Full counting statistics of Andreev scattering in an asymmetric chaotic cavity

We study the charge transport statistics in coherent two-terminal double junctions within the framework of the circuit theory of mesoscopic transport. We obtain the general solution of the circuit-theory matrix equations for the Green’s function of a chaotic cavity between arbitrary contacts. As an example we discuss the full counting statistics and the first three cumulants for an open asymmetric cavity between a superconductor and a normal-metal lead at temperatures and voltages below the superconducting gap. The third cumulant shows a characteristic sign change as a function of the asymmetry of the two quantum point contacts, which is related to the properties of the Andreev reflection eigenvalue distribution.

Figure 1

A chaotic cavity coupled to the leads by two junctions with transmission eigenvalues $\left\{T_n^{\left(1\right)}\right\}$ and $\left\{T_n^{\left(2\right)}\right\}$ (top). The discrete circuit-theory representation of the system is shown in the lower part. The leads and cavity are characterized by the corresponding matrix Green’s functions. The junctions, depicted as connectors, carry conserved matrix currents.

Figure 2

Conductance $G_S$ [panel (a)] and the Fano factor $F_S$ [panel (b)] of the S-cavity-N junction as a function of the junction asymmetry $1∕(1+g_1∕g_2)$. Results for three different couplings of a cavity to the leads are shown for comparison: coupling by quantum point contacts (solid curves), tunnel junctions (dashed curves), and coupling by tunnel junction from the S side and quantum point contact from the N side (dotted curves). Conductance is normalized to the normal-state value $G_N=g_1{g}_2∕(g_1+g_2)$, with $g_i=(2e^2∕h)N_i$ for quantum point contacts and $g_i=(2e^2∕h){\sum }_n{T}_n^{\left(i\right)}$ for tunnel junctions. Fano factors $F_N$ for the corresponding normal-state junctions are shown in panel (b) for comparison.

Figure 3

The third cumulant as a function of bias-to-temperature ratio [panel (a)], shown for three characteristic junction asymmetries: $C_I$ has the maximal positive slope at high biases for symmetric coupling $\eta =g_{\mathit{min}}∕g_{\mathit{max}}=1$, saturates at high biases for $\eta \approx 0.22$, and has maximal negative slope for $\eta \approx 0.046$. The high-bias slope of the third cumulant is shown in panel (b) as a function of the junction asymmetry, with the normal-state value given for comparison. Effective charge is $e^{*}=2e$ and $e^{*}=e$ for the superconducting and normal-state junction, respectively.

Figure 4

Distribution of the Andreev reflection probabilities ${\rho }_A\left(R_A\right)=2{\rho }_c\left(\mathcal{T}\right)d\mathcal{T}∕d{R}_A$ (solid curves) and the distribution of transmission eigenvalues ${\rho }_c\left(\mathcal{T}\right)$ (dash-dotted curves) for an open chaotic cavity, shown for different asymmetries of the junction $\eta =g_{\mathit{min}}∕g_{\mathit{max}}$ (from top to bottom).