Amplification of localized acoustic waves by the electron drift in a quantum well
We have investigated acoustic waves in a heterostructure with a layer embedded into a semiconductor providing acoustic waves localization near the layer and electron confinement inside the layer. For layer thicknesses smaller than wavelengths we have obtained and analyzed the dispersion relation for...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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Demidenko, A. A. Kochelap, V. A. 2017-05-27T09:29:28Z 2017-05-27T09:29:28Z 1999 Amplification of localized acoustic waves by the electron drift in a quantum well / A. A. Demidenko, V. A. Kochelap // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 1. — С. 11-24. — Бібліогр.: 32 назв. — англ. 1560-8034 PACS 63.22, 72.20, 73.20.D https://nasplib.isofts.kiev.ua/handle/123456789/117857 We have investigated acoustic waves in a heterostructure with a layer embedded into a semiconductor providing acoustic waves localization near the layer and electron confinement inside the layer. For layer thicknesses smaller than wavelengths we have obtained and analyzed the dispersion relation for the localized waves. For electrons into the layer we have supposed that parallel transport is semiclassical, while perpendicular electron motion is quantized. For two-dimensional confined electrons interacting with the acoustic waves we have solved the Boltzmann equation in a parallel electric field. The solutions have been found for electron-phonon interaction via deformation potential. The dispersion relation for coupled charge density and acoustic waves has been analyzed. We have established conditions of amplification of localized acoustic waves under the electron drift for two extreme cases: i) the only lowest two-dimensional subband is populated, ii) a large number of the subbands are populated. We have found that the amplification coefficient of the acoustic waves in THz-rigion is of the order of 100 cm⁻¹. We have discussed the results and compared them with acoustic waves amplification in bulk like semiconductors. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Amplification of localized acoustic waves by the electron drift in a quantum well Article published earlier |
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Amplification of localized acoustic waves by the electron drift in a quantum well |
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Amplification of localized acoustic waves by the electron drift in a quantum well Demidenko, A. A. Kochelap, V. A. |
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Amplification of localized acoustic waves by the electron drift in a quantum well |
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Amplification of localized acoustic waves by the electron drift in a quantum well |
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Amplification of localized acoustic waves by the electron drift in a quantum well |
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Amplification of localized acoustic waves by the electron drift in a quantum well |
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amplification of localized acoustic waves by the electron drift in a quantum well |
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Demidenko, A. A. Kochelap, V. A. |
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Demidenko, A. A. Kochelap, V. A. |
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1999 |
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English |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Article |
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We have investigated acoustic waves in a heterostructure with a layer embedded into a semiconductor providing acoustic waves localization near the layer and electron confinement inside the layer. For layer thicknesses smaller than wavelengths we have obtained and analyzed the dispersion relation for the localized waves. For electrons into the layer we have supposed that parallel transport is semiclassical, while perpendicular electron motion is quantized. For two-dimensional confined electrons interacting with the acoustic waves we have solved the Boltzmann equation in a parallel electric field. The solutions have been found for electron-phonon interaction via deformation potential. The dispersion relation for coupled charge density and acoustic waves has been analyzed. We have established conditions of amplification of localized acoustic waves under the electron drift for two extreme cases: i) the only lowest two-dimensional subband is populated, ii) a large number of the subbands are populated. We have found that the amplification coefficient of the acoustic waves in THz-rigion is of the order of 100 cm⁻¹. We have discussed the results and compared them with acoustic waves amplification in bulk like semiconductors.
|
| issn |
1560-8034 |
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https://nasplib.isofts.kiev.ua/handle/123456789/117857 |
| citation_txt |
Amplification of localized acoustic waves by the electron drift in a quantum well / A. A. Demidenko, V. A. Kochelap // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 1. — С. 11-24. — Бібліогр.: 32 назв. — англ. |
| work_keys_str_mv |
AT demidenkoaa amplificationoflocalizedacousticwavesbytheelectrondriftinaquantumwell AT kochelapva amplificationoflocalizedacousticwavesbytheelectrondriftinaquantumwell |
| first_indexed |
2025-11-26T02:05:55Z |
| last_indexed |
2025-11-26T02:05:55Z |
| _version_ |
1850607793856315392 |
| fulltext |
11© 1999, Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Semiconductor Physics, Quantum Electronics & Optoelectronics. 1999. V. 2, N 1. P. 11-24.
1. Introduction
The phenomena of sound amplification and generation
by the charge carrier drift [1] have been studied in detail
since the beginning of the 60�s [2], mainly in uniform bulk
semiconducting materials (see, for example, the reviews
[3-6]). Different mechanisms of electron-phonon inter-
actions were considered. These include: piezoelectric in-
teraction, interactions via the deformation potential, the
electrostriction mechanism [7], etc. During the same pe-
riod, the general laws of sound amplification and gener-
ation were established. Multilayer and spatially bound-
ed materials were also studied. In these cases, the specif-
ic features of the formation of sound waves were taken
into account, particularly, the possibility of their local-
ization in separate layers or near the surfaces [8, 9]. The
thickness of the layers was assumed to be larger than a
micrometer size, and electrons in all cases without any
exceptions were considered as three-dimensional.
Quantum heterostructures, which are now the most
popular object of studies in the semiconductor physics,
consist of layers having dimensions of the order of a
hundred of Angstroms or less, and their electron sub-
systems are essentially quantized [10]. These two condi-
tions are essential both for the electron-phonon interac-
tions in general (see, for example, [11]), and for the acous-
PACS 63.22, 72.20, 73.20.D
Amplification of localized acoustic waves by the electron
drift in a quantum well
A. A. Demidenko, V. A. Kochelap
Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Kyiv, 252028, Ukraine
Abstract. We have investigated acoustic waves in a heterostructure with a layer embedded into a
semiconductor providing acoustic waves localization near the layer and electron confinement in-
side the layer. For layer thicknesses smaller than wavelengths we have obtained and analyzed the
dispersion relation for the localized waves. For electrons into the layer we have supposed that
parallel transport is semiclassical, while perpendicular electron motion is quantized. For two-di-
mensional confined electrons interacting with the acoustic waves we have solved the Boltzmann
equation in a parallel electric field. The solutions have been found for electron-phonon interaction
via deformation potential. The dispersion relation for coupled charge density and acoustic waves
has been analyzed. We have established conditions of amplification of localized acoustic waves
under the electron drift for two extreme cases: i) the only lowest two-dimensional subband is pop-
ulated, ii) a large number of the subbands are populated. We have found that the amplification
coefficient of the acoustic waves in THz-rigion is of the order of 100 cm-1. We have discussed the
results and compared them with acoustic waves amplification in bulk like semiconductors.
Keywords: acoustic waves localization, acoustic waves amplification, electron drift.
Paper received 21.01.99; revised manuscript received 17.05.99; accepted for publication 24.05.99.
A. A. Demidenko et al.: Amplification of localized acoustic waves by ...
1 2 SQO, 2(1), 1999
tic-electronic phenomena in particular. The latter include
interactions of electrons with sound flows, dragging of
electrons by phonons, amplification and generation of
sound by electrons.
