The rich structural, electronic and bonding landscape of 1$T$-type TaTe$_2$ single-layers

Charge density waves (CDW) in single-layer 1$T$-MTe$_2$ (M= Nb, Ta) recently raised large attention because of the contrasting structural and physical behavior with the sulfide and selenide analogues. A first-principles study of fourteen different 1$T$-type TaTe$_2$ single-layers is reported. The importance of Te to Ta electron transfer and multicenter metal-metal bonding in stabilizing different structural modulations is highlighted. Analysis of the electronic structure of the optimized structures provides a rationale for what distinguishes 1$T$-TaTe$_2$ from the related disulfide and diselenide, what are the more stable structural modulations for 1$T$-type TaTe$_2$ single-layers, the possible role of Fermi surface nesting on some of these CDW instabilities, how the CDW affects the metallic properties of the non-distorted lattice and the possibility that some of these CDW phases exhibit exotic properties. All CDW phases studied exhibit band structures typical of metallic systems although some exhibit both very flat and dispersive bands at the Fermi level so that Mott effects could develop; one of the (4$\times$4) phases studied exhibits a Dirac cone at the Fermi level.