Improved Electronic Structure and Optical Performance of Bi 2 Te 3x Se x From First-principle Calculations Within TB-mBJ Exchange Potential

Using the first-principle calculations of density functional theory within the (FP-LAPW) method, we have investigated the structural, electronic and optical properties of Bi2Te3-xSex alloys with compositions x = 0, 1, 2 and 3 of Se. The generalized gradient approximation functional of Wu and Cohen (GGAPBE) is used to calculate ground state structural parameters of Bi2Te3-xSex , which are in good agreement with theoretical and experimental data. The electronic band structures and optical constants have been improved with Tran-Blaha modified Becker-Johnson (TB-mBJ) parameterization scheme. Also, we have analyzed in detail the performance of dielectric function, refractive index, reflectivity and optical conductivity of these alloys. Our results show that Bi2Te3-xSex alloys are promising candidates for optoelectronic applications especially in the Infrared and visible fields. Bi2Te3-xSex materials have a direct band gap and can be tuned from 0.1706 eV to 0.7819 eV by varying In composition so emission was tunable from 1.58 to 7.26 micrometers (infrared field), in addition for their direct band gap and in view of their attractive optical properties such conductivity, absorption and reflectivity these materials is considered as promising materials for optoelectronic applications.


Introduction
Bismuth telluride (Bi2Te3) and bismuth selenide (Bi2Se3) have technological interest owed to their thermoelectric utilization, narrow band gap and photosensitive 1 .During last decennaries, these compounds have been extensively studied to advance their thermoelectric and optical properties by different aspect technological like doping, high-pressure and variation in mesostructure [2][3][4][5] .Structural, electrical and optical properties of Bi 2 Se 3 and Bi 2 Se (3x-1) Te x thin films have been studied experimentally 6 .Synthesis and thermoelectric characterization of Bi2Te3 nanoparticles has been studied experimentally 7 .Yamini Sharma et al. 8 have studied the electronic structure, optical properties and Compton profiles of Bi 2 S 3 and Bi 2 Se 3 .Oriented Bi 2 Se 3 nanoribbons film: Structure, growth, and photoelectric properties have been studied experimentally by Yuan Yu et al. 9 .Thermoelectricity and superconductivity in pure and doped Bi2Te3 with Se is reported by H.A. Rahnamaye Aliabad and M. Kheirabadi 10 .Recently Kun Zhao et al. have calculated the pressureinduced anomalies in structure, charge density and transport properties of Bi2Te3 using density functional theory (DFT) in the framework of the projector augmented waves method 11 .
Preceding studies have reported that Bi 2 Te 3 , Bi 2 Te 2 Se, Bi 2 TeSe 2 and Bi 2 Se 3 , crystallizes in hexagonal crystal with the space group hR5 (R3m) 12 as demonstrated in Fig. 1.The structures can be detected as Bi and Te (Se) layers stacked along the c-axis and containing five atoms per unit cell (two Bi atoms and three Te or Se atoms) 10 .Increase both theoretical and experimental investigation on electronic structure and optical properties of Bi 2 Te 3-x Se x (x=0,1,2 and 3) exhibit a direct band gap 13 and The refractive index of the Bismuth Telluride is higher than any value previously reported for a semiconductor 14 .
In this work the principle of our inspection is concentrated on predicting the structural, electronic and optical properties of Bi 2 Te 3-x Se x alloys with a number of concentrations (x) (x=0,1,2 and 3).We have used in this calculation the full potential linearized augmented plane wave (FP-LAPW) method within density functional theory (DFT) 15 .For theoretical calculations of structural constants we used the generalized gradient approximation GGA with PBE functional 16 for electronic and optical properties we used Tran-Blaha modified Becke-Johnson exchange potential approximation (TB-mBJ) 17 .
After the abstract and introduction, the rest of the paper is formed as follows: in Section 2, a brief outline of the method of calculation is given.In Section 3 details of the obtained results and discussion related to structural, electronic and optical properties of Bi 2 Te 3-x Se x alloys are presented.The Tlemcen University, Algeria consistent calculations are evaluated to be converged when the total energy of the system is stable within 0.0001 Ryd.

