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Relaxation of Shallow Donor Electron Spin due to Interaction with Nuclear Spin Bath

We study the low-temperature dynamics of a shallow donor, e.g., $^{31}$P, impurity electron spin in silicon, interacting with the bath of nuclear spins of the $^{29}$Si isotope. For small applied magnetic fields, the electron spin relaxation is controlled by the steady state distribution of the nuclear spins. We calculate the relaxation times $T_1$ and $T_2$ as functions of the external magnetic field, and conclude that nuclear spins play an important role in the donor electron spin decoherence in Si:P at low magnetic fields.