# Beyond the ordinary acoustoelectric effect: superluminal phenomena in the
acoustic realm and phonon-mediated Bloch gain

A. Apostolakis<EMAIL_ADDRESS>Department of Condensed Matter Theory,
Institute of Physics CAS, Na Slovance 1999/2, 182 21 Prague, Czech Republic
Department of Physics, Loughborough University, Loughborough LE11 3TU, United
Kingdom  A. G. Balanov<EMAIL_ADDRESS>Department of Physics,
Loughborough University, Loughborough LE11 3TU, United Kingdom  F. V.
Kusmartsev  Department of Physics, Loughborough University, Loughborough LE11
3TU, United Kingdom  College of Art and Science, Khalifa University, PO Box
127788, Abu Dhabi, United Arab Emirates  Microsystem and Terahertz Research
Center, Chengdu, 610200, P.R. China  K. N. Alekseev  Department of Physics,
Loughborough University, Loughborough LE11 3TU, United Kingdom  Center for
Physical Sciences and Technology, Vilnius LT-10257, Lithuania

###### Abstract

It has been shown that coherent phonons can be used as a potent tool for
controlling and enhancing optoelectronic and transport properties of
nanostructured materials. Recent studies revealed that interaction of acoustic
phonons and fast-moving carriers in semiconductor heterostructures can be
accompanied by electron-phonon instabilities that cause ordinary and induced
Cherenkov effects. However, the development of such instabilities is still
poorly understood. Our study shows that other supersonic phenomena, beyond the
Cherenkov instability, are possible for non-equilibrium charge transport in
the miniband semiconductor superlattices (SLs) driven by an acoustic plane
wave. Using semiclassical nonperturbative methods and elements of the
bifurcation theory, we find the conditions for the onset of dynamical
instabilities (bifurcations) which are caused by the emission of specific SL
phonons by supersonic electrons, and their back action on the electrons.
Notably, the underlying radiation mechanism is connected either to normal or
anomalous Doppler effects in full accordance with the Ginzburg-Frank-Tamm
theory. The appearance of induced Doppler effects is also discussed in
relation to the formation of electron bunches propagating through the
spatially periodic structure of the SL. When the amplitude of the acoustic
wave exceeds certain threshold, the dynamical instabilities developed in the
system are manifested as drift velocity reversals, resonances in sound
attenuation and absolute negative mobility. We demonstrate that the discovered
superluminal Doppler phenomena can be utilized for tunable broadband
amplification and generation of GHz-THz electromagnetic waves, which creates a
ground for development of novel phononic devices.

††preprint: APS/123-QED

##  I INTRODUCTION

The mediating role of lattice vibrations (phonons) in nanocrystals has been
readily recognized as a fundamental tenet in condensed matter physics. In
particular, understanding in depth the transport properties of phonons and
their interactions with electrons is important for enhancing the efficiency of
thermoelectric nanostructures [1, 2] and developing acoustic metamaterials [3,
4] or novel spectroscopic schemes [5, 6]. In the past few years, there has
been an intensive research aimed to study the connection between physics of
coherent phonons excitation and nonequilibrium dynamics in electronic systems.
Prominent examples include phononic devices based on two-dimensional (2D)
materials [7, 8, 9, 10] which can exhibit high electron mobility [9, 10],
high-frequency phonon transducers [11, 12, 13] and quantum structures where
coherent acoustic phonons have been generated in the THz-GHz range [7, 14, 15,
16]. Recently, there has been increased interest in amplification of acoustic
phonons due to their interaction with fast moving carriers in semiconductor
superlattices (SLs) [16]. These experiments revealed the importance of
electron-phonon instabilities where ordinary and induced Cherenkov effects
come into play. In general, the Cherenkov effect is the well-know phenomenon
encountered in the electrodynamics when a charged particle passes through a
dielectric medium at speed greater than the phase velocity of light in the
medium. We likewise note new research efforts devoted to the radiation
dynamics of the superluminal particle [17, 18, 19, 20]. Interesting examples
include further developments in Ginzburg and Frank theory [21, 22] describing
the Doppler effects [20, 23], Smith-Purcell radiation in plasmonic crystals
[17] and nonperturbative generalization of Cherenkov radiation [19]. In the
acoustic realm though, Cherenkov emission can be induced when the average
electron velocity ($v_{d}$) in the presence of a static field moves faster
than the speed of sound $(v_{s})$. This supersonic condition ($v_{d}>v_{s}$)
is well satisfied in an electrically biased SL [16] where the propagating
acoustic wave interacts with the electrically driven electron current.

Figure 1: Schematic diagram of novel electroacoustical effects beyond the
ordinary charge drag. A strong acoustic plane wave of the energy amplitude $U$
propagates at $v_{s}=\omega_{s}/k_{s}$ along the growth direction ($x$–axis)
of a superlattice (SL) with a miniband width of $\Delta$ and lattice period
$d$. (a) For $U<\Delta/2$ charges are always transported along the direction
of the wave propagation. However, bunches of electrons begin to oscillate
strongly at $U=U_{cr_{1}}$, which is slightly below $\Delta/2$ [see Eq. (13)],
and then drift of the bunches quickly slows down with an increase of $U$. (b)
For $U>\Delta/2$ the electronic bunches move opposite to the propagating wave,
and their backward drift becomes fastest at $U_{cr_{2}}$, which value is
slightly above $\Delta/2$ [Eq. (13)]. Our analytical selection rules [Eq.
(13)] indicate that the critical amplitudes $U_{cr_{1}}$ and $U_{cr_{2}}$ mark
respectively superluminal anomalous and superluminal backward Doppler regimes
[Eq. (16)] of emission and absorption of specific phonons [Eq. (13)] by the
electrons. Additionally, at the wave amplitude, $U_{cr_{2}}$, an interaction
of the electrons with a high-frequency ($\omega$) electromagnetic wave can
result in its amplification.

The aforementioned study further confirmed that Cherenkov effects and possibly
other more exotic electromagnetic phenomena might have acoustic counterparts.
It was only recently, when semiclassical studies and quantum-mechanical
simulations [24, 25, 15] predicted that under the action of a strong acoustic
wave, the propagating deformation potential can induce quasiperiodic Bloch
oscillations of miniband electrons. The rise of these complex Bloch
oscillations was linked to global instabilities triggered with an increase of
the wave amplitude, which serves as control parameter [25]. Remarkably, the
development of these instabilities clearly distinguishes the case of SL from
the ordinary electroacoustic effect in bulk semiconductors with a quadratic
band [26, 27].
In this work, we study supersonic phenomena that should appear when a coherent
acoustic stimulus induces high-frequency electron dynamics in semiconductor
superlattice. In particular, we theoretically consider acoustically driven
miniband electrons which tunnel through the SL periodic potential under the
action of a longitudinal strain wave with energy amplitude $U$, as shown in
Fig. 1(a). In this case, the acoustic stimulus interacts with the electrons by
means of a deformation potential. We demonstrate that this interaction and the
induced miniband transport are controlled by the laws of physics typical for
superluminal particles with internal degrees of freedom [28]. In our case, the
internal degrees of freedom are associated with the propagating strain-induced
deformation potential. Analysis of the semiclassical phase space dynamics in
the reference frame moving at $v_{s}$ allowed to identify specific dynamical
instabilities (bifurcations) which are developed for certain critical values
of the strain-induced potential $U=U_{cr_{n}}$ ($n=1,2...$,
$U_{cr_{1}}<U_{cr_{2}}<...$). Tuning $U$ to these values induces the formation
of new bound states between the electrons and the phonons. We found out that
the phonon-assisted transport involving those bound states are, in essence, a
direct generalization of the normal and anomalous Doppler effects discussed in
the radiation theories introduced by Ginzburg and Frank for supreluminal
photons [22]. Our model parameters correspond to the regime related to the
recently discovered superlight inverse Doppler effect [20]. Moreover, our
results provide compelling arguments for the long-standing idea introduced by
Tamm [29] that the characteristics of the Ginzburg-Frank radiation theory can
be implemented to supersonic acoustical systems. The emitted phonons which are
involved in the radiation processes are found to behave in a manner similar to
a Smith-Purcell (SP) photons [30, 31].
Our study showed that stimulated emission creates electron bunches which move
along the spatially periodic structure of the SL. For both normal and
anomalous Doppler behavior, the electron bunching effects on directed charge
transport appear to be robust even when we take into account scattering
processes. Remarkably, the involvement of the anomalous Doppler effects for
$U$ $<\Delta/2$ decelerates the single electron with the overall drift
velocity remaining positive. Here $\Delta$ is the first minband width, within
which a charge tunneling transport is assumed. For $U>\Delta/2$, the reversal
of the drift results from the emergence of both anomalous and conventional
normal Doppler shifts.
Signatures of the Doppler effects are also found in attenuation of of the
acoustic wave, which is characterized by the calculated absorption
coefficient. We also demonstrate a possibility of absolute negative mobility
(ANM) for a SL structure in combined dc bias and acoustic wave drive with
sufficient large amplitude to induce superluminal Doppler effects. Notably,
ANM has already attracted great research interest in connection to spontaneous
generation of large static fields [32] and high-frequency stimulated emission
of photons [33] when the superlattice structure is driven by intense terahertz
electric fields.
Our analysis shows that the discovered phenomena can be exploited practically
in schemes for tunable broadband amplification and generation of GHz-THz
electromagnetic waves. Note that SLs under moderate electric fields have been
already shown as a system to provide optical gain due to the Bloch
oscillations in the presence of weak dissipation [34, 35]. This prediction was
based on semiclassical arguments and coined as Bloch gain, raising the
possibility of inversionless lasing in a dc biased SL. The use of phononic
waves though, opens new opportunities to enhance the performance of
superlattice oscillators. Here we put forward a scheme for the broadband
amplification of THz radiation in acoustically driven superlattice similar to
the Bloch gain in a electrically biased SL.
This paper is organized as follows. In Sec. II, we employ elements of the
bifurcation theory and we consider the conditions for which the superluminal
nature of electron kinematics arises. Therefore, we identify a class of
Cherenkov and Doppler resonances which are directly connected to global
bifurcations developing with an increase of the wave wave amplitude. In Sec.
III, we focus on the stimulated emission of the SL-phonons for choices of the
wave amplitude between the analytical bifurcations points which is confirmed
by numerical calculations determining the kinetic behavior of a classical
ensemble of particles. In Sec. IV a non-perturbative solution of the Boltzmann
transport equation is followed to examine the main transport characteristics
in SL at the presence of scattering. We discuss the feasibility of ANM, for an
electrically biased SL in Sec. V and the broadband amplification of an
electromagnetic (EM) wave for an acoustically pumped SL in Sec. VI. We
conclude in Sec. VII with a few remarks.

##  II  Superluminal phenomena in the realm of acoustoelectric interactions

This section discusses the nature of the acoustoelectric effects in SLs. The
nonlinear analysis of the acoustoelectric instabilities provides a fruitful
insight for the absorption of the acoustic wave, its scattering by miniband
electrons and the related quantum processes. This would allow us to show that
a wide range of superluminal phenomena have distinct acoustic counterparts.
We consider a longitudinal acoustic wave that propagates in the direction of
the superlattice axis ($x$) [Fig. 1(a)] generating a position and time-
dependent potential energy given by [36, 24]

$V(x,t)=-U\sin[(k_{s}(x+x_{i})-\omega_{s}t)],$ (1)

where $U=D\epsilon$ is defined by the deformation potential constant $D$ and
the strain magnitude $\epsilon$. The displacement $x_{i}$ defines the initial
phase of the driving wave, $\omega_{s}$ is the wave frequency, $v_{s}$ is the
speed of sound in the materials of SL, $k_{s}=\omega_{s}/v_{s}$ is the wave
number, and $t$ is time. In the simplest tight-binding scheme, it sufficient
to describe the dispersion of the first SL miniband as [37]

$\mathcal{E}(p_{x})=\dfrac{\Delta}{2}\left[1-\cos\left(\dfrac{p_{x}d}{\hbar}\right)\right],$
(2)

where $\mathcal{E}$ is the energy of an electron with quasi-momentum $p_{x}$,
$\Delta$ is the miniband width, and $d$ is the SL period. In general, the
dispersion relation should also include a contribution of the lateral (along
the superlattice layers) motion in the form
$\mathcal{E}_{\parallel}(p_{y},p_{z})=p^{2}_{y}/2m^{*}+p^{2}_{z}/2m^{*}$,
where $m^{*}$ is the effective electron mass. However, without account of
anisotropic scattering the electron motion in lateral directions is
independent from electron tunnelling along the superlattice axis, and
therefore the most interesting physical effects can be described by the
semiclassical Hamiltonian

$H(x,p_{x})=\mathcal{E}(p_{x})+V(x,t).$ (3)

Figure 2:  The modulus of electron velocity $|v(p_{x})|$ [Eq. (4a)]
represented by dashed lines corresponding to subsonic charge carriers and
solid curves to supersonic electrons that belong to the active zone and
therefore they can emit or absorb phonons while making transitions within the
miniband. The green horizontal line connects the hyperbolic points $p_{s_{i}}$
[Eq. (7)], with $i=1,2,3$, denoting the condition for absorption of the
acoustic wave for small $U$ [see Eq. (8)]. The arrows ($n=1$) and ($n=2$)
indicate which hyperbolic points are connected in order to satisfy the
selection rules described by Eq. (13).

It is easy to see from this Hamiltonian, the electron velocity,
$\partial\mathcal{E}/\partial p_{x}$ and the force acting on the electrons,
$-\partial H/\partial x$, due to the propagating deformation potential (1),
which read respectively

$\displaystyle v$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{E}}{\partial
p_{x}}=v_{0}\sin\left(\frac{p_{x}d}{\hbar}\right),$ (4a)
$\displaystyle\frac{dp_{x}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\partial H}{\partial
x}=k_{s}U\cos[(k_{s}(x+x_{i})-\omega_{s}t)],$ (4b)

with $v_{0}=\Delta d/(2\hbar)$ being the maximal miniband velocity. It has
been demonstrated that this simple model well describes the experiments on
charge transport in SL driven by a train of picosecond strain pulses [12, 15].
We consider mainly two different sets of parameters which are summarized in
Table 1 corresponding to
$\textrm{GaAs}/\textrm{Al}_{x}\textrm{Ga}_{1-x}\textrm{As}$ SLs used in recent
experiments [38, 16]. Our model (4) though has comfortably allowed us to
calculate similar results for a wide range of SL parameters.

