⚛️Solid State Physics Unit 3 – Lattice dynamics and phonons

Key Concepts and Definitions

• Lattice dynamics studies the vibrations and thermal properties of crystal lattices
• Phonons are quantized modes of vibration in a crystal lattice analogous to photons in electromagnetic waves
• Dispersion relation describes the relationship between the frequency and wavevector of phonons
  • Determines the group velocity and density of states of phonons
• Harmonic approximation assumes small displacements and linear restoring forces between atoms in a lattice
• Normal modes are independent collective motions of atoms in a lattice with a specific frequency and wavevector
• Brillouin zone is the primitive cell in the reciprocal lattice space
  • Phonon dispersion relations are typically plotted within the first Brillouin zone
• Debye model approximates the phonon dispersion relation as linear and isotropic up to a cutoff frequency

Crystal Lattice Basics

• Crystal lattices are periodic arrangements of atoms in three dimensions
• Bravais lattices are the 14 distinct lattice types that describe all possible crystal symmetries
• Primitive cell is the smallest repeating unit that can generate the entire lattice by translation
• Basis is the arrangement of atoms within the primitive cell
• Reciprocal lattice is the Fourier transform of the real-space lattice
  • Each point in the reciprocal lattice corresponds to a set of lattice planes in the real-space lattice
• Brillouin zones are the Wigner-Seitz cells of the reciprocal lattice
  • The first Brillouin zone contains all unique wavevectors closest to the origin

Harmonic Approximation and Normal Modes

• Harmonic approximation assumes small displacements of atoms from their equilibrium positions
• Potential energy of the lattice is expanded as a Taylor series truncated to the second order
• Equations of motion for the atoms are linear and can be solved exactly
• Normal modes are the eigenmodes of the dynamical matrix that couples the equations of motion
  • Each normal mode has a specific frequency and wavevector
• Acoustic modes have frequencies that approach zero as the wavevector goes to zero (long wavelength limit)
  • Correspond to rigid translations of the entire lattice
• Optical modes have non-zero frequencies at the long wavelength limit
  • Correspond to out-of-phase oscillations of atoms within the primitive cell

Phonon Dispersion Relations

• Phonon dispersion relations plot the frequency of phonons as a function of their wavevector
• Dispersion relations are typically plotted along high-symmetry directions in the Brillouin zone
• Acoustic branches start from zero frequency at the Brillouin zone center (Γ point)
  • Longitudinal acoustic (LA) and transverse acoustic (TA) branches correspond to different polarizations
• Optical branches have non-zero frequencies at the Γ point
  • Longitudinal optical (LO) and transverse optical (TO) branches correspond to different polarizations
• Group velocity of phonons is the slope of the dispersion relation ($\frac{d\omega}{dk}$)
• Density of states (DOS) measures the number of phonon modes per unit frequency range
  • Van Hove singularities occur at critical points in the Brillouin zone where the group velocity vanishes

Quantization of Lattice Vibrations

• Phonons are the quanta of lattice vibrations, analogous to photons in electromagnetic waves
• Each phonon has an energy $\hbar\omega$ and a wavevector $\vec{k}$
• Occupation number of a phonon mode follows the Bose-Einstein distribution ($\frac{1}{e^{\hbar\omega/k_BT}-1}$)
  • Phonons are bosons and can have any integer occupation number
• Zero-point energy is the quantum mechanical ground state energy of the lattice ($\frac{1}{2}\hbar\omega$ per mode)
• Creation and annihilation operators ($a^\dagger$ and $a$) raise or lower the occupation number of a phonon mode
• Displacement and momentum operators can be expressed in terms of the creation and annihilation operators
  • Satisfy the commutation relations of the quantum harmonic oscillator

Thermal Properties of Phonons

• Phonons are the primary heat carriers in insulators and semiconductors
• Specific heat of a crystal is determined by the phonon density of states and occupation numbers
  • At low temperatures, specific heat follows the Debye $T^3$ law due to acoustic phonons
  • At high temperatures, specific heat approaches the classical Dulong-Petit limit ($3k_B$ per atom)
• Thermal conductivity is determined by the phonon group velocities and scattering rates
  • Umklapp scattering dominates at high temperatures and limits the thermal conductivity
• Thermal expansion arises from the anharmonicity of the interatomic potential
  • Gruneisen parameter relates the thermal expansion to the phonon frequencies and volume dependence
• Phonon drag effect enhances the thermoelectric power of semiconductors at low temperatures
  • Phonons transfer momentum to charge carriers and increase the Seebeck coefficient

Experimental Techniques for Studying Phonons

• Inelastic neutron scattering measures the phonon dispersion relations and density of states
  • Neutrons have wavelengths comparable to interatomic distances and energies comparable to phonons
• Inelastic X-ray scattering probes phonons with higher energy and momentum resolution than neutrons
• Raman spectroscopy measures the frequency of optical phonons at the Γ point
  • Based on inelastic scattering of light by phonons that modulate the electronic polarizability
• Infrared spectroscopy probes the absorption of light by optical phonons that couple to the electric dipole moment
• Thermal conductivity measurements provide information on phonon scattering rates and mean free paths
• Heat capacity measurements reveal the phonon density of states and anharmonic effects
• Acoustic measurements (ultrasound, Brillouin scattering) probe the elastic properties and acoustic phonons

Applications and Advanced Topics

• Phonons play a crucial role in determining the thermal and electrical properties of materials
• Phonon engineering involves tailoring the phonon dispersion relations and scattering rates for specific applications
  • Phononic crystals and metamaterials exhibit novel phonon transport properties
  • Nanostructuring can reduce the thermal conductivity by scattering phonons at interfaces and boundaries
• Electron-phonon interactions are essential for understanding superconductivity and charge carrier mobility
  • Polarons are quasiparticles formed by the coupling of electrons to phonons
• Phonon-phonon interactions give rise to phonon scattering and thermal resistance
  • Anharmonic effects become significant at high temperatures or in strongly anharmonic materials
• Phonons in low-dimensional systems (nanotubes, graphene) exhibit unique properties due to confinement effects
  • Flexural modes and thermal ripples can dominate the thermal and mechanical properties
• Phonon lasers (sasers) have been demonstrated using optical or electrical pumping of phonon modes
  • Potential applications in high-frequency acoustics and heat management
• Phonon Hall effect is a topological phenomenon where phonons experience a transverse force in a magnetic field
  • Arises from the Berry curvature of the phonon bands and has potential applications in phononics