Singular Current Response from Isolated Impurities in $d$-Wave Superconductors

The current response of a $d$-wave superconductor containing a single impurity is calculated and shown to be singular in the low-temperature limit, leading in the case of strong scattering to a $1/T$ term in the penetration depth $\lambda (T)$ similar to that induced by Andreev surface bound states. For a small number of such impurities, we argue this low-$T$ upturn could be observable in cuprate superconductors.

Figure 1

Normalized change in penetration depth $\delta \lambda /[{\lambda }_0(N_0^{-1}/\mathcal{V}{T}_c)]$ due to single isolated impurity vs normalized temperature $T/T_c$ for three values of the cotangent of the impurity phase shift $c=0,0.1,0.3$. For this plot we took ${\eta }_{\mathbf{p}}\simeq 0.01T_c$.

Figure 2

Change in penetration depth $\delta {\lambda }_{\mathrm{t}\mathrm{o}\mathrm{t}}(\AA )$ due to isolated unitarity limit scatterers vs temperature $T(\mathrm{K})$ for three impurity concentrations $n_i={10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$. ${\Delta }_0/T_c=2.14$, ${\eta }_{\mathbf{p}}/T_c=0.001$, $T_c=100\mathrm{K}$. Dashed line: $T$ variation of pure penetration depth $(\log 2)T/{\Delta }_0$. Inset: $\delta {\lambda }_{\mathrm{t}\mathrm{o}\mathrm{t}}$ vs $n_i$ at $T=1\mathrm{K}$.