Long-range spin-triplet proximity effect in Josephson junctions with multilayered ferromagnets

We study the proximity effect in ${\text{SF}}^{\prime }\left(\text{AF}\right){\text{F}}^{\prime }\text{S}$ and ${\text{SF}}^{\prime }\left(\text{F}\right){\text{F}}^{\prime }\text{S}$ planar junctions, where S is a clean conventional ($s$-wave) superconductor, while ${\text{F}}^{\prime }$ and middle layers are clean or moderately diffusive ferromagnets. Middle layers consist of two equal ferromagnets with antiparallel (AF) or parallel (F) magnetizations that are not collinear with magnetizations in the neighboring ${\text{F}}^{\prime }$ layers. We use fully self-consistent numerical solutions of the Eilenberger equations to calculate the superconducting pair amplitudes and the Josephson current for arbitrary thickness of ferromagnetic layers and the angle between in-plane magnetizations. For moderate disorder in ferromagnets, the triplet proximity effect is practically the same for AF and F structures, like in the dirty limit. Triplet Josephson current is dominant for $d^{\prime }\approx \hslash v_F/2h^{\prime }$, where $d^{\prime }$ is the ${\text{F}}^{\prime }$ layer thickness and $h^{\prime }$ is the exchange energy. Our results are in a qualitative agreement with the recent experimental observations [T. S. Khaire, M. A. Khasawneh, W. P. Pratt, and N. O. Birge, Phys. Rev. Lett. 104, 137002 (2010)].

Figure 1

(Color online) Spatial dependence of singlet and triplet pair amplitudes $F_s$, $F_{t0}^{<}$, and $F_{t1}^{<}$, normalized to the bulk singlet amplitude $F_{sb}$, for $T/T_c=0.1$, $h^{\prime }/E_F=0.05$, $h/E_F=0.1$, $2d=500k_F^{-1}$, $d^{\prime }=20k_F^{-1}$, $\alpha =\pi /2$ and two values of the mean-free path: $l=\infty$ (dashed curves) and $l=200k_F^{-1}$ (solid curves). All amplitudes are calculated for the ground state, $\varphi =0$. The ${\text{SF}}^{\prime }\left(\text{AF}\right){\text{F}}^{\prime }\text{S}$ junction geometry is shown in the background; arrows and circles show orientation of magnetizations for each layer.

Figure 2

(Color online) Dependence of the Josephson critical current on the ${\text{F}}^{\prime }$ ferromagnetic layer thickness $d^{\prime }$, for $2d=500k_F^{-1}$, $T/T_c=0.1$, $h^{\prime }/E_F=0.05$, $h/E_F=0.3$, $l=200k_F^{-1}$, and for three types of junctions: SNS, ${\text{SF}}^{\prime }\left(\text{AF}\right){\text{F}}^{\prime }\text{S}$, and ${\text{SF}}^{\prime }\left(\text{F}\right){\text{F}}^{\prime }\text{S}$.

Figure 3

Dependence of the Josephson critical current on the middle layer thickness $2d$, for $d^{\prime }=20k_F^{-1}$, $T/T_c=0.1$, $h^{\prime }/E_F=0.05$, $h/E_F=0.3$, $l=200k_F^{-1}$, and for two values of the misorientation angle: $\alpha =\pi /2$ (thick and thin solid lines for AF and F structures) and for $\alpha =0$ (AF structure, dashed line). Arrows and circles show orientation of magnetizations in ferromagnetic layers.

Figure 4

The current-phase relation $I\left(\varphi \right)$ for ${\text{SF}}^{\prime }\left(\text{AF}\right){\text{F}}^{\prime }\text{S}$ junction: $d^{\prime }=20k_F^{-1}$, $2d=600k_F^{-1}$, $T/T_c=0.1$, $h^{\prime }/E_F=0.05$, $h/E_F=0.3$, $l=200k_F^{-1}$, and $\alpha =\pi /2$.