Interactions of electrons with acoustic vibrations in
quantum heterostructures were studied, first of all, as
one of mechanisms of electron scattering, which controls
the low-field mobility and relaxation of low-dimension-
al hot electrons [11-13]. Two types of lattice vibrations,
quasi-bulk and those localized in the region of the elec-
tron confinement, were taken into account. The role of
such interactions was found to increase with reduction
of the electron dimensionality. In the heterostructures
based on III-V semiconductors the acoustic phonons may
control the electron relaxation of two- and one-dimen-
sional electrons up to the liquid nitrogen temperatures
in the electric field range of 100 to 200 V/cm.
The interaction of electrons with sound flows was used
in [14-17] both to determine the peculiarities of the elec-
tron-phonon interaction and to test such specific low-
dimensional effects as the quantum Hall effect, etc. Gen-
erally, there are few works devoted to acoustic-electron-
ic phenomena. To a great extent, this is associated with
the following. For typical experimental conditions, bulk
(or surface) waves are excited. The electrons are local-
ized within the quantum wells and wires. The spatial
zones of overlapping of the sound waves and electrons
are very narrow and, as a consequence, the effect of elec-
trons on the propagation of such waves is weak.
The latter fact is even more critical for the amplifica-
tion and generation of acoustic waves. To demonstrate
it, let us consider a semiconductor structure having the
dimensions L
x
×L
y
×L
z
with the electrons localized in a
quantum well perpendicular to the Z axis. When calcu-
lating the amplification (absorption), it is necessary to
consider the interaction of electrons with each acoustic
mode. For a mode propagating along the X axis, evi-
dently, amplification (absorption) will be proportional
to (d/L
z
) where d is the width of the quantum well. As
d<<L
z
, amplification (absorption) of the bulk mode is
extremely low. Thus, we come to the conclusion that the
bulk modes can not be significantly amplified (or atten-
uated) by the electrons.
To achieve amplification of an acoustic wave, it
should be localized in the layer where electrons are quan-
tized. This type of localization is possible in two cases:
(1) due to a difference between elastic properties of the
quantum well layer and the adjacent layers [18-20], and
(2) due to the electron-phonon interaction, which local-
izes the acoustic wave even in the conditions of initially
uniform elastic properties of the structure [21-23]. In both
cases, the degree of localization of vibrations near the
quantum well layer increases with the vibration frequen-
cy, which should result in increasing the amplification
factor for high-frequency acoustic waves. It will be shown
below that efficient amplification of acoustic waves by
the electron drift in the quantum wells is achieved in the
frequency range above 100 GHz.
It should be noted that an experimental technique
for detecting acoustic vibrations with frequencies of the
order of hundreds GHz has been already developed [24,
25]. At present, such oscillations are generated using pulse
techniques [24, 25]: an irradiation with an ultrashort (sub-
picosecond) laser pulse results in thermal stresses in the
area of light absorption, which then induces a coherent
(non-monochromatic) high-frequency acoustic signal
propagating in the material.
In this paper, it is suggested to amplify the high-fre-
quency acoustic wave localized near the quantum well
by the electron drift. A calculation is presented of the
amplification (absorption) coefficient and of the wave
localization parameter for the electron-phonon interac-
tion via the deformation potential.
In section II, free localized acoustic waves and those
induced by the variable electron density ),( trn
r
are con-
sidered. Section III discusses the interaction of electrons
in quantum and classical wells (layers) with the acoustic
waves. The function ),( trn
r
is specified, and the self-con-
sistent solutions for the electron-phonon system under
study are obtained. Numerical estimations of the coeffi-
cient of amplification (absorption), parameter of local-
ization, and renormalization of the sound velocity are
presented. The main results are summarized in Section
IV, while Appendixes I-III present some auxiliary results.
II. Localized acoustic waves
Assume that a continuous isotropic medium contains a
layer with a thickness of 2d, perpendicular to the Z axis,
whose properties are different from those in the bulk.
Let the first Lame constans are λ inside the layer and λ
outside. Further on, all parameters outside the layer will
be denoted by the same letters as those inside, but with a
dash above. For the sake of simplicity, to prevent a mix-
ing of the longitudinal and transverse solutions, we as-
sume for the second Lame constant µµ = . We also as-
sume that the dielectric constant εε = , and the density
of the material is ρρ = . Let us denote h=
+
+
µλ
µλ
2
2
.
The concentration of charge carriers in the layer con-
sists of the equilibrium term N(z) and a nonequilibrium
deviation n(x, z, t) = n(z)eiqx-iωt The same dependence on
x and t is assumed for all variables. In this section, we
assume n(z), and, hence, the right-hand sides of the equa-
tions, to be specific functions of z. The dependence n(z)
will be specified in the next section. Here, we assume only
that n(z) is an even function of z vanishing at the layer
boundaries: n(±d) = 0. Inside the layer, the variable part
of the stress tensor contains an additional
term bnikik δσ =/ , where b is the constant of the deforma-
tion potential.
Consider a longitudinal elastic wave whose vector of
the elastic displacement U
r
has two components U
x
, U
z
depending on x, z, t. The problem is assumed to be uni-
form with respect to y.
A. A. Demidenko et al.: Amplification of localized acoustic waves by ...
13SQO, 2(1), 1999
The equations of motion of the medium inside the
layer (|z| < d), have the following form (the accent de-
notes the derivative with respect to z) [27]:
( )
( ) ,
,
//22//222
/22//222
nUqqiqUqUqk
niqUqqiqUqUqk
zLTzTzLT
zLTxLxTL
β
β
=−−−
=−−−
where:
q
V
V q
V
V k q q
bq
L
L
L T
T
T L T L T
T
2
2
2
2
2
2
2 2 2
2
2
2
= = + =
= = − =
+
ω λ µ
ρ
ω
µ
ρ
β
λ µ
, , ,
, ,, ,
Outside the layer (|z| > d) the equations are the same but
the right-hand sides are equal to zero )0( =n , and the
substitution 222 1
LLL q
h
qq =→ should be made.
Elimination of U
x
from (1), (2) yields the equation
for U
z
:
( ).)( /2///
2
22//22//// nkn
q
UkkUkkU T
T
zTLzTLz −−=+− +
β
(3)
From a general solution of Eq.(3), we choose the so-
lution matching in parity the right-hand side (solutions
of the opposite parity will not interact with the carriers):
( ) ( ),sinhsinh zyzkBzkAzU TLz ++= (4)
where
( ) ( ) ( ) ,
4
1
/2///
20
dznkn
q
e
kP
e
zy
i
T
T
z
zk
i
zk
i
i
∑ ∫
=
−
−−
′
= β
k k ki L T= ± ±, � are the roots of the characteristic poly-
nomial of Eq.(3)
( ) ( ) 222224
TLTL kkkkkkkP ++−= (5)
The calculation of y(z) is presented in Appendix 1:
( ) ( )
( ) ( ) .,
2
1sinh
0
sinhsinh0
22
//
2
−+
+
−
−
=
+ zkJ
k
zk
n
k
zk
k
zk
qq
n
q
zy
L
L
L
T
T
L
L
LTT
β
(6)
The definitions and properties of the integrals J± are
described in Appendix 1. Substitution of (4) in (2) fol-
lowed by integration gives the expression for U x :
( ) ( )U z
i
q
q
k
A k z k B k z z Cx
L
L T T= + + +
2
cosh cosh ,Ψ (7)
where ( ) ( ) .
1 /2
0
22
22
++−
−
=Ψ ∫ nyqdzzyqk
qq
z T
z
LT
LT
β
The constant C is determined by substitution of (4), (7)
into (1). According to (6)
( ) ( ) +
−
−
=+Ψ zkzk
k
q
qq
n
q
Cz TL
LLTT
coshcosh
0
2
2
22
//
2
β
( ) ( )+ −
−
q
k
n k z
q
k
J k z
L
L
L
L
2
2
2
0
1
2
cosh , . (8)
Outside the layer ( )z d> we choose the solutions de-
caying at z → ±∞ :
( )U
z
z
Ae Be
U
i
q
q
k
Ae k Be
z
k z k z
x
L
k z
T
k z
L T
L T
= +
= − +
− −
− −
,
,
2 (9)
The four constants A A B B, , , are determined by the
equations of boundary conditions (BC) at the layer
boundaries. The set of BC equations and its solution are
given in Appendix 2.
If the determinant of the system is not equal to zero,
then the equations determine the variables U U U Ux z x z, , ,
in the form of quadratures of n(z). In our model, carriers
interact only with the longitudinal waves, so we present
here the expressions for ( )zWU =div only, calculated in
Appendix 2:
( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ( )
( )
W z
q
k
A k z y C
bq J k d k z
F q
W z
q
k
Ae
bq J k d e
F q
L
L
L
L L L
L
L
k z L L
k z d
L
L
= − + ′ − − =
= −
+
= = −
+
+
− +
− −
2
2
2 2
2 2
2 2
cosh
, cosh
,
;
,
,
,
Ψ
λ µ ω
λ µ ω
(10)
where the factor ( )F q k k d k k dL L L Lω , cosh sinh= + is in-
troduced. In the absence of interaction (b = 0 or n = 0),
the set of BC equations is uniform, and the equality of
the determinant to zero (A 2.2) determines the law of
dispersion of free waves able to propagate in our compos-
ite medium. The condition kT = 0 determines the trans-
verse wave which in our model ( )µ µ= remains to be
(1)
(2)
A. A. Demidenko et al.: Amplification of localized acoustic waves by ...
1 4 SQO, 2(1), 1999
bulk-type. The condition ( )F qω , = 0 determines the dis-
persion law of the longitudinal localized wave which de-
cays outside the layer as e k zL− . Therefore, the condition
0222 >−= LL qqk should hold, and q qL> . The factor F
can turn to zero only if k q qL L
2 2 2 0= − < , i.e. q qL< .
Hence,
LLL qhqq =< . Thus, the localized wave is pos-
sible only when the medium inside the layer is «softer»
than that outside, i.e.
h = +
+
>λ µ
λ µ
2
2
1. (11)
In the limiting case of vacuum inside the layer, the
wave is converted into two independent Rayleigh waves
in the upper and lower half-spaces.
The dispersion equation for the localized free wave
can be rewritten as:
( )F q q q d q q q q d q qL L L Lω, cos sin .= − − − − − =2 2 2 2 2 2 2 2 0 (12)
This equation has an infinite number of solutions
(branches) ( )ω q . All branches are located on the plane
( )ω q between the straight linesω = V qL , i.e. the disper-
sion law of the longitudinal wave in the inner medium, and
the line ω = V qL corresponding to the bulk wave in the
outer medium.
The lowest (zero) branch (at ( )h qd− <<1 1) has the
following form:
( )ω ≈ − −
V q q d hL 1
1
2
12 2 2
. (13)
The origin points of the subsequent branches are de-
termined by the values of q and qL :
.2,1;
1
;
1
1
0,0 =
−
=
−
= ννπνπ
νν
dh
h
q
dh
q L ... (14)
The behavior of the ν-th branch near the point of its
origin can be also easily found:
( ) ( ) ( ) ;...1
2
1 2
0
22
000
+−−−−+= νννννω qqhdqqqqVL
.0νqq > (15)
At large q, all branches asymptotically approach the
straight line q qL= . Figure 1 shows, for illustration, the
zero branch and five subsequent branches for the case of
a strong difference in the elastic properties (h = 2).
Let us estimate quantitatively the frequencies corre-
sponding to the points of origin of non-zero branches
for the case of our interest, with thin layers ( 2 100d ~ Å)
and the difference in elastic properties of 10%, i.e. h = 1.1:
ω π
10 10
13 1
1
1
10= =
−
−V q
V
h d
sL
L ~ ,
which corresponds to the THz- range. Analysis of the
electron-phonon interactions at such high frequencies re-
quires a quantum approach, so for the case of thin lay-
ers we shall deal only with the zero branch ( )ω q . Non-
zero modes in the general case are considered in [19, 20].
For the zero branch and small thicknesses
( )d q qL
2 2 1− << instead of (12) we have:
( )F k q h dL= − − =2 1 0 (16)
At a fixed frequency, the solution of this equation is:
( )q q q h dL L0
2 2 21
1
2
1= + −
, (17)
which is inversion of Eq.(13). It is worth noting that the
wave localization near the layer kL , increases quadrati-
cally with the frequency. Obviously, it follows from (16)
that
( )( ) ( )k q h d
V
h dL
L
= − ≈ −2
2
21 1ω ω
. (18)
In the case of a non-zero interaction with the carri-
ers, the right-hand side ( )R q appears in Eq.(16), which
in the general case is complex. We assume it to be small
in comparison to each terms in the left-hand side. The
form of ( )R q will be specified below. As a result, at a
Fig. 1. The zero branch and five subsequent branches of the
dispersion law of the localized acoustic waves at h = 2 accord-
ing to Eq.(12).
A. A. Demidenko et al.: Amplification of localized acoustic waves by ...
15SQO, 2(1), 1999
fixed frequency ω, the index of spatial decay kL and the
wave vector q will be modified:
( ) ( )k q h d R qL = − +0
2
01 ; (19)
( ) ( ) ( )
q q R q q h d
R q
q
− = − +
0 0 0
0
0
1
2
. (20)
The determinant of the set of BC equations is now
different from zero, and the above solutions of the non-
uniform equations are valid.
In Eq.(19), the first term describes the localization
caused by nonuniformities of the medium, and the sec-
ond one corresponds to the localization due to interac-
tion with electrons reported previously [22, 23]. If the
medium is uniform (h = 1) and there is no interaction
(R = 0), then kL = 0 , the dependence of W (10) on z dis-
appears, i.e,. we have the case of a uniform plane bulk
wave propagating along the X axis.
III. Electron states
In the previous section, it was assumed that the electron
force acting on the lattice and the electron concentra-
tion are given. In this section, we will express these func-
tions in the explicit form.
The electron states in the quantum well are described
by the 3D Schrödinger equation:
( ) ( ),,,ˆ
,,,0 zrEzrH ikii
rr
rr κκ
Ψ=Ψ (21)
where the Hamiltonian $H0 includes the kinetic energy
of electrons and the potential of the heterostructure ( )V z
creating the quantum well. ( )Ψi r z, ,κ
r
and Ei k
, denote
the wave function and the electron energy in the i-th two-
dimensional subband with a planar wave vector
r
k ,
r
r is
the in-plane radius-vector:
( ) ( )
m
k
ze
S
zr ikii
rki
ki 2
;
1
,
22
,,
hr rr
r +=∈Ε=Ψ χ . (22)
Here, S is the area of the layer with the quantum well,
the wave function of transverse motion ( )χ i z is nor-
malized by unity, i∈ is the energy of the bottom of the i-
th subband, and m is the effective mass.
In the presence of an acoustic wave, we should in-
clude in the total Hamiltonian of the system the poten-
tial energy consisting of the electrostatic energy and that
of the electron-phonon interaction:
( ) ( ) ( )Φ
r r r
r z t e r z t bW r z t, , , , , , .= +φ (23)
The dependence of Ô on
r
r and t is assumed to be
proportional to eikr i t
rr
− ω . The self-consistent electrical po-
tential φ is determined by the Poisson equation
( )′′ − = −φ φ π
ε
q
e
n z2 4
,
where e is the charge of the carriers. Taking into account
that outside the layer ( )n z = 0 , and using the BCs at the
layer boundary-continuity of φ and ( )∂φ
∂
ε ε
z
= we ob-
tain:
( ) ( )( ) ( )[ ]
( ) ( )[ ]
φ π
ε
φ π
ε
= + − <
= − >
−
+ − −
+ −
−
4
2
4
2
e
q
e J q d J q d qz J q z z d
e
q
J q d qd J q d qd e z d
qd
q z
, , cosh , ; ,
, cosh , sinh ; .
(24)
When considering the kinetics of electrons in the
field ( )Φ
r
r z t, , , we restrict ourselves to the case when lon-
gitudinal motion of electrons can be approximated qua-
si-classically. The relevant kinetic equations can be sim-
plified for the next limiting cases: (1) when the single low-
est subband i = 1 is occupied (the quantum limit, QW);
(2) when the number of occupied subbands is so large
that transverse motion can be also described quasi-clas-
sically � the case of the classical well (CW). It is shown
in Appendix III how to introduce the electron distribu-
tion function in these cases. The charge carriers are sub-
jected to the action of the resulting self-consistent (three-
dimensional) effective field
.*
3 W
e
b
E D ∇−−∇= φ (25)
Quantum limit (QW)
If the lowest subband is occupied and the distance to the
next subbands are so large that the potential (23) does
not cause any inter-subband transitions, then the quasi-
classical distribution of electrons can be characterized by
a single function ( ) ( )F r V tD2
r r
, , ,
r
h
r
V
m
k=
1
, satisfying the
kinetic equation. This, according to Appendix III, can
be written in the standard form [32]:
( ) ( ) ( ) ( ) ( ) .
1 22222
coll
DDDDD
t
F
V
F
rmr
F
V
t
F
=⋅
Φ
−⋅+
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
rrr
r
(26)
The potential energy of the 2D carriers is introduced
here as:
( ) ( ) ( ) ( )Φ Φ2 1
2
D r t dz z r z t
r r
, , , .= ∫ χ (27)
The distribution function ( )F D2
determines the total
2D concentration of charge carriers:
A. A. Demidenko et al.: Amplification of localized acoustic waves by ...
1 6 SQO, 2(1), 1999
( ) ( )
( ) ( ) ( )∫∑
==
=+=
,
2
2,,
2
,,
2
2
2
2 D
k
D
ss
VFd
m
tVrF
S
trnNtrN
h
rr
rr
r π
(28)
where ( )n r t n es s
iqr i tr rr
, = − ω . Using ( )F D2 , one can determine
the energy of the electron-phonon interaction, i.e. the
change in the energy of electrons in the deformation
field, δFint . Obviously, in the adiabatic approximation
(V VL T>> , ), the change of energy of a single quasi-clas-
sical electron at the point
r
r due to deformation can be
written similarly to Eq.(27):
( ) ( ) ( )δ χE r t b dz z divU r z t
k1 1
2
,
, , , .r
r r r
= ∫
The change in the energy of all electrons is equal to
( ) ( ) ( )
( ) ( ) ( )
δ δ
χ
E d r E r t
S
F r V t
b d rdzN r t z divU r z t
k D
k
int
,
, , ,
, , , .
= =
=
∑∫
∫
2
2
2
1
2
1
2
r
r r r
r r r
r
(29)
Therefore, the density of the energy of electron-
phonon interactions is equal to
( ) ( ) ( )δε χint , , ,= bN r t z divU r z t
r r r
1
2
.
The density of the free energy of the entire system con-
sists of the elastic deformation energy EDε , the electron
energy eε , and of the electron-phonon interaction δε int :
intδεεεε ++= eED . (30)
Using this relation, one can calculate the stress ten-
sor:
( ) ( ).,
2
1
int trNzb
uuu ij
ED
ij
ijij
ED
ij
rχδσ
∂
∂δε
∂
∂ε
∂
∂ε +=+= (31)
The equilibrium two-dimensional concentration Ns
included in ( )N r t
r
, results only in a static deformation,
which will not be considered here. In section II, we in-
troduced a non-equilibrium addition to the 2D concen-
tration in the form ( )n z eiqr i t
rr− ω . According to (28) and (31),
the amplitude of this dependence, present in the equa-
tion of lattice motion, is
( ) ( )n z n zs= χ1
2
. (32)
Thus, we obtain a self-consisted set of equations:
equation for sound (1), (2) and the Poisson equation with
the right-hand sides determined by (32), the kinetic equa-
tion (26) with the potential energy ( )Φ 2D , determined
from equations (23), (27), and the relation (28) for the
2D carrier concentration.
The kinetic equation (26) for the 2D distribution func-
tion F F D= 2 will be solved in the approximation of the
relaxation time. Taking into account the assumed form
of dependence on rr and t, we will rewrite (26) as fol-
lows:
( ) ( )i q V F
e
m
E Å
F
V
F F V U N n es s
iqr i t
− + ⋅ + + ⋅ =
= − − − +
−
ω ∂
∂
τ
ω
r r r r
r
r r rr
*
&
,
0
0
1 (33)
where
r
rE
e r D
* = −
1
2
∂
∂
Φ is the effective field of the sound
wave acting on the 2D electrons (see Appendix III),
r
E0 is
the constant drift field, and τ is the relaxation time. The
argument of the equilibrium function F0 corresponds to
the shift of the center of the velocity distribution by the
velocity of the scattering impurities
r&
U (
r
U is the elastic
displacement vector of the medium). However, below we
will neglect this shift, because it is essential only at low
frequencies ω < −106 1s . Moreover, as shown in [28], this
contribution gives rise to a term in the electron current
which exactly compensates the ion current. At high fre-
quencies, we can neglect both, this shift and the ion cur-
rent, and assume the total current to be equal to the elec-
tron one. F0 also depends on the total 2D concentration
( )N r t
v
, parametrically. Thus, now the equilibrium func-
tion F0 is the Fermi function of the total ener-
gy 1
2
1 2
∈+=∈+ mVε , where 1∈ is the energy of the bot-
tom of the lowest subband occupied. The chemical po-
tential is calculated through the total 2D concentration
of electrons.
In the equilibrium state, according to (28), for de-
generate electrons (the degeneration temperature at the
parameter values chosen below is ~ 100 K) we have
N
m
d VF
m d
k T
s
F
B
=
=
+ −
+
=∫ ∫
∞
2
2
1
2
2
0 2
10
π π
ε
ε εh h
exp
º
( ) ,
212 FF
mm ε
π
ε
π
′=∈−=
hh
where ε F is the chemical potential. Therefore, the Fermi
energy is equal to the difference FF εε ′=∈− 1 related to
the equilibrium 2D concentration as
′ =ε π
F sm
N
h
2
. (34)
Then, the Fermi velocity VF and the Fermi electron
wavelength λF are
A. A. Demidenko et al.: Amplification of localized acoustic waves by ...
17SQO, 2(1), 1999
V
m
N
NF s F
s
= =h
2
2π λ π
, . (35)
The kinetic equation (33) will be solved in the linear
approximation with respect to small deviations E*
and ns , and linearly with respect to the drift field E0 .
