Transport Properties of an Electron-Hole Bilayer in Contact with a Superconductor Hybrid Junction

We investigate the transport properties of a junction consisting of an electron-hole bilayer in contact with normal and superconducting leads. The electron-hole bilayer is considered as a semimetal with two electronic bands. We assume that in the region between the contacts the system hosts an exciton condensate described by a BCS-like model with a gap $\mathrm{\Gamma }$ in the quasiparticle density of states. We first discuss how the subgap electronic transport through the junction is mainly governed by the interplay between two kinds of reflection processes at the interfaces: the standard Andreev reflection at the interface between the superconductor and the exciton condensate, and a coherent crossed reflection at the semimetal–exciton-condensate interface that converts electrons from one layer into the other. We show that the differential conductance of the junction shows a minimum at voltages of the order of $\mathrm{\Gamma }/e$. Such a minimum can be seen as a direct hallmark of the existence of the gapped excitonic state.

Figure 1

(a) Sketch of the electron-hole bilayer with the two normal ($N_{T/B}$) and two superconducting electrodes (${\mathrm{S}}_{T/B}$); the EC region in the middle has length $L$. Energy spectra are shown for the SM region (b), the EC region (c), and the SSM region (d).

Figure 2

$R_{NTB}$ reflection (blue surface) and $R_{ATT}$ (orange surface) reflection as a function of the length of EC region $L$ and injection energy $\mathcal{E}$; both surfaces are shown for injection energies up to $\mathrm{\Delta }$. Here, we have used $\mathrm{\Gamma }=1$, $\mathrm{\Delta }=2\mathrm{\Gamma }$, $G=100\mathrm{\Gamma }$, and normal incidence $\alpha =0$.

Figure 3

(a) Differential conductance as a function of the applied bias and for different lengths $L$ of the EC region, the various lines correspond to $\mathrm{\Gamma }=1$, $\mathrm{\Delta }=4\mathrm{\Gamma }$, $G=100\mathrm{\Gamma }$ and $H_{\mathrm{L}}=H_{\mathrm{R}}=0$. (b) Differential resistance as a function of the applied bias and different length of the EC region. In the figure we have set $\mathrm{\Gamma }=1$, $\mathrm{\Delta }=12\mathrm{\Gamma }$, $G=60\mathrm{\Gamma }$ and $H_{\mathrm{SM}-\mathrm{EC}}=H_{\mathrm{EC}-\mathrm{SSM}}=0.5$, The parameters are chosen in accordance to Ref. [9, 32]. In both panels, the different lines refer to $L=0$ (black solid line), $L={\xi }_{\mathrm{\Gamma }}/2$ (red dashed line), $L=0.7{\xi }_{\mathrm{\Gamma }}$ (blue dashed-dotted line), and $L={\xi }_{\mathrm{\Gamma }}$ (orange dashed-dashed-dotted-dotted line).