Abstract

Presently known technological applications of ZnSe-based devices partly motivated this study which illustrates how one can perform accurate calculations of properties of semiconductors in the framework of density functional theory (DFT). Unlike numerous, previous DFT results, ours for ZnSe are in agreement with available, corresponding experimental findings. In particular, our calculated band gap of 2.65 eV is in excellent agreement with experiment. In addition to the band structure, we also report the calculated eigenvalues at high symmetry points, the total and partial densities of states, and the curve of the total energy versus the lattice constant. We employed a local density functional approximation (LDA) potential and the linear combination of Gaussian orbitals. As explained below, the use of successively augmented basis sets, starting with the minimum basis set or one slightly larger, to perform self-consistent calculations leads to a generalized minimization of the energy content of the Hamiltonian down to the absolute minima of the occupied energies, i.e., the ground state. This process guarantees that our calculations strictly adhere to the two DFT theorems and produce results that possess the full physical content of DFT and agree with experiment.

Keywords

BZW-EF Method, Self-Consistent, Ground State, Basis Sets

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1. Introduction

ZnSe (zinc selenide) is a II - VI binary semiconductor that crystallizes predominantly in the cubic zinc-blende structure [1]-[4]. It is characterized by a direct band gap located at the Γ point of the Brillouin zone. It has a wide band gap and remarkable optoelectronic properties. The measured room-temperature band gap of ZnSe is 2.60 - 2.72 eV [5]-[7], while the low-temperature value is reported to be 2.82 eV [8]. ZnSe has attracted considerable attention for applications in blue and near-ultraviolet light-emitting diodes (LEDs), semiconductor lasers, photodetectors, and photovoltaic devices [6] [7]. The importance of its technological applications makes it an extensively studied semiconductor, both theoretically and experimentally [9]-[12].

A thorough understanding of the electronic properties of ZnSe is essential for optimizing its performance in technological applications. Fundamental parameters such as the electronic band structure, density of states (DOS), the magnitude and nature of the energy gap, and the orbital contributions to the valence and conduction bands play a crucial role in determining its optical and electrical properties [10] [13]-[16].

Several sources have reported results of experimental studies of ZnSe as indicated in Table 1.

Table 1. Experimental measurements of the band gap of ZnSe.

Theis [17] focused on the electronic band structure and critical optical transitions in II - VI semiconductors, emphasizing the effect of spin-orbit coupling on the valence bands and providing a theoretical basis for interpreting the fine optical properties of ZnSe. He observed a direct gap of 2.76 eV. Venkatachalam et al. [13] deposited ZnSe thin films via vacuum evaporation. X-ray diffraction confirmed a polycrystalline zinc-blende structure and optical measurements indicated a direct gap of 2.6 - 2.72 eV. Electrical measurements revealed thermally activated semiconductor behavior. Furthermore, Thirumavalavan et al. [14] extended the analysis using chemical bath deposition (CBD) and also examined morphological and dielectric properties. The measured direct gap was 2.65 eV, slightly lower than the one for evaporated films. Differences in film thickness and microstructure likely explain the small variations of the gap.

Ebina et al. [16] measured the E₀ and E₀ + Δ₀ transitions as a function of x and observed band-gap bowing, i.e., a nonlinear dependence of the gap on composition. Homann et al. [10] combined optical measurements and simulations to confirm the band gap bowing, providing a predictive model for designing semiconductors with tailored gaps.

From the content of the table, we conclude that zinc blende ZnSe exhibits a large direct band gap of 2.60 to 2.76 eV, depending on film thickness, deposition method, and microstructure.

In addition to the experimental works reported above, numerous theoretical studies of ZnSe have been performed. Some of them utilized ab-initio LDA or GGA potentials in first-principles calculations. Unlike the above experiments, the ab-initio calcullations report different band gaps which woefully underestimate the measured ones. We do not elaborate on theoretical results obtained with ad hoc potentials, i.e., those not resulting from the functional derivative of an exchange-correlation energy. In addition to the hybride ones, these ad s potentials include those obtained by adding or subtracting Hubbard’s u or other quantities to ab-initio LDA or GGA potentials.