Table 1: Basic parameters of superlattices studied. The frequency of sound is assumed $\omega_{s}=4\times 10^{11}$ rad/s with $\omega_{s}\sim 0.1/\tau$ and $k_{s}\sim 1/d$, unless stated otherwise. Further details are given in appendix E.  $\Delta$ (meV) |  $d$ (nm) |  $\tau$ (fs) |  $v_{s}$ (m/s) |  $D$ (eV) |
---|---|---|---|---|---
miniband width | lattice period | scattering time | velocity | Deformation |
|  |  | of sound | potential |
7 | 12.5 | 250 | 5000 | 10 |
20 | 11.4 |

Efficient interaction of electrons with phonons.— We turn now to the condition
for efficient interaction of the band electrons with acoustic phonons that was
initially introduced in the theory of ultrasound absorption in metals: In the
case the mean free path of electrons is large, the absorption of the sound
wave can be considered as scattering of an aggregate of phonons by those
electrons whose velocity in the direction of the acoustical wave vector is
equal to the phase velocity of sound [39, 40]. Indeed, for $v>v_{s}$ the
conservation laws should dictate that for the absorption of a phonon

$\displaystyle p_{f}$ $\displaystyle=$ $\displaystyle p_{i}+\hbar q,$ (5a)
$\displaystyle\mathcal{E}(p_{f})$ $\displaystyle=$
$\displaystyle\mathcal{E}(p_{i})+\hbar\omega_{q},$ (5b)

where $p_{f}$ ($p_{i}$) stands for the final (initial) momentum of the
electron, and $\hbar q>0$ is the quasimomentum of the absorbed phonon. For
generality we assume here that the phonon frequency $\omega_{q}$ can be
different from the frequency of the propagating deformation potential (1).
Combining Eqs. (5a), (5b) we obtain $\mathcal{E}(p_{i}+\hbar
q)=\mathcal{E}(p_{i})+\hbar\omega_{q}$. Since the momentum exchanged between
the low-frequency phonons and band electrons is very small, $\hbar q\ll
p_{i}$, the energy then $\mathcal{E}(p_{i}+\hbar q)$ can be expanded into
Taylor series as

$\mathcal{E}(p_{f})\approx\mathcal{E}(p_{i})+\hbar v(p_{f})q.$ (6)

By comparing Eq. (6) and Eq. (5b) we have

$v\approx v_{s}=\omega_{q}/q,$ (7)

which corresponds to Cherenkov effect [39, 41, 42]. It can be shown that the
same condition (7) needs to be satisfied in order an electron emits a phonon
with a small quasi-momentum $\hbar q$. In this derivation it was implicitly
assumed that influence of the electron potential energy in scattering events
is negligible, and also that the electrons oscillate far from edges of the
first Brillouin zone. Now we are going to show that the resonant condition (7)
naturally arises in the analysis of the fixed points of the dynamical system
(3).
Fixed points and their physical meaning.— We first make a canonic
transformation to the moving with the velocity $v_{s}$ reference frame, for
which new Hamiltonian, electron kinetic energy and coordinate take the forms

$H^{\prime}=\mathcal{E}^{\prime}(p_{x})+V(x^{\prime}),\quad\mathcal{E}^{\prime}(p_{x})=\mathcal{E}(p_{x})-v_{s}p_{x},$
$x^{\prime}(t)=x(t)+x_{0}-v_{s}t,$

where the expressions for $V(x)$ and $\mathcal{E}(p_{x})$ are still given by
Eqs. (1) and (2). While the Hamiltonian $H^{\prime}$ becomes time-independent,
the electron momentum $p_{x}$ is unchanged under the canonic transformation.
In Appendix A, we showed that the fixed points of this autonomous dynamic
system are

$\displaystyle x^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\pi}{2k_{s}}+\frac{m\pi}{k_{s}},$ (8a) $\displaystyle
p_{x}$ $\displaystyle=$
$\displaystyle(-1)^{l}\frac{\hbar}{d}\sin^{-1}\left(\frac{v_{s}}{v_{0}}\right)+l\frac{\hbar\pi}{d},$
(8b)

where $m$ and $l$ are arbitrary integer numbers.
To understand physical meaning of the fixed points we first consider the case
of small $U$ when $p_{x}(t)$ oscillations are well confined within the first
Brillouin zone $|p_{x}d/\hbar|<\pi$, evoking thereby only the conventional
acoustoelectric response. Importantly, Eq. (8b) for the $p-$components
originates from the condition $v(p_{x})=v_{s}$ [cf. Eq. (31a)], which is
identical to Eq. (7). Therefore, Cherenkov absorption or emission of phonons
can arise in the proximity of the stationary points. Figure 2 demonstrates
schematically the modulus of electron velocity in the momentum subspace and
the positions therein of the hyperbolic points (red circles):
$p_{s1}=(\hbar/d)\sin^{-1}(v_{s}/v_{0})$, $p_{s2}=\pi
d/\hbar-(\hbar/d)\sin^{-1}(v_{s}/v_{0})$ and
$p_{s3}=-\pi\hbar/d-(\hbar/d)\sin^{-1}(v_{s}/v_{0})$ which are connected by a
horizontal line corresponding to the condition [see Eq. (7)] for the direct
absorption of phonons by electrons. More detailed consideration shows that
both forward and backward Cherenkov [43, 42] effects can exist for supersonic
miniband electrons. For extended discussion see Appendix A.
The back action of such Cherenkov phonons on the miniband electrons is
definitely small. In what follows, we will discuss an opposite situation of
significant changes in the transport of the miniband electrons caused by their
interactions with such phonons that are able to carry large momenta and
therefore able to exert energy transitions both in the miniband (2) and in the
potential wells formed my the acoustic wave (1). Of course, this will require
relatively large amplitudes of the wave $U\gtrsim\Delta/2$, when effects of
the electron potential energy cannot be ignored any more.
The emergence of Doppler effects.— Next, we will reveal superluminal
mechanisms and the related instabilities (bifurcations) that are developed for
larger $U$. A series of global bifurcations with the increase of $U$ have been
analytically found previously for the model described by Eq. 4 [25]. It has
been shown that they are associated with the reconnection of the hyperbolic
points [see Eq. 8] by separatrices in the phase space of the dynamical system
(30) at the following critical values of $U$:

$\displaystyle U_{cr_{n}}$ $\displaystyle=$
$\displaystyle\frac{\Delta}{2}\sqrt{1-\left(\frac{v_{s}}{v_{0}}\right)^{2}}+\frac{\hbar
v_{s}q_{n}}{2}\quad(n\geq 1),$ (9a) $\displaystyle q_{n}$ $\displaystyle=$
$\displaystyle\frac{2}{d}\sin^{-1}\left(\frac{v_{s}}{v_{0}}\right)+\left(2n-3\right)\frac{\pi}{d}.$
(9b)

where $n$ is an integer designating the order of the bifurcation. From
physical point of view, Eqs. (9) imply multi-phonon resonances in a transition
scattering of the sound wave by electrons [44] placed in the SL periodic
potential. Specifically, the second term in Eq. (9a) defines an acoustic
phonon quantum $\hbar\omega(q_{n})=\hbar v_{s}q_{n}$ with the effective wave
number $q_{n}$ (9b), while the first term in Eq. (9a) is proportional to the
width of the active zone in energy band, which will be discussed later. The
emission and absorption of phonons with wavenumber $q_{n}$ [Eq. (9b)],
however, at critical (bifurcation) points result in both normal ( $n=1$) and
Umklapp ($n\geq 2$) scattering transitions of electrons in momentum space. As
can be seen in Fig. 2, the two hyperbolic points ($p_{s_{1}}$, $p_{s_{2}}$)
which are involved in the first bifurcation (Fig. 13) with critical value

$U_{cr_{1}}=\frac{\frac{\Delta}{2}[\cos(p_{s_{1}}d/\hbar)-\cos(p_{s_{2}}d/\hbar)]+v_{s}p_{s_{1}}-v_{s}p_{s_{2}}}{\sin(k_{s}x_{s_{2}})-\sin(k_{s}x_{s_{1}})},$
(10)

lie within the first BZ resulting in an electron normal scattering transition
in $p_{x}$–space. In contrast, an electron Umklapp scattering stems from the
second bifurcation with critical value

$U_{cr_{2}}=\frac{\frac{\Delta}{2}[\cos(p_{s_{2}}d/\hbar)-\cos(p_{s_{3}}d/\hbar)]+v_{s}p_{s_{2}}-v_{s}p_{s_{3}}}{\sin(k_{s}x_{s_{3}})-\sin(k_{s}x_{s_{2}})},$
(11)

in which the separatrix is reconnected (see Fig. 14) through ($p_{s_{1}}$,
$p_{s_{3}}$) with $p_{s_{3}}$ being located outside the first BZ. Note that
Eqs. (9) for $n$=1, 2 can be derived from Eq. (10) and (11), respectively, by
just rearranging their terms.
The active zone region.—The stationary points of hyperbolic type ($x_{s}$,
$p_{s}$) define intervals of supersonic electron motion in the momentum
subspace $p_{x}$. Within these intervals the momentary electron velocity
$v(p_{x})$ can exceed the sound velocity $v_{s}$. In detail, the pairs of
resonant points ($p_{s1},p_{s_{2}}$) and ($p_{s1},p_{s_{3}}$) determine
intervals within the first Brillouin zone, $0<p_{x}<\pi\hbar/d$ and
$-\pi\hbar/d<p_{x}<0$ respectively, in which the supersonic motion (with the
velocity in Fig. 2 represented by black solid lines) takes place. It follows
that $v(p_{x})>v_{s}$ holds for an energy interval that is centered at the
middle of the energy band $\mathcal{E}=\Delta/2$, and has the width

$\delta\mathcal{E}_{v>v_{s}}=\Delta\left[1-\left(\frac{v_{s}}{v_{0}}\right)^{2}\right]^{1/2}.$
(12)

Supersonic electrons belonging to the “active zone” defined by (12) can emit
phonons while following the phase trajectories defined by Eq. (33). When the
maximal miniband velocity $v_{0}$ is approaching the speed of sound $v_{s}$,
the active zone becomes narrow $\delta\mathcal{E}_{v>v_{s}}\rightarrow 0$.
Hypersonic limit.—We now consider the implications of the opposite limit
$v_{0}/v_{s}\gg 1$, which is well satisfied for typical semiconductor SLs [see
Table 2] and where the superluminal physics become more transparent. In this
case, the active zone practically coincides with the entire miniband,
$\delta\mathcal{E}_{v>v_{s}}\rightarrow\Delta$ and therefore practically
eliminating the subsonic regions (black dashed lines) between blue and red
circles in Fig. 2; see discussion in appendix E. Physically, it guarantees
that the miniband electron speed $v=v_{0}|\sin(p_{x}d/\hbar)|$ exceeds the
speed of sound $v_{s}$ almost for any momentum $p_{x}$. Additionally, the
hyperbolic points (8b), involved in the derivation of Eq. (9), are located at
$p_{s}\approx 0$ and very close to the boundaries of the first and successive
BZs. In what follows, we will mainly focus on the first two major
bifurcations, for which

$U_{cr_{1,2}}=\frac{\Delta}{2}\mp\frac{\hbar\omega_{q}}{2},\hskip
2.84544pt\omega_{q}=\pi v_{s}/d,$ (13)

since their effects play a fundamental role, as we will see, in the electron
kinetics. It is easy to see that the relative contribution of the terms
$\Delta/2$ and $\hbar\omega_{q}/2$ to the value $U_{cr}$ in equation (13) is
defined by the ratio $v_{0}/v_{s}$. Therefore, in the physically interesting
limit $v_{0}/v_{s}\gg 1$ the critical wave amplitude $U_{cr}$ asymptotically
approaches the value $U_{0}=\Delta/2$, i.e. the half of the miniband width.
The limit $v_{0}/v_{s}\rightarrow\infty$ itself can be reached either by
increasing $\Delta$, or by slowing down the speed of sound $v_{s}\rightarrow
0$. The later demonstrates that the appearance of the quantum in criterion
(13) is directly related to the propagation effects.

Figure 3: Diagrams of electron transitions within the potential of the
acoustic wave (left panel) and within the energy miniband (right panel)
manifested by anomalous and normal Doppler effects corresponding to the
bifurcations in increasing order. (a) The electron trapped by the wave $V(x)$
can either absorb the phonon $\hbar\omega_{q}$ and make a transition from the
top of the potential $V_{min}=U$ to its bottom $V_{max}=-U$ or follow the
reverse process by emitting a quantum. (b) The electron recoils, while
emitting a phonon, leading to a normal Doppler shift of the transition (green
arrow). Alternatively, the electron experiences an excitation (red arrow)
assisted by absorption of $\hbar\omega_{q}$. (c) One-step multiphonon
transitions involving odd number of normal Doppler shifts.The green lines
describe emission processes where the red lines absorption ones.

To get a deeper insight into the physical meaning of Eq. (13), consider an
electron that absorbs or emits a phonon with the quasi-momentum $\hbar q=\hbar
2\pi/\lambda_{q}=h/2d$. As a result of the radiation/absorption act, the
electron momentum becomes $p_{f}=p_{i}+\hbar q$, where $p_{f}$ ($p_{i}$)
stands for the final (initial) momentum and $q$ is positive or negative
depended on whether we have absorption or emission processes. Next, a
variation of the electron kinetic energy in the moving reference frame of the
acoustic wave is

$\delta\mathcal{E}^{\prime}=\delta\mathcal{E}-v_{s}\delta p,$ (14)

where $\delta\mathcal{E}=\mathcal{E}(p_{f})-\mathcal{E}(p_{i})$ and $\delta
p=p_{f}-p_{i}$. Assuming the electron is initially at the center of Brillouin
zone $p_{i}=0$, it is easy to find both the variation of the electron kinetic
energy $\delta\mathcal{E}=\Delta$ and the variation of its momentum $\delta
p=\hbar\omega_{q}/v_{s}$. Finally, equating $\delta\mathcal{E}^{\prime}$ to
the corresponding variation of the electron potential energy $\delta
V=V_{f}-V_{i}$, we get the condition of energy conservation in the moving
frame as

$\delta V=-\Delta\pm\hbar\omega_{q}.$ (15)

Now consider an electron transition from the top of the potential
$V(x^{\prime})$ at $x^{\prime}_{s}=-\pi/(2k_{s})$ to its bottom at
$x^{\prime}_{s}=\pi/(2k_{s})$, for which $\delta V=-U-(+U)=-2U$. Substituting
this $\delta V$ in (15) we obtain equation (13). Therefore, the values of the
critical amplitudes $U_{cr_{1,2}}$ follow from a delicate energy balance in
distribution of the acoustic wave energy ($2U$) between an excitation of the
electron within the miniband
($\mathcal{E}_{0}=0\rightarrow\mathcal{E}_{f}=\Delta$) and absorption or
emission of the quantum $\hbar\omega_{q}$. Since we have assumed that
$p_{i}=0$ (lattice temperature close to zero), the first bifurcation is
related to an absorption process in order to satisfy simultaneously Eqs.
(13)–(15). On the other hand, the same initial condition results in emission
of a phonon after the second bifurcation. While the energy of the quantum is
relatively small $\hbar\omega_{q}/\Delta\ll 1$, it brings a large momentum
inversely proportional to the lattice period $d$ [see Eq. (9b) for $n=1$].
Thereby this inelastic scattering event is able to kick the electron from the
bottom of the miniband directly to its upper edge, giving rise to the the
electron Bragg reflections. The condition of energy conservation (15) in the
moving frame can be rewritten in the alternative form

