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4. Phonons Crystal Vibrations

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- 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

- 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], [110], [111] in sc lattice.

longitudinal wave

transverse wave

→

Only K 1st BZ is physically significant.

K at zone boundary gives standing wave.

Group velocity:

1-D:

vG = 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:

Gap

Ka → π:

(M1 >M2 )

Transverse case:

TO branch, Ka → 0:

TA branch, Ka → 0:

p atoms in primitive cell → d p branches of dispersion.

d = 3 → 3 acoustical : 1 LA + 2 TA

(3p –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

KBr

Number of allowed K in 1st BZ = N

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 DOFs involve relative coordinates

→ phonons do not carry physical linear momenta ( except for K = Gmodes )

Reminder: K = G K = 0when restricted to 1st BZ .

Proof:

See 7th 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:

Normal Process: G = 0.

Umklapp Process: G0.

Inelastic scattering with a phonon absorbed:

Neutron scattering:

Conservation of momentum:

Conservation of energy: