Theory of high-field electron transport in the heterostructures AlxGa₁₋xAs/GaAs/AlxGa₁₋xAs with delta-doped barriers. Effect of real-space transfer
Steady-state electric characteristics of quantum heterostructures AlxGa₁₋xAs/GaAs/AlxGa₁₋xAs with -doped barriers have been analyzed in this work. It has been shown that at high doping the additional low-conductive channels are formed in the barrier layers. Current-voltage characteristics of the st...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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irk-123456789-1199252017-06-11T03:03:04Z Theory of high-field electron transport in the heterostructures AlxGa₁₋xAs/GaAs/AlxGa₁₋xAs with delta-doped barriers. Effect of real-space transfer Korotyeyev, V.V. Steady-state electric characteristics of quantum heterostructures AlxGa₁₋xAs/GaAs/AlxGa₁₋xAs with -doped barriers have been analyzed in this work. It has been shown that at high doping the additional low-conductive channels are formed in the barrier layers. Current-voltage characteristics of the structure were obtained in the wide interval of applied electric fields up to several kV/cm being based on the solution of Boltzmann transport equation. It has been found that in the electric fields higher than 1 kV/cm the effect of exchange of the carriers between the high-conductive channel of the GaAs quantum well and the channels in the AlGaAs barriers becomes essential. This effect gives rise to the appearance of the strongly nonlinear current-voltage characteristics with a portion of negative differential conductivity. The developed model of heterostructure is adequate to those recently fabricated and studied by Prof. Sarbey’s group. The obtained results explain some observation of this paper. It has been found that the effect of electron real-space transfer takes place at both low temperatures and room temperatures, which opens perspectives to design novel type nanostructured current controlled devices 2015 Article Theory of high-field electron transport in the heterostructures AlxGa₁₋xAs/GaAs/AlxGa₁₋xAs with delta-doped barriers. Effect of real-space transfer / V.V. Korotyeyev // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 1. — С. 1-11. — Бібліогр.: 31 назв. — англ. 1560-8034 PACS 72.20.Ht, 72.20.Dp, 73.23.-b, 85.35.-p http://dspace.nbuv.gov.ua/handle/123456789/119925 DOI: 10.15407/spqeo18.01.001 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Steady-state electric characteristics of quantum heterostructures AlxGa₁₋xAs/GaAs/AlxGa₁₋xAs with -doped barriers have been analyzed in this work. It has been shown that at high doping the additional low-conductive channels are formed in the barrier layers. Current-voltage characteristics of the structure were obtained in the wide interval of applied electric fields up to several kV/cm being based on the solution of Boltzmann transport equation. It has been found that in the electric fields higher than 1 kV/cm the effect of exchange of the carriers between the high-conductive channel of the GaAs quantum well and the channels in the AlGaAs barriers becomes essential. This effect gives rise to the appearance of the strongly nonlinear current-voltage characteristics with a portion of negative differential conductivity. The developed model of heterostructure is adequate to those recently fabricated and studied by Prof. Sarbey’s group. The obtained results explain some observation of this paper. It has been found that the effect of electron real-space transfer takes place at both low temperatures and room temperatures, which opens perspectives to design novel type nanostructured current controlled devices |
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Korotyeyev, V.V. |
| spellingShingle |
Korotyeyev, V.V. Theory of high-field electron transport in the heterostructures AlxGa₁₋xAs/GaAs/AlxGa₁₋xAs with delta-doped barriers. Effect of real-space transfer Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Korotyeyev, V.V. |
| author_sort |
Korotyeyev, V.V. |
| title |
Theory of high-field electron transport in the heterostructures AlxGa₁₋xAs/GaAs/AlxGa₁₋xAs with delta-doped barriers. Effect of real-space transfer |
| title_short |
Theory of high-field electron transport in the heterostructures AlxGa₁₋xAs/GaAs/AlxGa₁₋xAs with delta-doped barriers. Effect of real-space transfer |
| title_full |
Theory of high-field electron transport in the heterostructures AlxGa₁₋xAs/GaAs/AlxGa₁₋xAs with delta-doped barriers. Effect of real-space transfer |
| title_fullStr |
Theory of high-field electron transport in the heterostructures AlxGa₁₋xAs/GaAs/AlxGa₁₋xAs with delta-doped barriers. Effect of real-space transfer |
| title_full_unstemmed |
Theory of high-field electron transport in the heterostructures AlxGa₁₋xAs/GaAs/AlxGa₁₋xAs with delta-doped barriers. Effect of real-space transfer |
| title_sort |
theory of high-field electron transport in the heterostructures alxga₁₋xas/gaas/alxga₁₋xas with delta-doped barriers. effect of real-space transfer |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2015 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119925 |
| citation_txt |
Theory of high-field electron transport in the heterostructures AlxGa₁₋xAs/GaAs/AlxGa₁₋xAs with delta-doped barriers. Effect of real-space transfer / V.V. Korotyeyev // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 1. — С. 1-11. — Бібліогр.: 31 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT korotyeyevvv theoryofhighfieldelectrontransportintheheterostructuresalxga1xasgaasalxga1xaswithdeltadopedbarrierseffectofrealspacetransfer |
| first_indexed |
2025-07-08T16:55:15Z |
| last_indexed |
2025-07-08T16:55:15Z |
| _version_ |
1837098563096543232 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 1-11.
doi: 10.15407/ spqeo18.01.001
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
1
PACS 72.20.Ht, 72.20.Dp, 73.23.-b, 85.35.-p
Theory of high-field electron transport in the heterostructures
AlxGa1–xAs/GaAs/AlxGa1–xAs with delta-doped barriers.