$-\hbar\omega_{0}=\hbar(-q|v_{e}|\pm\omega_{q}),$ (16)

where $|v_{e}|=2v_{0}/\pi$ is the effective electron speed and
$\omega_{0}=|\delta V|/\hbar$ is the resonant frequency which corresponds to
the source excitation energy $\delta V$.
Remarkably, equation (16) describes both the normal and anomalous Doppler
effects, depending on the sign in from of $\omega_{q}$. Namely, entrance of
$+\omega_{q}$ corresponds to anomalous Doppler effect, while $-\omega_{q}$
implies a normal Doppler effect.
In our system the first resonance (at $U=$ $U_{cr_{1}}$) is associated with
the anomalous Doppler effect. The anomalous character of the process is
reflected in the fact that despite an electron absorbs the phonon
$\hbar\omega_{q}$ (red wavy arrow), it makes transition from the top of the
potential to its bottom (red vertical arrow) as illustrated in the left panel
of Fig. 3(a). Thus, the absorption of a quantum happens at the expense of
potential energy, whereas the electron is excited at the upper edge of the
miniband as indicated by the red arrow in the right panel of Fig. 3(a). Note
that the SL parameters considered in this work can also operate in the regime
where new phenomena such as the superlight inverse Doppler effect [20] are
possible and thus $|\delta V|>\hbar\omega_{q}$ whereas the corresponding
Doppler frequency shift has $\Delta\omega=\omega_{q}-\omega_{0}<$0\.
Regardless of the anomalous radiation act, just after the first bifurcation
the miniband electron (black circle with the right-handed arrow) continues to
move in the positive direction with effective particle velocity, $v_{e}$. For
detailed discussion of electron’s kinetic behavior see the following
subsection and Sec. IV. In similar way one could also analyze the radiating
processes far from the thermal equilibrium (at high temperature limit) by
considering that the electron could initially sit at the edge of Brillouin
zone ($p_{i}$=$\pi\hbar/d$) and subsequently inelastically scattered to center
of the first Brillouin zone at $p_{i}$=0. This process is depicted by the
green vertical arrows of Fig. 3(a). On these terms, the anomalous character of
the process would lie in electron emitting the phonon $\hbar\omega_{q}$ (green
wavy arrow) and making a transition from the bottom of the potential to its
top. Here, the emission of a quantum would happen instead at the expense of
the kinetic energy. In this sense our model can accommodate a further
generalization of the Ginsburg-Frank-Tamm theory [29].
After the second bifurcation (at $U=$ $U_{cr_{2}}$), which corresponds to a
normal Doppler effect, the realization of electronic transition from the top
to the bottom of the potential [green arrow, left panel Fig. 3(b)] is
conventionally accompanied by an emission of a phonon. That, however, makes
the electron to be excited [green arrow, right panel Fig. 3(b)] within the
miniband. Notably, such a transition can be associated with the backward
Cherenkov radiation [23, 43], where the electron (black circle with left-
handed arrow) moves in the opposite direction with respect to the emitted
phonon and the associated Doppler-type frequency shift has $\Delta\omega<0$.
On the contrary, for an arbitrary system such as Larmor oscillator [45], the
conventional normal Doppler effect holds $\omega_{0}=\omega_{q}-q|v_{e}|$ with
$\Delta\omega>0$. The appearance of the Cherenkov effect in periodic photonic
structures has demonstrated inherit connections between electron-photon
interactions manifested by the Smith-Purcell effect and the Cherenkov effect
[46]. Similarly, here is the SL which can a play a role of an effective
diffraction grating for Smith-Purcell-type phonons. We can identify that
resonances (13) describe the SP mechanism where the emission spectrum is
determined only by the period of the SL sample ($\omega_{q}\propto 1/d$) which
constitutes an effective diffraction grating. The emitted radiation at wave
number $q$ (cf. Eq. (2) in [31]) taken in the limits $\beta=v/v_{s}\gg 1$ is
reduced to a form containing the characteristic Doppler factor so that the
angle of emission is either $\theta=0$ or $\theta=\pi$. Finally, it is
possible to examine the phonon absorption (emission) processes for higher
order bifurcations at $U_{cr_{n}}$, $n\geq 3$, see Eq. (9a). For example, a
three-phonon process can occur if the electron is inelastically scattered by a
strong sound wave (at $U=U_{cr_{3}}$) which allows to perform a radiating
transition from the bottom to the top of the miniband [see the right panel of
Fig. 3(c)]. This higher-order process involves stimulated events which are
illustrated by the vertical arrows in the left panel of Fig. 3(c) and they are
facilitated on the boundaries of the first and the third Brillouin zone.Thus,
we obtain an realization of Umklapp processes which result in the inelastic
scattering of electron back into the regime of normal Doppler effect.

##  III  Electron bunches and induced Doppler effects

Figure 4:  The velocity $v_{m}$ of the electron bunches as a function of the
wave amplitude $U$ in the quasi-ballistic regime for $v_{e}/v_{s}$=22 (blue
curve) and $v_{e}/v_{s}$=10.5 (yellow curve); See Table 2. The dashed line at
$U_{0}=\Delta/2$ separates the forward motion of the electron bunches from the
backward one with respect to the propagation of acoustic wave for
$v_{e}/v_{s}\rightarrow\infty$. The comparison with the superluminal selection
rules [Eq. (18)] is indicated by the arrows. The inset shows the full
development of $v_{m}$ with an increase of $U$ around the first selection rule
(anomalous Doppler condition). The acoustic wave in both cases has a frequency
lying at $\omega_{s}=v_{s}/d$ with $d$ the lattice period of each SL
structure.

In this Section we discuss the role of the induced Doppler effects in
nonlinear mechanisms, similar to laser-plasma interactions [47, 28] and
relativistic microwave devices [48], leading to formation of traveling
electron bunches. In the presence of a small-amplitude acoustic wave some of
the electrons are captured by the propagating potential, forming a group of
trapped electrons (electron bunch) and perform oscillatory motion within
potential itself. However, if the acoustic stimuli are strong enough they can
enforce the electrons to perform complex Bloch oscillations (phonon-assisted
trajectories) [24, 25], which, as we found out, causes the electron bunches to
oscillate with large amplitude in $p_{x}$–space. To understand the role of
these effects in the acoustically driven charge transport in SL we analyze the
solution of the Boltzmann transport equation (BTE)

$\dfrac{\partial f}{\partial t}+v(p_{x})\frac{\partial f}{\partial
x}+eE(x,t)\dfrac{\partial f}{\partial p}=\textrm{St}[f],$ (17)

where $\textrm{St}[f]$ is the collision term specified later, $f(x,p_{x},t)$
is the electron distribution function, $E(x,t)=E_{s}\cos(k_{s}x-\omega_{s}t)$
is a force field derived from the potential energy function $V(x,t)$ with
$E_{s}=k_{s}U/e$ denoting the amplitude of the effective acoustoelectric
field.

We first consider the quasi-ballistic regime $a=k_{s}l\gg 1$, where
$l=v_{0}\tau$. Here $l$ denotes the mean free path of the carriers being
sufficiently larger than the sound wavelength and $\tau$ is the electron
scattering time. In this case, the collision term in (17) can be neglected as
$\textrm{St}[f]\rightarrow 0$. In addition, the value of the sound frequency
considered in our calculations is, $\omega_{s}=4\times 10^{11}$ rad/s with
$k_{s}\sim 1/d$, which lies far away from the frequency ranges of the phonon
stop bands [49, 50]. To characterize the kinetic properties the electrons
bunches, we introduce the average momentum $p_{a}=p_{\phi}\hbar/d$, where
$p_{\phi}=\textrm{arg}(m_{1})$ is a circular mean angle. Here
$m_{1}=\langle\textrm{exp}(ip_{x}(t)d/\hbar)\rangle_{x_{i}}$ is the first
trigonometric moment $m_{1}$ of the distribution function [51, 52]. The
operator $\langle.\rangle_{x_{i}}$ performs averaging over an ensemble of
time-dependent electron trajectories starting with the same initial momentum
$p_{i}=0$ (low temperature limit) and different initial positions
$x_{i}\in[-\lambda/2,\lambda/2)$, where $\lambda=2\pi/k_{s}$ means the
acoustic wave length.
Figure 4 presents the dependencies of the time-averaged electron velocity
$v_{m}=\langle v_{0}\sin(p_{a}(t)d/\hbar)\rangle_{\Delta t}$ on the wave
amplitude $U$ which are numerically calculated (see Appendix B for more
details) for two different ratios $v_{e}/v_{s}$.  Both graphs demonstrate
series of local minima of $v_{m}$, which are reached at the critical values of
$U=U_{cr_{n}}$. By rearranging terms, Eq. (13) can be rewritten in the form

$U_{cr_{n}}=\frac{\Delta}{2}\left[1+(2n-3)\frac{v_{s}}{v_{e}}\right],$ (18)

which, as it was mentioned above, in the limit $v_{e}/v_{s}\rightarrow\infty$
reveals a localization condition

$U_{cr_{n}}\longrightarrow U_{0}=\frac{\Delta}{2}.$ (19)

These values are highlighted by arrows with the numbers indicating the index
$n$ in (18), while the dashed line at $U_{0}=\Delta/2$ separates the forward
motion of the electron bunches from the backward one with respect to the
propagation of acoustic wave for $v_{e}/v_{s}\rightarrow\infty$. The graphs
evidences that for larger $v_{e}/v_{s}$, the $U$-value corresponding to
reversing of the electron velocity $v_{e}$ comes closer to $U_{0}$, thus
confirming the localization criterion (19).

Figure 5:  The temporal dynamics of the reduced electron distribution function
$\tilde{f}(p_{x},t)$ before the first bifurcation (a) $2U/\Delta=0.95$ and
just after (b) $2U/\Delta=0.98$ with the red region in the colormap indicating
the formation of the electron bunch. (c) $p_{x}$-space trajectories for
$2U/\Delta=0.98$ ($U_{cr_{1}}<U<U_{0}$) and different initial conditions:
$x_{0}=0$, trapped trajectory (dashed line) in the acoustic wave potential and
$x_{0}\approx-\pi/(2k_{S})$, trajectory (solid line) which experiences
anomalous Doppler instabilities. The wavy arrows indicate the absorption (red)
and the emission (green) of phonons. (d) The temporal oscillations of miniband
energy (black curve) and potential energy (blue curve) for $2U/\Delta=0.98$
($U_{cr_{1}}<U<U_{0}$) and the initial condition $x_{0}\approx-\pi/(2k_{S})$.
(e), (f) The temporal dynamics of the reduced electron distribution function
$\tilde{f}(p_{x},t)$ indicating the electron bunch (red) for different wave
amplitudes: (c) $2U/\Delta=1$ ($U=U_{0}$) and (d) $2U/\Delta=$1.04
($U_{cr_{1}}<U<U_{cr_{2}}$). The white lines designate the mean of electron
quasi-momentum distribution (mean angle $p_{\phi}=p_{a}d/\hbar$) as a function
of time. The calculations have been performed using the parameters of the SL
structure with $v_{e}/v_{s}=22$.  Figure 6: The numerically obtained
oscillations of (a) the averaged momentum $p_{a}$, (b) the coherence, $A$, of
the electron distribution, (c) the averaged electron energy $E_{a}$ and (d)
the averaged potential energy $V_{a}$ for $2U/\Delta=0.95$ ($U<U_{cr_{1}}$,
blue curves), $2U/\Delta=$0.96, 0.98 ($U_{cr_{1}}<U<\Delta/2$, black and red
curves). The calculations have been performed using the parameters of the SL
structure with $v_{e}/v_{s}=22$. Dashed vertical line is discussed in text.

The Doppler effects as described in the previous section are elementary
processes whereas here the induced Doppler effects correspond to collective
processes triggered by an instability in which an elementary radiation act can
induce sequentially another one. This becomes more evident, as we shall see,
by the behavior of the electron dynamics between the bifurcations. Figure 5(a)
shows the temporal dynamics of the electron distribution function by solving
Eq. (17) for $U<U_{cr_{1}}$, while 5(b) for the case $U_{cr_{1}}<U<U_{0}$. The
color in (a), (b) relates to the values of reduced electron distribution
function
$\tilde{f}(p_{x},t)=1/\lambda\int_{-\lambda/2}^{\lambda/2}f(x,p_{x},t)dx$,
where it changes from black for the lowest value to red – for the highest one
which indicates the presence of an enhanced concentration of electron
trajectories (electron bunches) around the center of Brillouin zone. As $U$
increases beyond $U_{cr_{1}}$, the electron dynamics become more complicated
which is manifested in appearance of quite sophisticated patterns in
$\tilde{f}(p_{x},t)$, compare Fig. 5(a) for the case of $U<U_{cr_{1}}$ and
Fig. 5(b) for the case of $U>U_{cr_{1}}$. While the former figure demonstrate
rather regular pattern, the latter shows signatures of turbulent behavior,
which evidences e.g in occurring isolated color spots on the map. Such
dramatic change in electron distribution can be attributed to emergence of a
specific phonon-assisted trajectories (complex Bloch oscillation) which is
shown as a solid curve in Fig. 5(c), recounting a wave packet moving slowly in
the proximity of $p_{x}=0$ and quickly at $p_{x}=\hbar\pi/d$. This type of
trajectories appear for $U>U_{cr_{1}}$ in addition to another low-amplitude
regular $p_{x}$-trajectory [dashed curve, Fig. 5(c)] which existed also for
$U<U_{cr_{1}}$. The latter one (dashed curve) indicates a wave packet moving
quickly around the center of the Brillouin zone, corresponding to the confined
electron motion within the propagating moving potential. These two types of
trajectories differ by their initial values $x_{i}$. The motion of the
electrons starting from certain initial conditions such as the solid
trajectory in Fig. 5(c) experiences Doppler instability accompanied either by
phonon absorption (red wavy arrow) or emission (green wavy arrow) which give
rise to oscillations of electron bunches with larger amplitude. Figure 5(d)
illustrates the balance between kinetic $\mathcal{E}$ and potential energy $V$
for phonon absorption or emission processes. In an absorption process, a wave
packet with a small crystal momentum $p_{x}\sim 0$, gains a considerable
portion of energy while moving [red dashed arrow, Fig. 5(d)] from the bottom
to the top of the miniband at the expense of potential energy [red curve, Fig.
5(d)]. In contrast, during the emission process, a wave packet with a large
$p_{x}$ (near $p_{x}=\pi\hbar/d$) loses a considerable part of its energy
[green dashed arrow, Fig. 5(d)] while a making a transition from the top to
the bottom of the potential (green arrow, Fig. 5(d)). Each radiation act is
preceded (followed) by extended presence of electron around the center of the
Brillouin zone. Considering now an electron ensemble, the electron wave
packets are concentrated around a single trajectory [white curve in Fig. 5(b)]
with mean value $p_{a}$ that oscillates further close to the center of BZ in
comparison to the trajectory in Fig. 5(a), causing the electron bunches to
slow down; See the sudden drop of $v_{m}$ in the inset of Fig. 4. At $U=U_{0}$
the electrons eventually become localized. In this case, the averaged momentum
$p_{a}$ [white curve in Fig. 5(e)] oscillates almost periodically around
$p_{x}$=0 resulting in a zero $v_{m}$. For $U>U_{0}$ the electronic bunches
counter-propagate with respect to the propagating sound wave. Such inversion
of motion becomes apparent in Fig. 5(f), where the electron bunch is shifted
below $p_{x}=0$. The inversion of electrons drift can be explained by an
increase of electron trajectories which are subjected to anomalous Doppler
shifts. The backward drift of electrons becomes maximal at $U_{cr_{2}}$,
coinciding with the first manifestation of normal Doppler effect as discussed
in Sec. II.
To gain a deeper insight in the emission processes and their implications in
electron bunching, we examine in detail the dynamics of averaged parameters in
the vicinity of the first bifurcation at $U\approx U_{cr_{1}}$. Namely, we
analyze the characteristics, which beside the averaged momentum $p_{a}$ (Fig.
5) include the coherence of electron distribution $A=|m_{1}|$, the averaged
electron energy, $E_{a}=\Delta/2\langle
1-\cos\\{p_{x}(t)d/\hbar\\}\rangle_{x_{i}}$, and the averaged potential energy
$V_{a}=-U\langle\sin(k_{s}x-\omega_{s}t)\rangle_{x_{i}}$. Note that the
electron distribution converges to a Dirac distribution centered on a single
trajectory at mean value $p_{a}$ for $A=0$ whereas for the opposite limit
$A=1$ there is no well defined mean of the electron momentum. Figure 6(a)
shows the dynamics of $p_{a}$ in time as $U$ growth. For $U<U_{cr_{1}}$= 9.55
meV ($2U/\Delta=0.955$), $p_{a}$ demonstrates a low-amplitude erratic
fluctuations (blue curve), which are substituted by more regular close to
quasi-periodic oscillations once $U$ exceeds $U_{cr_{1}}$ (red curve). Further
growth of $U$ leads to increase of the amplitude of the oscillations that
remains quite regular (black curve). Such regularization of the electron bunch
dynamics and appearance of pronounced large-amplitude oscillations of $p_{a}$
is associated with the Doppler instability evoked by he absorption or the
emission of phonons, which, as it has been shown above, take place for
$U>U_{cr_{1}}$. Repeating inelastic absorption or emission events perturb also
the coherence parameter $A$ which demonstrates larger oscillations for larger
$U$, indicating that electron bunches are created and then deformed almost
periodically (compare blue, red and black curves in Fig. 6(b)). This further
explains the abrupt change in electron distribution and the formation of
distinct color spots with time variation as was demonstrated in Fig. 5 (e),
(f). The effect of the Doppler instability related to phonon exchange can be
clearly seen in the time realizations of the kinetic $E_{a}$ and potential
component $V_{a}$ of the electrons energy, which are presented in Fig. 6 (c)
and (d), respectively. For $U<U_{cr_{1}}$, the energy the exchange between
$E_{a}$ and $V_{a}$ is very small (see blue curves in (c) and (d)). However,
if $U$ even slightly exceeds $U_{cr_{1}}$, the interplay between $E_{a}$ and
$V_{a}$ dramatically intensify, and become larger for larger $U$ (see red and
black curves in (c) and (d)). The vertical dashed lines in (c) and (d)
illustrates the energy transformation. Namely, it indicates a moment when the
electron bunch having a large potential energy $V_{a}$ start to lose it due to
phonon exchange events with simultaneous increasing the kinetic component
$E_{a}$. We note that the concept of a wave-like bunching of the electrons
[53, 54, 55] in momentum space has already been used to explain the
amplification of a THz field which might arise due to the interaction of the
bunches with the THz field itself. Extended analysis of the effects of the
scattering processes on miniband transport and sound absorption will be
presented in the following section.

