4 Phonons Crystal Vibrations Vibrations of Crystals with
4. Phonons Crystal Vibrations • • • Vibrations of Crystals with Monatomic Basis Two Atoms per Primitive Basis Quantization of Elastic Waves Phonon Momentum Inelastic Scattering by Phonons
Harmonic approximation: quadratic hamiltonian : elementary excitations Electrons, polarons & excitons are quasi-particles
Vibrations of Crystals with Monatomic Basis • • First Brillouin Zone Group Velocity Long Wavelength Limit Derivation of Force Constants from Experiment
Entire plane of atoms moving in phase → 1 -D problem Force on sth plane = (only neighboring planes interact ) Equation of motion: → → Dispersion relation
![Propagation along high symmetry directions → 1 -D problem E. g. , [100], [111]](http://slidetodoc.com/presentation_image/28a53639c0e2147384cdfafb5383ce43/image-5.jpg "Propagation along high symmetry directions → 1 -D problem E. g. , [100], [111]")
Propagation along high symmetry directions → 1 -D problem E. g. , [100], [111] in sc lattice. longitudinal wave transverse wave
First Brillouin Zone → Only K 1 st BZ is physically significant. K at zone boundary gives standing wave.
Group Velocity Group velocity: 1 -D: v. G = 0 at zone boundaries
Derivation of Force Constants from Experiment If planes up to the pth n. n. interact, Force on sth plane = If ωK is known, Cq can be obtained as follows: Prob 4. 4
Two Atoms per Primitive Basis →
Ka → 0: Ka → π: (M 1 >M 2 ) Transverse case: TO branch, Ka → 0: TA branch, Ka → 0: Gap
p atoms in primitive cell → d p branches of dispersion. d = 3 → 3 acoustical : 1 LA + 2 TA (3 p – 3) optical: (p– 1) LO + 2(p– 1) TO E. g. , Ge or KBr: p = 2 → 1 LA + 2 TA + 1 LO + 2 TO branches Ge Number of allowed K in 1 st BZ = N KBr
Quantization of Elastic Waves Quantization of harmonic oscillator of angular frequency ω → Classical standing wave: Virial theorem: For a power-law potential V ~ xp For a harmonic oscillator, p = 2,
Phonon Momentum Phonon DOFs involve relative coordinates → phonons do not carry physical linear momenta ( except for K = G modes ) Reminder: K = G K = 0 when restricted to 1 st BZ. Proof: See 7 th ed. Scattering of a phonon with other particles behaves as if it has momentum K E. g. , elastic scattering of X-ray: G = reciprocal lattice vector ( whole crystal recoil with momentum G / Bragg reflection) Inelastic scattering with a phonon created: Inelastic scattering with a phonon absorbed: Normal Process: G = 0. Umklapp Process: G 0.
Inelastic Scattering by Phonons Neutron scattering: Conservation of momentum: Conservation of energy:
- Slides: 14