We study the low-energy spectrum of a single hole confined in a planar Ge quantum dot (QD) within the effective-mass formalism. The QD is sandwiched between two GeSi barriers of finite potential height grown along the [001] direction. To treat this finite barrier problem, we adopt an independent-band approach in dealing with the boundary conditions. The effects of different system parameters are investigated, including the width of the out-of-plane confining well, the size of the dot, and silicon concentration in the confining layers. The more accurate finite-barrier model results in a nontrivial dependence of the anisotropic $g$ factor on the silicon concentration in the barrier, and the choice of boundary conditions can have a non-negligible impact on its predicted value. On the other hand, the corresponding $g$ factor predicted by the hard-wall model only depends on the silicon concentration monotonically. The comparison shows that the hard-wall model falls short in capturing the interplay between strain and band offset. Furthermore, while the ideal model of a planar dot with a square-well heterostructure already has an intrinsic spin-orbit coupling, realistic effects arising from the experimental setup may give rise to additional contributions. We investigate the impact of the top-gate electric field and the residual tensile strain on the qubit states. The results indicate that these effects are important contributions to the total spin-orbit coupling, which enables fast electric control.

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• Spin-orbit coupling
• Heterostructures
• Quantum dots
• Qubits
• Semiconductors
• Effective mass theory
• k dot p method