$\phi$-state and inverted Fraunhofer pattern in nonaligned Josephson junctions

A generic nonaligned Josephson junction in the presence of an external magnetic field is theoretically considered and an unusual flux-dependent current-phase relation (CPR) is revealed. We explain the origin of the anomalous CPR via the current density flow induced by the external field within a two-dimensional quasiclassical Keldysh-Usadel framework. In particular, it is demonstrated that nonaligned Josephson junctions can be utilized to obtain a ground state other than 0 and $\pi$, corresponding to a so-called $\phi$ junction, which is tunable via the external magnetic flux. Furthermore, we show that the standard Fraunhofer central peak of the critical supercurrent may be inverted into a local minimum solely due to geometrical factors in planar junctions. This yields good consistency with a recent experimental measurement displaying such type of puzzling feature [Keizer et al., Nature (London) 439, 825 (2006)].

Figure 1

Diagram of considered setups in this paper. An external magnetic field H (not shown) is applied to the junction in the $z$ direction. The junction lengths and widths are $d$ and $W$, respectively. (a) The planar Josephson junction that has experimentally been studied in, e.g., Ref. 13. The widths of the superconducting leads are assumed to be $W{1}_L$ and $W-(W{1}_L+W{2}_L)$. (b) The usual stacked geometry of a Josephson junction with displaced superconducting leads. The superconducting leads' sizes are $W{1}_L$ and $W{2}_R$ at the top and bottom of the junction, respectively. (c) Qualitative view of the current density flow inside the normal strip subject to an external magnetic field, which is used to describe the origin of the addressed unusual CPR.

Figure 2

Critical supercurrent as a function of external magnetic flux $\Phi$ through the normal part of the junction. The corresponding pair potential spatial map is given with $\phi =0$. Throughout the paper we have assumed that the junction width is fixed at $W=10{\xi }_S$. The first and second columns show the critical current $I_c/I_0$ vs normalized external magnetic flux $\Phi /{\Phi }_0$ and the corresponding pair potential spatial maps with thicknesses $d={\xi }_S$ and $4{\xi }_S$, respectively. Each row indicates different values of $W{1}_L$ and $W{2}_L$, namely, the first superconducting lead size and the separation of the superconducting leads, respectively (see Fig. 1).

Figure 3

Critical supercurrent against external magnetic flux and corresponding pair potential spatial maps of standard (stacked) Josephson junctions with displaced superconducting leads including various lead sizes. For the pair potential maps, the superconducting phase difference and external magnetic flux are fixed at $\phi =0$ and $\Phi =4{\Phi }_0$, respectively. The junction thickness and width are set to $d=2{\xi }_S$ and $W=10{\xi }_S$, respectively.

Figure 4

(a) $W{1}_L=3{\xi }_S$, $W{2}_L=4{\xi }_S$, $W{1}_R=3{\xi }_S$, and $W{2}_R=4{\xi }_S$; (b) $W{1}_L=6{\xi }_S$, $W{2}_L=4{\xi }_S$, $W{1}_R=0$, and $W{2}_R=4{\xi }_S$; and finally, (c) $W{1}_L=2{\xi }_S$ and $W{2}_L=6{\xi }_S$. The top panels represent the CPRs for various values of $\Phi /{\Phi }_0=0$, $3.92$, and $6.28$. The current density spatial maps in the bottom row show the results for $\phi =0$ and $\Phi =4{\Phi }_0$. The superconducting leads' sizes are set equal at $4{\xi }_S$ for all cases as schematically depicted on top of each column.