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

We revisit the problem of surface states in semiconductors with inverted-band structures, such as $\ensuremath{\alpha}$-Sn and HgTe. We unravel the confusion that arose over the past decade regarding the origin of the surface states, their topological nature, and the role of strain. Within a single minimalistic description, we reconcile different solutions found in the 1980s with the results obtained from modern-day numerical simulations, allowing us to unambiguously identify all branches of surface states around the $\mathrm{\ensuremath{\Gamma}}$ point of the Brillouin zone in different regimes. We also show that strain is a smooth deformation to the surface states, following the usual continuity principle of physics, and not leading to any drastic change of the physical properties in these materials, in contrast to what has recently been advanced in the literature. We consider biaxial in-plane strain that is either tensile or compressive, leading to different branches of surface states for topological insulators and Dirac semimetals, respectively. Our model can help in interpreting numerous experiments on topological surface states originating from inverted-band semiconductors.