Abstract. In this study, the authors will show that a semimetal-insulator phase transition
may take place in the GaSb/InAs heterostructures due to the exciton pairing between the
electrons and holes in the system. In the heterostructure, the top of the valence band of
the InAs layer is higher than the bottom of the conduction band of the GaSb layer. When
the Fermi energy level of the system is higher than the bottom of the conduction band of
the GaSb layer and lower than the top of the valence band of the InAs layer, the system
has both electrons and holes and is considered to be a semimetal system. The theoretical
calculations for the band gap and the simulation by CASTEP of the electronic structure
and density of state in the heterostructrure also indicate that there is a dielectric gap near
the Fermi energy level and the semimetal-insulator occurred. These results are in good
agreement with the experimental data.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2016-0037
Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 98-105
This paper is available online at
SEMIMETAL-INSULATOR PHASE TRANSITION
IN A GaSb/InAs HETEROSTRUCTURE
Nguyen The Lam
Faculty of Physics, Hanoi Pedagogical University No. 2
Abstract. In this study, the authors will show that a semimetal-insulator phase transition
may take place in the GaSb/InAs heterostructures due to the exciton pairing between the
electrons and holes in the system. In the heterostructure, the top of the valence band of
the InAs layer is higher than the bottom of the conduction band of the GaSb layer. When
the Fermi energy level of the system is higher than the bottom of the conduction band of
the GaSb layer and lower than the top of the valence band of the InAs layer, the system
has both electrons and holes and is considered to be a semimetal system. The theoretical
calculations for the band gap and the simulation by CASTEP of the electronic structure
and density of state in the heterostructrure also indicate that there is a dielectric gap near
the Fermi energy level and the semimetal-insulator occurred. These results are in good
agreement with the experimental data.
Keywords: Semimetal-insulator phase transition, GaSb/InAs heterostructure, electronic
structure GaSb/InAs.
1. Introduction
The heterojunction is the interface between two layers of the different semiconductors.
These semiconductors have unequal band gaps. Heterojunctions have a lot of advantages for
application of the electronic energy bands in devices such as semiconductor lasers, solar cells,
diode and transistors. The behavior of semiconductor heterojunctions depend directly on the
alignment of the energy bands at the interface. Semiconductor heterojunctions can be organized
into three types: straddling gap (type I), staggered gap (type II) and broken gap (type III). The
GaSb/InAs hetrostructure has staggered gap and is applied for many devices [1, 2]. In this
heterostructure, GaSb has a energy gap Eg = 760 meV and InAs has a energy gap Eg = 360
meV, and the top of the valence band of the InAs layer is higher than the bottom of the conduction
band of the GaSb layer [3].
The semimetal system has characteristics such that the top of the valence band is higher
than the bottom of the conduction band. Thus, when the Fermi level of the system is higher than
the bottom of the conduction band, and lower than the top of the valence band, the system has both
electrons and holes. In the case of GaSb/InAs hetrostructure, the top of the valence band of the
InAs layer is higher than the bottom of the conduction band of the GaSb layer. When, the Fermi
Received November 27, 2015. Accepted July 26, 2016.
Contact Nguyen The Lam, e-mail address: nguyenthelam2000@yahoo.com
98
Semimetal-insulator phase transition in a GaSb/InAs heterostructure
energy level of the system is higher than the bottom of the conduction band in the GaSb layer
and lower than the top of the valance band in the InAs layer, the system has both electrons and
holes and is considered to be a semimetal system. The overlap of electron (hole) wave functions
between adjacent InAs (GaSb) layers result in the formation of electron (hole) minibands in the
conduction (valence) band. Thus, in the hetrostructure, the two-dimensional carrier gas in the
interface between two semiconductors plays an important role in this phase transition. Because of
the exciton pairing of electrons and holes in two bands (layers) near the Fermi energy level, there
is a dielectric gap in the energy spectrum and the semimetal-insulator phase transition takes place.
This type of phase transition also takes place in single layer materials such as graphene [4, 5].
Figure 1. The semimetal-insulator phase transition
In this paper, the energy spectrum for conduction and valence bands of the bulk GaSb,
InAs and GaSb/InAs heterojunctions are calculated using CASTEP, a software for first principles
electronic structure calculations. Within the density functional formalism, it can be used to
simulate a wide range of materials including crystalline solids, surfaces, molecules, liquids
and amorphous materials [6]. Electronic, magnetic, thermodynamic and optical properties of
materialsl can be simulated by CASTEP.
