Calculations of DOS and electronic structure in Bi2S3 and Bi2Se3 by using FP-LAPW method

  • 1. Department of Physics, Gauhati University, Guwahati 7810142, India
  • 2. Department of Physics, Condensed Matter Theory Research Group, Mizoram University, Aizawl 796 004, Mizoram, India

Key words: DFTFP-LAPWDOSenergy band structureenergy band gap

Abstract: The electronic structures for Bi2S3 and Bi2Se3 have been investigated by the first principles full potential-linearized augmented plane wave (FP-LAPW) method with generalized gradient approximation. The calculated density of states and band structures show the semiconducting behavior of Bi2S3 and Bi2Se3 with a narrow indirect energy band gap of 1.4 eV in Bi2S3 and 0.8 eV in Bi2Se3.

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1.   Introduction
  • Today, semiconductors can be grown with various compositions from a monoatomic layer to nano-scale islands, rows, and arrays, in the art of quantum technology; numbers of these conceivable new electronic devices are manufactured[1]. Narrow gap semiconductors Bi$_2$S$_3$ and Bi$_2$Se$_3$ are classic room temperature thermoelectric materials[2]; these are the chalcogenides of poor metal having important technological applications in optoelectronic nano devices[3], field-emission electronic devices[4], photo-detectors and photo-electronic devices[5], photovoltaic convertors, and thermoelectric cooling technologies based on the Peltier effect[6, 7]. They have orthorhombic crystal structures at room temperature with $x$-, $y$- and $z$-positions of atoms[8] as given in Table 1 and space group Pnma (number 62). The crystal structures of Bi$_2$S$_3$ and Bi$_2$Se$_3$ are illustrated in Figures 1(a) and 1(b). In this report, we would like to present a systematic study of the DOS and energy band structures of Bi$_2$S$_3$ and Bi$_2$Se$_3$ using the FP-LAPW method.

2.   Theory and computational methods
  • The first principles FP-LAPW[9] method based on density functional theory (DFT) is used for calculations of the DOS and band structure of Bi$_2$S$_3$. The version of GGA as prescribed by Perdew, Burke and Ernzerhof[10] was used for the exchange and correlation potentials. The calculated total energy ($E$) within GGA as a function of the volume ($V$) were used for determination of theoretical lattice constants. Equilibrium lattice constants are calculated by fitting the calculated total energy to the Murnaghan's equation of state[11]. A series of total energy calculations as a function of volume can be fitted to an equation of states according to Murnaghan.
    $E(V) = E_0+\frac{\dfrac{V_0}{V}B'_0}{B'_0-1}+\frac{B_0V_0}{B'_0-1}, $(1)
    where $E_0$ is the minimum energy at $T =$ 0 K, $B_0$ is the bulk modulus at the equilibrium volume, and $B'_0$ is a pressure derivative of the bulk modulus at the equilibrium volume. The equilibrium volume is given by the corresponding total energy minimum, as shown in Figures 2(a) and 2(b)[12]. The equilibrium lattice constant was optimized using the experimental values of $a=$ 11.269 , $b=$ 3.9717 and $c=$ 11.129 for Bi$_2$S$_3$, and $a=$ 11.83 , $b=$ 4.09 and $c=$ 11.62 for Bi$_2$Se$_3$[8]. The calculation was accomplished by using the WIEN2K code[13]. In the FP-LAPW procedure, wave functions, charge density and potential are expanded in spherical harmonics within non overlapping atomic spheres of radius $R_{\rm mt}$ and, in the remaining space of the unit cell, plane waves are considered. The maximum multi-polarity $l$ for the waves inside the atomic spheres was confined within $l_{\rm max}$ $=$ 10. The wave functions in the interstitial region were expanded in plane waves with a cut-off of to $K_{\rm max}$ $=$ 2.5 a.u.$^{-1}$ (where $K_{\rm max}$ is the maximum value of the wave vector $\boldsymbol{K}$ $=$ $k$ $+$ $\boldsymbol{G}$). For Bi: 6s, 6p, S: 3s, 3p and Se: 4s, 4p states were treated as the valance state and all other lower states were treated as the core state. The potential and charge density were expanded up to a cut-off of $G_{\rm max}$ $=$ 12 a.u.$^{-1}$. The muffin-tin radii are set to $R_{\rm mt}$ $=$ 2.4~a.u. for Bi, 2.2 a.u. for S in Bi$_2$S$_3$, $R_{\rm mt}$ $=$ 2.3 a.u. for Bi and 2.2 a.u. for Se in Bi$_2$Se$_3$. A mesh of 1500 $k$-points was used after performing the $k$- optimization. The calculated lattice constants found by volume optimization are $a=$ 11.2263 , $b=$ 3.567~ and $c=$ 11.0868 for Bi$_2$S$_3$, and $a=$ 11.4753 , $b=$ 3.9647 and $c=$ 11.2716 for Bi$_2$Se$_3$ which are shown in Figures 2(a) and 2(b).

