Thermoelectric power in carbon nanotubes and quantum wires of nonlinear
optical, optoelectronic, and related materials under strong magnetic
field: Simplified theory and relative comparison
K. P. Ghatak
a兲
Department of Electronic Science, The University of Calcutta, 92, Acharyya Prafulla Road, Kolkata 700
009, India
S. Bhattacharya
Department of Computer Science, St. Xavier’s College, 30 Park Street, Kolkata 700 016, India
S. Bhowmik and R. Benedictus
Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft,
The Netherlands
S. Choudhury
Department of Electronics and Communication Engineering, Sikkim Manipal Institute of Technology,
Majitar, Rangpo, East Sikkim 737 132, India
共Received 10 October 2007; accepted 27 October 2007; published online 6 February 2008兲
We study thermoelectric power under strong magnetic field 共TPM兲 in carbon nanotubes 共CNTs兲 and
quantum wires 共QWs兲 of nonlinear optical, optoelectronic, and related materials. The corresponding
results for QWs of III-V, ternary, and quaternary compounds form a special case of our generalized
analysis. The TPM has also been investigated in QWs of II-VI, IV-VI, stressed materials, n-GaP,
p-PtSb
2
, n-GaSb, and bismuth on the basis of the appropriate carrier dispersion laws in the
respective cases. It has been found, taking QWs of n-CdGeAs
2
, n-Cd
3
As
2
, n-InAs, n-InSb, n-GaAs,
n-Hg
1−x
Cd
x
Te, n-In
1−x
Ga
x
As
y
P
1−y
lattice-matched to InP, p-CdS, n-PbTe, n-PbSnTe, n-Pb
1−x
Sn
x
Se,
stressed n-InSb, n-GaP, p-PtSb
2
, n-GaSb, and bismuth as examples, that the respective TPM in the
QWs of the aforementioned materials exhibits increasing quantum steps with the decreasing
electron statistics with different numerical values, and the nature of the variations are totally
band-structure-dependent. In CNTs, the TPM exhibits periodic oscillations with decreasing
amplitudes for increasing electron statistics, and its nature is radically different as compared with the
corresponding TPM of QWs since they depend exclusively on the respective band structures
emphasizing the different signatures of the two entirely different one-dimensional nanostructured
systems in various cases. The well-known expression of the TPM for wide gap materials has been
obtained as a special case under certain limiting conditions, and this compatibility is an indirect test
for our generalized formalism. In addition, we have suggested the experimental methods of
determining the Einstein relation for the diffusivity-mobility ratio and the carrier contribution to the
elastic constants for materials having arbitrary dispersion laws. © 2008 American Institute of
Physics. 关DOI: 10.1063/1.2827365兴
I. INTRODUCTION
Since Iijima’s discovery,
1
carbon nanotubes 共CNTs兲
have been recognized as fascinating materials with nano-
meter dimensions uncovering new phenomena in different
areas of low-dimensional science and technology. The re-
markable physical properties of these quantum materials
make them ideal candidates to reveal new phenomena in na-
noelectronics. The CNTs find wide applications in
conductive
2,3
and high strength composites,
4
chemical
sensors,
5
field emission displays,
6,7
hydrogen storage
media,
8,9
nanotweezeres,
10
nanogears,
11
nanocantilever
devices,
12
nanomotors
13
and nanoelectronic devices.
14,15
Single-walled carbon nanotubes 共SWNTs兲 appear to be ex-
cellent materials for single molecule electronics,
16–18
nano-
tube based diodes,
19
single electron transistors,
20
random ac-
cess memory cells,
21
logic circuits,
22
and in other
nanosystems. In this context, it may be noted that with the
advent of molecular-beam epitaxy, fine line lithography, and
other experimental techniques, low-dimensional structures
having quantum confinement of one, two, and three dimen-
sions 共such as quantum wells, wires, and dots兲 have attracted
much attention not only for their potential in uncovering new
phenomena in nanoscience, but also for their interesting ap-
plications in nanotechnology.
