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We revisit the problem of surface states in semiconductors with inverted bandstructures, such as $\alpha$-Sn and HgTe. We unravel the confusion that aroseover the past decade regarding the origin of the surface states, theirtopological nature, and the role of strain. Within a single minimalisticdescription, we reconcile different solutions found in the 1980s with theresults obtained from modern-day numerical simulations, allowing us tounambiguously identify all branches of surface states around the $\Gamma$-pointof the Brillouin zone in different regimes. We also show that strain is asmooth "deformation" to the surface states, following the usual continuityprinciple of physics, and not leading to any drastic change of the physicalproperties in these materials, in contrast to what has recently been advancedin the literature. We consider biaxial in-plane strain that is either tensileor compressive, leading to different branches of surface states for topologicalinsulators and Dirac semimetals, respectively. Our model can help ininterpreting numerous experiments on topological surface states originatingfrom inverted-band semiconductors.

Revisiting the physical origin and nature of surface states in inverted-band semiconductors