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.

Figure 1

Band structure of germanene along high-symmetry directions of the hexagonal 2D Brillouin zone. Right inset: Bands near $K$ around the Fermi level (used as energy zero). Left inset: The gap opening versus the strength of SOI.

Figure 2

(a) Top and (b) perspective view on the atomic structure of the hydrogen-passivated edge of a zigzag germanene nanoribbon.

Figure 3

Band structure of the nonmagnetic $N=16$ zigzag germanene nanoribbon (red curves) and the projected bulk band structure (gray shading) in the background. Inset: Spin polarization of particles in spin-degenerate topological states close to the Fermi level and their corresponding spin orientation (in red and blue) near the edges of the nanoribbon.

Figure 4

Wave-function square of the edge states with vanishing energy at $X$. Ge and H atoms are shown as small purple and white circles. The center of the ribbon is omitted.

Figure 5

Band structure of a nonmagnetic $N=16$ zigzag germanene nanoribbon with varying SOI enhanced by a factor ${\beta }_{\mathrm{SOI}}=1$ (black curves), 3 (red curves), 5 (green curves), or 10 (blue curves).

Figure 6

Band structure of the (a) antiferromagnetic and (b) ferromagnetic $N=16$ zigzag germanene nanoribbon (red curves) and the projected bulk band structure (gray shading) in the background.

Figure 7

Real-space localization of the spin density of the antiferromagnetic germanene nanoribbon at one of the edges.

Figure 8

Electronic bulk band gap of germanene as a function of the strength of an external homogeneous electric field perpendicular to the 2D crystal sheet.

Figure 9

One-dimensional energy bands mainly derived from edge states for germanene nanoribbons with $N=16$. Only states around the Fermi level, used as energy zero, and the $X$ point of the BZ boundary are plotted. The two colors (red and blue) of the bands indicate different spin orientations. The field strength varies: (a) $E_z=0$, (b) $E_z=1\times {10}^6$ V/cm, (c) $E_z=E_c=3.4\times {10}^6$ V/cm, and (d) $E_z=5\times {10}^6$ V/cm. The spatial localization of the wave functions at the edges and their spin orientation are illustrated by arrows.