Ab initio calculations of the band structure, total and partial densities of states and the spatial distribution of the electron charge density of crystalline Na 2GeS 3 are performed in the framework of density functional theory in the local density approximation for an exchange-correlation potential. According to the calculation results, sodium thiogermanate is a direct-gap crystal with the top of the valence band and the bottom of the conduction band at the point of the Brillouin zone. The calculated band gap is E g= 2.51 eV. The nature of the components of the electronic states in different subbands of the valence band is determined. The calculated total density of states in the valence band of the crystal is compared with the known experimental X-ray photoelectron spectrum of Na 2GeS 3 glass. Based on the maps of the electron density distribution, the nature of the chemical bonds and high mobility of Na + ions in Na 2GeS 3 crystal is analyzed.
In the recent years, an intense work is underway to search for electrode materials for sodium-ion batteries that could in future replace lithium-ion power sources [
Investigation of the phase diagram of GeS2-Na2S system by differential thermal analysis and X-ray diffraction analysis showed that in this system (depending on the GeS2/Na2S ratio), three crystalline phases―Na2GeS3, Na4Ge4S10, and Na6Ge2S7―could be formed [
In this paper, we report on the density functional theory (DFT) calculations of the energy band structure, total and partial local densities of states and the spatial distribution of the electron density for crystalline Na2GeS3. The calculated total density of electronic states of the crystal is compared with the experimental X-ray photoelectron spectrum (XPS) of Na2GeS3 glass taken from [
Sodium thiogermanate crystallizes in the monoclinic structure, which symmetry is described by the space group P21/c with the lattice parameters а = 6.952 Å, b = 15.230 Å, c = 5.720 Å, b = 115.24°, Z = 4 [
structure is composed of endless
tion of two [GeS4] tetrahedra linked by shared vertices having a-cis relative orientation (
Calculations of the energy band structure are performed within the density functional theory [
local density approximation (LDA) in order to the exchange and correlation effects to be taken into account [
The cut-off energy Ecut = 200 Ry for the atomic orbitals for the self-consistent calculation was chosen to obtain a convergence in the cell total energy not worse than 0.001 Ry/atom. The basis had about 22320 atomic orbitals. The density of the k-point grid in the reciprocal space for the self-consistent calculation was selected from the same reasons. The electron density was calculated by interpolation between the sites of the 5 × 3 × 6 grid in the reciprocal space. The total and partial electron densities of states were determined by the modified tetrahedra method, for which the energy spectrum and wave functions were calculated on a k-grid containing 90 points. Integration over the irreducible part of the Brillouin zone was performed using the method of special k-points [
Calculation of the band structure of Na2GeS3 crystal was performed for the high-symmetry points Г, B, D, Z, C, Y, A, E and along the lines between them in the Brillouin zone (
The valence bands have a weak dispersion and consist of three energy separated band bunches of (VBI, VBII, VBIII, numbering from the top subband), which correspond to discrete bands in the density of states N(E) spectra. The profiles of the total density of states for Na2GeS3 as well as the contributions from the individual states for different atoms are presented in
The average occupied subband VBII (from −7.79 to −5.07 eV) is of hybrid character and is formed by an overlap of the Ge4s- and S3p-states, playing thus a prominent role in the formation of Ge-S covalent bonds in the [GeS4] tetrahedral forming the endless chains.
The upper valence subband (VBI) can be conditionally divided into two parts: the lower one (from −4.42 to −2.18 eV) of 12 filled branches is mixed with the hybridized 3p-states of sulfur and 4p-states of germanium with the insignificant contribution of s- and p-states of sodium; the upper one (from −2.08 eV to the valence band top) consisting of 20 dispersive branches is formed mainly by the p-states of the lone pair of sulfur atoms which are mixed with the germanium d-states and the alkali metal s-, p-states. The top of the valence band at the Г point is formed exclusively by pz-atomic orbitals of sulfur.
The low-energy electronic structure of unoccupied electronic states of sodium thiogermanate is formed mainly by “mixing” of free states of p-electrons of sulfur atoms as well as s-and p-electrons of germanium and sodium atoms.
The energy spectrum of the valence electrons directly determines such an important characteristic of the crystal as the spectral dependence of the absorption coefficient. However, no papers on the experimental measurement of the intrinsic absorption edge of crystalline or glassy Na2GeS3 have been published so far. The lack of the absorption edge data does not enable us to compare the calculated band gap value for Na2GeS3 with the experimental Eg.
