Theoretical and experimental study of second harmonic generation from the surface of the topological insulator Bi2Se3

We develop a theoretical model that describes the second harmonic generation of light from the surface of the topological insulator Bi${}_{2}$Se${}_{3}$ and experimentally demonstrate that the technique is sensitive to the surface electrons. By performing a crystal symmetry analysis of Bi${}_{2}$Se${}_{3}$(111) we determine the nonlinear electric susceptibility tensor elements that give rise to second harmonic generation. Using these results, we present a phenomenological model that shows that the relative magnitudes of these tensor elements can be determined by measuring the polarization and intensity of the radiated second harmonic light as a function of the in-plane crystal orientation and incident laser polarization. We describe optical techniques capable of isolating second harmonic light and, using these techniques, we measure the first-order linear optical and second-order nonlinear optical responses as a function of crystal orientation and laser polarization on bulk single crystals of Bi${}_{2}$Se${}_{3}$(111). The experimental results are consistent with our theoretical description. By comparing the data to our theoretical model we determine that a portion of the measured second harmonic light originates from the accumulation region of Bi${}_{2}$Se${}_{3}$(111), which we confirm by performing surface doping-dependent studies. Our results show that second harmonic generation is a promising tool for spectroscopic studies of topological surfaces and buried interfaces.