Thus we assume the latter to be weak (the heating of the
electron gas is neglected, and the electron drift velocity
V Ed = =µ 0
e
m
τ
E0 is constant). The right-hand side of
(33) can be expanded as
( ) ( )
( ) ( )
F N n F N
F
N
n
F N
F
N
n F N
m
n
F
s s s
s
s
s
F
F
s
s s s
0 0
0
0
0
0
2
0
ε ε ∂
∂
ε ∂
∂ε
∂ε
∂
ε π ∂
∂ε
+ = + =
= +
′
′ = − h
.
(36)
The non-equilibrium distribution function can be
written as
F F F F= + +0 1
0
1
1,
where ( )F V1
0
r
is the non-equilibrium additive associated
with the drift in the external field E0 , independent of
r
r
and t, ( )F V r t1
1
r r
, , is a small non-equilibrium additive due
to the effective field E* and the non-equilibrium concen-
tration ns . It is easy to obtain for F1
0 that
F m
F
V Vd1
0 0= − ⋅
∂
∂ε
r r
. (37)
Using (36) and (37) and applying the standard lin-
earization procedure for Eq.(33), we obtain for F1
1 :
( )
( )
( ) ( ) ( )
F
F
e
V
Q
V
Q Q
i q V V
Q
E
m Q
i q V
Q
n
F
e m V V
Q Q
V E
V V
Q
n
d
d
d
s
d
d
s
1
1 0
2 3
2
3
2
0
2 2
2
2
1 1
1
1 1
= −
+ +
+
⋅
⋅ +
+ +
⋅
+
+ ⋅ +
⋅ +
⋅
∂
∂ε
τ
τ
π τ
∂
∂ε
τ π
r
r
r r r
r
h
r r
r r r r
h
r r
*
*
,
(38)
where Q denotes the following expression:
( )Q i qV= − −1 τ ω r r
. (39)
The linear density of the 2D current is:
( )∫
= .
2
2 2
2
VdVFV
m
ej
rr
h
r
π (40)
Taking into account the possibility of substituting in
the integrand
r
rV
F F
m V
∂
∂ε
∂
∂ε
∂
∂
2
0
2
0 1→ −
,
and the delta-like behavior of the derivative −
∂
∂ε
F0
,
the current density Ix in the longitudinal case, when the
vectors E E*,
r
0 and
r
q are directed along the X axis, can
be written as:
( )[ ]
[ ]
−+
+
+−++=
130211
2322121121
*
2
ipII
V
V
I
Ven
IIipII
V
V
IEj
F
dFs
F
ds
x
π
π
σ
(41)
where the angle integrals are denoted as:
( )
I
d
a ip
nm
n
m
=
+∫ cos
cos
φ φ
φ
π
0
2
. (42)
Here the following notations are introduced:
σ τ τ ωτs
s
F
e N
m
p ql q V a i s p a= = = = − = +
2
2 21; ; ; . (43)
At arbitrary p and ωt , the integrals in (41) can be
easily obtained from the integral I
s01
2
= π
. They are
equal to:
( )
I
ip
a
s
I
a
p
a
s
I
p
is
I
p
a
p s
p a I
a
s
I
pa
is
I
s
p
a
11 21 2 12 3
22 2 2 3
2 2
02 3 13 5
23 5
2
2
2
1
2
1
2
2 2
2
2 3
2
2
= −
= −
=
= − + + = =
= − +
π π π
π π π π
π
; ; ;
; ;
.
;
The continuity equation of the in-plane current has
the following form:
Lsx Venj = . (44)
Using (44), the current (41) can be presented as:
** Ej x σ= . (45)
The general form of the effective 2D conductance σ *
at arbitrary tql ω, can be easily obtained from the above
equations. Let us write down σ * for the two limiting cas-
es: 1) 1<<ql , then it should be all the more so for
A. A. Demidenko et al.: Amplification of localized acoustic waves by ...
1 8 SQO, 2(1), 1999
1<<= ql
V
V
F
Lωτ :
.
1
2
12
4
*
+
=<
i
qlV
VLq
V
F
L
L
γ
π
εσ (46)
2) 1>ql :
.
2
4
*
−=> i
V
V
L
V
F
L
q
L γ
π
εσ (47)
Here the drift parameter is introduced
L
d
V
V
−= 1γ (48)
and the two-dimensional screening length L:
s
eD
L
πσ4
2= ,
D is the diffusion coefficient, which in the 2D case, ac-
cording to the Einstein relation, is equal to:
2
2
FV
D
τ
= ,
hence
.
2 2
2
me
L
εh= (49)
Thus, L depends only on the effective mass m. For
GaAs m m= 0 0,067 , where m0 is the free electron mass
and L cm= ⋅ −05 10 6. . The effective electric field E* (45)
is determined by the integral of Eq.(25) over the layer
thickness taking into account (10) and (32), i.e.
( )E
iq
e D
* = − Φ 2 where ( )Φ 2D
is determined from equa-
tions (27), (23).
The wave function of the transverse motion for the
lowest subband is assumed to be:
( )χ π
1
1
2
z
d
z
d
= cos . (50)
Calculating the integral in (27) with the wave func-
tion (50) and the expressions for W divU=
r
and the po-
tential φ obtained earlier, taking into account the for-
mulae (A, 1.5) for the effective field in the case of a thin
layer we obtain:
( )E i
e
G
q
F q
ns
* ,= − −
4
2
1 2π
ε
ω (51)
where a constant coefficient G is introduced:
G
b
e VL
= ε
π ρ
2
2 44
. (52)
Substituting (44), (47), (51) into (45) and dividing by
n
s
, we obtain the dispersion equation at ql > 1:
(Lq + 1 + iη)F(q)=qGω
2
(1+iη) (53)
with the following notation used for simplicity:
γη
F
L
V
V
= . (54)
In the absence of the interaction, the right-hand side of
(53) is equal to zero, and the equation determines two non-
interacting waves: the 2D drift wave discussed previously
in [29] and the localized acoustic wave which was consid-
ered above. In the presence of interaction, both of these
waves do not represent the exact solutions, and mixed acous-
tic and charge waves emerge. If the interaction is weak, it
can be treated in terms of the perturbation theory, as it was
made above in section II for the acoustic wave. The right-
hand side of R(q), introduced in (19) and (20), according to
(53), is equal to:
22
2
2
2
)1(
1
1
1
)1(
)(
η
ηη
ω
η
ηω
++
+++
=
++
+=
Lq
i
Lq
LqqG
iLq
iG
qqR . (55)
For further consideration, it is reasonable to estimate
the relative role of the two terms in (19), (20). Let the
difference of the elastic moduli be 10 per cent, i.e.
