  We develop the theory of single-electron silicon spin qubit relaxation in the
presence of a magnetic field gradient. Such field gradients are routinely
generated by on-chip micromagnets to allow for electrically controlled quantum
gates on spin qubits. We build on a valley-dependent envelope function theory
that enables the analysis of the electron wave function in a silicon quantum
dot with an arbitrary roughness at the interface. We assume the presence of
single-layer atomic steps at a Si/SiGe interface and study how the presence of
a gradient field modifies the spin-mixing mechanisms. We show that our
theoretical modeling can quantitatively reproduce results of experimental
measurements of qubit relaxation in silicon in the presence of a micromagnet.
We further study in detail how a field gradient can modify the EDSR Rabi
frequency of a silicon spin qubit. While this strongly depends on the details
of the interface roughness, interestingly, we find that adding a micromagnet on
top of a spin qubit with an ideal interface can even reduce the EDSR frequency
within some interval of the external magnetic field strength.
