# High-temperature magneto-inter-chirality oscillations in 2D systems with
strong spin-orbit coupling

M. E. Raikh  Department of Physics and Astronomy, University of Utah, Salt
Lake City, UT 84112

###### Abstract

Conventional Shubnikov–de Haas oscillations of conductivity in 3D are washed
out as the temperature exceeds the spacing between the Landau levels. This is
due to smearing of the Fermi distribution. In 2D, in the presence of two or
more size-quantization subbands, there is an additional type of magneto-
oscillations, usually referred to as magneto-inter-subband oscillations, which
do not decay exponentially with temperature. The period of these oscillations
is determined by the condition that the energy separation between the subbands
contains an integer number of Landau levels. Under this condition, which does
not contain the Fermi distribution, the inter-subband scattering rate is
maximal. Here we show that, with only one subband, high-temperature
oscillations are still possible. They develop when the electron spectrum is
split due to the spin-orbit coupling. For these additional oscillations, the
coupling enters both the period and the decay rate.

###### pacs:

73.50.-h, 75.47.-m

##  I Introduction

Experimental study of the quantum resistance oscillations is usually carried
out by plotting the Fourier transform of the field-dependent correction to the
resistance versus the inverse magnetic field, $B$. In two dimensions, the
slope of the straight line obtained yields the information about the density
of electrons occupying the corresponding size-quantization subband. If two or
more subbands are occupied, the low-temperature Fourier transform exhibits two
or more peaks. As the temperature, $T$, is elevated, the peaks quickly vanish.
This is the result of smearing of the Fermi distribution. At elevated
temperatures the Shubnikov–de Haas peaks give way to the additional peaks
corresponding to the sum and difference of the fundamental frequencies. The
peak corresponding to the difference does not decay with $T$ since the effects
of smearing coming from both subbands get cancelled as a result of the
subtraction. The underlying mechanism of the slow decaying peak corresponding
to the difference of the fundamental frequencies is the inter-subband
scattering of electrons by the disorder. This effect is confidently observed
experimentally, see e. g. Refs. [Coleridge, ; Leadly, ; Bykov, ; Vitkalov, ;
Vitkalov1, ; minkov, ], and understood theoretically in tiniest details.
Polyanovsky ; shahbazyan ; Golub ; Grigoriev ; Raichev

A natural question arises: whether the high-temperature oscillations are
possible if there is only one subband? One option is the spin splitting of the
single-subband spectrum. As argued in Ref. [Bykov, ], the situation mimicking
the inter-subband scattering can be realized if the applied magnetic field is
tilted with respect to the 2D plane. Then the condition for magneto-
oscillations, which plays the role of inter-subband scattering, is that the
Zeeman splitting in the parallel field contains an integer number of Landau
levels created by the normal component of the net magnetic field.

A question addressed in the present paper: what if magnetic field is strictly
normal to the 2D plane? Below we demonstrate that strong enough spin-orbit
coupling alone is sufficient for the disorder-induced magneto-oscillations.
This is despite, in the presence of spin-orbit coupling, the Landau levels are
not even-spaced. A ”price” paid for inability to align the staircases of
Landau levels from two spin-orbit branches is that both the period of
additional oscillations and their decay with temperature contain the spin-
orbit coupling strength.

##  II Hamiltonian and eigenfunctions

At a zero magnetic field, the Hamiltonian of spin-orbit coupled 2D gas has the
form

${\hat{H}}=\frac{\hbar^{2}{\hat{\bm{k}}}^{2}}{2m}+\alpha({\hat{\bm{\sigma}}}\times{\hat{\bm{k}}})\cdot{\bm{z}}_{0},$
(1)

where ${\bm{z}}_{0}$ is the normal to the 2D plane, ${\hat{\bm{k}}}$ is the
momentum operator, while ${\hat{\bm{\sigma}}}$ is the vector of Pauli
matrices. Parameters $m$ and $\alpha$ stand for the effective mass and the
spin-orbit constant, respectively. The spectrum of the Hamiltonian Eq. (1)
consists of two branches

