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

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 k·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.