Coherent transport properties of a three-terminal hybrid superconducting interferometer

We present an exhaustive theoretical analysis of a double-loop Josephson proximity interferometer, such as the one recently realized by Strambini et al. for control of the Andreev spectrum via an external magnetic field. This system, called $\omega$-SQUIPT, consists of a T-shaped diffusive normal metal ($N$) attached to three superconductors ($S$) forming a double-loop configuration. By using the quasiclassical Green-function formalism, we calculate the local normalized density of states, the Josephson currents through the device, and the dependence of the former on the length of the junction arms, the applied magnetic field, and the $S/N$ interface transparencies. We show that by tuning the fluxes through the double loop, the system undergoes transitions from a gapped to a gapless state. We also evaluate the Josephson currents flowing in the different arms as a function of magnetic fluxes, and we explore the quasiparticle transport by considering a metallic probe tunnel-coupled to the Josephson junction and calculating its $I-V$ characteristics. Finally, we study the performances of the $\omega$-SQUIPT and its potential applications by investigating its electrical and magnetometric properties.

Figure 1

Scheme of the $\omega$-SQUIPT in a current-biased setup. $I$ is the current flowing through the circuit, and $V$ is the voltage drop across the device. ${\mathrm{\Phi }}_L$ and ${\mathrm{\Phi }}_R$ represent the magnetic fluxes piercing the left and right loop, respectively. $L_L, L_C$, and $L_R$ refer to the left, center, and right arms length of the T-shaped normal metal, respectively. Finally, $S_L, S_C$, and $S_R$ refer to the left, center, and right superconducting leads.

Figure 2

Evolution of the DOS at the Fermi energy $N_F({\mathrm{\Phi }}_L,{\mathrm{\Phi }}_R)$ for increasing pair-breaking scattering both in the $S$ leads, ${\mathrm{\Gamma }}_S$ (left column), and in the $N$ weak link, ${\mathrm{\Gamma }}_N$ (right column). The values of ${\mathrm{\Gamma }}_N/E_{\text{Th}}$ and ${\mathrm{\Gamma }}_S/{\mathrm{\Delta }}_0$ are reported in each panel. The weak link is of an intermediate length $E_{\text{Th}}/{\mathrm{\Delta }}_0=0.5$ and the $S/N$ interfaces are transparent.

Figure 3

DOS in the center of the three-terminal junction ($x=0$) calculated at zero fluxes, ${\mathrm{\Phi }}_L={\mathrm{\Phi }}_R=0$. (a) Dependence of the DOS on ${\mathrm{\Gamma }}_S/{\mathrm{\Delta }}_0$ (fixed $E_{\text{Th}}/{\mathrm{\Delta }}_0=0.5$ and ${\mathrm{\Gamma }}_N/E_{\text{Th}}={10}^{-3}$). (b) Dependence of the DOS on ${\mathrm{\Gamma }}_N/E_{\text{Th}}$ (fixed $E_{\text{Th}}/{\mathrm{\Delta }}_0=0.5$ and ${\mathrm{\Gamma }}_S/{\mathrm{\Delta }}_0={10}^{-3}$). (c) Dependence of the DOS on the Thouless energy $E_{\text{Th}}/{\mathrm{\Delta }}_0$ (fixed ${\mathrm{\Gamma }}_S={\mathrm{\Gamma }}_N={10}^{-3}{\mathrm{\Delta }}_0$).

Figure 4

DOS calculated in the middle of the three-terminal junction ($x=0$) for equal fluxes ${\mathrm{\Phi }}_L={\mathrm{\Phi }}_R\equiv \mathrm{\Phi }$ with ${\mathrm{\Gamma }}_S={\mathrm{\Gamma }}_N={10}^{-3}{\mathrm{\Delta }}_0$. Each panel corresponds to a different Thouless energy: (a) $E_{\text{Th}}/{\mathrm{\Delta }}_0=5$; (b) $E_{\text{Th}}/{\mathrm{\Delta }}_0=1$; (c) $E_{\text{Th}}/{\mathrm{\Delta }}_0=0.5$; and (d) $E_{\text{Th}}/{\mathrm{\Delta }}_0=0.1$.

Figure 5

Spatial dependence of the DOS evaluated at different fluxes, with $E_{\text{Th}}/{\mathrm{\Delta }}_0=0.5$ and ${\mathrm{\Gamma }}_S={\mathrm{\Gamma }}_N={10}^{-3}{\mathrm{\Delta }}_0$. Each central box indicates the value of the flux ${\mathrm{\Phi }}_L$ associated with the near plots; the top plots show the case of equal fluxes ${\mathrm{\Phi }}_R={\mathrm{\Phi }}_L$ and the bottom plots show the single flux case with ${\mathrm{\Phi }}_R=0$. (a) ${\mathrm{\Phi }}_L=0$; (b) ${\mathrm{\Phi }}_L=0.25{\mathrm{\Phi }}_0$; (c) ${\mathrm{\Phi }}_L=0.33{\mathrm{\Phi }}_0$; and (d) ${\mathrm{\Phi }}_L=0.5{\mathrm{\Phi }}_0$.

