Accurate, First-Principle Study of Electronic and Related Properties of the Ground State of Li2Se

We present results from ab-initio, self-consistent calculations of electronic and related properties for the ground state of cubic lithium selenide (Li2Se). We employed a local density approximation (LDA) potential and performed computations following the Bagayoko, Zhao, and Williams (BZW) method, as enhanced by Ekuma and Franklin (BZW-EF). This method verifiably leads to the ground state of materials without employing over-complete basis sets. We present the calculated electronic energies, total and partial densities of states, effective masses, and the bulk modulus. The present calculated band structures show clearly that cubic Li2Se has a direct fundamental energy band gap of 4.065 eV at the Γ point for the room temperature experimental lattice constant of 6.017 Å. This result is different from findings of previous density functional theory (DFT) calculations that uniformly reported an indirect band gap, from Γ to X, for Li2Se. We predicted a direct band gap of 4.363 eV, at the computationally determined equilibrium lattice constant of 5.882 Å, and a bulk modulus of 35.4 GPa. For the first time known to us, we report calculated electron and hole effective masses for Li2Se. The experimental confirmation of the large, direct gap we found will point to a potential importance of this material for ultraviolet technologies and applications. Due to a lack of experimental results, most of our calculated ones in this paper are predictions for Li2Se.

ionic conductivity and large fundamental energy band gaps, leading to their promising applications in power sources, fuel cells, solid state gas-detectors, ultraviolet space technology devices and photocatalysis [1] [2] [3] [4]. Compared to the extensive studies on the alkali metals oxides and sulfides, the alkali metal selenides have received much less attention. Recently, several research works have focused on lithium selenide (Li 2 Se) for its superionic (SI) properties. The fast Li 1+ -ion transport of the SI Li 2 Se solid makes it a prime candidate for solid-state electrolytes in next generation lithium battery technologies [5]. Up until now, however, experimental studies of electronic and related properties of Li 2 Se are very few. As far as we know, most of the research work on Li 2 Se has been confined to studies of its structural properties. No experimental measurements regarding the electronic and related properties of Li 2 Se are available. Theoretically, only the following three research groups have performed first-principle calculations of electronic band structures of Li 2 Se. We summarized their findings in Table 1 below. Eithiraj et al. calculated the electronic structure of Li 2 Se, using a Tight-Binding and Linear Muffin-Tin Orbital (TB-LMTO) method [6] [7] and the local density approximation (LDA) potential of von Barth and Hedin [8] [9].
Their results show that Li 2 Se is an indirect band gap semiconductor, with a gap of 2.748 eV, from Γ to X. Alay-e-Abbas et al. calculated the band structures of Li 2 Se using the Full Potential Linearized Augmented Plane Wave (FP-LAPW) method, as implemented in the WIEN2K program package [10], and density functional theory (DFT) potentials [11] [12]. Specifically, these authors employed a local density approximation (LDA), the Perdew-Burke-Ernzerhof [13] generalized gradient approximation (PBE-GGA), the Wu and Cohen [14] GGA (WU-GGA), which entails fourth-order gradient expansion of exchange energy function, and the Engel and Vosko [15] GGA (EV-GGA) potentials. The band structures calculated within the LDA, PBE-GGA, WC-GGA and EV-GGA potentials exhibit Γ to X indirect band gap values of 2.78 eV, 2.93 eV, 2.82 eV, and 4.08 eV, respectively.  [16]. In their work, they utilized the recently modified Becke and Johnson (mBJ) potential [17], which is a "hybrid" potential whose amount of "exact exchange" is controlled by a parameter c, to improve the calculated, electronic band structure. The first-principles WC-GGA, EV-GGA and mBJ calculations by Ali et al. show that Li 2 Se has a Γ to X indirect band gap of 2.80 eV, 4.12 eV, and 4.19 eV, respectively [16]. Our motivation for this work partly stems from current and potential applications of Li 2 Se for the next generation of battery technologies. Accurate, calculated electronic and related properties are important in informing and in guiding the development of new applications. While previous DFT calculations agreed on the indirect nature of the band gap, the resulting numerical values range from 2.748 eV to 4.19 eV. Such a wide range points to the need for further theoretical studies of electronic and related properties of lithium selenide. The current lack of experimental studies of electronic and related properties of Li 2 Se is an added motivation for this work. With our distinctive computational method, we have correctly described and predicted electronic and related properties of more than 30 semiconductors [18]. These past successes portend an accurate DFT description of this material, using our BZW-EF method. We describe below, in Section 2, the general computational approach and our distinctive method. We subsequently present our findings in Section 3. We then provide discussions and a conclusion in Sections 4 and 5, respectively.

