Donor clusters in silicon. Results of ESR measurements

Measurements of ESR spectra of Si:P and Si:As samples with donor concentrations in the range 3×${10}^{16}$/${\mathrm{cm}}^3$ to 9×${10}^{17}$/${\mathrm{cm}}^3$ have been obtained in the range 1.4-16 K at 9.5 GHz. The higher temperature $\frac{d{\chi }^{\prime \prime }}{\mathrm{dH}}$ spectra were integrated to obtain ${\chi }^{\prime \prime }\mathrm{}(N,T,H)\mathrm{}$. The experimental results and simulated spectra based on Poisson statistics, but neglecting cluster topological effects and cluster-cluster interactions, are not in good agreement and demonstrate the problems in using peak heights of the central pair line or triad lines to determine cluster statistics. However, Poisson statistics predictions for the ratio of the integrated intensity of the outer lines to the total integrated intensity are shown to be obeyed and determine the characteristic $V_c$ for cluster formation. $V_c\approx 8.2\times {10}^6$ ${\mathrm{\AA }}^3$ for Si:P and 5.6×${10}^6$ ${\mathrm{\AA }}^3$ for Si:As. A search on a 2.5×${10}^{17}$/${\mathrm{cm}}^3$ Si:P sample outside the isolated donor transitions for the $\frac{\pm 5A}{6}$ hyperfine transitions ($A$ is the Fermi contact hyperfine constant) of the upper $S=\frac{1}{2}$ state of donor triads predicted by Shimizu has revealed very weak extra "bumps" in $\frac{d{\chi }^{\prime \prime }}{\mathrm{dH}}$ at $H_0\pm 35.5$G in good agreement with Shimizu's prediction. However, the intensity of these "bumps" is very much weaker than expected neglecting topological effects and cluster-cluster interactions. An analysis of the many-electron hyperfine interaction based on a linear combination of Slater determinants is given and applied to the donor triad case. The results are shown to agree with Shimizu's results when overlap terms are neglected. Monte Carlo calculations on donor clusters have yielded the topological distribution function $P\left(\theta \right)$ [$tan\left(2\mathrm{}\theta \right)=\sqrt{3}\frac{(J_{\mathrm{bc}}-J_{\mathrm{ac}})}{({2J}_{\mathrm{ab}}-J_{\mathrm{bc}}-J_{\mathrm{ac}})}$] for donor triads and demonstrate it is sharply peaked about $\theta \approx 0°$. The role of cluster-cluster interactions on the ESR spectrum of ${\chi }^{\prime \prime }\mathrm{}(N,T,H)\mathrm{}$ is considered and these interactions are discussed for donor triads.