Theoretical investigations of phonon confinement effects in nanomaterials have become increasingly important for understanding size-dependent thermal and vibrational properties. As material dimensions approach the nanoscale, the mean free paths of phonons become comparable to or larger than the physical dimensions of the structure, leading to significant modifications in lattice dynamics. Computational methods such as density functional theory (DFT) and molecular dynamics (MD) simulations have proven indispensable for predicting these effects in systems like silicon nanowires and graphene nanoribbons, where quantum confinement alters thermal conductivity and Raman spectra.

In bulk materials, phonons propagate without significant boundary scattering, but in nanostructures, their dispersion relations and scattering mechanisms are strongly influenced by finite-size effects. DFT calculations provide a first-principles approach to studying these modifications by solving the electronic structure and deriving interatomic force constants. These force constants serve as inputs for lattice dynamics calculations, enabling the prediction of phonon dispersion relations and group velocities. For silicon nanowires with diameters below 20 nm, DFT-based studies reveal a reduction in acoustic phonon group velocities due to boundary scattering, leading to decreased thermal conductivity. The anisotropy of phonon confinement in nanowires further complicates the thermal transport, with longitudinal modes being less affected than transverse modes.

Molecular dynamics simulations complement DFT by capturing anharmonic effects and temperature-dependent phonon-phonon interactions that are computationally expensive for DFT. Classical MD using empirical potentials, such as the Stillinger-Weber potential for silicon or the Tersoff potential for carbon, has been widely employed to study thermal conductivity in nanostructures. Non-equilibrium MD simulations of silicon nanowires show that thermal conductivity decreases by up to 80% when the diameter is reduced from 10 nm to 2 nm. This reduction is attributed to increased phonon-boundary scattering and the suppression of long-wavelength phonons, which dominate heat conduction in bulk silicon. Similarly, equilibrium MD simulations based on the Green-Kubo formalism predict a strong diameter dependence of thermal conductivity in graphene nanoribbons, with armchair edges exhibiting higher conductivity than zigzag edges due to differences in edge phonon scattering.

Raman spectroscopy serves as a sensitive probe of phonon confinement, and theoretical models have been developed to predict size-dependent shifts in Raman peaks. In silicon nanowires, the confinement of optical phonons leads to an asymmetric broadening and a downward shift of the Raman peak position. DFT calculations show that for diameters below 5 nm, the Raman shift can deviate by more than 5 cm-1 from the bulk value due to the relaxation of momentum conservation rules. A similar effect is observed in graphene nanoribbons, where the G-band Raman peak softens as the ribbon width decreases below 10 nm. Tight-binding models and first-principles calculations attribute this softening to edge-induced phonon renormalization and the confinement of vibrational modes in the transverse direction.

Theoretical studies also highlight the role of surface reconstructions and edge terminations in modifying phonon properties. In hydrogen-terminated silicon nanowires, DFT calculations demonstrate that surface hydrogenation reduces the surface-induced phonon localization, partially restoring bulk-like thermal transport characteristics. For graphene nanoribbons, edge functionalization with oxygen or hydrogen alters the phonon density of states, leading to variations in thermal conductivity by a factor of two or more depending on the edge chemistry. Ab initio MD simulations further reveal that edge disorder introduces additional phonon scattering centers, further suppressing thermal transport.

Finite-length effects in nanostructures introduce additional complexity, as phonon confinement occurs in multiple dimensions. In ultrashort silicon nanowires with lengths comparable to the phonon mean free path, ballistic transport dominates, and classical Fourier's law breaks down. Modal analysis based on lattice dynamics calculations shows that the contribution of different phonon branches to thermal conductivity becomes highly length-dependent, with optical phonons playing a non-negligible role in short nanowires despite their low group velocities in bulk silicon. For graphene nanoribbons, length effects are less pronounced due to the high intrinsic phonon mean free paths, but edge roughness becomes a limiting factor for thermal transport in structures longer than a few hundred nanometers.

Theoretical frameworks have also been developed to account for temperature-dependent phonon confinement effects. Quasi-harmonic approximations combined with DFT calculations enable the prediction of thermal expansion coefficients and Grüneisen parameters in nanostructures, which differ significantly from bulk values due to surface stress effects. In silicon nanowires, the negative thermal expansion observed at low temperatures is enhanced by phonon confinement, while graphene nanoribbons exhibit anisotropic thermal expansion behavior depending on edge chirality. Anharmonic lattice dynamics calculations further reveal that the temperature dependence of thermal conductivity in nanostructures deviates from the traditional T-1 behavior observed in bulk materials due to the modified phonon scattering phase space.

Recent advances in computational methods have enabled the study of coupled electron-phonon effects in nanostructures, particularly in the context of thermoelectric applications. DFT calculations incorporating electron-phonon coupling show that phonon confinement can enhance the Seebeck coefficient in silicon nanowires by modifying the density of states near the band edges. Similarly, in graphene nanoribbons, phonon-drag effects become significant at low temperatures, contributing to anomalous thermoelectric behavior that is not captured by classical Boltzmann transport theory alone.

The predictive power of these theoretical approaches has been instrumental in guiding the design of nanomaterials with tailored thermal and vibrational properties. By quantifying the impact of size, shape, and edge effects on phonon transport, computational studies provide insights that are difficult to obtain experimentally. Future developments in multiscale modeling and machine learning-assisted force field development promise to further enhance the accuracy and efficiency of these predictions, enabling the exploration of more complex nanostructured systems and their applications in nanoelectronics, thermoelectrics, and photonics.