On the theory of phonons: A journey from their origins to the intricate mechanisms of their transport

Jose Antonio Pascual-Gutierrez, Purdue University

Abstract

In recent years, a number of factors have fueled interest in the nanoscale thermal transport. The success of the microelectronics industry in delivering on its promises to continuously fabricate smaller and faster devices has significantly increased chip-level heat fluxes, which has focused interest in transistor-scale heat transfer. In addition, dramatic advances in nanofabrication technology in the last decade have made it possible to consider band-gap engineering and atomic-level tailoring to achieve desired physical properties. As we speak, large-scale research and development efforts are being focused on thin film photovoltaic technology. These advances have been and are fueling interest in developing a fundamental understanding of heat transport in the nanometer regime and the genesis of new theoretical concepts. Phonons in nanostructures such as semiconductor superlattices, quantum wells, quantum wires and carbon nanotubes have elicited significant recent interst. As the size of the sample becomes smaller and the surface-to-volume ratio increases, new surface phenomena emerges which are not well understood. Additionally, extreme reduction of the size of a structure unveils new phenomena triggered by changes in the energy spectrum of the lattice. In order to provide insight into size-confinement effects, we study via lattice dynamics the onset of phonon-spectrum changes in silicon [111]-films and [111]-wires. It is determined that for film thicknesses below 10 nm and for wire diameters smaller than 5 nm, size-confinement effects are observed. These thresholds are supported by computations of the density of states, volumetric specific heats, and phonon group velocities. It is found that the volumetric specific heat for very low temperatures can be two orders of magnitude greater than that of bulk silicon for films with thicknesses under one nanometer and as much as three orders of magnitude greater for wires with diameters in the sub-nanometer range. Nevertheless, confinement effects appear only below 100 K, when the predominant phonon wavelengths become comparable to the physical dimension of the structure. Group velocity computations confirm that sizeinvariance is approached above 10 nm for silicon thin films and above about 5 nm for wires. However, wires and films exhibit different group velocity asymptotes. This may be explained by the fact that the primitive cells defined for theses nanosystems expand to the physical limits of the structure, and geometry-specific modes are captured which would not be present in the bulk material. Motivated by the electronic industry’s thermal management issues and aided by the increased availability of computing power, scientists have begun to develop computational tools to model nano-heat transfer. The Boltzmann transport equation (BTE) has proven to be the well suited to simulate phonon transport. However, the complexity of the phonon BTE has prompted scientist to resort to so-called relaxation time approximations. Here we present the use of perturbation theory of quantum mechanics through Fermi’s Golden Rule to compute the scattering rates of three-phonon processes. Also, a detailed algorithm for identifying triads of phonons involved in threephonon interactions from actual dispersion curves is developed. The Environment- Dependent Interatomic Potential (EDIP) is employed to evaluate the anharmonic interatomic force constants (IFCs) needed to evaluate Fermi’s Golden Rule. The correctness of the computed scattering rates is verified by their introduction in a linearized phonon BTE and the computation of the phonon distribution under the assumption of an infinitesimal temperature gradient. Using these phonon distributions the thermal conductivity of bulk silicon is computed and compared with experimental data. Good agreement is found. We find that nearly 96% of the heat transfer is by acoustic modes, and of that 96%, about two-thirds is carried by the transverse acoustic modes. Additionally, computations of phonon mean free paths clearly indicate that thermal conductivity reduction due to boundary scattering is expected to occur at about 300 nm, which corresponds to the mean free path of phonon modes carrying most of the energy. The full-fledged phonon BTE is an integro-differential equation that cannot be solved through discretization techniques. The Monte Carlo approach provides an ideal path-to-solution to solve the full-fledged phonon BTE. Here we develop a novel Monte Carlo approach that uses scattering rates obtained from the perturbation theory of quantum mechanics to solve the full-fledged phonon BTE. Three-phonon processes consistent with energy and momentum conservation rules are accounted for. The Monte Carlo code is able model both ballistic and diffuse transport and any situation in between. Also, the code is shown to capture the transient regime well. Thermal conductivities for silicon in the quasi-diffuse limit are computed and it shows that the values so obtained match the thermal conductivities resulting form the direct solution of the full-fledged phonon BTE in its linear form, and are in reasonable agreement with experimental data in the 200 K to 500 K temperature range. The current Monte Carlo approach represents a significant advance over existing computational techniques and facilitates the study of nanoscale heat transport with granularity not hitherto possible.

Degree

Ph.D.

Advisors

Viskanta, Purdue University.

Subject Area

Mechanical engineering

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