23–25
In QWs, the restriction of the motion of the carriers in
the two directions may be viewed as carrier confinement by
two infinitely deep one-dimensional 共1D兲 rectangular poten-
tial wells, leading to quantization 共known as quantizing size
effect兲 of the wave vectors along the two orthogonal direc-
tions, allowing 1D electron transport representing new physi-
cal features not exhibited in bulk semiconductors. The low-
dimensional heterostructures based on various materials are
a兲
Author to whom correspondence should be addressed. Electronic mail:
kamakhyaghatak@yahoo.co.in.
JOURNAL OF APPLIED PHYSICS 103, 034303 共2008兲
0021-8979/2008/103共3兲/034303/21/$23.00 © 2008 American Institute of Physics103, 034303-1
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widely investigated because of the enhancement of carrier
mobility. An enormous range of important applications in the
quantum regime, together with a rapid increase in computing
power, have generated much interest in the analysis of nano-
structured devices for investigating their properties.
26–29
Ex-
amples of such new applications include various quantum
wires,
30–34
quantum resistors,
35
resonant tunneling diodes
and band filters,
36,37
quantum switches,
38
quantum
sensors,
39–41
quantum logic gates,
42,43
quantum transistors
and subtuners,
44–46
heterojunction field-effect transistors
共FETs兲,
47
high-speed digital networks,
48
high-frequency mi-
crowave circuits,
49
optical modulators,
50
optical switching
systems,
51
and other devices. Though extensive work has
already been done for both the CNTs and QWs, it appears
from the literature that the TPM for both the CNTs and the
QWs has yet to be investigated in detail.
It is well known that the TPM is a very important
relation
52
since the entropy 共a very important thermodynamic
property that cannot be experimentally determined兲 can be
known from this relation by knowing the experimental val-
ues of the electron concentration. The TPM is more accurate
than any two of the individual relations for the electron con-
centration or the entropy, which is considered to be the two
most widely used quantities in investigating the thermody-
namics of the electronic materials. Besides, in recent years
with the advent of the quantum Hall effect,
53
there has been
considerable interest in studying the TPM for various com-
pounds having different band structures.
54–63
It is worth re-
marking that the analysis of the thermopower generates in-
formation about the band structure, the density-of-states
function, and the effective mass of the carriers.
59
The classi-
cal TPM 共G兲 equation is given by G=共
2
k
B
/ 3e兲共k
B
and e
are Boltzmann’s constant and the magnitude of the carrier
charge, respectively兲 and is well known in the literature.
59
In
this conventional form, it appears that the TPM depends only
on the fundamental constants and is independent of tempera-
ture and carrier concentration under the condition of carrier
nondegeneracy. Askerov et al.
63
showed that the TPM is in-
dependent of scattering mechanisms and depends only on the
dispersions laws of the carriers.
It is well known from the fundamental work of
Zawadzki
61
that the TPM for electronic materials having de-
generate electron concentration is essentially determined by
their respective energy band structures. It has, therefore, dif-
ferent values in different materials and varies with the dop-
ing, the magnitude of the reciprocal quantizing magnetic
field under magnetic quantization, the quantizing electric
field as in inversion layers, the nanothickness as in quantum
wells and quantum-well wires, and with the superlattice pe-
riod as in quantum confined superlattices of small gap semi-
conductors with graded interfaces having various carrier en-
ergy spectra. Some of the significant features that have
emerged from these studies are as follows:
共a兲 The TPM decreases monotonically with the increase in
electron concentration.
共b兲 The TPM decreases with doping in heavily doped
semiconductors forming band tails.
共c兲 The nature of variations is significantly influenced by
the spectrum constants of various materials having dif-
ferent band structures.
共d兲 The TPM oscillates with inverse quantizing magnetic
field due to the Shubnikov–de Haas effect.
共e兲 The TPM decreases with the magnitude of the quantiz-
ing electric field in inversion layers.
共f兲 The TPM exhibits composite oscillations with signifi-
cantly different values in superlattices and various
other quantized structures.
