Temperature-induced 0-ir coexistence in clean superconductor-ferromagnet-superconductor Josephson junctions

A clean ballistic superconductor-ferromagnet metal-superconductor (SFS) junction is studied within the quasiclassical theory of superconductivity. We show that the temperature-induced coexistence of the 0 and $\ensuremath{\pi}$ states and the transition $0\text{\ensuremath{-}}\ensuremath{\pi}$ results from the temperature-induced transition between stable and metastable states of the junction. The $\ensuremath{\pi}$ shift occurs in small but finite critical intervals ${{Z}_{c}}$ of the magnetic barrier influence parameter $Z=2dh/\ensuremath{\hbar}{v}_{0}$, where $h$ is the exchange energy and $d$ the barrier thickness. The width of ${{Z}_{c}}$ and the coexistence temperature ${T}_{0\ensuremath{\pi}}$ depend also on the reduced barrier thickness $\overline{d}=d/{\ensuremath{\xi}}_{0}$, where ${\ensuremath{\xi}}_{0}$ is the zero-temperature superconducting coherence length. The interval ${{Z}_{c}}$ increases with $\overline{d}$. For $T\ensuremath{\ne}{T}_{0\ensuremath{\pi}}$ the junction is in the 0 or $\ensuremath{\pi}$ state. In contrast to the dirty case, the critical current ${I}_{c}$ depends monotonously on temperature, changing the sign but not the magnitude at the transition point. The nonvanishing critical current ${I}_{c}({T}_{0\ensuremath{\pi}})$ is related to the existence of the second harmonic in the current-phase relation, which could be observed experimentally.