Ghaleb et al. [19] studied the structural, electronic, and optical properties of cubic ZnSe in the sphalerite phase using density functional theory (DFT). They used an ab-initio LDA potential and the calculated band gap of ZnSe was 1.33 eV. Optical properties derived from the dielectric function indicate strong absorption in the ultraviolet region, highlighting the suitability of ZnSe for optoelectronic applications. Asadi et al. [20] employed the full-potential linearized augmented plane wave (FP-LAPW) formalism and reached a band gap value of 1.17 eV. Jafarova. et al. [11] utilized the local spin density approximation (LSDA) and found 1.57 eV for the band gap of ZnSe. Calculations employing ad-hoc DFT potentials do not have predictive capability; their results vary with the number and nature of the adjustable parameters employed in the construction of the ad-hoc potential.

Other results obtained with generalized gradient approximation (GGA) potentials, as shown in Table 2, are either underestimates or overestimates of the band gap of ZnSe. The full potential linearized augmented plane wave plus local orbitals approach (FP-APW + lo) that obtained a gap of less 2 eV [21] [28] employed an additional potential besides that of Perdew-Burke-Ernzerhof (PBE). The use of the ab-initio PBE plus the Hubbard (PBE + U) produced a band gap of 2.5 eV [25]. While this value is close to the experimental one of 2.65 eV, PBE + U is an ad-hoc potential with no fully predictive capability. The Heyd-Scuseria-Ernzerhof 2006 (HSE06) hybrid functional is a DFT approach that includes a fraction of the exact Hartree-Fock exchange; it leads to an improved accuracy of band gap calculations compared to the mainstream ones that employed LDA or GGA functionals. The calculations with hybrid potentials obtained gaps of 2.0 - 2.85 eV [13] [19] [20], with the upper limit in agreement with the low-temperature experimental value of 2.82 eV [8]. Like PBE + U ones, hybrid potentials are essentially ad-hoc, with limited predictive capability due to the dependence of the output on the construction of the potential.

Table 2. Previous, calculated band gaps (Eg in eV) of ZnSe reported in the literature.

For this reason, these results, while very useful, do not resolve the fundamental question of the serious band-gap underestimation by most mainstream DFT calculations using ab initio LDA or GGA potentials. With several fitting parameters, the empirical pseudopotential calculations shown in Table 2 led to the correct room-temperature experimental band gap of ZnSe.

The above overview of the literature points to the need for our work. Indeed, numerous, calculated values of the band gap disagree with corresponding, experimental ones. The disagreement between sets of calculated band gaps, as evident above and in Table 2, adds to our motivation for this work. At the outset, we must answer the question of why our LDA calculations are expected to yield an accurate description of the electronic and related properties of ZnSe. Past, accurate descriptions and predictions [28] of semiconductor properties, using the distinctive features of our calculations, portend the same for ZnSe. This distinctive feature, the Bagayoko, Zhao, and Williams (BZW) method, as enhanced by Ekuma and Franklin (BZW-EF), strictly adheres to the conditions of validity of DFT as elucidated by Bagayoko [29].

We are aware of some explanations of the failures of many previous calculations to lead to correct values of the band gaps of semiconductors or insulators. Prominent among them are the self-interaction [30] and the derivative discontinuity [31]-[33] of the exchange correlation energy. Bagayoko [29], using strictly DFT theorems and the Rayleigh theorem for eigenvalues, demonstrated that self-consistent calculations that do not adhere to well-defined, intrinsic features of DFT cannot claim to produce eigenvalues and other quantities that possess the full, physical content of DFT. Hence, disagreements between their results and experiment may arise mostly from the fact that their findings do not fully possess the physical content of DFT. Our perusal of the articles that reported the results in Table 2 did not lead to any indication that these calculations adhered totally to these features of DFT: Specifically, we could not find any calculation that methodically searched for and verifiably attained the absolute minima of the occupied energies, using increasingly larger basis sets [29] [34] [35], i.e., basis sets such that, except for the first, smaller one, each basis set is entirely included in the one immediately following it. The point here is that popular explanations of band gap underestimation by DFT calculations notwithstanding, our distinctive computational method is likely to describe ZnSe accurately. The rest of this paper is organized as follows.

