We report the observation of optical second harmonic generation (SHG) in single-layer indium selenide (InSe). We measure a second harmonic signal of $>10^3$ $textrm{cts/s}$ under nonresonant excitation using a home-built confocal microscope and a standard pulsed pico-second laser. We demonstrate that polarization-resolved SHG serves as a fast, non-invasive tool to determine the crystal axes in single-layer InSe and to relate the sharp edges of the flake to the armchair and zigzag edges of the crystal structure. Our experiment determines these angles to an accuracy better than $pm$ $0.2^{circ}$. Treating the two-dimensional material as a nonlinear polarizable sheet, we determine a second-order nonlinear sheet polarizability $| chi_{textrm{sheet}}^{(2)}|=(17.9 pm 11.0)times 10^{-20}$ $textrm{m}^2 textrm{V}^{-1}$ for single-layer InSe, corresponding to an effective nonlinear susceptibility value of $| chi_textrm{eff}^{(2)}| approx (223 pm 138) times 10^{-12}$ $textrm{m} textrm{V}^{-1}$ accounting for the sheet thickness ($textrm{d} approx 0.8$ $textrm{nm}$). We demonstrate that the SHG technique can also be applied to encapsulated samples to probe their crystal orientations. The method is therefore suitable for creating high quality van der Waals heterostructures with control over the crystal directions.
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