Electrostatics of an inhomogeneous quantum Hall liquid

The distribution of electron density in the quantum Hall liquid is consider ed in the presence of macroscopic density gradient caused by side electrode s or inhomogeneous doping. In this case different Landau levels are occupie d in different regions of a sample. These regions are separated by "incompr essible liquid." It is shown that the applicability of the approach by Chkl ovskii et al. is substantiality restricted if the density gradient is not v ery large and disorder is important. Due to the fluctuations of the remote donor's density the liquid in the transition region cannot be considered as completely incompressible. In the typical situation, when the gap between Landau levels is not much larger than the energy of disorder, the transitio n region is a wide band where electron density, averaged over the fluctuati ons, is independent of magnetic field. The band is a random mixture od regi ons occupied by electrons of upper level, by holes of lower level, and by i ncompressible liquid. The width of this band is calculated and an analytica l expression for the fraction of incompressible liquid in different parts o f this band is given. [S0163-1829(99)06543-1].