The structural phases and vibrational properties of Mo_1−xW_xTe₂ alloys

Transition metal dichalcogenides (TMDs) are van der Waals (vdW) compounds that follow the general formula of MX₂, where M is a transition metal from Groups IVB–VIB, and X is a Group VIIA chalcogen, such as S, Se, and Te. This chemical versatility leads to unique electronic properties, such as semiconducting behavior [1], superconductivity [2–4], and topological electronic states [5–8]. Furthermore, two-dimensional (2D) TMD monolayers can be vertically stacked without the need for lattice matching, which allows these dissimilar electronic phases to be combined in a single heterostructure [9].

The chemical versatility intrinsic to TMDs and the novel interactions achieved through vdW stacking are complimented further by the structural polymorphism of TMDs. A prototypical example is molybdenum ditelluride (MoTe₂), which can be grown in a semiconducting 2H phase (space group P6₃/mmc) or a semimetallic 1T' phase (space group P2₁/m) [10–12]. The hexagonal 2H phase, characterized by a trigonal prismatic coordination, is thermodynamically stable under ambient conditions, while the monoclinic 1T' phase is stable above 900 ^∘C. The 1T' phase can be stabilized at room temperature by rapid cooling [13], control of the tellurization rate of Mo films [10], or choosing appropriate precursors for chemical vapor deposition [14, 15]. When cooled below  ∼250 K, 1T'-MoTe₂ transitions into an orthorhombic T_d phase (space group Pmn2₁) with broken inversion symmetry as evidenced by electrical, structural, and optical measurements [11, 16–18].

Interest in MoTe₂ has surged due to the unique electronic properties of its structural phases as well as the possibility of engineering controlled transitions between these phases. For instance, type-II Weyl semimetal states were theoretically predicted and experimentally observed in both T_d-MoTe₂ and T_d-WTe₂ [6, 7, 19–23]. The broken inversion symmetry of the T_d phase is a necessary condition for the type-II Weyl state [6, 19, 20] and allows for fundamental studies of interesting topological physics. However, efforts to directly observe the Weyl state using angle-resolved photoemission spectroscopy are frustrated by the presence of overlapping band-crossings and insufficient experimental resolution [20]. A more practical application driving investigations of MoTe₂ is the development of atomically thin phase change memory. MoTe₂ has a small energy difference between the 2H and 1T' phases, making the prospect of engineering on-demand transitions with low power consumption realistic [24–28]. Unfortunately, reversible and rapid phase changes in MoTe₂ have yet to be demonstrated. This may suggest that the energy difference between the 2H and 1T' phases must be reduced further in order to successfully perform phase change operations.

The limitations of MoTe₂ highlighted above can be addressed by alloy engineering. Substitutional doping of Mo with W atoms results in $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ alloys which have properties advantageous for both fundamental investigations and practical applications. $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ alloys have been theoretically predicted [20] and experimentally confirmed [29] to be type-II Weyl semimetals. Importantly, the separation of the Weyl nodes in the alloys can be tuned with composition [20], which facilitates the observation of topological electronic states. Additionally, the ground state energy difference between the semiconducting 2H and semimetallic 1T' or T_d phases in $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ alloys can also be tuned with composition [25, 27], thereby reducing the energy required to trigger a semiconductor-semimetal phase transformation. The desirable combination of tunable phase transitions with the low-dimensionality of TMDs makes $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ highly promising for phase change memory applications.

Application of the $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ alloy system in the 2D limit necessitates an understanding of the compositional dependence of phase transformations and the impact of disorder upon the bulk material properties. The literature on this subject is very limited. In a pioneering work on $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ polycrystalline powders, Revolinsky and Beerntsen found that the alloys crystallize in the 2H phase for $x\leqslant 0.15$, the T_d phase for $x\geqslant 0.65$, and a two-phase region of 2H  +  T_d in between [30]. On the contrary, Champion detected a two-phase 2H  +  T_d region only for $x=0.25$ composition, whereas a higher (lower) W content resulted in $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ powders in a T_d (2H) phase, respectively [31]. However, no detailed structural studies were reported in these two papers to shed light on the co-existence of the 2H and T_d phases, especially considering a noticeable difference in their symmetry. Recently, Rhodes et al reported a simplified phase diagram without two-phase regions for single-crystalline $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ alloys grown by the chemical vapor transport method (CVT) with iodine or TeCl₄ as the transport agent [32]. Finally, Lv et al and Yan et al suggested a more complicated phase diagram with a mixed 1T'  +  T_d region that exists for compositions $0.5<x<0.7$ in $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ alloys also grown using CVT, but with Br₂ as the transport agent [33, 34]. This significant disagreement between the studies about phase boundaries between the phases in the MoTe₂-WTe₂ system, combined with the absence of thorough studies of compositional disorder on optical properties, calls for a fresh look at this alloy system.

