Ab initio calculations of electronic properties, frequency dispersion of dielectric coefficients and the edge of the optical absorption of TlInS$_2$ $\langle$Sn$\rangle$ single crystals

Calculations of the band structure and density of states for supercells of TlInS$_2$ and TlInS$_2$ $\langle$Sn$\rangle$ semiconductor crystals with monoclinic singony within the density functional theory (DFT) are presented. In DFT calculations, the Coulomb repulsion (the Hubbard parameter $(U)$) was taken into account in order to correctly describe the band gap $(E_g)$ of crystals. It is shown that the maximum of the valence band and the minimum of the conduction band of TlInS$_2$ are located at the center (point ) of the Brillouin zone, which points to the direct energy of the band gap of the TlInS$_2$ and TlInS$_2$ $\langle$Sn$\rangle$ crystals. Thrue results of calculating the band structure and features of the distribution of the density of electronic states in TlInS$_2$ and TlInS$_2$ $\langle$Sn$\rangle$ are discussed.
TlInS$_2$ single crystals doped with 0.1 mol.% tin (TlInS$_2$ $\langle$0.1 mol.% Sn$\rangle$) were synthesized and then grown by the Bridgman–Stockbarger method. The frequency dispersion of the dielectric coefficients and conductivity of a TlInS$_2$ and TlInS$_2$ $\langle$0.1 mol.% Sn$\rangle$ single crystals in the frequency range $f$ = 5 $\cdot$ 10$^4$–3.5 $\cdot$ 10$^7$ Hz is studied. It is shown that relaxation losses occur in TlInS$_2$ $\langle$Sn$\rangle$. A hopping mechanism of ac charge transfer in TlInS$_2$ $\langle$Sn$\rangle$ has been established. In TlInS$_2$ $\langle$Sn$\rangle$, the parameters of localized states, such as the density of states near the Fermi level and their energy spread, the average hopping time and distance, and the concentration of deep traps, are estimated.
The optical properties of TlInS$_2$ and TlInS$_2$ $\langle$0.1 mol.% Sn$\rangle$ single crystals have been studied.
The values of $E_g$ for direct optical transitions in TlInS$_2$ and TlInS$_2$ $\langle$0.1 mol.% Sn$\rangle$ crystals were obtained from the optical absorption spectra. It has been shown that the introduction of 0.1 mol.% Sn, which replaces indium atoms, into TlInS$_2$ reduces the value of $E_g$, for example, at 150 K from 2.539(TlInS$_2$) to 2.486 eV (TlInS$_2$ $\langle$0.1 mol.% Sn$\rangle$). From optical measurements, the average temperature coefficient of the band gap $\partial E_g/\partial T=-7\cdot10^{-4}$ eV/K for TlInS$_2$ $\langle$Sn$\rangle$ was calculated. The decrease in the band gap of the TlInS$_2$ $\langle$Sn$\rangle$ single crystal with respect to TlInS$_2$ is 16 meV at 300 K and 53 meV at 150 K.