Quantitatively Predicting Angle-Resolved Polarized Raman Intensity of Anisotropic Layered Materials

Angle-resolved polarized Raman (ARPR) spectroscopy provides insights into optical anisotropy and symmetry-related electron-photon/electron-phonon couplings of anisotropic layered materials (ALMs). However, since their discovery over ten years ago, ARPR responses in ALM flakes has exhibited a puzzling dependence on flake thickness, excitation wavelength, and dielectric environment, complicating their understanding and prediction. By taking black phosphorus (BP) (⩾20 nm) flakes and four-layer Td-WTe₂ as examples, this study introduces intrinsic Raman tensors (R^int) and proposes strategies to predict the ARPR intensity profiles of thick and atomically-thin ALM flakes by considering birefringence, linear dichroism and multilayer interference inside multilayered structures with experimentally determined complex refractive indexes along in-plane axes and complex tensor elements of R^int for the corresponding phonon modes. The tensor elements of effective Raman tensors (R^eff), which are directly linked to the polarization vectors of incident and scattered light outside the ALM surface, are derived to quantitatively predict ARPR intensity for these ALM flakes, showing intricate dependence on ALM thickness, dielectric substrates, and excitation wavelengths. This framework can be extended to other ALM flakes from atomically-thin layers to bulk limit, facilitating comprehensive prediction of their ARPR intensity regardless of layer-dependent electronic properties.