Andreev States near Short-Ranged Pairing Potential Impurities

We study Andreev states near atomic scale modulations in the pairing potential in both $s$- and $d$-wave superconductors with short coherence lengths. For a moderate reduction of the local gap, the states exist only close to the gap edge. If one allows for local sign changes of the order parameter, however, resonances can occur at energies close to the Fermi level. The local density of states (LDOS) around such pairing potential defects strongly resembles the patterns observed by tunneling measurements around Zn impurities in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+x}$ (BSCCO). We discuss how this phase impurity model of the Zn LDOS pattern can be distinguished from other proposals experimentally.

Figure 1

Left: LDOS in a $d$-wave superconductor at the impurity site for varying potential scattering strengths $V$ (clean case: dashed curve). As $V$ increases the resonant state sharpens and moves to lower energy. Right: real-space LDOS pattern at the resonant energy ${\omega }_B$ with a delta-function impurity potential ($V=5t$) at the center [30].

Figure 2

(a) Real part of the determinant, $\mathrm{Re}D(\omega )$, with $\tilde{\alpha }=0.4$ (top), $\tilde{\alpha }=0.6$ (middle), and $\tilde{\alpha }=0.8$ (bottom). (b) LDOS at the impurity site for the same values of $\tilde{\alpha }$ as in (a); the dashed curve displays the clean result.

Figure 3

LDOS for various strengths of the ${\tau }_1$ scatterer at the impurity site (a), on the nearest neighbor site (b), and the next-nearest neighbor site (c). In each figure, the dashed curve shows the bulk LDOS. (d) The real-space LDOS distribution at the resonance energy ${\omega }_B$ when $\alpha =1.4$.

Figure 4

LDOS at $(0,0)$ (a) and $(2,0)$ (b) when two impurities are positioned at $(-1,1)$ and $(1,-1)$ for both the ${\tau }_3$ (solid line) and ${\tau }_1$ (dashed line) channel. In both cases, the ${\tau }_3$ LDOS has been rescaled for clarity. (c)–(d) LDOS at the impurity site for a single ${\tau }_3$ (c) and ${\tau }_1$ (d) scatterer of fixed strength ($V=5t$ for ${\tau }_3$, and $\alpha =1.4$ for ${\tau }_1$) but varying ${\Delta }_0$.