Ultralong Coherence of Phosphorus Donors in High-Purity $^{28}$Si Silicon

We report on electron spin coherence measurements for phosphorus donors in high purity, highly-enriched $^{28}$Si, with residual $^{29}$Si of less than 50 ppm. At this low $^{29}$Si density, spectral diffusion processes by nuclear spin flip-flops are suppressed, and therefore other relaxation processes become prominent. By examining a series of $^{28}$Si crystals with a donor concentration of 1$\times $10$^{14}$ to 3$\times $10$^{15}$/cm$^{3}$, we identified three decoherence mechanisms, all related to dipole interactions between donors: (1) instantaneous diffusion, caused by flips of donor spins induced by the applied microwave pulses; (2) spectral diffusion caused by T$_{1}$-induced flips of neighboring donors; (3) spectral diffusion caused by donor spin flip-flops. We demonstrate how all three mechanisms can be suppressed, leading to measured coherence times extrapolating to T$_{2}\sim $10 sec. The work was funded by DOE and LPS.