Probing the Penetration Depth of Topological Surface States by Magnetic Impurity Scattering in V-doped Sb$_2$Te$_3$
Topological insulators host Dirac surface states (SS) protected by time-reversal symmetry. Inter-surface hybridization can gap the SS and give rise to the quantum spin Hall effect in films that are sufficiently thin compared to the SS penetration depth. However, quantifying the SS penetration depth typically requires painstaking synthesis of multiple films with varying thickness. Here we introduce a direct method to probe the SS penetration depth in bulk crystals, by studying the interplay between SS and magnetic impurities in \SVT. Using scanning tunneling microscopy and spectroscopy, we find that even sparse magnetic impurities ($\lesssim0.25\%$ vanadium) can gap the Dirac SS. However, a single V impurity induces only localized states, and does not form an impurity band, so the gapped Dirac dispersion is preserved away from the impurity. In high magnetic fields, we observe an energy shift of the $0^\text{th}$ Landau level and a suppression of quasiparticle lifetime at the Dirac point, indicating \newtext{magnetic} scattering of the SS. Crucially, by employing V impurities at different depths as precise scattering probes, we reveal the SS penetration depth on the sub-nanometer scale in a bulk crystal.