Electrical Conduction inn-Type Germanium at Low Temperatures (1962)

A thorough study has been made of the electrical conductivity (including Hall effect) of high-purity $n$-type Ge in the temperature range 4-25\ifmmode^\circ\else\textdegree\fi{}K. Both the Ohmic region and non-Ohmic or "hot electron" region were studied, the latter by both dc and high-speed pulse techniques. From measurements in the Ohmic region, accurate values for the impurity concentrations, in particular the compensating impurities, and for the donor activation energies are obtained; these must be known in order to convert the pulse data to cross sections for certain elementary processes that were investigated. As a subsidiary result, it is confirmed that for P and As donors there is a difference between the thermal and optical activation energies which is proportional to the singlet-triplet splitting of the ground state. By comparing experimental results for the electric field dependence of the drift mobility with the (hot electron) theory for acoustic phonon scattering including anisotropy, it is shown that the electrons appear to be hotter at lower temperatures than the theory would predict. One must conjecture a "forward scattering effect" that reduces the average collision rate for hot electrons moving in the heavy-mass direction, i. e., they can scatter only through small angles at low temperatures because of a death of the appropriate phonons. The conjecture is confirmed by a measurement of the mobility anisotropy parameter $K$ (=4.2 for phonon scattering), obtained from data on the magnetic field dependence of the breakdown field. The major aim of the work was to determine values for the (velocity-and temperature-dependent) cross sections for two recombination processes: the recombination of a single electron with an ionized donor (the inverse of thermal ionization) and the Auger recombination of an electron with an ionized donor (the inverse of ionization of a donor by collision). Cross sections have been determined for these processes, and for the inverse processes as well. Criticisms made of some of our early work by Ascarelli and Brown are shown to be invalid. In particular the ad hoc assumption by Ascarelli and Brown that Auger processes do not exist is what led them to false conclusions; their data, reinterpreted, give values for the Auger recombination probability in agreement with ours. Both the direct and Auger recombination cross sections are very large (\ensuremath{\sim}${10}^{\ensuremath{-}12}$ ${\mathrm{cm}}^{2}$ and \ensuremath{\sim}${10}^{\ensuremath{-}24}$ n ${\mathrm{cm}}^{2}$, respectively); they agree well, as regards absolute magnitude, and temperature and energy dependence, with the "giant trap" cascade mechanism proposed first by M. Lax. Moreover, the inverse of the giant-trap mechanism would be a mechanism for impact ionization of neutral impurities that would have a rapid dependence on lattice temperature and would have no threshold for the energy of the colliding electron. This has been observed. Lastly, the hot electron behavior for orientations of the applied electric field other than along a $〈100〉$ direction have been studied; the electrons in different valleys then have different "temperatures" and densities. It is shown that, though this is the case, no net transfer of carriers from one valley to the other occurs. A simple theoretical treatment for the breakdown field in terms of impurity content and for the variation of the breakdown field with crystallographic orientation and with magnetic field has been formulated, using as a basis Price's criterion for breakdown, which is in excellent agreement with the data.