Here, we provide a comprehensive examination of the $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ alloy system by combining x-ray diffraction (XRD), scanning transmission electron microscopy (STEM), density-functional theory (DFT), and polarization-resolved Raman spectroscopy to explore the properties of the 1T', T_d, and 2H structural phases in bulk $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ crystals grown by iodine-assisted CVT. Measurement of bulk flakes has the added advantage of minimizing the well-known degrading oxidation effects of Te-based TMDs [35]. XRD and STEM measurements indicate that the $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ alloys with the 1T' crystal structure are stable at elevated temperatures with W content $x\leqslant 0.04$, while alloys with the T_d structure are stable for $x\geqslant 0.63$. The alloys with intermediate compositions $0.04<x<0.63$ form a two-phase, 1T'  +  T_d, mixture. Polarized Raman measurements offer further insight into the transition from 1T' to the two-phase, 1T'  +  T_d, field and ultimately to the T_d single-phase region as a function of x. We use Raman tensor analysis to assign the phonon mode symmetry for all compositions and find that the tensor elements must be complex in order to capture the polarization dependence. This observation is consistent with prior studies of MoTe₂ [18, 36] as well as studies of other layered TMD materials [37]. The Raman peaks for certain phonon modes show particular sensitivity to x and lattice symmetry. For example, the Raman peak at 128 $\text{c}{{\text{m}}^{-1}}$ for the 1T'-Mo${{}_{1-\text{x}}}$W_xTe₂ alloys broadens at $\text{x}=0.09$ and splits into a doublet for compositions $x\geqslant 0.29$, which implies a loss of inversion symmetry [18] due to the substitution of Mo by W. We also observe the activation of a new Raman mode at 178 $\text{c}{{\text{m}}^{-1}}$ that is unique to $0.02<\text{x}<1$ alloys. Based on our DFT calculations, we assign this feature as a disorder-activated infrared mode in MoTe₂. Furthermore, the separation between the two modes near 260 $\text{c}{{\text{m}}^{-1}}$ in MoTe₂ is highly composition-dependent and can be used to infer x. In $\text{2H-M}{{\text{o}}_{1-\text{x}}}$W_xTe₂, we observe minor changes in the ${{A}_{1g}}$ and $E_{2g}^{1}$ mode frequencies, linewidths, and relative intensities. We also identify a new Raman mode at 202 $\text{c}{{\text{m}}^{-1}}$ that originates from a double-resonance Raman process [38] and is apparently enhanced by alloy disorder. The comprehensive structural and spectroscopic data assembled here provide an important roadmap for the future study and application of 2D $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ alloys.

Polycrystalline $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ alloys ($x=0...1$) were prepared by reacting stoichiometric amounts of molybdenum (99.999%), tungsten (99.9%), and tellurium (99.9%) powders at 750${{~}^{\circ}}$C in vacuum-sealed quartz ampoules. $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ crystals were then grown by the CVT method using approximately 1.5 g of poly-Mo${{}_{1-\text{x}}}$W_xTe₂ charge and a small amount of iodine (99.8%, 5 mg $\text{c}{{\text{m}}^{-1}}$³) sealed in evacuated quartz ampoules. It was found that the temperature required for high-yield synthesis of $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ crystals is lower for higher x. Therefore, the growth temperatures used in this study were 1000${{~}^{\circ}}$C for $x\leqslant 0.12$, 950${{~}^{\circ}}$C for $0.12<x\leqslant 0.63$, and 900${{~}^{\circ}}$C for $x\geqslant 0.71$. The ampoules were ice-water quenched after 7 d of growth. To study phase transformation in $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂, as-grown crystals were vacuum-sealed in small ampoules (internal volume $\approx 1$ cm³) and annealed at 750${{~}^{\circ}}$C for 72 h followed by cooling to room temperature at a rate of 10 ${{~}^{\circ}}$C/hr.