Structural properties
With a view to calculate the ground states properties of Bi 2 Te 3 , Bi 2 Te 2 Se, Bi 2 TeSe 2 and Bi 2 Se 3 the total energies are calculated for different volumes about the equilibrium cell volume V 0 .The calculated total energies are adapted to the Murnaghan's equation of state 25 to find the ground state properties like the equilibrium lattice constant a, c, the bulk modulus B and its pressure derivative B'.The calculated equilibrium parameters (a, c, B and B') for all structures are given in Table 1, which also contains works of previous calculations as well as the experimental data.Previous research have reported that Bi2Te3, Bi2Te2Se, Bi2TeSe2 and Bi2Se3, have the greatest possibility of crystallizing in trigonal system (hexagonal ) with the space group hR5 (R3m) (N°166) and in orthorhombic system with the space group Pnma (N°62).
In Fig. 2, we depicted the calculated lattice constants a and c as a function of selenide concentration (x) (x=0,1,2 and 3) and with comparative data 10 .We have determined the total bowing parameter (b) by fitting the non linear variation optimized lattice constants a and c as Se concentration (x) by the quadratic function.Our results are presented by the relation (1) and that comparative data 10 are presented with the relation (2). (1) (2) Our results exhibit a negligible bowing parameter for the lattice constant a which has a bowing parameter b = 0.00442 Å and b=0.01233Å for the lattice constant c.
The bulk modulus of Bi 2 Te 3-x Se x for various x concentrations is reported in Table 1.The variation of the bulk modulus versus Se composition (x) is presented in Fig. 3 the bowing parameter of the bulk modulus is determined by fitting the non linear variation, Relation (3) exhibit the result: (3)

Electronic properties
In this part, we have calculated the electronic band structure for Bi 2 Te 3-x Se x using GGA-PBE 16 and (TB-mBJ) of Tran-Blaha modified Becke-Johnson 17 approaches.
Table 2, show the computed energy band gaps Eg of Bi 2 Te 3-x Se x using (GGA-PBE) and (TB-mBJ) approaches

