Crystal Lattice Vibrations: Phonons Introduction to Solid State Physics http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html

Lattice dynamics above T=0 Crystal lattices at zero temperature posses long range order – translational symmetry (e.g., generates sharp diffraction pattern, Bloch states, …). At T>0 ions vibrate with an amplitude that depends on temperature – because of lattice symmetries, thermal vibrations can be analyzed in terms of collective motion of ions which can be populated and excited just like electrons – unlike electrons, phonons are bosons (no Pauli principle, phonon number is not conserved). Thermal lattice vibrations are responsible for: → Thermal conductivity of insulators is due to dispersive lattice vibrations (e.g., thermal conductivity of diamond is 6 times larger than that of metallic copper). → They reduce intensities of diffraction spots and allow for inellastic scattering where the energy of the scatter (e.g., neutron) changes due to absorption or creation of a phonon in the target. → Electron-phonon interactions renormalize the properties of electrons (electrons become heavier). → Superconductivity (conventional BCS) arises from multiple electron-phonon scattering between time-reversed electrons. PHYS 624: Crystal Lattice Vibrations: Phonons

Vibrations of small amplitude: 1D chain Classical Theory: Normal Modes 2 3 1 4 Quantum Theory: Linear Harmonic Oscillator for each Normal Mode PHYS 624: Crystal Lattice Vibrations: Phonons

Normal modes of 4-atom chain in pictures PHYS 624: Crystal Lattice Vibrations: Phonons

Adiabatic theory of thermal lattice vibrations Born-Oppenheimer adiabatic approximation: Electrons react instantaneously to slow motion of lattice, while remaining in essentially electronic ground state → small electron-phonon interaction can be treated as a perturbation with small parameter: PHYS 624: Crystal Lattice Vibrations: Phonons

The non-adiabatic term can be neglected at T<100K! Adiabatic formalism: Two Schrödinger equations (for electrons and ions) The non-adiabatic term can be neglected at T<100K! PHYS 624: Crystal Lattice Vibrations: Phonons

Newton (classical) equations of motion Lattice vibrations involve small displacement from the equilibrium ion position: 0.1Å and smaller → harmonic (linear) approximation N unit cells, each with r atoms → 3Nr Newton’s equations of motion PHYS 624: Crystal Lattice Vibrations: Phonons

Properties of quasielastic force coefficients PHYS 624: Crystal Lattice Vibrations: Phonons

Solving equations of motion: Fourier Series PHYS 624: Crystal Lattice Vibrations: Phonons

Example: 1D chain with 2 atoms per unit cell PHYS 624: Crystal Lattice Vibrations: Phonons

1D Example: Eigenfrequencies of chain PHYS 624: Crystal Lattice Vibrations: Phonons

1D Example: Eigenmodes of chain at q=0 Optical Mode: These atoms, if oppositely charged, would form an oscillating dipole which would couple to optical fields with Center of the unit cell is not moving! PHYS 624: Crystal Lattice Vibrations: Phonons

2D Example: Normal modes of chain in 2D space Constant force model (analog of TBH) : bond stretching and bond bending PHYS 624: Crystal Lattice Vibrations: Phonons

3D Example: Normal modes of Silicon L — longitudinal T — transverse O — optical A — acoustic PHYS 624: Crystal Lattice Vibrations: Phonons

Symmetry constraints →Relevant symmetries: Translational invariance of the lattice and its reciprocal lattice, Point group symmetry of the lattice and its reciprocal lattice, Time-reversal invariance. PHYS 624: Crystal Lattice Vibrations: Phonons

Acoustic vs. Optical crystal lattice normal modes →All harmonic lattices, in which the energy is invariant under a rigid translation of the entire lattice, must have at least one acoustic mode (sound waves) ←3 acoustic modes (in 3D crystal) PHYS 624: Crystal Lattice Vibrations: Phonons

Normal coordinates →The most general solution for displacement is a sum over the eigenvectors of the dynamical matrix: In normal coordinates Newton equations describe dynamics of 3rN independent harmonic oscillators! PHYS 624: Crystal Lattice Vibrations: Phonons

Quantum theory of small amplitude lattice vibrations: First quantization of LHO PHYS 624: Crystal Lattice Vibrations: Phonons

Second quantization representation: Fock-Dirac formalism PHYS 624: Crystal Lattice Vibrations: Phonons

Quantum theory of small amplitude lattice vibrations: Second quantization of LHO →Second Quantization applied to system of Linear Harmonic Oscillators: →Hamiltonian is a sum of 3rN independent LHO – each of which is a refered to as a phonon mode! The number of phonons in state is described by an operator: PHYS 624: Crystal Lattice Vibrations: Phonons

Phonons: Example of quantized collective excitations →Creating and destroying phonons: →Arbitrary number of phonons can be excited in each mode → phonons are bosons: →Lattice displacement expressed via phonon excitations – zero point motion! PHYS 624: Crystal Lattice Vibrations: Phonons

Quasiparticles in solids Electron: Quasiparticle consisting of a real electron and the exchange-correlation hole (a cloud of effective charge of opposite sign due to exchange and correlation effects arising from interaction with all other electrons). Hole: Quasiparticle like electron, but of opposite charge; it corresponds to the absence of an electron from a single-particle state which lies just below the Fermi level. The notion of a hole is particularly convenient when the reference state consists of quasiparticle states that are fully occupied and are separated by an energy gap from the unoccupied states. Perturbations with respective to this reference state, such as missing electrons, are conveniently discussed in terms of holes (e.g., p-doped semiconductor crystals). Polaron: In polar crystals motion of negatively charged electron distorts the lattice of positive and negative ions around it. Electron + Polarization cloud (electron excites longitudinal EM modes, while pushing the charges out of its way) = Polaron (has different mass than electron). PHYS 624: Crystal Lattice Vibrations: Phonons

Collective excitation in solids In contrast to quasiparticles, collective excitations are bosons, and they bear no resemblance to constituent particles of real system. They involve collective (i.e., coherent) motion of many physical particles. Phonon: Corresponds to coherent motion of all the atoms in a solid — quantized lattice vibrations with typical energy scale of Exciton: Bound state of an electron and a hole with binding energy Plasmon: Collective excitation of an entire electron gas relative to the lattice of ions; its existence is a manifestation of the long-range nature of the Coulomb interaction. The energy scale of plasmons is Magnon: Collective excitation of the spin degrees of freedom on the crystalline lattice. It corresponds to a spin wave, with an energy scale of PHYS 624: Crystal Lattice Vibrations: Phonons

Classical theory of neutron scattering Bragg or Laue conditions for elastic scattering! PHYS 624: Crystal Lattice Vibrations: Phonons

Classical vs. quantum inelastic neutron scattering in pictures Lattice vibrations are inherently quantum in nature → quantum theory is needed to account for correct temperature dependence and zero-point motion effects. Phonon absorption is allowed only at finite temperatures where a real phonon be excited: PHYS 624: Crystal Lattice Vibrations: Phonons