Chemical compositions with an accuracy of 0.01 were determined by energy-dispersive x-ray spectroscopy (EDS) using a JEOL JSM-7100F field emission scanning electron microscope (FESEM) equipped with an Oxford Instruments X-Max 80 EDS detector⁶. We examined the θ–$2\theta$ XRD patterns derived from a Norelco Philips Diffractometer with the Bragg-Brentano geometry. Lattice parameters were refined using the MDI-JADE 6.5 software package. For the powder XRD study, $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ crystals were finely ground using an agate mortar. An Aberration-Corrected High Angle Annular Dark Field Scanning Transmission Electron Microscopy (Cs-corrected HAADF-STEM) FEI Titan 80-300 TEM/STEM operating at 300 kV was employed for the characterization of $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ samples. The flakes were crushed in ethanol and a drop of solution was deposited onto an amorphous Carbon (a-C) coated TEM grid (Agar Inc.). HAADF-STEM images were collected at a camera length of 100 mm corresponding to inner and outer collection angles of 70.6 and 399.5 mrad respectively.

For Raman measurements, the as-grown $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ crystals were mechanically exfoliated onto Si/SiO₂ substrates. Polarization-dependent Raman measurements were performed on bulk flakes in a back-scattering geometry at room temperature in atmosphere using a linearly-polarized 532 nm continuous wave laser. The polarization of the excitation beam was controlled with a motorized achromatic half-wave plate and was focused onto the sample using a 0.75 NA microscope objective. The back-scattered Raman emission was collected by the same objective, and then sent through a motorized analyzer and a long-pass filter. The excitation polarization and collection analyzer were oriented in both co-polarized ($\parallel$) and cross-polarized ($\bot$) configurations, and then rotated together while the sample remained fixed. The filtered light was focused into a multimode fiber to scramble polarization and then directed to a spectrometer for analysis. Raman peaks were fit to Lorentzian functions to extract mode frequencies, linewidths, and amplitudes. For angle-dependent Raman maps, measurements were normalized by the feature with the greatest intensity, the  ∼163 $\text{c}{{\text{m}}^{-1}}$ peak. The angle-dependent peak intensities were fit using the Raman tensors to assign peak symmetries.

All simulations were based on density-functional theory (DFT) using the projector-augmented wave method as implemented in the plane-wave code VASP [39]. The simulations were performed using the vdW-DF-optB88 exchange-correlation functional [40], which provides an excellent description of the lattice constants of bulk 1T'-MoTe₂, T_d-MoTe₂, and T_d-WTe₂. An energy cutoff of 600 eV and k-point mesh of 10  ×  18  ×  5 for the 1  ×  1  ×  1 unit cells of bulk 1T'-MoTe₂, T_d-MoTe₂, and T_d-WTe₂ resulted in an accuracy of the total energies of 1 meV/unit cell. The 5s²5p⁴ and 4d⁵5s electrons were considered as the valence electrons for Te and Mo, respectively. Including the semicore 4s²p⁶ electrons for Mo had a negligible effect on the results, as, for instance, the lattice parameters of bulk 1T'-MoTe₂ changed by less than 10%. $\Gamma$-point phonon frequencies of bulk 1T'-MoTe₂, T_d-MoTe₂, and T_d-WTe₂ were estimated from density-functional perturbation theory simulations of the $1\times 1\times 1$ unit cells of respective materials. Irreducible representations of normal modes were obtained from the PHONOPY program [41] and the Bilbao Crystallographic Server [42]. The phonon dispersion of T_d-WTe₂ in the entire Brillouin zone was estimated by computing normal mode frequencies on a uniform three-dimensional mesh of $51\times 51\times 51$ $\vec{q}$-points between (0, 0, 0) and ($2\pi /a$, $2\pi /b$, $2\pi /c$) (figure S1 in the supporting information (stacks.iop.org/TDM/4/045008/mmedia)). The phonon dispersion of T_d-WTe₂ in the entire Brillouin zone was computed using the finite difference method on the 106 atom 3  ×  3  ×  1 supercell.

Figure 1(a) summarizes heat-treatment schedules, compositions, and crystal phases of $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ ($x=0...1$) samples examined in this study. The high-temperature phases of the alloys were preserved by quenching of the growth ampoules in an ice-water bath. This process is known to prevent reversal of the 1T' phase to the 2H phase, which is thermodynamically stable in MoTe₂ under ambient conditions [43]. Notably, XRD θ-2θ scans from the as-grown, un-milled $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ flakes produce only 00l-type reflections and miss all asymmetric reflections, thus limiting the ability to reliably determine phase composition in the alloys. Therefore, we collected the scans from finely ground flakes to register all possible hkl reflections to distinguish 1T', T_d, and 2H phases and their mixtures. For example, R. Clarke et al have established that a 1T'  →  T_d transition in MoTe₂ and the two-phase region can be observed by specifically monitoring h0l reflections as a function of temperature, e.g. $\bar{1}$ 0 12 and 1 0 12 reflections of 1T'-MoTe₂ coalesce into a single 1 0 12 reflection of the low-temperature T_d phase [17]. A similar approach was applied to construct a phase diagram of $\text{M}{{\text{o}}_{1-\text{x}}}$Nb_xTe₂ alloys that undergo an orthorhombic to monoclinic phase transition with increasing x [44].

