Asymptotic theory for the inverse problem in magnetic force microscopy of superconductors

An asymptotic theory is formulated, which allows us to recover the London penetration depth λ in superconductors from magnetic force microscopy measurements. An ad hoc interpretation of the forward problem allows us to accomplish a complex variable based power series reversion scheme. The asymptotic series expansion of λ can be generated in terms of measurable quantities. Simulations for analytically tractable situations have been performed which confirm the stability of our approach, even for cases where noise corrupted data is considered. The method does not restrict to any particular symmetry and is suited for covering the whole temperature range λ(T), with obvious implications on the knowledge of the superconducting pairing state. By comparing with the analytic model of Coffey [Phys. Rev. B 57, 11 648 (1998)] we discuss the limitations and possible extensions of the existing theory.