h - 1 = 0.1, and the half-thickness of the layer is d =
= 100 Å. Then, as estimations show, the first term dom-
inates up to frequencies ω ≈ 1013 s-1. Keeping only the
first term in (20) and using Lqq ≈0 from (17), we ex-
press the left-hand side of (20) as
22
/
/
αωωαω
i
VV
VV
V
i
V
qq
LL
LL
LL
L +
−
−=−+=− (56)
where LV is the initial and /
LV is the renormalized sound
velocity outside the layer. According to (56), we have
for the renormalization of velocity:
L
L
LL
L
L
qq
Lq
Lq
hqdG
V
VV
V
V
=
++
++×
×−−=
−
≡
∆
,
)1(
1
)1)((
22
2
2
/
η
η
ω
. (57)
The absorption (amplification) coefficient of the
acoustic wave, α(ω), can be found as
A. A. Demidenko et al.: Amplification of localized acoustic waves by ...
19SQO, 2(1), 1999
{ }
)(
,
)1(
)1())((2
Im2)(
22
2
ω
η
ηω
ωα
Lqq
Lq
hLqqdGq
q
=
++
−=
=≡
(58)
It is evident, that if the Cherenkov criterion is met
V
d
> V
L
(59)
(h < 0) we get the amplification of the acoustic waves
localized near the quantum well: α < 0. Since ω∞Lq ,
the absorption (amplification) coefficient increases
monotonically with the frequency. When the drift ve-
locity rises, α(ω) also increases. At large supercritical
values, V
d
>>V
L
, it turns out that
0
0
N
E
V
V
F
d µ
α ∝∝ .
Formally, the equation (58) has the drift maximum
at η = ± (1+Lq) equal to:
Lq
Lq
hqdqG
+
−±=
1
)1)((2
max ωα . (60)
The lower sign corresponds to amplification. The
maximum of (58) is reached at )1(1 Lq
V
V
V
V
L
d
F
L +−=
− ,
which corresponds to drift velocities exceeding V
F
. Ob-
viously, under these conditions the anisotropic part of
the distribution function (34) is not small, and our anal-
ysis is no longer applicable. Therefore, in estimations
below, we will restrict ourselves to the drift fields E
o
such
that V
d
<V
F
.
For quantitative estimations, we will take the param-
eters of a single-crystal GaAs: ρ = 5.317 g/cm3; V
L
=
= 4.713⋅105 cm/s; ε = 12.85; m = 0.067m
0
. Assume also
that b = 15 eV, d = 10-6 cm; h = 1.1 Then, G =
= 10-26 s2, L = 0.5⋅10-6 cm. Let the frequency be ω =
= 100 GHz = 6.28⋅1011 s-1, then q
L
≈ 1.33⋅106 cm-1, Lq
L
≈
≈ 0.665. Also, let the equilibrium concentration of elec-
trons be N
s
= 2⋅1012 cm-2. Then, according to (35), V
F
=
= 5.6⋅107 cm/s, λ
F
= 180 �. Beyond any doubt, these pa-
rameters correspond to the quantum limit for electrons;
however, the wavelength of the sound wave 2π/q = 470 �
is greater than λ
F
, i.e., the acoustic wave and its interac-
tion with electrons can be considered using the classical
approach. If we assume the mobility to be µ =
= 105 cm2/V⋅s, then τ = 4⋅10-12 s and l = τV
F
= 1.6⋅10-4 cm,
i.e. it is a fortiori the case of ql >> 1. The condition (59)
is fulfilled in the fields higher than 4.7 V/cm. For the
parameters listed above and the field of 100 V/cm (V
d
=
= 107 V/cm < V
F
) we find for the acoustic wave amplifi-
cation α = − 60 cm-1.
The renormalization of the sound velocity, accord-
ing to (56), is negative and in our case is of the order of
10-4. The spatial decay index of the localized acoustic wave
in the transverse direction outside the layer is determined
by Eq.(19). At the chosen numerical values the wave in-
tensity decays by a factor of e at distances of about 5
wavelengths.
Consider also the case of a uniform medium (in terms
of the elastic properties). Then h = 1 and the first terms
in Eqs. (19) and (20) disappear. The real part of R(q)
determines the localization of the wave near the layer:
{ } { }
22
2
2
)1(
1
)(ReRe
η
ηω
++
++==
Lq
Lq
qGqRkL .
The localization increases rapidly with ω. For the
absorption (amplification) coefficient, according to (20),
(55), we have:
[ ]22
2
22
2
)1(
)1(
)()(2
2
Im2
η
ηηωα
++
++=
=
Lq
Lq
LqqG
q
R
. (61)
For the parameters chosen above, we find α = 3 cm-1. If
we select a frequency of 160 GHz, the value of the amplifi-
cation coefficient will be equal to that considered above for
h = 1.1.
The index of spatial decay in the transverse direction
Lk (19) is now complex. The phase front of the wave is
not flat: outside the layer the phase velocity is slightly
declined to the layer at V
d
< V
L
and in the opposite direc-
tion at V
d
> V
L
. The renormalization of the velocity
L
L
V
V∆
is now by the order of magnitude lower than in the case
of h = 1.1.
Classical potential well (CW)
If a large number of subbands in the well are occupied,
then transverse motion of electrons in the layer can be
considered quasi-classically. The potential energy of the
electron is the sum of the potential energy of the well and
the potential energy of the electron in the field of the
sound wave (23). The potential energy forming the well
is assumed to be zero inside the layer and infinite outside
of it. Also, we suggest that the dragging field XE ||0
r
is
applied. The 3D distribution function of electrons
),,,( tvzxf
r
, where v
r
is the 3D electron velocity, does not
depend on z inside the layer and is equal to zero outside
the layer, which is possible in the case of the mirror re-
flection of electrons from the boundaries. The bulk con-
centration is determined by Eqs. (A 1.4). The kinetic equa-
tion for the function f is written in the standard form in
the relaxation time approximation. Let us integrate this
equation with respect to z from -d to d. The integrals from
odd functions of z disappear; in particular, the term pro-
portional to z-projection of the effective field (25) and
A. A. Demidenko et al.: Amplification of localized acoustic waves by ...
2 0 SQO, 2(1), 1999
dz
df
become equal to zero. Introducing the distribution
function
∫−
=≡
d
d
tvxdfdztvzxftvxF ),,0,(2),,,(),,(
rrr
,
one can obtain for it the equation similar to (33) where,
however, v
r
is the 3D velocity. The solution of the ki-
netic equation for the function F is completely the same
as in the quantum case.
Finally, for the flat current in the X direction at
XqE ||||0
rr
we obtain
[ ]
−++
+
+−++= ∗
)(
2
)(
2
3
/
13
/
02
/
11
/
23
/
22
/
12
/
11
/
21
/
ipII
V
V
I
neV
EIIipII
V
V
IJ
F
dsF
x
F
d
sx σ
(62)
where the angle integrals are denoted as
∫− +
=
1
1
/
)( m
n
nm
ipxa
dxx
I , (63)
ss N
m
e
N
m
e
dd
ττσσ
22
22 === , Fermi velocity V
F
is
three-dimensional:
3/1
8
3
=
π
N
m
h
VF unlike that in (35).
Other notations are the same. The effective field in the X
direction is
Φ−= ∫−
∗
d
d
x dztzx
de
iq
E ),,(
2
1
(64)
where the potential energy Φ is determined by Eqs. (23)
and (24). Since the classical well is, generally speaking,
wider than the quantum one, we present here the result
of integration in (64) for an arbitrary thickness d. Using
(25), (A 2.3), (A 1.5), we obtain for the effective field:
s
L
L
qd
x n
qqd
qqd
qF
q
G
dq
e
qd
e
iE
−
−
−
−−−=
−
∗
2
22
22
2
22
2 sin
)(2
11
2
4 ω
ε
π
. (65)
At qd << 1 this equation is transformed into (51).
