Force between a magnetic tip and an inhomogeneous superconductor at zero field

The force exerted by a semi-infinite inhomogeneous superconductor with a planar interface to vacuum on a magnetic tip is studied theoretically in the absence of external magnetic fields. It is shown that the force has a contribution from inhomogeneities due to material defects with unique characteristics. Defects are taken into account in the London limit by allowing the mass parameter to vary spatially. The contribution from defects to the force is calculated analytically to first order in the deviation of the mass parameter from its constant value for the homogeneous superconductor, assuming that the tip is a point dipole perpendicular to the interface, and that it does not spontaneously create vortex matter. Random point defects and linear localized defects are considered phenomenologically. For each defect type the force dependence on the dipole position coordinates is obtained, and the force magnitudes are estimated numerically. The predictions for the dependence of the linear defect force on the dipole lateral position are found to agree qualitatively with experiment.