# Symmetry analysis of strain, electric and magnetic fields in the $\text{Bi}_2\text{Se}_3$-class of topological insulators

## TL;DR

This paper uses group theory to derive a comprehensive Hamiltonian for Bi2Se3 topological insulators, analyzing how strain, electric, and magnetic fields influence their electronic and surface state properties, with implications for device applications.

## Contribution

It provides the most general Hamiltonian including higher-order terms and analytically explores strain effects on electronic structure and surface states of Bi2Se3.

## Key findings

- Strain causes anisotropic Dirac cones with elliptical constant energy contours.
- Strain modifies spin-momentum locking and can induce a perpendicular spin component.
- Surface state band gap varies oppositely to the bulk gap under strain.

## Abstract

Based on group theoretical arguments we derive the most general Hamiltonian for the $\text{Bi}_2\text{Se}_3$-class of materials including terms to third order in the wave vector, first order in electric and magnetic fields, first order in strain and first order in both strain and wave vector. We determine analytically the effects of strain on the electronic structure of $\text{Bi}_2\text{Se}_3$. For the most experimentally relevant surface termination we analytically derive the surface state spectrum, revealing an anisotropic Dirac cone with elliptical constant energy counturs giving rise to different velocities in different in-plane directions. The spin-momentum locking of strained $\text{Bi}_2\text{Se}_3$ is shown to be modified and for some strain configurations we see a non-zero spin component perpendicular to the surface. Hence, strain control can be used to manipulate the spin degree of freedom via the spin-orbit coupling. We show that for a thin film of $\text{Bi}_2\text{Se}_3$ the surface state band gap induced by coupling between the opposite surfaces changes opposite to the bulk band gap under strain. Tuning the surface state band gap by strain, gives new possibilities for the experimental investigation of the thickness dependent gap and optimization of optical properties relevant for, e.g., photodetector and energy harvesting applications. We finally derive analytical expressions for the effective mass tensor of the Bi$_2$Se$_3$ class of materials as a function of strain and electric field.