The general form of the effective conductance σ* at ar-
bitrary ql and ωτ is obtained from (45), (44), (62). Let us
present σ* for the cases:
1,
13
12
4
<<
+
⋅= ∗
∗
< ql
i
qlV
V
V
F
L
L
γπ
εσ
qL (66)
1,
2
2
4
>
−⋅= ∗
∗
> qli
V
VV
F
LL γπ
π
εσ
qL
(67)
Here, by analogy with the QW case, a parameter with
the dimensionality of length is introduced:
2
33
1
DL
d
=∗L (68)
L
3D
is the 3D Debye screening length:
3/1
3/2
2
2
2
3
8
3
4
−
= N
me
L D ππ
εh
. (69)
We should note that the expression (66) for ql << 1 corre-
sponds to the hydrodynamic approximation, when the bulk
current density is determined by the equation
neDnVeEJ d ∇−+= ∗
rrr
σ . The results (66), (67) are analo-
gous to those obtained earlier in [30] for the bulk waves.
Using the expression (51) for the effective field at small
thicknesses, one can easily come to the expression for
α(ω) in the form (58), where the substitution L → L *
(68) should be made.
IV. Discussion of the results
Above we have analyzed the acoustic waves localized near
the layer with a quantum well for charge carriers. The
problem was considered for a simple model of an elastic
isotropic medium. It was assumed that only one elastic
constant � the first Lame constant � in the layer may be
different from that in the surrounding material. In such
model, the localization appears if the layer is «softer»
than the ambient. If the elastic properties of the medium
are uniform, the localization arises due to the deforma-
tion electron-phonon interaction. In both cases, the lo-
calization increases with the frequency of the acoustic
waves.
We have considered the electron-phonon interaction
via the deformation potential. The electrons in the quan-
tum well interact with all possible acoustic waves; how-
ever, they affect most of the localized waves. The elec-
tron drift along the layer results in a slight renormaliza-
tion of the wave velocities and leads to their attenuation
or amplification. The necessary condition for amplifica-
tion of the localized waves is the same as the well-known
condition for Cherenkov radiation (59). A high mobili-
ty of carriers in the quantum wells results in low values
of the electric fields required to satisfy this condition. If
the amplification condition is satisfied, the magnitude
of amplification rises essentially with frequency (α ∝ ω5).
The significant factor here is the increase of degree of
wave localization with frequency: 2ω∝Lk . Along with
the acoustic wave, the oscillations of the electrostatic po-
tential and the charge density of the 2D electrons are
amplified. Different relations between the frequency ω
A. A. Demidenko et al.: Amplification of localized acoustic waves by ...
21SQO, 2(1), 1999
and electron relaxation time τ, between the wave vector
q and the free length l were analysed. It was found that
in the most important case of ql >> 1 at frequencies in
the range of 100 GHz the amplification coefficient of
the localized acoustic wave can reach the values of tens
of inverse cm. The equations obtained indicate that am-
plification continues to rise with frequency. However,
at higher frequencies the quantum treatment of longitu-
dinal electron motion is needed.
From the very beginning, we did not take into ac-
count in the equations the lattice attenuation of sound,
since in the lowest approximation in respect to dissipa-
tion, which is the only approximation considered here,
all dissipation mechanisms are additive. In the perfect
crystals of the GaAs type the losses of acoustic waves
are much less than 10 cm-1 up to the frequencies above 1
THz [31]. Thus, drift-induced amplification obtained is
much greater than the lattice sound attenuation, at least
in perfect crystals. The amplification factor does not
depend strongly on whether only the lowest subband is
occupied (the quantum limit), or many subbands are
occupied (the classical well).
It is of interest to compare the results obtained with
the calculation of the amplification coefficient in the bulk
material [30]:
−×
+
×=
L
d
D
D
L
D
V
V
qL
qL
V
mb
1
))(1(
)(
6 2
3
4
3
23
2
3 ρπ
ωα
h
where all notations are the same as in the previous sec-
tion. The amplification coefficient for the electron drift
in the quantum well (58) for the practically important
conditions qL < 1, |η| < 1 can be rewritten in a similar
manner:
−××
×−=
L
d
s
L
V
V
qLqd
N
q
h
V
mb
1))((
)1(
2
2
2/1
232/52/3
2
ρπ
ωα
h
.
In these equations, we have separated three factors:
the first one depends on the elastic properties, the effec-
tive mass of electrons and the electron-phonon interac-
tion; the second one is associated mainly with the screen-
ing, and the third one relates to the electron drift. Com-
paring the equations, we can conclude that the first fac-
tor in our case is lower by a coefficient of (h - 1). The
second factor in the frequency range ω ~ 100 GHz is of
the order of unity. The third factor is of the same form
in both cases. However, the mobilities of charge carriers
in quantum heterostructures can be much higher than
in the bulk material, so that the same supercritical con-
dition 1 � V
d
/V
L
can be reached at much lower electric
fields.
Thus, we have shown that in the case of a parallel
transport of electrons in the quantum well amplifica-
tion of the high-frequency acoustic waves, localized near
the well, is observed. This effect can be used for sources
of intense and strongly collimated high-frequency acous-
tic oscillations. Such short-wave coherent oscillations can
find a broad range of applications: generation of micro-
wave radiation, non-destructive testing of microelectronic
devices, modulation of optical signals, deflectors for la-
ser beam control, etc.
The authors are grateful to Dr. V. N. Piskovoy and
Dr. V. I. Pipa for discussions of some problems consid-
ered in this paper, and to E. V. Mozdor for assistance in
manuscript preparation. This work was supported by the
State Fund of Fundamental Research of Ukraine, Grant
No. 2.4/679 and, in part, by STCU grant No. 437.
Appendix I
Let us define integrals:
∫ −=
z
vqzqzv dzzneezqI
0
)()( )(),(
),(),(),( )()()( zqIzqIzqJ vvv −±=±
where
v
v
v
dz
nd
n =)(
.
Integration by parts yields:
),()0()(),( )1()1()1()( zqqIneznzqI vvqzvv −−− +−= .
Then
),,(cosh)0(2
)(2),(
)1()1(
)1()(
zqqJqzn
znzqJ
vv
vv
−
−
−
−
+
+−
−=
[ ]),(cosh)0(2)(2
sinh)0(2),(
sinh)0(2),(
)2()2()2(
)1()1(
)1()(
zqqJqznznq
qznzqqJ
qznzqJ
vvv
vv
vv
−
−
−−
−−
+
−
−
+−+
+−=+
+−=
.
The derivatives of the characteristic polynomial (1.4a)
are:
)(2)( 22/
LTLL qqkkP −±=± ; )(2)( 22/
LTTT qqkkP −=± m .
