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

physical review materials

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