Electron relaxation and mobility in the inverted band quantum well CdTe/Hg₁₋xCdxTe/CdTe
Electron relaxation processes at nitrogen temperatures in CdTe/Hg₁₋xCdxTe/CdTe quantum well (QW) with an inverted band structure is modelled. In this structure, scattering by longitudinal optical phonons, charged impurities, acoustic phonons and interfaces were taken into account. It was found tha...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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| Zitieren: | Electron relaxation and mobility in the inverted band quantum well CdTe/Hg₁₋xCdxTe/CdTe / E.O. Melezhik, J.V. Gumenjuk-Sichevska, F.F. Sizov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2014. — Т. 17, № 1. — С. 85-96. — Бібліогр.: 45 назв. — англ. |
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| author | Melezhik, Ye.O. Gumenjuk-Sichevska, J.V. Sizov, F.F. |
| author_facet | Melezhik, Ye.O. Gumenjuk-Sichevska, J.V. Sizov, F.F. |
| citation_txt | Electron relaxation and mobility in the inverted band quantum well CdTe/Hg₁₋xCdxTe/CdTe / E.O. Melezhik, J.V. Gumenjuk-Sichevska, F.F. Sizov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2014. — Т. 17, № 1. — С. 85-96. — Бібліогр.: 45 назв. — англ. |
| collection | DSpace DC |
| container_title | Semiconductor Physics Quantum Electronics & Optoelectronics |
| description | Electron relaxation processes at nitrogen temperatures in CdTe/Hg₁₋xCdxTe/CdTe quantum well (QW) with an inverted band structure is modelled. In this
structure, scattering by longitudinal optical phonons, charged impurities, acoustic phonons
and interfaces were taken into account. It was found that for undoped and lightly doped
QWs (concentration of background n-type charged impurities in the well is 10¹⁴ – 10¹⁶ cm-³
or less), for x close to the band inversion value 0.16, the electron mobility grows
considerably when the QW width decreases. This mobility is higher for samples with
smaller concentrations of charged impurities.
|
| first_indexed | 2025-11-24T09:08:39Z |
| format | Article |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 85-96.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
85
PACS 73.21.Fg, 84.40.-x
Electron relaxation and mobility in the inverted band quantum
well CdTe/Hg1-xCdxTe/CdTe
Ye.O. Melezhik*, J.V. Gumenjuk-Sichevska**, F.F. Sizov
V.E. Lashkaryov Institute of semiconductor physics of NASU,
03028, Nauki av., 41, Kiev, Ukraine, +38(044)-525-18-10
*Corresponding author e-mail: emelezhik@gmail.com; **e-mail gumenjuk@gmail.com
Abstract. Electron relaxation processes at nitrogen temperatures in CdTe/Hg1-
xCdxTe/CdTe quantum well (QW) with an inverted band structure is modelled. In this
structure, scattering by longitudinal optical phonons, charged impurities, acoustic phonons
and interfaces were taken into account. It was found that for undoped and lightly doped
QWs (concentration of background n-type charged impurities in the well is 1014 – 1016 cm-3
or less), for x close to the band inversion value 0.16, the electron mobility grows
considerably when the QW width decreases. This mobility is higher for samples with
smaller concentrations of charged impurities.
Keywords: quantum well, mobility, HgCdTe, terahertz range.
Manuscript received 11.10.13; revised version received 15.01.14; accepted for
publication 20.03.14; published online 31.03.14.
Introduction
Terahertz spectral range attracts great interest of
researchers worldwide. Nowadays, terahertz detection
systems are widely used not only in astronomy, but also
beginning to find their application in biology, medicine
and security. These areas of usage need light-weight,
compact, high-speed and sensitive detectors which do
not need deep cooling. Most of the existing THz
detectors (e.g., superconducting hot electron bolometers,
SIS (superconductor-insulator-superconductor)
structures, etc.) have high sensitivity (noise-equivalent
power NEP~10–15–10–19 W/Hz1/2) and high speed
(response time ~ 10–9–10–11 s) but need deep cooling
up to T = 0.1-4 K [1]. Semiconductor bolometer InSb-
based detectors have high sensitivity (NEP~10–17
W/Hz1/2) within the spectral range ( 1 – 4 m). But
these detectors also need deep cooling and are
characterized by relatively high response time ( 10–6 s)
[2]. Response of existing un-cooled THz detectors in
most cases is limited by times 10 ms (except of
Schottky barriers and FET-based THz detectors [3]) and
their NEP is of the order of 10–9 – 10–10 W/Hz1/2.
Heterostructures based on the narrow band-gap
semiconductors (like solid solutions of Hg1–xCdxTe) hold
a high promise to be prospective materials for creation
of THz detectors. High-quality Hg1–xCdxTe quantum
wells (QWs) are characterized by low effective masses
of localized electrons, high electron mobilities even at
temperatures T 77 K, and possess great potential for
detection of terahertz radiation.
The problem of growth of Hg1–xCdxTe QWs and
investigation of their properties is widely addressed in the
literature. Photoluminescence [4, 5] and photo-
conductivity [6] of Hg1–xCdxTe QWs grown on different
substrates (Si, GaAs, ZnTe, CdTe) have been studied. It is
shown in [7] that high-quality HgTe QW structures can be
used for all-electric detection of radiation ellipticity in a
wide spectral range, from far-infrared to mid-infrared
wavelengths. Measurements of electrical conductivity, the
Hall coefficient, and photoluminescence of ion-milled
Hg1–xCdxTe films were performed in [8]. Recently
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 85-96.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
86
existence of 2D semimetal in the quantum wells of HgTe
was revealed and measurements of cyclotron resonance in
this structure were performed in [9].
A lot of works are devoted to electron mobility in
GaAs 2D heterostructures. In [10], limits of mobility
improving in 2D GaAs structure were theoretically
exhibited for extremely low temperatures. At T = 1 K the
mobility achieved 108 cm2/V s and was limited by
background impurities of the reduced concentration
1012 cm−3. At liquid helium temperatures, the mobility
was dropped significantly and limited by acoustic
phonon scattering.
There are much less experimental data on mobility
in mercury–cadmium–telluride 2D structures. Mobilities
were studied experimentally in SL with thin wells [11]
with the direct band structure, but they had not get any
mobility increasing in comparison with 3D. In [12] it
was shown that inverted band structure should be
realised in structures with QW thicker then ~6.5 nm.
The high mobility (2.8105 cm2/(Vs)) was
observed in the QW CdTe/HgTe/CdTe of 16 nm width
with inverted band structure at the temperature 3K [13].
On the other hand, for bulk MCT it was shown [14] that
electron mobility increases in the neighborhood of
composition value of 0.16 up to 106 cm2/(Vs) at the
liquid nitrogen temperature.
Here, we try to define limiting mechanisms of 2D
mobility in CdTe/HgCdTe/CdTe hetero structures at
moderate cooling. We modeled electron relaxation
processes in CdTe/Hg1–xCdxTe/CdTe QWs with thick
wells and inverted band scheme to find the possibility to
increase the in–plane mobility in these structures. Our
studies are aimed to estimate the optimal QW parameters
for the creation of high-speed and moderately cooled
THz detectors, namely field-effect transistors with high
mobilities in shallow channel.
1. The QW band structure and properties
1.1. Band scheme and properties of QW
Inside CdTe/Hg1–xCdxTe/CdTe QW for compositions
0 < x < 0.16, the inverted band scheme is realized [15],
while the direct band scheme is realized in the barriers
(Fig. 1). All calculations are carried out for the liquid
nitrogen temperature T = 77 K. In this chapter, we
describe some properties of the system under
consideration.
The concentration of charged impurities in these
QWs can be about 1014 – 1016 cm–3 [6, 7, 16]. Due to the
inverted band scheme and high density of states for
heavy holes, the Fermi level lies higher than the bottom
of the conduction band in the well.
For undoped and lightly doped Hg1–xCdxTe at
liquid nitrogen temperatures, there are two dominant
relaxation mechanisms for localized electrons in bulk
crystals – scattering by longitudinal optical (LO)
phonons and scattering by charged impurities [14, 16].
With the growth of temperature, the role of scattering by
optical phonons increases, while scattering by charged
impurities becomes less important. In heterostructures,
additional important scattering on interface roughness
appears. In this paper, we discuss all these mechanisms.
Also, we estimate scattering by acoustic phonons to
prove that it is minor scattering mechanism for our
system.
The scattering of localized electrons in the QW
with an inverted band scheme qualitatively differs from
the scattering in the direct band heterostructures. In
direct band semiconductors, the conduction band is
formed by the levels with Г6 symmetry, while the light-
hole and the heavy-hole bands are formed by Г8 levels.
In semiconductors with the inverted band scheme, Г6
and Г8 levels change their positions – and now the
conduction band is formed by Г8 levels, the heavy-hole
band has the same symmetry as the conduction band and
touches the conduction band at k = 0, while the light-
hole band is formed by Г6 levels and lies below the
conduction band.
There are contradicting experimental results in
literature, which describe the valence-band offset Δ in
the HgTe/CdTe QWs. The most cited values are 0.35 eV
[19, 20] and 0.55 eV at x=0, they are supposed to change
linearly with the composition. In our calculations, we
use the value 0.55(1–x) eV [21, 22, 23] (see Fig. 1).
1.2. Interface levels and wave functions
In direct band QWs, all levels of localized electrons lie
above the bottom of the conduction band. In general, the
Fermi level lies in the band gap of such structures. Thus,
electron levels are located much higher than the Fermi
level, and their occupancy influences the electron
scattering processes only slightly.
Fig. 1. Band scheme of QW CdTe/Hg1–xCdxTe/CdTe along the
z axis of the QW, where L is the QW width, Δ – valence band
offset, VS – conduction band offset and EA – negative band gap
inside the QW, EB – positive band gap of the barrier. VS = (EB –
Δ – EA).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 85-96.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
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Fig. 2. Electron wave functions for the QW width L=50 nm. The horizontal axis corresponds to z coordinate; peaks of the wave
functions are at the hetero-interfaces. The vertical axis represents the magnitude of the wave function in arbitrary units (a. u.).
The upper row corresponds to the composition x=0 while the lower row corresponds to the composition x=0.12.
CdTe/Hg1–xCdxTe/CdTe QWs with the composition
0 < x < 0.16 have an inverted band scheme inside the
well and a direct band scheme in barriers [15, 17]. Thus
in such structures there exist one or two electron levels
which lie below the bottom of conduction band of the
well (see Fig. 3) due to the mixture of the states of
different symmetry and due to the change of the sign of
effective mass when crossing the interface [15]. These
levels are built from evanescent states in each of the host
layers and are localized on the interfaces (see Fig. 2).
They are called “interface” levels [15] and the energy of
the ground interface level reduces at small well widths
[18, p. 81]. Consequently, one or several levels of
localized electrons are located below the Fermi level
(see Fig. 3). Such position of these levels drastically
changes the occupation of them and influences the
electrons scattering.
20 30 40 50 60
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
E
F
E
ne
rg
y,
e
V
L, nm
n=0
n=1
n=2
n=3
n=4
Fermi level
E
C
Fig. 3. Dependencies of energy spectra and Fermi level on
the well width L. The temperature is taken to be T = 77 K, the
composition x = 0.12. EC = 0 is the bottom of the conduction
band and EV = –76.6 meV – top of the valence band in the
well.
