Topological phase transition and two dimensional topological insulators in Ge-based thin films

We discuss possible topological phase transitions in Ge-based thin films of Ge(Bi$_x$Sb$_{1-x}$)$_2$Te$_4$ as a function of layer thickness and Bi concentration $x$ using the first principles density functional theory framework. The bulk material is a topological insulator at $x$ = 1.0 with a single Dirac cone surface state at the surface Brillouin zone center, whereas it is a trivial insulator at $x$ = 0. Through a systematic examination of the band topologies we predict that thin films of Ge(Bi$_x$Sb$_{1-x}$)$_2$Te$_4$ with $x$ = 0.6, 0.8 and 1.0 are candidates for two-dimensional (2D) topological insulators, which would undergo a 2D topological phase transition as a function of $x$. A topological phase diagram for Ge(Bi$_x$Sb$_{1-x}$)$_2$Te$_4$ thin films is presented to help guide their experimental exploration.