Ab-Initio Self-Consistent Density Functional Theory Description of Rock-Salt Magnesium Selenide (MgSe)

We report results on electronic, transport, and bulk properties of rock-salt magnesium selenide (MgSe), from density functional theory (DFT) calculations. We utilized a local density approximation (LDA) potential and the linear combination of atomic orbitals formalism (LCAO). We followed the Bagayoko, Zhao, and Williams (BZW) method, as enhanced by Ekuma and Franklin (BZW-EF), to perform a generalized minimization of the energy, down to the actual ground state of the material. We describe the successive, self-consistent calculations, with augmented basis sets, that are needed for this generalized minimization. Due to the generalized minimization, our results have the full, physical content of DFT, as per the second DFT theorem [AIP Advances, 4, 127104 (2014)]. Our calculated, indirect bandgap of 2.49 eV, for a room temperature lattice constant of 5.460 Å, agrees with experimental findings. We present the ground-state band structure, the related total and partial densities of states, DOS and PDOS, respectively, and electron and hole effective masses for the material. Our calculated bulk modulus of 63.1 GPa is in excellent agreement with the experimental value of 62.8 ± 1.6 GPa. Our predicted equilibrium lattice constant, at zero temperature, is 5.424 Å, with a corresponding indirect bandgap of 2.51 eV. We discuss the reasons for the agreements between our findings and available, corresponding, experimental ones, particularly for the band gap, unlike the previous DFT results obtained with ab-initio LDA or GGA potentials.


Introduction
Group II-VI semiconductors and their alloys have attracted attention from scientific and technological communities, in recent years, due to their various applications. Specifically, magnesium selenium (MgSe) is an attractive binary semiconductor utilized as a key component in II-VI semiconductor alloys having n and p doping abilities [1]. MgSe can exist in four crystalline phases that include rock salt (rs), zinc blende (zb), wurtzite (wz) and nickel arsenide (NiAs) [2] [3]. First principle, self-consistent local density approximation calculations of properties of MgSe show that it is stable in the rock salt phase [3] [4]. Results from several groups show the crystallization of MgSe in cubic (zinc blende) structure from the epitaxial growth on a GaAs substrate by molecular beam epitaxy or Metallo-organic vapor phase epitaxy [2] [5] [6] [7] [8] [9].
Rock salt (rs) MgSe has applications in high-temperature and high-power blue and ultraviolet wavelength optics [2] [3] [10]. It is utilized in photovoltaic devices and in electro-optic crystals with ultra-fast response time spectroscopy. Magnesium selenide (rs-MgSe) has applications in blue-green light-emitting laser diodes (LEDs) [2] [9] [11] [12] and in integrated optical devices [13]. Studies by Falke et al. show that ZnMgSe alloy has applications in semiconductor optics and in active light-emitting components [14].
Many theoretical studies report an indirect bandgap for MgSe, from Γ to X.
Drief et al. [15] performed first-principle calculations to determine structural, electronic, elastic and optical properties of MgSe using the Full Potential Linea-  [19] showed an indirect bandgap of 1.75 eV for rock salt MgSe. A density functional theory (DFT) calculation by Khan et al. [20], with a modified Becke and Johnson GGA potential and the FP-LAPW method, found an indirect bandgap of 2.83 eV for rs-MgSe. The above calculations and others are summarized in Table 1.
The presumed, experimental band gap for bulk rock-salt MgSe is 2.47 eV, as reported by Heyd et al. [30]. The presumption stems from the fact that these authors compared their calculated band gaps, for bulk rs-MgSe, to this value of 2.47 eV, without any mention of films or nanoparticles. We should note that  [32]. In Table 2, below, we show these results and others, obtained for films or nanocrystals of rs-MgSe. Ab-initio LDA potentials led to results that vary from 1.57 eV to 1.95 eV, with an outlier of 3.82 eV. Ab-initio GGA calculations produced band gaps between 1.75 and 1.82 eV. These calculated values, except for the outlier, are much lower than the reported, experimental findings in Table 2. The outlier of 3.82 eV is much larger than the experimental gaps in Table 2. The calculated values obtained with ab-initio DFT potentials are also much smaller than the reported bands for films and nanocrystals. Despite the known enhancement of the band  gaps of films and nanocrystals, in comparison to that of the bulk, by quantum confinement, it is not expected to be as large as 0.88 eV, the difference between the lowest of the measured gaps (2.45 eV) and the lowest of the calculated ones (1.57 eV). The resolution of this disagreement between experiment and theory is our central motivation for this work. Indeed, the importance of the band gap resides in part in the fact that several properties of materials depend on it, including optical, dielectric, and transport properties. For transport properties, this dependence is partly through the effective masses that are inversely proportional to mobility.

Computational Method and Details for Replication
In this work, our implementation of the linear combination of atomic orbitals  (3)  Our self-consistent calculations were performed utilizing a room temperature, experimental lattice constant of 5.460 Å [50]. We first performed ab-initio, self-consistent calculations for the ionic species, Mg 2+ and Se 2− . We employed a set of even-tempered Gaussian exponents to expand the radial parts of the atomic wave functions in terms of the Gaussian functions. The number of even-tempered Gaussian exponents used for the s, p, and d orbitals for Mg 2+ were 18, 18 and 16, respectively. Likewise, the s, p and d orbitals for Se 2− . were 18, 18 and 16, respectively. The maximum and minimum exponents used for Mg 2+ were 1.1 × 10 6 and 0.1822, respectively, whereas the corresponding ones for Se 2− were 0.24 × 10 6 and 0.135, respectively. The computational error for the valence charge was 3.289 × 10 −4 per electron. Self-consistency was reached after 60 iterations, with 81 k-points in the irreducible Broullouin zone, when the difference between the potentials from two consecutive iterations was less than or equal to 10 −5 .