It is important to note that for compositions x that
are close to zero, ground level electrons are localized at
interfaces of the QW, which leads to the importance of
interface scattering. On the other hand, ground level
electrons are localized inside the quantum well for the
compositions x within the range 0.1 < x < 0.16 (see Fig.
2). In this case, the probability to find electron at
interfaces is low and interface scattering is also
supposed to be small. In this paper, we consider
CdTe/Hg1–xCdxTe/CdTe QW, as an example, for
composition x = 0.12.
Principal features of electron relaxation in the
CdTe/Hg1–xCdxTe/CdTe QW are determined by the
influence of the ground levels occupation on the
scattering processes.
For undoped QW with the composition x = 0.12 at
the temperature T = 77 K, as shown in Fig. 3, the Fermi
level is evaluated by alignment of concentrations of
electrons and heavy holes. Fermi level varies from 1.8
meV from the Γ8 band bottom of electrons in the well for
QW of 20-nm width to 8.6 meV for QW of 60-nm
width. The evaluated concentration of electrons in QW
varies from 1017 cm–3 for QW of 20 nm width to 41016
cm–3 for QW of 60 nm width.
1.3. Influence of the misfit strain on electron spectra
To calculate the energy structure of CdTe/
Hg1–xCdxTe/CdTe quantum wells, it is important to
consider the band shift and band splitting due to the
strains in the heterostructure. These strains arise from
lattice mismatch between the materials of the well and the
matrix. To estimate such strains, we use the simple
approach that assumes that the in-plain lattice parameters
of the well material are forced to be the same as
appropriate parameters of the barrier material [24, 25].
Consequently, the biaxial strain in the layer plane
arises inside the well (x, y). The value of this strain can
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 85-96.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
88
be estimated from the simple formula
HgCdTe
HgCdTeCdTe
yyxx a
aa
ee
[25, 26], where aCdTe and
aHgCdTe are the lattice parameters of the appropriate
layers in the heterostructure. The values are as follows:
aCdTe = 0.6482 nm, aHgTe = 0.64605 nm at T=300 K [27].
For x = 0.12, aHgCdTe = 0.64631 nm.
With the values of the lattice parameters of the well
and barrier, we get the strain components exx = eyy =
0.0029274, ezz = 0.
According to [28, 34], biaxial strain leads to two
effects. The first effect is the alteration of the gap value
between Г6 and Г8 bands. The change of this gap is
given by [29]:
3
1
)(
i
iig eaCE (1.1)
where (C – a) is the difference of deformation potentials
of conduction and valence bands, respectively.
(C – a)HgTe = –3.69 eV [29], (C – a)CdTe = – 3.16 eV [30].
For Hg1–xCdxTe with x = 0.12, the value of this
parameter is found from the linear approximation
(C – a)HgCdTe = –3.63 eV. Effective band gap for x = 0.12
is –50.9 meV without strain. Taking into account strain
effect, the band gap reaches –72.1 meV.
The second effect of biaxial tensile strain [31] is in
splitting of electron and heavy-holes Г8 states. In this
case, the top of Г8 heavy-hole band shifts under the
bottom of the Г8 conduction band. The value of such
splitting in the point k = 0 is given by [28, 34]:
||2 xxbeE
g
(1.2)
From [32, 33], Eq. (1.2) and the Hooke’s law in the
simplest form we can estimate the modulus of
deformation potential b to be of the order of 1.54 eV. In
further calculations, we assume that the difference
between the estimated value of b for HgTe and the value
of this potential for Hg1–xCdxTe with the composition
x = 0.12 is negligible. The calculated splitting between
light and heavy ΔГ8 bands is about 9 meV. Thus, the
strain introduces the gap between electrons and heavy
holes. Consequently the gapless case can not be realized
even for x = 0.16.
2. Boltzmann transport
Presented in this section is the general theory that is the
basis for all further calculations of electron relaxation
times and mobilities. We start from the Hamiltonian of
the localized electrons being scattered and use the
Boltzmann transport equation (BTE) to obtain relaxation
times for longitudinal optical phonon scattering and
charged impurities scattering. It is assumed that external
electric field is applied in the plane of QW. In our
calculations, only electron-lattice interactions are taken
into account while electron-electron interactions are
neglected.
The Hamiltonian of localized electron can be
written as:
defHrFeHH
0 (2.1)
where H0 is unperturbed Hamiltonian, while Hdef
includes the local fluctuations of the electrostatic
potentials due to the defects and lattice vibrations, F
is
a weak, constant and homogeneous electric field. We use
BTE in the simplest form:
)()(
,
kfWkfW
f
Fe
f
m
k
t
f
dt
df
nknknnknkn
kn
nknr
n
nn
(2.2)
where ),( kn and ),( kn describe the initial and final
states of electron during the scattering process, k and
k are two-dimensional electron wave-vectors,
1
1
)(
exp)(
Tk
EkE
kf
B
Fn
n is the Fermi-Dirac
distribution function for localized electrons in the QW,
energy E depends on the wave-vector k via the
nonparabolic dispersion law, for example in the well this
law has the form
2222 )(3/8
2
1
)( PkkEEkE nAAn [17], where
P is the Kane matrix element equal to 8.310–8 eVcm.
Using the principle of detailed balance, one can obtain
that
knnkknkn WW . The distribution function
can be written as sum of symmetric and asymmetric
parts:
f = f S + f A (2.3)
The sum over the symmetric part f S in Eq. (2.2) is
equal to zero. We deal with the homogeneous system
that is in the steady state under a uniform electric field.
Changing the variable in the derivative in the left-hand
side of Eq. (2.2), we obtain:
.)(
)(
)(
1)(
)]()([
)(
)(
,
,
knnkkn
A
n
knnkA
n
A
n
kn
A
n
kn
nnknnk
n
n
n
Wkf
W
kf
kf
kf
kfkfW
kE
kf
Fk
m
e
(2.4)
In the present work, we use the simple qualitative
approach that operates with k = 0 wave functions to
calculate electron relaxation times and mobilities for
different scattering mechanisms. This approach neglects
effects of s-p hybridization. Nevertheless, the k = 0
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 85-96.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
89
wave functions correctly describe many principal
features of the band structure under consideration,
particularly, they correctly describe the localization of
electrons in the well and selection rules for interband
transitions. Moreover, the energy spectra obtained in
terms of the envelope functions approach [17] behave in
the same way as such spectra obtained in terms of the
88 kp method [37]. In our calculations, we use
nonparabolic dispersion law in order to describe
correctly the energy dependence of the density of states.
Our approach allows us to calculate and analyze the
transition matrix elements for all the considered
scattering mechanisms.
3. Two-dimensional electron relaxation
on LO phonons
Accounted in the calculations of this chapter is only one
scattering mechanism – scattering by longitudinal optical
phonons. This mechanism was mentioned above as one
of the dominant at the liquid nitrogen temperatures for
the bulk crystals of MCT. We calculated relaxation
times of two-dimensional electrons to reveal the
influence of changes in parameters of the QW on the
electron scattering processes. In a strict sense, we cannot
introduce the momentum-scattering time for inelastic
processes. But the relaxation time can be considered to
be a good qualitative estimate for the momentum
relaxation time, since phonon emission and absorption
result in large changes in the electron momentum.
The momentum relaxation time n(k) of localized
electrons can be introduced from [34] and Eq. (2.4) by
means of the momentum relaxation rate for electron gas
excited initially into the ),( kn
state:
knnk
n
W
k
kk
k
kn
cos
)(
1
,
(3.1)
where is the angle between the initial and final electron
wave-vectors in the process of scattering.
In [35] it was also proved that the relaxation time
approach could be used when phonon energy is higher
than the thermal energy kBT. Gelmont et al. compared
relaxation-time approximation and Monte-Carlo
simulations for polar-optical phonon scattering in GaN
and obtained a good agreement between these
techniques. Comparison between relaxation-time
approximation and Monte-Carlo simulation of electron
mobility in Hg1–xCdxTe at liquid nitrogen temperatures
for polar-optical phonon scattering was made in [36],
also a good agreement was obtained.
For 2D case for steady-state transport under an
uniform electric field, we have 0/ tf , 0 fr
. For
low-field SA ff , ff S , and the asymmetric part
of the occupancy function can be found from Eq. (2.4)
and Eq. (3.1):
nkn
A
n fF
e
kf
)( (3.2)
For the longitudinal optical phonon scattering,
transition probabilities
knnkW can be written from
Fermi’s golden rule:
])()([
||
2 2
LOnn
pheknnk
wkEkE
knHnkW
(3.3)
Where He – ph is the Hamiltonian of the electron-phonon
interaction, and LOw is the phonon energy.
Detailed calculations of the transition probabilities
and matrix elements for the longitudinal optical phonon
scattering were carried in [34]. To calculate relaxation
times for this scattering mechanism, we use the final
formula (6.141) from [34] where the states occupancy in
the QW is taken into account:
,
))/((
])()([
))(1(
),,(
2
1
2
1
)(
112
)(/1
22
2
,
0
2
Lmq
wqkEkE
qkfqd
mnnG
TN
L
we
k
LOnn
n
mn
LO
LO
n
(3.4)
where 0 = 20.5 – 15.6x + 5.7x2 and
= 15.2 –15.6x + 8.2x2 are static and high-frequency
dielectric permittivities [27, p. 126],
1exp/1)(
Tk
w
TN
B
LO
LO
is the concentration of
LO phonons at a given temperature, m is the quantum
number of the phonon mode, L is the QW width, n and
n’ are the quantum numbers of initial and final electron
levels during the scattering process, k
is the initial
wave-vector of electron in the layer plane of the QW.
In HgCdTe compounds, two types of longitudinal
optical phonons exist the first one is related to HgTe and
the other one – to CdTe matrices. These phonons have
slightly different energies – 1LOw = 17.1 meV and
2LOw = 20.8 meV, respectively [27]. In our
calculations, we assume that the relative concentration of
HgTe-related phonons is proportional to (1-x), while the
concentration of CdTe-related phonons is proportional to
x. Relaxation rates in Eq. (3.4) should be calculated for
each of these types separately, and then the summary
rate should be found from the relation
xkxkk LO
n
LO
nn )(/1)1()(/1)(/1 21
One should note that the relaxation time in Eq.
(3.4) depends on the electron kinetic energy via wave-
vector k
. This relaxation time determines relaxation of
the population of the state ),( kn
.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 85-96.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
90
In equation (3.4), G(n, n, m) is the square of the
overlap integral for electron and phonon wave functions:
22/
2/
)()()(
2
),,(
zzzdz
L
mnnG phnn
L
L
(3.5)
where )(zn and )(zn are the envelope wave functions
of the initial and final states of the localized electrons in
the scattering process and )(zph is the wave function
of the LO phonon, which describes the symmetry of the
phonon electrostatic field:
km
L
mz
km
L
mz
zph
2,sin
12,cos
)( (3.6)
The overlap integral G(n, n, m) describes the
parity selection rules in the process of electron
scattering. The integrand should be symmetric to get a
non-zero value of G. The relaxation time in Eq. (3.4) for
electrons from a level n takes into account the scattering
of these electrons within the level as well as transitions
of these electrons to other levels in the QW.
