Landau quantization of a two-dimensional electron with the nonparabolic dispersion law, pseudospin components and chirality terms

Abstract

The Landau quantization of a two-dimensional electron in a perpendicular magnetic field on the basis of a Hamiltonian with two pseudospin components is considered. The diagonal elements of the Hamiltonian have non-parabolic but circular symmetric dispersion laws, whereas the off diagonal elements contain the chirality terms of different degrees. The solution of the matrix form Schrödinger equation was found following the method proposed by Rashba in his theory of spin–orbit coupling, taking into account different degrees of chirality and deviations on the parabolic dispersion law. The Landau quantization Hamiltonians were obtained by substituting the canonical momentum operators by the kinetic momentum operators. Two concrete examples were discussed. One of them concerns the Mexican hatlike dispersion law in the biased bilayer graphene with second order chirality, when the Landau quantization levels except two are characterized by two quantum numbers ( n − 2 ) and n for n ≥ 2 , corresponding to different pseudospin projections. They differ by 2 as the degree of chirality is. There are two energy levels E ± ( n − 2 , n ) with the same numbers ( n − 2 ) and n . The lower energy levels E − ( n − 2 , n ) have a linear decreasing behavior with dependence on the magnetic field strength H with different slopes and minima for different values of n ≥ 2 . At the intersection point H t h , two energy levels E − ( 1 , 3 ) and E − ( 0 , 2 ) have the same energy forming two degenerate LLLs. Touching the minima at different values of H , the energy branches gradually transform in the increasing quadratic dependences proportional to ( 2 n + 1 ) 2 H 2 . The similar results were obtained in the case of cosine-type dispersion law in the frame of one-band model.