Effect of real-space transfer
V.V. Korotyeyev
V. Lashkaryov Institute of Semiconductor Physics, Department of Theoretical Physics,
41, prospect Nauky, 03028 Kyiv, Ukraine; e-mail: koroteev@ukr.net
Abstract. Steady-state electric characteristics of quantum heterostructures
AlxGa1–xAs/GaAs/AlxGa1–xAs with -doped barriers have been analyzed in this work. It
has been shown that at high doping the additional low-conductive channels are formed in
the barrier layers. Current-voltage characteristics of the structure were obtained in the
wide interval of applied electric fields up to several kV/cm being based on the solution of
Boltzmann transport equation. It has been found that in the electric fields higher
than 1 kV/cm the effect of exchange of the carriers between the high-conductive channel
of the GaAs quantum well and the channels in the AlGaAs barriers becomes essential.
This effect gives rise to the appearance of the strongly nonlinear current-voltage
characteristics with a portion of negative differential conductivity. The developed model
of heterostructure is adequate to those recently fabricated and studied by Prof. Sarbey’s
group. The obtained results explain some observation of this paper. It has been found that
the effect of electron real-space transfer takes place at both low temperatures and room
temperatures, which opens perspectives to design novel type nanostructured current
controlled devices.
Keywords: quantum heterostructures, real-space transfer, electron transport.
Manuscript received 30.10.14; revised version received 17.11.14; accepted for
publication 19.02.15; published online 26.02.15.
1. Introduction
In recent decades, modulation-doped quantum well
heterostructures received great attention due to their
unique electronic properties allowing numerous
applications in science, medicine and industry. One of
the important areas for these applications is the high-
frequency and high-power electronics [1-4], which
relates to fabrication of solid-state modulators and
generators of electromagnetic radiation in the sub-THz
and THz spectral ranges [5, 6]. Quantum wells made of
different semiconductor materials such as GaN/AlGaN
[7-9], GaAs/AlGaAs [10], InGaAs/InAlAs [11] show
good perspectives to realize these goals. For generation
of high-frequency radiation, several mechanisms of the
current instabilities are used, all of them are based on the
action of strong applied electric fields [12-16].
Among them, it should be noted the effect of the
electron real-space transfer (RST) that was observed in
the quantum heterostructures. Usually, this effect is
detected in measurements of current-voltage (I-V)
characteristics as an interval with negative differential
conductivity [17, 18]. The phenomena of RST is referred
to as transfer of hot electrons in the direction
perpendicular to heterolayers due to the heating effect in
the electric fields applied in the direction along hetero-
layers. There are number of papers published in the past
where RST was measured and explained [19-22].
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 1-11.
doi: 10.15407/ spqeo18.01.001
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
2
Recently, the effect of RST was observed in the
novel, high-quality n-AlGaAs/GaAs/AlGaAs hetero-
structures with δ-like doped layers in the AlGaAs
barriers [23]. The values of impurity concentrations and
position of δ-layer seem to be quite enough to form the
low-mobility side well (SW) and ensure the effective
exchange of carriers between SW and structural high-
mobility GaAs quantum well (QW). At electron heating,
SW will be effectively filled up by electrons from high-
mobility QW, as a result, the total conductivity of
structure is decreased that can lead to the non-linear
current-voltage characteristics with well-pronounced
saturation or even dropping behavior. Apparently, this
redistribution of electrons can also change the high-
frequency conductivity and transparency of
heterostructure for electromagnetic radiation, as well.
This idea can be used in development of very fast,
electrically-tuned modulator of high-frequency
radiation [24].
The aim of this paper is to offer the analytical
theory of the RST phenomenon in the heterostructures
with δ-doped barriers. To properly describe the high-
field electron transport under condition of the real-space
transfer, it is necessary to account the electron scattering
mechanisms at high kinetic energies and modification of
electrostatic potential at charge redistribution induced by
the applied strong lateral field. In particularly, the
calculation model of electrostatic characteristics of
heterostructure is formulated in Section 2. The model of
the high-field electron transport in the high-mobility
QW, and the model of electron redistribution between
QW and SW are discussed in Section 3. The results of
self-consistent calculations of potential profiles of the
structure, distributions of electrons at different applied
electric fields, field dependences of the drift velocity and
electron temperature, and the current-voltage
characteristics are presented in Section 4. Finally, the
main results are summarized in Section 5.
2. Electrostatics of quantum well heterostructure
with δ-doped barriers
Discussion of the electrostatic model for the
heterostructure under consideration is separated by two
interrelated parts. The first one relates to the description
of well GaAs layer, and the second one does to
description of barrier AlxGa1–xAs layers.