In these structures, the electron ground level
always lies below the bottom of the conduction band of
the well, while the first level appears inside the band
gap at large widths of QW [17]. With the decrease of
the QW width, the excited levels climb up, while the
ground level goes deeper under the Fermi level. That’s
why, 2D electron gas of the ground level is always
degenerated, and the degree of degeneracy is
determined by the energy interval between the ground
level and the Fermi one. The average kinetic energy of
electrons on the degenerated level is much higher than
the classical value kT for 2D electrons. We calculated
how the average kinetic energy of electrons of different
levels depends on the QW width, using the classical
formula:
.
)(
)()(
kf
kfkE
E
nk
nnkn
k
(3.7)
As can be seen from Fig. 3, the ground electron
level is always placed below the Fermi level, while the
first excited one is placed below it at L > 20 nm.
Dependencies of the average electron kinetic energy on
the well width L for three lowest levels of the QW are
presented in Fig. 4. One can see that the average kinetic
energy of the ground level changes from 3.82 kT to
5.34 kT, while varying the QW width. The average
kinetic energy of the first level varies from 1.18 kT to
3.24 kT while this energy for the second level is around
1.04 kT to 1.13 kT.
Relaxation times for electrons from the bottom of
three ground levels of the QW are presented in Fig. 5(a)
(higher levels are not considered because the level of
their occupancy is negligible). Relaxation times for
electrons with an average kinetic energy at these levels
are presented in Fig. 5(b).
20 30 40 50 60
0
1
2
3
4
5
A
ve
ra
g
e
ki
n
e
tic
e
n
e
rg
y
in
u
ni
ts
o
f
kT
L, nm
n=0
n=1
n=2
Fig. 4 Dependencies of the average kinetic energies of
electrons at the ground (n=0), first (n=1) and second (n=2)
levels of CdTe/Hg1-xCdxTe/CdTe QW on the well width L. The
temperature is taken to be T=77 K, the composition x=0.12.
EC=0 is the bottom of the conduction band and EV= - 76.6 meV
– top of the valence band in the well.
20 30 40 50 60
1E-11
1E-10
1E-9
1E-8
R
e
la
xa
tio
n
tim
e
, s
L, nm
n=0
n=1
n=2
a)
20 30 40 50 60
1E-12
1E-11
1E-10
R
e
la
xa
tio
n
ti
m
e,
s
L, nm
n=0
n=1
n=2
b)
Fig. 5. LO phonon scattering dependences for relaxation times
of electrons with zero in-plane kinetic energy from the ground
(n=0), first (n=1) and second (n=2) levels (a) and relaxation
times of electrons with the average in-plane kinetic energy
from these levels (b) in the CdTe/Hg1-xCdxTe/CdTe QW on the
well width L. Temperature T=77 K, composition x=0.12.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 85-96.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
91
The relaxation time of electrons from the bottom of
the ground level grows considerably when QW width
decreases (see Fig. 5 (a)). Its increase is caused by the
growth of the degree of degeneracy of the ground
miniband,which leads to decreasing the number of free
states where electrons with low kinetic energies can
scatter to. Not only the ground, but the first miniband
becomes degenerate at large QW widths, when this
miniband descends, below the bottom of the conduction
band of the well. However, degeneracy of the first level
disappears at the small QW widths, when this level is
high (see Fig. 3). All other 2D levels at Fig. 3 are
nondegenerate.
The drift mobility of electrons in the QW in the
applied in-plane electric field F
can be calculated by
averaging all the possible electron velocities in this QW.
Taking into account the quantization of the wave-vector
k along the QW axis and using Eq. (3.2), we can rewrite
the latter equation:
,
)(
)(
)(
||
1
)(
)(
||
1
2
2
2
2
*
kdf
kdfF
km
k
k
e
F
kdf
kdf
kf
kf
m
k
F
k
nn
nk
n
k
nn
k
nn
k
n
LO
nn
Ak
Ak
LO
, (3.8)
where relaxation times are given in Eq. (3.4); effective
masses and distribution function depend on the level
number n and the modulus of the in-plane wave-vector k .
High level of degeneracy and high average kinetic
energies of electrons of ground and first levels (at large
well widths) lead to the substantial contribution of lateral
transitions to the processes of electron scattering on LO
phonons. During this lateral transition, electrons with
huge kinetic energies are scattered to the levels energy
change to which from the initial level is much higher
than the phonon energy.
4. Electron mobility for charged impurities scattering
We calculated the mobility of electrons on charged
impurities in the QW CdTe/Hg1-xCdxTe/CdTe by
modifying the approach of [18], where it is used the
approximation T = 0 K, which allows to get
simplifications in the calculations. Here, it is considered
more general case. Using Fermi’s golden rule and the
principle of detailed balance, one can obtain:
.
))()((
2
2
knHnk
kEkEWW
def
nnknknknnk
(4.1)
We will find the solutions of the equations (2.4) and
(4.1) in the form of the equations (2.3), (3.2). It is shown
in [18] that Eq. (2.3) will be the solution of Eq. (2.4), if
relaxation times fulfill the linear equation [18]:
.
)0(2
jijj
k
i
i
k K
E
f
m
k
(4.2)
We will deal especially with elastic scattering,
because scattering by charged impurities changes only
the direction of the electron momentum, while the
magnitude of this momentum remains unchanged.
Transitions between different levels during the scattering
process are restricted under this assumption.
Consequently, coefficients Ki j will be equal to zero,
and Eq. (4.2) will be separated by several independent
equations for each energy level.
To calculate the relaxation time, we introduce the
additional wave-vector qkk
. From the
momentum conservation law, we obtain:
,)cos1(
2
sin2 222
kkqk
where is the
angle between the initial and final wave-vectors of
electron in the scattering process. We obtain the formula
for the relaxation time on electrons of the n-th level
scattered by charged impurities:
,)()cos1]([
)(][
1
2
1
)0(
0
2
)0(
0
3
0
dkdE
E
f
EE
qnkHnk
kdE
E
f
EE
sm
n
n
n
nn
qq
average
def
n
n
n
nn
n
n
(4.3)
where
2
sin20
kq . One should note that in the latter
formula nn Ef /)0( is not the Dirac delta function as it
was at T=0. To find the matrix element
0
2
qq
average
def qnkHnk , we follow the same
procedure as that described in [18] for T = 0. However,
we modify the screening function that describes the
screening in 2DEG to account the non-zero temperature.
According to the semi-classical approach, when q 0
[18, p. 210], the screening function can be found as:
qqq /1)( 0 (4.4)
where
F
e
E
ne
q
0
2
0
2
and
ke s
rn
2
)(
1]}/)exp[(1{ TkEE BFn . Evaluating the derivative
of the electron concentration for non-zero temperature,
we obtain the final equation for the screening function:
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 85-96.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
92
)0(
21
1)(
2
0
2
n
n f
me
q
q
. (4.5)
In the limiting case of T = 0 K, fn(0) = 1, and Eq.
(4.5) reduces to appropriate formula from [18]. Having
the explicit Eq. (4.5) for the )( q and substituting the
formula for the matrix element from [18] into Eq. (4.3),
we obtain the final formula for the relaxation time of the
localized electrons on the n-th level, which are scattered
by charged impurities at non-zero temperature:
.,,
2
sin2
2
sin2
2
)cos1(
][)(
)(][
1
max
min
2
2
000
)0(
0
)0(
0
2
l
l
z
z
iimpi
ll
n
n
nnn
n
n
n
nn
l
l
n
n
nzkgdz
qk
eeZ
d
E
f
EEkdE
kdE
E
f
EE
c
m
(4.6)
The concentration of l-th impurity species (charge
elZl) is denoted by cl, and the length over which these
impurity species are found in the heterostructure is
denoted by lll
z zzL minmax . The sum is carried out
over all kinds of the charged impurities in the system.
For CdTe/Hg1–xCdxTe/CdTe QW, one should use the
value Z = 1 or 2, having in mind Hg+ interstitials in n-
type material or Hg2- in p-type material [38, 39].
Distribution of the charged impurities in the system
is described by the form-factor gimp:
dzeznzqg izzq
niimp
||2 )(),.( (4.7)
where )(zn is the envelope wave function of the n-th
level electrons. The mobility of electrons for each of
these levels can be expressed as:
)( Fn
n
n E
m
e
(4.8)
The average mobility of electrons for only one
scattering mechanism (charged impurities scattering) can
be found then as:
k
nn
k
n
CI
nn
CI
kdf
kdf
2
2
(4.9)
In the n-type QW with the composition x = 0.12
and the concentration of background charged impurities
1014 – 1015 cm–3, the Fermi level varies from 5.3 up to
11.2 meV (see Fig. 3). According to our estimations,
background impurities affect electron scattering much
stronger than the remote delta-doped layer of the
equivalent charge density. Thus, background charged
impurities play an important role in the scattering for
these QWs. In further calculations, we will treat the case
of the background impurities scattering, because these
impurities are always present in the QW.
5. Estimation of minor scattering mechanisms
5.1. Scattering by acoustic phonons
In HgCdTe quantum wells can exist two channels of
electron relaxation via the acoustic phonons scattering.
These are scattering by the deformation potential
interaction mechanism and scattering by the
piezoelectric interaction mechanism [34]. However, for
the quantum wells, growth direction of which is [001]
axis, the piezoelectric interaction is absent [40, p. 48].
So, in this chapter, we will estimate the relaxation time
of electrons that scatter on acoustic phonons via
deformation potential interaction.
In layered heterostructures, acoustic waves consist
of extended and confined modes. Extended modes
propagate through the whole heterostructure in any
direction. Confined modes are localized in the layer
plane of the well and propagate along this plane.
However, due to the small differences between elastic
properties of the matrix and quantum well materials, this
confinement is weak. Thus, confined modes can be
neglected, while extended modes can be approximated
by plane waves. The energy of acoustic phonon can be
estimated as
L
sE Lacoustic
2
~ [34, 6.135], where
sL = 105 cm/s is the sound velocity in the layer plane. For
L = 20 nm, the acoustic phonon energy is two orders of
magnitude lower than the mean kinetic energy of the
ground level electrons (see Fig. 4). Therefore, the
scattering can be assumed to be elastic.
Thus, for estimations we used the simple model
[34, 6.137] that assumes the well to be infinitely deep,
acoustic phonons to be bulk-like and non-parabolicity of
the dispersion law to be neglected:
,)(1
2
1
1
)(2
),(
1
00,0
3
*2
00
kf
LCs
TkmaC
kn
nnnn
LL
B
s
(5.1)
where )(1 00 kfn is the factor that describes the
occupancy of the state electron scatters to; assuming the
scattering to be elastic [34], we can state )()( 00 kfkf nn .
The deformation potential constant (C – a)HgCdTe =
–3.63 eV is (see Eq. (1.1)); m* is the electron effective
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 85-96.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
93
mass, n0 and n are the numbers of electron levels and
00 /VsMC LL = 5.06251017 eVs/cm4 is the mean
elastic constant of the material (taken to be the same as in
HgTe), M0 and V0 are the mass and the volume of the
primitive cell of HgTe.
According to our estimates, the relaxation time of
the ground level electrons with the average kinetic
energy is close to 610–10 s for L = 20 nm and 3.810–10
s for L = 60 nm; the relaxation time of first level
electrons with the average kinetic energy is around
2.610–10 s for L = 60 nm. These times are much larger
than appropriate times for longitudinal optical phonon
scattering (see Fig. 5 (b)). Consequently, the acoustic
phonons scattering is small and can be neglected in the
further treatment.
5.2. Scattering on interface roughness
Very little is known about the microscopic structure of
the interface defects so they usually use very simple
models for the calculation of scattering by these defects.
