Semimetal-insulator phase transition in a GaSb/InAs heterostructure

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. 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