Effective Hamiltonian, energy spectrum, and phase transition induced by in-plane magnetic field in symmetric HgTe quantum wells

Abstract

The effective $6\ifmmode\times\else\texttimes\fi{}6$ matrix Hamiltonian for two-dimensional states in HgTe/CdTe quantum wells is derived. The use of the extended basis (in contrast to the previously studied $4\ifmmode\times\else\texttimes\fi{}4$ matrix Hamiltonian) allows us to describe quantum wells with arbitrary orientation of interfaces and investigate the influence of in-plane magnetic field. The Hamiltonian is applied to calculation of energy spectra of both two-dimensional subbands and edge states in the range of well widths corresponding to the topological insulator phase. It is found that the in-plane magnetic field opens a gap in the edge-state spectrum, and the increase in the field causes disappearance of one of the edge-state branches. A strong (about 10 T) magnetic field induces a phase transition from the gapped state to a gapless two-dimensional state, with energy spectrum similar to that of bulk HgTe. An analytical expression for the transition field through the parameters of the effective Hamiltonian is obtained.