Rapid advances in solid-state ionics have increased interest of researchers in the group IA-VI compounds M2Ch [M: Li, Na, K, Rb; Ch: O, S, Se, Te] over past few decades. At room temperature these compounds crystallize into a stable face centered cubic anti-fluorite (anti-CaF2) structure (space group Fm-3m). Rb2Te is an exception, which is metastable in anti-CaF2 structure at room temperature and transforms irreversibly to anti-PbCl2 structure type upon warming. These materials are potential candidates for their use in power sources, fuel cells, gas-detectors and ultra violet space technology devices. Presently these materials are being used as power sources for portable electronics devices such as laptops, cellular phones and digital cameras 1-5. These ionic compounds also play important role in the development of photocathode, in supporting catalytic reactions and enhancing oxidation of semiconductor surfaces 6-9, and therefore are ideal for theoretical and experimental studies.
Solid State properties of M2O crystal have been extensively studied experimentally 10-14, whereas the sulfides, selenides and tellurides of alkali metal have received less experimental attention. Various researchers have undertaken several theoretical studies of this crystal family. In case of group-IA oxides first 15-16 theoretical studies for structural properties have been reported by using Hartree-Fock linear combination of atomic orbital method (LCAO-HF). Electronic band structures of alkali metal oxides at ambient conditions have been discussed using the self-consistent pseudpotential method 17 (PP) and linear muffin-tin orbitals in its tight-binding representation 18 (TB-LMTO). Lattice dynamics (LD) 19, molecular dynamics (MD) 19 simulations, local density approximations (LDA) with pseudopotentials and plane wave basis set 20 for high temperature properties, aspherical ion model 21, stoichiometric and defective properties using density functional theory 22 have also been reported for dilithium monoxide. Simulation for electron energy-loss near edge structure at the lithium K edge has been carried out 23 with full potential linearized augmented plane wave (FP-LAPW) method. Moakafi et al. 24 have performed a detailed investigation of structural, electronic, and optical properties of alkali metal oxides at normal and under pressure conditions using FP-APW+lo method. Most of the studies on M2S compounds have been dedicated to study elastic properties 25-26 and the structural phase transformation 27-29. Buhrer et al. 30 fitted a shell model to inelastic neutron data of Na2S at 50K. Lichanot et al. 31 investigated structural and elastic properties of Li2S and Na2S compounds at Hartree Fock level. Schön et al implemented the LCAO-HF 32 method to analyze high-pressure structural phase transition of M2S compounds. Self-consistent PP method, the LCAO-HF and TB-LMTO studies have been reported by Zhuravlev et al. 33, Azavant et al. 34 and Eithiraj et al. 35, respectively, for electronic band structure of alkali metal sulfides. Khachai et al. 36 reported structural, electronic and optical properties of group-IA sulfides with and without applied pressure by using FP-APW+lo method. Alkali metal selenides and tellurides have received even lesser attention from researchers. For these compounds most of the work done by researchers has been confined to study the cohesive energies 37-40, elastic properties 38 and bulk modulus 39-40 of these compounds.
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Like all other crystal families, alkali metal chalcogenides follow specific trends in their structural and electronic behavior. Detailed computational investigations for structural, electronic and optical properties of alkali metal oxides and sulfides have been reported recently 24, 36. However, no such computational work carried out for selenides and tellurides of alkali metals exists, to the best of our knowledge. The lack of some reference data on alkali metal selenides and tellurides makes it difficult to generalize trends in structural and electronic properties of this crystal family. Keeping in view the technological importance of these materials we have performed a detailed study of solid-state properties for 16 anti-CaF2 structure type alkali metal chalcogenide compounds using the most accurate and state-of-the-art computational method based on density functional theory (DFT). In order to compare performance of different exchange-correlation energy functionals the generalized gradient approximation (GGA) functional proposed by Wu and Cohen 41 (WC) has been utilized in addition to Perdew-Burke-Ernzerhof 43 (PBE) GGA and local density approximation (LDA) 44 for structural properties of these compounds. The reason for choosing WC GGA is based on the fact that its implementation 42 for ionic solids has shown to improve results of structural parameters over other parameterization schemes. Correlation between different structural parameters have been analyzed and compared with other theoretical results. Furthermore, behavior and trends in the electronic properties such as energy band gaps, upper valence bandwidth (UVBW), lowest energy band gaps, minimal valence transition energies and density of states are discussed. In addition to LDA, PBE and WC parameterization schemes Engel and Vosko 45 GGA functional has been used for studying electronic properties. A comprehensive data for M2Ch [M: Li, Na, K, Rb; Ch: O, S, Se, Te] compounds has been presented that may provide reference for future experimental as well as theoretical studies related to these ionic materials and their mixed binary/ternary systems.
