Transitions in Xenes between excitonic, topological and trivial insulator phases: influence of screening, band dispersion and external electric field

Using a variational approach, the binding energies $E_b$ of the lowest bound excitons in Xenes under varying electric field are investigated. The internal exciton motion is described both by Dirac electron dispersion and in effective-mass approximation, while the screened electron-hole attraction is modeled by a Rytova-Keldysh potential with a 2D electronic polarizability $\alpha_{2{\rm D}}$. The most important parameters as spin-orbit-induced gap $E_g$, Fermi velocity $v_F$ and $\alpha_{2{\rm D}}$ are taken from ab initio density functional theory calculations. In addition, $\alpha_{2{\rm D}}$ is approximated in two different ways. The relation of $E_b$ and $E_g$ is ruled by the screening. The existence of an excitonic insulator phase with $E_b>E_g$ sensitively depends on the chosen $\alpha_{2{\rm D}}$. The values of $E_g$ and $\alpha_{2{\rm D}}$ are strongly modified by a vertical external electric bias $U$, which defines a transition from the topological into a trivial insulator at $U=E_g/2$, with the exception of plumbene. Within the Dirac approximation, but also within the effective mass description of the kinetic energy, the treatment of screening dominates the appearance or non-appearance of an excitonic insulator phase. Gating does not change the results: the prediction done at zero electric field is confirmed when a vertical electric field is applied. Finally, Many-Body perturbation theory approaches based on the Green's function method, applied to stanene, confirm the absence of an excitonic insulator phase, thus validating our results obtained by ab initio modeling of $\alpha_{2{\rm D}}$.