Dimensional reduction of the Luttinger Hamiltonian and g -factors of holes in symmetric two-dimensional semiconductor heterostructures

The spin-orbit interaction of holes in zinc-blende semiconductors is much stronger than that of electrons. This makes the hole systems very attractive for possible spintronics applications. In three dimensions (3D), the dynamics of holes is described by well-known Luttinger Hamiltonian. However, most recent spintronics applications are related to two-dimensional (2D) heterostructures where dynamics in one direction is frozen due to quantum confinement. The confinement results in dimensional reduction of the Luttinger Hamiltonian, 3D→2D. Due to the interplay of the spin-orbit interaction, the external magnetic field, and the lateral gate potential imposed on the heterostructure, the reduction is highly nontrivial and as yet unknown. In the present work we perform the reduction and hence derive the general effective Hamiltonian which describes spintronics effects in symmetric 2D heterostructures. In particular, we do the following: (i) derive the spin-orbit interaction and the Darwin interaction related to the lateral gate potential, (ii) determine the momentum-dependent out-of-plane g-factor, (iii) point out that there are two independent in-plane g-factors, (iv) determine momentum dependencies of the in-plane g-factors.