Hole spin in a Ge quantum dot with finite barriers

Holes confined in Ge quantum dots are a good candidate for spin qubit. They have strong spin-orbit coupling, originating mainly from the mixing of bands at finite momentum even close to the Γ point and is sensitive to the momentum change. Fast electrical control of hole spins is thus intrinsically possible.

Here we present a theoretical study of single-hole spectra in a planar quantum dot based on a Ge/GeSi heterostructure, with the main objective of exploring the effects of the finite GeSi barrier on the hole spin properties. Calculations are performed using the LCAO technique with single-band QD orbitals. We particularly focus on the Zeeman splitting of the lowest heavy-hole states, which form the hole spin qubit. A uniform magnetic field is applied in the in-plane or out-of-plane directions. We study the variation of qubit Zeeman splitting with changing silicon concentration in the barrier material at different thicknesses of the Ge layer, as well as different dot sizes and magnetic field directions. While the large difference between in-plane and out-of-plane g factor is known, anisotropy can also be seen when changing the magnetic field direction in the x-y plane. This in-plane anisotropy comes from the change of orbital contribution in different directions, therefore, it is sensitive to the size of the dot. Our calculations highlight the importance of barrier parameters and the significance of accurate treatment of the boundary conditions in optimizing hole spin qubits.