Long-Range Superharmonic Josephson Current

We consider a long superconductor-ferromagnet-superconductor junction with one spin-active region. It is shown that an odd number of Cooper pairs cannot have a long-range propagation when there is only one spin-active region. When the temperature is much lower than the Thouless energy, the coherent transport of two Cooper pairs becomes the dominant process and the superharmonic current-phase relation is obtained ($I\propto \sin 2\varphi$).

Figure 1

The first two harmonics in the Josephson current-phase relation. The first one ($I_1$) consists of two normal Andreev reflections from $SF$ interfaces. The second one ($I_2$) has two contributions: the one with four normal Andreev reflections (short-range), the other with two normal and two anomalous Andreev reflections (long-range, total phase acquired in $F$ layer is zero). The solid (dotted) lines represent electron (hole) excitations in the $F$ layer. The red (vertical) arrows represent spin projections, while the black (horizontal) arrows denote the excitation velocity direction. The right $SF$ boundary (hatched) is spin-active.

Figure 2

The long-range component of the second (thick solid curve) and the fourth (thin solid curve) harmonic of the Josephson current-phase relation. For $Z^{\prime }\gg 1$, both the curves converge to constant values (dotted lines), ${\overline{I}}_2$ and ${\overline{I}}_4$.

Figure 3

The spectral weights of the current-phase relation for $Z^{\prime }=0.5$ and $Z=1000$, and for three values of the relative angle between magnetizations: $\theta =\pi /10$, $\pi /4$, $\pi /2$. The inset depicts the generic CPR given by Eq. (13). The Josephson current is normalized by $e{R}_N/(4E_T)$, where $E_T=\hslash v_F/L$ is the Thouless energy.