Two-dimensional arrays of superconducting and soft magnetic strips as dc magnetic metamaterials

We have theoretically investigated the magnetic response of two-dimensional (2D) arrays of superconducting and soft magnetic strips, which are regarded as models of dc magnetic metamaterials. The anisotropy of the macroscopic permeabilities depends on whether the applied magnetic field is parallel to the wide surface of the strips ($\mu_{\parallel}$) or perpendicular ($\mu_{\perp}$). For the 2D arrays of superconducting strips, $0<\mu_{\perp}/\mu_0\ll \mu_{\parallel}/\mu_0\simeq 1$, whereas for the 2D arrays of soft magnetic strips, $\mu_{\parallel}/\mu_0\gg\mu_{\perp}/\mu_0\simeq 1$, where $\mu_0$ is the vacuum permeability. We also demonstrate that strong anisotropy of the macroscopic permeability can be obtained for hybrid arrays of superconducting and soft magnetic strips, where $\mu_{\parallel}/\mu_0\gg 1\gg \mu_{\perp}/\mu_0>0$.