Mixed state of a dirty two-band superconductor: application to MgB_2
  We investigate the vortex state in a two-band superconductor with strong intraband and weak interband electronic scattering rates. Coupled Usadel equations are solved numerically and the distributions of the pair potentials and local densities of states are calculated for two bands at different values of magnetic fields. The existence of two distinct length scales corresponding to different bands is demonstrated. The results provide qualitative interpretation of recent STM experiments on vortex structure imaging in MgB_2. 
A. E. Koshelev and A. A. Golubov