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arXiv:2602.09913

Disentangling orbital and confinement contributions to $g$-factor in Ge/SiGe hole quantum dots

L. Sommer, I. Seidler, F. J. Schupp, S. Paredes, N. W. Hendrickx, L. Massai, K. Tsoukalas, A. Orekhov, E. G. Kelly, S. W. Bedell, G. Salis, M. Mergenthaler, P. Harvey-Collard, A. Fuhrer, T. Ihn

cond-mat.mes-hallquant-phSubmitted Feb 10, 2026• Announced Feb 11, 2026• Revised Feb 27, 2026 (v2)
v1 · Feb 10, 2026v2 · Feb 27, 2026
arXivHTMLPDF

Abstract

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 $g$-factor, a key parameter for qubit control. We investigate the out-of-plane $g$-factor in Ge quantum dots using excitation (single-particle) and addition (many-body) spectra. Excitation spectra allow us to distinguish the pure Zeeman $g$-factor from orbital contributions to the magnetic field splitting of states despite the strong spin-orbit coupling. This distinction clarifies discrepancies between $g$-factors extracted with the two methods, for different orbital states and different hole numbers. Furthermore, we find gate-tunability of $g$-factors at the level of 15%, highlighting its relevance for all-electric qubit manipulation.

Journal reference
Nano Letters 26, 8117 (2026)
DOI
10.1021/acs.nanolett.6c00999

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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.
The spin-orbit interaction permits to control the state of a spin qubit via electric fields. For holes it is particularly strong, allowing for fast all electrical qubit manipulation, and yet an in-depth understanding of this interaction in hole systems is missing. Here we investigate, experimentally and theoretically, the effect of the cubic Rashba spin-orbit interaction on the mixing of the spin states by studying singlet-triplet oscillations in a planar Ge hole double quantum dot. Landau-Zener sweeps at different magnetic field directions allow us to disentangle the effects of the spin-orbit induced spin-flip term from those caused by strongly site-dependent and anisotropic quantum dot g tensors. Our work, therefore, provides new insights into the hole spin-orbit interaction, necessary for optimizing future qubit experiments.
Hole spin qubits provide one of the leading platforms for spin-based quantum computing due to their large intrinsic spin-orbit interaction (SOI), which enables fast electrical manipulation. The SOI of planar quantum dots has mostly been investigated in theoretical studies by examining the SOI already present in the two-dimensional hole gas (2DHG). Here, we study the SOI created by the in-plane confinement by deriving non-perturbative effective Hamiltonians numerically for hole spin qubits. We find that the quantum geometry of the 2DHG naturally emerges, leading to a meaningful non-perturbative definition of pseudospin valid far away from the ΓΓΓ-point. The SOI of the 2DHG and of the in-plane confinement have different forms; therefore, they cannot be turned off simultaneously, ruining the perfect spin-orbit switch functionality of spin qubits. We construct effective Hamiltonians using the symmetry approach for various low-dimensional hole systems: (i) a heavy-hole confined in a SiGe/Ge/SiGe heterostructure, (ii) a light-hole confined in SnGe/Ge, (iii) a gate-defined nanowire in SiGe/Ge/SiGe, and (iv) a hole confined in a Ge/Si core/shell nanowire. The non-perturbative effective Hamiltonians provide results with excellent agreement with the full Hamiltonians.
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 $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ 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 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.
Spin qubits in semiconductor quantum dots constitute one of the leading platforms for quantum information processing, combining long coherence times, high-fidelity gate operations, and compatibility with semiconductor industry technologies. In particular, holes confined in planar Ge/SiGe heterostructures have recently emerged as promising qubit candidates owing to their low effective mass, high mobility, low percolation density, absence of valley states, and strong spin-orbit interaction, which enables fast all-electrical spin control. At the same time, spin-orbit interaction also gives rise to a highly anisotropic and electrically tunable gyromagnetic response. Understanding and controlling this response is therefore essential for high-fidelity single- and multi-qubit operations. The realization of scalable quantum processors also requires efficient qubit readout and long-range connectivity. Circuit quantum electrodynamics (cQED), based on the coupling between artificial atoms and microwave photons confined in superconducting resonators, provides a natural framework to address these challenges. However, the