Quasiparticle electronic band structure of the alkali metal chalcogenides

The electronic energy band spectra of the alkali metal chalcogenides M$_2$A (M: Li, Na, K, Rb; A: O, S, Se, Te) have been evaluated within the projector augmented waves (PAW) approach by means of the ABINIT code. The Kohn-Sham single-particle states have been found in the GGA (the generalized gradient approximation) framework. Further, on the basis of these results the quasiparticle energies of electrons as well as the dielectric constants were obtained in the GW approximation. The calculations based on the Green's function have been originally done for all the considered M$_2$A crystals, except Li$_2$O.


Introduction
The alkali metal chalcogenides M 2 A (M: Li, Na, K, Rb; A: O, S, Se, Te) are found to crystallize in the cubic anti-fluorite (anti-CaF 2 -type) structure at ambient conditions. They draw considerable attention of researchers due to their possible applications in power sources, fuel cells, gas-detectors and ultraviolet space technology devices [1].
The properties of the crystals M 2 O have been extensively studied experimentally [2], whereas the sulfides, selenides and tellurides of alkali metal have received less experimental attention. The electronic energy band spectra of the M 2 A crystals have been evaluated using the full potential linearized augmented plane waves plus local orbitals (FP APW+lo) method based on DFT [1]. However, it is well known that the resulting band gap values in this approach are much underestimated. A proper way of calculating single-particle excitation energies or quasiparticle energies is provided by the Green's function theory. Here, the GW approximation (GWA) is used, which is the simplest working approximation beyond the Hartree-Fock approach taking screening into account [3].
Calculations of the electron energy spectrum beyond the local (LDA) or quasilocal (GGA) approximations were made only for the crystal Li 2 O [4]. The Kohn-Sham ground-state data have been evaluated [4] on the norm-conserving pseudopotential basis. On this basis there were obtained quasiparticle corrections to the eigenenergies using the GWA.
Then, the Bethe-Salpeter equation, which includes the screened electron-hole interaction as well as the unscreened electron-hole exchange term [4], was solved and the lowest exciton eigenvalue at 6.6 eV was found. This value is well compared with the optical absorption energy at about 6.6 eV. The GW corrections open the gap at the Γ point by 2.1 eV yielding a minimum direct gap of 7.4 eV. Therefore, the difference between the GW energy and the excitonic energy gives the exciton binding energy of 0.8 eV.
The above listed applications of the alkali metal chalcogenides do not exhaust the potential capabilities of these crystals. In fact, the recently registered patents suggest a possible use of these crystals, doped with d -or f -transition elements, in spintronics [5]. Finally, it is worth to mention an interesting theoretical prediction of the occurrence of a ferromagnetic half-metallic ordering in these crystals caused by the doping with nonmagnetic elements C, Si, Ge, Sn and Pb [6].
The compounds considered here have large lattice constants. As a result, the hybridization between respective orbitals of an impurity and a parent atom would be weak. Thus, the alkali metal atom, such as K, Na, Li or Rb can be substituted with each of the 3d , 4d and 5d transition metal elements and the rareearth 4 f elements [5]. The transition metal element is incorporated in the alkali chalcogenide compound in the form of a solid solution. The substitution of the alkali metal with the d or f transition element is performed at up to about 25% through a non-equilibrium crystal growth process at a low temperature to provide a ferromagnetic characteristic thereto.
Taking into account the importance of these materials due to their practical application, we reach the conclusion that a more precise calculation of the parameters of the electron energy spectra for them is an actual problem. And now let us turn to the solution.

Calculation
The first stage is to calculate the electron energy spectrum and eigenfunction in the generalized gradient approximation (GGA). For this purpose, the Kohn-Sham equations (2.1) are solved in a self-consistent way [7,8]: where Σ(r, r , ε q p nk ) is the non-local self-energy operator. The wave functions can be expanded as follows: |ψ q p nk 〉 = n a n n |ψ GGA nk 〉.