3.   Results and discussions
  • In Figures 2(a) and 2(b), we show the total energy curve as a function of the unit cell volume for Bi$_2$S$_3$ and Bi$_2$Se$_3$. The total and partial DOS plots of Bi$_2$S$_3$ and Bi$_2$Se$_3$ are shown in Figures 3-6. From Figure 3, we found that the contributions to the total DOS were from Bi-6p and S-3p electron states in Bi$_2$S$_3$ and from Bi-6p and Se-4p electron states in Bi$_2$Se$_3$. The core region, which is below $-6$ eV, is formed by 6s and 6p electron states of Bi and a sharp peak at around -10.5 eV is observed in Bi$_2$S$_3$ (Figure 4(a)), while a sharp peak at around -10.2 eV is observed in Bi$_2$Se$_3$ (Figure 4(b)). The conduction region, which is above the Fermi level, is mainly contributed by Bi-6p, 6p$_{x}$, 6p$_{y}$ state electrons in Bi$_2$S$_3$ (Figure 5(a)) and by Bi-6p, 6p$_{x}$ state electrons in Bi$_2$Se$_3$ (Figure 5(b)). In the valence region (below Fermi level), we have observed that the S-3p electron state mainly contributes to the total DOS giving a sharp peak at around $-0.8$ eV in Bi$_2$S$_3$ (Figure 6(a)), and the Se-4p electron state mainly contributes to the total DOS giving sharp peaks at around $-0.5$ eV and $-1.2$ eV in Bi$_2$Se$_3$ (Figure 6(b)).

    From the band structure plots, we observed an indirect band gap of the order of 1.4 eV in Bi$_2$S$_3$ (Figure 7(a)) and an indirect band gap of the order of 0.8 eV in Bi$_2$Se$_3$ (Figure 7(b)). Band structure plots were also found with a higher number of bands at the regions where peaks of the DOS were observed. In Figures 8 and 9, we have compared band structures with DOS plots of Bi$_2$S$_3$ and Bi$_2$S$_3$, and found that higher DOS regions correspond to having more bands.

4.   Conclusions
  • In conclusion, we have observed a qualitative agreement between theoretical and experimental lattice constants. The calculated band gap is very close to the experimental value. Band gaps of the order of 1.4 eV and 0.8 eV suggest that Bi$_2$S$_3$ and Bi$_2$Se$_3$ are semiconductors with a low energy gap. Since the Fermi level is very close to the valance band, it indicates that the semiconductors are p-type. The calculated band gaps also suggest that the compounds may be used as a suitable candidate for thermoelectric applications. Semiconductor Bi$_2$S$_3$, with a band gap of 1.4 eV, belongs to a family of solid state materials with applications in thermoelectric cooling technologies based on the Peltier effect[14, 15]. Semiconductor Bi$_2$Se$_3$, with a band gap of 0.8 eV, has useful applications in the field of thermoelectric devices as solid state coolers or generators[16, 17]. However, the band gaps when checked with experimental values (1.3 eV in Bi$_2$S$_3$ and 0.35 eV in Bi$_2$Se$_3$)[14] seem to have differences. We propose to check these discrepancies with mBJ potential inclusion.

    Acknowledgements DD is grateful to the Department of Physics, Gauhati University and the Department of Physics, Mizoram University for extending all the necessary facilities to do this work. RKT acknowledges a research grant from UGC.

Figure (9)  Table (3) Reference (17) Relative (20)

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