomes sign-alternating, see the blue line in Fig. 3. This happens because of
the exponentially small number of ”circling” electrons at small
$\omega_{c}\tau_{s}\ll 1$. In this case, the initial region of $\rho_{xx}(B)$
dependence remains dominated by the drift velocity spreading of the
”wandering” electrons, while the memory effect becomes active only at larger
magnetic fields, when the number of ”circling” electrons becomes noticeable
enough. The effect of the sign-alternative magnetoresistance is also an
important property of bulk systems with the extended defects.

We further take into account the effect of the electron scattering on the
short-range impurities. In the limit $\tau_{i}\ll\tau_{s}$, opposite to the
considered above, the expression for the $\mathcal{D}$ turns into the ordinary
diffusion coefficient $\mathcal{D}=\langle
v_{x}^{2}\rangle\tau_{i}/(1-i\omega_{c}\tau_{i})$ for an electron gas in the
the external magnetic, here the single parameter $\tau_{i}$ describes the
scattering on short-range impurities. In this limit, naturally, there is no
talking about the formation of ”circling” electrons, as the electron
encounters EOD much rarely than the short-range impurities, hence the memory
effect becomes suppressed. Fig. 4 demonstrates the behavior of $\rho_{xx}(B)$
for different values of the ratio $\tau_{i}/\tau_{s}^{0}$; here we employed
the $\theta$-dependent model $1/\tau_{s}=\sin{\theta}/\tau_{s}^{0}$ of the
electron scattering on EOD. As follows from Fig. 4, the memory-effect induced
magnetoresistance looses its key role only when the scattering time on scalar
impurities becomes comparable with that on EOD ($\tau_{i}\approx\tau_{s}$). It
is right up to this moment that the magnetoresistance is dominated in strong
fields by the contribution due to ”circling” electrons. For instance, for
$\tau_{i}/\tau_{s}^{0}=10$ the magnitude of the effect at
$\omega_{c}\tau_{s}\approx 4$ is still large, approaching $20\%$.

###  5.2 Discussion of effects from tilted magnetic fields and rosette
trajectories

It is of interest to study the effect of a weak tilting of the magnetic field
$\mathbf{B}$ from the EOD direction $\mathbf{e}_{z}$. Let us denote the angle
between these directions as $\alpha\ll 1$, the mean distance between the EOD
as $L=n_{s}^{-1/2}$ and the diameter of EOD as $d\propto\sqrt{\sigma_{s}}$
(the diameter corresponds to the scattering cross-section $\sigma_{s}$). As
for the considered above cases $\mathbf{B}||\mathbf{e}_{z}$ and
$\mathbf{B}\perp\mathbf{e}_{z}$, we imply that the typical magnitudes of the
magnetic field $B$ correspond to the cyclotron radius
$R_{c}=v_{\perp}/\omega_{c}$ in the interval $L\ll R_{c}\ll l_{s}$. Here
$v_{\perp}=v_{F}\sin\theta$ is the absolute value of the projection of the
electron velocity on the $xy$ plane (defined as in the above sections) and
$l_{s}=v_{\perp}\tau_{s}\sim L^{2}/d$ is the scattering length corresponding
to the velocity components in the $xy$ plane due to collisions with EOD.

The main effect of a small tilting of the magnetic field is the appearance of
very rare collisions of the electrons on the circling orbits with EOD due to a
slow drift of the orbit center in the $xy$-plane. The wandering electrons are
weakly affected by a deviation of $\mathbf{B}$ from the EOD direction $z$. The
tilt of the magnetic field can be considered small when the shift of the
center of circling electron trajectory projection on the $xy$ plane after one
cyclotron period is small compared to the distance between EOD, i.e. $\alpha
R_{c}\ll L$. Such drift of electron cyclotron circles in the $xy$ plane
results in collision of each circling electron with a EOD after a certain
number of rotations. This number can be estimated as $N^{*}/p^{*}$, where
$N^{*}$ is the number of rotations by which the electron center makes the
distance $L$, $N^{*}=L/(\alpha R_{c})$, and $p^{*}$ is the probability to
collide with a EOD when an electron moves on the trajectory segment
corresponding to one cyclotron period and located on the closest distance from
a EOD (as compared with other segments). This probability can be estimated as
$p^{*}\approx 1$ for very small tilting angles $\alpha R_{c}\ll d$, while for
the angles satisfying $d\ll\alpha R_{c}\ll L$ it takes $p^{*}\approx d/(\alpha
R_{c})$.

