Heat-Pulse Propagation inp-Type Si and Ge under Uniaxial Stress

Abstract

Heat-pulse propagation has been studied in $p$-type silicon and $p$-type germanium as a function of uniaxial stress (up to ${10}^9$ dyn ${\mathrm{cm}}^{-2\mathrm{}}$) and of pulse temperature. The coupling between thermal phonons and the stress-split ground state of acceptors has been calculated using the effective-mass approximation for the relevant acceptor-hole wave functions. In addition to the $s$-like parts of the expansion for the envelopes usually considered, $d$-like parts of the expansion have been included here. The important phonon scattering rates due to stress-split acceptor states were derived. The rates were obtained for resonance absorption (for phonons with energies close to the splitting energy), and for several second-order processes. In addition, Rayleigh scattering of phonons by isotopic impurities has been evaluated. A black-body model for phonon emission from the heat-pulse generator was assumed. The total calculated scattering rate agrees with the observed stress dependence of the heat-pulse amplitudes when the effects of internal strains, and, in the case of high acceptor contents, of phonon multiple scattering are taken into account. A fit of the experimental results to the calculated results yields two distinct sets of values for the "static" ($a^{*}q\ll 1$) and "dynamic" ($a^{*}q\gtrsim 1$) deformation-potential constants ($a^{*}$ is the effective Bohr radius of the impurity and $q$ is the wave number of the relevant wave component). This observation resolves the apparent conflict in previously reported values of these constants, obtained from separate studies in the static (e.g., piezoreflectance) and dynamic (thermal-conductivity) regimes. The present theory does not yield such distinctions for the two regimes. It is concluded, therefore, that the effective-mass approximation is not adequate for describing the full range of frequency-dependent stress effects. It should be emphasized that the heat-pulse study is particularly well suited for investigating these differences because the constants for the two regimes are derived within the framework of the same experiment.