Meanwhile, the calculations of total and partial densities of states significantly facilitate the interpretation of the experimental X-ray photoelectron spectra (XPS) which reflect the distribution of the total density of states in the valence band. In
of glassy sodium thiogermanate by vibrational spectroscopy [
We identify the nature of the peaks observed in the experimental XPS spectrum of the glass (curve 2 in
Investigation of the distribution of the total charge density of the valence electrons r(r) in the form of contour maps provides helpful information about the nature of both intra-chain and inter-chain chemical bonds in Na2GeS3 which is rather difficult to be obtained experimentally for such a complex crystal structure. Besides, from the contour maps of the electron density one can obtain additional information about the character of the ionic conductivity.
The choice of suitable sectional planes is crucial for the construction of the electron density contour maps for a crystal. The tetrahedral short-range structure of Na2GeS3 encumbers there presentation of the contour maps in a 2D format. In this case, it is most convenient to represent the electronic configurations for a single [GeS4] tetrahedron in the plane passing through two sulfur atoms and one germanium atom, i.e. in the plane along the S-Ge-S bond lines. Since the [GeS4] tetrahedra linked in the chain contain two bridged (S-Ge-S) and two non- bridged (end-point) (Ge-S) sulfur atoms, it is reasonable to built two planes: one through the germanium atom and the two bridging sulfur atoms, and the second one through the germanium atom and the two non-bridged sulfur atoms. The distribution of the electron density r(r) over the germanium atom and the two non-bridged and bridged sulfur atoms belonging to the same [GeS4] tetrahedron is illustrated by the electron density maps shown in
tetrahedra. As a result, specific chains of
The common contours, encompassing the maxima of the electron density on the Ge and S atoms, characterize the covalent component of the chemical bond. The deformation of contours towards the germanium and sulfur atoms indicates these directions as being capable of formation of sp3-hybrid bonding orbitals. Besides the covalent component of the chemical bonding, Na2GeS3 is characterized by ionic and weak van der Waals bonding components. The ionic component is determined by partial transfer of the charge density from the germanium and sodium atoms to the more electronegative sulfur atoms.
The ionic component of the chemical bond is characterized by charges localized on the atoms themselves, an asymmetry (polarization) of the covalent bond which is revealed in the charge shift at the Ge-S bond, and deformation of electron density contours. Since the distribution of the total charge in one (isolated) chain forms an almost closed shell (
Sodium atoms localized the in inter-chain space, have no common contours of the electron density r(r) with the neighboring anions (sulfur atoms). From the contour maps of the electron density of Na2GeS3 crystal (
Na+ ions) and the presence of pass-through “tunnels” between the
charge density along the c axis indicate the direct participation of sodium ions in the ionic conductivity providing high conductivity of the sodium ions along these channels and, as a consequence high ionic conductivity.
Ab initio DFT calculations based on the linear combination of atomic orbitals are performed using the SIESTA software to obtain the electron energy spectrum and the density of states of sodium thiogermanate. Analysis of the total and local partial densities of electronic states of atoms contained in Na2GeS3 enabled us to determine the genesis of the components of the electronic states in different subbands of the valence band.
It is determined that Na states give a negligible contribution to the formation of the valence band of crystalline Na2GeS3 while the major contribution comes from the 3s-states of sulfur in the valence band lower part as well as hybridized Ge4s-, 4p- and S3р-states in other filled subbands. The top of the valence band of Na2GeS3 is formed exclusively by lone pair sulfur states while the bottom of the conduction band results from mixing of free S3p-, Ge4s- and Na3s-, 3p-states.
The calculated total density of electronic states of the crystal is compared with the known experimental X-ray photoelectron spectrum of the glass. The similarity of the shape of the N(E) spectrum of the crystal and the XPS spectrum of glass shows that the short-range order is preserved at the crystal-to-glass transition in sodium thiogermanate.
From the maps of the electron density distribution of Na2GeS3 crystal, it follows that the chemical bond between
the cation (germanium) and the anion (sulfur) in the
the chains are linked together by weak van der Waals forces. The negligible charge density around Na+ ions and the presence of pass-through “tunnels” between the infinite chains provide a considerable mobility of sodium ions along these channels and, as a consequence, high ionic conductivity of the crystal and glass.