Method and Computational Details
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The calculations for group IA-VI ionic crystals were performed using the FP-APW+lo method based on DFT as implemented in the WIEN2K code 46. As in every Kohn-Sham calculation, determination of accurate exchange correlation energy (Exc) ensures the accuracy of results; Exc is estimated with an approximate exchange correlation functional. For a rough determination of Exc the exchange-correlation functional in the standard LDA which can be calculated accurately for a homogeneous electron gas 44, is effectively used. In a real system however, where inhomogeneous valence electron density varies relatively slowly with space, great improvement in determining Exc is achieved by using GGA exchange-correlation functionals. Since in the WC GGA expression fourth-order gradient expansion of exchange energy functional is performed by slowly varying the density, WC GGA is more suitable for studying structural properties of solids as compared to other GGA schemes 42. On the other hand, due to their simple forms the LDA, PBE and WC functionals are not flexible enough to accurately determine total exchange-correlation energy and its charge derivative. However, the modified GGA form of EV GGA functional that produces more accurate exchange-correlation potential at the expense of less accurate exchange energy has proved to improve results of electronic band structure for alkali metal oxides and sulfides 24, 36.
In the present work, the exchange-correlation potential for the Kohn-Sham equation have been obtained by using WC functional for M2O and M2S compounds and are compared with earlier calculated PBE and LDA results. Structural properties of M2Se and M2Te compounds are calculated using all above-mentioned schemes. For electronic band structure calculations of M2O and M2S compounds only WC functional has been utilized, whereas all LDA, PBE, WC and EV schemes have been employed while calculating electronic band structure of M2Se and M2Te compounds.
In the FP-APW+lo method a muffin-tin model for crystal potential is assumed and the electrons are paired into two groups namely the "core electrons" whose charge densities are confined within the muffin-tin spheres and the "valence electrons" that reside outside the surface of muffin tin spheres. The wave function, charge density and potential are expanded inside the non-overlapping spheres of muffin-tin radius (RMT) around each species, linear combination of radial solution of the Kohn-Sham equation times the spherical harmonic has been utilized. Plane wave basis set has been utilized in the remaining interstitial space of the unit cell. The core electron states are treated fully relativistically by solving the Dirac equation, whereas the valence electrons have been treated scalar relativistically.
The muffin-tin sphere radius, RMT, for each species of the compounds under study is so chosen, such that there is no charge leakage from the core and total energy convergence is ensured. The RMT values for Li, Na, K, Rb, O, S, Se and Te were taken to be 1.6, 2.1, 2.7, 2.9, 1.8, 2.2, 2.8 and 3.1, respectively. These values have been selected after performing several tests using different muffin tin radii as well as different sets of k-points to ensure convergence of energy. The maximum value of angular momentum (lmax) for the wave function expansion inside the atomic spheres has been taken at 10. In the interstitial region the plane wave cut-off value of Kmax x RMT = 9 has been used for these calculations. A mesh of 72 k-points was taken for the Brillion zone integrations in the corresponding irreducible wedge.
Results and Discussion
While calculating structural properties of M2O and M2S compounds, volume optimization was performed using WC GGA. Equilibrium lattice parameters, ao, and bulk moduli, B0, have been calculated by fitting Murnaghan equation of states 47 to the total energy versus volume curve. Table 1 shows the results obtained in present work for lattice constants and bulk moduli of alkali metal oxides and sulfides along with other theoretical and experimental results. Except for Li2O and Li2S crystals, the results obtained by using WC GGA for Na, K and Rb oxides and sulfides are in better agreement with experiment. In Table 2 calculated values of ao and B0 using WC GGA, PBE GGA and LDA functional for M2Se and M2Te compounds have been listed. Clearly, our results for lattice parameters are in good agreement with the available experimental data. It is observed that for DFT calculations the PBE overestimates and LDA underestimates the values of lattice constant for heavier cations of this crystal family. However results of WC parameterization scheme clearly show that ao for Na, K and Rb chalcogenides are in between the result of PBE and LDA and in better agreement with experimental data.
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The underestimation and overestimation of ao as compared to experimental values using WC functional is between 0% to 2.133 % and 0.217% to 0.64%, respectively. In case of lithium chalcogenides the results of PBE GGA are much closer to experimental values. This may be a consequence of weak van der Waals interactions between the lithium cation and chalcogen anions and also because of smaller contribution of lithium cations to electron density, which makes the interstitial region of Li chalcogenide compounds somewhat more inhomogeneous as compared to other members of this family.
