Strain tuning of topological band order in cubic semiconductors

We theoretically explore the possibility of tuning the topological order of cubic diamond/zinc-blende semiconductors with external strain. Based on the tight-binding model, we analyze the evolution of the cubic semiconductor band structure under hydrostatic or biaxial lattice expansion, by which a generic guiding principle is established that lattice expansion can induce a topological phase transition of small band-gap cubic semiconductors via a band inversion, and further breaking of the cubic symmetry leads to a topological insulating phase. Using density functional theory calculations, we demonstrate that a prototype topological trivial semiconductor, InSb, is converted to a nontrivial topological semiconductor with a 2$\%$–3$\%$ biaxial lattice expansion.

Figure 1

The band order of cubic diamond/zinc-blende semiconductors at the $\Gamma$ point with ${\lambda }_{\mathrm{SO}}>0$. (a) The normal band order. (b), (c) The inverted band order can be obtained from the normal band order by either increasing the spin-orbit coupling ${\lambda }_{\mathrm{SO}}$ (b) or by decreasing the coupling potentials $V_{ss}$ and $V_{pp}$ (c). The band inversion strength is defined as $\Delta E=E_{{\Gamma }_6}-E_{{\Gamma }_8}$, which has positive value in normal band order and negative value in inverted band order.

Figure 2

The band structures of InSb at the equilibrium lattice constant (a) and with 1$\%$ (b), 2$\%$ (c), and 3$\%$ (d) hydrostatic lattice expansions. The ${\Gamma }_6$, ${\Gamma }_7$, and ${\Gamma }_8$ states are denoted by red, black, and blue colors, respectively. The size of dots is proportional to the probability of $s$-orbit projection.

Figure 3

The evolution of band order of InSb with nonhydrostatic expansion (a) in the $ab$ plane and (b) along the $c$ axis. The black solid circles are the results without lattice optimization, orange down-diamonds and purple up-diamonds are optimized by LDA and GGA potential, respectively. (c) and (d) are normal band order and inverted band order, which correspond to the states marked by dotted circles in (a). The size of dots is proportional to the probability of $s$-orbit projection.