In this article, we have studied the TPM in CNTs and
also in QWs of nonlinear optical, III-V, ternaries, quaterna-
ries, II-VI, IV-VI, stressed materials, n-GaP, p-PtSb
2
,
n-GaSb, and bismuth on the basis of their respective carrier
energy spectra. In this context, it may be noted that the non-
linear optical materials are also known as tetragonal com-
pounds due to their crystal structure.
64
These materials are
being used increasingly in light-emitting diodes, Hall pick-
ups, and thermal detectors.
65–67
Rowe and Shay
68
demon-
strated that the quasicubic model
69
can be used to explain the
observed splitting and symmetry properties of the conduction
and valence bands at the zone center of the k space of the
aforementioned compounds. The s-like conduction band is
singly degenerate and the p-like valence bands are triply de-
generate. The latter splits into three subbands because of the
spin-orbit and the crystal-field interactions. The large contri-
bution of the crystal-field splitting occurs from the noncubic
potential.
70
The experimental data on the absorption
constants,
71
the effective mass,
72
and the optical third-order
susceptibility
73
have produced strong evidence that the con-
duction band in the same compound corresponds to a single
ellipsoid of revolution at the zone center in k space.
Considering the crystal potential in the Hamiltonian, and
special features of the nonlinear optical compounds, Kildal
74
proposed the energy spectrum of the conduction electrons
under the assumptions of the isotropic momentum matrix
element and the isotropic spin orbit splitting constant, re-
spectively, although the anisotropies in the two aforemen-
tioned band parameters are the significant physical features
of said compounds.
75
In Sec. II A, we have formulated the expressions of the
TPM in CNTs by formulating the respective expressions of
the electron statistics for both 共n,n兲 and 共n ,0兲 tubes, respec-
tively. In Sec. II B, we have studied the TPM in QWs of
nonlinear optical materials by formulating the generalized
dispersion relation of the conduction electrons, considering
the anisotropies of the effective electron masses and the spin-
orbit splitting of the valance band together with the proper
inclusion of crystal-field splitting in the Hamiltonian through
the k· p formalism. In Sec. II C, it has been shown that the
corresponding results for the QWs of III-V, ternary, and qua-
ternary materials form special cases of our generalized analy-
sis as derived in Sec. II B. The III-V materials are being used
increasingly in integrated optoelectronics,
76
passive filter
devices,
77
distributed feedback lasers, and Bragg reflectors.
78
We have used n-Hg
1−x
Cd
x
Te and n-In
1−x
Ga
x
As
y
P
1−y
lattice
matched to InP as examples of ternary and quaternary com-
pounds, respectively. The n-Hg
1−x
Cd
x
Te is a classic narrow-
gap compound and is an important optoelectronic material
034303-2 Ghatak et al. J. Appl. Phys. 103, 034303 共2008兲
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because its band gap can be varied to cover a spectral range
from 0.8
m to over 30
m by adjusting the alloy
composition.
79
This compound finds extensive applications
in infrared detector materials
80
and photovoltaic detector
arrays
81
in the 8 − 12
m wave bands. The above uses have
spurred an Hg
1−x
Cd
x
Te technology for the production of
high-mobility single crystals, with specially prepared surface
layers that are also ideally suitable for narrow subband phys-
ics because the relevant material constants are within the
reach of experiment.
82
The quaternary compounds also find
extensive applications in optoelectronics, high electron mo-
bility transistors, visible heterostructure compound lasers, in-
frared light-emitting diodes, lasers for fiber optic systems,
83
tandem solar cells,
84
avalanche photodetectors,
85
long-
wavelength light sources, detectors in optical fiber
communications,
86
and new types of optical devices.
87
The
expressions for electron concentration per unit length 共n
1D
兲
and the TPM for QWs whose energy band structures are
defined by the two-band model of Kane and that of parabolic
energy bands have further been formulated under certain lim-
iting conditions for the purpose of relative assessment. To
perform numerical computations, n-CdGeAs
2
and n-Cd
3
As
2
have been used as examples of nonlinear optical and tetrag-
onal compounds.
