Three-Dimensional Topological Insulators in $\mathrm{I}\mathrm{\text{−}}\mathrm{III}\mathrm{\text{−}}{\mathrm{VI}}_2$ and $\mathrm{II}\mathrm{\text{−}}\mathrm{IV}\mathrm{\text{−}}{\mathrm{V}}_2$ Chalcopyrite Semiconductors

Using first-principles calculations within density functional theory, we investigate the band topology of ternary chalcopyrites of composition $\mathrm{I}\mathrm{\text{−}}\mathrm{III}\mathrm{\text{−}}{\mathrm{VI}}_2$ and $\mathrm{II}\mathrm{\text{−}}\mathrm{IV}\mathrm{\text{−}}{\mathrm{V}}_2$. By exploiting adiabatic continuity of their band structures to the binary 3D-HgTe, combined with direct evaluation of the $Z_2$ topological invariant, we show that a large number of chalcopyrites can realize the topological insulating phase in their native states. The ability to host room-temperature ferromagnetism in the same chalcopyrite family makes them appealing candidates for novel spintronics devices.

Figure 1

Comparison of the zinc blende and the chalcopyrite structure. (a) The zinc-blende compound HgTe. (b) The chalcopyrite compound $AB{C}_2$. The internal displacement of the anion is defined as $\delta u=(R_{AC}^2-R_{BC}^2)/a^2$, where $R_{AC}$ and $R_{BC}$ are the bond lengths between the anion $C$ and its two nearest $A$ and $B$ cations.

Figure 2

Band structures of HgTe in comparison with ${\mathrm{AuTlTe}}_2$ and ${\mathrm{AuTlS}}_2$. (a)–(c) Band structures of HgTe (a), HgTe in an ideal chalcopyrite supercell (b), and HgTe in a chalcopyrite cell with $\eta =1.016$ and $\delta u=-0.018$ (c). The size of red dots denotes the probability of $s$-orbital occupation on the cations. The inset in (c) shows the energy gap opened by chalcopyrite distortion. (d),(e) Band structures of ${\mathrm{AuTlTe}}_2$ (d) and ${\mathrm{AuTlS}}_2$ (e). Both display an inverted band order similar to HgTe, indicating a nontrivial band topology. $\Delta E$ in (e) indicates the band-inversion strength.

Figure 3

Inverted band strength for various chalcopyrites as a function of the lattice constant. Open symbols mark the compounds whose lattice constant was reported in the literature. For the rest of the compounds their equilibrium lattice constants are obtained by first-principles total energy minimization. When $\Delta E<0$, squares mark the compounds that are topological insulators and diamonds mark the systems that are topological metals. Shaded areas indicate materials that are expected to be closely ($\pm 2\%$) lattice matched to either GaAs, InAs, or InSb.

Figure 4

Band structures of ${\mathrm{CdSnAs}}_2$ without and with a hydrostatic strain. The application of a hydrostatic strain causes the $s$-like ${\Gamma }_6$ bands (marked by red dots) to jump below the valence band top, providing the necessary band inversion that leads to a nontrivial topological order.