##  IV  NONLINEAR ELECTRIC TRANSPORT AND THE SOUND ATTENUATION EFFECTS

Figure 7: (a) The dependence of the drift velocity $v_{d}$ as a function of
the acoustic wave amplitude $U$. The left panel demonstrates $v_{d}(U)$ for
the wave frequency $\omega_{s}=0.1\tau^{-1}$ whereas the right panel for
$\omega=0.5\tau^{-1}$ (dashed yellow curve) and $\omega_{s}=\tau^{-1}$ (blue
curve). (b) The dependence of the absorption coefficient of the sound wave
$A_{s}=\Gamma/(2k_{s}U)$ upon $U$ calculated for different frequencies:
$\omega_{s}=0.1\tau^{-1}$ (left panel), $\omega_{s}=0.5\tau^{-1}$ (right
panel, blue curve) and $\omega_{s}=\tau^{-1}$ (right panel, yellow curve). (c)
Zoom in of (b)–right panel, showing the different regimes of phonons
emissions. The vertical lines designate the critical values of $U$. The
calculations have been performed using the parameters of the SL structure with
$v_{0}/v_{s}=13$; See Tables 2, 3.

Up to now, we have considered the quasiballistic regime, i.e.
$\alpha=k_{s}l\gg 1$, to study the role of the acoustoelectric effects in the
quasi-classical description of the electron transport. However, the effects of
coherent phonon dynamics on the miniband electrons and the related
superluminal phenomena might appear over a wide range of the parameter
$\alpha$, as it was previously shown for electron phonon interactions in
metals [56, 40]. A more general approach is to take into account the
scattering effects, when the mean free path, $l=v_{0}\tau$, is comparable with
to the sound wavelength $\lambda$. In this case, to describe the directed
electron transport in a superlattice we use the Boltzmann transport equation
(17) with $\textrm{St}[f]=(f_{0}-f)/\tau$, where $\tau$ is the relaxation time
and $f_{0}(x,p_{x})$ is the Fermi distribution in the nondegenerate limit.
Solution of BTE yields the drift velocity of electrons

$v_{d}=\int_{0}^{T_{s}}\frac{dt}{T_{s}}\int_{-\infty}^{t}e^{\frac{-(t-t_{i})}{\tau}}v_{x}(t,t_{i})\frac{dt_{i}}{\tau}.$
(20)

$t_{i}$ is the time of last collision at position $x_{i}$, $v_{x}(t,t_{i})$ is
a trajectory in time governed by Eqs. (4) and $T_{s}$ $=2\pi/\omega_{s}$ is
the period of the acoustic plane wave. The relaxation rate approximation,
$\tau$, can facilitate both elastic scattering processes [57] and additional
inelastic scattering processes due to long wave phonons. Eq. (20) can be
obtained using the time-dependent path-integral approach [58, 59, 60] which
allows to investigate generic features of electron transport and unravel the
acoustolectirc effects that might arise in the system. To analyze the
deformation propagation in a SL we additionally take into account the back
action of electrons on the phonon wave by considering the sound attenuation
effects. In general, the electron contribution to the attenuation of the
ultrasound in materials arises because energy is transferred between the wave
and electrons. Therefore, the absorption coefficient of the sound is given by
[36, 61, 62]

$\Gamma=\int\left<\dot{H}(p_{x},t)f(p_{x},x,t)\right>_{T_{s}}dp_{x}.$ (21)

Here $\dot{H}=dH/dt$, whereas the angle brackets designate averaging over
time, namely the period of the sound wave $T_{s}=2\pi/\omega_{s}$. By solving
Eq. (21) we find (see Appendix C) the absorption coefficient of the sound in
the SL miniband, which is presented for our convenience in a normalized form

$A_{s}=\left<\left(\tilde{v}(t)-\dfrac{v_{s}}{2v_{0}}\right)\cos(k_{s}x-\omega_{s}t)\right>_{T_{s}},$
(22)

where the absorption coefficient $\Gamma$ is related to $A_{s}$ as
$\Gamma=(2k_{s}U)A_{s}$ and $\tilde{v}(t)=\left<{v}_{x}(t)\right>/v_{0}$ is
the electron velocity averaged over the distribution function, $f$, satisfying
the BTE [Eq. (17)]. The dependence of the drift velocity $v_{d}$ on $U$ for an
acoustic wave with $\omega_{s}=0.1\tau^{-1}$ $(\alpha\sim 1.3)$ and a
superlattice with a miniband width of 7 meV ($v_{0}/v_{s}=13$) is presented in
the left-hand panel of Fig. 7(a). One can recognize two characteristic values
of $U$. First, the drift velocity is drastically suppressed beyond
$U=U_{cr_{1}}$ and subsequently the $v_{d}(U)$ characteristic exhibits an
observable change in slope at $U=U_{cr_{2}}$. The black solid curve in Fig.
7(a) reminds the classical Esaki-Tsu $v_{d}(E_{dc})$ [37], which describes the
response of miniband electrons to an electric field, $E_{dc}$, applied along
the growth direction of the superlattice. However, in our case the specific
changes in $v_{d}$ with variation of $U$ are directly associated with changes
in the electron bunches dynamics, revealing the appearance of the superluminal
effects as discussed in Sec. III. Hence, the critical amplitudes $U_{cr_{1}}$
and $U_{cr_{2}}$ designated by the vertical lines in Fig. 7(a) correspond to
the thresholds for triggering the superluminal anomalous and superluminal
backward Doppler effects that are associated with emission and absorption of
phonons in the presence of scattering. Figure 7(b) demonstrates the behavior
of $A_{s}$ with change of $U$. The increase of $U$ first enhances the
absorption of the acoustic wave, since its its interaction with the electrons
becomes stronger. However, once $U$ reaches $U_{cr_{2}}$ the absorption starts
to drop gradually. Such reduction of $A_{s}$ relates to the rise of normal
Doppler effects resulting from the recoil momentum given to the radiating
system by the emitted phonons. The changes in $v_{d}(U)$ and $A_{s}(U)$
indicating transitions between different superluminal regimes which become
more evident for acoustic driving with higher frequency $\omega_{s}$. In the
right panel of Fig. 7(a) the dependencies $v_{d}(U)$ are shown for
$\omega_{s}=0.5\tau^{-1}$ $(\alpha\sim 6.6)$ by dashed yellow curve and
$\omega_{s}=\tau^{-1}$ $(\alpha\sim 13.2)$ by solid blue curve. Both
dependencies have pronounced features at the same values of $U$. However, in
contrast to the low-frequency case presented in the left panel Fig. 7(a) the
high-frequency dependencies indicate more features also related to
$U_{cr_{n}}$ (9) for $n>2$. Importantly, these high-frequency (short-
wavelength) acoustic excitations can induce the reverse of the electron drift
($v_{d}<0$) due to mechanisms discussed in Section II. The features in the
graph of $v_{d}(U)$ are also reflected in the dependence $A_{s}(U)$ shown in
in the right-hand panel of Fig. 7(b). For example, a kink in $A_{s}(U)$ at
$U_{cr_{1}}$ in Fig. 7(b) coincides with the maximum of $v_{d}(U)$ in Fig.
7(a). A local minimum of $A_{s}$ at $U=U_{cr_{2}}$ corresponds to the reverse
of drift velocity $v_{d}$. Other abrupt changes in $v_{d}$ are observed when
$A_{s}(U)$ attains the minima [see Figure 7(b)] at $U_{cr_{n}}$ for n=3 and 4.
The appearance of these kinks, i.e. local minima in $v_{d}(U)$, $A_{s}(U)$ can
attributed to the transition between different supeluminal regimes of phonons
emission. Figure 7(c) illustrates these regimes considering a zoomed part of
yellow curve $A_{s}(U)$ from the right panel of the Fig. 7(b). They include:
(i) an anomalous Doppler regime ($U_{cr_{1}}<U<U_{cr_{2}}$) indicated by the
blue shaded area in Fig. 7(c) where a significant portion of electron
trajectories experiences anomalous Doppler instabilities which start to move
backwards leading to the suppression of drift velocity and considerable
reduction in the sound absorption coefficient. (ii) Coexistence of anomalous
and normal Doppler effects ($U_{cr_{2}}<U<U_{cr_{3}}$). In this regime, an
increase of $U$ enables more charged particles to experience normal Doppler
instabilities resulting in emission of phonons at the expense of electron
trajectories which are subjected only to anomalous Doppler instabilities.
Hereafter, the overall drift of electron bunches remains negative whereas the
absorption of the sound wave is further reduced. (iii) multiphonon Doppler
processes ($U>U_{cr_{3}}$) in which the electrons are inelastically scattered
by the acoustic wave allowing their trajectories to enter successively higher
Brillouin zones. These more complicated trajectories are responsible for the
further suppression of the absorption coefficient.

##  V  Absolute negative mobility

Figure 8:  Drift velocity-electric field characteristics with variation of the
wave amplitude $U=0.3,0.5,2$ meV and at the critical values of the
instabilities $U_{cr_{1}}=9.55$ meV and $U_{cr_{2}}=10.46$ meV. $E_{c}=2.3$
kV/cm is the critical field for the onset of NDV in the Esaki-Tsu
characteristic. The solid and dashed black lines indicate the $v_{d}(E_{dc})$
characteristics for small wave amplitude at $U=0.3$ meV and $U=0.5$ meV
respectively. The frequency of the acoustic wave is fixed at
$\omega_{s}=0.1\tau^{-1}$ ($\alpha$ $\sim$ 3.5) The calculations have been
performed using the parameters of the SSL structure with $v_{0}/v_{s}=35$; See
Table 2.

In previous Section IV, we studied the superluminal mechanisms governing the
directed electron transport in a strongly coupled SL subjected to a sole
acoustic plane wave. Next, we consider the absolute negative mobility (ANM)
and the superluminal Doppler effects which go hand in hand in case a constant
electric field $E_{dc}$ is additionally applied along the SL. The ANM has been
already reported [33] in DC-biased SLs under high frequency irradiation
$\propto$ $E_{\Omega}\cos(\Omega t)$, shown as inversion of electron current
in the vicinity of $E_{dc}=0$. In our model, the acoustic drive oscillates
both in time and space and thus the semiclassical equations of motion for the
miniband electrons are

$\displaystyle v$ $\displaystyle=$ $\displaystyle
v_{0}\sin\left(\frac{p_{x}d}{\hbar}\right),$ (23a)
$\displaystyle\frac{dp_{x}}{dt}$ $\displaystyle=$ $\displaystyle
eE_{dc}+k_{s}U\cos[(k_{s}(x+x_{i})-\omega_{s}t)],$ (23b)

where $e>0$ is the elementary charge. Figure 8 shows the dependencies of drift
velocity $v_{d}$ as a function of the electric field $E_{dc}$ numerically
calculated for different amplitudes of $U$. For our calculations we chosen the
parameters close to ones from recent experiments [16], $\Delta=20$ meV and
$\omega_{s}=0.1\tau^{-1}$ ($\alpha$ $\sim$ 3.5). For a weak acoustic
excitation (black solid and black dashed curves) $v_{d}(E_{dc})$ is almost
identical to the Esaki-Tsu dependence [37]. Therefore, the supersonic
condition ($v_{d}>v_{s}$) for the Cherenkov emission can exist for $U\ll
U_{cr_{1}}$, however it does not necessary produce ANM or result in the
suppression of the drift velocity. This was confirmed in the recent
experiments [16], which have shown that the current-voltage characteristics
are almost identical with or without illumination of the SL sample by the
femtosecond pulse train resulting the coherent acoustic wave generation due to
stimulated Cherenkov phonon emission. However, as the value of $U$ increases
the dependence become less and less steeper, compare pink curve for $U$= 2
meV, yellow curve $U=U_{cr_{1}}$=9.55 meV and blue curve for
$U=U_{cr_{2}}$=10.46 meV. The mechanisms of such a suppression [12, 15] are
similar to the mechanisms governing the electron transport in a DC-biased SL
under electromagnetic irradiation [33]. They relate to the fact that the
electron tunnelling probability is affected by the energy $\hbar\omega$ of the
excitation quantum of frequency $\omega$, which could be either photon
($\hbar\Omega$) or phonon ($\hbar\omega_{s}$) interacting with the electron.
Here, the drift velocity can be expressed in terms of the phonon-assisted
replicas of the Esaki-Tsu drift velocity [63, 12],

$v_{d}\sim\sum_{n=-\infty}^{\infty}J_{n}^{2}(\beta)v_{d_{0}}(E_{dc}d+n\hbar\omega_{s}/e),$
(24)

where $J_{n}$ is the first-kind Bessel function of order $n$,
$v_{d_{0}}(E_{dc})$ is the dependence of the electron drift velocity on the DC
bias $E_{dc}$ for SL in the absence of photon/phonon excitations,
$\beta=edE_{s}/(\hbar\omega_{s})$ and $E_{s}=$ $k_{s}U/e$ is voltage
equivalent of the magnitude of the acoustic wave driving the SL. For a range
of $\beta$ the term with $J_{0}$ produces a dominant effect, so implying
suppression of the $v_{d}$ with growth of $U$. The validity of Eq. (24) for
the phonon-assisted transport is limited either to small amplitude $\beta$
(black curve, Fig. 8) or to the quasistatic limit $a=k_{s}l\ll 1$ [64]. Thus,
we resort instead to the fully numerical integration of the model (23) and
calculation of drift velocity via BTE solution (20) which indicate that ANM
takes also place for a propagating acoustic excitation. In particular, we
found that for selected values of the SL parameters, ANM is realized when $U$
exceeds $U_{0}=10$ meV. Since $U_{cr_{1}}<U_{0}<U_{cr_{2}}$, it indicates the
gradual onset of the ANM due to the absorption and emission of phonons under
the conditions of the anomalous Doppler effect. The inset of Fig. 8
illustrates a zoomed part of the main panel of Fig. 8 in vicinity of
$E_{dc}=0$ and it reveals a negative drift of the electrons ($v_{d}<0$) for
$E_{dc}=0$ when $U=U_{cr_{2}}$=10.46 meV (blue curve) and positive $v_{d}$
when $U=U_{cr_{1}}$=9.55 meV (yellow curve) or $U=$2 meV (pink curve). It
follows that the condition for near zero mobility of an acoustically driven SL
can be given by

