Lattice dynamics and Raman spectra of isotopically mixed diamond

Abstract

We present coherent-potential-approximation (CPA) calculations and first- and second-order Raman spectra for diamonds with varying concentrations of $^{12}\mathrm{C}$ and $^{13}\mathrm{C}$. The calculations are based on the valence-force model of Tubino, Piseri, and Zerbi [J. Chem. Phys. 56, 1022 (1972)]. Contrary to previous claims, we find that this model does not give a sharp peak in the density of states (DOS) near the Raman mode. Alternative dispersion curves that do give such a peak are discussed. Raman results are reported for high-quality, single-crystal synthetic diamonds with isotopic compositions ranging from nearly pure $^{12}\mathrm{C}$ to nearly pure $^{13}\mathrm{C}$. A measurable deviation in the Raman frequency away from a simple ${\mathit{M}}^{\mathrm{\ensuremath{-}}1/2}$ (``virtual-crystal'') dependence and an observable broadening of the first- and second-order spectra are qualitatively consistent with CPA predictions for the effects of isotopic disorder. Quantitative agreement between theory and experiment is achieved only if the reference DOS contains the above-mentioned peak. This supports the interpretation of the controversial second-order Raman peak in naturally abundant diamond (1.1 at. % $^{13}\mathrm{C}$) at 2667 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ as a DOS effect. At all compositions, the effects of isotopic disorder are relatively weak because of the small mass difference between $^{12}\mathrm{C}$ and $^{13}\mathrm{C}$. For 1.1 at. % $^{13}\mathrm{C}$, the maximum broadening predicted in the CPA is less than 1 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$, nearly two orders of magnitude smaller than a previous estimate. For the lowest-frequency modes most relevant to the thermal conductivity, the CPA scattering rate reduces to the usual ${\mathrm{\ensuremath{\omega}}}^{4}$ dependence first derived by Klemens for phonon-isotope scattering. Using Callaway's theory, we show that this term can easily account for the recently observed 50% enhancement in room-temperature thermal conductivity upon elimination of $^{13}\mathrm{C}$ impurities, provided that sufficient normal scattering also occurs.