Geometrical barriers and the growth of flux domes in thin ideal superconducting disks

When an ideal (no bulk pinning) flat type-II superconducting disk is subjected to a perpendicular magnetic field H_a, the first vortex nucleates at the rim when H_a = H_0, the threshold field, and moves quickly to the center of the disk. As H_a increases above H_0, additional vortices join the others, and together they produce a domelike field distribution of radius b. In this paper I present analytic solutions for the resulting magnetic-field and sheet-current-density distributions. I show how these distributions vary as b increases with H_a, and I calculate the corresponding field-increasing magnetization.