All estimations are carried out for the well of the 20-nm
width, which is the minimal width presented in our
calculations.
To estimate the scattering by interface roughness,
we use the model presented in the book of Bastard [18].
In this model, the interface defect is assumed to have the
thickness h of one monolayer in the growth direction and
have extensions lx and ly in the layer plane. It is assumed
that these defects are located far away from each other,
so scattering events on different defects are independent
from each other. Calculations are made at T = 0 K in the
Electric Quantum Limit, which assumes that only the
lowest electron sub-band is occupied. In our system, at
small well widths electrons scatter from the ground
level, therefore this assumption is reasonable. We used
the approach from [18], which allows to take into
account the sharpness of the electron envelope function
at the interface:
,
)cos1]([sin
sin
2
sin)1(cos
2
sin
)
2
sin2(
)(16
)(
1
24
22
0
2
0
2
03
022
F
yFxF
F
h
defb
Froug
k
lklk
k
d
dzz
m
NVh
E
(5.2)
where kF is the Fermi wave vector, 0(z) – envelope
wave function of the ground level electrons, Vb – barrier
height, m0 – effective mass of the ground level electrons,
(k) is defined in Eq. (4.5), is the angle between
electron wave vectors before and after the scattering.
To estimate the relaxation rate from the latter
equation, one should know the average concentration of
interface defects Ndef and their average lateral
dimensions lx and ly. Lateral dimensions of the defects
were taken from [41], which is often used for
estimations of the interface scattering in quantum wells.
We have found no data about the concentration of
interface roughness defects in HgCdTe QWs.
Nevertheless, the authors [42] created HgCdTe
heterostructures with fixed charges at the interfaces with
the concentration within the range 1010 – 51010 cm–2.
We assume that the concentration of roughness defects
Ndef is the same as the concentration of interface fixed
charges in the samples of [42]. Thus, we obtain that for
ground level electrons with the average kinetic energy,
the relaxation time for interface scattering in the well of
20-nm width is close to 10–9 – 710–9 s. This relaxation
time is an order of magnitude higher than the appropriate
time for the longitudinal optical phonon scattering (see
Fig. 5 (b), thus the interface scattering can be considered
as negligible in the further treatment.
5.3. Influence of the value of the valence-band offset
Values of the valence-band offset Δ in the literature are
contradicting (see section 1.1). Then, it is important to
estimate the influence of variation of this parameter on
the mobility of localized electrons. We compared the
mobilities for charged impurities scattering and for the
LO phonons scattering for two QW widths – 20 and 60
nm. There are the values compared as to the valence-
band offset Δ1 = 0.55(1–x) eV and Δ2 = 0.35(1–x) eV.
According to our calculations, different values of the
valence-band offset lead to the mobilities that differ by
approximately 1.5 times at small well widths for each of
the scattered mechanisms. For the wide wells, variation
of the mobility with the change in the valence-band
offset is weaker.
Thus, we can see that the uncertainty in the value
of the valence-band offset strongly affects the magnitude
of the electron mobilities, and it is principally impossible
to obtain precise numerical results for relaxation times
and mobilities. Also, it is impossible to estimate the
strain values precisely, because they depend on the other
technological heterostructure layers and on the
heterostructure growth conditions.
You can also aim the reduction of stress in the
layers. Our calculations show that for the QW with the
concentration of n-type charged impurities of 1015 cm–3,
the value of strain sufficiently affects the mobility. For
example, in the well of 20-nm width, the mobility in the
absence of strains is in times greater than such mobility
in the presence of the strain (see section 6). Thus, the
absence of the strain leads to growth in the electron
mobility.
6. Results for electron mobility
in CdTe/Hg1-xCdxTe/CdTe quantum well
The average mobility of electrons in the CdTe/
Hg1–xCdxTe/CdTe QW that includes both the LO
phonons scattering (Eq. (3.9)) and charged impurities
scattering (Eq. (4.9)) can be found as:
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 85-96.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
94
,
2
2
kdf
kdf
k
nn
k
nn
(6.1)
where the summary mobility of the n-th level electrons
is found from the relation
CI
n
LO
nn /1/1/1 (6.2)
Results for the electron mobility for LO-phonon
scattering only (Eq. (3.9)) and results for the average
mobility that includes two scattering mechanisms (Eq.
(6.1)) are plotted in Fig. 6.
The electron mobility that takes into account only
the LO phonons scattering, grows substantially, when
the QW width decreases. Therefore, at small charged
impurity concentrations the LO phonons scattering
dominates, and the average electron mobility also grows
at small well widths. When the concentration of charged
impurities is greater, the growth of average mobility is
partially suppressed. For example, for the concentration
of n-type charged impurities 1016 cm–3, the average
mobility grows approximately two times only. These
results that describe variation of the average electron
mobility with the quantum well width can
be possibly used to optimize growth procedure for
CdTe/Hg1–xCdxTe/CdTe QWs with different properties
within the same technological line.
It is important to note that usage of more pure
samples of Hg1–xCdxTe with lower concentrations of
charged impurities allows one to obtain higher electron
mobilities for the charged impurities scattering. In this
case, the average electron mobility also is higher. Also,
one should note that in this paper we considered the
model sample where strain is not relaxed. In the case of
using the samples with buffer layers that partially relax
the strain, the strain-induced increase of the band gap
will be smaller, and the mobility of localized electrons
should increase by several times.
20 30 40 50 60
2E6
4E6
6E6
8E6
1E7
1.2E7
1.4E7
M
o
bi
liy
, c
m
2 /(
V
s)
Well width L, nm
A
B
C
Fig. 6. Electron mobility in the CdTe/Hg1–xCdxTe/CdTe QW.
The mobility includes longitudinal optical phonon scattering
(LOP) and charged impurities scattering (CI). Line A is for LOP
and CI scattering with the concentration 1015 cm–3, line B is for
LOP and CI scattering with the concentration 1014 cm–3, line C is
for pure LOP scattering mechanism. The parameters used are as
follows: temperature T = 77 K; composition x = 0.12.
Unfortunately, the authors do not know the
experimental data on the mobility of electrons in the
CdTe/Hg1–xCdxTe/CdTe quantum well with inverted
band structure at the nitrogen temperature. Therefore,
it is useful to compare electron mobilities in the
CdTe/Hg1–xCdxTe/CdTe QW with inverted band scheme
in the well (Fig. 6) with electron mobilities in other
Hg1–xCdxTe structures at T = 77 K. The electron mobility
in bulk undoped Hg1–xCdxTe with the direct band
scheme at the composition x = 0.2 is 2.5105 cm2/(Vs)
[27, p. 95]. The electron mobility in bulk HgTe (which
has the inverted band scheme) is 2.2105 cm2/(Vs) [27,
p. 95].
According to our calculations, the electron mobility
in the QW CdTe/Hg1–xCdxTe/CdTe with the
concentration of n-type charged impurities 1015 cm–3 is
not higher than 6–7105 cm2/(Vs) for the composition
x = 0.2, and it is close to 1–1.2106 cm2/(Vs) for the
composition x = 0. It was shown in [13] that the electron
mobilities in the QW CdTe/HgTe/CdTe (at T = 3 K) can
reach 2.8105 cm2/(Vs). From the comparison of these
mobilities one can see that the maximal mobility of
electrons in Hg1–xCdxTe could be reached in quantum
wells with the inverted band scheme inside the well.
One should note that, at very small QW widths
(about 10 nm and less [12]), the width of interfaces
becomes comparable with the width of the quantum
well. So the scattering by interface roughness [34,
p. 249] and the scattering by interface phonons [34, p.
287] become more important and the growth of mobility
can be suppressed.
Apparently, we can expect the highest mobility in
mercury–cadmium–telluride structures with the inverted
band scheme quantum wells at moderate concentrations of
charged impurities. Therefore, these structures are of
interest for the creation of semiconductor THz detectors
based on hot electrons operating under moderate cooling
(up to liquid nitrogen temperature) or ambient temperatures.
It is known that narrow gap semiconductors are promising
materials for direct THz detectors due to high electron
mobility, high carrier concentration and low effective
masses of electrons [3, 43]. In our case, we should expect
the electron relaxation time of about 10-11 s and scattering
mean free path of about 50 μm. For this relaxation time,
the QW structure provides a strong interaction with THz
radiation by the free-carrier Drude absorption mechanism
[44] in the frequency range of 100-700 GHz. The high
conductivity of degenerate electron gas on the metal type
structure provides resistance to the order of a few tens of
ohms, which will ensure a good matching with the planar
metallic antenna. A large mean free path carriers will
implement those structures technologically.
Therefore, the quantum wells with the inverted
band scheme HgCdTe are the promising structures for
THz detector design. Thus, it is possible to realize
detectors on the effect of the electron gas heating, such
as hot electron bolometers or field effect transistors with
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 85-96.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
95
a shallow channel on the basis of the in-plane transport
in the quantum well [45].
7. Conclusions
We have modelled electron relaxation processes in the n-
type CdTe/Hg1-xCdxTe/CdTe wide QWs with the
inverted band structure at the liquid nitrogen
temperatures and for x close to the band inversion value
0.16, for their potential usage as THz range detectors. It
was found that the longitudinal optical phonons
scattering and charged impurities scattering are
dominant mechanisms, while the acoustic phonons
scattering and interface roughness scattering can be
neglected. It was shown that the electron mobility could
reach values of the order of magnitude of 5*106 – 107
cm2/(Vs) for the well widths 60 – 20 nm in such
structure. These values could be estimated at the
concentration of the diluted charged impurity of 1014 –
1015 cm-3.
Usage of relaxing buffer layers allows one to reach
the partial relaxation of the strains in the quantum well
and to increase the mobility by several times. This
mobility is an order of magnitude higher than for the
QW with the direct band structure and bulk mercury-
cadmium-telluride. On the one hand, it places high
demands on the quality of the structure. On the other
hand, it provides the possibility to implement fast direct
terahertz detectors, for example, field effect transistors
with the shallow channel based on the quantum well or
hot electron bolometer operating at moderate cooling.
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 85-96.
PACS 73.21.Fg, 84.40.-x
Electron relaxation and mobility in the inverted band quantum
well CdTe/Hg1-xCdxTe/CdTe
Ye.O. Melezhik*, J.V. Gumenjuk-Sichevska**, F.F. Sizov
V.E. Lashkaryov Institute of semiconductor physics of NASU,
03028, Nauki av., 41, Kiev, Ukraine, +38(044)-525-18-10
*Corresponding author e-mail: emelezhik@gmail.com; **e-mail gumenjuk@gmail.com
Abstract. Electron relaxation processes at nitrogen temperatures in CdTe/Hg1-xCdxTe/CdTe quantum well (QW) with an inverted band structure is modelled. In this structure, scattering by longitudinal optical phonons, charged impurities, acoustic phonons and interfaces were taken into account. It was found that for undoped and lightly doped QWs (concentration of background n-type charged impurities in the well is 1014 – 1016 cm-3 or less), for x close to the band inversion value 0.16, the electron mobility grows considerably when the QW width decreases. This mobility is higher for samples with smaller concentrations of charged impurities.
Keywords: quantum well, mobility, HgCdTe, terahertz range.
Manuscript received 11.10.13; revised version received 15.01.14; accepted for publication 20.03.14; published online 31.03.14.