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The θ–2θ scans of three representative $\text{M}{{\text{o}}_{1-\text{x}}}$W_xTe₂ samples with $x=0.04$, 0.33, and 0.71, produced by milling as-grown flakes in an agate mortar, are shown in figure 1(b). Figures 1(c) and (d) show enlarged portions of the scans around 2θ angles of 35° and 48°, respectively, which illustrate the distinct changes in the lineshape with increasing x. The scans for $x=0.04$ and 0.71 were unambiguously assigned to the 1T' and T_d phase, respectively. The $x=0.33$ scan can only be fitted by combining reflections from both 1T' and T_d phases, which indicates a two-phase coexistence. Calculated lattice parameters, Bragg reflection angles for the three samples, and a detailed explanation of the protocol used for identifying structural phases are presented in tables S1, S2, and supporting note 1 in the supporting information. By analysis of the powder XRD scans, we established that the quenched Mo${{}_{1-\text{x}}}$W_xTe₂ alloy samples synthesized in this study are in the monoclinic 1T' phase for $x\leqslant 0.04$, the orthorhombic T_d phase for $x\geqslant 0.63$, and in the 1T'  +  T_d two-phase state for the compositions x between 0.04 and 0.63.

We further verified these observations by performing HAADF-STEM measurements of 1T'-$\text{M}{{\text{o}}_{0.96}}$${{\text{W}}_{0.04}}$Te₂ and T_d-$\text{M}{{\text{o}}_{0.29}}$${{\text{W}}_{0.71}}$Te₂ crystals, shown in figures 1(e) and (f) with the overlapped atomic models and their corresponding fast fourier transforms (FFT) in the insets. Both the 1T' and the T_d phases exhibit a 'buckled' structure with visible shifts for Te atoms and a zig-zag pattern for Mo/W atoms. The presence of the two phases was observed on a sample with $x=0.33$, proving the 1T'  +  T_d coexistence in Mo${{}_{1-\text{x}}}$W_xTe₂ alloys for $0.04<x<0.63$, although we were not able to map the spatial distribution of 1T' and T_d phases.

In order to study temperature-induced phase transformations in Mo${{}_{1-\text{x}}}$W_xTe₂ alloys, the samples were annealed in vacuum-sealed ampoules at 750 ${{}^{\circ}}\text{C}$ for 72 h followed by slow cooling to room temperature (squares in figure 1(a)). We found that alloys with $x\leqslant 0.09\pm 0.01$ could be converted to the hexagonal 2H phase, as schematically depicted in figure 1(a) by the pink-colored area. HAADF-STEM and XRD data of the $\text{M}{{\text{o}}_{0.91}}$${{\text{W}}_{0.09}}$Te₂ sample converted from 1T' into 2H phase are provided in figure S2 of the supporting information. The vacuum annealing did not change the crystal structures of the alloys with larger x. Thus, an upper limit for Mo${{}_{1-\text{x}}}$W_xTe₂ alloys to experience a reversible phase transformation between semiconducting 2H and metallic 1T' phases is $x\approx 0.09$. Approximately the same boundary between metallic and semiconducting phases was recently reported for Mo${{}_{1-\text{x}}}$W_xTe₂ alloys [32, 33]. Additional studies are required to determine compositional dependencies of the phase transition temperatures in Mo${{}_{1-\text{x}}}$W_xTe₂ alloys, which are beyond the scope of this paper.

