First principles calculations of electric‑field‑driven topological phase transitions in silicene, germanene and stanene.

The emergence of two-dimensional topological materials, particularly the group-14 monolayers known as silicene, germanene, and stanene has opened promising pathways for next-generation nanoelectronics and spintronics. Their buckled honeycomb structure and strong spin-orbit coupling allow for bandgap engineering via a perpendicular electric field, leading to topological phase transitions (TPTs) from non-trivial to trivial insulating states. However, precise determination of the critical electric fieldEzcrat which these transitions occur remains challenging, with tight-binding models often underestimating these values. Here, we present a first-principles framework that combines density-functional theory (DFT), maximally localized Wannier functions, and evolution of the Wannier charge centers (WCC) to accurately characterize TPTs in silicene, germanene, and stanene through the ℤ2topological invariant. In contrast to earlier work, at each electric-field strength we run fully self-consistent ab initio simulations to obtain the screened electronic structure, accounting for the material's dielectric response from both electrons and ions. From these converged results we construct a Wannier tight-binding Hamiltonian at each electric field strength, which then enables a gauge-invariant calculation of the ℤ2topological invariant. This methodology yields significantly more accurate numerical predictions ofEzcr-0.020 and 0.250 V/Å for silicene and germanene, respectively-and provides deeper insight into the interplay between electronic structure and topological order. Compared to previous approaches, our framework delivers a marked quantitative improvement for predicting topological phase boundaries, essential for guiding the design of topological field-effect transistors and electrostatically controlled quantum devices based on two-dimensional materials.