The sought function y(z) (1.4) is equal to:
)(
)(2
)( 222 z
qqq
zy
L
Φ
−
−=
ΤΤ
β
;
A. A. Demidenko et al.: Amplification of localized acoustic waves by ...
2 2 SQO, 2(1), 1999
[ ]
[ ].),(),(
1
),(),(
1
)(
)1(2)3(
)1(2)3(
zkJkzkJ
k
zkJkzkJz LL
L
Τ−ΤΤ−
Τ
−Τ−
−−
−−=Φ
κ
Using the recurrent equations written above, taking
into account that n ′(0) = 0, we obtain the expression (6)
in the main text. The integrals ),(),( )0( zqJzqJ ±± ≡ are
determined as:
∫ ∫−−
± ±=
z z
qzqzqzqz dzzneedzzneezqJ
0 0
)()(),( . (A 1.1)
The properties of these integrals are:
),(),( zqqJzqJ
dz
d
+− = ; )(2),(),( znzqqJzqJ
dz
d += −+ ;
∫ −+ =
z
zqJ
q
dzzqJ
0
),(
1
),( . (A 1.2)
Expansion at small
q
z
1<< yields:
[ ] ...
12
)0()0()0(),(
4
2//2 +++=−
z
nqnqzqnzqJ
[ ] ...
3
)0()0()0(2),(
3
2// +++=+
z
nqnznzqJ (A 1.3)
For a classical potential well (CW):
>
<==
=
dz
dz
d
n
n
zn
s
||0
||const
2
)0(
)( . (A 1.4)
Then
qd
qz
nzqJ s
sinh
),( =+ ; qd
qz
nzqJ s
)1(cosh
),(
−
=− . (A 1.5)
For the quantum well (QW):
>
<
=
dz
dz
d
z
d
n
zn
s
||0
||
2
cos
)(
2 π
. (A 1.6)
In this case, if, for the narrow well, (qd)2 is neglected
in comparison to π2, J
+
and J
-
are determined by the
same equations (A 1.5).
Appendix II
The boundary conditions at z = d (at z = - d the condi-
tions are identical because the functions are even):
1) ;xx UU =
2) zz UU = ;
3) xzxz σσ = ;
4) zzzz σσ = ; z = d.
Taking into account that µµ = , and also that n (±d)
= 0, the 3rd and 4th conditions can be simplified:
3) z
U
z
U xx
∂
∂
=
∂
∂
; 4) WWh = ; z = d
The set of BC equations may be expressed in the fol-
lowing form:
[ ]
[ ]
[ ]
−−=+
−=−
=−−+
+−=
=+++
−
−
−−
−
Τ
− Τ
.)()(
1
cosh
11
;)()(
1
sinh
);(sinhsinh
;)(
coshcosh
/
2
2/
2
22
Cddy
q
dk
k
Ae
k
A
dykd
q
dkAeA
dydkBdkAeBeA
Cd
dkBkdk
k
q
AekBe
k
q
A
L
L
L
dk
L
T
T
L
dk
TL
dkdk
TTL
L
dkdk
L
L
L
TL
L
ψ
ψ
ψ
The determinant of the system (A 2.1) is:
Fe
kk
k dk
LL
T L−−=∆ ;
dkkdkkF LLLL sinhcosh += . (A 2.2)
Assuming 0≠∆ , all amplitudes can be determined
from the system (A 2.1). For UdivzW ≡)( it is sufficient
to know only A and A , which are determined from the
3rd and 4th equations (A 2.1). As a result, we get:
+
+
−=
=−−+−=
−+
zk
F
dkJ
k
k
dkJ
q
q
CyzkA
k
q
zW
L
L
L
L
L
L
T
L
L
L
cosh
),(),(
2
)(cosh)(
2
2
/
2
β
ψ
−+ − ),(
2
)(
2
zkJ
k
q
zn L
L
L
;
(A 2.1)
(A 2.3)
A. A. Demidenko et al.: Amplification of localized acoustic waves by ...
23SQO, 2(1), 1999
.
)sinh(),()cosh(),(
2
)(
)|(|
2
2
||
2
dzk
LLLLL
T
zk
L
L
L
L
e
F
dkdkJdkdkJq
q
eA
k
q
zW
−−
−+
−
×
×−−=
==
β
Since we are interested in the range of wave vectors
near the resonance, where F is small, non-resonant terms
can be neglected in (A 2.3) up to frequencies ω ≈ 1013s-1.
Neglecting also the small terms (qd < 1) in the numera-
tors of the fractions in (A 2.3), (A 2.4), we get the equa-
tions (10) from the main text.
Appendix III
The quasi-classical N-particle distribution function f N
w
in a smooth field is obtained from the Wigner function
∫ −−+=
=
pi
NN
N
N
W
N
eqqñd
V
tpqf
rrr
h
rr
h
rrh γγγγπ
π
)
2
1
,
2
1
(
)2(
)2(
1
),,(
3
3
where ),(ˆ /qqpN is the density matrix in the X-presenta-
tion, Nrrrq
rrr
..., 21≡ , controlled by the equation
HH
t
p
i NN
N ˆˆˆˆˆ
ρρ −=
∂
∂
h , (A 3.1)
),(ˆ q
q
iHH
∂
∂−≡ h , N is the number of electrons. The
limit of smooth (in respect to the de Broglie wavelength),
fields is obtained at 0→h
r
:
[ ]collN
NNN f
p
f
q
H
q
f
p
H
t
f
=
∂
∂
∂
∂
+
∂
∂
∂
∂
+
∂
∂
, (A 3.2)
where H = H(p, q) is the classical Hamiltonian.
In the case of 2D electrons, it is convenient to use a
mixed presentation for the density matrix: // ,,, kiki
p , where
i, i/ are the numbers of subbands, /,kk
rr
are the 2D longi-
tudinal wave vectors. In the coordinate presentation for
the longitudinal variables
∑ −−=
kk
rki
kiki
rki
ii
eerr
rr
rrrrrr //
// ,
/
,
),( ρρ .
The equation for this matrix has the form
[ ],),(),(),(),(
),(
////
//////////
/
,
///
,
/////
,
//
/
,
∑ −=
=
∂
∂
ri
iiiiiiii
ii
rrHrrrrrrH
rr
t
i
r
rrrrrrrr
rr
h
ρρ
ρ
where ∑ ∫≡ //
// rd
r
r
r . If the only one subband i = 1 is
occupied and the transitions between the subbands are
not essential, we have // ,,
~
iiii
H δ . Then
[ ]∑ −=
=
∂
∂
//
),(),(),(),(
),(
///
11
//
11
///
11
//
11
/
11
r
rrHrrrrrrH
t
rr
i
r
rrrrrrrr
rv
h
ρρ
ρ
.
The latter equation is the complete analog of (A 3.1)
with 2D r
r
and /r
r
and the matrix elements of the 3D
Hamiltonian H
0
+ Ô. Here H
0
corresponds to (21), and
the matrix elements are calculated for the functions χ
1
(z):
),(ˆ
),()()(
2
)2(
1
12
22
111
rH
zrzV
dz
d
m
H
D
r
r
rh
+=∈
>=Φ+++∆−=< χχ
11
2
)2( ),(
2
)(ˆ χχ zr
m
rH r
D rhr
Φ+∆−≡ . (A 3.4)
Using (A 3.3) and (A 3.4), in the limit 0→h we ob-
tain the standard equation (26) for the single-electron
quasi-classical distribution function, where the velocity
of 2D electrons is pHV D rr
∂∂= /)2( , and the force acting
on these electrons is equal to
r
zr
r
r
H
D
D
r
r
r
r
∂
Φ∂
−>=Φ<
∂
∂−=
=
∂
∂−
)2(
11
)2(
|),(|
ˆ
χχ . (A 3.5)
Ô
(2D)
is given by the integral (27).
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