The ability of a material to convert heat into electricity is measured by the dimensionless thermoelectric figure of merit , where is the Seebeck coefficient, the electrical conductivity, the temperature, and the thermal conductivity. Good thermoelectric materials should simultaneously have a high Seebeck coefficient, a high electrical conductivity, and a low thermal conductivity. Due to the strong interconnection between the parameters that control , it has been traditionally proved difficult to achieve values above unity, which translates to low conversion efficiencies. Recent advancements in nanofabrication, however, have led to breakthrough experiments on nanostructured thermoelectric devices. In this thesis, the thermal and thermoelectric properties of silicon- and graphene-based nanostructures are numerically investigated. The Seebeck coefficient, electrical conductivity, and thermal conductivity in nanostructures are computed, and the thermoelectric figure of merit is extracted.

For graphene-based nanostructures, we employ the force constant method for the calculations of the phononic properties, and the tight-binding model for the electronic properties. Both ballistic and diffusive transport regimes are considered employing the Landauer approach and the non-equilibrium Green's function technique, respectively. The ballistic to diffusive crossover of the thermoelectric properties of graphene nanoribbons with armchair edges has been studied. Our results indicate that in armchair graphene nanoribbons the power factor increases with the width due to the contribution of the second conduction subband. However, the small band-gap of wide ribbons degrades the Seebeck coefficient which results in a low power factor. Including the high thermal conductance of graphene, we show that the ballistic value remains below . The introduction of edge roughness degrades the transport of electrons much more than that of phonons. The diffusive values of armchair ribbons, therefore, are smaller than the ballistic ones, and the thermoelectric performance decreases with increasing the channel length. On the other hand, by introducing ordered antidots, the zero band-gap graphene can be converted into a narrow band-gap semiconductor. We show that the size and the circumference of the antidots, and the distance between them can strongly influence the thermal properties of graphene antidot lattices. By appropriate selection of the antidot parameters, the thermal conductance can be significantly reduced and achieved. In the case of zigzag graphene ribbons, positively charged substrate background impurities and extended line defects in the length direction of the nanoribbon create an asymmetry in the density of modes around the Fermi level, which improves the Seebeck coefficient. In contrast to armchair ribbons, here, the introduction of edge roughness degrades the phonon thermal conductivity much more than the electronic thermal conductivity. In zigzag graphene nanoribbons these effects can theoretically result in large values of around .

For silicon-based nanostructures, we employ atomistic calculations of the phonon modes using the modified-valence-force-field method. We consider ultra-narrow silicon nanowires of side sizes of to as well as ultra-thin silicon layers of thicknesses between and . Our results indicate that nanowires have the highest phonon group velocity and thermal conductance, whereas nanowires have the lowest. We also find that the ballistic thermal conductance in the thin layers is anisotropic, with the channels exhibiting the highest and the channels the lowest thermal conductance with a ratio of about two. The nanowires and thin layers are thus the most suitable channels for thermoelectric devices in terms of the thermal conductance. The effects of scattering mechanisms, such as phonon-phonon scattering and surface roughness scattering are investigated employing the Boltzmann transport equation for phonons. The thermal conductivity of quasi-1D nanowires diverges as the diameter is reduced. We attribute this to the fact that in ultra-narrow nanowires the density-of-states and the transmission function of long-wavelength phonons acquires a finite value, as compared to zero in the bulk materials, which increases their importance in carrying heat. At the same roughness conditions, boundary scattering is more specular for the ultra-narrow nanowires, and becomes more diffusive as the diameter is increased. This results in a striking anomalous increase in the thermal conductivity as the diameter is reduced below . Taking the electronic power factor of ultra-narrow silicon nanowires from literatures, we show that in the best case the value at is around . The largest contribution towards achieving this relatively high value is attributed to the significant reduction in the thermal conductivity due to boundary scattering of phonons. In the case of fully diffusive boundaries, the values can increase above unity for both -type and -type nanowires.