Electron spin-lattice relaxation rates have been measured on silicon, double-doped with phosphorus donors and the various shallow acceptors, in the low-temperature, low-magnetic-field region where concentration-dependent relaxation mechanisms dominate. By optically neutralizing the charged impurities, the concentration-dependent relaxation rate due to neutral donor pairs, ${\left(\frac{1}{T_1}\right)}^{\mathrm{n}-\mathrm{n}}$, is isolated. The compensation-dependent mechanism arising from neutral-impurity and ionized-impurity complexes, ${\left(\frac{1}{T_1}\right)}^{\mathrm{n}-\mathrm{i}\mathrm{c}}$, is determined from the compensated samples in their equilibrium charge state. A theory is presented for the (n-n) mechanism in which the Zeeman energy of close exchange-coupled donor pairs relax via an exchange reservoir to the lattice, and the isolated donors relax because of spin diffusion (within narrow spin packets) from the fast centers. The number of effective fast centers, and hence also ${\left(\frac{1}{T_1}\right)}^{\mathrm{n}-\mathrm{n}}$, depends on temperature and magnetic field; on the former because high-exchange-energy pairs populate a nonmagnetic singlet state as the temperature is lowered, and on the latter through the relative heat capacities of the Zeeman and exchange systems. Taking into account the distribution of exchange energies in the random impurity distribution leads to agreement of the calculated and experimental values of ${\left(\frac{1}{T_1}\right)}^{\mathrm{n}-\mathrm{n}}$ with respect to magnitude and with respect to temperature, magnetic field, and concentration dependences. Random spin diffusion plays an important role in the relaxation, and a simplistic theory is presented which works well for the relaxation results as well as for separate spectral-diffusion experiments. The ${\left(\frac{1}{T_1}\right)}^{\mathrm{n}-\mathrm{i}\mathrm{c}}$ are found to be magnetic-field-independent and approximately linearly dependent on temperature, both of which are properly accounted for theoretically. The concentration dependence of ${\left(\frac{1}{T_1}\right)}^{\mathrm{n}-\mathrm{i}\mathrm{c}}$ is not in agreement with any theory. Resonant spin-spin interaction of donor electrons with holes bound to the various acceptors is encountered in low-field relaxation measurements on optically pumped samples, and the results agree qualitatively with the theory.

• Received 31 July 1967

DOI:https://doi.org/10.1103/PhysRev.168.271