Electronic structure of ZrS${}_x$Se${}_{2-x}$ by Tran-Blaha modified Becke-Johnson density functional

The electronic properties of the layered transition metal dichalcogenide ZrS${}_x$Se${}_{2-x}$ semiconductors for $x$ $=$ 0 and 2 as well as for the ternary compound with $x$ $=$ 1 have been calculated by density functional theory for six different exchange-correlation energy approximations by the wien2k code. The results show that, among the functions we tested, a new semilocal potential that combines the modified Becke–Johnson potential and the local spin density approximation correlation potential as proposed by Tran and Blaha [F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009)] (TB-MBJ) remains superior for estimating the band gap. Thus, the calculations have been performed within the TB-MBJ method both with and without spin-orbit interaction. Calculations by all methods reveal that the valence band maximum and conduction band minimum are located at the Γ and $M$ points, respectively, which are in agreement with experimental data. Moreover, in the three compounds the band gap decreases linearly from ZrS${}_2$ to ZrSe${}_2$. When considering spin-orbit (SO) coupling, the degeneracy of the valance bands is removed. The size of the SO splitting increases by the atomic number of chalcogenide from ZrS${}_2$ to ZrSe${}_2$.

Figure 1

Band structure and DOS of ZrS${}_2$ within the TB-MBJ method by considering (a) without and (b) with spin-orbit interaction in the calculation, where the band gaps are 1.41 and 1.40 eV, respectively. The valence band maximum and conduction band minimum are located at Γ and $M$ points, respectively. The SO splitting of the top valence band at the Γ point is about 0.087eV (see b). The inset shows the Brillouin zone.

Figure 2

Band structure and DOS of ZrSe${}_2$ within the TB-MBJ method by considering (a) without and (b) with spin-orbit interaction in the calculation, where the band gaps are 0.92 and 0.81 eV, respectively. The valence band maximum and conduction band minimum are located at Γ and $M$ points, respectively. The splitting of the top valence band at the Γ point is 0.472 eV (see b).

Figure 3

Band structure and DOS of ZrSeS within the TB-MBJ method by considering (a) without and (b) with spin-orbit interaction in the calculation, where the band gaps are 1.18 and 1.13 eV, respectively. The valence band maximum and conduction band minimum are located at Γ and $M$ points, respectively. The splitting of the top valence band at the Γ point is 0.236 eV (see b).