Introduction
Terahertz spectral range attracts great interest of researchers worldwide. Nowadays, terahertz detection systems are widely used not only in astronomy, but also beginning to find their application in biology, medicine and security. These areas of usage need light-weight, compact, high-speed and sensitive detectors which do not need deep cooling. Most of the existing THz detectors (e.g., superconducting hot electron bolometers, SIS (superconductor-insulator-superconductor) structures, etc.) have high sensitivity (noise-equivalent power NEP~10–15–10–19 W/Hz1/2) and high speed (response time ( ~ 10–9–10–11 s) but need deep cooling up to T = 0.1-4 K [1]. Semiconductor bolometer InSb-based detectors have high sensitivity (NEP~10–17 W/Hz1/2) within the spectral range (( ( 1 – 4 (m). But these detectors also need deep cooling and are characterized by relatively high response time (( (10–6 s) [2]. Response of existing un-cooled THz detectors in most cases is limited by times ( ( 10 ms (except of Schottky barriers and FET-based THz detectors [3]) and their NEP is of the order of 10–9 – 10–10 W/Hz1/2.
Heterostructures based on the narrow band-gap semiconductors (like solid solutions of Hg1–xCdxTe) hold a high promise to be prospective materials for creation of THz detectors. High-quality Hg1–xCdxTe quantum wells (QWs) are characterized by low effective masses of localized electrons, high electron mobilities even at temperatures T ( 77 K, and possess great potential for detection of terahertz radiation.
The problem of growth of Hg1–xCdxTe QWs and investigation of their properties is widely addressed in the literature. Photoluminescence [4, 5] and photoconductivity [6] of Hg1–xCdxTe QWs grown on different substrates (Si, GaAs, ZnTe, CdTe) have been studied. It is shown in [7] that high-quality HgTe QW structures can be used for all-electric detection of radiation ellipticity in a wide spectral range, from far-infrared to mid-infrared wavelengths. Measurements of electrical conductivity, the Hall coefficient, and photoluminescence of ion-milled Hg1–xCdxTe films were performed in [8]. Recently existence of 2D semimetal in the quantum wells of HgTe was revealed and measurements of cyclotron resonance in this structure were performed in [9].
A lot of works are devoted to electron mobility in GaAs 2D heterostructures. In [10], limits of mobility improving in 2D GaAs structure were theoretically exhibited for extremely low temperatures. At T = 1 K the mobility achieved 108 cm2/V s and was limited by background impurities of the reduced concentration 1012 cm−3. At liquid helium temperatures, the mobility was dropped significantly and limited by acoustic phonon scattering.
There are much less experimental data on mobility in mercury–cadmium–telluride 2D structures. Mobilities were studied experimentally in SL with thin wells [11] with the direct band structure, but they had not get any mobility increasing in comparison with 3D. In [12] it was shown that inverted band structure should be realised in structures with QW thicker then ~6.5 nm.
The high mobility (2.8(105 cm2/(V(s)) was observed in the QW CdTe/HgTe/CdTe of 16 nm width with inverted band structure at the temperature 3K [13]. On the other hand, for bulk MCT it was shown [14] that electron mobility increases in the neighborhood of composition value of 0.16 up to 106 cm2/(V(s) at the liquid nitrogen temperature.
Here, we try to define limiting mechanisms of 2D mobility in CdTe/HgCdTe/CdTe hetero structures at moderate cooling. We modeled electron relaxation processes in CdTe/Hg1–xCdxTe/CdTe QWs with thick wells and inverted band scheme to find the possibility to increase the in–plane mobility in these structures. Our studies are aimed to estimate the optimal QW parameters for the creation of high-speed and moderately cooled THz detectors, namely field-effect transistors with high mobilities in shallow channel.
1. The QW band structure and properties
1.1. Band scheme and properties of QW
Inside CdTe/Hg1–xCdxTe/CdTe QW for compositions 0 < x < 0.16, the inverted band scheme is realized [15], while the direct band scheme is realized in the barriers (Fig. 1). All calculations are carried out for the liquid nitrogen temperature T = 77 K. In this chapter, we describe some properties of the system under consideration.
The concentration of charged impurities in these QWs can be about 1014 – 1016 cm–3 [6, 7, 16]. Due to the inverted band scheme and high density of states for heavy holes, the Fermi level lies higher than the bottom of the conduction band in the well.
For undoped and lightly doped Hg1–xCdxTe at liquid nitrogen temperatures, there are two dominant relaxation mechanisms for localized electrons in bulk crystals – scattering by longitudinal optical (LO) phonons and scattering by charged impurities [14, 16]. With the growth of temperature, the role of scattering by optical phonons increases, while scattering by charged impurities becomes less important. In heterostructures, additional important scattering on interface roughness appears. In this paper, we discuss all these mechanisms. Also, we estimate scattering by acoustic phonons to prove that it is minor scattering mechanism for our system.
The scattering of localized electrons in the QW with an inverted band scheme qualitatively differs from the scattering in the direct band heterostructures. In direct band semiconductors, the conduction band is formed by the levels with Г6 symmetry, while the light-hole and the heavy-hole bands are formed by Г8 levels. In semiconductors with the inverted band scheme, Г6 and Г8 levels change their positions – and now the conduction band is formed by Г8 levels, the heavy-hole band has the same symmetry as the conduction band and touches the conduction band at k = 0, while the light-hole band is formed by Г6 levels and lies below the conduction band.
There are contradicting experimental results in literature, which describe the valence-band offset Δ in the HgTe/CdTe QWs. The most cited values are 0.35 eV [19, 20] and 0.55 eV at x=0, they are supposed to change linearly with the composition. In our calculations, we use the value 0.55((1–x) eV [21, 22, 23] (see Fig. 1).
1.2. Interface levels and wave functions
In direct band QWs, all levels of localized electrons lie above the bottom of the conduction band. In general, the Fermi level lies in the band gap of such structures. Thus, electron levels are located much higher than the Fermi level, and their occupancy influences the electron scattering processes only slightly.
Fig. 1. Band scheme of QW CdTe/Hg1–xCdxTe/CdTe along the z axis of the QW, where L is the QW width, Δ – valence band offset, VS – conduction band offset and EA – negative band gap inside the QW, EB – positive band gap of the barrier. VS = (EB – Δ – EA).
CdTe/Hg1–xCdxTe/CdTe QWs with the composition 0 < x < 0.16 have an inverted band scheme inside the well and a direct band scheme in barriers [15, 17]. Thus in such structures there exist one or two electron levels which lie below the bottom of conduction band of the well (see Fig. 3) due to the mixture of the states of different symmetry and due to the change of the sign of effective mass when crossing the interface [15]. These levels are built from evanescent states in each of the host layers and are localized on the interfaces (see Fig. 2). They are called “interface” levels [15] and the energy of the ground interface level reduces at small well widths [18, p. 81]. Consequently, one or several levels of localized electrons are located below the Fermi level (see Fig. 3). Such position of these levels drastically changes the occupation of them and influences the electrons scattering.
20
30
40
50
60
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
E
F
Energy, eV
L, nm
n=0
n=1
n=2
n=3
n=4
Fermi level
E
C
Fig. 3. Dependencies of energy spectra and Fermi level on the well width L. The temperature is taken to be T = 77 K, the composition x = 0.12. EC = 0 is the bottom of the conduction band and EV = –76.6 meV – top of the valence band in the well.
It is important to note that for compositions x that are close to zero, ground level electrons are localized at interfaces of the QW, which leads to the importance of interface scattering. On the other hand, ground level electrons are localized inside the quantum well for the compositions x within the range 0.1 < x < 0.16 (see Fig. 2). In this case, the probability to find electron at interfaces is low and interface scattering is also supposed to be small. In this paper, we consider CdTe/Hg1–xCdxTe/CdTe QW, as an example, for composition x = 0.12.
Principal features of electron relaxation in the CdTe/Hg1–xCdxTe/CdTe QW are determined by the influence of the ground levels occupation on the scattering processes.
For undoped QW with the composition x = 0.12 at the temperature T = 77 K, as shown in Fig. 3, the Fermi level is evaluated by alignment of concentrations of electrons and heavy holes. Fermi level varies from 1.8 meV from the Γ8 band bottom of electrons in the well for QW of 20-nm width to 8.6 meV for QW of 60-nm width. The evaluated concentration of electrons in QW varies from 1017 cm–3 for QW of 20 nm width to 4(1016 cm–3 for QW of 60 nm width.
1.3. Influence of the misfit strain on electron spectra
To calculate the energy structure of CdTe/
Hg1–xCdxTe/CdTe quantum wells, it is important to consider the band shift and band splitting due to the strains in the heterostructure. These strains arise from lattice mismatch between the materials of the well and the matrix. To estimate such strains, we use the simple approach that assumes that the in-plain lattice parameters of the well material are forced to be the same as appropriate parameters of the barrier material [24, 25].
Consequently, the biaxial strain in the layer plane arises inside the well (x, y). The value of this strain can be estimated from the simple formula
HgCdTe
HgCdTe
CdTe
yy
xx
a
a
a
e
e
-
=
=
[25, 26], where aCdTe and aHgCdTe are the lattice parameters of the appropriate layers in the heterostructure. The values are as follows: aCdTe = 0.6482 nm, aHgTe = 0.64605 nm at T=300 K [27]. For x = 0.12, aHgCdTe = 0.64631 nm.
With the values of the lattice parameters of the well and barrier, we get the strain components exx = eyy = 0.0029274, ezz = 0.
According to [28, 34], biaxial strain leads to two effects. The first effect is the alteration of the gap value between Г6 and Г8 bands. The change of this gap is given by [29]:
(1.1)
where (C – a) is the difference of deformation potentials of conduction and valence bands, respectively.
(C – a)HgTe = –3.69 eV [29], (C – a)CdTe = – 3.16 eV [30]. For Hg1–xCdxTe with x = 0.12, the value of this parameter is found from the linear approximation
(C – a)HgCdTe = –3.63 eV. Effective band gap for x = 0.12 is –50.9 meV without strain. Taking into account strain effect, the band gap reaches –72.1 meV.
The second effect of biaxial tensile strain [31] is in splitting of electron and heavy-holes Г8 states. In this case, the top of Г8 heavy-hole band shifts under the bottom of the Г8 conduction band. The value of such splitting in the point k = 0 is given by [28, 34]:
|
|
2
xx
be
E
g
=
D
G
(1.2)
From [32, 33], Eq. (1.2) and the Hooke’s law in the simplest form we can estimate the modulus of deformation potential b to be of the order of 1.54 eV. In further calculations, we assume that the difference between the estimated value of b for HgTe and the value of this potential for Hg1–xCdxTe with the composition x = 0.12 is negligible. The calculated splitting between light and heavy ΔГ8 bands is about 9 meV. Thus, the strain introduces the gap between electrons and heavy holes. Consequently the gapless case can not be realized even for x = 0.16.
2. Boltzmann transport
Presented in this section is the general theory that is the basis for all further calculations of electron relaxation times and mobilities. We start from the Hamiltonian of the localized electrons being scattered and use the Boltzmann transport equation (BTE) to obtain relaxation times for longitudinal optical phonon scattering and charged impurities scattering. It is assumed that external electric field is applied in the plane of QW. In our calculations, only electron-lattice interactions are taken into account while electron-electron interactions are neglected.
