| Although the study of thermal transport in condensed matter has a very long history, it continues to be an active field of work due to its importance in many applications. The research subject reported in this thesis is on theoretical investigations of thermal energy transport in systems whose linear dimension is less than the wavelength of thermal phonons. Such situations occur in mesoscopic and nanoscopic scale dielectric structures which can now be fabricated in a number of laboratories. Due to the small system dimensions, phonons must be treated as waves. Thermal energy transport, therefore, must be treated as phonon wave propagation through the system.; After reviewing the general physics of thermal energy transport in the classical regime, we derive, for dielectric materials, a formula for thermal energy flux in devices involving multi-terminals each connected to a thermal reservoir at local equilibrium. The energy flux is driven by a temperature bias and traverses the system by virtue of phonon wave scattering. A multi-terminal thermal conductance formula is derived in terms of phonon transmission coefficient. Using our theoretical formulation, we investigate thermal transport properties of both two-terminal and four-terminal dielectric devices by solving the quantum scattering problem using a mode matching numerical technique.; For thermal transport in a T-shaped dielectric nanostructure with two-terminals at low temperature, due to quantum interference the transmission coefficient of phonons becomes quite complicated. We found that the value of phonon transmission coefficients at zero energy may be unity or zero depending on a geometrical ratio of the nanostructure. The transmission has an oscillation behavior with quasi-periodicity and irregularity. The thermal conductance is found to increase monotonically with temperature---a result that we conclude to be generally true for any two-terminal device. We confirm the existence of the universal quantum of thermal conductance which exists at the low temperature limit, and such a quantum is robust against all the system parameters.; The physical behavior of four-terminal thermal conductance for mesoscopic dielectric systems with arbitrary shapes of scattering region is also investigated in detail. If we make a two-terminal measurement in the four-terminal device, the two-terminal conductance is a monotonically increasing function of temperature, and is equal to the universal quantum of thermal conductance masked by a geometric factor. If we make a four-terminal measurement, the four-terminal conductance has a non-monotonic dependence. In the low temperature limit, we predict that the four-terminal conductance diverges inversely proportional to temperature.; Finally, we discuss an interesting theoretical problem on the general behavior of thermal conductance for multi-terminal systems when thermal carriers satisfy fractional exclusion statistics. Our analysis allows us to conclude that results for fractional exclusion statistics are quite different from those of the Bose-Einstein statistics. |
