Robust wavefront dislocations of Friedel oscillations in gapped graphene

Friedel oscillation is a well-known wave phenomenon which represents the oscillatory response of electron waves to imperfection. By utilizing the pseudospin-momentum locking in gapless graphene, two recent experiments demonstrate the measurement of the topological Berry phase by corresponding to the unique number of wavefront dislocations in Friedel oscillations. Here, we study the Friedel oscillations in gapped graphene, in which the pseudospin-momentum locking is broken. Unusually, the wavefront dislocations do occur like that in gapless graphene, which requires immediate verification in the current experimental condition. The number of wavefront dislocations is ascribed to the invariant pseudospin winding number in gapped and gapless graphene. This study deepens the understanding of the correspondence between topological quantity and wavefront dislocations in Friedel oscillations and implies the possibility to observe the wavefront dislocations of Friedel oscillations in intrinsic gapped two-dimensional materials, e.g., transition metal dichalcogenides.

Figure 1

Schematic measurement of electronic density oscillation by STM. (a) In the real-space atomic structure of graphene, there is a single-atom vacancy (red dot) on one sublattice site; by scanning the STM tip (blue dot) one can probe the vacancy-induced density oscillations which are contributed by the intravalley and intervalley scatterings. Focusing on the intervalley scattering, we plot the corresponding physical process; that is, the STM tip emits the electron waves in one valley $\mathbf{K}$, as shown by the propagator $G(\mathbf{K},\mathbf{0},\mathbf{r})$, leading to scattering by the vacancy back to the STM tip through the other valley ${\mathbf{K}}^{\prime }$, as shown by the propagator $G({\mathbf{K}}^{\prime },\mathbf{r},\mathbf{0})$. When the STM tip is shifted around the vacancy in real space as shown by the blue arrow along the circle, the contributing momentum states to the propagator change their momentum around $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$, as shown in (c). (b) The Brillouin zone used to define $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$. (c) Intervalley scattering (green double arrow) in momentum space. The contributing momentum states in each valley are parallel to $\mathbf{r}$ and are denoted by using the same arrows in (a).

Figure 2

Friedel oscillation pattern around a single-atom vacancy at the origin. Intervalley scattering contribution to density oscillations $\delta {\rho }_A(\mathbf{r},\varepsilon )$ and $\delta {\rho }_B(\mathbf{r},\varepsilon )$ on sublattice $A$ (left panel) and on sublattice $B$ (middle panel), respectively. And the sum $\delta {\rho }_A(\mathbf{r},\varepsilon )+\delta {\rho }_B(\mathbf{r},\varepsilon )$ as the total electronic density modulation (right panel). Following Ref. [13], the energy is integrated over energy up to Fermi energy. Here, Fermi level $\varepsilon =0.11t_0\simeq 0.4$ eV, $\mathrm{\Delta }=0.017t_0\simeq 50$ meV, and the color scale is normalized by a numerical factor $C\times {10}^{-3}$.

Figure 3

Pseudospin texture pattern. (a) Pseudospins on the constant-energy contour. (b) Pseudospin as a function of ${\theta }_k$. Here, $\mathrm{\Delta }=50$ meV, $\varepsilon =100$ meV, and $\eta =1$.