Influence of edge and field effects on topological states of germanene nanoribbons from self-consistent calculations

We investigate the edge states of real group IV honeycomb crystals by means of ab initio methods. We fully take spin-orbit coupling into account. Thereby germanene, the germanium-based counterpart to graphene, is used as the model system. We focus on the topological properties depending on the non-, antiferro-, and ferromagnetic edges of the ribbon. We show that, in contrast to model studies, the dispersion of the edge states is strongly nonlinear, destroying gapless Dirac bands. Applying an electric field we further study the topological phase transition to a trivial insulator above a critical field strength. The results are critically discussed and compared with tight-binding predictions.