2.1. Electrostatics of the well layer
Let us consider the symmetric heterostructure
AlxGa1–xAs/GaAs/AlxGa1–xAs with layers grown along z-
axis. The structural GaAs quantum well has the width d.
The reference point for the z-variable corresponds to the
middle of QW. The electrostatic profile, )(z , and wave
functions of electron, )(z , inside QW can be obtained
by means of the self-consistent solution of Schroedinger
and Poisson equations as follows:
,)(
4
=
)(
,0=)()()(
)(
2
22
0
2
2
12
2
*
2
zn
e
dz
zd
zEzezV
dz
zd
m
D
e
(1)
where V(z) is the heterostructural built-in potential
energy, bVzV =)( at /2|<| dz and V(z) = 0 at
/2>|| dz (Vb is the conduction band offset GaAs and
AsGaAl x1x materials and one depends on chemical
composition, x), )(z is the electrostatic potential
induced by the finite values of the electron
concentration, D
en2 , inside QW. Generally speaking, the
several discrete energy levels of size quantization
corresponding to different minima of the conduction
band can exist in QW as a subject to x and d. In the
model reported here, we restrict ourselves by the
consideration of these parameters for the heterostructure
(see Section 3) at which only the lowest level of size
quantization, E1, corresponding to the -valley of GaAs
is found in QW and populated by electrons with the
effective mass, m
*
. This medium has the dielectric
constant 0 . The difference between dielectric constants
of the well and barrier materials will be neglected. It is
convenient to rewrite the system (1) in the following
compact dimensionless form:
,0)(
,0
22
1
z
Ea
D
e
V
(2)
where accents denote the spatial derivative with respect
to the dimensionless variable = z/d. The introduced
dimensionless parameters are as follows:
22
B
*2= dTkmaV , Tkdne D
e
D
e B0
222 4= ,
TkzVze B)()(= and TkEE B11 = , where T is
the equilibrium lattice temperature.
The solution of the system of nonlinear differential
equations (2) can be found numerically, using the
following boundary condition:
)(0=(0));(=(1/2)
);(0=(0));(0=(1/2)
dcv
ba
b
(3)
with TkVv bb B= . The condition (a) assumes that
electrons are strongly localized inside QW
(approximation of deep QWs), conditions (b) and (d)
are the results of the symmetry properties of the wave
function in the ground state and spatial symmetry of
the heterostructure relatively to the point z = 0,
respectively. The condition (c) is due to the choice of
the reference system. Finally, the unknown parameter
E1 is looked for from the normalization condition of the
wave function:
1=)(
21
1
d (e). (3)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 1-11.
doi: 10.15407/ spqeo18.01.001
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
3
It is easy to see that dimensionless electric field at
the interface GaAs/AlxGa1–xAs is defined as
2=(1/2) 2D
e (this parameter is important for
consideration of electrostatics in the barrier).
Thus, at a given electron concentration D
en2 , we
can completely solve the electrostatic and quantum-
mechanic problems inside this QW, in particular to
determine the position of quantization levels, spatial
dependences of both wave function and potential profile.
2.2. Electrostatics of the barrier layer
Let two donor-doped δ-layers are placed at the distance
id from the middle of QW. The impurity concentration
in each δ-layer is assumed to be Ni. The donor
concentration is supposed to be so large that wave
functions of localized electrons in the donor states are
overlapped and conductive channels are formed in the
barriers. In AlxGa1–xAs, the impurities of Si are shallow
donors, as it was reported in the ref. [25]. For example, for
the donor states with the energy 15 meV the electron
localization radius aH is close to 5 nm. Therefore, the
criteria of Mott transition 0.351/2 Hi aN (see ref. [26])
can be fulfilled at Ni > 211cm105 . Moreover, at
sufficient electron heating by the applied electric fields
(see Section 4), the major part of electrons can transfer
from the donor states to the conduction band.
Thus, one can assume that a positive charge of
ionized impurity layer creates the electrostatic potential
well for three-dimensional free electrons in the barrier
layer. The electrostatic potential in the barrier region
( 2> dz ) has to satisfy the Poisson equation written as
follows:
ii
D
e dzNzn
e
dz
zd
)(
4
=
)( 3
0
2
2
(4)
with three-dimensional concentration of electrons in the
barrier )(3 zn D
e . According to the symmetry of the
heterostructure under the consideration, it is sufficient to
solve the electrostatic problem for z > 0. The estimations
carried out bellow show that the effect of the electron
quantization inside SWs is insignificant, and therefore,
electrons can be treated as the three-dimensional ones.
Indeed, SWs formed in the barrier are wide and shallow,
meanwhile, electrons localized there have very low
mobilities and very short relaxation times, which leads
to the essential broadening of the levels. Assuming that
electrons are nondegenerate and obey the Maxwell-
Boltzmann statistics, we can write that
,)(exp
4
2
=)( BF33
3/2
B
*
3
e
ebD
e TkzeE
Tkm
zn
(5)
where *
bm and eT is the effective mass of electrons and
electron temperature in Γ-valley of AlxGa1–xAs, respec-
tively. Fermi level EF is measured from the bottom of the
unperturbed conduction band of barrier material and is
constant across the entire heterostructure. Note, the
distribution function (5) assumes that the drift velocity in
the region of SW is small.
