Efficient method for calculating Raman spectra of solids with impurities and alloys and its application to two-dimensional transition metal dichalcogenides

Raman spectroscopy is a widely used, powerful, and nondestructive tool for studying the vibrational properties of bulk and low-dimensional materials. Raman spectra can be simulated using first-principles methods, but due to the high computational cost calculations are usually limited only to fairly small unit cells, which makes it difficult to carry out simulations for alloys and defects. Here, we develop an efficient method for simulating Raman spectra of alloys, benchmark it against full density-functional theory calculations, and apply it to several alloys of two-dimensional transition metal dichalcogenides. In this method, the Raman tensor for the supercell mode is constructed by summing up the Raman tensors of the pristine system weighted by the projections of the supercell vibrational modes to those of the pristine system. This approach is not limited to 2D materials and should be applicable to any crystalline solids with defects and impurities. To efficiently evaluate vibrational modes of very large supercells, we adopt mass approximation, although it is limited to chemically and structurally similar atomic substitutions. To benchmark our method, we first apply it to Mo$_x$W$_{(1-x)}$S$_2$ monolayer in the H-phase, where several experimental reports are available for comparison. Second, we consider Mo$_x$W$_{(1-x)}$Te$_2$ in the T'-phase, which has been proposed to be 2D topological insulator, but where experimental results for the monolayer alloy are still missing. We show that the projection scheme also provides a powerful tool for analyzing the origin of the alloy Raman-active modes in terms of the parent system eigenmodes. Finally, we examine the trends in characteristic Raman signatures for dilute concentrations of impurities in MoS$_2$.