Superconducting phase transistor in diffusive four-terminal ferromagnetic Josephson junctions

We study diffusive magnetic Josephson junctions with four superconducting terminals in the weak proximity limit where the leads are arranged in cross form. Employing the linearized Keldysh-Usadel technique, the anomalous Green's function and Josephson current are analytically obtained based on a quasiclassical theory using the Fourier series method. The derived results may be reduced to nonmagnetic junctions by setting the exchange field equal to zero. We find that increments of the magnetic barrier thickness may cause a reversal of the supercurrent direction flowing into some of the leads, whereas the direction of current flow remains invariant at the others. The reversal direction can be switched by tuning the perpendicular superconducting phases. In the nonmagnetic case, we find that the supercurrent flowing between the leads in one direction can be tuned by changing the superconducting phase difference in the perpendicular direction. These findings suggest the possibility of constructing a nanoscale superconducting phase transistor whose core element consists of the proposed four-terminal Josephson junction with rich switching aspects.

Figure 1

Experimental schematic setup of the cruciate Josephson junction. The junction is assumed to lie in the $xy$ plane with interfaces located at $x=0,L$ and $y=0,W$. The four spin-singlet superconductors have different superconducting phases: ${\theta }_{\mathrm{up}}$, ${\theta }_{\mathrm{down}}$, ${\theta }_{\mathrm{left}}$, and ${\theta }_{\mathrm{right}}$. The exchange field $\mathbf{h}$ is assumed to be oriented in the $z$ direction, perpendicular to the sandwiched layer plane.

Figure 2

Top left: Supercurrent in the $x$ direction as a function of left condensation phase, ${\theta }_{\mathrm{left}}$, at left superconductor gate, i.e., $x=0$. Top right: Supercurrent in the $x$ direction vs left superconducting phase, ${\theta }_{\mathrm{left}}$, at right superconductor gate, i.e., $x=L$. Bottom left: Supercurrent in the $y$ direction as a function of left condensation phase at down superconductor gate, i.e., $y=0$. Bottom right: Supercurrent in the $y$ direction vs left superconducting phase at up gate, i.e., $y=W$. Here other superconductor phases, namely ${\theta }_{\mathrm{up}}$ and ${\theta }_{\mathrm{down}}$, are assumed to be zero.

Figure 3

Critical supercurrent as a function of the normalized junction length $L/{\xi }_S$ at different superconducting gates and for various values of ${\theta }_{\mathrm{up}}$, the superconducting phase of the up terminal. Top left: At the left superconductor gate, i.e., $x=0$. Top right: At the right superconductor gate, i.e., $x=L$. Bottom left: At the down superconductor gate, i.e., $y=0$. Bottom right: At the up gate, i.e., $y=W$. The other superconductor phases are fixed at zero.