  We study the relaxation of an electron spin qubit in a Si quantum dot due to
electrical noise. In particular, we clarify how the presence of conduction-band
valleys influences spin relaxation. In single-valley semiconductor quantum
dots, spin relaxation is through the mixing of spin and envelope orbital states
via spin-orbit interaction. In Si, the relaxation could also be through the
mixing of spin and valley states. We find that the additional spin relaxation
channel, via spin-valley mixing and electrical noise, is indeed important for
an electron spin in a Si quantum dot. By considering both spin-valley and
intravalley spin-orbit mixings and Johnson noise in a Si device, we find that
the spin relaxation rate peaks at the hot spot, where the Zeeman splitting
matches the valley splitting. Furthermore, because of a weaker field
dependence, the spin relaxation rate due to Johnson noise could dominate over
phonon noise at low magnetic fields, which fits well with recent experiments.