$E_{s}=\alpha E_{c}.$ (25)

Here, we have rewritten localization condition (19) in terms of the amplitude,
$E_{s}$ of the effective acoustoelectric field, whereas $E_{c}=\hbar/(ed\tau)$
is the critical electric field for the onset of the negative differential
velocity (NDV) in Esaki-Tsu characteristic [37]. In contrast, the localization
condition [33, 63] for the photon-assisted transport is $J_{0}(\beta)=0$ with
the first root given at $\beta=E_{\Omega}/(E_{c}\Omega\tau)\sim 2.4$.
Furthermore, the dynamic localization and ANM are not possible for any ac
field amplitude when $\Omega\tau<1$ and therefore the smallest amplitude of an
ac field which can induce ANM is $E_{\Omega}=2.4E_{c}$ at $\Omega=1/\tau$
[65]. In Sec. IV and Appendix E we noticed that Doppler effects start to have
more prominent implications in directed transport once the sound wavelength
becomes comparable with the mean free path of the charge carrier, i.e.
$\alpha/(2\pi)\gtrsim 0.5$. This is further confirmed by Fig. 9 showing that
the increase of sound frequency $\omega_{s}$ and therefore of $\alpha$ results
in the appearance of ANM ($\alpha=3.9$, red curve) similar to Fig. 8 in which
we have considered a slightly smaller $\alpha=3.5$. Hereafter, Eq. (25)
indicates values that are comparable with the minimum amplitude of ac field
which can induce ANM but with considerably lower oscillating frequencies. On
the other hand, for $\alpha$ close to zero, i.e low sound frequencies
$\omega_{s}$ or SL structures with extremely narrow minibands [66] and
therefore smaller $v_{0}$, onset of ANM is not feasible. In this limit, the
acoustic wave driving acts practically like a quasistatic ac signal, whereas
the electron drift can be described well by (24).

Figure 9:  Dependence of the drift velocity $v_{dc}$ on the dc bias $E_{dc}$
for the amplitude of the acoustic wave $U=U_{cr_{2}}=3.92$ meV and for
different sound frequencies : positive slope for $\omega_{s}\tau=0.1$
($a=1.3$, black curve), ANM for $\omega_{s}\tau=0.3$ ($a=3.9$, red curve) and
$\omega_{s}\tau=0.5$ ($a=6.6$, blue curve). $E_{c}=2.1$ kV/cm is the critical
field for the onset of NDV in the Esaki-Tsu characteristic. The calculations
have been performed using the parameters of the SSL structure with
$v_{0}/v_{s}=13$; See Table 2.

##  VI  Broadband amplification of EM waves

In this Section we consider amplification of high-frequency electromagnetic
waves in acoustically driven SLs [see Fig. 1(b)] and the corresponding role of
the Doppler effects, which result in the gain similar to the Bloch gain in
electrically biased SLs [34, 35]. In this study, we consider the model (23),
in which the Eq. (23b) is substituted by

$\frac{dp_{x}}{dt}=eE_{\omega}\cos(\omega
t)+k_{s}U\cos[(k_{s}(x+x_{i})-\omega_{s}t)],$ (26)

with the term $E(t)=E_{\omega}\cos(\omega t)$ designating a weak probe field
of the amplitude $E_{\omega}$ and frequency $\omega$. Note that in practice,
the frequency $\omega$ can be favorably tuned by an external resonant cavity
[67]. The absorption of the probe ac field $E(t)$ is determined by the real
part of the dynamical conductivity [68]

$\sigma_{r}(\omega)=\dfrac{2\langle j(t)\cos(\omega t)\rangle}{E_{\omega}},$
(27)

where $j(t)=eNv_{d}(t)$ is the time-dependent current generated by the SL
driven by the acoustic wave which is calculated using the time-dependent drift
velocity

$v_{d}(t)=\int_{-\infty}^{t}e^{\frac{-(t-t_{i})}{\tau}}v_{x}(t,t_{i})\frac{dt_{i}}{\tau},$
(28)

similar to the semiclassical approach presented in Sec. IV, whereas the
ballistic trajectory $v_{x}(t,t_{i})$ is now governed by Eqs. (4a), (26).
Within this framework, the absorption corresponds to $\sigma_{r}(\omega)>0$
and gain to $\sigma_{r}(\omega)<0$. The Drude conductivity of the SL is
$\sigma_{0}=2j_{p}/E_{c}$ where $j_{p}=eNv_{0}$ is the peak current density,
directly proportionally to electron density $N$. Here, $\sigma_{r}(\omega)$ is
estimated for a superlattice with a miniband widh (20 meV) sufficiently
smaller than the optic phonon energy in GaAs [69] and a pump frequency
$\omega_{s}\tau=0.1$. The magnitude of the acoustic wave’s frequency
corresponds to $\alpha$=3.5. The proper choice of $\alpha$–parameter is
important for the appearance of electron bunches with a negative drift and
ANM, which indicate strong involvement of Doppler effects, and for controlling
interaction between bunches and the probe field, see Appendix E for more
details.
Figure 10 illustrates how the absorption profile $\sigma_{r}(\omega)$ is
affected by the variation of $U$. For small wave amplitude $U$=1.2 meV,
$\sigma_{r}(\omega)$ (green dash-dotted curve) almost follows the free-carrier
absorption (red curve) in the absence of a pump field, demonstrating a power-
like decay as probe frequency $\omega$ increases, i.e.
$\sigma_{r}\varpropto(1+\omega^{2}\tau^{2})^{-1}$. Comparison of the
absorption profiles calculated for $U=U_{cr_{1}}$ (yellow), $U=U_{0}$ (black)
and $U=U_{cr_{2}}$ (blue) shows that as $U$ grows, the low-frequency
absorption gradually decreases, and for $U$ close to $U_{cr_{2}}$ a low-
frequency gain is realized in the system, see blue curve for $U=U_{cr_{2}}$.
This transition originates from the emergence of phonon-assisted Bloch
oscillations which are associated with the anomalous Doppler effect. As we
showed earlier in Section II, at $U\approx U_{cr_{1}}$ the electrons starts to
demonstrate the Doppler instabilities that obey the first selection rule [see
Eq. (13)]. As a result electron bunches are formed, which include the
frequency-modulated Bloch oscillations with maximal frequency
$\omega_{B}^{max}=k_{s}Ud/\hbar$ [25, 24]. Appearance of such trajectories
decreases absorption for low frequencies $\omega$. Indeed, the absorption
profile calculated for $U=U_{cr_{1}}$ (yellow curve) shows significantly lower
low-frequency absorption as compared to one for $U=1.2$ meV (green curve),
with the second maximum of absorption in the vicinity of $\omega_{B}^{max}$,
which for the given parameters corresponds to $\omega\tau\approx 3.3$. Further
increase of $U$ can localize electrons drift, see the inset illustrating the
averaged ballistic electron trajectories calculated for $U=U_{cr_{1}}$
(yellow) and for $U_{0}$ (black). This localization leads to further decrease
of absorption for low-frequency range of electromagnetic probe field, compare
black and yellow curves in the main panel of Fig. 10. For large enough $U$,
the Doppler effects become more and more prominent, and the drift of electron
bunches can eventually be reversed. These mechanisms create a condition for
appearance of a low-frequency gain ($\sigma_{r}(\omega)<0$) as it is evidenced
by the absorption profile calculated for $U=U_{cr_{2}}$ (blue line). The
numerical calculations reveal that for the chosen parameters and
$U=U_{cr_{2}}$, the gain is possible for the wide range of frequencies up to
$\omega/(2\pi)=$ 480 GHz. Further analysis showed that $\sigma_{r}(\omega)$
can obtain a significant THz gain value ($\sigma_{r}\approx-0.07\sigma_{0}$)
at $U=U_{cr_{2}}$ comparable with the maximum gain of a merely dc-biased
superlattice (i.e. canonical Bloch oscillator):
min$\\{\sigma_{r}(\omega)/\sigma_{0}\\}=-1/8$ at $E_{dc}/E{c}=\sqrt{3}$ [68]
and larger than expected from SLs driven by monochromatic and polychromatic
fields with oscillating frequencies at the sub-THz range [70]. Normally, we
relate the magnitude of the gain $\alpha_{g}$ in units c$\textrm{m}^{-1}$ to
dynamical conductivity as $a_{g}=a_{0}(\sigma_{r}/\sigma_{0})$ with
$a_{0}=1/(c\sqrt{\epsilon}\epsilon_{0})\times 2j_{p}/E_{c}$ [71, 70]. This
means that one can acquire the value $a_{g}=37$ c$\textrm{m}^{-1}$ for
$a_{0}\approx 527$ c$\textrm{m}^{-1}$ given a temperature of few millikelvins,
moderate doping $N=10^{16}$ c$\textrm{m}^{-3}$ and relative permittivity
$\varepsilon=13$ corresponding to GaAs.

Figure 10: Absorption and gain profiles, $\sigma_{r}(\omega)$, at fixed
$\omega_{s}=0.1\tau^{-1}$ and different values of wave amplitude ($U$=1.2 meV,
$U_{cr_{1}}=9.55$ meV, $U_{0}=10$ meV, and $U_{cr_{2}}=10.46$ meV). The red
curve signifies the free-carrier absorption whereas the horizontal dashed line
marks the zero absorption. Inset: Averaged position of ballistic electron
trajectories, $x_{a}(t)$, calculated for $U=U_{cr_{1}}$ (yellow curve) and
$U=U_{0}$ (black curve). The calculations have been performed using the
parameters of the SL structure with $v_{0}/v_{s}=35$; See Table 2.

##  VII CONCLUSION

In this work we theoretically studied the physical mechanisms of acousto-
electric effects associated with miniband charge transport in semiconductor
superlattice. Our analysis reveals that superluminal effects, previously
described within the framework of Ginzburg-Frank-Tamm theory for electro-
magnetic radiation [21, 22, 29], play an important role in acoustically
(strain) driven electron transport in minibands. In particularly, our results
suggest the realization of superluminal anomalous Doppler and backward normal
Doppler effects which are related to the development of dynamical
instabilities for certain magnitudes of acoustic stimuli propagating through a
superlattice structure. A kinetic model, based on a semiclassical
nonperturbative approach demonstrates that these effects induced by coherent
phonons manifest themselves in excitation of complex Bloch oscillations and
formation of electron bunches which can counter-propagate or be localized with
respect to the propagating deformation pulse. Such character of electron
transport is reflected in characteristic kinks in the dependence of electron
drift velocity upon the acoustic wave amplitude, and can even lead to the
absolute negative mobility. Remarkably, our calculations indicate that the
anomalous Doppler effect for supersonic miniband electrons (effectively
superluminal) enables realization of a highly tunable gain similar to the
Bloch gain occurring in the voltage driven superlattices [34]. These findings
open additional avenues for development of efficient acoustoelectronic devices
for microwave and THz ranges by providing alternative means for manipulations
of electromagnetic waves in various superlattice electronic devices (SLEDs)
[72, 73] including harmonic multipliers [74, 75] and heterostructure
millimiter-wave detectors [76]. Since semiconductior superlattices serve as
building blocks for quantum cascade lasers (QCLs), our theory provides with
helpful guidelines for the investigation of unusual acoustoelectric phenomena
in QCLs [77], which can be employed for tuning QCL’s broadband THz emission
[78].
In a wider context, similar phenomena can be expected in other miniband
systems subjected to slowly propagating excitations such as ultracold atoms in
optical lattices [79] or in the physical systems with similar Hamiltonians,
e.g. the driven Harper models [80]

##  VIII Acknowledgments

A.A. acknowledges financial support by the Czech Science Foundation (GAČR)
through grant No. 19-03765. Research of K.N.A. was partially supported by
Marius Jakulis Jason Foundation. F.V.K. acknowledges financial support from
FSU-2021-030/8474000371.

##  Appendix A  Nonlinear dynamics in the rest frame of the acoustic wave

The system dynamics describing the motion of an electron under the influence
of a propagating wave potential is described by the equations (4). To
conceptualize how the acoustoelectric phenomena discussed in terms of waves
and quanta are related to bifurcations mechanisms, we analyze the equations of
motion in the moving reference frame $x^{\prime}(t)=x(t)+x_{i}-v_{s}t$. In
this frame, the electron is subjected to a time-independent potential
$V(x^{\prime})=-U\sin(k_{s}x^{\prime})$ whereas the kinetic energy of an
electron is translated into
$\mathcal{E}^{\prime}(p_{x})=\mathcal{E}(p_{x})-v_{s}p_{x}$. After this
transformation, the Hamiltonian becomes

$H^{\prime}=\mathcal{E}^{\prime}(p_{x})+V(x^{\prime})$ (29)

and therefore the equations of motion can be cast as

$\displaystyle\dot{x}^{\prime}$ $\displaystyle=$ $\displaystyle
v_{0}\sin\left(\frac{p_{x}d}{\hbar}\right)-v_{s},$ (30a)
$\displaystyle\dot{p}_{x}$ $\displaystyle=$ $\displaystyle
k_{s}U\cos(k_{s}x^{\prime}),$ (30b)

which, in contrast to Eqs. (4a), (4b), It follows that the equilibria of the
dynamical system (30) should meet the following conditions

$\displaystyle v_{0}\sin\left(\dfrac{p_{x}d}{\hbar}\right)=v_{s},$ (31a)
$\displaystyle\cos(k_{s}x^{\prime})=0.$ (31b)

This dictates the locations of the fixed points

$\displaystyle x^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\pi}{2k_{s}}+\frac{m\pi}{k_{s}},$ (32a) $\displaystyle
p_{x}$ $\displaystyle=$
$\displaystyle(-1)^{l}\frac{\hbar}{d}\sin^{-1}\left(\frac{v_{s}}{v_{0}}\right)+l\frac{\hbar\pi}{d},$
(32b)

where $m$ and $l$ are arbitrary integer numbers. From a viewpoint of a wider
miniband ($v_{0}>v_{s}$) and therefore higher conduction current, a simple
stability analysis reveals that all these fixed points are always either
hyperbolic or elliptic points. After dividing (32b) by (30a) and performing
integration we obtain the phase trajectory equation

$\displaystyle x=\dfrac{(-1)^{j}}{k_{s}}\sin^{-1}\left\\{\sin
k_{s}x_{i}-\frac{\hbar
v_{s}}{Ud}\left[\frac{v_{0}}{v_{s}}\bigg{(}\cos\dfrac{p_{x}d}{\hbar}\right.\right.$
$\displaystyle-\Biggl{.}\Biggl{.}\cos\dfrac{p_{i}d}{\hbar}\bigg{)}+\dfrac{p_{x}d}{\hbar}-\dfrac{p_{0}d}{\hbar}\bigg{]}\bigg{\\}}+j\dfrac{\pi}{k_{s}},$
(33)

where $j$ is an integer number, and ($x_{i}$,$p_{i}$) is an initial condition.
Cherenkov effects near fixed points.—

Figure 11:  Schematic representation in the $\bar{q}$–$\bar{v}$ space of the
forward and the backward Cherenkov effects when a charged particle is moving
with a supersonic velocity $v(p_{x})$ at the center and the borders of the
superlattice Brillouin zone. (a) Forward Cherenkov absorption in the proximity
of $p_{s_{1}}$. (b) Forward Cherenkov emission in the proximity of
$p_{s_{1}}$. (c) Backward Cherenkov absorption in the proximity of
$p_{s_{2}}$. (d) Backward Cherenkov emission in the proximity of $p_{s_{2}}$.
The angle of $\theta$ is the opening angle of the Vavilov-Cherenkov cone
indicating an almost perpendicular orientation of the velocity vector
$\bar{v}$ with respect to abscissa-located phonon wave vector $\bar{q}$ for
$U\ll U_{c1}$. The insets demonstrate the projection of Cherenkov effects in
the active zone described by Fig. 2(a).

