Low-temperature magnetic susceptibility of Si: P in the nonmetallic region

Abstract

Measurements of the phosphorus donor susceptibility ${\ensuremath{\chi}}_{D}(T)$ in Si: P have been made using a superconducting quantum interference device at phosphorus concentrations from 1 \ifmmode\times\else\texttimes\fi{} ${10}^{17}$ to 4 \ifmmode\times\else\texttimes\fi{} ${10}^{18}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ over the temperature range from 2 mK to 4 K and in applied magnetic fields down to 10 Oe. Above $T\ensuremath{\sim}30$ mK a plot of $\mathrm{ln}{\ensuremath{\chi}}_{D}$ vs $\mathrm{ln}T$ shows a roughly linear characteristic with a slope which diminished with increasing concentration. For $T<30$ mK ${\ensuremath{\chi}}_{D}$ is found to level off in an unexpected fashion in that the temperature at which this occurs is relatively independent of concentration. The leveling is less pronounced at 10 Oe for the most dilute sample; however, this sample exhibits an anomalously large saturation effect in 50 Oe. The leveling of ${\ensuremath{\chi}}_{D}$ cannot reasonably be attributed to hyperfine effects or to magnetic ordering as a consequence of interdonor exchange interactions. Susceptibility calculations are presented which are based on a priori exchange couplings between many-valley orbital ground states for the donors. In the spherical approximation for the envelope functions, the exchange integral is the product of a rapidly varying interference factor and an isotropic factor which can be obtained from variational calculations for the hydrogen molecule. Calculations of ${\ensuremath{\chi}}_{D}(T)$ have been carried out using both a pair approximation modified to take account of larger clusters and a computer-simulation cluster analysis which includes an error estimate based on a molecular-field approximation. The molecular-field approximation is described in detail here for the first time. The two methods of calculation are in mutual agreement and in good accord with the data down to $T\ensuremath{\sim}30$ mK, where modified pair corrections are small. The calculations yield values of envelope function radius and donor concentration in good agreement with other experimental and theoretical estimates.

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