88
The TPM has also been investigated nu-
merically by taking n-InAs, n-InSb, and n-GaAs as examples
of III-V compounds and n-Hg
1−x
Cd
x
Te and
n-In
1−x
Ga
x
As
y
P
1−y
lattice-matched to InP as examples of ter-
nary and quaternary materials in accordance with the three-
and the two-band models of Kane together with the parabolic
energy bands, respectively, for the purpose of relative com-
parison among the above-mentioned models.
The II-VI compounds find extensive applications in in-
frared detectors,
89
ultrahigh-speed bipolar transistors,
90
optic
fiber communications,
91
and advanced microwave devices.
92
These compounds possess the appropriate direct band gap to
produce light-emitting diodes and lasers from blue to red
wavelengths. The Hopfield model describes the dispersion
relation of both the carriers of II-VI materials where the
splitting of the two-spin states by the spin-orbit coupling and
the crystalline field has been taken into account.
93
In Sec.
II D, we shall study the TPM in QWs of II-VI compounds on
the basis of the Hopfield model by formulating the appropri-
ate carrier statistics and taking CdS as an example.
In Sec. II E, we shall study the TPM in QWs of IV-VI
materials, which are being widely used in thermoelectric de-
vices, superlattices, and other quantum effect devices.
94
The
dispersion relation of the carriers in IV-VI compounds has
been formulated by Dimmock
94
by including the contribu-
tions of the transverse and longitudinal effective masses of
the external bands, which arises from the k · p perturbations
with the other bands, taken to the second order together with
the special anisotropic properties of the energy band struc-
tures of the above-mentioned compounds. For the purpose of
numerical computations, we have used PbTe, PbSnTe, and
Pb
1−x
Sn
x
Se as examples of IV-VI compounds. In recent
years, there has been considerable interest in studying the
various electronic properties of stressed compounds because
of their important physical features.
95
In Sec. II F, we have
formulated the TPM in QWs of stressed compounds, taking
stressed n-InSb as an example.
The n-GaP, platinum antimonide, and gallium anti-
monide occupy significant positions in the realm of quantum
effect devices due to their important physical properties and
the corresponding dispersion relations.
96,110,111
We have stud-
ied the TPM for the QWs of n-GaP, p-PtSb
2
, and n-GaSb in
Secs. II G, II H, and II I, respectively, by considering the
appropriate carrier energy spectra.
It is well known that the carrier energy spectra in bis-
muth differ considerably from simple spherical surfaces of
the degenerate electron gas, and several models have been
developed to describe the energy band structure of Bi. Earlier
works
97,98
demonstrated that the ellipsoidal parabolic model
or the one-band model could describe the carrier properties
of Bi. Shoenberg
97
indicated that the de Haas–Van Alphen
and cyclotron resonance experiments supported the one-band
model, although the latter work showed that Bi could be
described by the two-band nonparabolic ellipsoidal Lax
model since the magnetic field dependence of many physical
parameters of Bi supports the above model.
99
The magneto-
optical results
100
and the ultrasonic quantum oscillation
data
101
favor the Lax ellipsoidal nonparabolic model,
102
whereas Kao,
102
Dinger and Lawson,
103
and Koch and
Jensen
104
indicated that the Cohen model
105
is in better
agreement with the experimental results. It may be noted that
the Hybrid model of Bi as proposed by Takaoka et al.
106
also
explains many important physical properties. Besides, Mc-
Clure and Choi
107
presented a new model of Bi that fits the
data for a large number of magneto-oscillatory and resonance
experiments. In Sec. II J, we have formulated the TPM in
QWs of Bi in accordance with the aforementioned energy
band models for the purpose of relative assessment. In the
Sec. II K, we have suggested the experimental methods of
determination of the Einstein relation for the diffusivity-
mobility ratio and the carrier contribution to the elastic con-
stants for materials having arbitrary carrier energy spectra.