$E_{1,2}({\bm{k}})=\frac{\hbar^{2}k^{2}}{2m}\mp\alpha|{\bm{k}}|.$ (2)

The corresponding wave functions have the form

$\Psi_{{\bm{k}}}^{(1,2)}({\bm{r}})=e^{i{\bm{k}}\cdot{\bm{r}}}\chi_{{\bm{k}}}^{(1,2)},$
(3)

where the spinors $\chi_{{\bm{k}}}^{(1,2)}$ are defined as

$\chi_{{\bm{k}}}^{(1)}=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{i\Phi_{{\bm{k}}}}\\\
-1\end{pmatrix},\hskip
8.53581pt\chi_{{\bm{k}}}^{(2)}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
e^{-i\Phi_{{\bm{k}}}}\end{pmatrix}$

with $\Phi_{{\bm{k}}}$ being the azimuthal angle of the vector ${\bm{k}}$. It
is crucial that the lower branch of the spectrum Eq. (2) has a minimum at

$k=k_{0}=\frac{m\alpha}{\hbar^{2}}.$ (4)

Near this minimum the spectrum can be simplified as

$E_{1}({\bm{k}})=-\Delta+\frac{\hbar^{2}(k-k_{0})^{2}}{2m},$ (5)

with the depth of the minimum
$\Delta=\frac{m\alpha^{2}}{2\hbar^{2}}=\frac{\hbar^{2}k_{0}^{2}}{2m}$.

Switching on the magnetic field implies the replacement of ${\hat{\bm{k}}}$ by
the kinetic momentum ${\hat{\bm{k}}}+\frac{e}{c}{\bm{A}}$, where ${\bm{A}}$ is
the vector potential, which we choose in the form ${\bm{A}}={\bm{y}}_{0}Bx$.

In the absence of spin-orbit coupling, the eigenstates are the conventional
Landau levels. The corresponding wave functions are
$e^{ip_{y}y}\psi_{n}(x-p_{y}l^{2})$, where $p_{y}$ is the conserving
$y$-component of the momentum, $l=(\hbar c/eB)^{1/2}$ is the magnetic length,
and $\psi_{n}$ is the harmonic oscillator wave function. Since the operator
${\hat{\bm{k}}}$ couples only the neighboring Landau levels, the eigenstates
in the the presence of both magnetic field and spin-orbit coupling represent
the following vectors of the harmonic oscillator wave functionsRashba

$\Psi\left(x,y\right)=e^{ip_{y}y}\begin{pmatrix}i\lambda
b_{n}^{-\lambda}\psi_{n-1}(x-p_{y}l^{2})\\\
b_{n}^{\lambda}\psi_{n}(x-p_{y}l^{2})\end{pmatrix},$ (6)

where $\lambda$ denotes the chirality and takes the values $\pm 1$.
Normalization coefficients $b_{n}$ are given by

$b_{n}^{\lambda}=\Bigl{(}\frac{1}{2}+\frac{\lambda}{4c_{n}}\Bigr{)}^{1/2},$
(7)

and the coefficients $c_{n}$ are defined as

$c_{n}=\Big{(}\gamma^{2}n+\frac{1}{4}\Big{)}^{1/2}.$ (8)

Spin-orbit constant, $\alpha$, enters into Eqs. (7) and (8) via the parameter
$\gamma$ given by

$\gamma=\Biggl{[}\frac{2m\alpha^{2}}{\hbar^{2}(\hbar\omega_{c})}\Biggr{]}^{1/2}=\Biggl{(}\frac{4\Delta}{\hbar\omega_{c}}\Biggr{)}^{1/2},$
(9)

where $\omega_{c}=\Big{(}\frac{eB}{mc}\Big{)}^{1/2}$ is the cyclotron
frequency.

