The Cost of a Parabola: Phonon Mode-Counting Errors of the Debye Spectrum in Lattice Thermal Conductivity Estimates for Silicon, Germanium, and Lead Telluride

The Debye model conserves the total phonon mode count but misplaces spectral weight. For silicon, germanium and lead telluride, a digitised representation of the published phonon density of states and the Debye spectrum, both normalised to three modes per atom, are inserted into the same relaxation-time expression for lattice thermal conductivity with an identical Umklapp scattering law, isolating the pure spectral-placement error. The Debye spectrum underestimates the conductivity by 32% (Si), 25% (Ge) and 38% (PbTe) at the Debye temperature, and by 48-55% near 0.1 times the Debye temperature. Includes the paper, the computation script and its full numerical output.