We now answer the question whether our moving particle with velocity
$v(p_{x})$ larger than $v_{s}$, is able to induce phenomena analogous to
Cherenkov-like conical flow and how the stationary points can help to probe
these effects. Strictly speaking, to answer this question one should then take
into consideration a realistic three-dimensional description by including a
quadratic isotropic dispersion law in the Hamiltonian [Eq. (3)] for the in-
plane kinetic momentum components $p_{\parallel}=(p_{y},p_{z})$. However, the
forward and reversed Cherenkov radiation, in the proximity of the stationary
points, is consistent with the assumption of the 1D emission in Eq. (5) being
finitely small. This implication is easily understood by considering an
electron with $v_{s}/v(\bar{p})\ll 1$ with $\bar{p}$ the total electron
momentum. In that case, the Cherenkov mechanism 111This is nothing more than
the kinetic description of Cherenkov interactions of sound wave with electrons
in metals stemming from Landau absorption mechanism [87]. acts mainly in a
direction $\hat{q}_{x}$ almost perpendicular to electron velocity $\bar{v}$
with its x-component being the scalar projection $\bar{v}$ onto
$\bar{q}=q\hat{q}_{x}$:

$comp_{\bar{q}}\bar{v}=|v|\cos\theta=\dfrac{q}{|q|}v_{0}\sin(p_{x}),$ (34)

Figure 12:  (a), (b) Phase space trajectories (left panel) and the related
electron trajectories (right panel) in real space that can determine the
nature of Cherenkov effects in the vicinity of the center and the border
Brillouin zone for $U$=1.2 meV. The yellow trajectories correspond to the
backward Cherenkov whereas the blue ones to the forward Cherenkov. (c) The
complex meandering trajectory and the corresponding trajectory in real space
representing the phonon-assisted Bloch oscillation which arises for $U=10$ mev
$>U_{cr_{1}}$. The positions of the elliptic points are depicted by black
solid circles and the hyperbolic points by red crosses. The calculations have
been performed using the parameters of the SL structure with $v_{0}/v_{s}=35$;
See Table 2.

where $\theta$ is the angle between the electron velocity and wavevector
$\bar{q}$ of the emitted phonon in the proximity of the stationary points.
Hereafter we consider in detail the nature of Cherenkov effects around the
center of the first Brillouin zone: (i) when $p_{f}>$ $p_{i}=-p_{s_{1}}$, so
that $q>0$. This results in an absorption of phonon and a positive electron
velocity that surpasses sound velocity, which is forward Cherenkov radiation
[Fig. 11(a)]. (ii) When $p_{f}<$ $p_{i}=p_{s_{1}}$, so that $q<0$. This
results in an emission of phonon and a negative electron velocity that its
norm surpasses sound velocity, which is again forward Cherenkov radiation.
Consider now what happens at the border of the first Brillouin zone: (iii)
When $p_{f}<$ $p_{i}=p_{s_{2}}$, so that $q<0$. This results in an emission of
phonon and a positive electron velocity with $v>v_{s}$, which is backward
Cherenkov. Finally, (iv) when $p_{f}>$ $p_{i}=p_{s_{2}}$ which may lie outside
the first BZ (Umklapp process), so that $q>0$ and in this sense is an
absorption of phonon [Fig. 11(d)].

Figure 13:  Separatrix structure in the proximity of the first bifurcation
point $U_{cr_{1}}$ for (a) $U$=9 meV; (b) $U=$9.52 meV; (c) $U$=9.6 meV. The
positions of the elliptic points are depicted by black solid circles and the
hyperbolic points by red crosses. The calculations have been performed using
the parameters of the SL structure with $v_{0}/v_{s}=35$; See Table 2.

To find the physically equivalent point inside the first BZ, we have to
subtract the module of the reciprocal superlattice vector $|G_{0}|=2\pi/d$
from $p_{f}$. Thus, $p_{f}^{\prime}=p_{f}-|G_{0}|$ will be found in the
proximity of $p_{s_{3}}$ with $v<0$ which results in a backward Cherenkov
effect. Practically, we have forward and backward Cherenkov-like conical flow
in the center and at the edges of BZ respectively. Similar description one can
attain from the elliptic points at the center and edges of the BZ, i.e. normal
versus backward Cherenkov. In essence, the counter-wise rotation of the phase-
space trajectories around the elliptic points [left-handed panel of Fig.
12(a)] with $p_{e_{1}}=(\hbar/d)\sin^{-1}(v_{s}/v_{0})$ and $p_{e_{2}}=\pi
d/\hbar-(\hbar/d)\sin^{-1}(v_{s}/v_{0})$ designate the nature of the Cherenkov
instabilities. Therefore, for absorption of an infinitely small quasi-particle
($q$) the clockwise trajectory (blue curve) in the proximity of $p_{e_{1}}$ is
related to an electron trajectory in the real space that moves in the positive
direction of $x$, $v(p_{e_{1}}+q)>0$ (forward Cherenkov effect). On the
contrary, the electron trajectory which moves momentarily in the negative
direction, $v(p_{e_{1}}+q)<0$ (backward Cherenkov effect), corresponds to the
counterclockwise trajectory (yellow curve). Those two trajectories are
directly associated with the motion of electrons confined by the propagation
potential and they coexist for small $U$ with unbounded trajectories [left-
handed panel of Fig. 12(b)] in the moving reference frame. The latter phase-
space trajectories are related to unconfined motion [right-handed panel of
Fig.12(b)] of electrons in real space and their different direction has origin
again in the nature of different Cherenkov effects.

Figure 14:  Separatrix structure in the proximity of the second bifurcation
point $U_{cr_{2}}$ for (a) $U$=10 meV; (b) $U=$10.4 meV; (c) $U$=10.5 meV. The
positions of the elliptic points are depicted by black solid circles and the
hyperbolic points by red crosses. The calculations have been performed using
the parameters of the SL structure with $v_{0}/v_{s}=35$; See Table 2.

Global instabilities (bifurcations).— The aforementioned trajectories and the
associated Cherenkov effects can well exist both for small and large wave
amplitudes. Considering though substantially larger values of $U$ triggers a
series of global instabilities which are attributed to instances where the
manifold of one hyperbolic point touches another hyperbolic point. It has been
shown that the conservation of energy, $H^{\prime}=$ const [see Eq. (29)], can
be used to determine the bifurcation points analytically [25]

$U_{cr}=\frac{\frac{\Delta}{2}[\cos(p_{s_{i}}d/\hbar)-\cos(p_{s_{j}}d/\hbar)]+v_{s}p_{s_{i}}-v_{s}p_{s_{j}}}{\sin(k_{s}x_{s_{j}})-\sin(k_{s}x_{s_{i}})},$
(35)

where $(x_{s_{i}},p_{s_{i}})$, $(x_{s_{j}},p_{s_{j}})$ are a set of
coordinates of hyperbolic points which are involved in a specific bifurcation.
Using these coordinates one can attain the explicit formulations for critical
values of the wave amplitude, i.e. Eq (9)-(11). The bifurcations cause
dramatic transformation of the phase space (see Figs. 13, 14) and result in
the emergence of meandering trajectories such as the one depicted in the left-
handed panel of Fig. 12(c) which corresponds to frequency-modulated
oscillations in real space shown in the right-handed panel of Fig. 12(c). This
trajectory in contrast to the ones depicted in Fig. 12(a),(b) follows from a
non-conventional acoustoelectric effect which is an effective counterpart of
the superluminal anomalous Doppler effect. The discussion under which
conditions this phenomenon arises and how relates to global instabilities are
presented in detail in Sec. II.

##  Appendix B  Ensemble-averaged physical quantities

By resorting to ensemble-averaged physical quantities in Sec. III helped us to
better understand the role of the Doppler effects in the global transport in
the dissipationless limit. In this appendix, we will discuss in detail the
proper definition of these quantities starting from a straightforward approach
in which we simply average across the initial positions $x_{i}$ of electrons,
or, equivalently across the initial phases of the acoustic wave

$\displaystyle v_{a}(t)$ $\displaystyle=$ $\displaystyle
v_{0}\left\langle\sin\left\\{\dfrac{p_{x}(t)d}{\hbar}\right\\}\right\rangle_{x_{i}},$
(36a) $\displaystyle E_{a}(t)$ $\displaystyle=$
$\displaystyle\dfrac{\Delta}{2}\left\langle
1-\cos\left\\{\dfrac{p_{x}(t)d}{\hbar}\right\\}\right\rangle_{x_{i}},$ (36b)
$\displaystyle V_{a}(t)$ $\displaystyle=$
$\displaystyle-U\left\langle\sin(k_{s}x-\omega_{s}t)\right\rangle_{x_{i}},$
(36c)

where $v_{a}(t)=\langle\dot{x}\rangle_{x_{i}}$ is the averaged electron
velocity, $E_{a}(t)=\langle\mathcal{E}(p_{x})\rangle_{x_{i}}$ is the averaged
electron energy and $V_{a}(t)=\langle V(x,t)\rangle_{x_{i}}$ is the averaged
potential energy. For example, the value of $\langle
v_{a}(t){x}\rangle_{x_{i}}$ is determined by averaging $\dot{x}$ over an
ensemble of electron trajectories with different $x_{i}$ from the interval
$[-\lambda/2,\lambda/2)$. Here, $\lambda=2\pi/k_{s}$ designates the space
period of the propagating wave. We are ready now to define the dissipationless
approach ($\tau\rightarrow\infty$) for the evaluation of (20). Thus, one can
formally calculate the time-averaged velocity $v_{m}=\langle
v_{a}(t)\rangle_{\Delta t}$ by using

$v_{m}=\frac{1}{\lambda}\int_{-\lambda/2}^{\lambda/2}d{x_{i}}\int_{0}^{\Delta
t}v(t,t_{i})\frac{dt}{\Delta t}.$ (37)

and the initial conditions

$x(t_{i},t_{i})=x_{i},\hskip 8.5359ptp_{x}(t_{i},t_{i})=0.$ (38)

In this approach $[v(t,t_{i}),p_{x}(t,t_{i})]$ plays the role of the Vlasov
phase and $t_{i}$ designates the initial time as described in [82, 83] and
references therein.
However, this approach described by Eq. (36) faces shortcomings when we are
seeking to define precisely the averaged momentum $p_{a}$, i.e. the center of
the mass of the electron distribution (CMED), $f(p_{x},x,t)$, in the
quasimomentum space. Let us now the reduce the dimensionality of
$f(p_{x},x,t)$ by integrating over all initial phases (from $-\lambda/2$ to
$\lambda/2$) of the acoustic wave

$\tilde{f}(p_{x},t)=\langle f(p_{x},x,t)\rangle_{x_{i}},$

We may obtain the $x_{i}$-averaged quantities [Eq. (36a)-(36c)] that were
introduced earlier by averaging against $\tilde{f}$,
$\langle.\rangle_{x_{i}}=d/(\pi\hbar)\int_{-\pi\hbar/d}^{\pi\hbar/d}(.)\tilde{f}(p_{x},t)d{p_{x}}$.
One can easily understand the implications of such a choice for $\langle
p_{x}\rangle_{x_{i}}$ by considering the $\tilde{f}(p_{x},t)$ centered around
a symmetric and narrow peak at $p_{x}=\pi\hbar/d$. In that case, the
expectation value $\langle p_{x}\rangle_{x_{i}}=0$ contradicts the real value
of $\pi\hbar/d$. To predict with precision the CMED we resort to a circular
mean angle $p_{\phi}=p_{a}d/\hbar$ which is determined by the first
trigonometric moment $m_{1}$ of the distribution 222This method relies on
superlattice balance equations [51] and their connection to rotationally
symmetric distributions in directional statistics [52]

$p_{\phi}=\textrm{arg}\hskip 2.84544ptm_{1},\hskip
14.22636ptm_{1}=\langle\textrm{exp}(ip_{x}d/\hbar)\rangle_{x_{i}},$ (39)

while the (circular) variance of the distribution is obtained from absolute
value, $|m_{1}|=A$, of the first trigonometric moment

$\mathcal{V}=1-A.$ (40)

Here the $\mathcal{V}$ takes values between 0 and 1. The value of the lower
bound, $\mathcal{V}=0$, implies a Dirac $\delta$–distribution function
centered at mean angle $p_{\phi}$, whereas the value of the upper bound,
$\mathcal{V}=1$, indicates a distribution function which has no defined mean.
To analyze though the effects of electron bunching in Sec. III, we shall use
$A$ which describes the coherence of the electron distribution, i.e. how
concentrated is $\tilde{f}$ around its mean. The averaged velocity and
averaged energy can be obtained in terms of the variables $p_{\phi}$, $A$

$v_{a}=\dfrac{A\Delta d}{2\hbar}\sin(p_{\phi}),\hskip
14.22636ptE_{a}=\dfrac{A\Delta}{2}[1-\cos(p_{\phi})].$ (41)

From Eqs. (36), (41) one finds that

$\displaystyle A\sin(p_{\phi})=\langle\sin(p_{x}d/\hbar)\rangle_{x_{i}},$
(42a) $\displaystyle A(1-\cos(p_{\phi}))=\langle
1-\cos(p_{x}d/\hbar)\rangle_{x_{i}},$ (42b)

In fact, our numerical calculations further confirmed that the ensemble
average approach [Eq. (36a)] and the averaging against circular distribution
[see Eq. (41)] of electron velocity are equivalent.

Figure 15: The temporal dynamics of $p_{x}$–space trajectories subjected to
normal and anomalous Doppler instabilities (first panel), the electron bunches
(second panel), the averaged electron energy $E_{m}$ (third panel) and the
averaged potential energy $V_{m}$ (fourth panel) for (a) $2U/\Delta=1.06$, (b)
$2U/\Delta=1.08$, (c) $2U/\Delta=1.1$ with $U_{c2}<U<U_{cr_{3}}$ and (d)
$2U/\Delta=1.14$, (e) $2U/\Delta=1.17$, (f) $2U/\Delta=1.2$ with
$U_{cr_{3}}<U<U_{cr_{4}}$. The black and blue curves in the first panel were
calculated for $x_{i}=-\pi/(2k_{s})$ and $x_{i}=-2/k_{s}$ respectively. The
white lines in the second panel indicate the center of electron distribution
as a function of time. The calculations have been performed using the
parameters of the SL structure with $v_{e}/v_{s}$=22.

