B(H) Constitutive Relations Near H_c1 in Disordered Superconductors

We provide a self-contained account of the B vs. H constitutive relation near H_c1 in Type II superconductors with various types of quenched random disorder. The traditional Abrikosov result B ~ [ln (H - H_c1)]^{-2}, valid in the absence of disorder and thermal fluctuations, changes significantly in the presence of disorder. Moreover, the constitutive relations will depend strongly on the type of disorder. In the presence of point disorder, B ~ (H - H_c1)^{3/2} in three-dimensional (thick) superconductors, as shown by Nattermann and Lipowsky. In two-dimensional (thin film) superconductors with point disorder, B ~ (H - H_c1). In the presence of parallel columnar disorder, we find that B ~ exp[-C / (H - H_c1)] in three dimensions, while B ~ exp[-K / (H - H_c1)^{1/2}] in two dimensions. In the presence of nearly isotropically splayed disorder, we find that B ~ (H - H_c1)^{3/2} in both two and three dimensions.