Crystal Lattice Vibrations: Phonons

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

2. 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.

3. Vibrations of small amplitude: 1D chain Classical Theory:Normal Modes 2 3 1 4 Quantum Theory: Linear Harmonic Oscillator for each Normal Mode

4. Normal modes of 4-atom chain in pictures

5. 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:

6. Adiabatic formalism: Two Schrödinger equations (for electrons and ions) The non-adiabatic term can be neglected at T<100K!

7. 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

8. Properties of quasielastic force coefficients

9. Solving equations of motion: Fourier Series

10. Example: 1D chain with 2 atoms per unit cell

11. 1D Example: Eigenfrequencies of chain

12. 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!

13. 2D Example: Normal modes of chain in 2D space • Constant force model (analog of TBH): bond stretching and bond bending

14. 3D Example: Normal modes of Silicon L — longitudinal T —transverse O —optical A —acoustic

15. Symmetry constraints →Relevant symmetries:Translational invariance of the lattice and its reciprocal lattice, Point group symmetry of the lattice and its reciprocal lattice, Time-reversalinvariance.

16. 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)

17. 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!

18. Quantum theory of small amplitude lattice vibrations: First quantization of LHO →First Quantization:

19. Second quantization representation: Fock-Dirac formalism

20. 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:

21. 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!

22. 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).

23. 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

24. Classical theory of neutron scattering Bragg or Laue conditions for elastic scattering!

25. Classical vs. quantum inelastic neutron scattering in pictures • Lattice vibrations are inherentlyquantum 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: