Theory of invariants-based formulation of ${\bf k}\cdot{\bf p}$ Hamiltonians with application to strained zinc-blende crystals

TL;DR

This paper develops a systematic invariants-based method combining group theory and ${\bf k}\cdot{\bf p}$ theory to accurately derive spin-dependent effective Hamiltonians for strained zinc-blende crystals, including all symmetry-allowed contributions.

Contribution

It introduces a comprehensive approach to identify all invariants in ${\bf k}\cdot{\bf p}$ Hamiltonians for zinc-blende crystals, accounting for strain effects on spin-dependent terms.

Findings

Derived all invariants for the effective Hamiltonian in zinc-blende crystals.

Identified all symmetry-allowed spin-dependent contributions including strain effects.

Provided explicit constants for spin-splitting and g-factor modifications.

Abstract

Group theoretical methods and ${\bf k}\cdot{\bf p}$ theory are combined to determine spin-dependent contributions to the effective conduction band Hamiltonian. To obtain the constants in the effective Hamiltonian, in general all invariants of the Hamiltonian have to be determined. Hence, we present a systematic approach to keep track of all possible invariants and apply it to the ${\bf k}\cdot{\bf p}$ Hamiltonian of crystals with zinc-blende symmetry, in order to obtain all possible contributions to effective quantities such as effective mass, g-factor and Dresselhaus constant. Further spin-dependent contributions to the effective Hamiltonian arise in the presence of strain. In particular, with regard to the constants $C_3$ and $D$ which describe spin-splitting linear in the components of ${\bf k}$ and ${\boldsymbol\varepsilon}$, considering all possible terms allowed by symmetry is crucial.