implementation of hybrid semiconductor-superconductor devices in planar Ge has so far been hindered by large microwave losses originating from the heterostructure. In this thesis, we demonstrate, for the first time in planar Ge, strong coupling between a hole charge qubit in a double quantum dot and a high-impedance microwave resonator, reaching a cooperativity of C â 100. In addition, we leverage the frequency tunability of the resonator to perform detailed spectroscopy of a multi-level energy spectrum of low-lying states arising from strong Coulomb-correlation effects. To further improve the quality of the hole-photon interface, we investigate the source of microwave losses in different Ge/SiGe heterostructures, identifying defects at the lattice-mismatched Si/Ge interface as the dominant source of resonator losses in reverse-graded heterostructures. To mitigate this limitation, a deep tapered-etching approach is introduced, enabling the fabrication of superconducting resonators directly on etched heterostructures. This method yields internal quality factors exceeding Qint > 10 000 for high-impedance Josephson junction array resonators and Qint > 100 000 for 50 â Nb resonators. Finally, this thesis explores the spin properties of multi-hole quantum dots beyond the conventional single-orbital picture. By reconstructing and electrically tuning the g tensors of holes confined in different orbital states, we show that confinement can substantially modify both their magnitude and orientation. These results open new opportunities for engineering qubit properties through orbital occupation. Furthermore, highly coherent singlet-triplet oscillations are demonstrated in the seven-hole regime, yielding a quality factor of approximately 330 at in-plane magnetic fields as low as 2.5 mT. Together, these results establish a foundation for high-fidelity hole-spin-qubit operation based on g-tensor engineering and for the integration of planar Ge quantum dots with superconducting quantum circuits.
Two-dimensional (2D) electron/hole gases confined in semiconductor heterostructures suffer from weak Rashba spin-orbit coupling (SOC) for manipulating spin degreee of freedom via an electric rather than a magnetic field. Here, we show that complementary metal-oxide-semiconductor technology-accessible strain could substantially enhance the linear Rashba SOC of the top hole subband in Ge/SiGe quantum wells (QWs) to a level comparable to that of 2D Rashba materials through enhancing the mixture of the light-hole and heavy-hole bands. We further show that strongly enhanced Rashba SOC boosts the Rabi frequency of hole spin qubits confined in Ge/SiGe QWs by two orders of magnitude to an unprecedented 40 GHz, more than one order of magnitude faster than other qubit platforms. We also demonstrate that the hole spin rotation with Rabi frequency > 25 GHz enters a new regime being immune to gate control-induced electric noise, opening a new avenue to simultaneously improve the gate speed and gate fidelity. Our findings provide a new routine to substantially enhance the Rashba SOC in 2D semiconductor hole gases to a level that is great for spintronic applications.
High-purity germanium (Ge) has re-emerged as a versatile semiconductor platform for spin-based quantum information processing because it combines mature materials processing, access to spin-free isotopes, high mobilities, small effective masses, and strong but engineerable spin--orbit coupling. However, ``Ge qubits'' are not a single technology. Donor spin qubits, acceptor spin qubits, gate-defined hole spin qubits, and gate-defined electron spin qubits exploit different parts of the Ge band structure and therefore make distinct trade-offs among coherence, controllability, fabrication complexity, and scalability. Here we compare these four Ge-based spin-qubit modalities on a common physical and architectural footing. We review the shared Ge materials physics, including isotopic purification, the multivalley LLL-point conduction band, the spin-3/23/23/2 valence band, heavy-hole/light-hole mixing, strain, interfaces, disorder, and phonons. We also introduce a common framework for estimating phononic-crystal-modified T1T_1T1​ using a calibrated reference relaxation rate, a geometry-dependent strain-density-of-states suppression factor, and parasitic relaxation channels. The comparison shows that gate-defined Ge hole-spin qubits currently offer the strongest combination of all-electrical control, demonstrated multiqubit operation, and scalability. Donor, acceptor, and gate-defined electron qubits remain important complementary directions for memory, hybrid, and exploratory architectures. Overall, Ge supports a diverse qubit ecosystem, with gate-defined hole-spin qubits presently providing the clearest path toward scalable Ge-based quantum processors.