The Hamiltonian of localized electron can be written as:
(2.1)
where
includes the local fluctuations of the electrostatic potentials due to the defects and lattice vibrations, H0 is unperturbed Hamiltonian, while Hdef
is a weak, constant and homogeneous electric field. We use BTE in the simplest form:
)
(
)
(
,
^
¢
¢
®
^
¢
®
¢
¢
¢
¢
¢
-
¢
=
=
Ñ
×
-
Ñ
×
+
¶
¶
=
^
^
^
^
^
å
k
f
W
k
f
W
f
F
e
f
m
k
t
f
dt
df
n
k
n
k
n
n
k
n
k
n
k
n
n
k
n
r
n
n
n
r
h
r
r
r
h
(2.2)
where
and
)
,
(
^
¢
¢
k
n
and describe the initial and final states of electron during the scattering process, k( are two-dimensional electron wave-vectors,
is the Fermi-Dirac distribution function for localized electrons in the QW, energy E depends on the wave-vector k via the nonparabolic dispersion law, for example in the well this law has the form
÷
ø
ö
ç
è
æ
+
+
+
=
^
^
2
2
2
2
)
(
3
/
8
2
1
)
(
P
k
k
E
E
k
E
n
A
A
n
[17], where P is the Kane matrix element equal to 8.3(10–8 eV(cm. Using the principle of detailed balance, one can obtain that . The distribution function can be written as sum of symmetric and asymmetric parts:
f = f S + f A
(2.3)
The sum over the symmetric part f S in Eq. (2.2) is equal to zero. We deal with the homogeneous system that is in the steady state under a uniform electric field. Changing the variable in the derivative in the left-hand side of Eq. (2.2), we obtain:
.
)
(
)
(
)
(
1
)
(
)]
(
)
(
[
)
(
)
(
,
,
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å
k
n
nk
k
n
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n
k
n
nk
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n
A
n
k
n
A
n
k
n
n
n
k
n
nk
n
n
n
W
k
f
W
k
f
k
f
k
f
k
f
k
f
W
k
E
k
f
F
k
m
e
h
(2.4)
In the present work, we use the simple qualitative approach that operates with k( = 0 k( = 0 wave functions to calculate electron relaxation times and mobilities for different scattering mechanisms. This approach neglects effects of s-p hybridization. Nevertheless, the wave functions correctly describe many principal features of the band structure under consideration, particularly, they correctly describe the localization of electrons in the well and selection rules for interband transitions. Moreover, the energy spectra obtained in terms of the envelope functions approach [17] behave in the same way as such spectra obtained in terms of the 8(8 k(p method [37]. In our calculations, we use nonparabolic dispersion law in order to describe correctly the energy dependence of the density of states. Our approach allows us to calculate and analyze the transition matrix elements for all the considered scattering mechanisms.
3. Two-dimensional electron relaxation
on LO phonons
Accounted in the calculations of this chapter is only one scattering mechanism – scattering by longitudinal optical phonons. This mechanism was mentioned above as one of the dominant at the liquid nitrogen temperatures for the bulk crystals of MCT. We calculated relaxation times of two-dimensional electrons to reveal the influence of changes in parameters of the QW on the electron scattering processes. In a strict sense, we cannot introduce the momentum-scattering time for inelastic processes. But the relaxation time can be considered to be a good qualitative estimate for the momentum relaxation time, since phonon emission and absorption result in large changes in the electron momentum.
The momentum relaxation time (n(k()
of localized electrons can be introduced from [34] and Eq. (2.4) by means of the momentum relaxation rate for electron gas excited initially into the state:
^
^
^
¢
¢
¢
¢
®
^
^
^
^
q
¢
-
=
t
å
k
n
nk
n
W
k
k
k
k
k
n
cos
)
(
1
,
(3.1)
where ( is the angle between the initial and final electron wave-vectors in the process of scattering.
In [35] it was also proved that the relaxation time approach could be used when phonon energy is higher than the thermal energy kBT. Gelmont et al. compared relaxation-time approximation and Monte-Carlo simulations for polar-optical phonon scattering in GaN and obtained a good agreement between these techniques. Comparison between relaxation-time approximation and Monte-Carlo simulation of electron mobility in Hg1–xCdxTe at liquid nitrogen temperatures for polar-optical phonon scattering was made in [36], also a good agreement was obtained.
For 2D case for steady-state transport under an uniform electric field, we have
0
/
=
¶
¶
t
f
,
. For low-field
, , and the asymmetric part of the occupancy function can be found from Eq. (2.4) and Eq. (3.1):
n
k
n
A
n
f
F
e
k
f
r
r
r
h
r
Ñ
t
=
^
)
(
(3.2)
For the longitudinal optical phonon scattering, transition probabilities
^
^
¢
¢
®
k
n
nk
W
can be written from Fermi’s golden rule:
]
)
(
)
(
[
|
|
2
2
LO
n
n
ph
e
k
n
nk
w
k
E
k
E
k
n
H
nk
W
h
m
h
^
¢
^
^
-
^
¢
¢
®
¢
-
d
´
´
¢
¢
p
=
^
^
(3.3)
Where He – ph
is the Hamiltonian of the electron-phonon interaction, and is the phonon energy.
Detailed calculations of the transition probabilities and matrix elements for the longitudinal optical phonon scattering were carried in [34]. To calculate relaxation times for this scattering mechanism, we use the final formula (6.141) from [34] where the states occupancy in the QW is taken into account:
,
)
)
/
(
(
]
)
(
)
(
[
))
(
1
(
)
,
,
(
2
1
2
1
)
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1
1
2
)
(
/
1
2
2
2
,
0
2
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ï
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ü
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+
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+
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+
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ì
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ç
è
æ
e
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e
=
t
^
¢
^
^
¢
¢
¥
^
ò
å
L
m
q
w
q
k
E
k
E
q
k
f
q
d
m
n
n
G
T
N
L
w
e
k
LO
n
n
n
m
n
LO
LO
n
h
r
r
r
r
r
(3.4)
where (0 = 20.5 – 15.6x + 5.7x2 and
(( = 15.2 –15.6x + 8.2x2 are static and high-frequency dielectric permittivities [27, p. 126], is the concentration of LO phonons at a given temperature, m is the quantum number of the phonon mode, L is the QW width, n and n’ are the quantum numbers of initial and final electron levels during the scattering process,
is the initial wave-vector of electron in the layer plane of the QW.
In HgCdTe compounds, two types of longitudinal optical phonons exist the first one is related to HgTe and the other one – to CdTe matrices. These phonons have slightly different energies –
1
LO
w
h
= 17.1 meV and = 20.8 meV, respectively [27]. In our calculations, we assume that the relative concentration of HgTe-related phonons is proportional to (1-x), while the concentration of CdTe-related phonons is proportional to x. Relaxation rates in Eq. (3.4) should be calculated for each of these types separately, and then the summary rate should be found from the relation
x
k
x
k
k
LO
n
LO
n
n
´
t
+
+
-
´
t
=
t
^
^
^
)
(
/
1
)
1
(
)
(
/
1
)
(
/
1
2
1
One should note that the relaxation time in Eq. (3.4) depends on the electron kinetic energy via wave-vector
. This relaxation time determines relaxation of the population of the state
)
,
(
^
k
n
r
.
In equation (3.4), G(n, n(, m) is the square of the overlap integral for electron and phonon wave functions:
2
2
/
2
/
)
(
)
(
)
(
2
)
,
,
(
÷
÷
ø
ö
ç
ç
è
æ
j
×
j
×
j
×
=
´
¢
¢
-
ò
z
z
z
dz
L
m
n
n
G
ph
n
n
L
L
(3.5)
where
)
(
z
n
j
and
are the envelope wave functions of the initial and final states of the localized electrons in the scattering process and
is the wave function of the LO phonon, which describes the symmetry of the phonon electrostatic field:
ï
ï
î
ï
ï
í
ì
=
÷
ø
ö
ç
è
æ
p
+
=
÷
ø
ö
ç
è
æ
p
=
j
k
m
L
mz
k
m
L
mz
z
ph
2
,
sin
1
2
,
cos
)
(
(3.6)
The overlap integral G(n, n(, m)describes the parity selection rules in the process of electron scattering. The integrand should be symmetric to get a non-zero value of G. The relaxation time in Eq. (3.4) for electrons from a level n takes into account the scattering of these electrons within the level as well as transitions of these electrons to other levels in the QW.
In these structures, the electron ground level always lies below the bottom of the conduction band of the well, while the first level appears inside the band gap at large widths of QW [17]. With the decrease of the QW width, the excited levels climb up, while the ground level goes deeper under the Fermi level. That’s why, 2D electron gas of the ground level is always degenerated, and the degree of degeneracy is determined by the energy interval between the ground level and the Fermi one. The average kinetic energy of electrons on the degenerated level is much higher than the classical value kT for 2D electrons. We calculated how the average kinetic energy of electrons of different levels depends on the QW width, using the classical formula:
.
)
(
)
(
)
(
k
f
k
f
k
E
E
n
k
n
n
k
n
k
S
S
=
(3.7)
As can be seen from Fig. 3, the ground electron level is always placed below the Fermi level, while the first excited one is placed below it at L > 20 nm. Dependencies of the average electron kinetic energy on the well width L for three lowest levels of the QW are presented in Fig. 4. One can see that the average kinetic energy of the ground level changes from 3.82 kT to 5.34 kT, while varying the QW width. The average kinetic energy of the first level varies from 1.18 kT to 3.24 kT while this energy for the second level is around 1.04 kT to 1.13 kT.
Relaxation times for electrons from the bottom of three ground levels of the QW are presented in Fig. 5(a) (higher levels are not considered because the level of their occupancy is negligible). Relaxation times for electrons with an average kinetic energy at these levels are presented in Fig. 5(b).
20
30
40
50
60
0
1
2
3
4
5
Average kinetic energy in units of kT
L, nm
n=0
n=1
n=2
Fig. 4 Dependencies of the average kinetic energies of electrons at the ground (n=0), first (n=1) and second (n=2) levels of CdTe/Hg1-xCdxTe/CdTe QW on the well width L. The temperature is taken to be T=77 K, the composition x=0.12. EC=0 is the bottom of the conduction band and EV= - 76.6 meV – top of the valence band in the well.
20
30
40
50
60
1E-11
1E-10
1E-9
1E-8
Relaxation time, s
L, nm
n=0
n=1
n=2
a
)
EMBED Origin50.Graph 20
30
40
50
60
1E-12
1E-11
1E-10
Relaxation time, s
L, nm
n=0
n=1
n=2
b
)
Fig. 5. LO phonon scattering dependences for relaxation times of electrons with zero in-plane kinetic energy from the ground (n=0), first (n=1) and second (n=2) levels (a) and relaxation times of electrons with the average in-plane kinetic energy from these levels (b) in the CdTe/Hg1-xCdxTe/CdTe QW on the well width L. Temperature T=77 K, composition x=0.12.
The relaxation time of electrons from the bottom of the ground level grows considerably when QW width decreases (see Fig. 5 (a)). Its increase is caused by the growth of the degree of degeneracy of the ground miniband,which leads to decreasing the number of free states where electrons with low kinetic energies can scatter to. Not only the ground, but the first miniband becomes degenerate at large QW widths, when this miniband descends, below the bottom of the conduction band of the well. However, degeneracy of the first level disappears at the small QW widths, when this level is high (see Fig. 3). All other 2D levels at Fig. 3 are nondegenerate.
