Odd spin-triplet superconductivity in a multilayered superconductor-ferromagnet Josephson junction

Abstract

We study the dc Josephson effect in a diffusive multilayered ${\text{SF}}^{\ensuremath{'}}{\text{FF}}^{\ensuremath{'}}\text{S}$ structure, where S is a superconductor and F and ${\text{F}}^{\ensuremath{'}}$ are different ferromagnets. We assume that the exchange energies in the ${\text{F}}^{\ensuremath{'}}$ and F layers are different ($h$ and $H$, respectively) and the middle F layer consists of two layers with parallel or antiparallel magnetization vectors $M$. The $M$ vectors in the left and right ${\text{F}}^{\ensuremath{'}}$ layers are generally not collinear to those in the F layer. In the limit of a weak proximity effect we use a linearized Usadel equation. Solving this equation, we calculate the Josephson critical current for arbitrary temperatures, arbitrary thicknesses of the ${\text{F}}^{\ensuremath{'}}$ and F layers (${L}_{h}$ and ${L}_{H}$) in the case of parallel and antiparallel $M$ orientations in the F layer. The part of the critical current ${I}_{c\text{SR}}$ formed by the short-range singlet and $S=0$ triplet condensate components decays on a short length ${\ensuremath{\xi}}_{H}=\sqrt{D/H}$, whereas the part ${I}_{c\text{LR}}$ due to the long-range triplet $|S|=1$ component decreases with increasing ${L}_{H}$ on the length ${\ensuremath{\xi}}_{N}=\sqrt{D/\ensuremath{\pi}T}$. Our results are in a qualitative agreement with the experiment [T. S. Khaire, M. A. Khasawneh, W. P. Pratt, Jr., and N. O. Birge, Phys. Rev. Lett. 104, 137002 (2010)].