2. Content
2.1. Basic equations
Following [7] the Hamiltonian for the GaSb/InAs heterojunction obtains the form
H =
∑
~κσi
ε(~k)a+iσ(
~k)aiσ(~k) +
1
2
∑
σ1σ2~k1~k2~q
λa+iσ1(
~k1 + ~q)a
+
jσ2
(~k2 − ~q)aiσ2(~k2)ajσ1(~k1) (2.1)
where aiσ(~k) and a
+
iσ(
~k) are the annihilation and creation electron operators respectively with spin
σ or -σ in the i-th band, λ is the screen Coulomb interaction and i, j = 1, 2 are the band index.
The Green functions are defined by
Gσ1σ2ij (
~k, t) =< Tt(aiσ1(
~k, t)a+jσ2(
~k, 0)) > (2.2)
where Gσσ
ij
(~k, t) is the Green function for exciton pairs with the same orientation of spin for both
electrons and holes and Tt is the time-ordering operator. It may lead to the appearance of a charge
density wave (CDW) and a structural phase transition. Gσ−σ
ij
(~k, t) is the Green function for exciton
pairs with different orientation of spin for electrons and holes.
99
Nguyen The Lam
Using the Green function method [8] and combining Hamiltonian (2.1) with the definitions
(2.2), in the self-consistent field approximation, we have
[−iωn + ε1(~k)]Gσσ11 (~k, ωn) + ∆Gσσ12 (~k, ωn) = 1
∆∗Gσσ11 (~k, ωn) + [−iωn + ε2(~k)]Gσσ12 (~k, ωn) = 0 (2.3)
where, Gσσ
11
(~k, ωn), Gσσ12 (
~k, ωn) are Fourier transformation coefficients of the Gσσ11 (
~k, t),
Gσσ
12
(~k, t) respectively and the gap ∆ is defined
∆ = T
∑
~k,ωn
λ.Gσσ
12
(~k, ωn) (2.4)
and ωn = (2n + 1)πT with n is an integer and T is the temperature.
To evaluate the model, we choose the energy spectrum in the simple form
ε1(~k) = δµ + εGaSb(~k) (2.5)
ε2(~k) = δµ − εInAs(~k) (2.6)
where, εGaSb(~k) is the energy of the conduction band of the GaSb layer and εInAs(~k) is the energy
of the valence band of the InAs layer. σµ is the shift of Fermi level in the GaSb and InAs layers.
Solving the system of equations (2.3) for Gσσ
11
(~k, ωn), Gσσ12 (
~k, ωn), we obtain
Gσσ
12
(~k, ωn) =
∆∗
∆2 − [−iωn + ε1(~k)].[−iωn + ε2(~k)]
(2.7)
From (2.7), the excitation spectrum for one particle is given as
ω±σ(~k) = δµ ± E(~k) (2.8)
where
E(~k) = [ε1(~k).ε2(~k) + ∆
2]
1
2 (2.9)
Substituting (2.7) into (2.4), we get the equation for the dielectric gap.
1 = λ
∑
~k
1
4E
[
th
E + δµ
2T
+ th
E − δµ
2T
]
(2.10)
2.2. Numerical results and discussion
We used CASTEP to calculate the energy bands,εGaSb(~k)for the GaSb and εInAs(~k) for the
InAs. Both GaSb and InAs crystals have a Zincblende structure and are built in order to calculate
the energy band. They are shown in Figure 2.
100
Semimetal-insulator phase transition in a GaSb/InAs heterostructure
Figure 2. The Zincblende structure of GaSb and InAs semiconductors
where the lattice spacing a1 = 6.0954 A˚ for the GaSb and a2 = 6.05838 A˚ for the InAs
The lattice spacing is a1 = 6.0954 A˚ for GaSb and a2 = 6.05838 A˚ for InAs [9]. The energy
band structures for the bulk GaSb and InAs are shown in Figures 3 and 4. In the energy band for
the bulk GaSb, we see that, there is a energy gap Eg(GaSb) = 0.735 eV (the band gap Eg of GaSb,
which is measured by experimental, is 0.76 eV [3]) and the Fermi level is chosen, such that, it is
higher than the bottom of the conduction band in the GaSb layer. In Figure 4, there is an energy
gap Eg(InAs) = 0.342 eV (the band gap Eg of InAs, which is measured by experimental data, is
0.36 eV [3]) and the Fermi level is lower than the top of the valence band in the InAs layer. These
calculations are also close to other experimental data in which, the GaSb has band gap Eg = 0.68
eV and the InAs has band gap Eg = 0.36 eV at 300 K [10].