###  $p$–space bunching between higher order bifurcations

Figure 15 complements Figs. 5,6 by summarizing the properties of wave-like
bunching of electrons in momentum space between consecutive higher-order
bifurcations. Specifically, we consider three values of $U$ between
$U_{cr_{2}}$ and $U_{cr_{3}}$, viz., $2U/\Delta$=1.06, $2U/\Delta$=1.08,
$2U/\Delta$=1.1 and another three values between $U_{cr_{3}}$ and
$U_{cr_{4}}$, namely, $2U/\Delta$=1.14, $2U/\Delta$=1.17 and $2U/\Delta$=1.2.
When $U>U_{cr_{2}}$ new type of trajectories rise, such as the ones depicted
by the dashed curves in the first panel of Figs. 15(a)-(c), resulting in
nonlinear bunching of the electron distribution function $\tilde{f}$ [see
second panel Figs. 15(a)-(c)]. The amplitude of these $p_{x}$–nonlinear
oscillations exceeds the size of the second Brillouin zone caused due to an
anomalous Doppler shift and a normal Doppler shift near the end of the first
and the second Brillouin zone respectively. As $U$ changes between
$U_{cr_{2}}$ and $U_{cr_{3}}$, the nonlinear bunching becomes more prominent
at the value of $2U/\Delta=1.08$ [second panel Fig.15(c)], coinciding with a
local maximum of $v_{m}(U)$ in Fig. 4. This behavior implies a balanced
mixture of anomalous and normal Doppler emissions similar to Smith-Purcell
effects in photonics crystals [46]. Once $U$ lies between $U_{cr_{3}}$ and
$U_{cr_{4}}$ more complicated trajectories emerge involving three-phonon
emissions, e.g. black dashed curve in the left panel of the Fig. 15(d).
Finally, calculating the averaged electron energy and potential energy further
confirms the absorption-emission events since the amplitude of their
oscillations is enhanced with the variation of $U$, implying the emission
(absorption) of larger number of phonons. Note $E_{a}(t)$ and $V_{a}(t)$ in
third and fourth panel of Figs. 15(a)-(f).

##  Appendix C  Derivation of the sound absorption coefficient

Consider a coherent acoustic wave that propagates through the superlattice. We
want to study its evolution and the attenuations effects due to the
interaction with the electrons and the crystal under the Hamiltonian (3).
Then, the derivative of $H$ with respect to time is given by

$\dot{H}=\frac{\partial\mathcal{E}}{\partial p_{x}}\dot{p_{x}}+\frac{\partial
V}{\partial x}\dot{x}+\frac{\partial H}{\partial t}$ (43)

By averaging (43) over $T_{s}=2\pi/\omega_{s}$, we obtain

$\langle\dot{H}\rangle_{T_{s}}=2k_{s}U\left<\left(\dfrac{v_{x}(p_{x})}{v_{0}}-\dfrac{v_{s}}{2v_{0}}\right)\cos(k_{s}x-\omega_{s}t)\right>_{T_{s}}.$
(44)

To calculate the absorption $\Gamma$ coefficient, we need to substitute the
right-hand side of (44) in the definition (21). The last step requires the
exact solution of Boltzmann equation in the general case by using the path-
integral approach [Eq. 20].

##  Appendix D  Small-signal gain in the quasistatic limit

Figure 16:  (a) Color map showing the dependence of the drift velocity $v_{d}$
on $v_{0}/v_{s}$ and $\omega_{s}\tau$. The open circles mark the exact values
of $v_{0}$ for which absorption was calculated numerically in the figure
below. (b) Absorption $\sigma_{r}(\omega)$ of the oscillating field $E(t)$ as
a function of the wave amplitude $U$ ($1^{\textrm{st}}$ lower scale) or the
strain magnitude $\epsilon$ ($2^{\textrm{nd}}$ lower scale) for
$\omega\tau<0.1$ and different values of $v_{0}/v_{s}=:13,35,55$. The upper
scale denote the values of the acoustoelectric field [see Eq. (45)] for a SL
structure with $v_{0}/v_{s}=35$ and $E_{c}=2.3$ kV/cm. The vertical lines
correspond to the $U=U_{cr_{2}}$ for different SL parameters; see Table 2.

Here we discuss how $a$–parameter, effectively the product
$(v_{0}/v_{s})\times(\omega_{s}\tau)$, is important for the appearance of
electron bunches with a negative drift but also whether it can influence the
interaction of electrons with the electric probe field in the quasistatic
limit. By employing again the formulation for the amplitude, $E_{s}=k_{s}U/e$,
of a effective acoustoelectric field, we obtain a generalized condition for
the onset of the succesive bifurcations derived by Eq. (18) which reads

$\left(\dfrac{E_{s}}{E_{c}}\right)_{n}=\alpha+(2n-3)\left(\dfrac{v_{s}}{v_{e}}\right)\left(\dfrac{\Delta\hbar}{2\tau}\right)k_{s}d.$
(45)

Then from the above equation, it follows that $\alpha$ can also determine the
amplitude of the acoustoelectric field for which the electron localization
[see Eq. (19)] takes place. In addition, the second term of Eq. (45) depends
on $v_{s}/v_{e}$ with $v_{e}=2v_{0}/\pi$, which controls the size of the
active region [Eq. 12], the scattering induced broadening $\hbar/\tau$
incorporated in Eq. (20) and the product $k_{s}d$ which is associated with
propagation of the acoustic wave. To investigate all the former
considerations, we calculated first the drift velocity $v_{d}$ as function of
$v_{0}/v_{s}$ and $\omega_{s}\tau$ at $U=U_{cr_{2}}$ [Fig. 16(a)] where the
negative drift velocity is possible. The upper-right blue area indicates
values ($v_{0}/v_{s}$, $\omega_{s}\tau$) that result in an enhancement of the
backward drift due to the complex Bloch oscillations which are linked to the
anomalous Doppler effect. In contrast, the dark red area designates the region
with no reversal of drift velocity to negative values. As a next step, we
calculate the absorption, $\sigma_{r}(\omega,U)$, of an ac probe field as a
function of an arbitrary amplitude, $U$, of the acoustic wave by resorting to
an exact solution of the BTE. We assume that frequency of the acoustic wave is
$\omega_{s}\tau=0.1$ whereas the signal field frequency is much smaller than
the inverse relaxation time ($\omega\tau\ll 0.1)$. Figure 16(b) demonstrates
how the absorption changes with the variation of $U$ for different
superlattice parameters and therefore different values of the parameter $a$.
By increasing the wave amplitude the incoherent absorption remains positive
(yellow curve) for $v_{0}/v_{s}$=13 $(a\sim 1.3)$ even if $U$ exceeds the
critical value $U_{cr_{1}}$. On the contrary, for a larger miniband width,
$\Delta=$ 20 meV $(a\sim 3.5)$, gain is feasible (blue curve) if $U$ reaches
the values, where the sign of $\sigma_{r}$ starts to be sensitive to the
Doppler frequency shifts. Remarkably, gain exhibits an abrupt change and
attains a maximum value close to $U=U_{cr_{2}}$ similar to the characteristic
changes in drift velocity [see Fig. 7]. These effects imply a sensitive
dependence of absorption on the sign of drift velocity and $\alpha$ parameter.
It then follows that the true parameter that determines the mode of absorption
of the probe field is $\alpha$ rather than just $\omega_{s}\tau$. This occurs
because $\omega_{s}$ and $k_{s}$ enter in the kinetic equations into a
combination. Interestingly, for an even larger $\alpha\sim 5.5$, the
$\sigma_{r}(\omega)$ dips anew to negative values when $U=U_{cr_{3}}$ [green
curve, Fig. 16(a)].

##  Appendix E  Key quantities and practical considerations

Table 2: Parameters of superlattices studied $\Delta$ (meV) |  $d$ (nm) |  $\tau$ (fs) | $v_{0}/v_{s}$ | $\sigma_{0}$ |
---|---|---|---|---|---
miniband width | lattice period | scattering time | $v_{e}/v_{s}$ | Drude |
|  |  |  | conductivity |
7 | 12.5 | 250 | 13 (10.5) | 5.04 |
20 | 11.4 | 250 | 35 (22) | 11.98 |
60 | 6 | 200 | 55 (35) | 7.96 |

In this appendix we provide further details regarding the superlattices
structures considered to calculate the physical quantities related to
acoustoelectric effects. Table 2 outlines the chosen parameters taken from
realistic samples [38, 16, 68] and depicts the critical ratio $v_{0}/v_{s}$
that is largely responsible for the demonstration of different types of
supersonic behavior and effectively controls the size of the active zone. The
entries in the parentheses denote the ratio which is proportional to the
effective electron speed ($v_{e}=2v_{0}/\pi$) and it is involved in the
expressions of the selections rules [Eq. 18] allowing us to make a straight-
forward comparison with their effects on direct-transport in the quasi-
ballistic limit. Furthermore, the fourth column gives the Drude conductivity
which is used to scale the dynamical conductivity and therefore complementing
Figs. 10, 16.
One can characterize in a direct way the fraction of supersonic electrons by
calculating the probability density $g(v(p_{x}))$ of miniband electron
velocities for the particles starting from $x_{i}=0$ and the initial momenta
$p_{i}$ uniformly distributed within the interval $(-\pi\hbar/d,\pi\hbar/d)$,
i.e. within the first BZ. In this case, the miniband electron velocities obey
a shifted arcsine distribution [85] whose probability density function is
$g(v)=1/(\pi\sqrt{(v_{0})^{2}-v^{2}})$ with $-v_{0}<v(p_{x})<v_{0}$. Figure 17
illustrates a colormap of $g$ calculated versus $v(p_{x})/v_{s}$ and
$v_{0}/v_{s}$. For a typical SL structure with $v_{0}/v_{s}$=13, whose
parameters are given in table 2, the probability density $g$ demonstrates
values of small order $\sim 0.01$ in the subsonic (subluminal) region shown by
the area between the dashed horizontal lines. Figure 17 reveals that $g$ in
the subsonic region is reduced significantly (transition between yellow and
dark-purple colors) with the increase of $v_{0}/v_{s}$, ensuring that $v$
would surpass $v_{s}$ almost for any $p_{x}$ at the limit $v_{0}\gg v_{s}$.

Figure 17:  Colormap showing the probability density $g$ of a particle’s
velocity $v(p_{x})$ within $(-v_{0},v_{0})$ for $p_{x}$ uniformly distributed
over $(-\pi\hbar/d,\pi\hbar/d)$ and different ratios $v_{0}/v_{s}$.

Table 2 summarizes the critical values associated with the physical processes
corresponding to bifurcations [cf. Eq. 9]. The bolded entries represent the
critical values of $U$ which are all in excellent agreement with numerical
simulations, as portrayed by Fig. 7. The wave amplitudes are directly
proportional to the deformation potential $D$ and the maximum strain,
$\epsilon$, that the acoustic wave creates. For the simulations in this work
we consider $D=10$ eV [38] and $\epsilon<0.5$ $\%$ which lies within an easily
accessible experimental range [38, 86]. These values are sufficiently large to
reach the critical values of $U$ described in Table 2.

Table 3: Critical values of the wave amplitude for the SL structures in table 2 | $U_{cr_{1}}$ | $U_{cr_{2}}$ | $U_{cr_{3}}$ | $U_{cr_{4}}$ |
---|---|---|---|---|---
$v_{0}/v_{s}$ | meV | meV | meV | meV |
13 | 3.1 | 3.92 | 4.75 | 5.58 |
35 | 9.55 | 10.46 | 11.37 | 12.27 |
55 | 29.14 | 30.87 | 32.59 | 34.31 |