Experimental values of bulk moduli are available only for Li2O, Li2S and Na2S compounds. A comparison of our results with experimental and other theoretical values of B0 shows that WC scheme provides better results for Li2O. However for Li2S the value of Bo obtained using LDA is closer to the experimental value whereas all of WC, PBE and LDA schemes predict lower values of Bo for Na2S. It can be seen that the lattice parameter of group IA-VI compounds increases whereas the bulk modulus decreases with increasing cation radii. It has been reported 39 that the product of bulk modulus and unit cell volume for this crystal family is almost constant. In present study it has been observed that the product of unit cell volume and bulk modulus deviate far too much from the mean value of the product of unit cell volume and bulk modulus to be recognized as a constant. The deviation in our results of bulk moduli from experimental values of Li2O, Li2S and Na2S is expected because second derivatives are more sensitive to numerical inaccuracies.
Cohesive energies of these compounds calculated in this study obtained by using WC GGA are presented in Table 3. The cohesive energies were calculated by subtracting corresponding atomic energies from equilibrium total energy per unit cell. When compared to other theoretical results, values obtained in present work appear to be in good agreement with the available experimental data. We assume that the large disagreement in some theoretical and experimental values of Bo and cohesive energies is the consequence of less accurate correlation parts of the energy functionals.
It is observed that the cohesive energies M2Ch compounds decrease with increasing lattice parameter as shown in Fig. 1. For common anions of alkali metals a linear increase in cohesive energy with increasing crystallographic ratio is observed (See Fig. 2). The crystallographic ratio is defined as ratio of anion radius and interionic distance. The anion radius values for O, S, Se and Te have been taken from Ref. 37 and the interionic distance were calculated as in Ref. 38.
Electronic band structures
Band structures parameters of alkali metal oxides and sulfides have been calculated by using WC GGA only whereas for alkali metal selenides and tellurides WC GGA, PBE GGA, LDA and EV GGA schemes were used. The reason for using EV GGA for electronic band structure values in addition to other parameterization schemes has been the fact that the electronic band structure values of solids are very sensitive to accurate determination of correlation potential. As mentioned earlier, the simple form of LDA functional and the shortcomings of PBE and WC functionals, arising due to their dedication to accurate determination of exchange energy part, make LDA, PBE and WC less accurate for band structure calculations. On the other hand, the modified form of EV GGA being more accurate in determining correlation potential at the expense of less accurate exchange energy proves to be promising for band structure calculations of solids. Since the band structures diagrams for M2O and M2S compounds obtained with EV GGA already exist in literature 24, 36, therefore in this work we present only the band structures diagrams of alkali metal selenides and tellurides (Fig. 3).
A close look at Fig. 3 clearly shows that lithium, potassium and rubidium chalcogenides are indirect band gap semiconductors and sodium chalcogenides are direct band gap semiconductors. The valence band maximum for Li and Na chalcogenides are located at the Î“-point and for K and Rb chalcogenides at X-point. The conduction band minimum occurs at X-point for Li chalcogenides and at Î“-point for Na, K and Rb chalcogenides. Also, Î“-point band degeneracy is evident from Fig. 3 for all materials. In order to better elucidate the occurrence of various band structures mentioned above we will discuss electronic density of states (DOS) in next section.
In Table 4 we summarize the calculated values for the energy band gaps (Î“- Î“, X- Î“ and X-X) and upper valence bandwidths (UVBW) of M2Ch compounds [M: Li, Na, K, Rb; Ch: O, S, Se, Te]. It is clear that the values for energy gaps of all alkali metal chalcogenides obtained with LDA, PBE and WC functionals are lower in value than that obtained with EV functional as mentioned earlier.
As shown in Fig. 4, the lowest energy band gap for common anions of Li, Na, K and Rb decreases as one goes from Li chalcogenides to Rb chalcogenides. Increasing atomic number of cations may be the cause for this, because the conduction band minimum tends to push its way downwards to the valence band as number of electrons of cations increase. It can be seen that the indirect band gap values for lithium chalcogenides decrease as one goes from O to Te. However, for Na, K and Rb chalcogenides the direct Î“-Î“ band gap value increases on going from O to S and then decreases from S to Te. Exactly similar behavior is observed for indirect X-Î“ band gap of these materials. The X-X band gap decreases when one goes from O to Te compounds of Li and Na whereas X-X band gap increases if we go from O to S and then decreases from S to Te compounds of K and Rb.
The minimal vertical transition energies values (MVT) for Li2O, Li2S, Li2Se and Li2Te are 12.95 eV, 8.38 eV, 8.19 eV and 7.24 eV, respectively and for Na2O, Na2S, Na2Se and Na2Te are 13.20 eV, 8.52 eV, 8.98 eV and 7.46 eV, respectively. For K2O, K2S, K2Se and K2Te are 12.16 eV, 8.82 eV, 9.31 eV and 7.86 eV, respectively whereas for Rb2O, Rb2S, Rb2Se and Rb2Te are 9.63 eV, 8.60 eV, 9.03 eV and 7.81 eV, respectively.