Figure 6

Evolution of the DOS at the Fermi energy $N_F({\mathrm{\Phi }}_L,{\mathrm{\Phi }}_R)$ for different values of $S/N$ interface opacities $r_R$ and $r_C$ reported in the $x$ and $y$ axis, respectively. Here $r_L=1, E_{\text{Th}}/{\mathrm{\Delta }}_0=0.5$, and ${\mathrm{\Gamma }}_S={\mathrm{\Gamma }}_N={10}^{-3}{\mathrm{\Delta }}_0$.

Figure 7

Energy spectrum of the DOS as a function of equal fluxes ${\mathrm{\Phi }}_L={\mathrm{\Phi }}_R\equiv \mathrm{\Phi }$ and calculated for different values of $S/N$ interface opacity $r_R$ and $r_C$ reported in the $x$ and $y$ axis, respectively. Here $r_L=1, E_{\text{Th}}/{\mathrm{\Delta }}_0=0.5$, and ${\mathrm{\Gamma }}_N={\mathrm{\Gamma }}_S={10}^{-3}{\mathrm{\Delta }}_0$.

Figure 8

Outgoing supercurrent of the left arm in the case of equal fluxes ${\mathrm{\Phi }}_R={\mathrm{\Phi }}_L=\mathrm{\Phi }$. (a) Supercurrent spectral density $S_L\left(E\right)$ in the case of $E_{\text{Th}}/{\mathrm{\Delta }}_0=5$ and with ${\mathrm{\Gamma }}_S={\mathrm{\Gamma }}_N={10}^{-3}{\mathrm{\Delta }}_0$. Related cuts at different fluxes $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ are reported in (b). Panel (c) shows the supercurrent ${\mathcal{I}}_L$ at a fixed temperature $T=0.02T_C$. ${\mathcal{I}}_L$ has a periodic behavior as a function of $\mathrm{\Phi }$, with nodes due to the three-terminal junction at $\mathrm{\Phi }/{\mathrm{\Phi }}_0=0,1/3,1/2,\text{and}2/3$; see also the cuts present in (b).

Figure 9

Color plot of the supercurrents in each arm for different magnetic fluxes ${\mathrm{\Phi }}_R$ and ${\mathrm{\Phi }}_L$. Here we have fixed $E_{\text{Th}}/{\mathrm{\Delta }}_0=5$ and ${\gamma }_S={\gamma }_N={10}^{-3}{\mathrm{\Delta }}_0$, and $T=0.02T_c$. The supercurrents flowing out of the left, central, and right arm are plotted in (a), (b), and (c), respectively.

Figure 10

$I-V$ characteristics of the tunnel contact between the probe and the $\omega$-SQUIPT junction at different values of flux $\mathrm{\Phi }$, with ideal $S/N$ interfaces. Here, ${\mathrm{\Gamma }}_S={\mathrm{\Gamma }}_N={10}^{-4}{\mathrm{\Delta }}_0$. This quantity is reported both in linear (left column) and Log $y$ (right column) scale. Panels (a) and (b) refer to the $\omega$-SQUIPT with a normal metallic probe NP and $E_{\text{Th}}/{\mathrm{\Delta }}_0=0.5$ and 5, respectively. Panels (c) and (d) refer to the $\omega$-SQUIPT with a superconducting probe SCP and $E_{\text{Th}}/{\mathrm{\Delta }}_0=0.5$ and 5, respectively.

Figure 11

Left column: Flux to voltage characteristic $V_I\left(\mathrm{\Phi }\right)$ of the $\omega$-SQUIPT. Right column: Transfer function $\mathcal{F}$ associated with the flux voltage characteristics. Here, the $S/N$ interfaces are transparent and ${\mathrm{\Gamma }}_S={\mathrm{\Gamma }}_N={10}^{-4}{\mathrm{\Delta }}_0$. Parts (a) and (b) correspond to the case of $\omega$-SQUIPT with a normal probe NP and $E_{\text{Th}}/{\mathrm{\Delta }}_0=0.5$ and 5, respectively. Parts (c) and (d) refer to the $\omega$-SQUIPT with a superconducting probe SCP and $E_{\text{Th}}/{\mathrm{\Delta }}_0=0.5$ and 5, respectively.

Figure 12

Working principle of the $\omega$-SQUIPT as a magnetometer. Here the interfaces are asymmetric, with $r_R=r_C=0.1$ and $r_L=5$. Panel (a) presents the DOS at Fermi energy $N_F$ as a function of ${\mathrm{\Phi }}_L$ and ${\mathrm{\Phi }}_R$. Panel (b) shows the DOS of the $\omega$-SQUIPT at Fermi energy $N_F$ in the case of equal fluxes ${\mathrm{\Phi }}_L={\mathrm{\Phi }}_R=\mathrm{\Phi }$ (orange solid line). For the sake of comparison, we plot also the result in the case of a conventional two-terminal device (dark red dashed curve) with equal contact resistances $r=0.1$.