In this way, we arrive to the estimate for the scattering time of the in-plane
velocity components $v_{x,y}$ of the “almost circling electrons” due to rare
collisions with the EOD $\tau^{*}\sim N^{*}T_{c}/p^{*}$. In the case $\alpha
R_{c}\ll d$ we obtain $\tau^{*}\sim L/(\alpha v_{\perp})$, whereas at
$d\ll\alpha R_{c}\ll L$ we have $\tau^{*}\sim L/(d\omega_{c})$. The described
effect of the rare collisions of the circling electrons with EOD plays the
role in the electron dynamics analogous to the scattering of the circling
electrons on short-range impurities in the bulk, thus it also determines the
conditional probability $\tilde{P}^{\prime}(\tau_{1}|T_{c})$ by the similar
way as scattering on the bulk impurities. Thus, the equation for the diffusion
coefficients in (20) should be modified by changing $1/\tau_{i}$ by
$1/\tau_{i}+1/\tau^{*}(\theta)$ in order to take into account both scattering
mechanisms.

It is worth noting that in the leading order with respect to $\mathcal{P}$
there is another microscopic mechanism for the memory effect. Namely, upon the
external magnetic field the electron moving along the cyclotron circle has a
finite probability to encounter the same scatterer many times in a row. This
memory effect results in the formation of the so-called rosette trajectories,
which effectively lead to the localization of the electron in real space
[21,22]. Our analysis based on [21,22] shows that, despite the magnitude of
this phenomenon is controlled by the same parameter
$\mathcal{P}=e^{-2\pi/\omega_{c}\tau}$ as the appearance of the ”circling”
electrons, the main effect of the rosette-induced localization concerns the
renormalization of the transport time [21] and it does not critically affect
the effect of negative magnetoresistance, considered in detail in this paper.

Both these two groups of effects, the magnetoresistanstance at tilted magnetic
fields and the extended collisions (rosettes), require further studies in
order to obtained the details of the picture presented in this section.

##  6 Conclusion

We demonstrated that in bulk non-magnetic materials with an isotropic Fermi
surface and extended one-dimensional defects, such as skrew dislocations or
charge stripes, the strong magnetoresistance effect (up to $100\%$ change)
arises when the magnetic field becomes co-aligned with the defect axis. This
feature stems from the memory effect, which is the existence of collisionless
electron trajectories in a static pattern of EOD. Indeed, upon the external
magnetic field there is always a finite number of electrons which do not
experience the scattering on EOD during one cycloton period. Essentially, when
the magnetic field is directed along EOD-axis these electrons will keep moving
along the spiral trajectories winding around EOD without any subsequent
scattering on them. This memory effect results in the enhanced negative
magnetoresistance in the plane perpendicular to EOD-axis. Applying the
magnetic field in the transverse direction (perpendicular to EOD-axis)
destroys the trajectory correlations responsible for the memory effect, thus
reducing the magnitude of the magnetoresistance. We analyzed the electric
kinetics in this geometry and demonstrated that the magnetoresistivity still
can arise from the velocity direction dependence of the microscopic electron
scattering rates on EOD. However, the obtained magnitude of the effect does
not exceed several percents, thus indicating the appearance of the strong
anisotropy of the magnetoresistance. Based on the correlation function
approach we derived the general analytical expression which describes the
electrical resistivity with account for the memory effect. The noteworthy
advantage of the developed theory is that it allowed us to take into account
the electron scattering on short-range impurities and the deviation of the
magnetic field direction from the EOD-axis as well. Both this factors result
in the suppression of the magnetoresistance anisotropy. We demonstrated that
the magnetoresistance in classicaly strong magnetic fields acquires an
exponential character and it reaches large values for
$\omega_{c}\tau_{s}\approx 1$, where $\omega_{c}$ is a cyclotron frequency and
$\tau_{s}$ is a scattering time on 1D defects. We found out that the memory
effect gets suppressed when the scattering times corresponding to EOD and
short-range impurities become comparable.

