Prediction of entropy stabilized incommensurate phases in the system MoS$_{2}$-MoTe$_{2}$

A first principles phase diagram calculation, that included van der Waals interactions, was performed for the system (1-X)$\cdot$MoS$_{2}$-(X)$\cdot$MoTe$_{2}$. Surprisingly, the predicted phase diagram has at least two ordered solid-solution phases, at $X \approx 0.46$, even though all calculated formation energies are positive, in a ground-state analysis that examined all configurations with 16 or fewer anion sites. The lower-temperature {\bf $I$}-phase is predicted to transform to a higher-temperature {\bf $I^{\prime}$}-phase at $T \approx 500K$, and {\bf $I^{\prime}$} disorders at $T \approx 730K$. Both these transitions are predicted to be first-order, and there are broad miscibility gaps on both sides of the ordered regions. Both the {\bf $I$}- and {\bf $I^{\prime}$}-phases are predicted to be incommensurate: {\bf $I$}-phase in three dimensions; and {\bf $I^{\prime}$}-phase in two dimensions.