Determination of carrier densities in lightly doped silicon crystals from the Hall effect

The evaluation of carrier densities from the Hall effect depends very sensitively on the values of the Hall factors $r_n$ and $r_p$. This sensitivity is particularly pronounced in the intermediate temperature region of $p$-type silicon crystals. This is due to the fact that there are generally two different carrier concentrations which lead to the same Hall coefficient. A general theory of Hall factors is presented, taking details of the band structure and phonon-scattering mechanisms of silicon crystals into account. A somewhat more stringent approach to the solution of the Boltzmann equation is used in this paper than that of taking the relaxation-time ansatz for granted. Of special importance is the third split-off valence band, which introduces an additional temperature dependence to the Hall factors. Problems connected with interpreting the solution of the Hall formula are examined and resolved. The theory is applied to measurements on lightly doped $n$- and $p$-type silicon single crystals with doping concentrations of about ${10}^{13}$ ${\mathrm{cm}}^{-3\mathrm{}}$ in the temperature region between 90 and 450 K. The deformation-potential constants are determined and are found to be in fair agreement with previously published values.