II. THEORETICAL BACKGROUND
A. Investigation of the TPM in „n, n… and „n,0… CNTs
For the 共n , n兲 and 共n,0兲 tubes, the energy dispersion
relations are given by
108
E
m
共k
y
兲 = ⫾ t
冑
1+4cos
m
n
cos
k
y
a
2
+ 4 cos
2
k
y
a
2
,
共1兲
−
a
⬍ k
y
⬍
a
⬘
and
E
m
共k
x
兲 = ⫾ t
冑
1+4cos
冑
3k
x
a
n
cos
m
n
+ 4 cos
2
m
n
,
共2兲
034303-3 Ghatak et al. J. Appl. Phys. 103, 034303 共2008兲
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−
冑
3a
⬍ k
x
⬍
冑
3a
,
where the notations are defined in Refs. 108 and 109. Using
Eqs. 共1兲 and 共2兲, the electron concentration per unit length
共n
1D
兲 can be written as
n
1D
=
4
a
关A
1
共E
F
,m,n 兲 + B
1
共E
F
,m,n 兲兴 共3兲
and
n
1D
=
4
a
冑
3
关A
2
共E
F
,m,n 兲 + B
2
共E
F
,m,n 兲兴, 共4兲
where
A
1
共E
F
,m,n 兲 = cos
−1
关
1
2
共
冑
A
2
−1+共E
F
/t兲
2
− A兲
兴
,
A = cos
冉
m
n
冊
,
E
F
is the Fermi energy,
B
1
共E
F
,m,n 兲 =
兺
r=1
S
0
Z
r
A
1
共E
F
,m,n 兲,
r is the set of real positive integers whose upper limit is S
0
,
Z
r
=2共k
B
T兲
2r
共1−2
1−2r
兲
共2r兲
2r
E
F
2r
,
T is the temperature,
共2r兲 is the zeta function of order 2r,
A
2
共E
F
,m,n 兲 = cos
−1
冋
共4A兲
−1
再
冉
E
F
t
冊
2
−1−4A
2
冎
册
,
and
B
2
共E
F
,m,n 兲 =
兺
r=1
S
0
Z
r
A
2
共E
F
,m,n 兲.
The TPM in this case can, in general, be expressed as
60
G =
2
k
B
2
T
3en
1D
冉
n
1D
E
F
冊
. 共5兲
Combining Eqs. 共3兲–共5兲, the expressions of the TPM for
the 共n, n兲 and 共n,0兲 tubes can be written as
G =
2
k
B
2
T
3e
冉
A
1
⬘
共E
F
,m,n 兲 + B
1
⬘
共E
F
,m,n 兲
A
1
共E
F
,m,n 兲 + B
1
共E
F
,m,n 兲
冊
共6兲
and
G =
2
k
B
2
T
3e
冉
A
2
⬘
共E
F
,m,n 兲 + B
2
⬘
共E
F
,m,n 兲
A
2
共E
F
,m,n 兲 + B
2
共E
F
,m,n 兲
冊
, 共7兲
where the primes denote the differentiation of the respective
differentiable functions with respect to E
F
.
B. Investigation of the TPM in QWs of nonlinear
optical materials
The form of the k ·p matrix for the nonlinear optical
materials can be written as
H =
冋
H
1
H
2
H
2
+
H
1
册
, 共8兲
where
H
1
=
冤
E
g
0
P
储
k
z
0
0
共−2⌬
⬜
/3兲
共
冑
2⌬
⬜
/3兲
0
P
储
k
z
共
冑
2⌬
⬜
/3兲
−
共
␦
+
1
3
⌬
储
兲
0
00 00
冥
共9兲
and
H
2
=
冤
0
− f
,+
0
f
,−
f
,+
000
0000
f
,+
000
冥
, 共10兲
in which E
g
is the band gap, P
储
and P
⬜
are the momentum
matrix elements parallel and perpendicular to the direction of
the c-axis, respectively,
␦
is the crystal-field splitting con-
stant, ⌬
储
and ⌬
⬜
are the spin-orbit splitting constants parallel
and perpendicular to the direction of the c axis, respectively,
f
⫾
=共P
⬜
/
冑
2兲共k
x
⫾ik
y
兲, and i=
冑
−1.
