Friedel oscillations and superconducting gap enhancement by impurity scattering

Experiments observe an enhanced superconducting gap over impurities as compared to the clean-bulk value. In order to shed more light on this phenomenon, we perform simulations within the framework of Bogoliubov–de Gennes theory applied to the attractive Hubbard model. The simulations qualitatively reproduce the experimentally observed enhancement effect; it can be traced back to an increased local density of states at the Fermi energy in the metal close to the impurity site. In addition, the simulations display significant differences between a thin [two-dimensional (2D)] and a very thick [three-dimensional (3D)] film. In 2D pronounced Friedel oscillations can be observed, which decay much faster in 3D and therefore are more difficult to resolve. Also, this feature is in qualitative agreement with the experiment.

Figure 1

(a) STM topography of an Al(111) surface with a buried defect after mild ${\mathrm{Ar}}^{+}$ bombardment appearing as a depression. (b) Superconducting gap $\mathrm{\Delta }$ obtained by fitting $dI/dU$ spectra recorded as a function of lateral position. (c) Examples of local $dI/dU$ spectra of maximal and minimal $\mathrm{\Delta }$ recorded at positions indicated in (a). Tunneling parameters (a)–(c): $I=0.5$ nA, $U=0.4$ mV. (d) Change of the $z$ position, i.e., tip elevation, with distance to the surface defect appearing as a protrusion together with grayscale encoded $dI/dU$ recorded and the fitted superconducting gap $\mathrm{\Delta }$. Tunneling parameters: $I=8$ nA, $U=2$ mV; further experimental details are given in the Appendix.

Figure 2

(a) Spatial map of the response of the pairing amplitude of a 2D superconductor to an impurity located at the origin. (b) Response of the pairing amplitude along the surface diagonal multiplied by the distance from the impurity $r$ in units of the lattice constant $a$. (Parameters: filling $n=0.2$, $U=1.6$, $V_{\text{imp}}=-0.06t$, ${\lambda }_F/2\approx 2.85a$.)

Figure 3

(a) Spatial map of the logarithm of the pairing amplitude in the surface layer of a 3D superconductor with an impurity at the origin. (b) Response of the pairing amplitude to an impurity on the surface of a 3D superconductor. The response is computed from data shown in (a) averaged over the surface angle. The prefactors of the power-law fits are chosen such that they agree with the data at $r=1a\approx 1.48{\lambda }_F/2$. (Parameters: filling $n=0.12$, $U=3.2$, $V_{\text{imp}}=-0.06t$, ${\lambda }_F/2\approx 2.14a$, $C\approx 1.8\times {10}^{-4}{\mathrm{\Delta }}_{\text{BCS}}$.)

Figure 4

Response of the superconducting gap $\delta {\mathrm{\Delta }}_{\text{imp}}$ (left) and the normal-state LDOS at the Fermi energy $\delta {\rho }_F$ (right) to an impurity (adatom) located on the surface of the cubic lattice (same situation as in Fig. 3). Shown is the dependency on the impurity potential $V_{\text{imp}}$ (horizontal axis) and the number of nearest neighbors $N_{\text{nn}}$ of the impurity site (vertical axes). $\delta {\mathrm{\Delta }}_{\text{imp}}$ and $\delta {\rho }_F$ quantify the average response of the gap and the normal-state LDOS at the Fermi energy on all sites neighboring the impurity. (Parameters: filling $n=0.12$, $U=3.2$.)

Figure 5

Differential conductance $dI/dU$ recorded at 25 mK on bare Al(111) (blue dots) and atop a single Fe atom on Al(111) (red dots) together with fits (solid lines) to thermally smeared BCS density of states. Blue and red solid lines display the fits to the experimental data points. Feedback conditions: $U=400$ $\mu \mathrm{V}$, $I=500$ pA.