Energy level positions corresponding to the eigenfunctions Eq. (6) are equal
to

$E_{n}^{\lambda}=\hbar\omega_{c}\big{(}n+\lambda c_{n}\big{)}.$ (10)

Naturally, for vanishing spin-orbit coupling $\gamma\rightarrow 0$, the values
$c_{n}$ approach $\frac{1}{2}$, as it should be for the textbook Landau
levels. It is seen from Eq. (10) that the spacings between the neighboring
levels

$\frac{E_{n+1}^{\lambda}-E_{n}^{\lambda}}{\hbar\omega_{c}}=1+\lambda\big{(}c_{n+1}-c_{n}\big{)}$
(11)

depends on the chirality, $\lambda$. This fact is crucial for the period and
the decrement of the inter-chirality oscillations.

As it is seen from Eq. (9), the dimensionless parameter $\gamma$ represents a
square root of the number of the cyclotron quanta that fit into the minimum of
the depth $\Delta=\frac{m\alpha^{2}}{2\hbar^{2}}$ in the zero-field spectrum
Eq. (5), see Figure.

##  III Disorder-induced inter-chirality scattering, $E_{F}\gg\Delta$

In the presence of disorder, $V({\bf r})$, the discrete energies,
$E_{n}^{\lambda}$, get broadened. This broadening is captured by the
chirality-dependent self-energy, $\Sigma^{\lambda}(E)$. For a given chirality,
the density of states is expressed via $\Sigma^{\lambda}(E)$ as

$\displaystyle
g_{\lambda}(E)=\frac{1}{2\pi^{2}l^{2}}{\text{I}}{\text{m}}\sum_{n}\frac{1}{E-E_{n}^{\lambda}-\Sigma^{\lambda}(E)}$
$\displaystyle=\frac{1}{2\pi^{2}l^{2}}\sum_{n}\frac{{\text{I}}{\text{m}}\Sigma^{\lambda}}{\big{(}E-E_{n}^{\lambda}\big{)}^{2}+{\big{(}\text{I}}{\text{m}}\Sigma^{\lambda}\big{)}^{2}}.$
(12)

For a weak magnetic field
$E\gg{\text{I}}{\text{m}}\Sigma^{\lambda}\gg\hbar\omega_{c}$ the summation can
be replaced by integration yielding

$g_{\lambda}^{(0)}(E)=\Big{(}\frac{m}{2\pi\hbar^{2}}\Big{)}\frac{1}{1+\frac{\lambda\gamma}{2}\big{(}\frac{\hbar\omega_{c}}{E}\big{)}}.$
(13)

Oscillating correction to the density of states comes from the first harmonics
of the Poisson expansion of Eq. (III)

$\displaystyle
g_{\lambda}^{(1)}(E)=2g_{\lambda}^{(0)}(E)\cos\Bigg{[}2\pi\frac{E}{\hbar\omega_{c}}+2\pi\gamma\lambda\Big{(}\frac{E}{\hbar\omega_{c}}\Big{)}^{1/2}\Bigg{]}\times$
$\displaystyle\exp\Bigg{[}-2\pi\frac{{\text{I}}{\text{m}}\Sigma^{\lambda}(E)}{\hbar\omega_{c}}\Bigg{]}.$
(14)

The form of the argument of the cosine follows from the combination $n+c_{n}$
in the expression Eq. (10) for $E_{n}^{\lambda}$ with
$n=\frac{E}{\hbar\omega_{c}}$. Higher harmonics in $g_{\lambda}^{(1)}$ can be
neglected under the condition
${\text{I}}{\text{m}}\Sigma^{\lambda}\gg\hbar\omega_{c}$, which allows to
replace the summation over the Landau levels by integration. Since the meaning
of ${\text{I}}{\text{m}}\Sigma^{\lambda}$ is the inverse scattering time, this
condition implies that the scattering time is much shorter than the cyclotron
period.

