Topological Phase Diagrams of Bulk and Monolayer ${\mathrm{TiS}}_{2-x}{\mathrm{Te}}_x$

With the use of ab initio calculations, the topological phase diagrams of bulk and monolayer ${\mathrm{TiS}}_{2-x}{\mathrm{Te}}_x$ are established. Whereas bulk ${\mathrm{TiS}}_{2-x}{\mathrm{Te}}_x$ shows two strong topological phases [1;(000)] and [1;(001)] for $0.44<x<0.56$ and $0.56<x<1$, respectively, the monolayer is topologically nontrivial for $0.48<x<0.80$. Because in the latter doping range the topologically nontrivial nature survives down to a monolayer, ${\mathrm{TiS}}_{2-x}{\mathrm{Te}}_x$ is a unique system for studying topological phases in three and two dimensions simultaneously.

Figure 1

(a) Unit cell of ${\mathrm{TiS}}_2$ and ${\mathrm{TiTe}}_2$. First Brillouin zone of (b) bulk and (c) monolayer ${\mathrm{TiS}}_{2-x}{\mathrm{Te}}_x$.

Figure 2

Electronic band structures of bulk (a) ${\mathrm{TiS}}_2$ and (b) ${\mathrm{TiTe}}_2$, in the presence of spin-orbit coupling. The $\mathrm{S}/\mathrm{Te}$ $p$ and Ti $d$ band characters are indicated by different colors. Band hierarchy at the $\Gamma$ and $A$ points around $E_F$ in bulk (c) ${\mathrm{TiS}}_2$ and (d) ${\mathrm{TiTe}}_2$. The symmetry at the $\Gamma /A$ point is labeled, and even/odd parity is indicated by $+/-$.

Figure 3

(a) Topological phase diagram of bulk ${\mathrm{TiS}}_{2-x}{\mathrm{Te}}_x$ as a function of $x$, given by the energy differences between ${\Gamma }_{56}^{-}/{\Gamma }_4^{-}$ and ${\Gamma }_4^{+}/{\Gamma }_{56}^{+}$ (${\Delta }_{\Gamma }^1/{\Delta }_{\Gamma }^2$, solid lines) and between $A_{56}^{-}/A_4^{-}$ and $A_4^{+}/A_{56}^{+}$ (${\Delta }_A^1/{\Delta }_A^2$, dotted lines). Different topological phases are indicated by different colors. (b) Topological phase diagram of monolayer ${\mathrm{TiS}}_{2-x}{\mathrm{Te}}_x$ as a function of $x$.

Figure 4

Electronic band structures of monolayer (a) ${\mathrm{TiS}}_2$ and (b) ${\mathrm{TiTe}}_2$, in the presence of spin-orbit coupling. The $\mathrm{S}/\mathrm{Te}$ $p$ and Ti $d$ band characters are indicated by different colors. Band hierarchy at the $\Gamma$ and $A$ points around $E_F$ in bulk (c) ${\mathrm{TiS}}_2$ and (d) ${\mathrm{TiTe}}_2$. The symmetry at the $\Gamma /A$ point is labeled, and even/odd parity is indicated by $+/-$.

Figure 5

(a) Energy diagram of the $p_{xy}$ and $p_z$ states at the $\Gamma /A$ point near $E_F$ in ${\mathrm{TiS}}_2/{\mathrm{TiTe}}_2$ (I) without spin-orbit coupling, (II) with spin-orbit coupling but without hybridization between the two $\Gamma /A_4^{-}$ states, and (III) with spin-orbit coupling and hybridization. (b) Energy differences ${\Delta }_{\Gamma }/{\Delta }_A$ between the $p_{xy}$ $\Gamma /A_{56}^{-}$ and $\Gamma /A_4^{-}$ states and ${\Delta }_{\Gamma }^{\prime }/{\Delta }_A^{\prime }$ between the $p_{xy}$ and $p_z$ $\Gamma /A_4^{-}$ states for bulk and monolayer ${\mathrm{TiS}}_2$ and ${\mathrm{TiTe}}_2$.