When a three-dimensional electron gas is subjected to a very strong magnetic field, it can reach a quasi-one-dimensional state in which all electrons occupy the lowest Landau level. This state is referred to as the extreme quantum limit (EQL) and has been studied in the physics of pulsars and bulk semiconductors. Here we present a theory of the EQL phase in electron accumulation layers created by an external electric field E at the surface of a semiconductor with a large Bohr radius such as InSb, PbTe, SrTiO3 (STO), and particularly in the LaAlO3/SrTiO3 (LAO/STO) heterostructure. The phase diagram of the electron gas in the plane of the magnetic field strength and the electron surface concentration is found for different orientations of the magnetic field. We find that in addition to the quasi-classical metallic phase (M), there is a metallic EQL phase, as well as an insulating Wigner crystal state (WC). Within the EQL phase, the Thomas?Fermi approximation is used to find the electron density and the electrostatic potential profiles of the accumulation layer. Additionally, the quantum capacitance for each phase is calculated as a tool for experimental study of these phase diagrams.

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