# Spin-orbit coupling: atom versus semiconductor crystal

## TL;DR

This paper presents a novel approach to understanding spin-orbit coupling in semiconductors, emphasizing the role of crystal symmetry and degeneracy, and clarifies the nature of exciton-photon interactions.

## Contribution

It introduces a symmetry-based method for analyzing valence band splitting due to spin-orbit coupling without relying on atomic-like angular momentum concepts.

## Key findings

- Valence band eigenstates are determined by crystal symmetry and degeneracy.
- Atomic and semiconductor spin-orbit eigenstates share the same structure despite different potentials.
- A simple derivation of exciton-photon interaction based on spin conservation.

## Abstract

We reconsider a key point in semiconductor physics, the splitting of the valence band states induced by the spin-orbit interaction, through a novel approach which uses neither the group theory formalism, nor the usual $\textbf{L}\cdot\textbf{S}$ formulation valid for atoms but conceptually incorrect for periodic lattices, the angular momenta $\textbf{L}$ and $\textbf{J}$ having no meaning due to the absence of spherical symmetry. We show that for zinc-blende structures, the valence band eigenstates resulting from spin-orbit coupling are uniquely determined by: (i) the equivalence of the ($x,y,z$) crystal axes, (ii) the three-fold degeneracy of the valence band. The fact that these two conditions are also fulfilled by atomic $p$ states allows us to understand why the spin-orbit eigenstates for three-fold atomic and valence electrons have exactly the same structure, albeit the drastic differences in the potential and electronic symmetries. We also come back to the commonly accepted understanding of the exciton-photon interaction in terms of bright and dark excitons having total angular momenta $J=(1,2)$ respectively and present a simple derivation of this interaction which only relies on spin conservation.