Figure 3. The energy band of the bulk GaSb is calculated by CASTEP
This is energy gap Eg = 0.735 eV and the Fermi level is chosen,
such that, it is higher than the bottom of the conduction band in the GaSb laye.
From our obtained branch of the conduction band in the energy band of the GaSb layer and
from the branch of the valence band in the energy band of the InAs layer, we can determine the
energy spectrum εGaSb(~k)andεInAs(~k) numerically. Substituting theεGaSb(~k) and εInAs(~k) into
the equation (2.10) we get the screen Coulomb interaction λ = -0.08 eV [11]. Solving the equation
(2.10), we obtain the dependence of the dielectric gap on the temperature (see Figure 5). The
existence of the dielectric gap in the energy spectrum has shown that, there is the phase transition
from the semimetal state to insulator (or semiconductor) state.
101
Nguyen The Lam
Figure 4. The energy band of the bulk InAs is calculated using CASTEP
These is energy gap Eg = 0.342 eV and the Fermi level is chosen,
such that, it is lower than the top of the valence band in the InAs layer
Figure 5. The dependence of the energy gap on temperature
Here, the screen Coulomb interaction λ = -0.08 eV
In fact, the dielectric gap may depend on other parameters, one being doping. But in this
paper, we only study the dependence of the dielectric gap on temperature. The dependence of this
energy gap on the doping need is a subject for further study.
To confirm this phase transition, we have also calculated the electronic structure and
the density of state for the GaSb/InAs heterostructure using CASTEP. The crystal of the
heterostructure is built in order to make the calculations and is shown in Figure 6. Because
CASTEP requires a period structure, the GaSb layer is separated into two parts.
From the electronic structure (Figure 7) and density of state (Figure 8) of the GaSb/InAs
heterostructure, we see that, there is a dielectric gap. This calculated energy gap is 1.68 eV. In
other hand, due to the 2D carrier gas in the layers, the motion of carriers on the third dimension
(perpendicular to the layers) is quantization. Thus, the conduction and valence bands are divided
102
Semimetal-insulator phase transition in a GaSb/InAs heterostructure
into many mini-bands (Figure 7). The experiments have shown that, a semimetal-insulator phase
transition occurred in the GaSb/InAs superlattice. This superlattice is heterostructure and made
up to many different material layers. In the GaSb/InAs superlattice, the ratios of thickness of
GaSb/InAs are 20.5/27 (A˚) and 17.5/24 (A˚), the band gaps of Eg the GaSb/InAs superlattice are
0.299 eV and 0.34 eV respectively [12]. In this paper, the ratio of thickness of GaSb/InAs is
12.1538/12.1538 (A˚) and band gap Eg is 1.68 eV.
Figure 6. The crystal of the GaSb/InAs heterostructure for calculating the energy band
and density of state, where the thickness of the GaSb and InAs layers are 12.1538 A˚
Figure 7. The energy band of the GaSb/InAs heterostructure is calculated using CASTEP
There is an energy gap Eg = 1.68 eV
103
Nguyen The Lam
The theoretical calculation used for the renormalization method [13] or other method [14],
and the simulation of GaSb/InAs [15] have shown that, there is a dielectric gap in the GaSb/InAs
superlattice and semimetal-insulator phase transition has occurred.
Figure 8. The density of state of the GaSb/InAs heterostructure is calculated by CASTEP
There is an energy gap Eg = 1.68 eV
3. Conclusion
Using CASTEP, we have calculated the energy band for the single GaSb and InAs layers.
With these energy bands, we have shown that, there is a semimetal-insulator phase transition in
the GaSb/InAs heterostructure which is due to the exciton pairing between the electrons and the
holes in two bands, which are near the Fermi level. The calculations of the energy band and the
density of state for the GaSb/InAs heterostructrue have shown that, there is a dielectric gap in the
energy band of the heterostructrue which is in good agreement with the experimental data.
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