Harper - Hofstadter problem for 2 D electron gas with k - linear Rashba spin - orbit coupling

– The Harper-Hofstadter problem for two-dimensional electron gas with Rashba spin-orbit coupling subject to periodic potential and perpendicular magnetic field is studied analytically and numerically. The butterfly-like energy spectrum, spinor wave functions as well as the spin density and average spin polarization are calculated for actual parameters of semiconductor structure. Our calculations show that in two-dimensional electron gas subject to periodic potential and uniform magnetic field the effects of energy spectrum splitting caused by large spin-orbit Rashba coupling can be observed experimentally. Introduction. – The problem of quantum states of an electron subject to both periodic potential and homogeneous magnetic field remains to be actual for several last decades. There are various approaches and models for periodic potential and different approximations which are used for investigations of these states (see, ref. [1]). However, the spin-orbit interaction is usually excluded from the models as well as the Zeeman term. Such approach can be justified as long as the amplitude of the periodic potential is big enough to make the Landau level splitting much greater than the typical spin-orbit coupling energy and the Zeeman term. At the same time, under the realistic experimental conditions with 2D electron gas subject to the potential of lateral superlattice the potential amplitude V0 can be of the same order as the spin-orbit (SO) Rashba coupling. For example, in recent papers [2,3] where a pioneering step into experimental observation of 2D magnetic Bloch states has been done, the magnitude of V0 was around 1-5 meV. In semiconductors structures with large SO coupling [4] the typical splitting energy can be of the same order as V0. In this paper we study magnetic Bloch states of 2D electrons subject to both periodic potential of a lateral superlattice and perpendicular magnetic field under the conditions when the SO coupling and Zeeman term should be taken into consideration. The energy band structure is calculated in the magnetic Brillouin zone (MBZ) and the magnetic Bloch states are constructed. Also, the spin density distribution in the elementary cell is obtained and the average spin polarization in a state with given quasimomentum are calculated. It is shown that the consideration of spin-orbit coupling is necessary for interpretation of realistic experiments.