Scanning SQUID susceptometry of a paramagnetic superconductor

Scanning SQUID susceptometry images the local magnetization and susceptibility of a sample. By accurately modeling the SQUID signal we can determine physical properties such as the penetration depth and permeability of superconducting samples. We calculate the scanning SQUID susceptometry signal for a superconducting slab of arbitrary thickness with isotropic London penetration depth $\ensuremath{\lambda}$ on a nonsuperconducting substrate, where both slab and substrate can have a paramagnetic response that is linear in the applied field. We derive analytical approximations to our general expression in a number of limits. Using our results, we fit experimental susceptibility data as a function of the sample-sensor spacing for three samples: (1) $\ensuremath{\delta}$-doped SrTiO${}_{3}$, which has a predominantly diamagnetic response, (2) a thin film of LaNiO${}_{3}$, which has a predominantly paramagnetic response, and (3) the two-dimensional electron layer at a SrTiO${}_{3}$/LaAlO${}_{3}$ interface, which exhibits both types of response. These formulas will allow the determination of the concentrations of paramagnetic spins and superconducting carriers from fits to scanning SQUID susceptibility measurements.