Spontaneous vortex state and $\varphi$-junction in a superconducting bijunction with a localized spin

A Josephson bijunction made of three superconductors connected by a quantum dot is considered in the regime where the dot carries a magnetic moment. In the range of parameters where such a dot, if inserted in a two-terminal Josephson junction, creates a $\pi$-shift of the phase, the bijunction forming a triangular unit is frustrated. This frustration is studied within both a phenomenological and a microscopic model. Frustration stabilizes a phase vortex centered on the dot, with two degenerate states carrying opposite vorticities, independently of the direction of the magnetic moment. Embedding the bijunction in a superconducting loop allows one to create a tunable “$\varphi$”-junction whose equilibrium phase can take any value. For large enough inductance, it generates noninteger spontaneous flux. Multiloop configurations are also studied.

Figure 1

Bijunction made of three superconductors and one quantum dot carrying a spin $S=1/2$. The inset show the equivalent triangular model, valid in the perturbative regime only. The asymmetry ratios between the junction's critical currents are indicated.

Figure 2

(a),(b) Two symmetric canted solutions in the frustrated situation, corresponding to a spontaneous phase vortex. The straight arrows represent the superconducting phase. (c) Nonfrustrated solution with two $\pi$-junctions ($S_1-S_2$ and $S_1-S_3$) and a 0-junction ($S_2-S_3$).

Figure 3

(Top left) Variation of the “floating” phase ${\varphi }_1$ with the phase $\varphi$ across $S_2-S_3$, for $g=1$ (staircase) and $0.9$. Total energy of the bijunction inserted in a single loop, as a function of the phase ${\varphi }_{23}=\varphi$ for (top right) $g_0=1$, $g=1$; (middle left) $g_0=1$, $g=0.9$; (middle right) $g_0=1$, $g=0.7$; (bottom left) $g_0=0.5$, $g=0.9$; (bottom right) $g_0=1.5$, $g=0.9$. Despite the $2\pi$-periodicity, the plot between $\varphi =\pm 2\pi$ shows the different barriers depending on the couple of degenerate phase vortex states.

Figure 4

Total energy of the bijunction, from the microscopic model, as a function of the phase ${\varphi }_{23}=\varphi$. Parameters are $\Delta =1$, $\Gamma =2$, $J=5$, $\varepsilon =0$, temperature $T=0.02$, and (a) ${\gamma }_{1,2,3}=(1/3,1/3,1/3)$, (b) ${\gamma }_{1,2,3}=(0.4,0.4,0.2)$.

Figure 5

Connecting the bijunction with one loop can stabilize an arbitrary flux. $J$ (blue arrow) denotes the current flowing in junction $S_2-S_3$.

Figure 6

Total energy of the bijunction inserted in a single loop, as a function of the phase ${\varphi }_{23}=\varphi$. The asymmetry between leads 2 and 3 is weak. Parameters are (left panels) $g_0=0.5$ and (right panels) $g_0=1.5$, and from top to bottom ${\Phi }_{\mathrm{ext}}=0,{\Phi }_0/4,{\Phi }_0/2$.

Figure 7

Connecting the bijunction with two loops stabilizes two symmetric spontaneous fluxes. The figure corresponds to zero external flux, large inductance, and symmetric bijunction. The blue arrows denote the currents circulating in the junctions.

Figure 8

Connecting the bijunction with three loops globally traps one flux quantum.