Ch4 . Phonons? Crystal Vibrations

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Solid State Physics
Ch4 . Phonons? Crystal Vibrations
Prof. J. Joo (jjoo_at_korea.ac.kr) Department of
Physics, Korea University http//smartpolymer.kore
a.ac.kr
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4.1 Vibrations of Crystals with Monatomic Basis
(1)

• ? Primitive cell ?? ??? ??(? ??? ?? monatomic)
• ? ? crystal ? elastic vibration? ????
• ?? ????? wave (? ????)? ????
• ???? ?? ?(F)? ???? ? vibration
• ?? ?? (? Matter)? ??? ??? ?? ???
  vibration
• ? Phonon??? Energy of a lattice vibration is
  quantized.
• The quantum of energy is Phonon
• ? Wave ?? ??? ??? ???? ??? ???..

?? ??? ??
??? ?? transverse (wave) displacement
??? ?? longitudinal (wave) displacement

two mode
one mode
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4.1 Vibrations of Crystals with Monatomic Basis
(2)

• lt ??? ?? Monatomic ?? ? ?? gt
• ? ?? crystal ? elastic ? vibration response ?
  ??? ?? linear ? ??
• ( ? F kx Hookes law )
• Fs ? us or E ? us2
• (?) (??)
• ?? Energy ? us3 ? ???? ??
  high temp.
• ? Consider,
• C ??? ?? (?? ?monatomic )
• M ?? ?? (?? ?monatomic )
• ? ? longitudinal ??? transverse ? ??? C ? ??
• ? S ???? ? Fs C(us1-us) C(us-1-us)
  C(usp-us)
• ?p1 C(usp-us)

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4.1 Vibrations of Crystals with Monatomic Basis
(3)

• ? ?? nearest-neighbor ????? dominant ? ??? ??, ?
  p1
• ? Fs C(us1-us) C(us-1-us)
• Note us ? ??? time dependent ? exp(-i?t)
• ?, us us0 exp(-i?t)
• ? -M?2us C(us1us-1-2us) ?
• ? Let the general solution for us as
• us u0 exp(iksa) exp(-i?t) ?? x
  x0sin(kx- ?t)
• ?... and
• usp u0 exp(ik(sp)a) exp(-i?t)

lattice const.
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4.1 Vibrations of Crystals with Monatomic Basis
(4)

• ? ??? ??? ???? ??
• ? At the boundary of the 1st B.Z. (kp/a)

? The plot of ? vs. k
?
Longitudinal
Transverse C? ??
slope is zero
k
p/a
-p/a
Special signification of phonon wavevectors lies
on the zone boundary
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4.2 First Brillouin Zone

• ?? ??? ???? ??? elastic wave? ????? ??? ??
• ? only the 1st B.Z.
• ??
• ? 1? 2?? physical property? ??
• ?? (?, 1st B.Z.)? ??? ??? ?? ??? ??? ??

?
Extended zone
1
2
k
0
p/a
2p/a
-p/a
-2p/a
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4.3 Group Velocity

• ? ?? elastic wave packet ? ????
• ? group velocity
• ?)

the velocity of energy propagation in the medium
no propagation!! ? the wave is standing wave
? zero net transmission velocity (note)
diffraction condition
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4.4 Long Wavelength Limit
?gtgta
? continium theory of elastic wave
?2a
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4.5 Two Atoms per Primitive Basis (1)

• Looking for phonon dispersion relation
• in crystals with 2 different atoms /
  primitive basis
• For each polarization mode, the dispersion
  relation (? vs. k)
• acoustic branch (LA, TA)
• and
• optical branch (LO, TO)
• Consider a cubic crystal with 2 atoms
• Consider the interaction
• between the n, n, atoms

basis associated with primitive cell
Longitudinal or transverse mode
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4.5 Two Atoms per Primitive Basis (2)

• Looking for a solution in the form of a traveling
  wave

eq.?
eq.?
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4.5 Two Atoms per Primitive Basis (3)

• ?
• ? ? ? ?? 4? ???
• ?? ???? ?? 2?? ??? ???? ??,
• ?, kaltlt1 (?? ????) and kap (zone boundary??)
• ?
• ?

?, ka?0
??
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4.5 Two Atoms per Primitive Basis (4)

• Transverse mode ? ??? ???? (Fig.8a)
• Ge at 80K
• For transverse optical branch,
• ? ? the atoms vibrate each other,
• but their C.O.M. is fixed
• ? we may excite a motion of this type
• with el. field of a light wave
• ? the branch is called the optical branch
• For transverse acoustical branch, at a small k,
  uv
• ? the atoms (and their C.O.M.) move together,
• as in the long wavelength acoustic
  vibrations
• ? acoustic branch

?? (eq.? and eq. ?)
TO TA in a diatomic linear lattice
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4.6 Quantization of Elastic Waves

• The energy of a lattice vibration is quantized
  
• the quantum of lattice vibration energy is
  called a phonon
• (similar to the photon of
  the EM wave)
• Note Phonon? ??? ?? ??
• Neutron scattering can map entire B.Z., but
  poor resolution
• (needs a special places such as Natl Lab.)
• 2. Raman scattering (polariton scattering)
• high resolution and for optical mode
• 3. Brillouin scattering (photon scattering)
• very high resolution and for acoustic mode

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4.7 Phonon Momentum

• A phonon of wave vector K
• Interacts with photons, neutrons, and electrons
• For most practical purposes, a phonon acts as if
  its momentum wave
  crystal momentum
• Note For a x-ray photon by a crystal (photon
  elastic scattering)
• momentum conservation
• reciprocal lattice vector
• incident photon wave vector
• scattered photon wave vector
• If the scattering of the photon is inelastic
• ? with the creation of a phonon (K),
• then the wave vector selection rule is
• ? if a phonon K is absorbed in the process,
• ? For an inelastic scattering,

?
?
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4.8 Inelastic Scattering by Phonons

• ? the energy conservation is
• ???? ??? neutron? ???
• the energy of the phonon created (), or
  absorbed (-)