Analysis of Lattice and Ionized Impurity Scattering inp-Type Germanium

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The scattering of holes in $p$-type germanium by acoustical and optical phonons, and by ionized impurities has been studied over a wide range of temperature from 7 to 300\ifmmode^\circ\else\textdegree\fi{}K, and for impurity concentrations from ${10}^{13}$ to ${10}^{17}$ ${\mathrm{cm}}^{\ensuremath{-}3}$. The mobility was analyzed on the basis of the following approximations: (1) The light- and heavy-hole valence bands are parabolic with spherical constant-energy surfaces and effective-mass ratios of 0.043 and 0.35; (2) the relaxation time for lattice scattering is the same for the heavy and light holes; and (3) the ionized impurity scattering is describable by the Brooks-Herring formula with the scattering limited to intraband transitions. The following results were obtained: (1) The expectation was substantiated that the lattice scattering at $T\ensuremath{\le}70\ifmmode^\circ\else\textdegree\fi{}$K can be attributed to acoustical phonon interactions; the mobility ${\ensuremath{\mu}}_{\mathrm{ac}}=3.37\ifmmode\times\else\texttimes\fi{}{10}^{7}{T}^{\ensuremath{-}\frac{3}{2}}$ ${\mathrm{cm}}^{2}$/V-sec was obtained by treating it as the sole adjustable parameter in the range 10 to 70\ifmmode^\circ\else\textdegree\fi{}K; (2) the stronger ${T}^{\ensuremath{-}2.3}$ dependence of mobility in the range 120 to 300\ifmmode^\circ\else\textdegree\fi{}K can be attributed to the advent of optical mode scattering if the ratio of optical to acoustical mode coupling constants ${(\frac{{\mathcal{E}}_{\mathrm{op}}}{{\mathcal{E}}_{\mathrm{ac}}})}^{2}$ is chosen to be 3.8; (3) the Brooks-Herring formula describes the ionized impurity scattering very well from 30 to 300\ifmmode^\circ\else\textdegree\fi{}K for impurity concentrations ${N}_{I}\ensuremath{\le}{10}^{15}$ ${\mathrm{cm}}^{\ensuremath{-}3}$, but overestimates the mobility for higher ${N}_{I}$ or lower $T$. The eventual failure of the analysis may be attributed to the neglect of carrier-carrier and interband ionized impurity scattering.