Non-Fraunhofer Interference Pattern in Inhomogeneous Ferromagnetic Josephson Junctions

Generic conditions are established for producing a non-Fraunhofer response of the critical supercurrent subject to an external magnetic field in ferromagnetic Josephson junctions. Employing the quasiclassical Keldysh-Usadel method, we demonstrate theoretically that an inhomogeneity in the magnitude of the energy scales in the system, including Thouless energy, exchange field and temperature gradient normal to the transport direction, influences drastically the standard Fraunhofer pattern. The exotic non-Fraunhofer response, similar to that observed in recent experiments, is described in terms of an intricate interplay between multiple “$0\mathrm{\text{−}}\pi$” states and is related to the appearance of proximity vortices.

Figure 1

Experimental setup of the inhomogeneous ferromagnetic Josephson junction. The arrow inside the $F$ layer shows the direction of the inhomogeneity in magnitude of either the Thouless energy, magnetic exchange field, or temperature normal to transport direction ($x$ direction). To model an inhomogeneity in the Thouless energy, the setup (a) is considered in this Letter. Although making the setup (b) might be easier to achieve technically, the two setups generate the same results in the diffusive limit. An inhomogeneity in the magnetic exchange field is modeled by $\mathbf{h}=h(0,0,y/W)$. The external field is applied to the system in the $z$ direction (not shown).

Figure 2

Normalized supercurrent through an inhomogeneous Josephson junction vs normalized external magnetic flux $\Phi /{\Phi }_0$ perpendicular to the junction. (a) An S/F/S junction where the $F$ region has a wedged shape with inclination angle $\alpha$ and exchange field $h=10{\Delta }_0$. The inset panel zooms in on the interference pattern at small and large values of $\Phi$ and $\alpha$, respectively. (b) The $F$ region is now geometrically rectangular, but the magnitude of exchange field is now inhomogeneous according to the texture $\mathbf{h}=h(0,0,y/W)$. (c) An S/N/S junction where the $N$ region has a wedged shape with inclination angle $\alpha$. The legends are the same as in (a). In all cases, the height of the trapezoidal region is fixed at $W=10{\xi }_S$ while the upper base $d$ is equal to $2{\xi }_S$ (see Fig. 1). Therefore, $\alpha =0$ makes a rectangular junction ($L=d=2{\xi }_S$) and $\pi /130$ ($L=2.483{\xi }_S$), $\pi /67$ ($L=2.938{\xi }_S$), $\pi /43$ ($L=3.463{\xi }_S$), and $\pi /20$ ($L=5.168{\xi }_S$) make wedged junctions.

Figure 3

(a) Normalized spatial map of the pair potential for a S/N/S junction where $\alpha =0$, $\pi /43$ ($L=3.463{\xi }_S$) in the left and right frames, respectively. (b) The normalized pair potential in a S/F/S junction. Arrows indicate the location of proximity vortices. The zoom-in frame of a S/F/S trapezoidal junction with $\alpha =\pi /43$ is shown using a different color map. The magnetic flux through the $N$ or $F$ region is assumed to be $\Phi =4{\Phi }_0$ and no superconducting phase difference is applied ($\varphi =0$). (c) The $0\mathrm{\text{−}}\pi$ crossover profile of a rectangular S/F/S junction where $\Phi =0$, $3{\Phi }_0/2$, and $W=10{\xi }_S$ vs normalized junction length $d/{\xi }_S$. (d) Current density spatial map of the magnetic Josephson junction in the absence of external magnetic field, $\Phi =0$. Top and bottom frames exhibit current density flowing through the junction where $\alpha =0$, $\pi /43$, respectively ($\alpha =\pi /43$ constitutes a “$0\mathrm{\text{−}}\pi$” junction). Arrows indicate current directions.