Electronic properties
Total density of electronic states (DOS) of Li 2 Se crystal is shown in figure 1. As can be seen, the wave functions of electrons in all energy bands are hybridized. This is indicated by the mark appearing next 33702-2        The widths of the corresponding bands are equal to 0.75 and 1.00 eV, respectively. Finally, the full width of the valence band, obtained in the GGA and GWA, equals 9.52 and 9.64 eV, respectively. Now, consider the results of the calculation presented in table 1. The values of the band energies obtained in the FP APW [1] approach by means of the WIEN2K code, and evaluated here in the PBE PAW framework with ABINIT code, are substantially underestimated. Let us first consider the properties of the Li 2 O crystal for which the experimental value of the energy of the optical absorption is known [4].
The value of the X − Γ gap, found in [1] within the DFT is 4.96 eV, and our value equals 5.07 eV. And the value of this parameter, calculated here within the GWA equals 7.55 eV. However, the experimental value of the optical absorption energy is equal to 6.6 eV. Now it is possible to estimate the binding energy of an exciton, which is simply equal to the difference of the last two values of the energy that is 0.95 eV. The corresponding value found recently from the Bethe-Salpeter equation is 0.98 eV [14]. Table 1 shows that all the Li 2 A, K 2 A and Rb 2 A crystals have an indirect band gap X − Γ, and Na 2 A crystals are characterized by a direct gap Γ − Γ. Table 1 shows that the values of the direct and indirect gaps in the Li 2 A crystals monotonously decrease with the replacement of the second element O → S → Se → Te. A similar behavior is also shown by the direct gap X − X in the crystals Na 2 A. Now, consider the results of the calculation presented in table 2. As can be seen, the most significant changes in the energy gaps ∆E are obtained for the crystal Li 2 O. We pay attention to the fact that the values of all the changes in the energy gaps ∆E obtained for each crystal are different. The greatest value of the ∆E change for the direct gap Γ − Γ is obtained for the Li 2 O crystal and the smallest value is found for the Li 2 Te crystal.
The macroscopic dielectric function ε LF M (ω), including local field effects, is related to the inverse of

33702-5
the microscopic dielectric matrix [12]: ε LF M (ω) = lim q→0 1 ε −1 00 (q,ω) . If local fields are neglected (no local fields, NLF), the irreducible polarizability is computed in the independent particle approximation. In this case, ε NLF M (ω) = lim q→0 ε 00 q, ω . The value ε M (0) is the static dielectric constant ε ∞ presented in table 3. The value of the dielectric constant for the Li 2 O crystal obtained here is 2.65, and the one evaluated in work [16] is 2.62. The corresponding experimental result equals 2.68 [4]. The convergence of the values of dielectric constants listed in table 3 served as an additional criterion of the choice of the plane wave basis in the Kohn-Sham problem, and in the calculation of the exchange Σ x and correlation Σ c parts [12] of the self-energy.

Conclusions
The electron energy spectra for M 2 A crystals have been originally calculated based on quasiparticle corrections within the GW approach. The results obtained herein show that the values of the interband gaps found without the quasiparticle corrections are usually underestimated by 20 − 50 percent (see table 2). All the Na 2 A crystals considered here are characterized by direct gaps Γ − Γ. The rest of the M 2 A crystals have indirect gaps X − Γ. The non-local self-energy operator Σ in equation (2.2) was evaluated without application of the plasmon pole model. The GW calculations have been carried out using the ABINIT code employing the contour deformation method [12,15]. As can be seen from table 2, the corrections ∆E are not weakly dependent on the wave vector. Therefore, the scissor operator is not a good approximation for all the crystals considered here. The long wave limits of the dielectric constants of the considered crystals have been evaluated for the first time. The last one found for Li 2 O crystal is well compared with the experimental value. Table 3 shows that the nine crystals listed therein have dielectric constants less than 3.0. We can assume that the exciton binding energy possessed by them is in the range from about 0.5 to 1.0 eV. Thus, the bandgap calculated in the GWA would exceed the experimental value of the optical absorption energy by the value of the binding energy of the exciton [4,14]. We hope that the results obtained here will stimulate the experimental study of these materials, which is important for practical applications.