The drift mobility of electrons in the QW in the applied in-plane electric field
F
r
can be calculated by averaging all the possible electron velocities in this QW. Taking into account the quantization of the wave-vector k along the QW axis and using Eq. (3.2), we can rewrite the latter equation:
,
)
(
)
(
)
(
|
|
1
)
(
)
(
|
|
1
2
2
2
2
*
^
^
^
^
^
^
^
×
S
×
Ñ
×
t
S
=
=
×
S
×
m
S
=
S
S
=
m
ò
ò
ò
ò
^
^
^
^
k
d
f
k
d
f
F
k
m
k
k
e
F
k
d
f
k
d
f
k
f
k
f
m
k
F
k
n
n
n
k
n
k
n
n
k
n
n
k
n
LO
n
n
A
k
A
k
LO
r
r
r
r
r
h
r
h
r
r
r
r
h
r
r
r
r
r
r
r
r
,
(3.8)
where relaxation times are given in Eq. (3.4); effective masses and distribution function depend on the level number n and the modulus of the in-plane wave-vector
^
k
.
High level of degeneracy and high average kinetic energies of electrons of ground and first levels (at large well widths) lead to the substantial contribution of lateral transitions to the processes of electron scattering on LO phonons. During this lateral transition, electrons with huge kinetic energies are scattered to the levels energy change to which from the initial level is much higher than the phonon energy.
4. Electron mobility for charged impurities scattering
We calculated the mobility of electrons on charged impurities in the QW CdTe/Hg1-xCdxTe/CdTe by modifying the approach of [18], where it is used the approximation T = 0 K, which allows to get simplifications in the calculations. Here, it is considered more general case. Using Fermi’s golden rule and the principle of detailed balance, one can obtain:
.
))
(
)
(
(
2
2
^
^
^
¢
^
®
¢
¢
¢
¢
®
¢
¢
´
´
¢
-
d
p
=
=
^
^
^
^
k
n
H
nk
k
E
k
E
W
W
def
n
n
k
n
k
n
k
n
nk
h
(4.1)
We will find the solutions of the equations (2.4) and (4.1) in the form of the equations (2.3), (3.2). It is shown in [18] that Eq. (2.3) will be the solution of Eq. (2.4), if relaxation times fulfill the linear equation [18]:
.
)
0
(
2
j
ij
j
k
i
i
k
K
E
f
m
k
t
S
=
¶
¶
S
-
^
^
^
(4.2)
We will deal especially with elastic scattering, because scattering by charged impurities changes only the direction of the electron momentum, while the magnitude of this momentum remains unchanged. Transitions between different levels during the scattering process are restricted under this assumption. Consequently, coefficients Ki ( j will be equal to zero, and Eq. (4.2) will be separated by several independent equations for each energy level.
To calculate the relaxation time, we introduce the additional wave-vector
^
^
^
=
-
¢
q
k
k
r
r
r
. From the momentum conservation law, we obtain: where ( is the angle between the initial and final wave-vectors of electron in the scattering process. We obtain the formula for the relaxation time on electrons of the n-th level scattered by charged impurities:
,
)
(
)
cos
1
](
[
)
(
]
[
1
2
1
)
0
(
0
2
)
0
(
0
3
0
q
ï
þ
ï
ý
ü
¶
¶
q
-
-
´
´
ï
î
ï
í
ì
+
´
´
¶
¶
-
p
-
=
t
^
=
^
^
^
^
òò
ò
^
^
d
k
dE
E
f
E
E
q
nk
H
nk
k
dE
E
f
E
E
sm
n
n
n
n
n
q
q
average
def
n
n
n
n
n
n
n
h
(4.3)
where
2
sin
2
0
q
=
^
^
k
q
. One should note that in the latter formula
is not the Dirac delta function as it was at T=0. To find the matrix element , we follow the same procedure as that described in [18] for T = 0. However, we modify the screening function that describes the screening in 2DEG to account the non-zero temperature. According to the semi-classical approach, when q( ( 0 [18, p. 210], the screening function can be found as:
^
^
+
=
x
q
q
q
/
1
)
(
0
(4.4)
where
F
e
E
n
e
q
¶
¶
e
p
=
0
2
0
2
and
1
]}
/
)
exp[(
1
{
-
-
+
´
T
k
E
E
B
F
n
. Evaluating the derivative of the electron concentration for non-zero temperature, we obtain the final equation for the screening function:
)
0
(
2
1
1
)
(
2
0
2
n
n
f
m
e
q
q
h
e
+
=
x
^
^
.
(4.5)
In the limiting case of T = 0 K, fn(0) = 1,
and Eq. (4.5) reduces to appropriate formula from [18]. Having the explicit Eq. (4.5) for the
and substituting the formula for the matrix element from [18] into Eq. (4.3), we obtain the final formula for the relaxation time of the localized electrons on the n-th level, which are scattered by charged impurities at non-zero temperature:
.
,
,
2
sin
2
2
sin
2
2
)
cos
1
(
]
[
)
(
)
(
]
[
1
max
min
2
2
0
0
0
)
0
(
0
)
0
(
0
2
ò
ò
ò
ò
å
ú
û
ù
ê
ë
é
q
´
´
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
ú
û
ù
ê
ë
é
+
q
e
p
q
-
q
´
´
¶
¶
-
´
´
¶
¶
-
p
-
=
t
^
^
p
^
^
l
l
z
z
i
imp
i
l
l
n
n
n
n
n
n
n
n
n
n
l
l
n
n
n
z
k
g
dz
q
k
e
eZ
d
E
f
E
E
k
dE
k
dE
E
f
E
E
c
m
h
h
(4.6)
The concentration of l-th impurity species (charge
elZl) is denoted by cl, and the length over which these impurity species are found in the heterostructure is denoted by . The sum is carried out over all kinds of the charged impurities in the system. For CdTe/Hg1–xCdxTe/CdTe QW, one should use the value Z = 1 or 2, having in mind Hg+ interstitials in n-type material or Hg2- in p-type material [38, 39].
Distribution of the charged impurities in the system is described by the form-factor gimp:
ò
¥
¥
-
-
-
^
^
×
c
=
dz
e
z
n
z
q
g
i
z
z
q
n
i
imp
|
|
2
)
(
)
,
.
(
(4.7)
where
)
(
z
n
c
is the envelope wave function of the n-th level electrons. The mobility of electrons for each of these levels can be expressed as:
)
(
F
n
n
n
E
m
e
t
=
m
(4.8)
The average mobility of electrons for only one scattering mechanism (charged impurities scattering) can be found then as:
(4.9)
In the n-type QW with the composition x = 0.12 and the concentration of background charged impurities 1014 – 1015 cm–3, the Fermi level varies from 5.3 up to 11.2 meV (see Fig. 3). According to our estimations, background impurities affect electron scattering much stronger than the remote delta-doped layer of the equivalent charge density. Thus, background charged impurities play an important role in the scattering for these QWs. In further calculations, we will treat the case of the background impurities scattering, because these impurities are always present in the QW.
5. Estimation of minor scattering mechanisms
5.1. Scattering by acoustic phonons
In HgCdTe quantum wells can exist two channels of electron relaxation via the acoustic phonons scattering. These are scattering by the deformation potential interaction mechanism and scattering by the piezoelectric interaction mechanism [34]. However, for the quantum wells, growth direction of which is [001] axis, the piezoelectric interaction is absent [40, p. 48]. So, in this chapter, we will estimate the relaxation time of electrons that scatter on acoustic phonons via deformation potential interaction.
In layered heterostructures, acoustic waves consist of extended and confined modes. Extended modes propagate through the whole heterostructure in any direction. Confined modes are localized in the layer plane of the well and propagate along this plane. However, due to the small differences between elastic properties of the matrix and quantum well materials, this confinement is weak. Thus, confined modes can be neglected, while extended modes can be approximated by plane waves. The energy of acoustic phonon can be estimated as
÷
ø
ö
ç
è
æ
p
L
s
E
L
acoustic
2
~
h
[34, 6.135], where sL = 105 cm/s is the sound velocity in the layer plane. For L = 20 nm, the acoustic phonon energy is two orders of magnitude lower than the mean kinetic energy of the ground level electrons (see Fig. 4). Therefore, the scattering can be assumed to be elastic.
Thus, for estimations we used the simple model [34, 6.137] that assumes the well to be infinitely deep, acoustic phonons to be bulk-like and non-parabolicity of the dispersion law to be neglected:
(
)
,
)
(
1
2
1
1
)
(
2
)
,
(
1
0
0
,
0
3
*
2
0
0
k
f
L
C
s
T
k
m
a
C
k
n
n
n
n
n
L
L
B
s
-
÷
ø
ö
ç
è
æ
d
+
S
´
´
-
=
t
h
(5.1)
where
(
)
)
(
1
0
0
k
f
n
-
is the factor that describes the occupancy of the state electron scatters to; assuming the scattering to be elastic [34], we can state . The deformation potential constant (C – a)HgCdTe =
–3.63 eV
is (see Eq. (1.1)); m* is the electron effective mass, n0 and n are the numbers of electron levels and = 5.0625(1017 eV(s/cm4 is the mean elastic constant of the material (taken to be the same as in HgTe), M0 and V0 are the mass and the volume of the primitive cell of HgTe.
According to our estimates, the relaxation time of the ground level electrons with the average kinetic energy is close to 6(10–10 s for L = 20 nm and 3.8(10–10 s for L = 60 nm; the relaxation time of first level electrons with the average kinetic energy is around 2.6(10–10 s for L = 60 nm. These times are much larger than appropriate times for longitudinal optical phonon scattering (see Fig. 5 (b)). Consequently, the acoustic phonons scattering is small and can be neglected in the further treatment.
5.2. Scattering on interface roughness
Very little is known about the microscopic structure of the interface defects so they usually use very simple models for the calculation of scattering by these defects. All estimations are carried out for the well of the 20-nm width, which is the minimal width presented in our calculations.
To estimate the scattering by interface roughness, we use the model presented in the book of Bastard [18]. In this model, the interface defect is assumed to have the thickness h of one monolayer in the growth direction and have extensions lx and ly in the layer plane. It is assumed that these defects are located far away from each other, so scattering events on different defects are independent from each other. Calculations are made at T = 0 K in the Electric Quantum Limit, which assumes that only the lowest electron sub-band is occupied. In our system, at small well widths electrons scatter from the ground level, therefore this assumption is reasonable. We used the approach from [18], which allows to take into account the sharpness of the electron envelope function at the interface:
,
)
cos
1
](
[
sin
sin
2
sin
)
1
(cos
2
sin
)
2
sin
2
(
)
(
16
)
(
1
2
4
2
2
0
2
0
2
0
3
0
2
2
q
-
q
ú
û
ù
ê
ë
é
q
ú
û
ù
ê
ë
é
-
q
q
x
q
´
´
÷
÷
ø
ö
ç
ç
è
æ
c
p
=
t
ò
ò
p
F
y
F
x
F
F
h
def
b
F
roug
k
l
k
l
k
k
d
dz
z
m
N
V
h
E
h
(5.2)
where kF is the Fermi wave vector, (0(z)is the angle between electron wave vectors before and after the scattering.
is defined in Eq. (4.5), ( – envelope wave function of the ground level electrons, Vb – barrier height, m0 – effective mass of the ground level electrons, ((k()
To estimate the relaxation rate from the latter equation, one should know the average concentration of interface defects Ndef and their average lateral dimensions lx and ly. Lateral dimensions of the defects were taken from [41], which is often used for estimations of the interface scattering in quantum wells. We have found no data about the concentration of interface roughness defects in HgCdTe QWs. Nevertheless, the authors [42] created HgCdTe heterostructures with fixed charges at the interfaces with the concentration within the range 1010 – 5(1010 cm–2. We assume that the concentration of roughness defects Ndef is the same as the concentration of interface fixed charges in the samples of [42]. Thus, we obtain that for ground level electrons with the average kinetic energy, the relaxation time for interface scattering in the well of 20-nm width is close to 10–9 – 7(10–9 s. This relaxation time is an order of magnitude higher than the appropriate time for the longitudinal optical phonon scattering (see Fig. 5 (b), thus the interface scattering can be considered as negligible in the further treatment.