This section, devoted to the introduction, is followed by a fuller description of our computational method, in Section 2. We subsequently present our results in Section 3 and discuss them in Section 4. Section 5 provides a short conclusion.

2. Method and Computational Details

More details about our computational approach are available in previous articles [29] [36]-[43]. We used the Ceperley and Alder LDA potential [44] with the parameterization of Vosko and his group [45], the linesr combination of atomic orbitals (LCAO), and the BZW method, as enhanced by Ekuma and Franklin (BZW-EF) [46]. Both the BZW and BZW-EF employ successive augmentation of the initial, small basis set, by one orbital at a time, to execute several self-consistent calculations. The difference between the BZW and BZW-EF methods follows. In the BZW method, the basis set is augmented by adding orbitals in the order of increasing, unoccupied energies (in atomic or ionic species). As explained by Bagayoko [29], in systems with two or more atoms, the polarization orbitals p, d and f has primacy over the spherical symmetry of s orbitals, for the valence electrons; In the BZW-EF method, a basis set is augmented in such a way that, for any principal quantum number n, the p, d, and f orbitals, when applicable, are added in that order before the corresponding s orbital of that quantum number. An orbital is applicable if it is occupied by at least one electron in one of the atomic or ionic species.

We used a computer program developed at the US Department of Energy’s Ames Laboratory in Iowa to perform non-relativistic calculations. We first performed the self-consistent calculations of the electronic energies for the atomic or ionic species in the material being investigated. These species are Zn²⁺ and Se²⁻. Our choice was based on the results of preliminary calculations for ZnSe using neutral atoms. The preliminary results showed a transfer of approximately two electrons from Zn to Se. We subsequently performed ab-initio, self-consistent calculations to determine the atomic basis sets to be used in the solid-state calculations for ZnSe.

If two consecutive calculations lead to the same occupied energies, these energies belong, at least, to a local minimum. If the next, consecutive calculations reproduces the same occupied energies, then the absolute minimum energies, i.e., the ground state, have been reached. Otherwise, the augmentation and calculations continue. The necessary and sufficient criterion for stopping the computation is to have three consecutive calculations that produce identical, occupied energies.

In accordance with the above, our calculations necessarily begin with a small basis set which accommodate all the electrons of the system under study. Calculation II follows, with a basis set comprising that of Calculation I plus an orbital representing an excited state. We compare the occupied energies of the two self-consistent calculations: invariably, some occupied energies from calculation II are lower than their corresponding values from calculation I. We augment the basis set of calculation II, with an orbital, to perform calculation III. We compare the occupied energies from calculations II and III. We continue this process until three consecutive calculations produce the same occupied energies, indicating that we reached the ground state. Among these last three (3) consecutive calculations, only the first, which has the smallest of the three basis sets, provides the DFT description of the ground state of the material [37]-[43]. The basis set for this calculation is the optimal one. The optimal basis set, once self-consistency is achieved, leads to the ground state charge density of the material. The use of basis sets which are larger than the optima one, and which contain the optimal one, lead to the ground state energies and to the ground state charge density upon reaching self-consistency. As explained by our group in the publications referenced above and others, the use of these larger basis sets also lowers some unoccupied energies: these lowered unoccupied energies are not due to a physical interaction in the Hamiltonian which does not change from its value obtained with the optimal basis set. Incidentally, since the occupied energies do not change once, we reach the optimal basis set, the further lowering of some unoccupied energies because of the use of larger basis sets containing the optimum is one plausible explanation of the almost universal underestimation of band gaps and energy gaps by mainstream calculations [29] [36]-[43]. Indeed, these calculations, to date, have used a single basis set which was deliberately chosen to be large in order to ensure completeness. Some of these basis sets were likely over-complete for the description of the ground state [29]. In the absence of a verifiable attainme of the ground state, many of these single basis set were likely incomplete. The central point in LCAO calculations using ab-initio DFT potentials is the following. Any reasonable basis set can produce self-consistent results for the accurate description of stationary states. There exists an infinite number of such states. So, unless a DFT calculation verifiably attains the ground state, as described above, its results cannot possess the full, physical content of DFT.