It is convenient to rewrite Eqs. (4) and (5) in the
form of the dimensionless equations as follows:
.expexp= F
3/23"
eee
D
e (6)
In the latter, signs “–” and “+” correspond to the
spatial regions iD1/2 ( ddD ii = ) and iD ,
respectively, ,= TTee TkE BFF = and
dimensionless screening parameter
31/2
0
1/2
B
3/2*223 2= Tkmde b
D
e . The solutions of
Eq. (6) have to satisfy the following boundary
conditions:
,)(=)()(
,)()(=)(
,)(2=(1/2)
,)(0=(1/2)
''
2'
dDD
cDD
b
a
iii
iii
D
e
(7)
where TkdNe ii B0
24= and i is the electrostatic
potential corresponding to the bottom of the side well.
Here, the condition (a) follows from the reference
system of potential energy, and condition (b) is the result
of continuity of the electric field at the interface
AsGaGaAs/Al x1x . The conditions (c) and (d) express
the continuity and discontinuity of the potential and
electric field, respectively, in the point of the impurity δ-
layer. Using (7), solutions of (6) can be found in the
form of the implicit functions. In the region
iD1/2 ,
1])/(exp[),(2= F
22'
ee
D
e G ,
1])/(exp[),(2
1/2=
F
220
ee
D
e G
d
.
(8)
In the region iD ,
eeG 2exp),(= F
' ,
.2),(
2explog2=
F iee
eie
DG
(9)
Here ee
D
eeG F
5/23
F exp2=),( , i is the
function of the electron temperature and Fermi level and
is defined as follows:
.),(2),(/2)(log
2=),(
FF
222
eie
D
ei
eFei
GG
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 1-11.
doi: 10.15407/ spqeo18.01.001
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
4
Note that solutions (9) far from the δ-layer are
=)( and 0=)(' . Finally, from the second
equation of (8), we can obtain the following relationship
between e and F :
.
1/exp,2
1/2=
F
22
F,
0
ee
D
e
ei
i
G
d
D
(10)
Thus, at the given electron concentration D
en2 in
QW, one can able to solve the electrostatic problem
completely, if we determine one of the parameters e or
F . For this purpose, it is necessary to consider electron
transport in the quantum well and specify mechanisms of
carrier exchange between QW and this barrier. Within
the frames of simple consideration of the RST effect, we
suppose that due to the strong electron-electron
interaction there is a temperature balance between 2D
electrons in QW and 3D electrons in SW. Under this
assumption, the common electron temperature and Fermi
level between these two groups of electrons are
established even for non-equilibrium electron gas
in QW.
3. Transport model for 2DEG in QW
As known, the basic electric characteristics of hot electrons
are determined by non-equilibrium distribution function
)( pf
, where p
is the electron momentum. In the
frameworks of the classical kinetic theory, the distribution
function is given by solution of Boltzmann transport
equation, which in the case of steady-state and spatially
uniform applied electric field F
can be written as
}{}{=
)(
fIfI
p
pf
Fe latee
, (11)
where }{ fI ee is the collision integral of the electron-
electron (e-e) scattering, and the collision integral
}{ fIlat describes the electron scattering by phonons and
crystal lattice imperfections. As mentioned above, we
consider the case of a high electron concentration, so
that e-e collisions dominate over all the scattering
process, and the electron subsystem is characterized by
collective momentum and energy budget. Thus, to solve
the kinetic equation (11), we can use the electron
temperature approach with the distribution function
taken in the form of shifted Fermi-Dirac distribution [27]
edF TkpvEEp
tpf
B1)(exp1
1
=),(
(12)
with *2 /2=)( mpp
and three unknown parameters:
Fermi level EF, drift velocity Vd and electron temperature
Te. The parameters EF, Te and Vd are to be found from
the concentration, momentum and energy conservation
equations that can be obtained by multiplying Eq. (11)
by 1, px and p
2
, respectively, followed by integrating
them over all p
(in our case, p
is the two-dimensional
vector). The similar approach of the solution of
Boltzmann transport equation was applied at the analysis
of the hot-electron kinetics in the bulk-like GaN [28]. In
our case, these equations are
,,,=
,,,=
,,,=
F
F
F
2
edEd
edP
edN
D
e
QF
QF
Qn
(13)
where the specific form of NQ is determined only by
the form of distribution function (12). The determination
of the functions PQ and EQ already requires the
specification of scattering mechanisms. Since, we
mainly focus on the investigations of electron transport
within the range of high fields in polar material, then the
main dominant scattering mechanism that should be
considered is the electron scattering by polar-optical
phonons (for 2D electrons, it will be confinement
phonons). Therefore, all dimensionless parameters in the
system (13) are written through the inherent parameters
for the optical phonon scattering (see Appendix). We
describe other possible electron scattering (acoustic
phonons, roughness, dislocations) that can be important
in the low-field range, introducing the single elastic and
isotropic scattering mechanism with an effective
momentum relaxation time τp
. The latter value
corresponds to the typical transport time of the low-field
mobility in these heterostructures [23]. The explicit
forms of the functions EPNQ ,, are given in Appendix.