The diagonalization of the above matrix leads to the ex-
pression of the electron dispersion law in bulk specimens of
nonlinear materials as
␥
共E兲 = f
1
共E兲k
s
2
+ f
2
共E兲k
z
2
, 共11兲
where
␥
共E兲 =
兵
E共E + E
g
兲
关
共E + E
g
兲共E + E
g
+ ⌬
储
兲 +
␦
共
E + E
g
+
1
3
⌬
储
兲兴
+
2
9
E共E + E
g
兲共⌬
储
2
− ⌬
⬜
2
兲
其
,
f
1
共E兲 =
兵
ប
2
E
g
共E
g
+ ⌬
⬜
兲
关
2m
⬜
*
共
E
g
+
2
3
⌬
⬜
兲兴
−1
其关
␦
共
E + E
g
+
1
3
⌬
储
兲
+ 共E + E
g
兲
共
E + E
g
+
2
3
⌬
储
兲
+
1
9
共⌬
储
2
− ⌬
⬜
2
兲
兴
, k
s
2
= k
x
2
+ k
y
2
,
f
2
共E兲 =
兵
ប
2
E
g
共E
g
+ ⌬
储
兲
关
2m
储
*
共
E
g
+
2
3
⌬
储
兲兴
−1
其
⫻
关
共E + E
g
兲
共
E + E
g
+
2
3
⌬
储
兲兴
,
E is the electron energy measured from the edge of the con-
duction band in the vertically upward direction in the ab-
sence of any quantization, ប ⬅h/ 2
, h is Planck’s constant,
and
m
储
*
and m
⬜
*
are the effective electron masses at the edge of
the conduction band parallel and perpendicular to the direc-
tion of the c axis, respectively.
The 1D electron energy spectrum in QWs of nonlinear
optical compounds can be expressed as
␥
共E兲 = f
1
共E兲
共n
x
,n
y
兲 + f
2
共E兲k
z
2
, 共12兲
where
共n
x
,n
y
兲=共n
x
/ d
x
兲
2
+共n
y
/ d
y
兲
2
, n
x
共=1,2,3,...兲, and
n
y
共=1,2,3,...兲 are the size quantum numbers along the x and
y directions, respectively, and d
x
and d
y
are the nanothick-
ness along the respective directions.
The carrier statistics in this case can be expressed as
034303-4 Ghatak et al. J. Appl. Phys. 103, 034303 共2008兲
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n
1D
=
2
兺
n
x
=1
n
x
max
兺
n
y
=1
n
y
max
关T
1
共E
F
,n
x
,n
y
兲 + T
2
共E
F
,n
x
,n
y
兲兴, 共13兲
where
T
1
共E
F
,n
x
,n
y
兲⬅
冉
␥
1
共E
F
兲 − f
1
共E
F
兲
共n
x
,n
y
兲
f
2
共E
F
兲
冊
1/2
,
T
2
共E
F
,n
x
,n
y
兲⬅
兺
r=1
s
o
Z
r
关T
1
共E
F
,n
x
,n
y
兲兴.
Therefore, combining Eqs. 共5兲 and 共13兲,weget
G =
冉
2
k
B
2
T
3e
冊
兺
n
x
=1
n
x
max
兺
n
y
=1
n
y
max
关T
1
⬘
共E
F
,n
x
,n
y
兲 + T
2
⬘
共E
F
,n
x
,n
y
兲兴
兺
n
x
=1
n
x
max
兺
n
y
=1
n
y
max
关T
1
共E
F
,n
x
,n
y
兲 + T
2
共E
F
,n
x
,n
y
兲兴
.
共14兲
C. Investigation of TPM for QWs of III-V, ternary, and
quaternary materials
共a兲 Under the conditions ⌬
储
=⌬
⬜
=⌬ 共the isotropic spin
orbiting constant兲,
␦
=0, and m
储
*
=m
⬜
*
=m
*
共the isotropic ef-
fective electron mass at the edge of the conduction band兲,
Eq. 共11兲 assumes the form
ប
2
k
2
2m
*
= I共E 兲, I共E兲⬅
E共E + E
g
兲共E + E
g
+ ⌬兲
共
E
g
+
2
3
⌬
兲
E
g
共E
g
+ ⌬兲
共
E + E
g
+
2
3
⌬
兲
.
