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A topological boundary can be formed at the interface between a trivial and a topological insulator. The difference in the topological index across the junction leads to robust gapless surface states. Optical studies of these states are scarce in the literature, the reason being the difficulty to isolate their response from that of the bulk. In this work, we propose to deposit a $\delta$ layer of donor impurities in close proximity to a topological boundary to help detecting gapless surface states. As we will show, gapless surface states are robust against this perturbation and they enhance intraband optical transitions as measured by the oscillator strength. These results allow to understand the interplay of surface and bulk states in topological insulators.

Several IV-VI semiconductor compounds made of heavy atoms, such as Pb$_{1-x}$Sn$_{x}$Te, may undergo band-inversion at the $L$ point of the Brillouin zone upon variation of their chemical composition. This inversion gives rise to topologically distinct phases, characterized by a change in a topological invariant. In the framework of the $\mathbf{k}\cdot\mathbf{p}$ theory, band-inversion can be viewed as a change of sign of the fundamental gap. A two-band model within the envelope-function approximation predicts the appearance of midgap interface states with Dirac cone dispersions in band-inverted junctions, namely, when the gap changes sign along the growth direction. We present a thorough study of these interface electron states in the presence of crossed electric and magnetic fields, the electric field being applied along the growth direction of a band-inverted junction. We show that the Dirac cone is robust and persists even if the fields are strong. In addition, we point out that Landau levels of electron states lying in the semiconductor bands can be tailored by the electric field. Tunable devices are thus likely to be realizable exploiting the properties studied herein.