Electron Spin Decoherence in Isotope-Enriched Silicon

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 $T_2=2T_1$ 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, ${}^{29}\mathrm{Si}$ 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.

Figure 1

Decay times ($T_2$ for the Hahn echo) of Si:P donor electron spins for various $C_N$. At high $C_N$, contact hyperfine interactions dominate and $T_2$ is dependent upon the magnetic field direction relative to the lattice orientation. At low $C_N$, $T_2$ is dependent upon $C_E$, and eventually dominated only by dipolar interactions (which includes dipolar-approximated electronuclear interactions). Experimental results are shown as square symbols, from Ref. 2, and a star symbol, from Ref. 3.

Figure 2

Spin echo resulting from decoherence induced by various concentrations of (a) ${}^{29}\mathrm{Si}$ ($C_N$), (b) background phosphorus donors ($C_E$), or (c) the combination of both. In (b), static ${}^{29}\mathrm{Si}$-induced Overhauser field variations between donors suppresses their decoherence-inducing flip-flops. (a)–(c) use $\xi$ as a scaling parameter for both concentration and inverse time. (d) $T_2$, defined as the $1/e$ decay time, for various $C_N$ and $C_E$. The star symbol indicates an experimentally obtained [3] result at $C_N\approx 50\mathrm{ppm}$ and is in good agreement with the corresponding 50 ppm theoretical curve.

Figure 3

(a)–(c) correlate with (a)–(c) of Fig. 2, respectively, but plotted in terms of error (one minus the spin echo) on a log-log scale in order to highlight the low-error behavior that is relevant for fault-tolerant quantum computation. (d) Plots $T_2[{10}^{-4}]$, the time at which the echo error is ${10}^{-4}$, for various $C_N$ and $C_E$.

Figure 4

(top) Maximum error from donor-induced (without ${}^{29}\mathrm{Si}$) SD for at least 50%, 90%, or 99% of the central spins due to random spatial variations of the bath spins. (bottom) Corresponding fits of low-error behavior of $1-\exp [-(t/T_q{)}^{n_q}]\approx (t/T_q{)}^{n_q}$ in the ${10}^{-4}$ error regime.