Band structure engineering of multinary chalcogenide topological insulators

Topological insulators (TIs) have been found in strained binary HgTe and ternary I-III-VI${}_2$ chalcopyrite compounds such as CuTlSe${}_2$ which have inverted band structures. However, the nontrivial band gaps of these existing binary and ternary TIs are limited to small values, usually around 10 meV or less. In this work, we reveal that a large nontrivial band gap requires the material to have a large negative crystal field splitting ${\Delta }_{\mathit{CF}}$ at the top of the valence band and a moderately large negative $s-p$ band gap $E_g^{s-p}$. These parameters can be better tuned through chemical ordering in multinary compounds. Based on this understanding, we show that a series of quaternary ${\mathrm{I}}_2$-II-IV-VI${}_4$ compounds, including Cu${}_2$HgPbSe${}_4$, Cu${}_2$CdPbSe${}_4$, Ag${}_2$HgPbSe${}_4$, and Ag${}_2$CdPbTe${}_4$, are TIs, in which Ag${}_2$HgPbSe${}_4$ has the largest TI band gap of 47 meV because it combines the optimal values of ${\Delta }_{\mathit{CF}}$ and $E_g^{s-p}$.

Figure 1

(a) The conduction and valence band splitting of cubic and tetragonal semiconductors. (b) A plot showing how the band structure of normal semiconductors transfers into the inverted and topological insulator band structures. Note that the subscript $v$ ($c$) represents that the state belongs to the valence (conduction) band in the normal band structure.

Figure 2

Crystal structure plot of (a) ternary I-III-VI${}_2$ chalcopyrite structure, (b) quaternary ${\mathrm{I}}_2$-II-IV-VI${}_4$ kesterite structure, and (c) quaternary ${\mathrm{I}}_2$-II-IV-VI${}_4$ stannite structure.

Figure 3

The calculated band structure along the high-symmetry lines $X_X:\frac{2\pi }{a}(100)\to \Gamma :(000)\to X_Z:\frac{2\pi }{a}(001)\to \Gamma :(000)\to L:\frac{2\pi }{a}(0.50.50.5)$ of (a) HgTe with a (001) tensile strain and ${\Delta }_{\mathit{CF}}=70$ meV, and (b) CuTlTe${}_2$ with ${\Delta }_{\mathit{CF}}=76$ meV at its equilibrium state. $X_X$, $X_Z$, and $L$ are the notations for the zinc-blende structure, and $X_Z$ and $L$ correspond to $T$ and N, respectively, in the chalcopyrite structure. Red and blue colors are used to show the two spin-dependent bands clearly.

Figure 4

(a) The calculated nontrivial band gap as a function of ${\Delta }_{\mathit{CF}}$ for HgTe. Here ${\Delta }_{\mathit{CF}}$ is changed by tuning the (001) strain $\epsilon$. (b) The calculated ${\Delta }_{\mathit{CF}}$ and (c) $E_g^{s-p}$ of Cu-III-Te${}_2$ with III Al, Ga, In, or Tl.

Figure 5

The calculated band structure along the high-symmetry lines $X_X\to \Gamma \to X_Z(T)\to \Gamma \to L(N)$ of (a) Cu${}_2$HgPbTe${}_4$ and (b) Ag${}_2$HgPbSe${}_4$ in their equilibrium states.