In nanostructured semiconductors such as quantum dots and quantum wires, the modification of phonon dispersion relations plays a critical role in altering thermal transport properties. Unlike bulk materials, where phonons propagate freely with well-defined dispersion, low-dimensional systems impose spatial confinement that fundamentally changes vibrational behavior. This confinement leads to discrete phonon modes, modified group velocities, and increased boundary scattering, all of which contribute to a significant reduction in thermal conductivity—a key factor in enhancing thermoelectric efficiency.

The phonon dispersion in quantum dots and wires is strongly influenced by their size, shape, and boundary conditions. In quantum dots, which are zero-dimensional structures, phonons are confined in all three spatial dimensions, resulting in a complete quantization of vibrational states. The absence of propagating modes means that heat is primarily carried through localized vibrations, drastically suppressing thermal conductivity. For quantum wires, which are one-dimensional, phonons are confined in two dimensions but can propagate along the wire axis. However, the reduced dimensionality alters the phonon density of states and introduces subband formation, leading to modified dispersion relations compared to bulk materials.

A key consequence of spatial confinement is the discretization of the phonon spectrum. In bulk semiconductors, acoustic phonons follow a linear dispersion near the Brillouin zone center, with frequencies proportional to the wavevector. In quantum wires, however, the transverse confinement quantizes the phonon wavevectors, creating a series of one-dimensional subbands. Each subband exhibits a modified dispersion relation, with a cutoff frequency below which no propagating modes exist. This leads to a reduction in available phonon states for heat transport, particularly for low-frequency modes that dominate thermal conduction in bulk materials.

The group velocity of phonons is also significantly altered in quantum-confined systems. In bulk materials, the group velocity is determined by the slope of the dispersion curve, with acoustic phonons having high velocities near the zone center. In quantum wires, the group velocity becomes highly anisotropic, with transverse modes having near-zero velocities due to confinement. Even longitudinal modes exhibit reduced velocities compared to bulk due to the modified dispersion. This reduction directly lowers thermal conductivity, as heat transport is proportional to the product of phonon group velocity, specific heat, and mean free path.

Boundary scattering further suppresses thermal conductivity in quantum dots and wires. In bulk materials, phonon-phonon scattering (Umklapp processes) is the dominant resistive mechanism at room temperature and above. In nanostructures, however, surfaces and interfaces introduce additional scattering mechanisms that limit the phonon mean free path. For quantum dots, the mean free path is constrained by the dot size itself, while in quantum wires, surface roughness and edge defects strongly scatter phonons. The increased boundary scattering reduces the effective phonon lifetime, further diminishing thermal transport.

The interplay between modified dispersion and scattering mechanisms leads to a strong size dependence of thermal conductivity in quantum-confined systems. Experimental studies on silicon nanowires, for example, have demonstrated a reduction in thermal conductivity by over an order of magnitude compared to bulk silicon when the wire diameter is below 100 nm. Similar effects have been observed in quantum dot superlattices, where the periodic potential introduced by the dots disrupts phonon propagation, creating a phonon bandgap that filters specific frequencies.

For thermoelectric applications, the suppression of thermal conductivity is highly desirable, as it enhances the figure of merit (ZT) without degrading electronic properties. Quantum dots and wires enable selective scattering of phonons while maintaining reasonable electrical conductivity through careful engineering of carrier concentrations and band alignments. The ability to tune phonon dispersion through size and interface design provides a powerful route to optimizing thermoelectric performance.

In summary, quantum confinement in dots and wires fundamentally alters phonon dispersion relations, leading to discrete vibrational modes, reduced group velocities, and enhanced boundary scattering. These effects collectively suppress thermal conductivity, making low-dimensional semiconductors promising candidates for high-efficiency thermoelectrics. The precise control of nanostructure dimensions and interfaces allows for tailored phonon engineering, offering a pathway to advanced thermal management and energy conversion technologies.