Thus, four non-linear equations (10) and (13)
allow us to completely solve the steady-state problem
of RST. In particular, at the given electron
concentration D
en2 , we can find the drift velocity,
electron temperature, position of the Fermi level and
corresponding value of applied electric field. Having
these characteristics, we can reproduce the electrostatic
potential profile (Eqs. (8) and (9)) and concentration
profile (Eq. (5)) in the barrier layer.
Finally, we need to shortly discuss electron
transport in impurity SWs. As mentioned above,
electrons inside SWs have low mobilities due to the
strong scattering by impurities, dislocations and optical
phonons at high temperatures. The typical values of
electron mobilities in SW SW are the order of few
hundreds cm
2
/V∙s [19, 23]. It allows us to use Ohm’s
law for the current flowing through SWs. The total
current is a sum of currents flowing in QW and SWs:
,2
=
3
2
FFne
FvFenJJJ
SW
z
D
e
d
D
eSWQWtot
(14)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 1-11.
doi: 10.15407/ spqeo18.01.001
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
5
where the factor 2 takes into account left and right
symmetric SWs, and dznn D
e
dz
D
e
33 =
is the total two-
dimensional electron concentration in SW. Accordingly,
the charge conservation law looks as
D
ei
z
D
e nNn 23 2=2 .
4. Results and discussion of steady-state problem
Now, let us apply the model of RST and the procedure
of calculations described in Sections 2 and 3 to
heterostructures AlxGa1–xAs/GaAs/AlxGa1–xAs with
parameters characteristic to those of the experimental
samples studied in the ref. [23]. In calculations, we will
use the following approximation for barrier height Vb =
1.155x + 0.37x
2
(in eV) and the effective mass
0
* )0.083(0.063= mxmb (m0 is the free electron mass)
[29]. At the beginning, it is useful to estimate the range
of values of d and x, where the single energy level model
for electrons in QW can be applied.
Two curves in Fig. 1 that are calculated for
unperturbed conduction band structure under the
condition (3(a)) bound four regions of parameters x and d,
where four different cases are realized. The region I
corresponds to the frameworks of the presented model –
there is the only one discrete level Γ1 in QW. Other
regions correspond to the parameters of the
heterostructure, when several energy levels corresponding
to the minima of Γ1 and satellite L-valleys can exist in
QW. For example, two discrete levels of Γ1 and L1 (region
II), two discrete levels of Γ1 and Γ2 (region III) and three
discrete levels of Γ1, Γ2 and L1 (region IV) are in QW.
Below, calculations are presented for the following
parameters: x = 0.2 and d = 12 nm at which the role of the
upper levels of size quantization can be neglected.
Fig. 1. The QW width vs alloy composition. Curve 1 found from
the equation 2*22 /2=)( dmxVb (condition when the second
level, Γ2, of Γ valley coincides with hight of barrier). Curve 2
found from the equation 2*22 /2=)( dmxV LLb (condition
when the first level L1 of L valley coincides with the barrier hight),
where 0
* 0.85= mmL and ΔΓL = 0.29 eV are the effective mass of
density of states for L valley and energy separation between Γ and
L valleys for GaAs (see ref. [29]), respectively.
Figs 2 and 3 provide the results of calculation of
the field dependences of transport characteristics and
spatial distributions of the electrostatic potential energy
and electron concentration.
As seen, the current-voltage characteristic of the
heterostructure Jtot (F) roughly consists of several
intervals with distinct behavior. The linear behavior
(Ohm’s law) is observed in the range of weak applied
fields of <0.1 kV/cm. In this case, the values of D
en2 and
eT are close to its equilibrium values. Note, at the
equilibrium conditions, i
D
e Nn 1.85=2 , i.e.
approximately 92% of all electrons are transferred from
both SWs to QW. Within the range of applied fields 0.1
to 1 kV/cm, both Jtot (F) and vd (F) have a sublinear
behavior, which can be explained by activation of the
electron-optical-phonon scattering mechanism.
However, in this range the effect of RST is still
inconspicuous, the electron concentration in QW is
decreased only by ≈8%. Starting from the field of
1 kV/cm, the effect of RST begins to develop, which
results in quasi-saturated behavior of the current-voltage
characteristic Jtot (F). At the same time, the current-
voltage characteristics that neglects the RST effect
shows a superlinear growth. Within the range 1 to
2 kV/cm, electrons in QW gain significant heating,
electron temperature is increased from 150 up to 350 K,
as a result the electron concentration in QW decreases
from 1.7 down to 1.3Ni (25-% decrease). At fields above
the threshold field, F ≈ 2 kV/cm, the well-developed
effect of RST leads to the dropping current-voltage
characteristic.