We now investigate the impact of composition, disorder, and crystal structure on the Raman-active phonon modes of Mo${{}_{1-\text{x}}}$W_xTe₂ alloys. We first examine the 1T'  →  T_d phase transition in Mo${{}_{1-\text{x}}}$W_xTe₂, and then explore the impact of alloy potential fluctuations on the 2H phase. In order to minimize rapid degradation of the detected Raman signal due to surface oxidation of Te-based TMD layers in air [35], we focus exclusively on bulk Mo${{}_{1-\text{x}}}$W_xTe₂ flakes that are mechanically exfoliated onto Si substrates with a 285 nm SiO₂ layer. Future measurements will assess layer-dependent properties, but will require exfoliation and encapsulation in an inert environment. The home-built confocal Raman microscope used in these measurements is oriented in a backscattering geometry and operated in two polarization configurations: one with the excitation polarization and analyzer co-polarized ($\parallel$) and the other with them cross-polarized ($\bot$). The excitation/analyzer orientation is fixed and the two are rotated together relative to the crystal lattice. All measurements are performed at room temperature in atmosphere on bulk flakes. By acquiring a series of these spectra at different orientations, we assemble polarized Raman maps that provide a concise visualization of the angle-dependent Raman spectra as a function of x, which are shown for all 1T' and T_d samples in figure S3 of the supporting information. Due to our experimental geometry, only A_g (A₁) and B_g (A₂) symmetry modes are accessible for the 1T' (T_d) crystal structure. These modes have distinct dependencies on laser-analyzer orientation and the orientation relative to the crystal axes [36, 45]. In short, the polarized Raman signal is given by $I\left(\theta \right)=\,|{{\hat{e}}_{s}}\cdot \overset{\scriptscriptstyle\leftrightarrow}{R}{{\hat{e}}_{i}}{{|}^{2}}$, where ${{\hat{e}}_{i}}$ and ${{\hat{e}}_{s}}$ are the incident and scattered fields and $\overset{\scriptscriptstyle\leftrightarrow}{R}$ is the Raman tensor. In bulk MoTe₂ and WTe₂, $\overset{\scriptscriptstyle\leftrightarrow}{R}$ is complex-valued for all modes, suggesting that optical absorption is significant [36, 46, 47]. We summarize the results of the Raman tensor analysis fitting to the Raman peaks of 1T'-MoTe₂ and WTe₂ in figures S4, S5, and supporting note 2 of the supporting information. In table 1 we summarize the experimentally-determined mode assignments for 1T'-MoTe₂ and T_d-WTe₂, as well as the results of our DFT calculations of the 1T'-MoTe₂, T_d-MoTe₂, and T_d-WTe₂. In the 1T' phase, the A_g and B_g modes are Raman-active while the A_u and B_u modes are only infrared-active. Interestingly, all modes are Raman active for the T_d phase.

Table 1. Theoretically predicted and experimentally measured wavenumbers, ${{\omega}_{\text{calc}}}$ and ${{\omega}_{\text{exp}}}$, respectively, alongside their associated symmetries.

1T'-MoTe2			Td-MoTe2		Td-WTe2
${{\omega}_{\text{calc}}}$ ($\text{c}{{\text{m}}^{-1}}$)	${{\omega}_{\text{exp}}}$ ($\text{c}{{\text{m}}^{-1}}$)	Symmetry	${{\omega}_{\text{calc}}}$ ($\text{c}{{\text{m}}^{-1}}$)	Symmetry	${{\omega}_{\text{calc}}}$ ($\text{c}{{\text{m}}^{-1}}$)	${{\omega}_{\text{calc}}}$ ($\text{c}{{\text{m}}^{-1}}$)	Symmetry
76.81	78	Ag	76.96	A1	79.0	80	A1
85.56		Ag	85.74	B1	84.6	91	A2
88.20		Bg	88.09	B2	84.7		B2
90.88		Bg	90.68	A2	86.2		B1
104.90	107	Bg	104.88	A2	107.7		B2
105.61	111	Bg	105.52	B2	107.7		B2
108.37		Au	108.37	A2	110.9	111	A2
108.67		Ag	108.71	A1	111.5	117	A1
110.80		Au	110.76	B2	112.6		B2
113.60	111	Ag	113.61	B1	115.3		B1
115.38		Bu	115.53	B1	120.6		B1
123.82		Bu	123.23	A1	127.2	132	A1
125.52		Ag	126.23	A1	127.4		B1
128.25	128	Ag	128.07	B1	128.5	134	A1
129.60		Bu	129.92	B1	130.4	137	A1
134.80		Bu	134.84	A1	131.3		B1
155.54		Ag	155.59	B1	146.3		A2
159.24	163	Ag	159.35	A1	146.6		A2
176.52		Au	176.64	A2	152.4		B1
176.97		Au	176.87	B2	153.7		A2
187.99		Bg	188.27	A2	155.4		A1
189.20	192	Bg	189.29	B2	156.2		B2
192.13		Bu	192.11	B1	165.6		B1
192.33		Bu	192.25	A1	166.2	164	A1
247.13		Ag	247.24	A1	201.2		A1
249.03		Ag	248.99	B1	201.5		B1
251.58	251	Ag	251.48	A1	204.6	212	A1
254.14	260	Ag	253.95	B1	206.1		B1
265.96		Bu	266.14	B1	227.9		A1
267.37		Bu	267.42	A1	228.4		B1