At temperature $T=0$ the inter-chirality magneto-oscillations originate from
the fact that inter-chirality scattering time is proportional to the product
of the two oscillating corrections $g_{1}^{(1)}(E)g_{-1}^{(1)}(E)$ taken at
the Fermi energy, $E=E_{F}$. From Eq. (III) we find

$\displaystyle
g_{1}^{(1)}(E_{F})g_{-1}^{(1)}(E_{F})=4g_{1}^{(0)}(E_{F})g_{-1}^{(0}(E_{F})$
$\displaystyle\times\cos\Biggl{[}2\pi\frac{E_{F}}{\hbar\omega_{c}}+2\pi\gamma\Bigl{(}\frac{E_{F}}{\hbar\omega_{c}}\Bigr{)}^{1/2}\Biggr{]}$
$\displaystyle\times\cos\Biggl{[}2\pi\frac{E_{F}}{\hbar\omega_{c}}-2\pi\gamma\Bigl{(}\frac{E_{F}}{\hbar\omega_{c}}\Bigr{)}^{1/2}\Biggr{]}e^{-2W}$
(15)

where the Dingle factor, $e^{-2W}$, is defined as

$e^{-2W}=\exp\Bigg{[}-2\pi\frac{{\text{I}}{\text{m}}\Sigma^{1}(E)+{\text{I}}{\text{m}}\Sigma^{-1}(E)}{\hbar\omega_{c}}\Biggr{]}.$
(16)

We now convert the product of cosines into a sum

$\displaystyle
2\cos\Biggl{[}2\pi\frac{E_{F}}{\hbar\omega_{c}}+2\pi\gamma\Bigl{(}\frac{E_{F}}{\hbar\omega_{c}}\Bigr{)}^{1/2}\Biggr{]}$
$\displaystyle\times\cos\Biggl{[}2\pi\frac{E_{F}}{\hbar\omega_{c}}-2\pi\gamma\Bigl{(}\frac{E_{F}}{\hbar\omega_{c}}\Bigr{)}^{1/2}\Biggr{]}$
$\displaystyle=\cos\Biggl{[}4\pi\frac{E_{F}}{\hbar\omega_{c}}\Biggr{]}+\cos\Biggl{[}4\pi\gamma\Bigl{(}\frac{E_{F}}{\hbar\omega_{c}}\Bigr{)}^{1/2}\Biggr{]}.$
(17)

It follows from Eq. (III) that the inter-chirality scattering rate is the sum
of two contributions both oscillating with magnetic field. Since the first
term oscillates much faster than the second one, they decay with temperature
at completely different rate. The way to incorporate finite temperature into
Eq. (III) is straightforward. One should replace $E_{F}$ by $E_{F}+Tz$ and
integrate the result together with the dimensionless Fermi function
$(e^{z}+1)^{-1}$. For the rapidly oscillating term, corresponding to the sum,
this yields

$\Bigl{(}g_{1}^{(1)}g_{-1}^{(1)}\Bigr{)}^{(+)}=(g^{(0)})^{2}\cos\Biggl{(}\frac{4\pi
E_{F}}{\hbar\omega_{c}}\Biggr{)}A\Biggl{(}\frac{T}{T_{1}}\Biggr{)}e^{-2W},$
(18)

where the conventional damping function $A(\chi)$ is defined as

$A(\chi)=\frac{\chi}{\sinh\chi},$ (19)

where the argument, $\chi$, is given by

$\chi=\frac{4\pi^{2}T}{\hbar\omega_{c}},$ (20)

i.e. $T_{1}=\frac{\hbar\omega_{c}}{4\pi^{2}}$ which is two times smaller than
$T_{1}$ for conventional Shubnikov-de Haas oscillations, while the period of
the magneto-interchirality oscillations is two times bigger than that for
conventional oscillations.

Most interesting is the result of the temperature averaging of the second term
in Eq. (III), corresponding to the difference. It reduces to the same function
$A(\chi)$, namely,

$\Bigl{(}g_{1}^{(1)}g_{-1}^{(1)}\Bigr{)}^{(-)}=(g^{(0)})^{2}\cos\Biggl{[}4\pi\gamma\Bigl{(}\frac{E_{F}}{\hbar\omega_{c}}\Bigr{)}^{1/2}\Biggr{]}A\Biggl{(}\frac{T}{T_{2}}\Biggr{)}e^{-2W},$
(21)

where $T_{2}$ is now given by

$T_{2}=\frac{\Big{(}\hbar\omega_{c}E_{F}\Big{)}^{1/2}}{2\pi^{2}\gamma}.$ (22)

