Thickness determination of anisotropic van der Waals crystals by raman spectroscopy: the case of black phosphorus.

The large foreseeable use two-dimensional materials in nanotechnology consequently demands precise methods for their thickness measurements. Usually, having a quick and easy methodology is a key requisite for the inspection of the large number of flakes produced by exfoliation methods. An effective option in this respect relies on the measurement of the intensity of Raman spectra, which can be used even when the flakes are encapsulated by a transparent protective layer. However, when using this methodology, special attention should be paid to the crystalline anisotropy of the examined material. Specifically, for the case of black phosphorus flakes, the absolute experimental determination of the thickness is rather difficult because the material is characterized by a low symmetry and also because the Raman tensors are complex quantities. In this work, we exploited Raman spectroscopy to measure the thickness of black phosphorous flakes using silicon as reference material for intensity calibrations. We found out that we can determine the thickness of a flake above 5 nm with an accuracy of about 20%. We tested the reproducibility of the method on two different setups, finding similar results. The method can be applied also to other van der Waals materials with a Raman band characterized by the same Raman tensor.