Unraveling the Scaling Characteristics of Flux Pinning Forces in Bi_1.6Pb_0.4Sr₂Ca_2−xNa_xCu₃O_10+δ Superconductors

The temperature and magnetic field dependence of the critical current density (J_c) and pinning force density (F_p) in Bi_1.6Pb_0.4Sr₂Ca_2−xNa_xCu₃O_10+δ (x = 0, 0.02, 0.04, 0.06, 0.08, and 0.10) high-temperature superconductors is reported. Polycrystalline samples were prepared by a conventional solid-state reaction route. The influence of Na substitution on J_c and F_p at temperatures ranging between 65 K and 25 K was investigated. The collective pinning model is successfully applied to describe the magnetic field dependence of J_c for all the samples investigated. Magnetic field–temperature phase diagrams were also constructed to illustrate the effects of Na substitution in different field regions. The power-law dependence of the maximum F_p (F_p,max) versus the irreversibility field (B_irr) at different temperatures shows nearly unchanged exponent values, suggesting temperature independence of the flux pinning mechanism in the Na-substituted samples. The dominant pinning mechanism is confirmed to be δl pinning, induced by fluctuations in the mean free path of charge carriers.