Method of Calculation
In the present study, we have used the (FP) full-potential (LAPW) linearized augmented plane wave (FP-LAPW) method within the framework of the density functional theory (DFT) 15 as implemented in the Wien2k code 18 .In calculating structural parameters, we have used the generalized gradient approximation (GGA) with PBE functional because Numerical tests have shown that the PBE-GGA gives total-energy dependent properties in good agreement with experiment [19][20] .
(GGA-BPE) 16 and the recently recommended and better predictable approach technique called Tran-Blaha modified Becker-Johnson (TB-mBJ) potential approximation 17 have been used to calculate electronic and optical properties of Bi 2 Te 3-x Se x by reason it is capable to illustrate correctly the electronic structure of the solids and the insulators 21,22,23 .Tran et al., have demonstrated that this form of approach develop over the GGA and LDA potentials for the determination of band gaps value 24 .
In this paper, we have selected the muffin-tin radii (MT) for each atoms (Bi, Te and Se) to be 2.50 atomic units (a.u.).The K max = 8 (RMT) −1 (Kmax is the plane wave cut-off and RMT is the smallest of all atomic sphere radii).The Fourier expanded charge density was truncated at G max =12(Ryd) 1/2 , the l-expansion of the non-spherical potential and charge density was carried out up to lmax = 10.The cut-off energy is set to −6 Ryd to isolate core from valence states.The self-     functions of the conduction and valence bands, respectively.The real part of the dielectric function can be evaluated from ε 2 (ω) using the Kramers-Kronig relations is given by: ( The knowledge of both the real and imaginary parts of the dielectric function allows the calculation of important optical functions such as the refractive index n(ω), extinction coefficient k(ω), absorption coefficient α(ω), optical conductivity σ(ω), loss energy function L(ω) and the reflectivity R(ω) [42][43][44] .
The conductivity and the dielectric constant are related to each other by the relation: (9)   for all studied composition (x) (x=0,1,2 and 3), theoretical data for all concentration 10 and experimental data only for Bi 2 Te 3 and Bi 2 Se 3 .Fig. 4 shows the variation of band gap energies Eg versus the concentration (x) for the Bi 2 Te 3-x Se x compared with other theoretical calculation of band gap with and without spin orbit coupling (SOC) 10 .The electronic band structures properties were determined for all structures with the predicted lattice constant via GGA-PBE and TB-mBJ approaches.However, the results obtained by TB-mBJ approach and GGA-PBE are plotted.Fig. 5 shows the electronic band structures calculated for along the high-symmetry lines of the first Brillouin zone.The valence band maximum (VB max ) and conduction band minimum (CB min ) are founded Γ at point, showing Bi 2 Te 3-x Se x have the direct band gap for all concentration (x) (x=0,1,2 and 3), which are satisfactory for optoelectronic materials.
The imaginary part of the dielectric function, ε 2 (ω) is obtained from the momentum matrix elements 40 , and the electronic structure calculation (densities of states).The real part ε 1 (ω) can be derived from the ε 2 (ω), using the Kramer-Kronig transformations 41 .The imaginary part of the dielectric function ε 2 (ω) is expressed as follows: (4)   where e is the electric charge, Ω is the unit cell volume, u is the vector defining the polarization of the incident electric field, ω is the frequency of the light, φ K C and φ K V are the wave Band structure calculated for Bi 2 Te 3-x Se x using (TB-mBJ) method and GGA-PBE.
ħ Improved Electronic Structure and Optical Performance of Bi 2 Te 3-x Se x From First-principle Calculations Within TB-mBJ Exchange Potential According to the Drude classical free-electron theory 44 σ(ω) is given by the relation (10)   The loss function L(ω) which is also an important optical parameter describing the energy loss of a fast electron traversing in the material, can be calculated from the dielectric constant as: (11)   The reflectivity is calculated from the following relation: (12)   In order to calculate the optical properties, one needs to use a dense mesh of uniformly distributed k-points.We present calculations with 120 k-points in this study.
The calculated optical parameters for radiation up to 20 eV, within the TB-mBJ approach, are presented in Figs.6-16.      1, 2 and 3 respectively, revealing the peak moves toward the higher energy side with x increasing.Fig. 7 indicate the zero frequency limits ε 1 (0) versus Se (selenide) concentration (x) with comparable data 10 .The bowing parameter of The static dielectric constants (the zero frequency limits) ε 1 (0) Improved Electronic Structure and Optical Performance of Bi 2 Te 3-x Se x From First-principle Calculations Within TB-mBJ Exchange Potential is determined by fitting the non linear variation, Relation (13) show the result: (13)   The imaginary part of the dielectric function of Bi 2 Te 3- x Se x alloys with different concentration (x) is illustrated by Fig. 8.Our investigations of the imaginary part of the dielectric function trajectory exhibit that the first critical point of the dielectric function at x=0, 1, 2 and 3 exist about 0.4489 eV, 0.5306 eV, 0.8843 eV and 1.02 eV, respectively.It is visible that the critical point change toward higher energies with the increase of Selenide concentration.With rising energy, we sign that the dielectric function ε 2 (ω) display a essential maximum located at 1.8367 eV, 2.0272 eV, 2.0816 and 2.4082 eV for x=0, 1, 2 and 3, respectively.When the Selenide composition increase, all the structures in ε 2 (ω) are shifted toward higher energies.
The refractive index can provide information for us about the behavior of light.When light passes through the different substances its velocity decreases by increasing of the refractive index of these substances.In Fig. 9, the refractive index shows an appreciable value in low-energy region and a considerable reduction in high-energy region.
In Fig. 9 and 10, we show the refractive index n(ω) and the extinction coefficient k(ω) of Bi 2 Te 3-x Se x with different composition (x) at the equilibrium lattice constant were calculated using TB-mBJ method.From Fig. 9 we can detect that the static refractive index n(0) for Bi 2 Te 3, Bi 2 Te 2 Se, Bi 2 TeSe 2 and Bi 2 Se3 are found to be 5.6685, 5.0061, 4.6903 and 4.2640, respectively.Fig. 11 exhibit the static refractive index n(0) versus composition (x).The bowing parameter of The static refractive index n(0) is determined by fitting the non linear variation, Relation ( 14) exhibit the result.The n(ω) present a crucial maximum located at 1.5374 eV, 1.7551 eV, 1.7007 eV and 2.1360 eV for x=0, 1, 2 and 3, respectively.(14)   The absorption coefficient α(ω) is presented in Fig. 12, is an important parameter of each optoelectronic devices.The spectrum of absorption demonstrates that the energy of the threshold using TB-mBJ method is around 1.2088 eV for Bi2Te3, 1.4059 eV for Bi2Te2Se, 1.4615 eV for Bi2TeSe2 and 1.7601 eV for Bi2Se3 with x varies from 0 to 3, this energy is called the threshold of absorption.Each peak corresponds to an electronic transition, the first peak is located at around 2.3537 eV ,2.8708 eV, 2.7891 eV and 3.4150 eV for Bi2Te3, Bi2Te2Se, Bi2TeSe2 and Bi2Se3 respectively, these energies are corresponding to the visible field except Bi2Se3 which is conform to the Ultraviolet range. 42,43.The maximum of all peaks in the absorption curve is located, at 7.9321 eV for Bi2Te3, surroundings 8.4219 eV for both Bi2Te2Se, and Bi2TeSe2 and 12.1220 eV for Bi2Se3.Be accordant to the corresponding wavelength of these energies, we can remark that these materials are good applicant to work into Ultraviolet fields; in addition the absorption curve becomes considerable according to the Selenide composition; it amount its maximum in the Ultraviolet field.These advantage us to glean that these components can produce as absorption components of the Ultraviolet waves.
The optical conductivity presented and described as a function of inter-band and intra-band transitions.Figure 13 shows the optical conductivity σ (ω) of Bi  Se3 at the energy of 2.6256 eV, 2.8980 eV, 3.4150 eV and 3.6055 eV respectively, also, it is shown that after 18.57 eV the reflectivity decreases dramatically and tends towards zero.The bowing parameter of The static reflectivity constants (the zero frequency limits) R(0) is determined by fitting the non linear variation, Relation (15) show the result: (15)   Our results exhibit a negligible bowing parameter for the static reflectivity constant R(0), which has a bowing parameter b = 0.0022.

