Abstract

We revisit theoretically the problem of surface states in semiconductors with
inverted-band structures, such as α-Sn and HgTe [1,2]. 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 simple and exactly solvable
models, we reconcile different solutions found in the 1980s [3,4] with the results
obtained from modern-day numerical simulations.
We consider biaxial in-plane strain that is either tensile or compressive, leading
to different branches of surface states for topological insulators (TIs) and Dirac
semimetals (or the nodal-line-semimetals), respectively. We show that in TI
regime strain is a smooth deformation to the surface states 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 investigate also the interplay of the effect of a strain of an arbitrary sign and
that of the sample finite size. We clarify the origin of various transitions which
happen at a given strain with the change of the sample thickness, in particular the
transition between the Dirac semimetal and quasi-3D (quantized) topological
insulator.
For the k-p Kane model we derive effective boundary conditions in the case
when the direct band materials form high barriers for the carriers of the inner
inverted-band semiconductor (for example, CdTe/HgTe/CdTe case). We show that
in this case the BCs have a universal and simple form which does not depend on
the order of different non-commuting operators in the Hamiltonian. Even in the
limit of very high barriers the BCs do not reduce to the trivial zero form, but
contain the information about the asymmetry of the offsets in the conduction and
valence bands. These boundary conditions allow to investigate the realistic case of
finite mass of the heavy-hole band, and to compare the results obtained within the
Kane and the Luttinger models.
[1]. A. Khaetskii, V. Golovach, and A. Kiefer, Revisiting the physical origin and nature of
surface states in inverted-band semiconductors, Phys. Rev. B 105, 035305 (2022).
[2]. A. Khaetskii, V. Golovach, and A. Kiefer, Interplay between strain and size quantization in
a class of topological insulators based on inverted-band semiconductors; submitted to PRB.
[3]. M.I. Dyakonov, A.V. Khaetskii, JETP Lett. 33, 110 (1981).
[4]. B. A. Volkov, O. A. Pankratov, JETP Lett. 42, 178 (1985).