Anisotropy and effective dimensionality crossover of the fluctuation conductivity of hybrid superconductor/ferromagnet structures

We study the fluctuation conductivity of a superconducting film, which is placed to perpendicular non-uniform magnetic field with the amplitude $H_0$ induced by the ferromagnet with domain structure. The conductivity tensor is shown to be essentially anisotropic. The magnitude of this anisotropy is governed by the temperature and the typical width of magnetic domains $d$. For $d\ll L_{H_0}=\sqrt{\Phi_0/H_0}$ the difference between diagonal fluctuation conductivity components $\Delta\sigma_\parallel$ along the domain walls and $\Delta\sigma_\perp$ across them has the order of $(d/L_{H_0})^4$. In the opposite case for $d\gg L_{H_0}$ the fluctuation conductivity tensor reveals effective dimensionality crossover from standard two-dimensional $(T-T_c)^{-1}$ behavior well above the critical temperature $T_c$ to the one-dimensional $(T-T_c)^{-3/2}$ one close to $T_c$ for $\Delta\sigma_\parallel$ or to the $(T-T_c)^{-1/2}$ dependence for $\Delta\sigma_\perp$. In the intermediate case $d\approx L_{H_0}$ for a fixed temperature shift from $T_c$ the dependence $\Delta\sigma_\parallel(H_0)$ is shown to have a minimum at $H_0\sim\Phi_0/d^2$ while $\Delta\sigma_\perp(H_0)$ is a monotonically increasing function.