Tuning topology in thin films of topological insulators by strain gradients

We theoretically show that the coupling of inhomogeneous strains to the Dirac fermions of three-dimensional topological insulators (3DTI) in thin film geometries results in the occurrence of phase transitions between topologically distinct insulating phases. By means of minimal $\mathbf{k}·\mathbf{p}$ models for strong 3DTI in the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ materials class, we find that in thin films of stoichiometric materials a strain-gradient induced structure inversion asymmetry drives a phase transition from a quantum spin-Hall phase to a topologically trivial insulating phase. Interestingly, in alloys with strongly reduced bulk band gaps strain gradients have an opposite effect and promote a topologically non-trivial phase from a parent normal band insulator. These strain-gradient assisted switchings between topologically distinct phases are expected to yield a strain gradient tunability of magnetism in magnetic topological insulator thin films.

Figure 1

Sketch of the surface band structure of a bent 3DTI thin film. (a) In the strain-free configuration the top and bottom surface Dirac cones acquire an hybridization gap at the Dirac point. (b) At a critical bending radius the two gapped Dirac cones are pulled toward each other and touch on a nodal line $\left|\mathbf{k}\right|=k_c$. (c) For even larger bending radii the surface band gap is reopened with the system that then corresponds to a conventional band insulator. (d) Corresponding topological phase diagram in terms of the strain-gradient induced structure inversion parameter $V$ and the ratio between the hybridization surface gap $\delta m$ and surface effective mass $\delta \beta$.

Figure 2

Behavior of the surface Dirac cones hybridization gap $\delta m$ (a) and the effective mass parameter $\delta \beta$ (b) as a function of the 3DTI thin film thickness $w$. We have used the $\mathbf{k}·\mathbf{p}$ parameters of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ of Ref. [23]. For $2.5\mathrm{nm}\lesssim w\lesssim 5$ nm the unstrained thin film realizes a QSH insulator.

Figure 3

(a) Evolution of the $\overline{\mathrm{\Gamma }}$ point energy levels for a narrow band gap 3DTI thin film with $A_1=0$ as the strain coupling term $u=u_0$ is increased. The ratio between the strain coupling terms has been fixed to $u_0/u_3=0.2$. (b) Topological phase diagram in the $A_1$ vs strain plane. (c) Evolution of the band structure across the topological phase transition for $A_1=2.2\mathrm{eV}\AA$.