  An ideal quantum dot spin qubit architecture requires a local magnetic field
for one-qubit rotations. Such an inhomogeneous magnetic field, which could be
implemented via a micromagnet, couples the qubit subspace with background
charge fluctuations causing dephasing of spin qubits. In addition, a
micromagnet generates magnetic field evanescent-wave Johnson noise. We derive
an effective Hamiltonian for the combined effect of a slanting magnetic field
and charge noise on a single-spin qubit and estimate the free induction decay
dephasing times T2* for Si and GaAs. The effect of the micromagnet on Si qubits
is comparable in size to that of spin-orbit coupling at an applied field of
B=1T, whilst dephasing in GaAs is expected to be dominated by spin-orbit
coupling. Tailoring the magnetic field gradient can efficiently reduce T2* in
Si. In contrast, the Johnson noise generated by a micromagnet will only be
important for highly coherent spin qubits.
