Theory of Fano resonances in graphene: The influence of orbital and structural symmetries on STM spectra

We theoretically investigate Fano factors arising in local spectroscopy of impurity resonances in graphene. It is demonstrated that Fano line shapes can strongly differ from the antiresonances usually found on metal surfaces. Graphene’s highly symmetric Fermi points make this effect particularly sensitive to the detailed atomistic structure and orbital symmetries of the impurity. After a model discussion based on an Anderson impurity coupled to an electron bath with linearly vanishing density of states, we present first-principles calculations of Co adatoms on graphene. For Co above the center of a graphene hexagon, we find that the two-dimensional $E_1$ representation made of $d_{xz},d_{yz}$ orbitals is likely responsible for the hybridization and ultimately Kondo screening for cobalt on graphene. Anomalously large Fano $q$ factors depending strongly on the orbitals involved are obtained. For a resonant $s$-wave impurity, a similarly strong adsorption site dependence of the $q$ factor is demonstrated. These anomalies are striking examples of quantum-mechanical interference related to the Berry phase inherent to the graphene band structure.

Figure 1

(Color online) (a) and (d) Hybridization functions $\Delta$, and (b) and (e) asymmetry factors $q$ for Co on graphene (upper panel) and Co on Cu (111) (lower panel) as obtained from our first-principles calculations. The $q$ factors have been calculated from the hybridization functions using Eq. (15). For the hybridization functions $\text{Re}\Delta$ is plotted as solid line, and $\text{Im}\Delta$ as dashed line. The energy $E=0$ corresponds to the Fermi level of the undoped system. The hybridization of Co on graphene is strongly energy and orbital dependent. The adsorption geometries of Co on graphene and Cu (111) are shown in (c) and (f), respectively. For the Cu (111) slabs, only the uppermost Cu layer is shown for clarity.

Figure 2

(Color online) Model of an Anderson impurity in the middle of the graphene hexagon. Electrons from neighboring can hop onto the impurity by the hopping matrix elements $V$. For an electron from the vicinity of the Brillouin-zone corner $K$ phase differences of its wave function at neighboring sites belonging to sublattice A (atoms marked by small blue circles) are given. The sum of these phase factors vanishes.