5.3. Influence of the value of the valence-band offset
Values of the valence-band offset Δ in the literature are contradicting (see section 1.1). Then, it is important to estimate the influence of variation of this parameter on the mobility of localized electrons. We compared the mobilities for charged impurities scattering and for the LO phonons scattering for two QW widths – 20 and 60 nm. There are the values compared as to the valence-band offset Δ1 = 0.55((1–x) eV and Δ2 = 0.35((1–x) eV. According to our calculations, different values of the valence-band offset lead to the mobilities that differ by approximately 1.5 times at small well widths for each of the scattered mechanisms. For the wide wells, variation of the mobility with the change in the valence-band offset is weaker.
Thus, we can see that the uncertainty in the value of the valence-band offset strongly affects the magnitude of the electron mobilities, and it is principally impossible to obtain precise numerical results for relaxation times and mobilities. Also, it is impossible to estimate the strain values precisely, because they depend on the other technological heterostructure layers and on the heterostructure growth conditions.
You can also aim the reduction of stress in the layers. Our calculations show that for the QW with the concentration of n-type charged impurities of 1015 cm–3, the value of strain sufficiently affects the mobility. For example, in the well of 20-nm width, the mobility in the absence of strains is in times greater than such mobility in the presence of the strain (see section 6). Thus, the absence of the strain leads to growth in the electron mobility.
6. Results for electron mobility
in CdTe/Hg1-xCdxTe/CdTe quantum well
The average mobility of electrons in the CdTe/
Hg1–xCdxTe/CdTe QW that includes both the LO phonons scattering (Eq. (3.9)) and charged impurities scattering (Eq. (4.9)) can be found as:
,
2
2
^
^
×
S
×
m
S
=
m
ò
ò
^
^
k
d
f
k
d
f
k
n
n
k
n
n
r
r
r
r
(6.1)
where the summary mobility of the n-th level electrons is found from the relation
CI
n
LO
n
n
m
+
m
=
m
/
1
/
1
/
1
(6.2)
Results for the electron mobility for LO-phonon scattering only (Eq. (3.9)) and results for the average mobility that includes two scattering mechanisms (Eq. (6.1)) are plotted in Fig. 6.
The electron mobility that takes into account only the LO phonons scattering, grows substantially, when the QW width decreases. Therefore, at small charged impurity concentrations the LO phonons scattering dominates, and the average electron mobility also grows at small well widths. When the concentration of charged impurities is greater, the growth of average mobility is partially suppressed. For example, for the concentration of n-type charged impurities 1016 cm–3, the average mobility grows approximately two times only. These results that describe variation of the average electron mobility with the quantum well width can
be possibly used to optimize growth procedure for CdTe/Hg1–xCdxTe/CdTe QWs with different properties within the same technological line.
It is important to note that usage of more pure samples of Hg1–xCdxTe with lower concentrations of charged impurities allows one to obtain higher electron mobilities for the charged impurities scattering. In this case, the average electron mobility also is higher. Also, one should note that in this paper we considered the model sample where strain is not relaxed. In the case of using the samples with buffer layers that partially relax the strain, the strain-induced increase of the band gap will be smaller, and the mobility of localized electrons should increase by several times.
20
30
40
50
60
2E6
4E6
6E6
8E6
1E7
1.2E7
1.4E7
Mobiliy, cm
2
/(Vs)
Well width L, nm
A
B
C
Fig. 6. Electron mobility in the CdTe/Hg1–xCdxTe/CdTe QW. The mobility includes longitudinal optical phonon scattering (LOP) and charged impurities scattering (CI). Line A is for LOP and CI scattering with the concentration 1015 cm–3, line B is for LOP and CI scattering with the concentration 1014 cm–3, line C is for pure LOP scattering mechanism. The parameters used are as follows: temperature T = 77 K; composition x = 0.12.
Unfortunately, the authors do not know the experimental data on the mobility of electrons in the CdTe/Hg1–xCdxTe/CdTe quantum well with inverted band structure at the nitrogen temperature. Therefore,
it is useful to compare electron mobilities in the CdTe/Hg1–xCdxTe/CdTe QW with inverted band scheme in the well (Fig. 6) with electron mobilities in other
Hg1–xCdxTe structures at T = 77 K. The electron mobility in bulk undoped Hg1–xCdxTe with the direct band scheme at the composition x = 0.2 is 2.5(105 cm2/(V(s) [27, p. 95]. The electron mobility in bulk HgTe (which has the inverted band scheme) is 2.2(105 cm2/(V(s) [27, p. 95].
According to our calculations, the electron mobility in the QW CdTe/Hg1–xCdxTe/CdTe with the concentration of n-type charged impurities 1015 cm–3 is not higher than 6–7(105 cm2/(V(s) for the composition x = 0.2, and it is close to 1–1.2(106 cm2/(V(s) for the composition x = 0. It was shown in [13] that the electron mobilities in the QW CdTe/HgTe/CdTe (at T = 3 K) can reach 2.8(105 cm2/(V(s). From the comparison of these mobilities one can see that the maximal mobility of electrons in Hg1–xCdxTe could be reached in quantum wells with the inverted band scheme inside the well.
One should note that, at very small QW widths (about 10 nm and less [12]), the width of interfaces becomes comparable with the width of the quantum well. So the scattering by interface roughness [34, p. 249] and the scattering by interface phonons [34, p. 287] become more important and the growth of mobility can be suppressed.
Apparently, we can expect the highest mobility in mercury–cadmium–telluride structures with the inverted band scheme quantum wells at moderate concentrations of charged impurities. Therefore, these structures are of interest for the creation of semiconductor THz detectors based on hot electrons operating under moderate cooling (up to liquid nitrogen temperature) or ambient temperatures. It is known that narrow gap semiconductors are promising materials for direct THz detectors due to high electron mobility, high carrier concentration and low effective masses of electrons [3, 43]. In our case, we should expect the electron relaxation time of about 10-11 s and scattering mean free path of about 50 μm. For this relaxation time, the QW structure provides a strong interaction with THz radiation by the free-carrier Drude absorption mechanism [44] in the frequency range of 100-700 GHz. The high conductivity of degenerate electron gas on the metal type structure provides resistance to the order of a few tens of ohms, which will ensure a good matching with the planar metallic antenna. A large mean free path carriers will implement those structures technologically.
Therefore, the quantum wells with the inverted band scheme HgCdTe are the promising structures for THz detector design. Thus, it is possible to realize detectors on the effect of the electron gas heating, such as hot electron bolometers or field effect transistors with a shallow channel on the basis of the in-plane transport in the quantum well [45].
7. Conclusions
We have modelled electron relaxation processes in the n-type CdTe/Hg1-xCdxTe/CdTe wide QWs with the inverted band structure at the liquid nitrogen temperatures and for x close to the band inversion value 0.16, for their potential usage as THz range detectors. It was found that the longitudinal optical phonons scattering and charged impurities scattering are dominant mechanisms, while the acoustic phonons scattering and interface roughness scattering can be neglected. It was shown that the electron mobility could reach values of the order of magnitude of 5*106 – 107 cm2/(Vs) for the well widths 60 – 20 nm in such structure. These values could be estimated at the concentration of the diluted charged impurity of 1014 – 1015 cm-3.
Usage of relaxing buffer layers allows one to reach the partial relaxation of the strains in the quantum well and to increase the mobility by several times. This mobility is an order of magnitude higher than for the QW with the direct band structure and bulk mercury-cadmium-telluride. On the one hand, it places high demands on the quality of the structure. On the other hand, it provides the possibility to implement fast direct terahertz detectors, for example, field effect transistors with the shallow channel based on the quantum well or hot electron bolometer operating at moderate cooling.
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�
Fig. 2. Electron wave functions for the QW width L=50 nm. The horizontal axis corresponds to z coordinate; peaks of the wave functions are at the hetero-interfaces. The vertical axis represents the magnitude of the wave function in arbitrary units (a. u.). The upper row corresponds to the composition x=0 while the lower row corresponds to the composition x=0.12.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
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| id | nasplib_isofts_kiev_ua-123456789-118360 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2025-11-24T09:08:39Z |
| publishDate | 2014 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Melezhik, Ye.O. Gumenjuk-Sichevska, J.V. Sizov, F.F. 2017-05-30T05:45:11Z 2017-05-30T05:45:11Z 2014 Electron relaxation and mobility in the inverted band quantum well CdTe/Hg₁₋xCdxTe/CdTe / E.O. Melezhik, J.V. Gumenjuk-Sichevska, F.F. Sizov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2014. — Т. 17, № 1. — С. 85-96. — Бібліогр.: 45 назв. — англ. 1560-8034 PACS 73.21.Fg, 84.40.-x https://nasplib.isofts.kiev.ua/handle/123456789/118360 Electron relaxation processes at nitrogen temperatures in CdTe/Hg₁₋xCdxTe/CdTe quantum well (QW) with an inverted band structure is modelled. In this structure, scattering by longitudinal optical phonons, charged impurities, acoustic phonons and interfaces were taken into account. It was found that for undoped and lightly doped QWs (concentration of background n-type charged impurities in the well is 10¹⁴ – 10¹⁶ cm-³ or less), for x close to the band inversion value 0.16, the electron mobility grows considerably when the QW width decreases. This mobility is higher for samples with smaller concentrations of charged impurities. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Electron relaxation and mobility in the inverted band quantum well CdTe/Hg₁₋xCdxTe/CdTe Article published earlier |
| spellingShingle | Electron relaxation and mobility in the inverted band quantum well CdTe/Hg₁₋xCdxTe/CdTe Melezhik, Ye.O. Gumenjuk-Sichevska, J.V. Sizov, F.F. |
| title | Electron relaxation and mobility in the inverted band quantum well CdTe/Hg₁₋xCdxTe/CdTe |
| title_full | Electron relaxation and mobility in the inverted band quantum well CdTe/Hg₁₋xCdxTe/CdTe |
| title_fullStr | Electron relaxation and mobility in the inverted band quantum well CdTe/Hg₁₋xCdxTe/CdTe |
| title_full_unstemmed | Electron relaxation and mobility in the inverted band quantum well CdTe/Hg₁₋xCdxTe/CdTe |
| title_short | Electron relaxation and mobility in the inverted band quantum well CdTe/Hg₁₋xCdxTe/CdTe |
| title_sort | electron relaxation and mobility in the inverted band quantum well cdte/hg₁₋xcdxte/cdte |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/118360 |
| work_keys_str_mv | AT melezhikyeo electronrelaxationandmobilityintheinvertedbandquantumwellcdtehg1xcdxtecdte AT gumenjuksichevskajv electronrelaxationandmobilityintheinvertedbandquantumwellcdtehg1xcdxtecdte AT sizovff electronrelaxationandmobilityintheinvertedbandquantumwellcdtehg1xcdxtecdte |