Emergent charge density wave order in the monolayer limit of 1T−TiTe2 and 1T−ZrTe2

A peculiar charge-density wave (CDW) phase, absent in the bulk, has been widely studied in monolayer $1T\text{\ensuremath{-}}{\mathrm{TiTe}}_{2}$ and newly observed in monolayer $1T\text{\ensuremath{-}}{\mathrm{ZrTe}}_{2}$, while its origin and physical properties remain unclear. Here, we study the distorted lattice and associated energy band renormalization for ${\mathrm{TiTe}}_{2}$ and ${\mathrm{ZrTe}}_{2}$ monolayers using first-principles calculations. Both systems are found to exhibit a soft phonon mode at the $M$ point leading to a $2\ifmmode\times\else\texttimes\fi{}2$ CDW order with similar distortion pattern as in ${\mathrm{TiSe}}_{2}$ case. Electronic structure results with semilocal functional indicate that the CDW phases of monolayer ${\mathrm{TiTe}}_{2}$ and ${\mathrm{ZrTe}}_{2}$ maintain semimetallicity owing to their smaller lattice distortion than in the ${\mathrm{TiSe}}_{2}$ semiconducting CDW state. The unfolded band structure for monolayer ${\mathrm{TiTe}}_{2}$ reveals CDW-reconstructed features consistent with experiments, including backfolded bands from $\mathrm{\ensuremath{\Gamma}}$ to $M$ and vice versa, as well as a stronger energy gapping of the outer valence band along $\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}M$ due to orbital-dependent $p\text{\ensuremath{-}}d$ hybridization. We also explore the role of exchange interaction in the CDW formation. The nonlocal exchange effect enlarges the lattice distortion amplitude and causes an overcorrection of the electronic structure for monolayer ${\mathrm{TiTe}}_{2}$, while it barely affects the CDW distortion of monolayer ${\mathrm{ZrTe}}_{2}$, despite inducing a metal-semiconductor transition. Comparison of the CDW properties of monolayer ${\mathrm{TiSe}}_{2}, {\mathrm{TiTe}}_{2}$, and ${\mathrm{ZrTe}}_{2}$ suggests that the judgment of relative CDW strength in these systems should involve coupled electron-lattice modifications. Our work paves the way for elucidating the CDW order in two-dimensional group-IV transition-metal dichalcogenides.