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arXiv:2104.07730 · Physical review. B./Physical review. B

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

Andrea Secchi, Laura Bellentani, Andrea Bertoni, Filippo Troiani

Published Jul 12, 2021• 11 citations
arXivHTMLPDF

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 $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ 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.

DOI
10.1103/physrevb.104.035302

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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=3/2j=3/2j=3/2 and j=1/2j=1/2j=1/2 multiplets. Here we investigate two-hole states in prototypical coupled Si and Ge quantum dots via different theoretical approaches. Multiband \mathbitk\ifmmode⋅\else\textperiodcentered\fi\mathbitp\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}\mathbitk\ifmmode⋅\else\textperiodcentered\fi\mathbitp 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 JJJ mixing, even in the limit of purely heavy-hole states. The light-hole component additionally induces MMM mixing and a weak coupling between spinors characterized by different permutational symmetries.
Two electron spins in quantum dots coupled through coherent tunneling are generally acknowledged to approximately obey Heisenberg isotropic exchange. This has not been established for two holes. Here we measure the spectra of two holes and of two electrons in two vertically stacked self-assembled InAs quantum dots using optical spectroscopy as a function of electric and magnetic fields. We find that the exchange is approximately isotropic for both systems, but that significant asymmetric contributions, arising from spin-orbit and Zeeman interactions combined with spatial asymmetries, are required to explain large anticrossings and fine-structure energy splittings in the spectra. Asymmetric contributions to the isotropic Hamiltonian for electrons are of the order of a few percent while those for holes are an order of magnitude larger.
Disentangling orbital and confinement contributions to ggg-factor in Ge/SiGe hole quantum dots
Spin qubits are typically operated in the lowest orbital of a quantum dot to minimize interference from nearby states. In valence-band hole systems, strong spin-orbit coupling links spin and orbital degrees of freedom, strongly influencing the hole ggg-factor, a key parameter for qubit control. We investigate the out-of-plane ggg-factor in Ge quantum dots using excitation (single-particle) and addition (many-body) spectra. Excitation spectra allow us to distinguish the pure Zeeman ggg-factor from orbital contributions to the magnetic field splitting of states despite the strong spin-orbit coupling. This distinction clarifies discrepancies between ggg-factors extracted with the two methods, for different orbital states and different hole numbers. Furthermore, we find gate-tunability of ggg-factors at the level of 15%, highlighting its relevance for all-electric qubit manipulation.
Intrinsic spin--orbit interaction in germanium provides electrically active flopping-mode qubits without requiring magnetic-field gradients to generate spin--charge hybridization. We study double quantum dots (DQDs) with a multiband Luttinger--Kohn framework that retains heavy-hole (HH) and light-hole (LH) states on equal footing. Modeling vertical confinement with a finite confinement potential along the growth direction brings selected subbands into a regime of appreciable HH--LH mixing. This admixture activates electric-dipole spin-coupling through LH--LH and LH--HH transitions, complementing the conventional HH--HH pseudospin-flip resonance. For the parameter sets considered, these additional channels can produce larger spin--photon figures of merit than the HH--HH transition. HH--LH mixing thus supplies a tunable ingredient for qubit--cavity coupling in germanium hole-spin qubits and extends the operating regimes available to circuit-QED architectures.
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Abstract Spin qubits in semiconductor quantum dots offer a gate-tunable platform for quantum information processing. While two-qubit interactions are typically realized through exchange coupling between neighboring spins, coupling spin qubits to photons via hybrid spin-circuit QED (cQED) devices enables long-range interactions and integration with other cQED platforms. Here, we investigate hole spin–photon coupling in compact single quantum dot setups. By incorporating ubiquitous strain inhomogeneities to our theory, we identify three main spin–photon coupling channels: a vector-potential-spin–orbit geometric mechanism–dominant for vertical magnetic fields–, an inhomogeneous Rashba term generalizing previous spin–orbit field models, and strain-induced g -tensor terms–most relevant for in-plane fields. Comparing Si, unstrained (relaxed) Ge, and biaxially strained Ge wells, we find that Si and unstrained Ge provide optimal coupling strengths (tens of MHz) thanks to their reduced heavy-hole, light-hole splitting. We demonstrate efficient switching of the spin–photon coupling while preserving sweet spot conditions by operating with an in-plane magnetic field. Finally, we evaluate quantum state transfer and two-qubit gate protocols, achieving ≳ 99 % fidelity for state transfer and > 90 % for two-qubit gates with realistic coherence times, establishing single-dot hole spins as a viable platform for compact spin-cQED architectures and highlighting unstrained Ge as a promising candidate for spin–photon interactions.