Spin-controlled coexistence of 0 and $\pi$ states in $SFSFS$ Josephson junctions

Using the Keldysh-Usadel formalism, we theoretically study the 0-$\pi$ transition profiles and current-phase relations of magnetic $SFSFS$ and $SFSFFS$ Josephson nanojunctions in the diffusive regime. By allowing the magnetizations of the ferromagnetic layers to take arbitrary orientations, the strength and direction of the charge supercurrent flowing through the ferromagnetic regions can be controlled via the magnetization rotation in one of the ferromagnetic layers. Depending on the junction parameters, we find opposite current flow in the ferromagnetic layers, revealing that, remarkably, such configurations possess well-controlled 0 and $\pi$ states simultaneously, creating a three-terminal 0-$\pi$ spin switch. We demonstrate that the spin-controlled 0-$\pi$ profiles trace back to the proximity induced odd-frequency superconducting correlations generated by the ferromagnetic layers. It is also shown that the spin-switching effect can be more pronounced in $SFSFFS$ structures. The current-phase relations reveal the important role of the middle $S$ electrode, where the spin-controlled supercurrent depends crucially on its thickness and phase differences with the outer $S$ terminals.

Figure 1

Schematic of the two types of Josephson junctions considered: (i) $SFSFS$ and (ii) $SFSFFS$. We assume the $SF$ interfaces are located in the $zy$ plane while the $x$ axis is along the direction of Josephson current flow. The length of the middle superconductor $S$ and ferromagnets $F$ are $d_S$, $d_{F1}$, $d_{F2}$, and $d_{F3}$, respectively. Throughout the paper, we denote the ferromagnetic regions by $F_1$, $F_2$, and $F_3$ as labeled. The exchange field of the magnetic layers is assumed to have arbitrary orientation, $\vec{h}=(h_x,h_y,h_z)=h_0(cos\gamma sin\beta ,sin\gamma sin\beta ,cos\beta )$, in which $h_0$ is the amplitude of the exchange field. To analyze the system properties without loss of generality, all magnetizations are considered to reside in the $zy$ plane where $\gamma =0$, and, consequently, the magnetization orientations can be described solely by $\beta$. We therefore define ${\beta }_{1,2,3}$ for each magnetic layer.

Figure 2

Total critical supercurrent ($I_{\mathrm{tot}}$), singlet ($I_{S0}$), and odd-frequency ($I_{Sy}$, $I_{Sz}$) components of the Josephson $SFSFS$ structure. The current and the average of its components are shown as a function of exchange field orientation, ${\beta }_2$ (see Fig. 1). The averages are taken over the labeled magnetic regions. In the top row we assume $d_{F1}=d_{F2}=0.4{\xi }_S$ and $d_S=0.2{\xi }_S$. While in the bottom row $d_{F1}=0.4{\xi }_S$, $d_{F2}=0.7{\xi }_S$, and the thickness of middle superconducting lead is kept unchanged. We set the superconducting phase of the left and middle superconductors to be ${\theta }_L={\theta }_M=0$ and ${\theta }_R=\pi /2$ (corresponding to the maximum supercurrent in this case, see text), respectively.

Figure 3

The Josephson current and its components [singlet ($I_{S0}$) and odd-frequency triplet ($I_{Sy,z}$)] vs position in an $SFSFS$ junction for four values of the superconducting phase in the right terminal: ${\theta }_R=$ $\pi /8$, $\pi /4$, $3\pi /8$, $\pi /2$. We set $d_{F1}=0.4{\xi }_S$, $d_{F2}=0.7{\xi }_S$, $d_S=0.2{\xi }_S$, and ${\theta }_L={\theta }_M=0$. The top row represents results where ${\beta }_2=\pi /8$ while the bottom row shows results for ${\beta }_2=7\pi /8$. Throughout the paper we assume the exchange field direction in $F_1$ is oriented in the $z$ direction, so ${\beta }_1=0$. Different magnetic and superconducting regions are separated by vertical lines and marked by $F_1$, $F_2$, and $S$.

Figure 4

Total maximum supercurrent in each $F$ region against ${\beta }_2$, the magnetization orientation of $F_2$ in the $SFSFS$ junction. Also, the magnetization in $F_1$ is fixed along the $z$ axis, i.e., ${\beta }_1=0$. Here a thick middle $S$ terminal is considered with $d_S=4.0{\xi }_S$ corresponding to several tens of nanometers and $d_{F1}=0.4{\xi }_S$, $d_{F2}=0.7{\xi }_S$. (a) The macroscopic phase of the middle $S$ terminal is fixed at ${\theta }_M=\pi /4$ while in (b) this quantity is set to zero, ${\theta }_M=0$. In both cases, the macroscopic phase of the left superconducting terminal is ${\theta }_L=0$ while ${\theta }_R$ varies in order to find the maximum supercurrent.

Figure 5

(a) Supercurrent in a $SFSFFS$ junction where ${\beta }_1=0$, ${\beta }_2=\pi$, and ${\beta }_3$ is varied. The geometrical parameters correspond to $d_S=0.2{\xi }_S$, $d_{F1}=0.4{\xi }_S$, and $d_{F2}=d_{F3}=0.3{\xi }_S$. (b) Josephson current in the same structure with the same magnetization orientation but now with $d_{F2}=0.45{\xi }_S$, $d_{F3}=0.15{\xi }_S$, and unchanged $d_S$. As done previously, we set ${\theta }_L={\theta }_M=0$ and ${\theta }_R=\pi /2$. The supercurrent is conserved within each magnetic layer (see Figs. 1 and 3). Thus the current is the same in the double ferromagnet regions as clearly seen in Fig. 6.

Figure 6

$SFSFFS$ junction (see Fig. 1). The spatial profiles of the Josephson supercurrent and its components [singlet ($I_{S0}$) and odd-frequency triplet ($I_{Sy,z}$)]. The two magnetizations in $F_1$ and $F_2$ are fixed in an antiparallel configuration (${\beta }_1=0$, ${\beta }_2=\pi$) and three different magnetic orientations, ${\beta }_3$, are considered for $F_3$. The macroscopic phases, as before, are set to ${\theta }_L={\theta }_M=0$ and ${\theta }_R=\pi /2$. In all cases, the width of $F_1$ and middle $S$ electrode are fixed at $d_{F1}=0.4{\xi }_S$ and $d_S=0.2{\xi }_S$, respectively. The other geometrical parameters correspond to (top row) $d_{F2}=d_{F3}=0.3{\xi }_S$ and (bottom row) $d_{F2}=0.45{\xi }_S$, $d_{F3}=0.15{\xi }_S$. The vertical lines identify the interfaces of the junction among the magnetic and superconducting regions labeled by $F_1$, $F_2$, $F_3$, and $S$, respectively.