The similar value of threshold field was reported by
Hess and co-workers in ref.[20, 21] where the electron
transport in the AlGaAs/GaAs/AlGaAs structure was
studied by both numerical and analytical methods. In
contrast of the presented paper, those investigations
relate to the wide heterostructures (dGaAs ≈ 40 nm), for
which the electron transport inside GaAs layer was
treated as the three-dimensional one.
Theoretical predictions of the emergence of
current-voltage characteristics with a negative
differential resistance (NDR) are the main results of the
study of RST effect in heterostructures. In principle,
physics of appearance of NDR under RST and Gunn
effects are similar. In the Gunn effect, electrons transfer
from one valley of high mobility to another of low
mobility in momentum space. The most important
distinction is the fact that RST-effect device can be
controlled to a greater degree than with a device based
on the Gunn effect. The Gunn effect has the intervalley
deformation potential as a driven parameter, which is a
material property and cannot be changed. In the case of
the RST-effect device, the driven parameters are the
mobility of the AsGaAl x1x layer and height of barriers,
which can be controlled by adjusting the doping of the
layers and chemical composition x.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 1-11.
doi: 10.15407/ spqeo18.01.001
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
6
Fig. 2. Main panel: solid and dashed lines are the current-voltage characteristics Jtot (F) and JQW (F), respectively. Dash-dotted
line is the current JQW calculated neglecting the RST effect for the concentration D
en2 = 1.85Ni . Panel (a) shows the field
dependences of the electron temperature Te (F) and the drift velocity Vd (F). Panel (b) shows the field dependences )(2 Fn D
e and
Fermi level EF (F). Parameters of the heterostructure are following: impurity concentration Ni = 211cm105 , position di =
24 nm, mobility in SW μSW = 500 cm2/V∙s and ambient temperature T = 77 K. Optical phonon energy, 0 = 36 meV and
characteristic velocity, V0 = 4.4∙107 cm/s. Ohmic dependence shown in panel (a) corresponds to the mobility ≈3∙104 cm2/V∙s
(momentum relaxation time τp = 1.5 ps).
Fig. 3. Potential energy profiles )(ze (panel a) and electron concentration profile )(3 zn D
e (panel b). Black and grey lines
correspond to the F = 1 kV/cm and 2 kV/cm, respectively (these fields are marked by points in the characteristics Jtot (F) in
Fig. 2. Dashed and dash-dotted lines depict the positions of the Fermi level and first level of size quantization, respectively.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 1-11.
doi: 10.15407/ spqeo18.01.001
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
7
Fig. 4. Dependences of Jtot (F) (panel a) and )(2 Fn D
e (panel b). Black and grey lines correspond to di = 24 nm and 12 nm,
respectively, T = 77 K. Dash-dotted lines are the currents JQW without RST.
Fig. 5. The same as in Fig. 4 for T = 300 K.
The profiles of potential energy and electron
concentration in the barriers are illustrated in Figs. 3a
and 3b, respectively, for two characteristic fields
F = 1 kV/cm and 2 kV/cm. One of them is the field of
the onset of RST, and the second one is the threshold
field, respectively. As seen, the position of the
quantization level inside QW weakly depends on the
amplitudes of the applied field. The values of E1 are
practically the same for both fields and approximately
equal to –0.17 eV ( 04.75 ). However, the position of
Fermi levels is appreciably changed with F. For
example, at F = 1 kV/cm, the electron gas is
degenerated, the Fermi level is equal to –0.14 eV
( 03.89 ) and lies above the quantization level. At
F = 2 kV/cm, EF ≈ E1, and the electron gas has
intermediate degeneracy.
In the barrier, the positive impurity charge and
negative electron charge form the macroscopic electrostatic
wells with an asymmetric profile. The bottoms of SWs
correspond to the position of δ-layer. With an increase of
the applied field and, consequently, with increasing the
electron population inside the barrier regions, the bottoms
of SWs are lifted up, and SWs become shallower. In the
limiting case, when all electrons from QW transfer to the
barriers, the macroscopic electrostatic potential given by
Poisson equation (4) is equal to zero, and the resulting
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 1-11.
doi: 10.15407/ spqeo18.01.001
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
8
profile of the heterostructure will be rectangular. In the
opposite limiting case, when all electrons transfer from the
barriers to QW, the resulting electrostatic profile in the
barrier region will be like to the double charge layer. Within
the region iD<<1/2 , the potential energy has a linear
spatial dependence 2= 2D
e and in the region
iD> , the potential energy is constant,
i
D
ei D 2==)( 2 , and electron confinement
disappears.
As shown in Fig. 3b, the profiles of electron
concentration also have an asymmetric form in
compliance with the potential profile of SWs. The
maximum of electron concentration is reached at the
z = di, and electrons are more strongly localized in the
direction of QW than in the direction of a depth of the
barrier. So, at z = 20 nm (at –5 nm away from di) the
electron concentration is decreased almost by one order.
The same reduction occurs at z > 120 nm (+100 nm
away from di).
