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

TL;DR

This paper derives a reduced 2D Hamiltonian for hole systems in semiconductor heterostructures, capturing complex spin-orbit interactions and g-factors essential for spintronics applications.

Contribution

It provides the first comprehensive derivation of the effective 2D Hamiltonian from the 3D Luttinger Hamiltonian, including spin-orbit and g-factor dependencies.

Findings

Derived the effective 2D Hamiltonian for holes in heterostructures.

Identified two independent in-plane g-factors and their momentum dependencies.

Included the effects of lateral gate potential and magnetic field in the model.

Abstract

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) dynamics of holes is described by well known Luttinger Hamiltonian. However, most of recent spintronics applications are related to two dimensional 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 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 not known. In the present work we perform the reduction and hence derive the general effective Hamiltonian which describes spintronics effects in symmetric two-dimensional (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.