The key computational details necessary for reproducing this work are provided below. Gaussian functions are in the radial parts of atomic wave functions. We employed a set of even-tempered Gaussian exponents, with a minimum of 0.148 and a maximum of 0.93061 × 10⁵ in atomic units, for Zn²⁺. Nineteen (21) Gaussian functions were used for s and p orbitals, and 20 for d orbitals. Likewise, to describe Se²⁻, the exponents in Gaussian functions ranged from 0.125 to a maximum of 0.8 × 10⁵. The numbers of Gaussian functions for s and p orbitals are 21 and that of the d orbitals is 20. A mesh of 60 k-points, with appropriate weights in the irreducible Brillouin zone, was used in the iterations for self-consistency. The error in calculating the valence charge was −0.0027647 for 44 electrons, or −6.28 × 10⁻⁵ per electron. The convergence criterion of the iterations is to have a difference not greater than 10⁻⁵ between the values of the potentials for two consecutive iterations. The number of iterations for this convergence was around 60. With the method described above and the associated calculation details, we studied ZnSe as presented below.

3. Results and Discussions

We show the basis sets for the successive calculations in Table 3. Columns 1, 2 and 3 of the table indicate respectively the number of a calculation, the valence state orbitals for Zn²⁺ and the valence state orbitals for Se²⁻. Columns 4 and 5 show the total number of valence functions and the direct band gap calculated at the Γ point, respectively. In this table, the occupied energies from calculations III, IV, V, and VI are identical. Therefore, Calculation III provides the DFT description of ZnSe, in accordance with the above description of our method.

Table 3. The successive, self-consistent calculations of the BZW-EF method for ZnSe.

In particular, the occupied energies from this calculation are those for the ground state of the material. The resulting charge density is that of the ground state of ZnSe.

The total number of valence functions and the resulting band gaps at the Γ point, for each calculation, are also listed. As per the BZW-EF method, in going from one calculation to the next, one orbital is added to the basis set of the former. We see the progressive increase in the size of the basis set from one calculation to the next.

The generalized minimization of the occupied energies requires the described successive calculations with augmented basis sets. The calculations stopped at VI because Calculations III, IV, V, and VI produced identical occupied energies, i.e., the ground state. Figure 1 illustrates the overall lowering of the occupied energies as the size of the basis set is augmented. Figure 2 shows that Calculation III and VI produced the same occupied energies. Calculation VI and V led to the same occupied energies as Calculation III, establishing that the occupied energies from Calculation III represent the ground state of the stystem.

Figure 1. Electronic energy bands of cubic ZnSe from Calculation II (solid Lines) and Calculation III (dashed lines), upon setting the Fermi level to zero; this level is indicated by the horizontal dashed-dotted line.

Figure 2. Electronic energy bands of cubic ZnSe from Calculations III (solid lines) and IV (dashed lines), upon setting the Fermi level to zero; this level is indicated by the horizontal dashed-dotted line.

For all the six calculations (I - VI), the valence band maximum and the conduction band minimum are located at the same Γ point, as expected for a cubic crystal structure. Consequently, ZnSe is a direct band gap material. As illustrated above in Figure 1 and Figure 2. The calculated band gap, from Calculation III, is 2.65 eV; this value is in excellent agreement with the experimental ones of 2.60 - 2.70 eV, at room temperature [10]-[12].

The basis set for Calculation III is the optimal basis set, i.e., the smallest basis set that leads to the ground state charge density upon reaching self-consistency.

Table 4 exhibits the calculated eigenvalues at high symmetry points in the Brillouin zone. They permit detailed comparisons with features of X-ray and ultra-violet spectroscopy findings relative to the width of a band, inter-band transition energies, and the dominant states at the Fermi level or at the minimum of the conduction band.

Table 4. Calculated eigenvalues (in electron volts-eV) for zinc blende Zinc Selenide at high symmetry points in the Brillouin zone.

