Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Mechanical Engineering & Applied Mechanics

First Advisor

Jennifer R. Lukes


Phononic crystals are periodic structured materials whose frequency spectrum is characterized by band gaps, which are regions in frequency space where acoustic or elastic waves cannot propagate. Nano scale phononic crystals have shown promise for reducing thermal conductivity and improving the thermoelectric figure of merit. Correctly calculating the thermal conductivity of nano phononic crystals has become increasingly important due to the growing research interest in the thermal properties of these materials. A widely used expression to calculate thermal conductivity, presented by Klemens and expressed in terms of the relaxation time by Callaway and Holland, originates from the Boltzmann transport equation. In its most general form, this expression involves a direct summation of the heat current contributions from individual phonons of all wavevectors and polarizations in the first Brillouin zone. In common practice, the expression is simplified by three assumptions commonly applied in bulk materials: first, the isotropic assumption that converts the summation over wavevector to an integral over wavevector magnitude; second, the assumption that phonon-phonon scattering rates for nano-phononic crystals can be described by the same empirical expressions commonly used for bulk materials and fitted to experimental data in bulk materials; third, the effective material assumption that the thermal transport can be modeled by treating the nano-phononic crystal as a single bulk effective medium with properties dictated by the nano-phononic crystal dispersion relation. The accuracy of nano-phononic crystal thermal conductivity predictions using these three assumptions need to be validated. In this dissertation, we propose to verify these assumptions one by one.

First, to investigate the isotropic assumption, the thermal conductivities of bulk Si, Si/Ge superlattices, and Si/Ge quantum dot superlattices have been calculated using both the isotropic and direct summation methods, and the results show that the differences between the two methods increase substantially with supercell size. These differences arise because the vibrational modes neglected in the isotropic assumption provide an increasingly important contribution to the thermal conductivity for larger supercells. To avoid the significant errors that can result from the isotropic assumption, direct summation is recommended for thermal conductivity calculations in superstructures. Second, to investigate the assumption of the empirical phonon-phonon scattering rates from bulk material, work to calculate the phonon-phonon scattering rates from the empirical equations has been done and compared against the results from an established normal mode analysis method, which provides more accurate results. The fundamental reasons behind the difference between the empirical method and the NMA method will be discussed. Finally, the effective material assumption will be briefly examined by using Green Kubo Modal Analysis method. Overall, this dissertation will provide direction in the correct thermal conductivity calculation for nano-phononic crystals.

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