The polarized Raman maps presented in figure S3 of the supporting information are instructional for an overview of the composition-induced evolution of the vibrational modes from the 1T' phase to the T_d phase. However, the sensitivity of the Raman spectrum to the orientation of the excitation and analyzer complicates further interpretation of the data in this form. To eliminate this orientation-dependence, we sum the parallel and perpendicular data over all angular orientations (figures 2(e) and (f)). The composition-dependent Raman spectra of the alloys show several important features. Most modes are present in all compositions and exhibit only small frequency shifts due to the similarity between the 1T' and T_d lattices (figures 2(a) and (d)). However, certain Raman modes (identified with boxes in figures 2(e) and (f)) exhibit unique behavior that is dependent on structural symmetry and composition x in the alloys.

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We direct our attention first to the Raman peak observed at 128 $\text{c}{{\text{m}}^{-1}}$ (box i of figures 2(e) and 3(a)). For $x=0$, this feature corresponds to an A_g symmetry mode in MoTe₂, as demonstrated by our polarization-resolved measurements, DFT calculations [36], and other literature observations [10, 18, 48]. This mode is a single peak for compositions $x\leqslant 0.04$, which is consistent with the inversion symmetric 1T' phase. For compositions $x=0.09$ and 0.12, however, the 128 $\text{c}{{\text{m}}^{-1}}$ mode broadens and is best fit by a pair of Lorentzian functions. Finally, for $x\geqslant 0.29$, the 128 $\text{c}{{\text{m}}^{-1}}$ mode splits into two well-resolved peaks. The separation between these two peaks is presented versus composition in figure 3(c) and illustrates the appearance and evolution of the doublet, which persists into the $x=1$ (pure WTe₂) case but with a smaller peak separation. Temperature-dependent electrical and XRD measurements have previously shown that MoTe₂ undergoes a temperature-induced phase transition from the 1T' to T_d crystal structure when cooled below 250 K [16, 17]. Recent temperature-dependent Raman measurements in [18] have also demonstrated that the A_g mode at 128 $\text{c}{{\text{m}}^{-1}}$ in MoTe₂ splits into a doublet with A₁ mode symmetry due to inversion-symmetry breaking upon transitioning into the T_d phase at low temperatures. Our XRD measurements identify the $0.04<x<0.63$ region as two-phase, and therefore we cannot interpret the appearance of the doublet as signifying a phase transition from 1T' to T_d. Instead, we attribute the doublet to the breakdown of inversion symmetry in the 1T'-Mo${{}_{1-\text{x}}}$W_xTe₂ alloys, which originates not from a 1T'  →  T_d phase transition, but instead from the random substitution of Mo atoms with W atoms. Alloying therefore provides a means of destroying inversion symmetry without eliminating the 1T' phase. From these observations, it is apparent that $x\geqslant 0.29$ W concentration is sufficient to drive a breakdown of inversion symmetry and suggests that Weyl physics may be observable even in this two-phase regime [6, 19].

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We find that other modes also display sensitivity to compositional disorder and the substitution of Mo for W atoms. Box ii of figure 2(e) isolates MoTe₂ and WTe₂ Raman modes that evolve with changing composition, as well as a mode at 178 $\text{c}{{\text{m}}^{-1}}$ that is not present in pure 1T'-MoTe₂ or T_d-WTe₂. We summarize the composition-dependent relative intensities for these three peaks in figure 3(d). The 'MoTe₂ peak' refers to the feature at 192 $\text{c}{{\text{m}}^{-1}}$ (black squares) that is present only in Mo-rich compositions (small x) and is assigned as a B_g symmetry mode in MoTe₂. The 'WTe₂ peak' is the large 212 $\text{c}{{\text{m}}^{-1}}$ feature (red circles) present only in W-rich compositions (large x) and is assigned as an A₁ symmetry mode in WTe₂. Finally, the 'disorder peak' refers to the 178 $\text{c}{{\text{m}}^{-1}}$ mode unique to the alloys. The polarization dependence of the disorder peak in the $x=0.29$ composition can be seen in figure S6 of the supporting information. The MoTe₂ and WTe₂ peaks appear to faithfully track the removal and addition of each atomic species, while the disorder mode appears at $x=0.02$, peaks at $x=0.33$, and disappears at $x=1$. The observed frequency agrees with an infrared-active, but Raman-forbidden, A_u mode at 177 $\text{c}{{\text{m}}^{-1}}$ predicted by our DFT calculations (table 1). We therefore suggest that the disorder mode originates from an infrared mode that is activated by the loss of translation symmetry in the lattice. The combined effects of lattice disorder and reduced Mo content at large x values drives the mode to reach its maximum intensity at $x=0.33$, which also is the point where the ratio of the normalized intensities of the MoTe₂ peak to the WTe₂ peak approach unity. We note that similar activations of infrared modes by alloy disorder have been previously observed, particularly in Ga_xAl_1-xAs [49].

