Magnetic force microscopy measurement of the penetration depth in superconductors from Meissner repulsion

We report a method to locally measure the penetration depth $λ$ in a superconductor by detecting the diamagnetic response using magnetic force microscopy (MFM). We extract $λ$ by fitting the height dependence of the levitation force in the Meissner state using an analytical model that approximates the MFM tip as a single-domain, truncated conical shell. We demonstrate on two YBa$_2$Cu$_3$O$_{6+x}$ single crystals with two MFM tips that the obtained values agree well with previous results. This approach is not affected by the tip width and can be applied to similar but not identical tips.