Subgap structure in asymmetric superconducting tunnel junctions

Conductance measurements between two superconducting electrodes with different gap energies ${\Delta }_1$ and ${\Delta }_2$ were performed with a low-temperature scanning tunneling microscope. The temperature dependence and tip-sample distance dependence of the spectra show a pronounced subgap structure which is interpreted as multiple Andreev reflections (MARs). Low temperature conductance peaks not seen in symmetric superconducting tunnel junctions arise at energies $\pm \mid {\Delta }_1-{\Delta }_2\mid$ when the superconducting gaps are sufficiently different. We propose an explanation of these findings by extending the full counting statistics of MARs to describe these asymmetric junctions.

Figure 1

(Color online) $dI∕dV$ spectra observed at $0.56\mathrm{K}$ between a superconducting sample and tip with nearly equal gaps (${\Delta }_1=1.47\mathrm{meV}$, ${\Delta }_2=1.27\mathrm{meV}$) showing Andreev reflections for different junction resistances. All spectra are normalized by $R$. The peak evolving at $V=0$ is due to the Josephson supercurrent. The dotted lines are a guide for the eye marking characteristic features in the spectra. The spectra are shifted vertically with respect to each other for better visibility.

Figure 2

(Color online) Full lines: $dI∕dV$ spectra measured between a Nb sample $({\Delta }_1=1.47\mathrm{meV})$ and a tip with a small gap $({\Delta }_2=0.32\mathrm{meV})$ at $0.56\mathrm{K}$ and different junction resistances. All spectra are normalized by $R$. Dashed lines: Results of the single-channel MAR theory of Ref. 9. The transmission coefficient $\tau$ is determined by the corresponding $R$. Vertical lines: A guide for the eye to mark a new feature at $\pm ({\Delta }_1-{\Delta }_2)$.

Figure 3

(Color online) (a) Temperature dependent $dI∕dV$ spectra observed at $R=50\mathrm{k}\Omega$ between tip and sample as in Fig. 2. All spectra are normalized by $R$. (b) Results of the MAR theory with $\tau =0.26$. Vertical lines: A guide for the eye marking the ${\Delta }_1\pm {\Delta }_2$ positions at $0\mathrm{K}$. (c) Spectrum of a typical superconducting tip measured against a normal conducting Cu(111) sample at $0.56\mathrm{K}$ with $V_{\mathrm{mod}}=50\mu {\mathrm{V}}_{\mathrm{r.m.s.}}$ and $R=5\mathrm{M}\Omega$ (full line) and BCS calculation using $\Delta =0.41\mathrm{meV}$ and $T=0.6\mathrm{K}$ (dashed line). (d) Dots: Temperature dependence of the superconducting tip gap ${\Delta }_2$ for the tip shown in (a). Full line: BCS calculation.

Figure 4

(Color online) (a), (b) Schematic representation of the most pronounced MARs in asymmetric $\mathit{SIS}$ tunnel junctions [(a) ${\Delta }_2∕{\Delta }_1\approx 0.8$, (b) ${\Delta }_2∕{\Delta }_1\approx 0.4$] and their threshold. (i) At $\mathit{eV}⩾{\Delta }_2$ an electron $\left(e\right)$ tunnels from the left SC into the right SC and is reflected into a hole $\left(h\right)$ by creating a Cooper pair in the right SC. (ii) At $\mathit{eV}⩾{\Delta }_1$ a hole tunnels from the right SC into the left SC and is reflected into an electron annihilating a Cooper pair in the left SC. (iii) Two-reflection process at $\mathit{eV}⩾({\Delta }_1+{\Delta }_2)∕3$ involving the tunneling of 3 particles and the creation of a Cooper pair in the right and the annihilation in the left SC. (iv) Special case of the two-reflection process shown in (iii), where the left Andreev reflection takes place just inside the gap of the left SC. This case is only possible in junctions with ${\Delta }_2∕{\Delta }_1⩽0.5$. (c) Calculated current contributions $I_n$ to the overall tunneling current $I_{\text{sum}}$ (full line) of tunneling processes at $n\text{th}$ order in a junction with ${\Delta }_2∕{\Delta }_1=0.21$ and a transmission coefficient of $\tau =0.2$ at a temperature of $0\mathrm{K}$ using a full counting statistics calculation. The arrows and the vertical lines mark energies where peaks in the spectrum occur, except at $({\Delta }_1+{\Delta }_2)∕3$ (see text).

Figure 5

(Color) Full counting statistics calculation of the $dI∕dV$-signal in an energy range between $0⩽\mathit{eV}⩽2{\Delta }_1$ between two SC with different gap ratios ${\Delta }_2∕{\Delta }_1$ at a temperature of $T=0\mathrm{K}$ and a transmission coefficient of the junction of $\tau =0.25$. The intensity is coded in color (see inset). Peaks at ${\Delta }_1$, ${\Delta }_2$, and ${\Delta }_1+{\Delta }_2$ are visible for all gap ratios, while the peak at ${\Delta }_1-{\Delta }_2$ only exists for a ratio ${\Delta }_2∕{\Delta }_1⩽0.5$ and the peak at $({\Delta }_1+{\Delta }_2)∕3$ diminishes for a ratio ${\Delta }_2∕{\Delta }_1<0.3$.