  The structural polymorphism in transition metal dichalcogenides (TMDs)
provides exciting opportunities for developing advanced electronics. For
example, MoTe$_2$ crystallizes in the 2H semiconducting phase at ambient
temperature and pressure, but transitions into the 1T$^\prime$ semimetallic
phase at high temperatures. Alloying MoTe$_2$ with WTe$_2$ reduces the energy
barrier between these two phases, while also allowing access to the T$_d$ Weyl
semimetal phase. The MoWTe$_2$ alloy system is therefore promising for
developing phase change memory technology. However, achieving this goal
necessitates a detailed understanding of the phase composition in the
MoTe$_2$-WTe$_2$ system. We combine polarization-resolved Raman spectroscopy
with X-ray diffraction (XRD) and scanning transmission electron microscopy
(STEM) to study MoWTe$_2$ alloys over the full compositional range x from 0 to
1. We identify Raman and XRD signatures characteristic of the 2H, 1T$^\prime$,
and T$_d$ structural phases that agree with density-functional theory (DFT)
calculations, and use them to identify phase fields in the MoTe$_2$-WTe$_2$
system, including single-phase 2H, 1T$^\prime$, and T$_d$ regions, as well as a
two-phase 1T$^\prime$ + T$_d$ region. Disorder arising from compositional
fluctuations in MoWTe$_2$ alloys breaks inversion and translational symmetry,
leading to the activation of an infrared 1T$^\prime$-MoTe$_2$ mode and the
enhancement of a double-resonance Raman process in 2H-MoWTe$_2$ alloys.
Compositional fluctuations limit the phonon correlation length, which we
estimate by fitting the observed asymmetric Raman lineshapes with a phonon
confinement model. These observations reveal the important role of disorder in
MoWTe$_2$ alloys, clarify the structural phase boundaries, and provide a
foundation for future explorations of phase transitions and electronic
phenomena in this system.
