Electron spin coherence of phosphorus donors in silicon: Effect of environmental nuclei

We report electron paramagnetic resonance (EPR) experiments of phosphorus donors in isotopically controlled silicon single crystals. By varying the concentration of the ${}^{29}\text{S}\text{i}$ isotope, $f$, from 0.075% to 99.2%, we systematically study the effect of the environmental nuclear spins on the donor-electron spin. We find excellent agreement between experiment and theory for decoherence times due to nuclear-induced spectral diffusion, clarifying that the nuclear-induced decoherence is dominant in the range of $f$ studied. We also observe that the EPR linewidth shows a transition from the square-root dependence to the linear dependence on $f$, in agreement with theoretical predictions.

Figure 1

(Color online) (a) Definition of angles. The static magnetic field $B_0$ was applied parallel to the $\left(1\overline{1}0\right)$ plane of silicon crystals. By rotating samples about the $\left[1\overline{1}0\right]$ axis, the direction of $B_0$ is varied from [001] $\left(0°\right)$ to [110] $\left(90°\right)$. (b) Two-pulse ESE sequence. The tipping angles ${\theta }_1$ and ${\theta }_2$ are $90°$ and $180°$ except for the case of ${}^{29}\text{S}\text{i-}0.08\mathrm{\%}$. See the text. (c) Log-log plot of ESE decay curves at $0°$. The data are for ${}^{29}\text{S}\text{i-}0.08\mathrm{\%}$, 1%, 5%, 10%, 50%, and 100% from right to left (slow-to-fast decays). [(d)–(f)] Experimental (top panels) and theoretical (bottom panels) ESEEM at early time for ${}^{29}\text{S}\text{i-}5\mathrm{\%}$, 10%, 50%, and 100% (from top to bottom) at three representative angles. The traces are offset for clarity.

Figure 2

(Color online) Two-pulse ESE decay curves for the three representative angles. The base lines are offset for clarity. Note the different time scales in different samples. The insets are color plots showing the full angular dependence of the echo decay curves.

Figure 3

(Color online) (a) Angular dependence of spectral diffusion time $T_{\text{SD}}$ and stretched exponent $n$ for ${}^{29}\text{S}\text{i-}0.08\mathrm{\%}$ (circle), 1% (square), 5% (diamond), 10% (triangle), 50% (down triangle), and 100% (left triangle). The fitting errors are within the sizes of the respective symbols, unless given explicitly. (b) ${}^{29}\text{S}\text{i}$ concentration dependence of $T_{\text{SD}}$ and $n$ at $0°$ (circle) and $55°$ (square). For $f>1\mathrm{\%}$, theory predicts the $f^{-0.87}$ power law, in agreement with experiment. $n$ falls on between 1.8 and 2.6, consistent with a theoretical prediction of about 2.3.

Figure 4

(Color online) (a) cw EPR spectra. Only the high-field lines of the doublet peaks are shown. The horizontal axis is given as the offset from the resonance field. The base lines are offset for clarity (from top to bottom, ${}^{29}\text{S}\text{i-}100\mathrm{\%}$, 50%, 10%, 5%, 1%, and 0.08%) and are determined as ${\text{log}}_{10}f$ [e.g., the baseline for ${}^{29}\text{S}\text{i-}10\mathrm{\%}$ is ${\text{log}}_{10}\left(0.1\right)=-1$]. (b) ${}^{29}\text{S}\text{i}$ concentration dependence of the FWHM of the spectra [see also Ref. 19]. (c) (Top) The thick gray (red online) line shows the cw spectrum of ${}^{29}\text{S}\text{i-}50\mathrm{\%}$. The thin black line is fit by the first derivative of Gaussian. (Bottom) The thick gray (red online) line shows the field-sweep pulsed spectrum and the thin black line is the integral of the cw spectrum. The lines are offset for clarity. (d) Similar data for ${}^{29}\text{S}\text{i-}1\mathrm{\%}$. The spectrum resembles Lorentzian. The thin black line in the top panel is fit by the first derivative of Lorentzian.