Local impurity effects in superconducting graphene

We study the local electronic properties of superconducting graphene in presence of impurities. In particular, we consider the case of magnetic and nonmagnetic impurities being either strongly localized or acting as a potential averaged over one unit cell. The spin dependent local density of states (LDOS) is calculated and possibilities for identifying the superconducting order parameter in graphene by means of scanning tunneling experiments are pointed out. We show that the vanishing effective mass of the electrons results in no coherence peaks at the superconducting gap edge. A method for identifying magnetic scatterers even by non-spin-polarized scanning tunneling spectroscopy is given.

Figure 1

(Color online) Among the various local impurities we discuss two limiting cases. The scalar impurity (left), $V_s$, corresponds to a uniform potential averaged over one unit cell, whereas the on-site impurity (right), $V_o$, acts on one sublattice only.

Figure 2

(Color online) Energy of the impurity resonance for the scalar impurity as a function of the magnetic impurity potential $V_1$ for different electrostatic potentials $V_0$. The gap-parameter is $\Delta =W/10$. The lower right inset shows the splitting of the impurity state due to intervalley scattering. We model intervalley scattering as $V=V_1\left({\sigma }_0\otimes {\tau }_0\otimes {\Lambda }_0+a{\sigma }_0\otimes {\tau }_1\otimes {\Lambda }_0\right)$ with the strength of the intervalley scattering parametrized by $a$ and $V_1=5\text{W}$. One split state is shifted to the gap edge; the other state remains an intragap state.

Figure 3

(Color online) Left panel: Density $N$ of the intragap bound states as a function of the distance $r$ from the impurity for purely magnetic scalar impurities and different potentials $V_1$. The impurity strength is given in units of the bandwidth $W$. Right panel: Friedel oscillations in the LDOS around a scalar impurity at $r=0$ with $V_0=0$ and $V_1=3\text{W}$. The different curves correspond to the energies $\omega =0.8$, 0.5 and $0.25\text{W}$. In both panels, the gap-parameter is $\Delta =W/10$.

Figure 4

(Color online) The LDOS (upper panel) and the LSDOS (lower panel), $\delta N_{\uparrow }\left(r,\omega \right)-\delta N_{\downarrow }\left(r,\omega \right)$, at $r=0$ and $r=1$ are shown for scalar of impurities with different potentials: a purely magnetic impurity with $V_0=0$ and $V_1=3\text{W}$ (left) as well as an impurity with $V_0=2\text{W}$ and $V_1=1\text{W}$.