Width of the $0\text{−}\pi$ phase transition in diffusive magnetic Josephson junctions

We investigate the Josephson current between two superconductors (S) which are connected through a diffusive magnetic junction with a complex structure $\left({\text{F}}_c\right)$. Using the quantum circuit theory, we obtain the phase diagram of 0 and $\pi$ Josephson couplings for ${\text{F}}_c$ being an insulator-ferromagnet-insulator (IFI) double barrier junction or an IFNFI structure (where N indicates a normal-metal layer). Compared to a simple SFS structure, we find that the width of the transition, defined by the interval of exchange fields in which a $0\text{−}\pi$ transition is possible, is increased by insulating barriers at the interfaces and also by the presence of the additional N layer. The widest transition is found for symmetric ${\text{F}}_c$ structures. The symmetric SIFNFIS presents the most favorable condition to detect the temperature-induced $0\text{−}\pi$ transition with a relative width, which is five times larger than that of the corresponding simple SFS structure.

Figure 1

(Color online) Quantum circuit model of the ${\text{SF}}_c\text{S}$ structure. The contact $F_c$ is discretized into $n$ nodes which are connected to each other by tunnel barriers of conductance $g_T$; $g_{S1\text{F}}$ and $g_{\text{FS}2}$ denote the conductances of S1F and FS2 interfaces, respectively. The inverse of the average level spacing, $\delta$, represents the leakage term due to a finite volume of a node; FIR represents a ferromagnetic insulating reservoir.

Figure 2

(Color online) (a) Normalized critical current, $I_c/I_0$, versus temperature for a SFS structure with the length $L_{\text{F}}/{\xi }_s=1.5$ and different exchange fields $h/T_c$. The inset shows the current-phase characteristic for $h/T_c=5.1$ in the vicinity of the $0\text{−}\pi$ transition temperature. (b) The corresponding $0\text{−}\pi$ phase diagram showing the phase boundaries up to the second transition. (c) Phase diagram of the first $0\text{−}\pi$ transition for different lengths of the junction $L_{\text{F}}/{\xi }_s$.

Figure 3

(Color online) (a) Normalized critical current, $I_c/I_0$, versus temperature of a symmetric SIFIS structure with $g_{S1\text{F}}=g_{\text{FS}2}=0.018g_T$ for different $h/T_c$, when $L_{\text{F}}/{\xi }_s=1.5$. The inset shows the corresponding $I\text{−}\varphi$ characteristic for $h/T_c=2.0$ in the vicinity of the $0\text{−}\pi$ transition temperature. (b) Phase diagram of $0\text{−}\pi$ transition for a symmetric SIFIS structure with $g_{S1\text{F}}=g_{\text{FS}2}=0.018g_T$ (solid line) and the corresponding asymmetric structure (dashed line) with $g_{S1\text{F}}=0.1g_{\text{FS}2}=0.1g_T$, when $L_{\text{F}}/{\xi }_s=1.5$. (c) The same as (b) but for a symmetric SIFNFIS structure of $g_{S1\text{F}}=g_{\text{FS}2}=0.018g_T$ with $L/{\xi }_s=1.5$ and various $L_{\text{N}}/{Ł}_{\text{F}}$.