  Silicon is promising for spin-based quantum computation because nuclear
spins, a source of magnetic noise, may be eliminated through isotopic
enrichment. Long spin decoherence times, $T_2$, have been measured in
isotope-enriched silicon but come far short of the $T_2 = 2 T_1$ limit. The
effect of nuclear spins on $T_2$ is well established. However, the effect of
background electron spins from ever present residual phosphorus impurities in
silicon can also produce significant decoherence. We study spin decoherence
decay as a function of donor concentration, $^{29}$Si concentration, and
temperature using cluster expansion techniques specifically adapted to the
problem of a sparse dipolarly coupled electron spin bath. Our results agree
with the existing experimental spin echo data in Si:P and establish the
importance of background dopants as the ultimate decoherence mechanism in
isotope-enriched silicon.
