Theory for measurements of penetration depth in magnetic superconductors by magnetic force microscopy and scanning SQUID microscopy

We investigate the magnetic field distribution near the surface of a magnetic superconductor when a magnetic source is placed close to the superconductor. The magnetic field distribution can be measured by magnetic force microscopy and scanning SQUID microscopy, from which one can extract both the penetration depth $\lambda_L$ and magnetic susceptibility $\chi$. When the magnetic moments are parallel to the surface, one extracts $\lambda_L/\sqrt{1-4\pi \chi}$. When the moments are perpendicular to the surface, one obtains $\lambda_L$. By changing the orientation of the crystal, one thus is able to extract both $\chi$ and $\lambda_L$.