Layer-Dependent Vibrational and Optical Properties of $\mathrm{M}\mathrm{o}0.58\mathrm{W}0.42{\mathrm{S}\mathrm{e}}_2$ Alloy

Semiconducting Mo${}_x$W${}_{1-x}$Se${}_2$ alloys provide a versatile platform for tailoring the optical properties of two-dimensional materials through both composition and layer thickness. Here, we systematically investigate mechanically exfoliated Mo${}_{0.58}$W${}_{0.42}$Se${}_2$ flakes ranging from monolayer (1L) to nine layers by combining Raman scattering (RS), photoluminescence (PL), reflectance contrast (RC) spectroscopy, and first-principles phonon calculations. Thirteen RS peaks are identified, including the low-frequency interlayer shear mode, whose thickness dependence is well described by a linear-chain model, yielding an interlayer force constant of $K_s=(2.996\pm 0.015)\times {10}^{19}$ N m${}^{-3}$. PL measurements reveal a crossover from the direct-bandgap 1L to indirect-bandgap multilayers. The thickness evolution of the indirect optical transition is quantitatively reproduced using a quantum-confinement model, yielding an out-of-plane reduced effective mass of ${\mu }_{\perp }=0.75m_0$. RC spectroscopy reveals four excitonic resonances. While the A and B excitons associated with the $K^{\pm }$ valleys remain nearly independent of layer thickness, the higher-energy C and D resonances originating from the band-nesting regions exhibit pronounced redshifts, reflecting substantial thickness-induced modifications of the electronic band structure. These results establish comprehensive spectroscopic fingerprints of flake thickness, interlayer coupling, and electronic structure in Mo${}_x$W${}_{1-x}$Se${}_2$ alloys and provide a reliable, non-destructive framework for their optical characterization.