For the obtained electrostatic profiles in the barrier,
it is still necessary to check the possibility of electron
quantization. The estimations can be easy carried out by
modeling triangular well with the potential profile:
zeFzV z=)( at idz > and =V for idz < (shown in
Fig. 3a with the dots). For the parameters of Al0.2Ga0.8As,
the values of energy levels En,b are given by the simple
formula 2/35
, 101.68= znbn FpE (see, for example, the
ref. [30]), where pn is zeros of the Airy functions. Here,
energies are given in units of eV and Fz – in V/cm. For the
potential well depicted by the black dots, Fz = 10
4
V/cm
(lateral field F = 1 kV/cm) as a result the first three levels
with respect to the bottom of the well are equal to 0.5,
0.88, 1.2 in the units of 0 . For the potential well
depicted by the grey dots, Fz = 2∙10
4
V/cm (lateral field
F = 2 kV/cm), as a result, the first three levels are equal to
0.8, 1.4, 1.2 in the same units. As seen, for both cases the
inter-level distances are of the order of 05.0
(≈0.02 eV), which is comparable with the values of the
electron temperatures. Taking into account that the inter-
level distance for the original potential wells will be even
less, the quantization effect for electrons in electrostatic
SWs can be neglected. Thus, the presented model of
electron transport in the heterostructure with δ-doped
barriers is applicable.
Let us briefly analyze the effect of RST at other
parameters. Figs. 4 and 5 present the comparison of
current-voltage characteristics and field dependences of
electron population inside QW for two different
positions of the δ-doped layer di = 12 nm and 24 nm, as
well as two different temperatures T = 77 K and 300 K.
As seen, the population of QW is increased with
decreasing the distance between QW and δ-layer. Under
the equilibrium conditions, in the case of heterostructure
with di = 12 nm (grey curve in Fig. 4b), almost 100% of
electrons transfer from the barriers to QW. Note that in
the range of sub-threshold fields, 0 to 2 kV/cm, QW in the
heterostructure with smaller di has a greater population
(grey curve lies above black curve). However, at fields
above the threshold, QW in the heterostructure with lower
di is depopulated more rapidly. It reflects in behavior of
the current-voltage characteristics: currents obtained for
the heterostructure with di = 12 nm exceed those
obtained for the heterostructure with di = 24 nm. The
threshold fields for both heterostructures are practically
the same (2 kV/cm). At the fields above threshold, the
current is more sharply decreased for di = 12 nm than for
di = 24 nm.
At room temperature (see Fig. 5), general behavior
of the dependences Jtot (F) and Fn D
e
2 are modified as
compared with the case of nitrogen temperature. In
particular, the high-temperature current-voltage
characteristic has a wider interval of the ohmic behavior
0 to 1.5 kV/cm, and the threshold field is increased up to
the value 2.3 kV/cm.
Under equilibrium conditions, the same structure at
T = 300 K and 77 K has different populations of QW,
and they are equal to i
D
e Nn 1.4=2 and 1.84∙Ni,
respectively (black curves in Figs. 5b and 4b).
Moreover, within the range of applied fields 0 to
2 kV/cm, electron redistribution between QW and SWs
is less at the ambient temperature T = 300 K than at
77 K. For example, at T = 77 K, D
en2 varies from 1.85 to
1.33∙Ni, which is 25% from the initial value, and at T =
300 K, D
en2 varies from 1.4 to 1.09∙Ni, which is only
18% from the initial value.
It can be explained by the two following reasons.
First one is associated with different temperature
dependences of the density of states for three-
dimensional and two-dimensional electrons. For 3D
electrons, the density of states is proportional to 3/2
eT
Fig. 6. Dependences of Vd (F) (panel a) and Te (F) (panel b)
for the 2D electron gas inside QW. Black and grey curves
correspond to the temperatures 77 and 300 K,
respectively. The calculated values of low-field mobility
are 3∙104 and 8∙103 cm2/V∙s at T = 77 and 300 K,
respectively. Other parameters of the heterostructure are
the same as in Fig. 2.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 1-11.
doi: 10.15407/ spqeo18.01.001
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
9
and increases with temperature more rapidly than for 2D
electrons, which is proportional to Te. The second reason
is decreasing the electron mobility in QW due to
activation of electron scattering by polar optical
phonons. It proves the field dependences of Te (F) and
Vd (F) that are obtained for the cases of room and
nitrogen temperatures (see Fig. 6).
5. Summary
In the conclusion, it has been developed the theory of the
real-space transfer effect in heterostructures
AlxGa1–xAs/GaAs/AlxGa1–xAs with narrow quantum
wells (close to 10 nm) and δ-doped barriers. Under the
assumption of the fast carrier exchange between well
and barrier layers, the field dependences of electron
population in the quantum well, the Fermi level and
current-voltage characteristics were obtained. It has been
ascertained that, for the heterostructure with x = 2 and
doping of 211cm105 , the RST effect is well pronounced
at the applied electric fields larger than 1 kV/cm. Within
the interval of fields 1 to 2 kV/cm, essential
depopulation of QW with a variation of the Fermi level
occurs, which leads to saturation of current-voltage
characteristics. These results qualitatively explain the
measured non-linear current-voltage characteristics of
the heterostructures experimentally studied in the paper
[23]. It should be noted that the RST effect can be also
detected in the electro-optical experiments. For example,
variation of the Fermi level by applied electric field can
be directly observed in the optical absorption edge
measurements through detecting the Burstein-Moss
shift [31].
