Full counting statistics of phonon-assisted Andreev tunneling through a quantum dot coupled to normal and superconducting leads

We present a theoretical investigation for the full counting statistics of the Andreev tunneling through a quantum dot (QD) embedded between superconducting (SC) and normal leads in the presence of a strong on-site electron-phonon interaction using nonequilibrium Green function method. For this purpose, we generalize the dressed tunneling approximation (DTA) recently developed in dealing with inelastic tunneling in a normal QD system to the Andreev transport issue. This method takes account of vibrational effect in evaluation of electronic tunneling self energy in comparison with other simple approaches and meanwhile allows us to derive an explicit analytical formula for the cumulant generating function at the subgap region. We then analyze the interplay of polaronic and SC proximity effects on the Andreev reflection spectrum, current-voltage characteristics, and current fluctuations of the hybrid system. Our main findings include: (1) no phonon side peaks in the linear Andreev conductance; (2) a negative differential conductance stemming from the suppressed Andreev reflection spectrum; (3) a novel inelastic resonant peak in the differential conductance due to phonon assisted Andreev reflection; (4) enhancement or suppression of shot noise for the symmetric or asymmetric tunnel-coupling system, respectively.

Figure 1

The imaginary part of the first diagonal element of the vibrational dressed retarded self-energy, ${\mathrm{\Sigma }}_{c11}^r\left(\omega \right)$, are plotted for different bias voltages, $V=0$ (b), 0.1 (c), and 0.2 (d), respectively, at different temperatures, $T=0$ and 0.1. The parameters used for calculation are taken as: ${\mathrm{\Gamma }}_N={\mathrm{\Gamma }}_S=0.1, g=1.0$. For comparison, the corresponding result for the system without EPI is plotted in (a).

Figure 2

The real (a) and imaginary (b) parts of the nondiagonal element of the vibrational dressed retarded self-energy, ${\mathrm{\Sigma }}_{c12}^r\left(\omega \right)$, at $V=0$ and different temperatures, $T=0$ and 0.1. The other parameters are the same as those in Fig. 1. The thin-blue lines denote results for the system with $g=0$.

Figure 3

The Andreev reflection spectrum for the symmetric N-QD-S system ${\mathrm{\Gamma }}_S={\mathrm{\Gamma }}_N$ with (a) ${\varepsilon }_d=0$, (b) 0.1, and (c) $-0.2$ at zero bias voltage and zero temperature. The calculated results correspond to the different approximations: DTA (solid-black line), PTA (dashed-blue line), and SPA (dashed-dotted-red line).

Figure 4

The bias voltage dependence of the Andreev reflection spectrum in the DTA for the symmetric systems with ${\varepsilon }_d=0$ and 0.05 at zero temperature.

Figure 5

The Andreev reflection spectrum in the DTA at zero bias voltage and zero temperature for the asymmetric N-QD-S system with ${\mathrm{\Gamma }}_S=10{\mathrm{\Gamma }}_N$, and different values of the EPI constant, $g=0$, 0.5, 0.75, and 1.0. (a) is for the QD with the level ${\varepsilon }_d=0$, and (b) is for ${\varepsilon }_d=0.05$.

Figure 6

The linear Andreev conductance $G$ for the symmetric system as a function of the resonant level ${\varepsilon }_d$ of the QD with $g=1$ at different temperatures $T=0$, 0.05, and 0.1. The thick lines correspond to the DTA results, while the thin lines correspond to the PTA results. Inset: the corresponding results for the system in the absence of EPI.

Figure 7

The linear Andreev conductance $G$ for the asymmetric system ${\mathrm{\Gamma }}_S=10{\mathrm{\Gamma }}_N$. Other parameters are the same as those in Fig. 6.

Figure 8

(a) The calculated total current (solid line), elastic current (dashed line), inelastic current (dotted-dashed line) and (b) the corresponding differential conductances as functions of bias voltage for a symmetric system with ${\varepsilon }_d=0$ at zero temperature. The inset shows the PTA result (thick line) for the Andreev current and the current for a noninteracting system (thin line).

Figure 9

The same figure as Fig. 8 but for the system with ${\varepsilon }_d=0.05$.

Figure 10

Schematic diagram of the phonon emission assisted Andreev reflection processes involved in Figs. 8 and 9, respectively. Solid (open) circles denote the states of quasiparticles (quasiholes), and the arrows stand for directions of their tunneling. See text for more detail.

Figure 11

The same figure as Fig. 8 but for the asymmetric system with ${\mathrm{\Gamma }}_S=10{\mathrm{\Gamma }}_N$ and a weak EPI constant $g=0.5$.

Figure 12

(a) The zero-temperature shot noise (solid black line), and its elastic (dashed red line) and inelastic (dotted-dashed blue line) parts (normalized by $4e^2\mathrm{\Delta }/h$) as functions of bias voltage for a symmetric N-QD-S with ${\varepsilon }_d=0$ and $g=0.75$. The total Andreev current is also plotted as a solid purple line. (b) The Fano factors for the system ${\varepsilon }_d=0$ with different EPI constants $g=1.0$ (black line), 0.75 (blue line), and 0.5 (red line). For comparison, the thin line denotes the Fano factor of the system without EPI.

Figure 13

This figure is the same as Fig. 12 except for the asymmetric system with ${\mathrm{\Gamma }}_S=10{\mathrm{\Gamma }}_N$, and $g=0.5$ in (a).