Cooper pairing of two-dimensional electrons in a quantizing magnetic field and the fractional quantum Hall effect

Abstract

A brief review of the current state of the theory of fractional quantum Hall effect (FQHE) is given along with the assumption of possible connection between the experimentally observed features of the Hall resistance RH of a two-dimensional (2D) electron system in a strong quantizing magnetic field for a fractional filling factor of the lowest Landau level ν=q/(2n+1) with q⩾2, which cannot be described by the Laughlin wave function antisymmetric relative to pair transpositions, and the Cooper pairing of 2D electrons. It is assumed that the electron–electron attraction essential for Cooper pairing can be due to the interaction of 2D electrons with the surface acoustic waves (2D phonons) and the surface 2D plasmons localized near the crystal interfaces (heterojunctions) in the vicinity of inversion layers in the metal–insulator–semiconductor (MIS) structures and heterostructures. The coexistence of coupled electron pairs and unpaired electrons under the FQHE conditions must lead to peculiarities of RH for values of ν described by the Halperin relation following from the symmetry properties of the “mixed” wave function of pairs (bosons) and electrons (fermions). This relation makes it possible in principle to describe all experimental data on FQHE. The summation of “ladder” diagrams diverging according to a power law for T→0 leads to a Bethe–Salpeter-type equation for the vertex part of the electron–electron interaction for a 2D system in a quantizing magnetic field taking into account electron–electron and electron–hole pairing in the Cooper and zero-sound channels. This equation is used to calculate the critical temperature Tc of the phase transition to the state with coupled Cooper pairs and to prove that the value of Tc in the ultra-quantum limit is independent of the effective mass of electrons, i.e., on the 2D density of states. The phase diagram of the 2D system is constructed for the variable electron concentration and magnetic field. It is shown that the region of Cooper pairing of 2D electrons in the case of strong attraction almost coincides with the FQHE region for ν<1, while the region of electron-hole pairing with the formation of charge-density waves (CDW) is expelled to the region with ν>1, which is in accord with the experimental data concerning the CDW-induced features of the longitudinal resistance Rxx for ν=(2n+1)/2 with n⩾2.