  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$_2$.