Our most important observation is that Eq. (21) describes high-temperature
oscillations, i.e. $T_{2}\gg T_{1}$. To demonstrate this, we rewrite the ratio
$\frac{T_{2}}{T_{1}}$ in the form

$\displaystyle\frac{T_{2}}{T_{1}}=2\frac{\left(\hbar\omega_{c}E_{F}\right)^{1/2}}{\hbar\omega_{c}\gamma}=2\Bigl{(}\frac{E_{F}}{\hbar\omega_{c}}\Bigr{)}^{1/2}\Biggl{[}\frac{\hbar^{2}(\hbar\omega_{c})}{2m\alpha^{2}}\Biggr{]}^{1/2}$
$\displaystyle=2\Biggl{(}\frac{\hbar^{2}E_{F}}{2m\alpha^{2}}\Biggr{)}^{1/2}=2\Biggl{(}\frac{E_{F}}{\Delta}\Biggr{)}^{1/2}.$
(23)

We see that the condition $T_{2}\gg T_{1}$ is met when the Fermi level is much
higher that the depth, $\Delta$, of the spin-orbit minimum in the energy
spectrum at zero magnetic field. It is also instructive to cast the ”slow”
cosine in Eq. (21) in the form

$4\pi\gamma\Biggl{(}\frac{E_{F}}{\hbar\omega_{c}}\Biggr{)}^{1/2}=4\pi\frac{\Bigl{(}E_{F}\frac{2m\alpha^{2}}{\hbar^{2}}\Bigr{)}^{1/2}}{\hbar\omega_{c}}=4\pi\frac{\varepsilon_{0}}{\hbar\omega_{c}}.$
(24)

It follows from Eq. (24) that the same condition
$E_{F}\gg\frac{2m\alpha^{2}}{\hbar^{2}}$, i.e. $\varepsilon_{0}\ll E_{F}$,
which ensures that $T_{2}\gg T_{1}$ also ensures that the ”slow” cosine in Eq.
(21) is indeed parametrically slower than the cosine in Eq. (18).

We assumed that the magnetic field is strictly normal to the 2D plane. In
addition to the Landau quantization, this field causes the Zeeman splitting
$\pm\Delta_{Z}$ of the electron spectrum. It is straightforward to incorporate
the Zeeman splitting into the above result for the period of slow
oscillations. Namely, in the presence of the Zeeman splitting, the oscillating
magnetoresistance $\cos
4\pi\Bigl{(}\frac{\varepsilon_{0}}{\hbar\omega_{c}}\Bigr{)}$ transforms into

$\cos\Biggl{[}4\pi\Bigl{(}\frac{\varepsilon_{0}}{\hbar\omega_{c}}\Bigr{)}\Bigl{(}1+\frac{\Delta_{Z}^{2}}{\Delta_{0}^{2}}\Bigr{)}^{1/2}\Biggr{]}.$
(25)

Characteristic Zeeman splitting $\Delta_{0}$ at which the frequency of
magneto-oscillations increases twice is given by

$\Delta_{0}\\!=E_{F}^{1/2}\\!\Biggl{(}\frac{2m\alpha^{2}}{\hbar^{2}}\Biggr{)}^{1/4}\\!\Bigl{(}\hbar\omega_{c}\Bigr{)}^{1/4}\\!=\\!E_{F}\Biggl{(}\frac{\varepsilon_{0}}{E_{F}}\Biggr{)}^{1/2}\Bigl{(}\frac{\hbar\omega_{c}}{E_{F}}\Bigr{)}^{1/4}.$
(26)

We see that the last two brackets are small by virtue of the assumptions made.
This leads us to the conclusion that, with Zeeman splitting, slow magneto-
oscillations survive, but become ”faster”. Obviously, the magnitude of the
magneto-oscillations falls off with $\Delta_{Z}$. This is because the larger
is $\Delta_{Z}$ the smaller is the probability of the inter-chirality
scattering.

