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We use ultrafast laser pulses to experimentally demonstrate that thesecond-order optical response of bulk single crystals of the topologicalinsulator Bi$_2$Se$_3$ is sensitive to its surface electrons. By performingsurface doping dependence measurements as a function of photon polarization andsample orientation we show that second harmonic generation can simultaneouslyprobe both the surface crystalline structure and the surface charge ofBi$_2$Se$_3$. Furthermore, we find that second harmonic generation usingcircularly polarized photons reveals the time-reversal symmetry properties ofthe system and is surprisingly robust against surface charging, which makes ita promising tool for spectroscopic studies of topological surfaces and buriedinterfaces.

Nonlinear optical probe of tunable surface electrons on a topological insulator