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

The working principle of magnetic force microscopy and scanning SQUID microscopy is introducing a magnetic source near a superconductor and measuring the magnetic field distribution near the superconductor, from which one can obtain the penetration depth. We investigate the magnetic field distribution near the surface of a magnetic superconductor when a magnetic source is placed close to the superconductor, which can be used to extract both the penetration depth λ _L and magnetic susceptibility χ by magnetic force microscopy or scanning SQUID microscopy. When the magnetic moments are parallel to the surface, one extracts λ _L/√1-4πχ. When the moments are perpendicular to the surface, one obtains λ _L. By changing the orientation of the crystal, one thus is able to extract both χ and λ _L.