The remaining task is to demonstrate that the term Eq. (21) indeed emerges as
a certain term in the expression for the conductivity. More specifically, one
has to trace how the product $\Bigl{(}g_{1}^{(1)}g_{-1}^{(1)}\Bigr{)}^{(-)}$
originates from the general Kubo formula

$\sigma_{xx}(E)=\frac{\pi
e^{2}\hbar}{\Omega}\langle{\text{T}}{\text{r}}\Bigl{[}{\hat{v}}_{x}\delta(E-{\hat{H}}-V({\bm{r}})){\hat{v}}_{x}\delta(E-{\hat{H}}-V({\bm{r}}))\Bigr{]}\rangle.$
(27)

Here $\langle...\rangle$ denotes the averaging over the disorder
$V({\bm{r}})$, ${\hat{v}}_{x}$ is the velocity operator, and $\Omega$ is the
2D normalization volume. Similarly to intersubband scatteringPolyanovsky ;
shahbazyan ; Golub ; Grigoriev ; Raichev , evaluation of $\sigma_{xx}$ is
possible neglecting the localization effect when the self-consistent Born
approximation (SCBA) applies. As shown in Ref. shahbazyan,  this requires that
the disorder is short-ranged, namely, that the correlation radius is much
smaller than $l$. In the framework of the SCBA, the $\delta$-functions in the
right-hand side of Eq. (27) can be averaged independently. Within a factor
$\frac{1}{\pi}$ each average $\delta$-function is expressed through
$g_{\lambda}(E)$ and thus, via Eq. (III), through the self-energy. This is how
the product of ${\text{I}}{\text{m}}\Sigma^{\lambda}$ with different $\lambda$
enters into $\sigma_{xx}$.

The imaginary part of the self-energy is determined by the level numbers, $n$,
for which $E_{n}^{\lambda}$ is close to $E$. Equating $E_{n}^{\lambda}$ to
$E$, we find from Eq. (10)

$n^{\lambda}(E)=\Biggl{[}\Biggl{(}\frac{E}{\hbar\omega_{c}}+\frac{\gamma^{2}}{4}\Biggr{)}^{1/2}\pm\frac{\lambda\gamma}{2}\Biggr{]}^{2}.$
(28)

We see that, at a given $E$, the corresponding numbers of the $n$-values
depend on the chirality, $\lambda$. As a result, the self-energy depends on
chirality. This fact is at the core of inter-chirality magneto-oscillations.

To conclude the Section, we relate $\Sigma^{\lambda}$ to the disorder
strength. The quantitative characteristics of this strength is the width,
$\Gamma$, defined asshahbazyan

$\Gamma^{2}=\frac{1}{2\pi l^{2}}\int d{\bm{r}}\langle V(0)V({\bm{r}})\rangle.$
(29)

This width enters the system of SCBA equations for the self-energy

$\displaystyle\frac{\Sigma^{\lambda}}{\Gamma^{2}}=\sum_{n}\frac{\kappa_{\lambda,\lambda}}{E-E_{n}^{\lambda}-\Sigma^{\lambda}}+\sum_{n}\frac{\kappa_{\lambda,-\lambda}}{E-E_{n}^{-\lambda}-\Sigma^{-\lambda}},$
(30)

where $\kappa_{\lambda,\lambda}$ and $\kappa_{\lambda,-\lambda}$ are the
squares of matrix elements defined as

$\kappa_{\lambda,\lambda}={\Big{|}}\langle\lambda,n|V({\bf
r})|\lambda,n\rangle{\Big{|}}^{2},~{}~{}\kappa_{\lambda,-\lambda}={\Big{|}}\langle\lambda,n|V({\bf
r})|-\lambda,n\rangle{\Big{|}}^{2}.$ (31)

Finally, we relate $\Gamma$ to the imaginary part of self-energy which enters
the Dingle factor:
${\text{I}}{\text{m}}\Sigma^{\lambda}(E)=\frac{\Gamma^{2}}{\hbar\omega_{c}}$.

