Thermal Conductivity of Solids IV: Resonance Fluorescence Scattering of Phonons by Donor Electrons in Germanium

Abstract

The scattering of phonons from donor electrons in germanium is computed and applied to the problem of heat conduction below the low-temperature maximum. The effective-mass approximation is used for the electronic wave function. The calculation is analogous to that of the dispersive scattering of light by atoms. Since the ground state is split into a singlet and a triplet with a separation of energy within the thermal phonon distribution, anomalous effects due to resonance scattering occur. Apart from these, our results are similar to previous work by Keyes. Taking into account the boundary and isotopic scattering as well as the electron-phonon interaction, excellent quantitative agreement with thermal conductivity data is obtained, with no adjustable parameters, for Sb and As donors in Ge. Certain discrepancies at higher temperatures can be traced to the inadequacy of the effective-mass approximation. In addition, anomalies in the "phonon drag" thermoelectric power can be understood on the basis of our theory. A by-product of our calculation is the lifetime broadening of the triplet state. The slightly different case of $n$-type Si is worked out while a more qualitative treatment of our resonant scattering mechanism is given for $p$-type semiconductors and samples under uniaxial strain.