We thank A.P. Dmitriev from Ioffe Institute and D.G. Polyakov from Karlsruhe
Institute of Technology for fruitful discussions. This work was supported by
Russian Science Foundation (analytical theory – project No. 18-72-10111) and
by Russian Foundation for Basic Research (numerical calculations – project No.
18-02-01016).

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##  Appendix A Microscopic derivation of the collision integral with 1D
defects

The collision integral entering in the Boltzmann kinetic equation reads for
elastic electron scattering processes as

${\rm St}[f_{\bf p}]=\sum\limits_{\bf p^{\prime}}\left(W_{\bf
p,p^{\prime}}f_{\bf p^{\prime}}-W_{\bf p^{\prime},p}f_{\bf p}\right),$ (24)

where $f_{\bf p}$ is the distribution function of electrons with momentum
${\bf p}$ and $W_{\bf p,p^{\prime}}$ is the scattering rate corresponding to
the transition from $\bf p^{\prime}$ to $\bf p$ states. According to the Fermi
golden rule $W_{\bf p,p^{\prime}}$ is given by

$W_{\bf p,p^{\prime}}=\frac{N_{3D}}{\Omega}\frac{2\pi}{\hbar}|V_{\bf
p,p^{\prime}}|^{2}\delta(\varepsilon_{\bf p}-\varepsilon_{\bf p^{\prime}}).$
(25)

Here $\Omega$ is the volume of the sample and $N_{3D}$ is a number of the
scatterers in this volume. The scattering matrix element $V_{\bf
p,p^{\prime}}$ in case of 1D defects in 3D material takes into account the
momentum conservation along the $\bm{e}_{z}$ axis of these defects

$V_{\bf p,p^{\prime}}=u_{\parallel}({\bf q})2\pi\delta(k_{z}-k_{z}^{\prime}).$
(26)

Here we introduced the transversal scattering matrix element
$u_{\parallel}({\bf q})$ depending on the perpendicular to EOD components of
wave vector change ${\bf q}\equiv\bm{k}_{\perp}-\bm{k}_{\perp}^{\prime}$.
Taking the square of the matrix element we use a standard change
$\delta^{2}(k_{z})\rightarrow\delta(k_{z})L_{z}/2\pi$ introducing the linear
size of the system along EOD direction $L_{z}$. Then the scattering integral
takes the following form

${\rm St}\left[f_{{\bf
p}}\right]=\frac{2\pi}{\hbar}\nu_{3}n_{2}\int\frac{d\Omega^{\prime}}{4\pi}2\pi\delta(k_{z}-k_{z}^{\prime})|u(\theta;\varphi-\varphi^{\prime})|^{2}\left(f_{\bf{p}^{\prime}}-f_{\bf{p}}\right),$
(27)

where $(\theta,\varphi)$ are the angular coordinates of the momentum $\bm{p}$,
$\nu_{3}=g_{3}(E_{F})/\Omega$ is the density of states on a Fermi level and
$n_{2}=n_{3}L_{z}$ is a 2D concentration of 1D defects. Let us further
introduce an effective 2D electron differential scattering cross-section as
$d\sigma/d\varphi=\nu_{2}|u|^{2}/\hbar v_{\perp}$, where
$\nu_{2}=m/2\pi\hbar^{2}$ is the 2D density of states. Using
$d\sigma/d\varphi$ we express the collision integral as follows

