Interacting holes in Si and Ge double quantum dots: from a multiband approach to an effective-spin picture

TL;DR

This paper investigates two-hole states in Si and Ge quantum dots, combining multiband theoretical approaches with entanglement measures to develop an effective spin model that accounts for complex band mixing and spin-orbit effects.

Contribution

It introduces a generalized Hubbard model and effective spin representation for holes in quantum dots, extending the simple two-spin Hamiltonian used for electrons.

Findings

High degree of J-mixing in ground and excited states

Light-hole component induces M-mixing and weak coupling between different spinor symmetries

Effective spin model justified for describing hole states in coupled quantum dots

Abstract

The states of two electrons in tunnel-coupled semiconductor quantum dots can be effectively described in terms of a two-spin Hamiltonian with an isotropic Heisenberg interaction. A similar description needs to be generalized in the case of holes due to their multiband character and spin-orbit coupling, which mixes orbital and spin degrees of freedom, and splits $J=3/2$ and $J = 1/2$ multiplets. Here we investigate two-hole states in prototypical coupled Si and Ge quantum dots via different theoretical approaches. Multiband $\boldsymbol{k}\cdot\boldsymbol{p}$ and Configuration-Interaction calculations are combined with entanglement measures in order to thoroughly characterize the two-hole states in terms of band mixing and justify the introduction of an effective spin representation, which we analytically derive a from generalized Hubbard model. We find that, in the weak interdot regime, the ground state and first excited multiplet of the two-hole system display -- unlike their electronic counterparts -- a high degree of $J$-mixing, even in the limit of purely heavy-hole states. The light-hole component additionally induces $M$-mixing and a weak coupling between spinors characterized by different permutational symmetries.