Computational Method
In ambient conditions, Li 2 Se crystallizes in a stable face center cubic (FCC) antifluorite (anti-CaF 2 -type) structure [19] (Space group 5 3 h O Fm m − , No. 225), with the Li atoms located at ± (0.25, 0.25, 0.25) and the Se atoms at (0, 0, 0) Wyckoff positions. In this work, we performed first-principle full-potential DFT calculations for the electronic properties of Li 2 Se, using the experimental lattice constant of 6.017 Å from Zintl et al. [19] and our predicted, equilibrium lattice constant. We utilized a linear combination of atomic orbitals (LACO) formalism and the BZW-EF method, which has been extensively described in several of our previous publications [18] [20] [21] [22] [23]. Our first-principle LCAO package is from the Ames laboratory of the US Department of Energy, in Ames, Iowa [24] [25]. We began the calculations with self-consistent computations for the atomic wave functions for Li 1+ and Se 2− atoms. The radial parts of the atomic wave functions were expanded in terms of Gaussian functions. The s, p orbitals for the cation Li + were described with 16 even-tempered Gaussian functions with respective minimum and maximum exponents of 0.2400 and 0.90 × 10 5 for the atomic potential and 0.1200 and 0.90 × 10 5 for the atomic wave functions. The self-consistent calculations for Li + led to the total charge of 2.0009, which is also the valence charge. For Se 2− the s, p and d orbitals were described with 24 even-tempered Gaussian functions with respective minimum and maximum Gaussian exponents of 0.2300 and 0.220 × 10 6 for the atomic potential and 0.1350 and 0.240 × 10 6 for the atomic functions, respectively. These Gaussian exponents led to the convergence of the atomic calculations. We utilized the Ceperley and Alder local density approximation (LDA) potential. In the iterations for self-consistency, we used a mesh of 60 k-points with proper weights in the irreducible Brillouin zone. We reached convergence for a given self-consistent calculation after 90 iterations; the criterion for convergence was that then, the difference between the potentials from the last two consecutive iterations was 10 −4 or less. Further, for the production of the final, self-consistent electronic band structures, we used a total of 81 k points in the Brillouin zone, with the same computational errors as for the self-consistent potential calculations. Based on the above points, our computational approach is the same as those of other DFT calculations. We underscore below the critically important, distinctive feature of our computational method, with multiple, self-consistent calculations with basis sets of different sizes.
Our ab initio self-consistent calculations for the solid, with the BZW-EF method, began with a small basis set containing the minimum basis set, which is the smallest one accounting for all the electrons in the system under study, i.e., Li 2 Se. Following this Calculation I, we augmented the basis set with one orbital representing an excited state and performed Calculation II. We graphically and numerically compared the occupied energies from Calculations I and II, with the Fermi levels set to zero. After augmenting the basis set of Calculation II with one orbital, we carried out Calculation III and compared the resulting occupied energies with those from Calculation II. In both of the preceding comparisons of occupied energies, at least some of the ones obtained with the larger basis set were lower than corresponding ones from the immediately preceding calculation (with a smaller basis set). We continued this process of augmenting the basis set and of performing self-consistent calculations until three consecutive ones led to the same occupied energies. The perfect superposition of these occupied energies is the criterion or proof that these calculations produced the absolute minima of the occupied energies, i.e., the ground state of the system.
Let N be the number of the first of these three calculations to reach the ground state. We dubbed the basis set of this calculation as the optimal basis set, i.e., the smallest basis set that leads to the ground state upon the attainment of self-consistency. Calculations (N + 1), (N + 2) and other with larger, augmented basis sets produced (a) the same charge density, (b) the same Hamiltonian, and (c) the same occupied energies as respectively obtained with Calculation N. We distinguish the Hamiltonian from the Hamiltonian matrix that changes with the size of the basis set. Despite (a) through (c) above, some unoccupied energies from calculations (N + 1), (N + 2) and others with larger, augmented basis sets, were generally lower than corresponding ones obtained in Calculation N. Given that the Hamiltonian did not change from that of Calculation N, any eigenvalues that deviate from (i.e., are lower than) their corresponding values resulting from Calculation N are clearly unphysical. Another proof of this assertion stems from  18 of the first DFT theorem: According to it, the spectrum of the ground state Hamiltonian is a unique functional of the ground state charge density. Hence, if an eigenvalue from Calculation (N + 1) or higher is different (lower than) its corresponding value obtained in Calculation N, then the new value no longer belongs to the spectrum of the Hamiltonian-as the charge density did not change. In summary, Calculation N is the only one providing the true DFT description of the material; the resulting eigenvalues possess the full physical content of the DFT, unlike eigenvalues resulting from self-consistent iterations with a single basis set. These iterations produce stationary solutions among an infinite number of such solutions. Our generalized minimization of the energy functional of the Hamiltonian, using successive, self-consistent calculations, verifiably reaches the true ground state of the system-instead of an arbitrary, stationary solution unwittingly confused with the ground state.    From this figure and the content of Table 2 above, we conclude that Li 2 Se is a direct band gap semiconductor; this result is in stark disagreement with the previously reported DFT band gaps, in Table 1 In Table 3, we list illustrative, calculated, electronic energies for Li 2 Se at high symmetry points (Γ, X, K, and L) in the Brillouin zone. These energies are expected to be useful in comparisons of our findings with future, experimental results. Such results include direct, optical transition energies and various X-ray and ultraviolet (UV) spectroscopic measurements.    The electrical conductivities, and transport and other related properties of materials require an accurate and detailed knowledge of effective masses. As per the content of Table 4, we have performed calculations of electron effective masses around the minimum of the conduction band, at the Γ point, and around the next, lowest conduction band minimum at the X-point. We have calculated the effective masses of the light and the two heavy holes at the top of the valence band, at the Γ-point. We list these calculated, effective masses in Table 4  We expect future measurements to confirm our predictions in Table 4.

