Phonon dispersion relations describe the relationship between the frequency of lattice vibrations (phonons) and their wavevector in a crystalline solid. These relations are fundamental to understanding the dynamic behavior of atoms in semiconductors and play a crucial role in determining thermal and electronic properties. The dispersion curves reveal essential features such as acoustic and optical phonon branches, zone-boundary frequencies, and anisotropy in different crystallographic directions.

The theoretical foundation of phonon dispersion relations is rooted in lattice dynamics, where the motion of atoms is treated as a system of coupled harmonic oscillators. The forces between atoms are approximated using interatomic potentials, and the equations of motion are solved under periodic boundary conditions. For a crystal with a basis of multiple atoms per unit cell, such as in compound semiconductors, the phonon spectrum splits into acoustic and optical branches. Acoustic phonons represent in-phase vibrations of neighboring atoms, while optical phonons involve out-of-phase motion, often leading to higher frequencies. The shape of the dispersion curves depends on the crystal structure, atomic masses, and bonding strength.

In diamond-structured semiconductors like silicon, the phonon dispersion exhibits three acoustic branches (one longitudinal and two transverse) and three optical branches. The longitudinal acoustic (LA) and transverse acoustic (TA) branches show a linear dependence on the wavevector near the Brillouin zone center, reflecting the sound velocity in the material. The optical branches are nearly flat, indicating weak dispersion due to the strong covalent bonding. In contrast, zincblende-structured materials like gallium arsenide (GaAs) display more pronounced splitting between longitudinal optical (LO) and transverse optical (TO) phonons at the zone center due to the polar nature of the bonds, leading to LO-TO splitting.

Experimental determination of phonon dispersion relations relies on techniques that probe the energy and momentum transfer between incident particles and phonons. Inelastic X-ray scattering (IXS) and inelastic neutron scattering (INS) are the most widely used methods. IXS provides high momentum resolution and is suitable for studying small Brillouin zones, making it ideal for semiconductors with large unit cells. INS, on the other hand, offers superior energy resolution and is particularly useful for materials with light atoms, such as silicon carbide (SiC), where hydrogen-like isotopes may complicate X-ray measurements. Both techniques involve measuring the energy shifts of scattered particles to reconstruct the phonon dispersion curves.

The phonon dispersion of silicon has been extensively studied, revealing characteristic features such as the degeneracy of the transverse acoustic and optical branches along high-symmetry directions. The maximum phonon frequency in silicon occurs at the zone boundary and is approximately 15.5 THz. In GaAs, the LO-TO splitting at the Γ-point is around 1 THz, a consequence of the ionic contribution to the bonding. Silicon carbide, with its many polytypes, exhibits variations in phonon dispersion depending on the stacking sequence. For example, 3C-SiC (zincblende) shows distinct phonon branches compared to 4H-SiC or 6H-SiC (hexagonal), where the additional zone-folding effects due to larger unit cells modify the dispersion relations.

The influence of phonon dispersion on electronic properties arises primarily through electron-phonon interactions, which govern carrier scattering rates and mobility. In polar semiconductors like GaAs, the Fröhlich interaction with longitudinal optical phonons dominates at high temperatures, limiting electron mobility. In non-polar materials like silicon, deformation potential scattering with acoustic phonons is more significant. The anisotropy of phonon dispersion also affects the effective masses of carriers, as the curvature of the bands near critical points determines the density of states.

Phonon dispersion relations are also critical for understanding thermal properties, although this article avoids detailed discussion of thermal conductivity. The group velocity of phonons, derived from the slope of the dispersion curves, directly impacts heat transport. Flat optical branches contribute little to thermal conduction due to their low group velocities, while steep acoustic branches are the primary heat carriers. The presence of avoided crossings or phonon bandgaps can further influence thermal transport by restricting phonon propagation.

In summary, phonon dispersion relations provide a comprehensive picture of lattice dynamics in semiconductors, reflecting the interplay between crystal structure, atomic bonding, and vibrational modes. Experimental techniques like IXS and INS enable precise mapping of these relations, revealing material-specific features that influence both thermal and electronic behavior. From silicon's covalent bonds to GaAs's polar interactions and SiC's polytypic complexity, phonon dispersion curves serve as a fingerprint of the underlying atomic arrangement and bonding chemistry. Understanding these relations is essential for designing materials with tailored vibrational properties for advanced electronic and optoelectronic applications.