Characteristic Length for Pinning Force Density in Nb₃Sn

The pinning force density, Fp, is one of the main parameters that characterize the resilience of a superconductor to carrying a dissipative-free transport current in an applied magnetic field. Kramer (1973) and Dew-Hughes (1974) proposed a widely used scaling law for this quantity, where one of the parameters is the pinning force density maximum, Fp,max, which represents the maximal performance of a given superconductor in an applied magnetic field at a given temperature. Since the late 1970s to the present, several research groups have reported experimental data on the dependence of Fp,max on the average grain size, d, in Nb₃Sn-based conductors. Fp,maxd datasets were analyzed and a scaling law for the dependence Fp,maxd=A×ln1/d+B was proposed. Despite the fact that this scaling law is widely accepted, it has several problems; for instance, according to this law, at T=4.2 K and d≥650 nm, Nb₃Sn should lose its superconductivity, which is in striking contrast to experiments. Here, we reanalyzed the full inventory of publicly available Fp,maxd data for Nb₃Sn conductors and found that the dependence can be described by the exponential law, in which the characteristic length, δ, varies within a remarkably narrow range of δ=175±13 nm for samples fabricated using different technologies. The interpretation of this result is based on the idea that the in-field supercurrent flows within a thin surface layer (thickness of δ) near grain boundary surfaces (similar to London's law, where the self-field supercurrent flows within a thin surface layer with a thickness of the London penetration depth, λ, and the surface is a superconductor-vacuum surface). An alternative interpretation is that δ represents the characteristic length of the exponential decay flux pinning potential from the dominant defects in Nb₃Sn superconductors, which are grain boundaries.