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

We study the proximity effect in ${\text{SF}}^{\ensuremath{'}}(\text{AF}){\text{F}}^{\ensuremath{'}}\text{S}$ and ${\text{SF}}^{\ensuremath{'}}(\text{F}){\text{F}}^{\ensuremath{'}}\text{S}$ planar junctions, where S is a clean conventional ($s$-wave) superconductor, while ${\text{F}}^{\ensuremath{'}}$ 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 ${\text{F}}^{\ensuremath{'}}$ 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 ${d}^{\ensuremath{'}}\ensuremath{\approx}\ensuremath{\hbar}{v}_{F}/2{h}^{\ensuremath{'}}$, where ${d}^{\ensuremath{'}}$ is the ${\text{F}}^{\ensuremath{'}}$ layer thickness and ${h}^{\ensuremath{'}}$ 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)].