${\rm
St}\left[f_{\bm{p}}\right]=\frac{\tilde{\nu_{2}}}{\nu_{2}}n_{2}v_{\perp}\int
d\varphi^{\prime}\left(\frac{d\sigma}{d\varphi^{\prime}}f_{\bf{p}^{\prime}}-\frac{d\sigma}{d\varphi^{\prime}}f_{\bf{p}}\right),$
(28)

where $\tilde{\nu_{2}}={\pi}\nu_{3}/{k_{F}}$ is an effective 2D density of
states. For the $(xy)$-plane isotropic scattering model implied in our paper
the cross-section $d\sigma/d\varphi$ does not depend on the polar scattering
angle, which brings us to the collision integral in the following form:

${\rm St}[f_{p}]=-\frac{\delta f_{p}-\langle\delta
f_{p}\rangle_{\varphi}}{\tau_{s}},\qquad\frac{1}{\tau_{s}}=\frac{\tilde{\nu}_{2}}{\nu_{2}}n_{2}v_{\perp}\sigma_{s},$
(29)

where $\sigma_{s}$ is the total scattering cross-section.

##  Appendix B Numerical solution of the Boltzmann equation with the account
for an angle-dependent scattering on 1D defects

We are interested in solving the kinetic Boltzmann equation in ${\bf
B}\parallel{\bf e}_{y}$ geometry, when scattering time $\tau_{s}$ depends on
incident velocities angles of electrons and the analytical solution is not
possible. We consider the model when
$1/\tau_{s}=(1/\tau_{s}^{0})\sin{\theta}$, where $\tau_{s}^{0}$ is some
constant. Starting from (6) and looking for its solution as a decomposition in
spherical harmonics $\delta f_{p}=\sum_{l,m}f_{l,m}Y^{m}_{l}(\theta,\varphi)$
one can obtain

$\displaystyle\alpha
q_{\gamma}+\frac{\omega_{c}}{2}\sum\limits_{l,m}f_{l,m}\left[\sqrt{(l-m)(l+m+1)}Y^{m+1}_{l}-\sqrt{(l+m)(l-m+1)}Y^{m-1}_{l}\right]=$
$\displaystyle=-\sum\limits_{l,m}f_{l,m}\left[\frac{1}{\tau_{i}}Y^{m}_{l}+(1-\delta_{m,0})\frac{1}{\tau^{0}_{s}}\sin{\theta}~{}\\!Y^{m}_{l}\right],$
(30)

where $\alpha=(eE_{0}/m\sqrt{2})\partial f_{p}^{0}/\partial v$ and ${\bf
E}=E_{0}{\bf e}_{\gamma}$ with $\gamma=x,y,z$ resulting in appropriate
combinations of angular harmonics $q_{\gamma}$ (for example,
$q_{x}=Y_{1}^{-1}-Y_{1}^{1}$). Multiplying both left and right hand sides of
(B) by complex conjugate of some $Y^{q}_{k}$ and integrating over the solid
angle we get

$\displaystyle
f_{k,q-1}\frac{\omega_{c}}{2}\sqrt{(k-q+1)(k+q)}-f_{k,q+1}\frac{\omega_{c}}{2}\sqrt{(k+q+1)(k-q)}+f_{k,q}\frac{1}{\tau_{i}}+$
$\displaystyle+\sum\limits_{l}f_{l,q}(1-\delta_{q,0})\frac{\beta_{l,k}^{q}}{\tau^{0}_{s}}\int\limits_{-1}^{1}dxP_{k}^{q}(x)P_{1}^{1}(x)P_{l}^{q}(x)=-\alpha\delta_{k,1}(\delta_{q,-1}-\delta_{q,1}),$
(31)

which is an algebraic system of linear equations on $f_{k,q}$. Here we have
used a concrete choice of ${\bf E}=E_{0}{\bf e}_{x}$, resulting in a specific
form of the right hand side. For other directions of electric field the right
hand side will change, though it is always determined by $k=1$ harmonics
linear combinations. The coefficient $\beta$ stems from the spherical
harmonics normalization conditions and the fact that
$\sin{\theta}=-P_{1}^{1}(\cos{\theta})$, thus

