Theory of nuclear-induced spectral diffusion: Spin decoherence of phosphorus donors in Si and GaAs quantum dots

We propose a model for spectral diffusion of localized spins in semiconductors due to the dipolar fluctuations of lattice nuclear spins. Each nuclear spin flip flop is assumed to be independent, the rate for this process being calculated by a method of moments. Our calculated spin decoherence time $T_M=0.64\mathrm{}\mathrm{ms}$ for donor electron spins in Si:P is a factor of 2 longer than spin echo decay measurements. For ${}^{31}\mathrm{P}$ nuclear spins we show that spectral diffusion is well into the motional narrowing regime. The calculation for GaAs quantum dots gives $T_M=10\mathrm{}-50\mu \mathrm{s}$ depending on the quantum dot size. Our theory indicates that nuclear induced spectral diffusion should not be a serious problem in developing spin-based semiconductor quantum computer architectures.