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Spin-phonon interactions are one of the mechanisms limiting the lifetime ofspin qubits made in semiconductor quantum dots. At variance with othermechanisms such as charge noise, phonons are intrinsic to the device and canhardly be mitigated. They set, therefore fundamental limits to the relaxationtime of the qubits. Here we introduce a general framework for the calculationof the spin (and charge) transition rates induced by bulk (3D) and stronglyconfined 1D or 2D phonons. We discuss the particular case of hole spin-orbitqubits described by the 6 bands kp model. We next apply this theory to a holequbit in a silicon-on-insulator device. We show that spin relaxation in thisdevice is dominated by a band mixing term that couples the holes to transverseacoustic phonons through the valence band deformation potential d, and optimizethe bias point and magnetic field orientation to maximize the number of Rabioscillations Q that can be achieved within on relaxation time T1. Despite thestrong spin-orbit coupling in the valence band, the phonon-limited Q can reacha few tens of thousands. We next explore the effects of phonon confinement in1D and 2D structures, and the impact of the encapsulation materials on therelaxation rates. We show that the spin lifetimes can depend on the structureof the device over micrometer-long length scales and that they improve when thematerials around the qubit get harder. Phonon engineering in semiconductorqubits may therefore become relevant once the extrinsic sources of relaxationhave been reduced.

Hole-phonon interactions in quantum dots: Effects of phonon confinement and encapsulation materials on spin-orbit qubits