$\displaystyle\beta_{l,k}^{q}=-\frac{1}{2}\sqrt{(2l+1)(2k+1)}\sqrt{\frac{(l-q)!}{(l+q)!}\frac{(k-q)!}{(k+q)!}}.$
(32)

The latter is defined for positive $q$’s and it should give us a unit when
multiplied by the integral of associated Legendre polynomials as in (B), but
where the middle one polynomial $-P_{1}^{1}(x)$ is eliminated and $k$ equals
$l$.

One can see that the first three terms in (B) couple coefficients $f_{k,q}$
with different $q$ (with their $\pm 1$ change), but leave $k$ the same. On the
contrast, the last term in left hand side leaves the $q$ index fixed, while
couples the coefficients $f_{k,q}$ with different $k$. The integral in the
last term, which is an overlap integral of three associated Legendre
polynomials, could be computed using the formula and the algorithm described
in [29], and one can prove that there is an infinite number of harmonics that
couple to the $k=1$ one. These harmonics are $l=1,3,5,7,\dots$ and etc.

When $q=0$ the last term in left hand side of (B) is zero and the system
decomposes to rather simple relations

$\displaystyle
f_{k,0}=\frac{\omega_{c}\tau_{i}}{2}\sqrt{k(k+1)}(f_{k,1}-f_{k,-1}).$ (33)

This is due to the fact that the electrons moving along the 1D defects axis
are not scattered by them, so the relaxation of their velocities is entirely
due to short-range impurities described by $\tau_{i}$.

Nevertheless, in the case $q\neq 0$, different $f_{l,q}$ start to affect the
value of $f_{1,q}$. This means that the real system (B) is infinite. To
conduct numerical calculations we set $l_{max}=15$, although the changes in
calculation results become negligible starting even from $l=7$ harmonics
accounting.

We have rearranged a second order tensor $f_{k,q}$ to a vector form in a
following manner $[f_{1,-1},f_{1,0},f_{1,1},f_{3,-3},f_{3,-2},\dots]^{T}$.
This obvious rule for $f_{k,q}$ listing implies a consequent enumeration of
all $k=1,3,5,7,\dots,l_{max}$ (as soon as only these harmonics influence the
$k=1$ one, in which we are interested) with an appropriate account for
corresponding $q$ values changing from $q=-k$ to $q=k$. The general formula
for a new composite vector index is $n=k+q+1+k(k-1)/2$, which takes the values
$n=1,\dots,N$, where $N=(l_{max}+1)(l_{max}+2)/2$. Thus, (B) is transformed to
a linear algebraic equation in a matrix form of $N\times N$ dimension with
inhomogeneous part in the right hand side proportional to $\alpha$. $N=136$ in
our case and it grows with a square rate at the $l_{max}$ increase, but note
that an appropriate matrix of linear coefficients contains a lot of zeros.

Solving this linear system of equations we get numerical values of
$f_{1,-1},f_{1,0},f_{1,1}$, which determine all the components of the
diffusion tensor. For example,
$\mathcal{D}_{xx}\propto(f_{1,-1}-f_{1,1})/\sqrt{2}$,
$\mathcal{D}_{yx}\propto(-i)(f_{1,-1}+f_{1,1})/\sqrt{2}$ and
$\mathcal{D}_{zx}\propto f_{1,0}$, when the ${\bf E}\parallel{\bf e}_{x}$
condition determines (B) in its right hand side. To calculate other components
we need to repeat calculations with right hand side of (B) related to ${\bf
E}\parallel{\bf e}_{y}$ and ${\bf E}\parallel{\bf e}_{z}$.

