Spin Resonance of Localized and Delocalized Electrons in Phosphorus-Doped Silicon between 20 and 30 °K

Electron spin resonance has been studied in phosphorus-doped silicon from 20 to 300 °K in a donor concentration range ($7.5\times {10}^{14}{\mathrm{cm}}^{-3\mathrm{}}<~N_D<~8\times {10}^{16}{\mathrm{cm}}^{-3\mathrm{}}$), where exchange interactions between donor atoms are negligible. In the lower limit of the temperature range used, most of the donor electrons are bound in the ground state of the donor atoms and the two lines of the hyperfine structure are observed. It is shown that the broadening of these lines at $T\approx 30$ °K is determined by the exchange scattering between the donor electrons and the conduction electrons, while their shift arises from the thermal excitation to the first excited level. From the broadening, we obtain the value of the exchange cross section (${\sigma }_{\mathrm{ex}}=0.35\mathrm{}\times {10}^{-12\mathrm{}}$ ${\mathrm{cm}}^2$). At higher temperatures, a single line is observed which narrows when the temperature increases (in the region $50{}_{}{}^{\circ }{}_{}{}^{}\mathrm{K}\lesssim T\lesssim 75{}_{}{}^{\circ }{}_{}{}^{}\mathrm{K}$). This narrowing is attributed to the motional averaging of the hyperfine interaction. In the intermediate temperature region ($75{}_{}{}^{\circ }{}_{}{}^{}\mathrm{K}\lesssim T\lesssim 150{}_{}{}^{\circ }{}_{}{}^{}\mathrm{K}$), it is shown that the dominant relaxation mechanism arises from the spin-orbit interaction in the first excited level modulated by the thermal motion of the electrons. In the high-temperature region ($150{}_{}{}^{\circ }{}_{}{}^{}\mathrm{K}\lesssim T\lesssim 300{}_{}{}^{\circ }{}_{}{}^{}\mathrm{K}$), where most of the electrons are excited into the conduction band, the value and the temperature dependence of the linewidth are well accounted for by the theories of Yafet and Elliott on spin relaxation of conduction electrons. The observed behavior of the spin resonance is quantitatively explained over the whole range of temperatures and concentrations studied, using only two adjustable parameters.