共15兲
Equation 共15兲 describes the dispersion relation of the
conduction electrons in III-V, ternary, and quaternary mate-
rials and is well known in the literature as the three-band
model of Kane, which should be used as such for studying
the electronic properties of such compounds where the spin-
orbit splitting constant is of the order of the band gap.
109
The 1D dispersion relation in this case is given by
ប
2
共n
x
,n
y
兲
2m
*
+
ប
2
k
z
2
2m
*
= I共E 兲. 共16兲
The 1D electron statistics can thus be written as
n
1D
=
冉
2
冑
2m
*
ប
冊
兺
n
x
=1
n
x
max
兺
n
y
=1
n
y
max
关T
3
共E
F
,n
x
,n
y
兲 + T
4
共E
F
,n
x
,n
y
兲兴,
共17兲
where
T
3
共E
F
,n
x
,n
y
兲⬅
冉
I共E
F
兲 −
ប
2
2m
*
共n
x
,n
y
兲
冊
1/2
and
T
4
共E
F
,n
x
,n
y
兲⬅
兺
r=1
s
0
Z
r
T
3
共E
F
,n
x
,n
y
兲.
The use of Eqs. 共5兲 and 共17兲 leads to the expression of
the TPM in this case as
G =
冉
2
k
B
2
T
3e
冊
兺
n
x
=1
n
x
max
兺
n
y
=1
n
y
max
关T
3
⬘
共E
F
,n
x
,n
y
兲 + T
4
⬘
共E
F
,n
x
,n
y
兲兴
兺
n
x
=1
n
x
max
兺
n
y
=1
n
y
max
关T
3
共E
F
,n
x
,n
y
兲 + T
4
共E
F
,n
x
,n
y
兲兴
.
共18兲
共b兲 Under the inequalities⌬ E
g
or ⌬ E
g
, Eq. 共15兲 as-
sumes the form
E共1+

0
E兲 =
ប
2
k
2
2m
*
,

0
⬅ 1/E
g
. 共19兲
Equation 共19兲 is known as the two-band model of Kane
and should be as such for studying the electronic properties
of such Kane materials 共e.g., InSb兲 whose energy band struc-
tures satisfy the aforementioned constraints.
The 1D electron statistics in this case is given by
n
1D
=
冉
2
冑
2m
*
ប
冊
兺
n
x
=1
n
x
max
兺
n
y
=1
n
y
max
关T
5
共E
F
,n
x
,n
y
兲 + T
6
共E
F
,n
x
,n
y
兲兴,
共20兲
where
T
5
共E
F
,n
x
,n
y
兲⬅
冉
E
F
共1+

0
E
F
兲 −
ប
2
2m
*
共n
x
,n
y
兲
冊
1/2
and
T
6
共E
F
,n
x
,n
y
兲⬅
兺
r=1
s
0
Z
r
T
5
共E
F
,n
x
,n
y
兲.
The use of Eqs. 共5兲 and 共20兲 leads to the expression of
the TPM in this case as
G =
冉
2
k
B
2
T
3e
冊
兺
n
x
=1
n
x
max
兺
n
y
=1
n
y
max
关T
5
⬘
共E
F
,n
x
,n
y
兲 + T
6
⬘
共E
F
,n
x
,n
y
兲兴
兺
n
x
=1
n
x
max
兺
n
y
=1
n
y
max
关T
5
共E
F
,n
x
,n
y
兲 + T
6
共E
F
,n
x
,n
y
兲兴
.
共21兲
共c兲 For

0
→ 0, Eq. 共19兲 assumes the well-known form of
the electron dispersion law of wide-gap materials as
E =
ប
2
k
2
2m
*
. 共22兲
Thus under the condition

0
→ 0, the expressions of n
1D
and TPM for QWs in this case can be written from Eqs. 共20兲
and 共21兲 respectively, as
n
1D
=2
冉
冑
2
m
*
k
B
T
h
冊
兺
n
x
=1
n
x
max
兺
n
y
=1
n
y
max
关F
−1/2
共
1
兲兴 共23兲
and
034303-5 Ghatak et al. J. Appl. Phys. 103, 034303 共2008兲
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