##  Appendix C Calculation of the diffusion coefficient $\mathcal{D}_{zz}$
within the correlation function approach

The calculation of $\mathcal{D}_{zz}$ by means of the correlation functions
requires to take into account an infinite series of scattering events on EOD.
Indeed, since the electron scattering on EOD does not lead to the relaxation
of $v_{z}$, it requires an electron to meet a short-range impurity to destroy
the correlation with $v_{z}(0)$. This, however, can happen after an arbitrary
number of scattering acts on EOD.

Figure 5: The electron diffusion trajectory.

The trajectory of an electron with the initial velocity ${\bf v}(0)$ is the
polygonal line with the kinks corresponding to the scattering events (see Fig.
5). The segment ($N-1,N$) between neighboring kinks is the cyclotron arc whose
orientation is determined by the velocity direction ${\bf v}_{N-1}$ after
$(N-1)$-th collision. The probability density regulating the location of kinks
on each trajectory is determined by the Poisson stochastic process, see (15).
At the same moment, the probability density to experience $N$ subsequent
collisions each at time $t_{i}$ is simply given by the product of single
scattering event probabilities; for instance
$P_{s}(t_{N}-t_{N-1})dt_{N}P_{s}(t_{N-1}-t_{N-2})dt_{N-1}\dots
P_{s}(t_{1})dt_{1}$ describes the electron path for which the first $N$
collisions were due to the scattering on EOD. We calculate the diffusion
coefficient by summing separately the contributions to the time integral (14)
stemming from different segments of the electron trajectories:

$\mathcal{D}_{\alpha\beta}=\sum_{N=1}^{\infty}\mathcal{D}_{\alpha\beta}^{(N)},$
(34)

where the $N$-th term $\mathcal{D}_{zz}^{(N)}$ collects the correlation of the
electron velocity with its value ${\bf v}(0)$ at the initial moment from all
trajectories at the time interval between $N-1$ and $N$ scattering events.
Basically, such correlation in $\mathcal{D}_{zz}^{(N)}$ is preserved only for
such trajectories which have $(N-1)$ subsequent scattering acts on EODs. The
probability density to experience $(N-1)$ scattering events on EOD in a row
with an arbitrary type of scattering at $t_{N}$ time moment is given by
$P_{0}(t_{N}-t_{N-1})dt_{N}P_{s}(t_{N-1}-t_{N-2})dt_{N-1}\dots
P_{s}(t_{1})dt_{1}$, the corresponding contribution to the diffusion
coefficient reads as

$\mathcal{D}_{zz}^{(N)}=\Bigl{\langle}\int\limits_{0}^{\infty}d\tau_{1}P_{s}(\tau_{1})\dots\int\limits_{\tau_{N-2}}^{\infty}d\tau_{N-1}P_{s}(\tau_{N-1})\int\limits_{\tau_{N-1}}^{\infty}d\tau_{N}P_{0}(\tau_{N})\int\limits_{\tau_{N-1}}^{\tau_{N}}dt\cdot
v_{z}(t)v_{z}(0)\Bigr{\rangle},$ (35)

here $\tau_{k}=t_{k}-t_{k-1}$, $k=2\dots N$ ($\tau_{1}=t_{1}$) and the average
$\langle\dots\rangle$ goes over the electron isotropic initial velocity
direction distribution. Since $v_{z}(t)=v_{z}(0)$ (until the electron scatters
a short-range defect), the series for the diffusion coefficient transforms
into the geometric progression, which can be easily evaluated

$\mathcal{D}_{zz}=\Bigl{\langle}v_{z}^{2}\tau_{0}\sum_{n=0}\left(\frac{\tau_{0}}{\tau_{s}}\right)^{n}\Bigr{\rangle}=\langle
v_{z}^{2}\tau_{i}\rangle.$ (36)

Naturally, we obtained essentially the same expressions for the diffusion
coefficients ${\mathcal{D}_{zz}}$ as we did using the kinetic equation
approach, see (11).