Probing the Electronic Structure at the Boundary of Topological Insulators in the ${\mathrm{B}\mathrm{i}}_2{\mathrm{S}\mathrm{e}}_3$ Family by Combined STM and AFM

We develop a numerical scheme for the calculation of tunneling current $I$ and differential conductance $\mathsf{d}I\mathrm{/}\mathsf{d}V$ of metal and CO-terminated STM tips on the topological insulators ${\mathrm{B}\mathrm{i}}_2{\mathrm{S}\mathrm{e}}_3$, ${\mathrm{B}\mathrm{i}}_2{\mathrm{T}\mathrm{e}}_2\mathrm{S}\mathrm{e}$ and ${\mathrm{B}\mathrm{i}}_2{\mathrm{T}\mathrm{e}}_3$ and find excellent agreement with experiment. The calculation is an application of Chen’s derivative rule, whereby the Bloch functions are obtained from Wannier interpolated tight-binding Hamiltonians and maximally localized Wannier functions from first-principle DFT+$GW$ calculations. We observe signatures of the topological boundary modes, their hybridization with bulk bands, Van Hove singularities of the bulk bands and characterize the orbital character of these electronic modes using the high spatial resolution of STM and AFM. Bare DFT calculations are insufficient to explain the experimental data, which are instead accurately reproduced by many-body corrected $GW$ calculations.