NUCLEAR MAGNETIC RESONANCE STUDIES OF THE METALLIC TRANSITION IN DOPED SILICON

The Si: P and Si: B systems have been studied using the methods of pulse and cw nuclear magnetic resonance. The purpose of this study is to investigate the transition of an impurity system in a solid from an array of isolated paramagnetic atoms or clusters of atoms to a superlattice of impurity atoms having strong wave-function overlap and metallic character. Knight shifts, line shapes, and nuclear spin relaxation times were measured for ${\mathrm{Si}}^{29}$ and ${\mathrm{B}}^{11}$ in $p$-type silicon and ${\mathrm{Si}}^{29}$ and ${\mathrm{P}}^{31}$ in $n$-type silicon. Phosphorus concentrations vary from ${10}^{17}$ to ${10}^{20}$ impurities/${\mathrm{cm}}^{3}$ and the temperature range investigated extends from 1.4 to 300\ifmmode^\circ\else\textdegree\fi{}K. Onset of metallic behavior in $n$-type silicon at 4\ifmmode\times\else\texttimes\fi{}${10}^{18}$ phosphorus impurities/${\mathrm{cm}}^{3}$ is indicated by the ${\mathrm{Si}}^{29}$ ${T}_{1}$ becoming proportional to ${T}^{\ensuremath{-}1}$ between 1.4 and 4.2\ifmmode^\circ\else\textdegree\fi{}K and by the existence of a Knight shift for ${\mathrm{Si}}^{29}$. Above a phosphorus concentration of approximately 3\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$, ${\mathrm{Si}}^{29}$ ${T}_{1}'\mathrm{s}$ and Knight shifts obey the Korringa relation. Broadening of the ${\mathrm{Si}}^{29}$ resonance line by 5 times the dipolar width and of the ${\mathrm{P}}^{31}$ resonance line by 100 times the dipolar width at concentrations of 1.4\ifmmode\times\else\texttimes\fi{}${10}^{20}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ is shown to be caused by fluctuations of the local Knight shift about the average Knight shift value. Such fluctuations are explained by a model of a Poisson distribution for the local ${\mathrm{P}}^{31}$ impurity density with a threshold local density of 3\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ for transition to metallic properties. This model agrees with the observed ${\mathrm{P}}^{31}$ resonance line shape and explains the transition to metallic behavior in $n$-type silicon. In $p$-type silicon, ${\mathrm{B}}^{11}$ and ${\mathrm{Si}}^{29}$ Knight shifts are measured for boron concentrations greater than 1\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$. The ${\mathrm{B}}^{11}$ ${T}_{1}'\mathrm{s}$ and Knight shifts agree with the Korringa relation within a 15% experimental error. However, both the ${\mathrm{B}}^{11}$ ${T}_{1}$ and Knight shift are independent of concentration for boron concentrations between 2\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ and 8.5\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$. Such concentration independence may be explained by postulating a clustering of boron atoms at an average local density in a cluster greater than 8.5\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$. Wave function probability densities are calculated from Knight shifts with a free carrier density of states assumed valid. To facilitate comparison, wave-function densities are normalized per unit volume of the crystal and are 2600 ${\mathrm{cm}}^{\ensuremath{-}3}$ at ${\mathrm{P}}^{31}$ and 100 ${\mathrm{cm}}^{\ensuremath{-}3}$ at ${\mathrm{Si}}^{29}$ in $n$-type silicon and 80 ${\mathrm{cm}}^{\ensuremath{-}3}$ at ${\mathrm{Si}}^{29}$ in $p$-type silicon.