Josephson effect in thin-film superconductor/insulator/superconductor junctions with misaligned in-plane magnetic fields

We study a tunnel junction consisting of two thin-film $s$-wave superconductors separated by a thin, insulating barrier in the presence of misaligned in-plane exchange fields. We find an interesting interplay between the superconducting phase difference and the relative orientation of the exchange fields, manifested in the Josephson current across the junction. Specifically, this may be written $I_{\mathrm{J}}^{\mathrm{C}}=(I_0+I_m\cos \phi )\sin \Delta \theta$, where $I_0$ and $I_m$ are constants, and $\phi$ is the relative orientation of the exchange fields, while $\Delta \theta$ is the superconducting phase difference. Similar results have recently been obtained in other superconductor-insulator-superconductor junctions coexisting with helimagnetic or ferromagnetic order. We calculate the superconducting order parameter self-consistently, and investigate quantitatively the effect which the misaligned exchange fields constitute on the Josephson current, to see if $I_m$ may have an appreciable effect on the Josephson current. It is found that $I_0$ and $I_m$ become comparable in magnitude at sufficiently low temperatures and fields close to the critical value, in agreement with previous work. From our analytical results, it then follows that the Josephson current in the present system may be controlled in a well-defined manner by a rotation of the exchange fields on both sides of the junction. We discuss a possible experimental realization of this proposition.

Figure 1

(Color online) Plot of the function $g\left[\Delta (0,h)\right]$ given by Eq. (11) to illustrate the possible solutions for the gap, given by where the curves intersect the dotted line. When $h∕{\Delta }_0>0.5$, there is more than one solution to the gap equation, but only one of these is stable. As shown in the inset, where we have plotted the field dependence of this stable solution, a first-order phase transition to the normal state is present at zero temperature.

Figure 2

(Color online) The phase diagram in the $h-T$ plane for a superconductor in the presence of an exchange field. A nonzero solution for the gap exists under the dotted line, indicating a possible SC phase. The exact regime where SC is energetically favored over the normal state was studied in Ref. 19; see their Fig. 1. Since the phase transition is first order, note that the ratio $\Delta (T,h)∕T_c\left(h\right)$ is not constant as in the pure BCS case, as shown in the inset.

Figure 3

(Color online) Field dependence $h$ of the superconducting order parameter $\Delta =\Delta (T,h)$ at finite temperatures. The sudden end of the curves clearly shows the sharp drop in the gap, indicating a discontinuous nature of the normal metal-superconductor phase transition.

Figure 4

(Color online) The tunneling scenario illustrated for two quasiparticles approaching the barrier separating the superconductors. For incoming momenta with a large component perpendicular to the barrier (green), tunneling occurs with greater probability than for incoming momenta with a small component perpendicular to the barrier (red). The sign of the component of momentum perpendicular to the barrier must be preserved in the process. For $s$-wave superconductors, the tunneling matrix element may be approximated by a constant, while it may not for anisotropic superconductors.

Figure 5

(Color online) Plot of the components $I_0$ and $I_m$ as a function of exchange field $h$ for several temperatures $T$. It is seen that $I_m$ becomes nonzero only as $h$ increases towards ${\Delta }_0$, such that the Josephson current is only sensitive to a rotation of the misorientation of the exchange fields in this regime.

Figure 6

(Color online) Plot of the total Josephson current in the parallel $I_{\mathrm{J}}\left(0\right)$ and antiparallel $I_{\mathrm{J}}\left(\pi \right)$ configurations of the exchange fields on both sides of the junction. It is seen that the Josephson current is actually enhanced with increasing field strength for the antiparallel configuration for low enough temperatures, in agreement with the result of Refs. 19, 24.

Figure 7

(Color online) Plot of the total Josephson charge current at $T∕{\Delta }_0=0.001$ as a function of $h$ up to the critical field $h_c=0.7{\Delta }_0$ in the presence of an adiabatic rotation of $\phi$, ranging from $\phi =0$ to $\phi =\pi$ in steps of $0.1\pi$ from bottom to top.

Figure 8

(Color online) Suggested experimental setup for achieving homogeneous exchange fields in the superconductor. The antiparallel alignment of the exchange fields is probably the most viable to realize in order to avoid the Fraunhofer diffraction of the resulting Josephson current.