##  IV Intra-chirality scattering, $E_{F}<0$ and $|E_{F}|<\Delta$

At negative energy, $E<0$, the states Eq. (10) exist only for chirality
$\lambda=-1$, however, as illustrated in the Figure, there are two values of
$n$ corresponding to a given value of energy, namely

$\displaystyle
n_{-}(E)=\Biggl{[}\frac{\gamma}{2}+\Bigl{(}\frac{\gamma^{2}}{4}-\frac{|E|}{\hbar\omega_{c}}\Bigr{)}^{1/2}\Biggr{]}^{2},$
$\displaystyle
n_{+}(E)=\Biggl{[}\frac{\gamma}{2}-\Bigl{(}\frac{\gamma^{2}}{4}-\frac{|E|}{\hbar\omega_{c}}\Bigr{)}^{1/2}\Biggr{]}^{2}.$
(32)

To find the oscillating component of the density of states, we substitute Eq.
(10) with $\lambda=-1$ into the expression Eq. (13) and apply the Poisson
expansion. For the first harmonics, this yields

$g_{-1}^{(1)}(E)\propto\int\limits_{0}^{\infty}\frac{{\text{I}}{\text{m}}\Sigma^{-1}(E)\hskip
5.69054ptdx\cos 2\pi x}{\biggl{[}E-\hbar\omega_{c}\Big{(}x-\gamma
x^{1/2}\Big{)}\biggr{]}^{2}+\Big{(}{\text{I}}{\text{m}}\Sigma^{-1}(E)\Big{)}^{2}}.$
(33)

To illuminate the structure of the integral Eq. (33), we make a substitution

$x=\gamma^{2}\Big{(}\frac{1}{2}+s\Big{)}^{2}$ (34)

after which it assumes the form

$g_{-1}^{(1)}(E)\propto\frac{{\text{I}}{\text{m}}\Sigma^{-1}(E)}{(\gamma\hbar\omega_{c})^{2}}\int\limits_{-\frac{1}{2}}^{\infty}\frac{ds(1+2s)\cos
2\pi\gamma^{2}\Big{(}\frac{1}{2}+s\Big{)}^{2}}{\Biggl{[}\frac{E}{\gamma^{2}\hbar\omega_{c}}+\frac{1}{4}-s^{2}\Biggr{]}^{2}+\Big{(}\frac{{\text{I}}{\text{m}}\Sigma^{-1}(E)}{\gamma^{2}\hbar\omega_{c}}\Big{)}^{2}}.$
(35)

With the definition Eq. (9) of the parameter $\gamma$, we realize that the
square bracket in the denominator of Eq. (35) turns to zero at two values of
the argument, namely

$s=s_{\pm}=\pm\Biggl{(}\frac{\Delta-|E|}{4\Delta}\Biggr{)}^{1/2}.$ (36)

These values contribute to the integral Eq. (35) independently under the
condition

${\text{I}}{\text{m}}\Sigma^{-1}\ll\Bigl{(}\Delta-|E|\Bigr{)}.$ (37)

When this condition is met, the integral (35) can be evaluated by expanding
the integrand around $s=s_{+}$ and $s=s_{-}$. As a result, the oscillating
component, $g_{-1}^{(1)}(E)$, represents a sum of two contributions

$\displaystyle
g_{-1}^{(1)}(E)\propto\frac{1}{\hbar\omega_{c}}\Biggl{[}\cos\Biggl{(}\frac{8\pi\Delta}{\hbar\omega_{c}}\Biggr{)}\Bigl{(}\frac{1}{2}+s_{+}\Bigr{)}^{2}\Biggr{]}\times$
$\displaystyle\exp\Big{[}-2\pi\frac{{\text{I}}{\text{m}}\Sigma^{-1}}{\hbar\omega_{c}}\Big{(}\frac{1+2s_{+}}{2|s_{+}|}\Big{)}\Big{]}+$
$\displaystyle\frac{1}{\hbar\omega_{c}}\Biggl{[}\cos\Biggl{(}\frac{8\pi\Delta}{\hbar\omega_{c}}\Biggr{)}\Bigl{(}\frac{1}{2}+s_{-}\Bigr{)}^{2}\Biggr{]}\times$
$\displaystyle\exp\Big{[}-2\pi\frac{{\text{I}}{\text{m}}\Sigma^{-1}}{\hbar\omega_{c}}\Big{(}\frac{1+2s_{-}}{2|s_{-}|}\Big{)}\Big{]}.$
(38)

