Self-consistent equilibrium of a two-dimensional electron system with a   reservoir in a quantizing magnetic field: Analytical approach
  An analytical approach has been developed to describe grand canonical equilibrium between a three dimensional (3D) electron system and a two dimensional (2D) one, an energy of which is determined self-consistently with an electron concentration. Main attention is paid to a Landau level (LL) pinning effect. Pinning means a fixation of the LL on a common Fermi level of the 2D and the 3D systems in a finite range of the magnetic field due to an electron transfer from the 2D to the 3D system. A condition and a start of LL pinning has been found for homogeneously broadened LLs. The electronic transfer from the 3D to the 2D system controls an extremely sharp magnetic dependency of an energy of the upper filled LL at integer filling of the LLs. This can cause a significant increase of inhomogeneous broadening of the upper LL that was observed in recent local probe experiments. 
V. G. Popov