Coupling to real and virtual phonons in tunneling spectroscopy of superconductors

Fine structures in the tunneling spectra of superconductors have been widely used to identify fingerprints of the interaction responsible for Cooper pairing. Here, we show that for scanning tunneling microscopy (STM) of Pb, the inclusion of inelastic tunneling processes is essential for the proper interpretation of these fine structures. For STM the usual McMillan inversion algorithm of tunneling spectra must therefore be modified to include inelastic tunneling events, an insight that is crucial for the identification of the pairing glue in conventional and unconventional superconductors alike.

Figure 1

Illustration of the inelastic tunneling processes from a sharp tip (right) into a superconductor (left) in real space. Filled states are shown in red color with energy along the vertical axis. The inelastic tunneling process is accompanied by the excitation of real phonons (green).

Figure 2

STM topography of Pb on Si(111) ($300\times 175{\text{nm}}^2$, 1 V, 1 nA). The thickness of the island was determined to $x\approx 30$ monolayers.

Figure 3

Second derivative $d^{2}I/d{U}^2\sim {\alpha }^2{F}_{\text{tun}}\left(eU\right)$ measured in the normal conducting state ($T=0.8$ K, $B=1$ T).

Figure 4

Differential conductance $dI/dU$ in the normal (a) and superconducting (b) state measured on the island marked by the arrow in Fig. 2. The curves are normalized to the zero bias conductance $\sigma \left(0\right)$ in the normal state.

Figure 5

Comparison of experimental data (red) and theoretical prediction in the superconducting state: Calculated elastic (blue), inelastic (green), and total (black) contributions to $d^{2}I/d{U}^2$ (the elastic current is convoluted with a Gaussian function with a standard deviation $\sigma =0.31\text{meV}$ simulating the experimental broadening due to the modulation voltage of the lock-in technique). Characteristic peak-dip features around the zero axis can only be explained taking into account elastic and inelastic channels ($I^{\text{tot}}=I^{\text{el}}+I^{\text{inel}}$).