7. Summary and Conclusions

In this thesis, we investigate the thermal and thermoelectric properties of silicon- and graphene-based nanostructures. We start from the investigation of the ballistic to diffusive crossover in the thermoelectric transport in armchair graphene nanoribbon (AGNR). Although in the case of AGNRs a bandgap is naturally present, the band-gap decreases with increasing the width, which results in a lower value of Seebeck coefficient for wider ribbons. As a result, the ballistic power factor only slightly increases with temperature and the ribbon's width. In contrast, by increasing the width the lattice thermal conductance strongly increases and so the ballistic $ZT$ value decreases with increasing $W$ . The introduction of edge roughness in order to reduce high thermal conductivity does not benefit $ZT$ because the electrical conductance is severely degraded by the roughness. As a result, the $ZT$ figure of merit decreases with increasing the channel length and its value is limited to values below 0.3.

We also analyze the ballistic thermoelectric properties of GALs. Our results indicate that the size of the antidots, the circumference of the antidots, and the distance between antidots can strongly influence the thermal properties of GALs. Results from ballistic calculations show that by appropriate selection of the geometrical parameters one can significantly reduce the thermal conductance of GALs and improve their thermoelectric figure of merit.

We present a theoretical design procedure for achieving high thermoelectric performance in zigzag graphene nanoribbon (ZGNRs) channels, which in their pristine form have very poor performance. We show that by introducing extended line defects in the length direction of the nanoribbon we can create an asymmetry in the density of modes around the Fermi level, which improves the Seebeck coefficient. ELDs increase the electronic conduction subbands, which increase the channel conductance as well. The power factor is therefore significantly increased. In addition, we show that by introducing edge roughness the phonon thermal conductivity is degraded effectively more than the electronic thermal conductivity, or the electronic conductance. These three effects result in large values of the thermoelectric figure of merit, and indicate that roughed ZGNRs with ELDs could potentially be used as efficient high performance thermoelectric materials.

Next, we investigate how dimensionality affects thermal and thermoelectric properties of low-dimensional silicon-based nanostructures. We study the effect of confinement on the phonon properties of ultra-narrow silicon nanowires of side sizes of $1~\mathrm{nm}$ to $10~\mathrm{nm}$ . We use the modified valence force field (MVFF) method to compute the phononic dispersion and extract the density of states, the transmission function, the sound velocity, and the ballistic thermal conductance. We find that the phononic dispersion and the ballistic thermal conductance are functions of the geometrical features of the structures, i.e., the transport orientation and confinement length scale. The phonon group velocity and thermal conductance can vary by a factor of two depending on the geometrical features of the channel. The $\langle110\rangle$ nanowire has the highest phonon group velocity and thermal conductance, whereas the $\langle111\rangle$ has the lowest. The $\langle111\rangle$ channel is thus the most suitable orientation for thermoelectric devices based on silicon nanowires since it also has a large power factor.

Next, we investigate the effect of confinement and orientation on the phonon transport properties of ultra-thin silicon layers of thicknesses between $1~\mathrm{nm}$ to $16~\mathrm{nm}$ . We consider the major thin layer surface orientations $\{100\}$ , $\{110\}$ , $\{111\}$ , and $\{112\}$ . For every surface orientation, we study thermal conductance as a function of the transport direction within the corresponding surface plane. We find that the ballistic thermal conductance in the thin layers is anisotropic, with the $\{ 110 \} / \langle 110\rangle$ channels exhibiting the highest and the $\{112\} / \langle111\rangle$ channels the lowest thermal conductance with a ratio of about two. We find that in the case of the $\{110\}$ and $\{112\}$ surfaces, different transport orientations can result in $\sim 50\%$ anisotropy in thermal conductance. The thermal conductance of different transport orientations in the $\{100\}$ and $\{111\}$ layers, on the other hand, is mostly isotropic. These observations are invariant under different temperatures and layer thicknesses. We show that this behavior originates from the differences in the phonon group velocities, whereas the phonon density of states is very similar for all the thin layers examined. We also show how the phonon velocities can be understood from the phonon spectrum of each channel. These findings could be useful in the design of the thermal properties of ultra-thin silicon layers for thermoelectric and thermal management applications.

In this work we also study the thermal conductivity of ultra-thin silicon nanowires using the atomistic modified valence-force-field method for the computation of the phonon bandstructure and the Boltzmann equation for phonon transport. We show that the problem of long-wavelength phonons as described by Ziman and others, which causes divergence in the thermal conductivity of quasi-1D channels with increasing length, is also present in silicon nanowires. The divergence occurs not only as the length is increased, but also 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, and causes the thermal conductivity to increase as the diameter is reduced below $5~\mathrm{nm}$ . We point out that this effect has two important consequences: The first is that a larger portion of heat is carried by low frequency phonons in ultra-narrow nanowires as compared to bulk, e.g. almost $80\%$ of the heat is carried by phonons with energies below $5~\mathrm{meV}$ . The second is that, counter-intuitively, at the same roughness conditions, the 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 $5~\mathrm{nm}$ .

Finally, the room temperature $ZT$ figure of merit of ultra-narrow silicon nanowires of diameters $D<12~\mathrm{nm}$ is calculated using atomistic simulations for both electrons and phonons. The $ZT$ values at $300~\mathrm{K}$ in the best case are slightly below unity ($\sim 0.75$ ), in agreement with experimental measurements. We show that the largest contribution towards achieving this relatively high value is attributed to the significant reduction in the thermal conductivity due to boundary scattering. Phonon confinement also causes a reduction in thermal conductivity and $ZT$ improvement, but its effect is weaker. For ultra-narrow nanowire diameters ( $D\sim 3~\mathrm{nm}$ ), the power factor is strongly reduced due to surface roughness scattering. We show, however, that the benefits from phonon-boundary scattering are still persistent in increasing $ZT$ , since for the same roughness amplitudes, boundary scattering reduces the thermal conductivity significantly more than it reduces the power factor (by $\sim 4X$ ). Finally, we calculate that in the case of fully diffusive boundaries for phonons, the $ZT$ values can increase above unity for both n-type and p-type nanowires.