Results
We list below, in Table 3, the valence orbitals in the basis set for the successive, self-consistent calculations, along with the resulting band gaps. Calculation IV is the first of the three consecutive ones (IV, V, and VI) that produce the same, occupied energies. It provides the DFT description of the material; its basis set is the optimal basis set. Our calculated band gap for rock salt MgSe, as produced by Calculation IV, is an indirect one, from Γ-X, of 2.49 eV. Figure 1 shows the electronic band structures of rock salt MgSe, as obtained from Calculation IV (solid lines) and Calculation V (dashed lines). As explained in the section on our method, the occupied energies from Calculations IV and V are the same, as shown by the perfect superposition of the valence bands from the two calculations. The unoccupied energies below 4 eV are also the same for the two calculations. For higher energies, the Rayleigh theorem for eigenvalues [ Table 4 lists the electronic energies at high symmetry points (L-point, Γ-point, X-point, and K-point) in the Brillouin zone, as obtained with the optimal basis set of Calculation IV. The minimum of the conduction band is at the X-point. The content of this table lends itself to comparison with findings from future experimental studies, including UV and X-ray spectroscopy ones. Figure 2 and Figure 3, respectively, show the total and partial densities of states energy, DOS and pDOS, respectively, derived from the bands resulting Table 3. The successive, self-consistent calculations of the BZW-EF method. The basis set is augmented from one calculation to the next. Columns 2 and 3 show the valence orbitals on Mg 2+ and Se 2− . A superscript of zero (0) indicates an orbital unoccupied in the ionic species. We utilized a room temperature experimental lattice constant of 5.460 Å. As explained in the text, Calculation IV provides the DFT description of the material.     from Calculation IV, in the energy range of −14 eV to 12 eV. The total valence band width is about 12.61 eV. The width of the lowest laying valence band is 1.75 eV while that of the uppermost group of valence bands is 5.0 eV. The content of Figure 3 shows that the lowest conduction band is mostly dominated by Mg-s and Mg-p states. The group of uppermost valence bands is mostly from Se-p states, with a feint contribution from Mg-p. The lowest laying valence band is entirely from Se-s, with no other meaningful contributions. Effective masses are needed for the calculations of several transport properties of materials. These properties include electrical conductivity, mobilities. The latter are inversely proportional to applicable effective masses. For this reason, we calculated electron and hole effective masses for rs-MgSe. We obtained electron effective masses at the conduction band minimum (CBM), at X, and at the lowest point on the conduction band at Γ. We produced the heavy and light hole effective masses at the valence band maximum (VBM) at Γ. The calculated effective masses in Table 5 include electron effective mass at the Γ point (M eΓ ), the longitudinal and transverse electron effective masses at the X point, M eXlong and M eXtrans , respectively, and the effective masses for heavy holes 1 and 2 (M hh1 , M hh2 ) and for the light hole (M lh ), in various directions, at the VBM at Γ. These effective masses are inunits of free-electron mass (m o ). At the center of the zone, the electron effective mass at Г is isotropic. While the electron effective masses at X and the light hole effective masses are mildly anisotropic, the ones for the heavy holes are strongly anisotropic. Along the directions of (Г-L) 111 and (Г-X) 100 , we obtained the same respective results for the effective masses of heavy hole 1 and heavy hole 2. Due to the splitting of the bands by the Coulomb crystal field, the effective masses of the heavy holes are different in the (Г-K) 110 direction. We could not find any experimental measurements or theoretical calculations for electron and hole effective masses of rock salt MgSe. The results below are predictions expected to be confirmed by experiment. Figure 4 is a graph of the total energy versus the lattice constant, within a range of 5.30 Å -5.60 Å, with a scale of 0.05 Å units on the horizontal axis. Our predicted value of the equilibrium lattice constant, at zero temperature, is 5.424 Å, with a corresponding zero-temperature, indirect bandgap of 2.51 eV. We calculated the bulk modulus to be 63.1 GPa. This result is in excellent agreement with the experimentally determined value of 62.8 ± 1.6 GPa [51].

Discussion
The large difference between our calculated band gap and the ones previously obtained with ab initio DFT potentials needs an explanation. The detailed description of our method unveiled the crucial difference between BZW-EF calculations, that perform a generalized minimization of the energy, and other DFT calculations. Indeed, most of the other ab initio DFT calculations employ a single basis set to obtain self-consistent results that are assumed to be those for the ground state. Such results, however, are stationary ones among an infinite number of such solutions [37] [41] [43] [44] [52]. In contrast, DFT BZW-EF calculations search for and verifiably reach the ground state of systems under study.
The Rayleigh theorem [41] [52] [53] for eigenvalues ascertain that the results of self-consistent calculations depend on the size of the basis set. As the basis set  [43]. This fact, in addition to our correct description of properties of over 40 semiconductors, portends the likelihood that our calculated effective masses for rs-MgSe will be confirmed by experiment.

Conclusion
We utilized our well-established BZW-EF computational method to calculate the ground state electronic and related properties of rock salt MgSe. Unlike the previous, ab initio DFT results, our calculated band gap of 2.49 eV agrees with experiment. The same is true for our calculated bulk modulus of 63.1 GPa. Our predictions for the effective masses and hybridization patterns, where the latter are derived from the calculated partial densities of states, are expected to be confirmed by experiment.