Results
The graph of the total energy versus the lattice constant is shown in Figure 4  GPa; it is the same as the finding of Ali et al. [16] and is slightly larger than the calculated result of 34.72 GPa of Alay-e-Abbas [11] [12]. No experimental measurements for the bulk modulus of Li 2 Se are available for comparison.

Discussion
The following discussion is guided by the fact that our calculations, as explained in the section on our method, 1) verifiably attained the ground state of the system 2) while avoiding over-complete basis sets. The latter feature guarantees that spuriously low, unoccupied energies are not in the spectrum of the ground state Additionally, if such basis sets are not significantly larger than the optimal one, they also reproduce low laying, unoccupied energies obtained with the optimal basis set. The content of Figure 1 shows that the low laying, unoccupied energies produced by Calculation IV are the same as those from Calculation III, up to +6 eV. The values of the band gaps resulting from Calculation VI, whose basis set contains 12 more functions than the optimal one, illustrate the point. This calculation not only reduced the band gaps from their values obtained in Calculation III, but also it resulted in an indirect (Γ-X) band gap. The latter feature is in qualitative agreement with the findings of the previous DFT calculations shown in Table 1; we presume that these single basis set calculations most likely utilized relatively large basis sets.

Conclusion
In summary, we performed first principle, self-consistent calculations of electronic, transport, and bulk properties of cubic antifluorite lithium selenide (Li 2 Se), using a local density approximation (LDA) potential. As per the BZW-EF method, our implementation of the linear combination of atomic orbitals entailed the performance of successive, self-consistent calculations with increasingly large basis sets. We obtain the basis set of a calculation, except for the first one that has a small basis set, by augmenting the basis set of the immediately preceding calculation with one orbital. This generalized minimization of the energy not only reached the ground state, but also does so without employing over-complete basis sets that tend to lower, unphysically, some unoccupied energies. This fact suggests that the widespread underestimation of the band gaps of semiconductors and insulators, by DFT calculations, may be due to this spurious lowering of unoccupied energies. Our calculated, indirect band gap of Li 2 Se, at room temperature, is 4.065 eV. This result is in stark contrast with those from previous DFT calculations that found an indirect band gap. The accurate results we obtained for more than 30 semiconductors are the basis for us to expect a future, experimental confirmation of our results for the energy bands, the densities of states, effective masses, and the bulk modulus of Li 2 Se.