Nuclear spin coherence of neutral $^{31}$P donors in isotopically enriched $^{28}$Si

In natural silicon the nuclear spin coherence of neutral $^{31}$P donors is limited to about 1 second by flip-flopping $^{29}$Si nuclear spins. Here we eliminate this process by using isotopically enriched $^{28}$Si with 50 ppm of $^{29}$Si. This allows us to examine other processes which may decohere the $^{31}$P nuclear spins. We use X-band pulsed ENDOR at 1.7 K to examine isotopically enriched Si crystals with donor concentrations from 10$^{14}$ to 4x10$^{15}$ P/cm$^{3}$ and find a dependence of $^{31}$P nuclear spin coherence time on donor concentration. The measured nuclear spin echo decays are fit by a stretched exponential function, exp(-(t/T$_{2})^{\mathrm{n}})$, with n ranging from 0.7 to 1. This differs from n of about 2 commonly seen for spectral diffusion due to indirect spin flip-flops. The measured T$_{2}$ times decrease significantly when the donor concentration increases, changing from 8 s at 10$^{14}$ to 0.2 s at 4x10$^{15}$ P/cm$^{3}$. From the observed donor concentration dependence at higher densities, we conclude that direct electron spin flip-flops are responsible for $^{31}$P donor nuclear spin decoherence.