Thermal Transport Properties ofn-Type Ge at Low Temperatures

Abstract

The thermal resistivity $w$ contributed to Ge at low temperatures by electrically active impurities has been investigated by measurements on a series of single crystals ($n$-type) between 1.3 and 150\ifmmode^\circ\else\textdegree\fi{}K. By comparing compensated [(Sb+Ga)-doped] with noncompensated (Sb- or As-doped) samples, it is found that at temperatures below its minimum, $w$ may be correlated with the concentration of neutral donors ${n}_{\mathrm{ex}}$ (excess of donor over acceptor concentration). The form of the relation indicates that there must be at least two types of electron-phonon interaction: one for electrons in semi-isolated impurity states (${n}_{\mathrm{ex}}<6\ifmmode\times\else\texttimes\fi{}{10}^{17}$ ${\mathrm{cm}}^{\ensuremath{-}3}$) and another for electrons in the well-defined band formed at higher concentrations. In the low concentration range there is an impurity-species effect; the thermal conductivity $k$ in As-doped material is proportional to ${T}^{3}$ and is larger than in Sb-doped material with the same concentration, where it is proportional to ${T}^{4}$. These results seem qualitatively compatible with the models proposed by Keyes and by Griffin and Carruthers and confirm their main point: This phonon-electron interaction involves occupied semi-isolated states rather than transitions. An attempt to analyze the high concentration range by including in the Callaway model the phonon-electron relaxation time introduced by Ziman suggests that the theory is incomplete. The conservation of crystal momentum in these interactions may require correction terms that have not been considered. The thermoelectric power $Q$, the electrical resistivity $\ensuremath{\rho}$, and the Hall constant (the last for the compensated samples, from 77 to 400\ifmmode^\circ\else\textdegree\fi{}K) were also measured. These data indicate that in $n$-type Ge with total impurity concentration greater than about ${10}^{18}$ ${\mathrm{cm}}^{\ensuremath{-}3}$, the effect of 80% or more compensation is to reinstate the impurity levels in the energy gap. In addition it is found that $Q$ shows an impurity-species effect at temperatures at which the donor states are neutral, but not at higher temperatures. In the high-concentration material both $Q$ and $\ensuremath{\rho}$ can be analyzed by semimetal theory.