Electronic density of states
In order to better understand the electronic structure of these materials and change of band gap outlined in previous section, total density of states (DOS) with contribution from individual cation and anion states of these materials have been computed. The DOS for all 16 compounds were computed using EV GGA in present work, but in Fig. 5 we present the DOS of alkali metal selenides and tellurides only because detailed representation of alkali metal oxides and sulfides already exist in literature 24, 36.
The lowest structure for Li2O, Li2S, Li2Se and Li2Te is located at -15.14 eV, -10.54 eV, -11.05 eV and -9.95 eV, respectively and for Na2O, Na2S, Na2Se and Na2Te at -14.30 eV, -9.85 eV, -10.39 eV and -8.96 eV, respectively. These structures are mainly due to the anion s states. In case of K2O, K2S, K2Se and K2Te the lowest structures at -14.79 eV, -9.38 eV, -9.87 eV and -8.43 eV are due to anions s states and structures at -12.40 eV, -12.46 eV, -12.65 eV and -12.92 eV are due to cation p states. For Rb2O, Rb2S, Rb2Se and Rb2Te the lowest structures at -14.56 eV, -9.11 eV, -9.57 eV and -8.37 eV are due to anion s states and structures at -9.88 eV, -10.05 eV, -10.22 eV and -10.41 eV are due to cation p states. For lithium and sodium chalcogenides main contribution to the lowest structures comes from the anions, whereas for potassium and rubidium chalcogenides the lowest band splits into individual contributions from cation and anion. The density of states for K and Rb chalcogenides has contribution from the cation. The O s states contribution to these structures lie below the K and Rb p states contribution on the energy scale. On the other hand the S, Se and Te s states contribution to these structures lie above the K and Rb p states contribution on the energy scale.
For the UVBW, it has been observed that the major contribution comes from the anion p states in all these materials. Major cation-contribution to the two peaks of UVBW comes from s and p states of Li, Na, K and Rb. In case of Li2S, Li2Se and Li2Te cation s states contribute more to the lower structure of UVBW and cation p states contribute more to the upper structure of UVBW. Since there is no d like states involved in UVBW of lithium chalcogenides, the indirect band gap in these materials decreases from Li2O to Li2Te. Almost same trend has been observed in UVBW of Na2O, Na2S, Na2Se and Na2Te compounds with the exception that, in their upper structures, cation d states contribute more as compared to cation s states. The d like states contribution to the upper structure of UVBW is one of the reason why the Î“-Î“ band gap first increases from O to S and then decreases from S to Te compounds of Na, K and Rb. Major difference between UVBW of these alkali metal chalcogenides is large amount of splitting in case of Li and Na chalcogenides, whereas sharp peaks are observed in the K and Rb chalcogenide compounds. This causes the X-X band gap of lithium and sodium chalcogenides to decrease as one goes from O to Te while the X-X band gap of potassium and rubidium chalcogenides first increases from O to S and then decreases from Se to Te. It is easy to see that the valence band is mainly formed by the p like states of anion. Conduction band for Li2O and Na2O get major contribution from p states of O, Li and Na. For the remaining compounds the conduction band is a complex mixture of p and d states of anions and s, p and d states of cations. It is evident that with increasing size of cations the band structure becomes more and more complex and the lowest structure starts to split. This may be because of the hybridization caused by increasing anion radius.
In this study we have investigated structural and electronic properties of alkali metal selenides and tellurides (M2X) [M: Li, Na, K, Rb, X: Se, Te] using LDA, PBE, WC and EV functionals for the first time in the framework of DFT, whereas structural and electronic properties of alkali metal oxides and sulfides (M2X) [M: Li, Na, K, Rb, X: O, S] have been reinvestigated using WC GGA. Our calculations lead to the following conclusions: (a) Structural parameters such as lattice constants and bulk moduli for all 16 compounds obtained using WC functional are in better agreement with experimental data as compared to other DFT calculations performed by using LDA and PBE functionals. (b) Lattice parameters increase whereas bulk moduli decrease with increasing cation radii. (c) Cohesive energies of these compounds increase linearly with increasing crystallographic ratio for the common anions of alkali metals. (d) Band gap values calculated using EV GGA for alkali metal selenides and tellurides are closer to available experimental data compared to other DFT parameterization schemes. (e) Oxides, sulfides, selenides and tellurides of lithium, potassium and rubidium are indirect band gap semiconductors, whereas oxides, sulfides, selenides and tellurides of sodium are direct band gap semiconductors. (f) The fundamental energy band gap for the common anions of all alkali metals decreases as we go from Li to Rb. Detailed data for these materials presented in this study may provide reference material for future developments in these compounds and their complex systems.