Spontaneous Currents in Superconducting Systems with Strong Spin-Orbit Coupling

We show that Rashba spin-orbit coupling at the interface between a superconductor and a ferromagnet should produce a spontaneous current in the atomic thickness region near the interface. This current is counterbalanced by the superconducting screening current flowing in the region of the width of the London penetration depth near the interface. Such a current-carrying state creates a magnetic field near the superconductor surface, generates a stray magnetic field outside the sample edges, changes the slope of the temperature dependence of the critical field $H_{c3}$, and may generate the spontaneous Abrikosov vortices near the interface.

Figure 1

Hybrid superconductor ($S$)-ferromagnet ($F$) systems where the spin-orbit coupling produces spontaneous currents. (a) $S/F$ bilayer, (b) $S/F/S$ sandwich, and (c) bulk superconductor with the cylindrical ferromagnetic core. The direction of the spontaneous current $\mathbf{J}$ is shown with green arrows, the corresponding profile of the spontaneous magnetic field in the absence of external magnetic field is plotted schematically in orange, the distribution of the order parameter $\psi$ is shown with the blue curve. The magenta curves show the magnetic field profiles in the presence of the external magnetic field ${\mathbf{H}}_0$ directed along the $z$ axis. The unit vector $\mathbf{n}$ is the vector in the direction of the broken inversion symmetry at the $S/F$ interfaces, which determines the energy of the spin-orbit coupling.

Figure 2

The behavior of the critical field $H_{c3}$ in the presence of the ferromagnetic layer with the spin-orbit coupling in the case $ϵ>0$. The red and blue curves correspond to the different relative orientation between the exchange field $\mathbf{h}$ and the external field ${\mathbf{H}}_0$, while the black dashed line shows the field $H_{c3}$ in the absence of the $F$ layer.

Figure 3

The dependence of the magnetic field $B_z$ inside the ferromagnetic cylinder as a function of the applied magnetic field $H_{0z}$ (red curve). The dashed blue curve corresponds to the case when there is no spin-orbit coupling in the system, the black dash-dotted curve shows the dependence $B_z(H_{0z})$ for the case $M_z=0$ and ${\alpha }_{so}=0$. The radius of the $F$ cylinder is $R=\lambda$, the parameter ${\alpha }_{so}{h}_z=0.1{\mathrm{\Phi }}_0/\pi$, $H_c={\mathrm{\Phi }}_0/\pi {\lambda }^2$, $M_z=4.5H_c$.