Spin-orbit coupling in the band structure of reconstructed $1T\text{−}{\mathrm{TaS}}_2$

Empirical tight-binding simulations of the band structure of $1T\text{−}{\mathrm{TaS}}_2$ reveal that the combination of spin-orbit coupling and the low-temperature $\sqrt{13}\times \sqrt{13}$ reconstruction generates a distinct and very narrow band at the Fermi level. Spin-orbit interaction is therefore important in any understanding of the correlation effects in this material. The states within the split-off band are primarily of $d_{x^2-y^2}$, $d_{xy}$ character and reside preferentially on the central atom of the “Star-of-David” cluster.

Figure 1

(Color online) (a) Band structure of $1T\text{−}{\mathrm{TaS}}_2$ both with [bold green (light gray) curves] and without (thinner black curves) the $\mathrm{Ta}$ $5d$ spin-orbit coupling. (b) Crystal-field diagram showing the mean energies of the $z^2$, $x^2-y^2∕xy$, and $yz∕zx$ orbitals. The energy scales have been adjusted to bring the Fermi levels into alignment.

Figure 2

(Color online) Unit cell of the $\sqrt{13}\times \sqrt{13}$ reconstruction of $1T\text{−}{\mathrm{TaS}}_2$ showing the “Star-of-David” clusters having inequivalent “a,” “b,” and “c” Ta atoms. The arrows indicate the displacements of the Ta atoms from their original positions.

Figure 3

(Color online) (a) Band structure of $1T\text{−}{\mathrm{TaS}}_2$ in the unreconstructed case [bold green (light gray) curves] and the Umklapp bands (thinner black curves) generated by translation through the reciprocal lattice vectors of the reconstructed Brillouin zone. (b) Reconstructed band structure with inclusion of the bond strengthenings within the Star-of-David as described in the text. (c) Reconstructed band structure with inclusion of Ta spin-orbit coupling revealing [bold green (light gray) curve] a single distinct split-off band. (d) Atom-projected densities of states showing that the states in the split-off band reside primarily on the “a” atoms at the center of the cluster. (e) Orbital-projected densities of states showing that the states in the split-off band are primarily of $x^2-y^2∕xy$ character. The energy scales have been adjusted to bring the respective Fermi levels into alignment.