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