We show the total (DOS) and partial (PDOS) densities of states in Figure 3 and Figure 4, respectively. We subsequently report the total energy curve, the equilibrium lattice constant, and the bulk modulus. The total density of state (DOS) and the partial ones (pDOS) were obtained using the bands from Calculation III.

Figure 3. Total density of states (DOS) of ZnSe, as derived from the bands from Calculation III. The vertical dashed indicates the position of the Fermi level. The insert panel is the magnified DOS near the band gap.

Figure 4. Partial densities of states (PDOS) of ZnSe, as derived from ground state band structure from Calculation III. The vertical dashed line indicates the position of the Fermi level.

The inset in Figure 3 shows an enlarged view of the DOS near the absorption edge, with a magnification factor of 26.

Figure 4 shows the partial densities of states (PDOS) for the s, p, and d states of Zn and the s, p, and d states of Se, respectively. According to the content of Figure 4, the valence band of ZnSe is mainly dominated by the Se-p states near the Fermi level. The structure in the energy range from −13 to −11 eV is mainly due to the Se-s state, with relatively tinyl contributions from Zn s, p, and d states. The upper group of valence bands is formed predominantly by hybridization of Se-p states with much weaker contributions from Zn orbitals. The minimum of the conduction band is mainly made up of 4s states of the Zn cation.

The total energy as a function of the lattice parameter is shown in Figure 5. The minimum total energy is obtained for a lattice parameter of 5.47 Å, which corresponds to the calculated equilibrium lattice constant. The range of lattice parameters over which the total energy values were calculated extends from 5.37 Å to 5.57 Å. To determine the bulk modulus, the first step consisted of obtaining a least-square fit of the total energy curve in the vicinity of its minimum. The bulk modulus is then obtained from the second derivative of this fitted curve at the equilibrium lattice constant. It is generally determined from the fitting of the energy-volume curve using an equation of state according to $B=V\frac{{\partial }^{2}E}{\partial V^2}$ . The bulk modulus is a physical quantity that characterizes the resistance of a material to uniform compression. Our calculated value is 59.20 GPa. This value is not far from the experimental result we found of 62.4 GPa [47]-[49]; it is slightly higher than a previous, theoretical value of 57.30 GPa [28].

Figure 5. The total energy versus the lattice constant for the zinc blende ZnSe, obtained from calculations employing the optimal basis set.

4. Conclusion

We have presented electronic and related properties of ZnSe, obtained with an abinitio density functional theory (DFT) potential. ZnSe is among Important semiconductors with a multitude of applications. Unlike previous DFT calculations that led to underestimating the measured band gap, our results fully agreed with experiment without invoking a correction for self-interaction or derivative discontinuity of the exchange-correlation energy. This feat is credited to the strict adherence of our calculations to the two foundational theorems of DFT. In particular, we kept the total number of particles constant, and we successively augmented the initial, small basis set to lower the occupied energies down to the ground state.

Acknowledgements

The authors gratefully acknowledge the financial and institutional support provided by the U.S. Fulbright Fellowship Program (Program #G-10005 PS00383052), the Malian Ministry of Higher Education and Scientific Research, and the U.S. National Science Foundation (NSF Award No. HRD-2009765). The computational resources used in this work were provided by the Louisiana Optical Network Initiative (LONI). The authors also thank Southern University and A & M College and Louisiana Space Grant Consortium (LaSPACE) for their valuable support in conducting this research.

Author Contributions

The respective contributions of the authors follow. Dr. Moussa Diawara performed the calculations and drafting the manuscript. Dr. Malozovsky guided Dr. Diawara in the running of our program package and assisted Dr. Diawara with some graphs. Dr. DIAKITE conducted a thorough search of the literature to provide most of the references cited in the manuscript. He also contributed to the finalization of the manuscript and to its submission to JMP. Dr. Bagayoko guided the overall effort and the editing of the draft manuscript. He is the director of the High-Performance Computer Laboratory where the calculations were done. He also handles the payment of the publication cost for the accepted manuscript.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References