Electrical Properties ofN-Type Germanium

Abstract

Measurements of conductivity and Hall effect from 11\ifmmode^\circ\else\textdegree\fi{}K to 300\ifmmode^\circ\else\textdegree\fi{}K on a set of $n$-type germanium samples covering the range from intrinsic to degenerate are reported. The purity and uniformity of the samples and extensiveness of the data permit a more thorough-going comparison with theory than has been possible in previous work.The theory of mobility is reviewed briefly. The treatment of impurity scattering by Brooks and Herring is presented, and their formula for the impurity mobility is used throughout. An analytical formula for obtaining the mobility from lattice and impurity mobilities is included. The effect of electron-electron collisions on the mobility is considered in a qualitative manner.The principal conclusions concerning the mobility are as follows: (1) Over the range 20.4\ifmmode^\circ\else\textdegree\fi{}K to 300\ifmmode^\circ\else\textdegree\fi{}K the lattice mobility varies as ${T}^{\ensuremath{-}1.64}$ rather than the theoretically predicted ${T}^{\ensuremath{-}1.5}$. (2) The impurity mobility increases with temperature less rapidly than the theoretical formula predicts, the exponent of $T$ in the numerator being apparently between 1.0 and 1.5. (3) The Erginsoy formula for neutral impurity scattering appears to fit the experimental data reasonably well for a value of effective mass about one-third the free electron mass. (4) Dislocation scattering is negligible, leading to the conclusion that the density of edge-type dislocations is less than ${10}^{6}$/${\mathrm{cm}}^{2}$.In fitting the concentration data, the parameters involved are activation energy, acceptor concentration, and effective mass. An effective mass in the neighborhood of one-quarter the free electron mass gives the best fit for all samples. The values of acceptor concentration obtained in this way for this effective mass agree well with those calculated from the low-temperature mobility values. The activation energy obtained for the purer samples is 0.0125 ev, in agreement with the value calculated from the hydrogen-like model for one-quarter the free electron mass.The variation of activation energy with concentration does not agree with that observed by Pearson and Bardeen for $p$-type silicon. The effects which have been proposed to explain the variation are: residual potential energy of attraction between free electrons and ionized donors, screening of trapping centers by the free electrons, polarization of neutral centers by free electrons. It is concluded that a combination of the three effects will probably be required to explain the results.The ratio of Hall to drift mobility is shown to agree with the theoretically predicted value within about 10 percent in the range 78\ifmmode^\circ\else\textdegree\fi{}K to 300\ifmmode^\circ\else\textdegree\fi{}K.