## References

  * Hu  _et al._ [2020] R. Hu, S. Iwamoto, L. Feng, S. Ju, S. Hu, M. Ohnishi, N. Nagai, K. Hirakawa, and J. Shiomi, Machine-learning-optimized aperiodic superlattice minimizes coherent phonon heat conduction, Physical Review X 10, 021050 (2020).
  * Maire  _et al._ [2017] J. Maire, R. Anufriev, R. Yanagisawa, A. Ramiere, S. Volz, and M. Nomura, Heat conduction tuning by wave nature of phonons, Science advances 3, e1700027 (2017).
  * Lu  _et al._ [2009] M.-H. Lu, L. Feng, and Y.-F. Chen, Phononic crystals and acoustic metamaterials, Materials today 12, 34 (2009).
  * Davis and Hussein [2014] B. L. Davis and M. I. Hussein, Nanophononic metamaterial: Thermal conductivity reduction by local resonance, Physical review letters 112, 055505 (2014).
  * Mante  _et al._ [2015] P.-A. Mante, Y.-R. Huang, S.-C. Yang, T.-M. Liu, A. A. Maznev, J.-K. Sheu, and C.-K. Sun, Thz acoustic phonon spectroscopy and nanoscopy by using piezoelectric semiconductor heterostructures, Ultrasonics 56, 52 (2015).
  * Sitters  _et al._ [2015] G. Sitters, D. Kamsma, G. Thalhammer, M. Ritsch-Marte, E. J. Peterman, and G. J. Wuite, Acoustic force spectroscopy, Nature methods 12, 47 (2015).
  * Greener  _et al._ [2018] J. D. Greener, A. V. Akimov, V. Gusev, Z. Kudrynskyi, P. H. Beton, Z. D. Kovalyuk, T. Taniguchi, K. Watanabe, A. Kent, and A. Patanè, Coherent acoustic phonons in van der waals nanolayers and heterostructures, Physical Review B (2018).
  * Tamagnone  _et al._ [2018] M. Tamagnone, A. Ambrosio, K. Chaudhary, L. A. Jauregui, P. Kim, W. L. Wilson, and F. Capasso, Ultra-confined mid-infrared resonant phonon polaritons in van der waals nanostructures, Science Advances 4, eaat7189 (2018).
  * Poyser  _et al._ [2018] C. L. Poyser, L. H. Li, R. P. Campion, A. V. Akimov, E. H. Linfield, A. G. Davies, J. E. Cunningham, and A. J. Kent, A high electron mobility phonotransistor, Communications Physics 1 (2018).
  * Andersen  _et al._ [2019] T. I. Andersen, B. L. Dwyer, J. D. Sanchez-Yamagishi, J. F. Rodriguez-Nieva, K. Agarwal, K. Watanabe, T. Taniguchi, E. A. Demler, P. Kim, H. Park, _et al._ , Electron-phonon instability in graphene revealed by global and local noise probes, Science 364, 154 (2019).
  * Huynh  _et al._ [2008] A. Huynh, B. Perrin, N. Lanzillotti-Kimura, B. Jusserand, A. Fainstein, and A. Lemaître, Subterahertz monochromatic acoustic wave propagation using semiconductor superlattices as transducers, Physical Review B 78, 233302 (2008).
  * Poyser  _et al._ [2015] C. Poyser, A. Akimov, A. Balanov, R. Campion, and A. Kent, A weakly coupled semiconductor superlattice as a harmonic hypersonic-electrical transducer, New Journal of Physics 17, 083064 (2015).
  * Wilson [2018] T. E. Wilson, Evidence for terahertz acoustic phonon parametric oscillator based on acousto-optic degenerate four-wave mixing in a silicon doping superlattice, Physical Review B 98, 220304 (2018).
  * Kent  _et al._ [2006] A. Kent, R. Kini, N. Stanton, M. Henini, B. Glavin, V. Kochelap, and T. Linnik, Acoustic phonon emission from a weakly coupled superlattice under vertical electron transport: observation of phonon resonance, Physical review letters 96, 215504 (2006).
  * Wang  _et al._ [2020] F. Wang, C. Poyser, M. Greenaway, A. Akimov, R. Campion, A. Kent, T. Fromhold, and A. Balanov, Ultrafast strain-induced charge transport in semiconductor superlattices, Physical Review Applied 14, 044037 (2020).
  * Shinokita  _et al._ [2016] K. Shinokita, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and C. Flytzanis, Strong amplification of coherent acoustic phonons by intraminiband currents in a semiconductor superlattice, Physical review letters 116, 075504 (2016).
  * Kaminer  _et al._ [2017] I. Kaminer, S. Kooi, R. Shiloh, B. Zhen, Y. Shen, J. López, R. Remez, S. Skirlo, Y. Yang, J. Joannopoulos, _et al._ , Spectrally and spatially resolved smith-purcell radiation in plasmonic crystals with short-range disorder, Physical Review X 7, 011003 (2017).
  * Yang  _et al._ [2018] Y. Yang, A. Massuda, C. Roques-Carmes, S. E. Kooi, T. Christensen, S. G. Johnson, J. D. Joannopoulos, O. D. Miller, I. Kaminer, and M. Soljačić, Maximal spontaneous photon emission and energy loss from free electrons, Nature Physics 14, 894 (2018).
  * Roques-Carmes  _et al._ [2018] C. Roques-Carmes, N. Rivera, J. D. Joannopoulos, M. Soljačić, and I. Kaminer, Nonperturbative quantum electrodynamics in the cherenkov effect, Physical Review X 8, 041013 (2018).
  * Shi  _et al._ [2018] X. Shi, X. Lin, I. Kaminer, F. Gao, Z. Yang, J. D. Joannopoulos, M. Soljačić, and B. Zhang, Superlight inverse doppler effect, Nature Physics 14, 1001 (2018).
  * Ginzburg and Frank [1947] V. Ginzburg and I. Frank, On doppler effect at superluminal velocity, in  _Dokl. Acad. Nauk, USSR_ , Vol. 56 (1947) p. 583.
  * Ginzburg [1960] V. L. Ginzburg, Certain theoretical aspects of radiation due to superluminal motion in a medium, Sov. Phys. Uspekhi 2, 874 (1960).
  * Lin and Zhang [2019] X. Lin and B. Zhang, Normal doppler frequency shift in negative refractive-index systems, Laser & Photonics Reviews 13, 1900081 (2019).
  * Greenaway  _et al._ [2010] M. Greenaway, A. Balanov, D. Fowler, A. Kent, and T. Fromhold, Using acoustic waves to induce high-frequency current oscillations in superlattices, Physical Review B 81, 235313 (2010).
  * Apostolakis  _et al._ [2017] A. Apostolakis, M. K. Awodele, K. N. Alekseev, F. Kusmartsev, and A. G. Balanov, Nonlinear dynamics and band transport in a superlattice driven by a plane wave, Physical Review E 95, 062203 (2017).
  * Parmenter [1953] R. H. Parmenter, The acousto-electric effect, Phys. Rev. 89, 990 (1953).
  * Gulyaev [2005] Y. V. Gulyaev, Acoustoelectronics (historical review), Physics-Uspekhi 48, 847 (2005).
  * Nezlin [1976] M. V. Nezlin, Negative-energy waves and the anomalous doppler effect, Sov. Phys. Uspekhi 19, 946 (1976).
  * Tamm [1960] I. E. Tamm, General characteristics of vavilov-cherenkov radiation, Science 131, 206 (1960), http://science.sciencemag.org/content/131/3395/206.full.pdf .
  * Smith and Purcell [1953] S. J. Smith and E. M. Purcell, Visible light from localized surface charges moving across a grating, Phys. Rev. 92, 1069 (1953).
  * Bolotovskii and Voskresenskii [1968] B. M. Bolotovskii and G. V. Voskresenskii, Emission from charged particles in periodic structures, Sov. Phys. Uspekhi 11, 143 (1968).
  * Alekseev  _et al._ [1998] K. N. Alekseev, E. H. Cannon, J. C. McKinney, F. V. Kusmartsev, and D. K. Campbell, Spontaneous dc current generation in a resistively shunted semiconductor superlattice driven by a terahertz field, Physical review letters 80, 2669 (1998).
  * Keay  _et al._ [1995] B. Keay, S. Zeuner, S. Allen Jr, K. Maranowski, A. Gossard, U. Bhattacharya, and M. Rodwell, Dynamic localization, absolute negative conductance, and stimulated, multiphoton emission in sequential resonant tunneling semiconductor superlattices, Physical review letters 75, 4102 (1995).
  * Ktitorov  _et al._ [1972] S. Ktitorov, G. Simin, and V. Y. Sindalovskii, Bragg reflections and the high-frequency conductivity of an electronic solid-state plasma, Sov. Phys. Solid State 13, 1872 (1972).
  * Ignatov and Romanov [1976] A. Ignatov and Y. A. Romanov, Nonlinear electromagnetic properties of semiconductors with a superlattice, physica status solidi (b) 73, 327 (1976).
  * Kazarinov and Skobov [1963] R. F. Kazarinov and V. G. Skobov, On the theory of the amplification of ultrasound by semi-metals in electric and magnetic fields, Sov. Phys. JETP 16, 1057 (1963).
  * Esaki and Tsu [1970] L. Esaki and R. Tsu, Superlattice and negative differential conductivity in semiconductors, IBM Journal of Research and Development 14, 61 (1970).
  * Fowler  _et al._ [2008] D. Fowler, A. Akimov, A. Balanov, M. Greenaway, M. Henini, T. Fromhold, and A. Kent, Semiconductor charge transport driven by a picosecond strain pulse, Applied Physics Letters 92, 232104 (2008).
  * Galperin  _et al._ [1969] Y. Galperin, S. Gantsevich, and V. Gurevich, Giant oscillations of sound absorption by metals in the case of open trajectories, SOV PHYS JETP 29, 926 (1969).
  * [40] A. Abrikosov,  _Fundamentals of the Theory of Metals_.
  * Ashley [1965] J. Ashley, Phonon emission in piezoelectric solids, Journal of Applied Physics 36, 528 (1965).
  * Rivera and Kaminer [2020] N. Rivera and I. Kaminer, Light–matter interactions with photonic quasiparticles, Nature Reviews Physics 2, 538 (2020).
  * Chen and Chen [2011] H. Chen and M. Chen, Flipping photons backward: reversed cherenkov radiation, Materials Today 14, 34 (2011).
  * Ginzburg and Tsytovich [1990] V. L. Ginzburg and V. N. Tsytovich,  _Transition radiation and transition scattering_ (Adam Hilger Bristol, 1990).
  * Gintsburg [1962] M. Gintsburg, Anomalous doppler effect in a plasma, Soviet Phys. JETP 14, 542 (1962).
  * Luo  _et al._ [2003] C. Luo, M. Ibanescu, S. G. Johnson, and J. Joannopoulos, Cerenkov radiation in photonic crystals, Science 299, 368 (2003).
  * Kuzelev and Rukhadze [1987] M. V. Kuzelev and A. A. Rukhadze, Stimulated radiation from high-current relativistic electron beams, Sov. Phys. Uspekhi 30, 507 (1987).
  * Grigoriev  _et al._ [2018] A. D. Grigoriev, V. A. Ivanov, and S. I. Molokovsky,  _Microwave electronics_ , Vol. 61 (Springer, 2018).
  * Narayanamurti  _et al._ [1979] V. Narayanamurti, H. Störmer, M. Chin, A. Gossard, and W. Wiegmann, Selective transmission of high-frequency phonons by a superlattice: the” dielectric” phonon filter, Physical Review Letters 43, 2012 (1979).
  * Tamura  _et al._ [1988] S. Tamura, D. Hurley, and J. Wolfe, Acoustic-phonon propagation in superlattices, Physical Review B 38, 1427 (1988).
  * Isohätälä and Alekseev [2012] J. Isohätälä and K. N. Alekseev, Devil’s staircase, spontaneous-dc bias, and chaos via quasiperiodic plasma oscillations in semiconductor superlattices, arXiv preprint arXiv:1201.6280 (2012).
  * Mardia and Jupp [2009] K. V. Mardia and P. E. Jupp,  _Directional statistics_ , Vol. 494 (John Wiley & Sons, 2009).
  * Kroemer [2000] H. Kroemer, On the nature of the negative-conductivity resonance in a superlattice bloch oscillator, arXiv preprint cond-mat/0007482 (2000).
  * Schomburg  _et al._ [2003] E. Schomburg, N. Demarina, and K. F. Renk, Amplification of a terahertz field in a semiconductor superlattice via phase-locked k-space bunches of bloch oscillating electrons, Physical Review B 67, 155302 (2003).
  * Demarina and Renk [2005] N. Demarina and K. F. Renk, Bloch gain for terahertz radiation in semiconductor superlattices of different miniband widths mediated by acoustic and optical phonons, Physical Review B 71, 035341 (2005).
  * Spector [1967] H. N. Spector, Interaction of acoustic waves and conduction electrons, in  _Solid state physics_ , Vol. 19 (Elsevier, 1967) pp. 291–361.
  * Fromhold  _et al._ [2004] T. Fromhold, A. Patane, S. Bujkiewicz, P. Wilkinson, D. Fowler, D. Sherwood, S. Stapleton, A. Krokhin, L. Eaves, M. Henini, _et al._ , Chaotic electron diffusion through stochastic webs enhances current flow in superlattices, Nature 428, 726 (2004).
  * Budd [1963] H. F. Budd, The generalized path variable method, Journal of the Physical Society of Japan 18, 142 (1963).
  * MacCallum [1963] C. MacCallum, Kinetic integral solutions of the boltzmann equation, Physical Review 132, 930 (1963).
  * Bass  _et al._ [1981] F. G. Bass, V. V. Zorchenko, and V. I. Shashora, Theory of galvanomagnetic and high-frequency phenomena in super-lattice semiconductors, Sov. Phys. Semiconductors 15, 263 (1981).
  * Blount [1959] E. I. Blount, Ultrasonic attenuation by electrons in metals, Physical Review 114, 418 (1959).
  * Gal’perin  _et al._ [1979] Y. M. Gal’perin, V. Gurevich, and V. Kozub, Nonlinear effects in the propagation of high-frequency sound in normal conductors, Physics-Uspekhi 22, 352 (1979).
  * Wacker [2002a] A. Wacker, Semiconductor superlattices: a model system for nonlinear transport, Physics Reports 357, 1 (2002a).
  * Lax and Vernon Jr [1965] E. Lax and F. L. Vernon Jr, Microwave phonon-assisted tunneling in superconductors, Physical Review Letters 14, 256 (1965).
  * Ignatov  _et al._ [1995] A. A. Ignatov, E. Schomburg, J. Grenzer, K. Renk, and E. Dodin, Thz-field induced nonlinear transport and dc voltage generation in a semiconductor superlattice due to bloch oscillations, Zeitschrift für Physik B Condensed Matter 98, 187 (1995).
  * Grahn  _et al._ [1991] H. Grahn, K. Von Klitzing, K. Ploog, and G. Döhler, Electrical transport in narrow-miniband semiconductor superlattices, Physical Review B 43, 12094 (1991).
  * Renk  _et al._ [2005] K. F. Renk, B. I. Stahl, A. Rogl, T. Janzen, D. Pavel’ev, Y. I. Koshurinov, V. Ustinov, and A. Zhukov, Subterahertz superlattice parametric oscillator, Physical review letters 95, 126801 (2005).
  * Hyart  _et al._ [2009] T. Hyart, J. Mattas, and K. N. Alekseev, Model of the influence of an external magnetic field on the gain of terahertz radiation from semiconductor superlattices, Phys. Rev. Lett. 103, 117401 (2009).
  * Blakemore [1982] J. Blakemore, Semiconducting and other major properties of gallium arsenide, Journal of Applied Physics 53, R123 (1982).
  * Hyart  _et al._ [2008] T. Hyart, K. N. Alekseev, and E. V. Thuneberg, Bloch gain in dc-ac-driven semiconductor superlattices in the absence of electric domains, Phys. Rev. B 77, 165330 (2008).
  * Wacker [2002b] A. Wacker, Gain in quantum cascade lasers and superlattices: A quantum transport theory, Physical Review B 66, 085326 (2002b).
  * Eisele  _et al._ [2018] H. Eisele, L. Li, and E. H. Linfield, High-performance gaas/alas superlattice electronic devices in oscillators at frequencies 100–320 ghz, Applied Physics Letters 112, 172103 (2018).
  * Hramov  _et al._ [2014] A. E. Hramov, V. Makarov, A. A. Koronovskii, S. Kurkin, M. Gaifullin, N. V. Alexeeva, K. Alekseev, M. T. Greenaway, T. M. Fromhold, A. Patanè, _et al._ , Subterahertz chaos generation by coupling a superlattice to a linear resonator, Physical review letters 112, 116603 (2014).
  * Pereira  _et al._ [2017] M. Pereira, J. Zubelli, D. Winge, A. Wacker, A. Rodrigues, V. Anfertev, and V. Vaks, Theory and measurements of harmonic generation in semiconductor superlattices with applications in the 100 ghz to 1 thz range, Physical Review B 96, 045306 (2017).
  * Hayton  _et al._ [2013] D. Hayton, A. Khudchenko, D. Pavelyev, J. Hovenier, A. Baryshev, J. Gao, T.-Y. Kao, Q. Hu, J. Reno, and V. Vaks, Phase locking of a 3.4 thz third-order distributed feedback quantum cascade laser using a room-temperature superlattice harmonic mixer, Applied Physics Letters 103, 051115 (2013).
  * Shao  _et al._ [2018] Z. Shao, Z. Yin, H. Song, W. Liu, X. Li, J. Zhu, K. Biermann, L. L. Bonilla, H. T. Grahn, and Y. Zhang, Fast detection of a weak signal by a stochastic resonance induced by a coherence resonance in an excitable gaas/al 0.45 ga 0.55 as superlattice, Physical review letters 121, 086806 (2018).
  * Dunn  _et al._ [2020] A. Dunn, C. Poyser, P. Dean, A. Demić, A. Valavanis, D. Indjin, M. Salih, I. Kundu, L. Li, A. Akimov, _et al._ , High-speed modulation of a terahertz quantum cascade laser by coherent acoustic phonon pulses, Nature communications 11, 1 (2020).
  * Opačak  _et al._ [2021] N. Opačak, S. Dal Cin, J. Hillbrand, and B. Schwarz, Frequency comb generation by bloch gain induced giant kerr nonlinearity, Physical Review Letters 127, 093902 (2021).
  * Greenaway  _et al._ [2013] M. Greenaway, A. Balanov, and T. Fromhold, Resonant control of cold-atom transport through two optical lattices with a constant relative speed, Physical Review A 87, 013411 (2013).
  * Kolovsky and Mantica [2012] A. R. Kolovsky and G. Mantica, Driven harper model, Physical Review B 86, 054306 (2012).
  * Note [1] This is nothing more than the kinetic description of Cherenkov interactions of sound wave with electrons in metals stemming from Landau absorption mechanism [87].
  * Skadron and Jones [1974] G. Skadron and M. Jones, Properties of the vlasov phase in plasma turbulence theory, The Physics of Fluids 17, 500 (1974).
  * Dupree [1966] T. H. Dupree, A perturbation theory for strong plasma turbulence, The Physics of fluids 9, 1773 (1966).
  * Note [2] This method relies on superlattice balance equations [51] and their connection to rotationally symmetric distributions in directional statistics [52].
  * Arnold and Groeneveld [1980] B. C. Arnold and R. A. Groeneveld, Some properties of the arcsine distribution, Journal of the American Statistical Association 75, 173 (1980).
  * Van Capel  _et al._ [2015] P. Van Capel, E. Péronne, and J. Dijkhuis, Nonlinear ultrafast acoustics at the nano scale, Ultrasonics 56, 36 (2015).
  * Landau [1946] L. D. Landau, On the vibrations of the electronic plasma, Zh. Eksp. Teor. Fiz. 10, 25 (1946).