Above we had established that in the case of inter-chirality scattering, slow-
decaying oscillations of the magneto-resistance originate from the difference
of the arguments of cosines in the product $g_{1}^{(1)}(E)g_{-1}^{(1)}(E)$. On
the contrary, for intra-chirality scattering, the slow-decaying oscillations
originate from the sum of the arguments of two cosines, which emerge in the
term $\propto g_{-1}^{(1)}(E)^{2}$ in the conductivity. Indeed, the product of
two cosines in Eq. (IV) contains a term

$\cos\Biggl{[}\Bigl{(}\frac{4\pi\Delta}{\hbar\omega_{c}}\Bigr{)}\Bigl{(}1+\frac{\Delta-|E|}{\Delta}\Bigr{)}\Biggr{]}.$
(39)

The latter expression leads us to the conclusion that when the Fermi level is
located close to the bottom of the spectrum with $\lambda=-1$, i.e. for
$\left(\Delta-|E_{F}|\right)\ll\Delta$, the intra-subband correction to the
conductivity oscillates as
$\cos\Bigl{(}\frac{4\pi\Delta}{\hbar\omega_{c}}\Bigr{)}$ as if the
concentration of electrons was $\frac{m\Delta}{\pi\hbar^{2}}$ (without spin-
orbit coupling) rather than much smaller concentration $\frac{m(\Delta-
E_{F})}{\pi\hbar^{2}}$ (in the presence of a strong spin-orbit coupling). In
other words, while the Fermi level is positioned near the minimum of the
spectrum, the conductivity knows about the net depth of the minimum, $\Delta$.

##  V Concluding remarks

The origin of the conventional oscillations of magnetoconductivity
$\delta\sigma\propto\cos\Bigl{(}2\pi\frac{E_{F}}{\hbar\omega_{c}}\Bigr{)}$ is
the fact that the Landau levels near the Fermi energy, $E_{F}$, are even-
spaced. In this regard, the message of the present paper is that, spin-orbit
coupling, while lifting the equality of the spacings of the Landau levels,
leads to additional oscillations
$\delta\sigma\propto\cos\Bigl{[}8\pi\frac{(\Delta
E_{F})^{1/2}}{\hbar\omega_{c}}\Bigr{]}$, where $\Delta$ is the depth of the
minimum in the energy spectrum, see Figure. These oscillations are much slower
than the conventional oscillations and decay, upon increasing temperature,
much slower than the conventional oscillations.

Recent experiments [SmB6, ; E1, ; E3, ; E2, ; RRD, ; WTe, ; Young+Gate, ] had
revived interest to the subject of magneto-oscillations in solids, see e.g.
Refs. [1, ; 2, ; P.A.Lee, ; Allocca, ; Allocca1, ; knolle, ]. In short, the
experiments [SmB6, ; E1, ; E3, ; E2, ; RRD, ; WTe, ; Young+Gate, ] indicate
that magneto-oscillations persist even when the Fermi level lies inside the
energy gap, so that the carriers participating in susceptibility and transport
are thermally excited. At the core of the proposed explanation is the
observation that, in materials exhibiting the non-conventional oscillations,
the minimum of the conduction band is located at finite momentum. A
consequence of this fact is a peculiar structure of the Landau levels, namely,
as magnetic field increases, one of the levels periodically approaches the
band-edge, and, subsequently, narrows. This is the situation very similar to
that in Sect. IV. In this regard, while in the present manuscript, we
considered the conventional ”metallic” arrangement when the Fermi level lies
above the band minimum of the lower branch of the spectrum Eq. (2) has a
minimum at finite $k=k_{0}$. Then our prime message is that the disorder-
induced scattering between the levels with different $n$ boosts the
oscillations.

## References

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