Magnetic levitation force and penetration depth in type-II superconductors

The superconducting levitation force F acting on a magnet placed above a type-II superconductor in both Meissner and mixed states is calculated as a function of temperature, based upon the London model. A simple relationship between the levitation force and the London penetration depth λ is found. In particular, in the limit of a/λ≫1, where a is the separation between the magnet and the superconductor, F varies linearly with λ, regardless of the shape of the magnet. The temperature dependences of λ and F are examined for various superconducting pairing states, including s-wave, d-wave, and s+id states. It is found that, at low temperatures, both λ and F show an exponential temperature dependence for s-wave, linear-T for d-wave, and ${\mathit{T}}^2$ dependence in a wide low-temperature range for the s+id state with a dominant d-wave component. The magnetic force microscope (MFM) is proposed to accurately measure the temperature-dependent levitation force. It is shown that the microscopic size of the MFM tip enables one to obtain the intrinsic temperature-dependent penetration depth of a single grain, in spite of the overall quality of the superconducting sample.