Given the non-destructive nature and wide-spread use of Raman spectroscopy, it is desirable to determine alloy composition using a Raman-based method. The MoTe₂ Raman modes present near 260 $\text{c}{{\text{m}}^{-1}}$ (box iii, figure 2(f)) provide a potential measure of the alloy composition, which we demonstrate in figures 3(e) and (f). We observe a pair of broad Raman modes near 260 $\text{c}{{\text{m}}^{-1}}$ in MoTe₂ that are assigned as A_g modes (box iii of figures 2(f) and 3(e)) and have been seen in prior studies [36, 48]. By fitting these two peaks in each spectrum to Lorentzians, we can track the peak separation with composition. We find that the separation between these two features increases with increasing x, and that we can use it to estimate global W content in a Mo${{}_{1-\text{x}}}$W_xTe₂ crystal (figure 3(f)). Our results indicate that this method will be effective for $x>0.09$, and is therefore most appropriate for higher W concentrations.

Finally, we examine the primary WTe₂ peak at 212 $\text{c}{{\text{m}}^{-1}}$ which is broad and asymmetric upon its appearance at $\text{x}=0.29$, but sharpens as $x\to 1$ (figure 3(b)). We magnify this feature in figure 4(a) for select compositions. The asymmetric lineshape of the 212 $\text{c}{{\text{m}}^{-1}}$ peak provides valuable information regarding the incorporation of W into the Mo${{}_{1-x}}$W${{}_{x}}$Te₂ lattice. We find that the asymmetry of this feature and its evolution with x can be well understood in the context of the phonon confinement model, also referred to as the spatial correlation model [49, 50]. The phonon confinement model accounts for relaxation of the $\vec{q}=0$ Raman selection rule by multiplying the Lorentzian function, used to represent standard Raman peaks in a pure crystal, with a Gaussian function of the form $\exp \left(-{{q}^{2}}L_{C}^{2}/4\right)$. Thus, the intensity I of Raman peaks in the phonon confinement model is given by [49]

$\begin{eqnarray}&&I\left(\omega \right)\propto {{{\int}^{}}_{\text{BZ}}}\exp \left(\frac{-{{q}^{2}}L_{C}^{2}}{4}\right)\frac{{{\text{d}}^{3}}q}{{{\left[\omega -\omega (q)\right]}^{2}}~+~{{\left({{\Gamma}_{0}}/2\right)}^{2}}}~,\end{eqnarray} \tag{ 1 }$

where $\vec{q}$ is in units of $\left(2\pi /a,2\pi /b,2\pi /c\right)$, $a=6.3109~{\mathring{\text{A}}}$, $b=3.5323~{\mathring{\text{A}}}$, and $c=14.4192~{\mathring{\text{A}}}$ are the DFT-relaxed lattice parameters of WTe₂, ${{\Gamma}_{0}}=3.80$ $\text{c}{{\text{m}}^{-1}}$ is the full width at half maximum of the W peak for composition $x=1$, $\omega (q)$ is the dispersion relation which we calculate from DFT and shift to match the experimental value of $\omega (0)$ (figure S1 of the supporting information), and L_C is the phonon correlation length. In a pure crystal, L_C is infinite due to the translational symmetry of the lattice and results in plane wave eigenstates. The Gaussian factor in equation (1), in this case, is zero for all $\vec{q}$ except the $\Gamma$ point, and therefore the $\vec{q}=0$ Raman selection rule is preserved. However, Mo${{}_{1-\text{x}}}$W_xTe₂ alloys exhibit potential fluctuations due to the substitutional doping on the transition metal sublattice. The random positioning of the dopant atoms destroys translational symmetry in the crystal, thereby yielding a finite L_C and relaxing the $\vec{q}=0$ Raman selection rule. We fit the 212 $\text{c}{{\text{m}}^{-1}}$ W peak for $x\geqslant 0.29$ in background-subtracted Raman spectra with this model (red lines in figure 4(a)) using experimentally-derived parameters and the DFT calculated phonon dispersion. The extracted phonon correlation length L_C is plotted versus x in figure 4(b), and is found to increase rapidly with x.

