Thermal conductivity of germanium crystals with different isotopic compositions

Abstract

We have measured the thermal conductivity of seven germanium crystals with different isotopic compositions in the temperature range between 2 K and 300 K. These samples, including one made of highly enriched ${}^{70}\mathrm{Ge}(99.99%)$, show intrinsic behavior at room temperature with the exception of a $p$-type sample with $|{N}_{d}\ensuremath{-}{N}_{a}|\ensuremath{\cong}2\ifmmode\times\else\texttimes\fi{}{10}^{16} {\mathrm{cm}}^{\ensuremath{-}3}$. The ``undoped'' samples exhibit a ${T}^{3}$ dependence at low temperatures, basically determined by boundary scattering. The maximum value of $\ensuremath{\kappa}$ (which falls in the range between 13 K and 23 K) is found to be a monotonically decreasing function of the isotopic mass variance parameter $g$. The maximum ${\ensuremath{\kappa}}_{m}$ measured for the most highly enriched ${}^{70}\mathrm{Ge}(99.99%)$ sample is 10.5 kW/mK, one order of magnitude higher than for natural germanium. The experimental data have been fitted with the full Callaway theory, modified by treating transverse and longitudinal modes separately, using three free adjustable parameters for each set of modes to represent anharmonic effects plus the calculated contributions from isotopic and boundary scattering. For the isotopically purest ${}^{70}\mathrm{Ge}(99.99%)$ sample, dislocation scattering, or a similar mechanism, must be added in order to fit the data. We have also checked the effect of various surface treatments on the thermal conductivity in the low temperature region. The highest values of $\ensuremath{\kappa}$ are found after polish etching with a SYTON suspension.