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Based on group theoretical arguments we derive the most general Hamiltonianfor the $\text{Bi}_2\text{Se}_3$-class of materials including terms to thirdorder in the wave vector, first order in electric and magnetic fields, firstorder in strain and first order in both strain and wave vector. We determineanalytically the effects of strain on the electronic structure of$\text{Bi}_2\text{Se}_3$. For the most experimentally relevant surfacetermination we analytically derive the surface state spectrum, revealing ananisotropic Dirac cone with elliptical constant energy counturs giving rise todifferent velocities in different in-plane directions. The spin-momentumlocking of strained $\text{Bi}_2\text{Se}_3$ is shown to be modified and forsome strain configurations we see a non-zero spin component perpendicular tothe surface. Hence, strain control can be used to manipulate the spin degree offreedom 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 betweenthe opposite surfaces changes opposite to the bulk band gap under strain.Tuning the surface state band gap by strain, gives new possibilities for theexperimental investigation of the thickness dependent gap and optimization ofoptical properties relevant for, e.g., photodetector and energy harvestingapplications. We finally derive analytical expressions for the effective masstensor of the Bi$_2$Se$_3$ class of materials as a function of strain andelectric field.

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