Two-dimensional arrays of superconducting strips as dc magnetic metamaterials

We theoretically investigate the magnetic response of two-dimensional arrays of superconducting strips, which are regarded as essential structures of dc magnetic metamaterials. We analytically obtain local distributions of the magnetic field for the ideal complete shielding state (i.e., $\Lambda/w\to 0$, where $2w$ is the strip width, $\Lambda=\lambda^2/d$ is the Pearl length, $\lambda$ is the London penetration depth, and $d$ is the strip thickness), and derive effective permeability by averaging the local field distributions. We also perform numerical calculations for a realistic case, taking finite $\Lambda/w>0$ into account. We investigate two types of strip arrays: a rectangular array and a hexagonal array. The resulting effective permeability has large anisotropy that depends on the dimensions and arrangement of the superconducting strips, and the hexagonal array is found to be more advantageous for obtaining large anisotropy than the rectangular array.