Under the fields higher than 2 kV/cm, the well-
developed effect of RST can lead to the emergence of
negative differential resistance. Moreover, NDR effect
can exist at both nitrogen and room temperatures. The
obtained results allow to suggest that heterostructure
AlxGa1–xAs/GaAs/AlxGa1–xAs with δ-doped barriers can
be used for desirable current control in different devices.
Acknowledgement
The work has been carried out in the framework of the
State Program “Nanotechnology and nanomaterials”
(2010–2014), project №1.1.7.18/14-M. The author
acknowledges the support by The State Fund for
Fundamental Researches (Grant F53.2/031). Also, the
author is sincerely grateful to Prof. V.A. Kochelap and
Dr. V.N. Poroshin for their interest and active discussion
of various aspects of this work.
Appendix: Summary of balance integrals
In the case of the degenerate 2DEG, the scattering
integrals are the bilinear functional in respect to the
distribution function:
.1,1
,
2
=
2
'pfpf'ppWpf
'pfp'pW'pd
S
I
op
opop
(15)
Here, p
is the longitudinal electron momentum.
For the intravalley electron scattering process by the
confinement optical phonon, the scattering probability is
given by [30]
),,(),(=),( p'pWp'pWp'pW opopop
,
)/()(
)()(
2
1
2
1118
=),(
22
02
1=
0
0
2
0
22
dkp'p
p'p
G
N
V
e
p'pW
k
k
op
(16)
where superscripts “+” and “–” correspond to the optical
phonon emission an absorption processes, respectively,
ω0 is the optical phonon frequency, 1
00 1)(exp=
N
expresses the equilibrium phonon distribution function
with TkB00 = (effects of the phonon heating are
neglected); κ∞ and κ0 are the high-frequency and low-
frequency permittivities, and 4)(8= 2 kkGk is the
form-factor with an odd k, which numerates the modes
of confinement optical phonons. Note, probability (16) is
written for rectangular infinitely-deep quantum well and
takes into account only the lowest subband of size
quantization.
Following the procedure described in Section (3)
and using explicit expression (16), the momentum and
energy balance integrals take the forms:
)(cos
),(1,)(cos)(cos11
)(cos),(
1
)1,()(cos1)(
4
=
0
00
0
2
1=0
3
0
kk
kk
k
k
op
P
ba
xfxfxx
dddx
x
rxfN
x
rxfN
ddx
G
P
Q , (17)
)(cos
),()1,(
2
1
),()1,(1)(
2
=
0
22
00
0
2
1=0
4
0
kk
kk
k
k
op
E
ba
xfxf
dddx
ba
xfNxfN
ddx
G
P
Q . (18)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 1-11.
doi: 10.15407/ spqeo18.01.001
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
10
In the latter, 1
0
2
0 1/1/2=
de
defines the characteristic electron-optical phonon
scattering time, 0
*
0 2= mP is the characteristic
electron momentum corresponding to the optical phonon
energy 0 . The introduced variables ka , kb and
kr
are the functions of dimensionless energies,
0= x , 20= dPkxk and they are determined as
follows: kk xxa 12= , xxbk 1)(2= and
2211= kkkk baxr . The distribution function
ed xx
xf
)(cos2exp1
1
=),(
F
(19)
is rewritten in the dimensionless variables ),( x , where
is the angle between the electron momentum and the
drift velocity. Other dimensionless parameters are:
0
*= PVm dd , 0B= ee Tk , 01FF = EE .
In the case of non-degenerate electron gas, the
balance integrals (17), (18) are essentially simplified:
,
2exp
1
121)(exp1)(
exp
2
=
10
10
0
2
1=
F
0
3
0
x
rxIxN
x
rxIxN
dx
G
P
Q
kede
kede
k
k
e
op
P
,
2exp
121)(exp1)(
exp=
22
00
22
00
0
2
1=
F
0
4
0
kk
ede
kk
ede
k
k
e
op
E
ba
xIxN
ba
xIxN
G
P
Q
(20)
where I0,1 denote the Bessel functions of the third kind.
In the simple case of the elastic and isotropic
electron scattering, the balance integrals are reduced to
the simple expressions:
pd
D
e
el
P vmnQ
*22 2)(2 and 0=el
EQ . The
latter reflects the fact that electrons cannot dissipate their
energy during the pure elastic scattering.
The momentum and energy balance integrals in
dimensionless form introduced in the system (13) are:
pd
op
PD
e
edP Q
Pn
Q
0
0
22F
)(2
2
=,,
,
op
ED
e
edE Q
Pn
Q
2
0
0
22F
)(2
1
=,,
,
where the dimensionless electric field is defined as
00= PeFF and relaxation time 0= pp .
Finally, the balance integral F,, edNQ is nothing
but definition of the two-dimensional electron
concentration, and for the distribution function (19) it
has the form:
ed
eD
eFedN
P
nQ
2
F2
2
02 exp1log
2
=,,
.
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