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We now comment on the Raman spectra of 2H-Mo${{}_{1-\text{x}}}$W_xTe₂ ($\text{x}=0\to 0.09$), which are shown in figure 5(a). The ${{A}_{1g}}$ (173 $\text{c}{{\text{m}}^{-1}}$), $E_{2g}^{1}$ (234 $\text{c}{{\text{m}}^{-1}}$), and $B_{2g}^{1}$ (289 $\text{c}{{\text{m}}^{-1}}$) modes are visible in all compounds (figure 5(a)) and exhibit only small changes. The shifts in mode frequency and linewidth for the ${{A}_{1g}}$ and $E_{2g}^{1}$ modes are summarized in figures 5(b) and (c). For $\text{x}=0.09$, we find that the ${{A}_{1g}}$ and $E_{2g}^{1}$ modes develop asymmetric tails on the low and high energy sides of the peak, respectively. This asymmetry originates from a finite phonon correlation length as discussed previously and we note that the direction of the tail for each mode is consistent with the phonon dispersions of 2H-MoTe₂ [51]. In addition, we identify a feature appearing at 202 $\text{c}{{\text{m}}^{-1}}$ for $0.02\leqslant x\leqslant 0.09$ alloys which we assign as a double-resonance Raman mode originating from the scattering of two longitudinal acoustic phonons from the M-point or an ${{E}_{1g}}(M)$ and a transverse acoustic mode, both also from the M point [38]. This feature has only been observed in few-layer 2H-MoTe₂ under resonant excitation [38], and its appearance in the bulk alloy samples is believed to originate from an enhancement in $\vec{q}\ne 0$ Raman scattering processes by compositional disorder in the lattice.

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We have used XRD, STEM, DFT, and Raman spectroscopy to characterize the different crystal phases spanned by the Mo${{}_{1-\text{x}}}$W_xTe₂ alloy system. XRD and STEM measurements determined that 1T'/T_d-Mo${{}_{1-\text{x}}}$W_xTe₂ alloys are in the 1T' phase for $\text{x}\leqslant 0.04$ and the T_d phase for $\text{x}\geqslant 0.63$. For compositions $0.04<\text{x}<0.63$, Mo${{}_{1-\text{x}}}$W_xTe₂ exists in a 1T'  +  T_d two-phase mixture. Raman measurements enable the assignment of phonon mode symmetries across the compositional phase space and permit the observation of a new disorder-activated mode unique to Mo${{}_{1-\text{x}}}$W_xTe₂ alloys. Furthermore, we find that inversion symmetry breaking in the 1T' phase can occur without transitioning to an orthorhombic configuration by monitoring the splitting of the 128 $\text{c}{{\text{m}}^{-1}}$ peak. Finally, we find that the asymmetry of the primary WTe₂ peak can be captured by the phonon confinement model, which in turn allows for the determination of the phonon correlation length. Our studies of the 2H phase show small changes in mode frequencies with x and provide evidence for disorder enhancement of double-resonance Raman scattering processes. These measurements are foundational for future studies seeking to explore the electronic, vibrational, or topological properties of Mo${{}_{1-\text{x}}}$W_xTe₂ alloys.

SMO, JJ, IRS, and PMV acknowledge support from the Office of Naval Research through Grant No. N-00014-15-1-2357, the George Mason University OSCAR Program, and the George Mason University Presidential Scholarship Program. RB thanks the National Research Council Research Associateship Programs for its support. SK acknowledges support from the US Department of Commerce, National Institute of Standards and Technology under the financial assistance award 70NANB16H043. AKS is funded by the Professional Research Experience Postdoctoral Fellowship under award No. 70NANB11H012. This research used computational resources provided by the Texas Advanced Computing Center under Contract TG-DMR150006. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which was supported by National Science Foundation grant number ACI-1053575. AVD, SK, IK, and AB acknowledge the support of Material Genome Initiative funding allocated to NIST.