Conclusion
Bi 2 Te 3, Bi 2 Te 2 Se, Bi 2 TeSe 2 and Bi 2 Se3 by means of Wien2K computational package, PBE-GGA, and TB-mBJ in the scheme of Density Function Theory, are studied to predict the structural, electronic and optical properties.The lattice constants a, c, the bulk modulus B and its pressure derivative B' of Bi 2 Te 3-x Se x as a function of selenide concentration (x) (x=0,1,2 and 3) are calculated.The band structure of Bi 2 Te 3- x Se x is calculated using GGA-PBE and TB-mBJ method, the result exhibit that for all concentrations have direct band gap.The real part ε 1 (ω), imaginary part of the dielectric function ε 2 (ω), refractive index n(ω), extinction coefficient k(ω), absorption coefficient α(ω), optical conductivity σ(ω), loss energy function L(ω) and the reflectivity R(ω) are calculated using both GGA-PBE and TB-mBJ.
Our results show that Bi 2 Te 3-x Se x alloys are promising candidates for optoelectronic applications especially in the Infrared and visible fields.Bi 2 Te 3-x Se x materials have a direct band gap and can be tuned from 0.1706 eV to 0.7819 eV by varying In composition so emission was tunable from 1.58 to 7.26 micrometers (infrared field), in addition for their direct band gap and in view of their attractive optical properties such conductivity, absorption and reflectivity these materials is considered as promising materials for optoelectronic applications as photodetector and Infrared Receivers.

a
Institute of Technology, Nour Bachir University Center, El-Bayadh, 32000, Algeria.b Theoretical Physics Laboratory, U.S.T.H.B, Algiers, Algeria c Physics Department, Faculty of Sciences, University Dr Moulay Tahar, Saida, Algeria.d Laboratory of Separation and Purification Technology, Department of Chemistry, Faculty of Sciences,

Figure 2 .
Figure 2. Calculated lattice constant a and c of Bi 2 Te 3-x Se x as a function of Se concentration (x).Figure 3. The variation of the bulk modulus of Bi 2 Te 3-x Se x versus Se composition (x)

Figure 3 .
Figure 2. Calculated lattice constant a and c of Bi 2 Te 3-x Se x as a function of Se concentration (x).Figure 3. The variation of the bulk modulus of Bi 2 Te 3-x Se x versus Se composition (x)

Figure 4 .
Figure 4.The variation of band gap energies Eg of Bi 2 Te 3-x Se versus the concentration (x).

Table 1 .
Lattice constants a (in Å), c (in Å), bulk modulus B (in GPa) and its pressure derivative B' of Bi 2 Te 3x Se x with their corresponding experimental values and other theoretical data.